LMFDB - Embedded newform 1600.3.h.k.1599.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,3,Mod(1599,1600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1600, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1600.1599");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1600 = 2^{6} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1600.h (of order $2$, degree $1$, not 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:	$43.5968422976$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 160)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1599.2
Root			$-0.618034i$ of defining polynomial
Character	$\chi$	$=$	1600.1599
Dual form			1600.3.h.k.1599.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.23607 q^{3} -1.23607 q^{7} -7.47214 q^{9} +O(q^{10})$	$q-1.23607 q^{3} -1.23607 q^{7} -7.47214 q^{9} +11.4164i q^{11} -5.41641i q^{13} -6.94427i q^{17} -29.8885i q^{19} +1.52786 q^{21} -19.1246 q^{23} +20.3607 q^{27} +21.0557 q^{29} +34.4721i q^{31} -14.1115i q^{33} -19.3050i q^{37} +6.69505i q^{39} -58.1378 q^{41} +62.7639 q^{43} +63.4853 q^{47} -47.4721 q^{49} +8.58359i q^{51} +98.1378i q^{53} +36.9443i q^{57} -19.2786i q^{59} -1.19350 q^{61} +9.23607 q^{63} -5.01316 q^{67} +23.6393 q^{69} +84.3607i q^{71} -70.7214i q^{73} -14.1115i q^{77} +124.498i q^{79} +42.0820 q^{81} +160.652 q^{83} -26.0263 q^{87} -46.2229 q^{89} +6.69505i q^{91} -42.6099i q^{93} -133.331i q^{97} -85.3050i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 4 q^{7} - 12 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 4 * q^7 - 12 * q^9	$ 4 q + 4 q^{3} + 4 q^{7} - 12 q^{9} + 24 q^{21} + 4 q^{23} - 8 q^{27} + 120 q^{29} + 260 q^{43} + 84 q^{47} - 172 q^{49} + 192 q^{61} + 28 q^{63} + 132 q^{67} + 184 q^{69} - 100 q^{81} + 580 q^{83} + 200 q^{87} - 328 q^{89}+O(q^{100}) $ 4 * q + 4 * q^3 + 4 * q^7 - 12 * q^9 + 24 * q^21 + 4 * q^23 - 8 * q^27 + 120 * q^29 + 260 * q^43 + 84 * q^47 - 172 * q^49 + 192 * q^61 + 28 * q^63 + 132 * q^67 + 184 * q^69 - 100 * q^81 + 580 * q^83 + 200 * q^87 - 328 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1151$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	−1.23607	−0.412023	−0.206011	−	0.978550i	$-0.566048\pi$
$3$	−1.23607	−0.412023	−0.206011	+	0.978550i	$0.566048\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.23607	−0.176581	−0.0882906	−	0.996095i	$-0.528140\pi$
$7$	−1.23607	−0.176581	−0.0882906	+	0.996095i	$0.528140\pi$
$8$	0	0
$9$	−7.47214	−0.830237
$10$	0	0
$11$	11.4164i	1.03786i	0.854818	+	0.518928i	$0.173669\pi$
$11$	11.4164i	1.03786i	−0.854818	+	0.518928i	$0.826331\pi$
$12$	0	0
$13$	− 5.41641i	− 0.416647i	−0.978060	−	0.208323i	$-0.933199\pi$
$13$	− 5.41641i	− 0.416647i	0.978060	−	0.208323i	$-0.0668006\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 6.94427i	− 0.408487i	−0.978920	−	0.204243i	$-0.934527\pi$
$17$	− 6.94427i	− 0.408487i	0.978920	−	0.204243i	$-0.0654734\pi$
$18$	0	0
$19$	− 29.8885i	− 1.57308i	−0.617539	−	0.786541i	$-0.711870\pi$
$19$	− 29.8885i	− 1.57308i	0.617539	−	0.786541i	$-0.288130\pi$
$20$	0	0
$21$	1.52786	0.0727554
$22$	0	0
$23$	−19.1246	−0.831505	−0.415752	−	0.909478i	$-0.636482\pi$
$23$	−19.1246	−0.831505	−0.415752	+	0.909478i	$0.636482\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	20.3607	0.754099
$28$	0	0
$29$	21.0557	0.726060	0.363030	−	0.931778i	$-0.381742\pi$
$29$	21.0557	0.726060	0.363030	+	0.931778i	$0.381742\pi$
$30$	0	0
$31$	34.4721i	1.11200i	0.831181	+	0.556002i	$0.187665\pi$
$31$	34.4721i	1.11200i	−0.831181	+	0.556002i	$0.812335\pi$
$32$	0	0
$33$	− 14.1115i	− 0.427620i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 19.3050i	− 0.521755i	−0.965372	−	0.260878i	$-0.915988\pi$
$37$	− 19.3050i	− 0.521755i	0.965372	−	0.260878i	$-0.0840119\pi$
$38$	0	0
$39$	6.69505i	0.171668i
$40$	0	0
$41$	−58.1378	−1.41799	−0.708997	−	0.705211i	$-0.750852\pi$
$41$	−58.1378	−1.41799	−0.708997	+	0.705211i	$0.750852\pi$
$42$	0	0
$43$	62.7639	1.45963	0.729813	−	0.683647i	$-0.239607\pi$
$43$	62.7639	1.45963	0.729813	+	0.683647i	$0.239607\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	63.4853	1.35075	0.675375	−	0.737474i	$-0.263982\pi$
$47$	63.4853	1.35075	0.675375	+	0.737474i	$0.263982\pi$
$48$	0	0
$49$	−47.4721	−0.968819
$50$	0	0
$51$	8.58359i	0.168306i
$52$	0	0
$53$	98.1378i	1.85166i	0.377945	+	0.925828i	$0.376631\pi$
$53$	98.1378i	1.85166i	−0.377945	+	0.925828i	$0.623369\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	36.9443i	0.648145i
$58$	0	0
$59$	− 19.2786i	− 0.326757i	−0.986563	−	0.163378i	$-0.947761\pi$
$59$	− 19.2786i	− 0.326757i	0.986563	−	0.163378i	$-0.0522391\pi$
$60$	0	0
$61$	−1.19350	−0.0195655	−0.00978275	−	0.999952i	$-0.503114\pi$
$61$	−1.19350	−0.0195655	−0.00978275	+	0.999952i	$0.503114\pi$
$62$	0	0
$63$	9.23607	0.146604
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−5.01316	−0.0748232	−0.0374116	−	0.999300i	$-0.511911\pi$
$67$	−5.01316	−0.0748232	−0.0374116	+	0.999300i	$0.511911\pi$
$68$	0	0
$69$	23.6393	0.342599
$70$	0	0
$71$	84.3607i	1.18818i	0.804399	+	0.594089i	$0.202487\pi$
$71$	84.3607i	1.18818i	−0.804399	+	0.594089i	$0.797513\pi$
$72$	0	0
$73$	− 70.7214i	− 0.968786i	−0.874851	−	0.484393i	$-0.839041\pi$
$73$	− 70.7214i	− 0.968786i	0.874851	−	0.484393i	$-0.160959\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 14.1115i	− 0.183266i
$78$	0	0
$79$	124.498i	1.57593i	0.615720	+	0.787965i	$0.288865\pi$
$79$	124.498i	1.57593i	−0.615720	+	0.787965i	$0.711135\pi$
$80$	0	0
$81$	42.0820	0.519531
$82$	0	0
$83$	160.652	1.93557	0.967786	−	0.251774i	$-0.0810140\pi$
$83$	160.652	1.93557	0.967786	+	0.251774i	$0.0810140\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−26.0263	−0.299153
$88$	0	0
$89$	−46.2229	−0.519359	−0.259679	−	0.965695i	$-0.583617\pi$
$89$	−46.2229	−0.519359	−0.259679	+	0.965695i	$0.583617\pi$
$90$	0	0
$91$	6.69505i	0.0735720i
$92$	0	0
$93$	− 42.6099i	− 0.458171i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 133.331i	− 1.37455i	−0.726398	−	0.687275i	$-0.758807\pi$
$97$	− 133.331i	− 1.37455i	0.726398	−	0.687275i	$-0.241193\pi$
$98$	0	0
$99$	− 85.3050i	− 0.861666i
$100$	0	0
$101$	49.7771	0.492842	0.246421	−	0.969163i	$-0.420745\pi$
$101$	49.7771	0.492842	0.246421	+	0.969163i	$0.420745\pi$
$102$	0	0
$103$	196.705	1.90976	0.954879	−	0.296995i	$-0.0959844\pi$
$103$	196.705	1.90976	0.954879	+	0.296995i	$0.0959844\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−14.9017	−0.139268	−0.0696341	−	0.997573i	$-0.522183\pi$
$107$	−14.9017	−0.139268	−0.0696341	+	0.997573i	$0.522183\pi$
$108$	0	0
$109$	174.859	1.60421	0.802106	−	0.597182i	$-0.203713\pi$
$109$	174.859	1.60421	0.802106	+	0.597182i	$0.203713\pi$
$110$	0	0
$111$	23.8622i	0.214975i
$112$	0	0
$113$	− 33.0557i	− 0.292529i	−0.989246	−	0.146264i	$-0.953275\pi$
$113$	− 33.0557i	− 0.292529i	0.989246	−	0.146264i	$-0.0467250\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	40.4721i	0.345916i
$118$	0	0
$119$	8.58359i	0.0721310i
$120$	0	0
$121$	−9.33437	−0.0771435
$122$	0	0
$123$	71.8622	0.584246
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−93.7345	−0.738067	−0.369034	−	0.929416i	$-0.620311\pi$
$127$	−93.7345	−0.738067	−0.369034	+	0.929416i	$0.620311\pi$
$128$	0	0
$129$	−77.5805	−0.601399
$130$	0	0
$131$	80.8065i	0.616844i	0.951250	+	0.308422i	$0.0998008\pi$
$131$	80.8065i	0.616844i	−0.951250	+	0.308422i	$0.900199\pi$
$132$	0	0
$133$	36.9443i	0.277776i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	192.164i	1.40266i	0.712838	+	0.701329i	$0.247409\pi$
$137$	192.164i	1.40266i	−0.712838	+	0.701329i	$0.752591\pi$
$138$	0	0
$139$	6.60990i	0.0475533i	0.999717	+	0.0237766i	$0.00756905\pi$
$139$	6.60990i	0.0475533i	−0.999717	+	0.0237766i	$0.992431\pi$
$140$	0	0
$141$	−78.4721	−0.556540
$142$	0	0
$143$	61.8359	0.432419
$144$	0	0
$145$	0	0
$146$	0	0
$147$	58.6788	0.399175
$148$	0	0
$149$	−199.803	−1.34096	−0.670481	−	0.741927i	$-0.733912\pi$
$149$	−199.803	−1.34096	−0.670481	+	0.741927i	$0.733912\pi$
$150$	0	0
$151$	53.2523i	0.352664i	0.984331	+	0.176332i	$0.0564233\pi$
$151$	53.2523i	0.352664i	−0.984331	+	0.176332i	$0.943577\pi$
$152$	0	0
$153$	51.8885i	0.339141i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	43.8034i	0.279003i	0.990222	+	0.139501i	$0.0445499\pi$
$157$	43.8034i	0.279003i	−0.990222	+	0.139501i	$0.955450\pi$
$158$	0	0
$159$	− 121.305i	− 0.762924i
$160$	0	0
$161$	23.6393	0.146828
$162$	0	0
$163$	188.705	1.15770	0.578850	−	0.815434i	$-0.303502\pi$
$163$	188.705	1.15770	0.578850	+	0.815434i	$0.303502\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	260.705	1.56111	0.780554	−	0.625088i	$-0.214937\pi$
$167$	260.705	1.56111	0.780554	+	0.625088i	$0.214937\pi$
$168$	0	0
$169$	139.663	0.826405
$170$	0	0
$171$	223.331i	1.30603i
$172$	0	0
$173$	− 28.6950i	− 0.165867i	−0.996555	−	0.0829337i	$-0.973571\pi$
$173$	− 28.6950i	− 0.165867i	0.996555	−	0.0829337i	$-0.0264290\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	23.8297i	0.134631i
$178$	0	0
$179$	95.5016i	0.533528i	0.963762	+	0.266764i	$0.0859545\pi$
$179$	95.5016i	0.533528i	−0.963762	+	0.266764i	$0.914046\pi$
$180$	0	0
$181$	176.885	0.977268	0.488634	−	0.872489i	$-0.337495\pi$
$181$	176.885	0.977268	0.488634	+	0.872489i	$0.337495\pi$
$182$	0	0
$183$	1.47524	0.00806143
$184$	0	0
$185$	0	0
$186$	0	0
$187$	79.2786	0.423950
$188$	0	0
$189$	−25.1672	−0.133160
$190$	0	0
$191$	231.967i	1.21449i	0.794515	+	0.607245i	$0.207725\pi$
$191$	231.967i	1.21449i	−0.794515	+	0.607245i	$0.792275\pi$
$192$	0	0
$193$	168.387i	0.872471i	0.899832	+	0.436236i	$0.143689\pi$
$193$	168.387i	0.872471i	−0.899832	+	0.436236i	$0.856311\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 31.0820i	− 0.157777i	−0.996883	−	0.0788884i	$-0.974863\pi$
$197$	− 31.0820i	− 0.157777i	0.996883	−	0.0788884i	$-0.0251371\pi$
$198$	0	0
$199$	− 123.777i	− 0.621995i	−0.950411	−	0.310998i	$-0.899337\pi$
$199$	− 123.777i	− 0.621995i	0.950411	−	0.310998i	$-0.100663\pi$
$200$	0	0
$201$	6.19660	0.0308289
$202$	0	0
$203$	−26.0263	−0.128208
$204$	0	0
$205$	0	0
$206$	0	0
$207$	142.902	0.690346
$208$	0	0
$209$	341.220	1.63263
$210$	0	0
$211$	34.2492i	0.162319i	0.996701	+	0.0811593i	$0.0258623\pi$
$211$	34.2492i	0.162319i	−0.996701	+	0.0811593i	$0.974138\pi$
$212$	0	0
$213$	− 104.276i	− 0.489557i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 42.6099i	− 0.196359i
$218$	0	0
$219$	87.4164i	0.399162i
$220$	0	0
$221$	−37.6130	−0.170195
$222$	0	0
$223$	202.987	0.910255	0.455127	−	0.890426i	$-0.349594\pi$
$223$	202.987	0.910255	0.455127	+	0.890426i	$0.349594\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	232.207	1.02294	0.511468	−	0.859302i	$-0.329102\pi$
$227$	232.207	1.02294	0.511468	+	0.859302i	$0.329102\pi$
$228$	0	0
$229$	−163.390	−0.713494	−0.356747	−	0.934201i	$-0.616114\pi$
$229$	−163.390	−0.713494	−0.356747	+	0.934201i	$0.616114\pi$
$230$	0	0
$231$	17.4427i	0.0755096i
$232$	0	0
$233$	− 11.2198i	− 0.0481537i	−0.999710	−	0.0240768i	$-0.992335\pi$
$233$	− 11.2198i	− 0.0481537i	0.999710	−	0.0240768i	$-0.00766464\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 153.889i	− 0.649319i
$238$	0	0
$239$	− 216.721i	− 0.906784i	−0.891311	−	0.453392i	$-0.850214\pi$
$239$	− 216.721i	− 0.906784i	0.891311	−	0.453392i	$-0.149786\pi$
$240$	0	0
$241$	−49.1409	−0.203904	−0.101952	−	0.994789i	$-0.532509\pi$
$241$	−49.1409	−0.203904	−0.101952	+	0.994789i	$0.532509\pi$
$242$	0	0
$243$	−235.262	−0.968158
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−161.889	−0.655419
$248$	0	0
$249$	−198.577	−0.797500
$250$	0	0
$251$	282.525i	1.12560i	0.826594	+	0.562798i	$0.190275\pi$
$251$	282.525i	1.12560i	−0.826594	+	0.562798i	$0.809725\pi$
$252$	0	0
$253$	− 218.334i	− 0.862982i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 199.941i	− 0.777981i	−0.921242	−	0.388991i	$-0.872824\pi$
$257$	− 199.941i	− 0.777981i	0.921242	−	0.388991i	$-0.127176\pi$
$258$	0	0
$259$	23.8622i	0.0921322i
$260$	0	0
$261$	−157.331	−0.602802
$262$	0	0
$263$	247.761	0.942056	0.471028	−	0.882118i	$-0.343883\pi$
$263$	247.761	0.942056	0.471028	+	0.882118i	$0.343883\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	57.1347	0.213987
$268$	0	0
$269$	13.4164	0.0498751	0.0249376	−	0.999689i	$-0.492061\pi$
$269$	13.4164	0.0498751	0.0249376	+	0.999689i	$0.492061\pi$
$270$	0	0
$271$	− 59.4690i	− 0.219443i	−0.993962	−	0.109721i	$-0.965004\pi$
$271$	− 59.4690i	− 0.219443i	0.993962	−	0.109721i	$-0.0349959\pi$
$272$	0	0
$273$	− 8.27553i	− 0.0303133i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	510.859i	1.84426i	0.386884	+	0.922128i	$0.373551\pi$
$277$	510.859i	1.84426i	−0.386884	+	0.922128i	$0.626449\pi$
$278$	0	0
$279$	− 257.580i	− 0.923228i
$280$	0	0
$281$	489.299	1.74128	0.870638	−	0.491924i	$-0.163706\pi$
$281$	489.299	1.74128	0.870638	+	0.491924i	$0.163706\pi$
$282$	0	0
$283$	−210.233	−0.742873	−0.371436	−	0.928458i	$-0.621135\pi$
$283$	−210.233	−0.742873	−0.371436	+	0.928458i	$0.621135\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	71.8622	0.250391
$288$	0	0
$289$	240.777	0.833139
$290$	0	0
$291$	164.807i	0.566345i
$292$	0	0
$293$	− 208.472i	− 0.711509i	−0.934579	−	0.355754i	$-0.884224\pi$
$293$	− 208.472i	− 0.711509i	0.934579	−	0.355754i	$-0.115776\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	232.446i	0.782646i
$298$	0	0
$299$	103.587i	0.346444i
$300$	0	0
$301$	−77.5805	−0.257742
$302$	0	0
$303$	−61.5279	−0.203062
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−229.564	−0.747766	−0.373883	−	0.927476i	$-0.621974\pi$
$307$	−229.564	−0.747766	−0.373883	+	0.927476i	$0.621974\pi$
$308$	0	0
$309$	−243.141	−0.786864
$310$	0	0
$311$	− 351.692i	− 1.13084i	−0.824802	−	0.565421i	$-0.808714\pi$
$311$	− 351.692i	− 1.13084i	0.824802	−	0.565421i	$-0.191286\pi$
$312$	0	0
$313$	418.551i	1.33722i	0.743612	+	0.668612i	$0.233111\pi$
$313$	418.551i	1.33722i	−0.743612	+	0.668612i	$0.766889\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 351.082i	− 1.10751i	−0.832678	−	0.553757i	$-0.813194\pi$
$317$	− 351.082i	− 1.10751i	0.832678	−	0.553757i	$-0.186806\pi$
$318$	0	0
$319$	240.381i	0.753545i
$320$	0	0
$321$	18.4195	0.0573817
$322$	0	0
$323$	−207.554	−0.642583
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−216.138	−0.660972
$328$	0	0
$329$	−78.4721	−0.238517
$330$	0	0
$331$	− 630.853i	− 1.90590i	−0.303127	−	0.952950i	$-0.598031\pi$
$331$	− 630.853i	− 1.90590i	0.303127	−	0.952950i	$-0.401969\pi$
$332$	0	0
$333$	144.249i	0.433181i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 284.780i	− 0.845045i	−0.906352	−	0.422523i	$-0.861145\pi$
$337$	− 284.780i	− 0.845045i	0.906352	−	0.422523i	$-0.138855\pi$
$338$	0	0
$339$	40.8591i	0.120528i
$340$	0	0
$341$	−393.548	−1.15410
$342$	0	0
$343$	119.246	0.347656
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−118.902	−0.342656	−0.171328	−	0.985214i	$-0.554806\pi$
$347$	−118.902	−0.342656	−0.171328	+	0.985214i	$0.554806\pi$
$348$	0	0
$349$	−19.0495	−0.0545831	−0.0272916	−	0.999628i	$-0.508688\pi$
$349$	−19.0495	−0.0545831	−0.0272916	+	0.999628i	$0.508688\pi$
$350$	0	0
$351$	− 110.282i	− 0.314193i
$352$	0	0
$353$	− 96.3344i	− 0.272902i	−0.990647	−	0.136451i	$-0.956430\pi$
$353$	− 96.3344i	− 0.272902i	0.990647	−	0.136451i	$-0.0435696\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 10.6099i	− 0.0297196i
$358$	0	0
$359$	15.0031i	0.0417914i	0.999782	+	0.0208957i	$0.00665179\pi$
$359$	15.0031i	0.0417914i	−0.999782	+	0.0208957i	$0.993348\pi$
$360$	0	0
$361$	−532.325	−1.47458
$362$	0	0
$363$	11.5379	0.0317849
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−439.348	−1.19713	−0.598566	−	0.801073i	$-0.704263\pi$
$367$	−439.348	−1.19713	−0.598566	+	0.801073i	$0.704263\pi$
$368$	0	0
$369$	434.413	1.17727
$370$	0	0
$371$	− 121.305i	− 0.326968i
$372$	0	0
$373$	346.689i	0.929461i	0.885452	+	0.464730i	$0.153849\pi$
$373$	346.689i	0.929461i	−0.885452	+	0.464730i	$0.846151\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 114.046i	− 0.302510i
$378$	0	0
$379$	338.768i	0.893846i	0.894572	+	0.446923i	$0.147480\pi$
$379$	338.768i	0.893846i	−0.894572	+	0.446923i	$0.852520\pi$
$380$	0	0
$381$	115.862	0.304100
$382$	0	0
$383$	−142.351	−0.371673	−0.185836	−	0.982581i	$-0.559499\pi$
$383$	−142.351	−0.371673	−0.185836	+	0.982581i	$0.559499\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−468.981	−1.21184
$388$	0	0
$389$	91.6331	0.235561	0.117780	−	0.993040i	$-0.462422\pi$
$389$	91.6331	0.235561	0.117780	+	0.993040i	$0.462422\pi$
$390$	0	0
$391$	132.807i	0.339659i
$392$	0	0
$393$	− 99.8823i	− 0.254154i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 384.354i	− 0.968147i	−0.875027	−	0.484074i	$-0.839157\pi$
$397$	− 384.354i	− 0.968147i	0.875027	−	0.484074i	$-0.160843\pi$
$398$	0	0
$399$	− 45.6656i	− 0.114450i
$400$	0	0
$401$	−416.545	−1.03877	−0.519383	−	0.854542i	$-0.673838\pi$
$401$	−416.545	−1.03877	−0.519383	+	0.854542i	$0.673838\pi$
$402$	0	0
$403$	186.715	0.463313
$404$	0	0
$405$	0	0
$406$	0	0
$407$	220.393	0.541507
$408$	0	0
$409$	−764.354	−1.86884	−0.934419	−	0.356177i	$-0.884080\pi$
$409$	−764.354	−1.86884	−0.934419	+	0.356177i	$0.884080\pi$
$410$	0	0
$411$	− 237.528i	− 0.577927i
$412$	0	0
$413$	23.8297i	0.0576991i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 8.17029i	− 0.0195930i
$418$	0	0
$419$	117.613i	0.280699i	0.990102	+	0.140350i	$0.0448227\pi$
$419$	117.613i	0.280699i	−0.990102	+	0.140350i	$0.955177\pi$
$420$	0	0
$421$	429.915	1.02118	0.510588	−	0.859826i	$-0.329428\pi$
$421$	429.915	1.02118	0.510588	+	0.859826i	$0.329428\pi$
$422$	0	0
$423$	−474.371	−1.12144
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1.47524	0.00345490
$428$	0	0
$429$	−76.4334	−0.178166
$430$	0	0
$431$	− 596.466i	− 1.38391i	−0.721940	−	0.691956i	$-0.756749\pi$
$431$	− 596.466i	− 1.38391i	0.721940	−	0.691956i	$-0.243251\pi$
$432$	0	0
$433$	− 678.551i	− 1.56709i	−0.621333	−	0.783546i	$-0.713409\pi$
$433$	− 678.551i	− 1.56709i	0.621333	−	0.783546i	$-0.286591\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	571.607i	1.30802i
$438$	0	0
$439$	324.774i	0.739804i	0.929071	+	0.369902i	$0.120609\pi$
$439$	324.774i	0.739804i	−0.929071	+	0.369902i	$0.879391\pi$
$440$	0	0
$441$	354.718	0.804350
$442$	0	0
$443$	−49.4064	−0.111527	−0.0557634	−	0.998444i	$-0.517759\pi$
$443$	−49.4064	−0.111527	−0.0557634	+	0.998444i	$0.517759\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	246.971	0.552507
$448$	0	0
$449$	186.859	0.416167	0.208084	−	0.978111i	$-0.433277\pi$
$449$	186.859	0.416167	0.208084	+	0.978111i	$0.433277\pi$
$450$	0	0
$451$	− 663.724i	− 1.47167i
$452$	0	0
$453$	− 65.8235i	− 0.145306i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	645.214i	1.41185i	0.708289	+	0.705923i	$0.249468\pi$
$457$	645.214i	1.41185i	−0.708289	+	0.705923i	$0.750532\pi$
$458$	0	0
$459$	− 141.390i	− 0.308039i
$460$	0	0
$461$	−10.7864	−0.0233978	−0.0116989	−	0.999932i	$-0.503724\pi$
$461$	−10.7864	−0.0233978	−0.0116989	+	0.999932i	$0.503724\pi$
$462$	0	0
$463$	−594.233	−1.28344	−0.641720	−	0.766939i	$-0.721779\pi$
$463$	−594.233	−1.28344	−0.641720	+	0.766939i	$0.721779\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	246.148	0.527083	0.263542	−	0.964648i	$-0.415109\pi$
$467$	246.148	0.527083	0.263542	+	0.964648i	$0.415109\pi$
$468$	0	0
$469$	6.19660	0.0132124
$470$	0	0
$471$	− 54.1440i	− 0.114955i
$472$	0	0
$473$	716.539i	1.51488i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 733.299i	− 1.53731i
$478$	0	0
$479$	722.610i	1.50858i	0.656541	+	0.754290i	$0.272019\pi$
$479$	722.610i	1.50858i	−0.656541	+	0.754290i	$0.727981\pi$
$480$	0	0
$481$	−104.563	−0.217388
$482$	0	0
$483$	−29.2198	−0.0604965
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−482.403	−0.990561	−0.495281	−	0.868733i	$-0.664935\pi$
$487$	−482.403	−0.990561	−0.495281	+	0.868733i	$0.664935\pi$
$488$	0	0
$489$	−233.252	−0.476999
$490$	0	0
$491$	235.967i	0.480585i	0.970700	+	0.240293i	$0.0772434\pi$
$491$	235.967i	0.480585i	−0.970700	+	0.240293i	$0.922757\pi$
$492$	0	0
$493$	− 146.217i	− 0.296586i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 104.276i	− 0.209810i
$498$	0	0
$499$	− 591.331i	− 1.18503i	−0.805558	−	0.592516i	$-0.798135\pi$
$499$	− 591.331i	− 1.18503i	0.805558	−	0.592516i	$-0.201865\pi$
$500$	0	0
$501$	−322.249	−0.643212
$502$	0	0
$503$	−708.843	−1.40923	−0.704615	−	0.709590i	$-0.748880\pi$
$503$	−708.843	−1.40923	−0.704615	+	0.709590i	$0.748880\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−172.632	−0.340498
$508$	0	0
$509$	801.489	1.57463	0.787317	−	0.616548i	$-0.211469\pi$
$509$	801.489	1.57463	0.787317	+	0.616548i	$0.211469\pi$
$510$	0	0
$511$	87.4164i	0.171069i
$512$	0	0
$513$	− 608.551i	− 1.18626i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	724.774i	1.40188i
$518$	0	0
$519$	35.4690i	0.0683411i
$520$	0	0
$521$	285.882	0.548718	0.274359	−	0.961627i	$-0.411534\pi$
$521$	285.882	0.548718	0.274359	+	0.961627i	$0.411534\pi$
$522$	0	0
$523$	−493.013	−0.942664	−0.471332	−	0.881956i	$-0.656227\pi$
$523$	−493.013	−0.942664	−0.471332	+	0.881956i	$0.656227\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	239.384	0.454239
$528$	0	0
$529$	−163.249	−0.308600
$530$	0	0
$531$	144.053i	0.271286i
$532$	0	0
$533$	314.898i	0.590803i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 118.046i	− 0.219826i
$538$	0	0
$539$	− 541.961i	− 1.00549i
$540$	0	0
$541$	906.774	1.67611	0.838054	−	0.545588i	$-0.183694\pi$
$541$	906.774	1.67611	0.838054	+	0.545588i	$0.183694\pi$
$542$	0	0
$543$	−218.642	−0.402656
$544$	0	0
$545$	0	0
$546$	0	0
$547$	111.420	0.203693	0.101847	−	0.994800i	$-0.467525\pi$
$547$	111.420	0.203693	0.101847	+	0.994800i	$0.467525\pi$
$548$	0	0
$549$	8.91796	0.0162440
$550$	0	0
$551$	− 629.325i	− 1.14215i
$552$	0	0
$553$	− 153.889i	− 0.278279i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 462.255i	− 0.829902i	−0.909844	−	0.414951i	$-0.863799\pi$
$557$	− 462.255i	− 0.829902i	0.909844	−	0.414951i	$-0.136201\pi$
$558$	0	0
$559$	− 339.955i	− 0.608149i
$560$	0	0
$561$	−97.9938	−0.174677
$562$	0	0
$563$	−640.555	−1.13775	−0.568876	−	0.822423i	$-0.692622\pi$
$563$	−640.555	−1.13775	−0.568876	+	0.822423i	$0.692622\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−52.0163	−0.0917394
$568$	0	0
$569$	−817.915	−1.43746	−0.718730	−	0.695289i	$-0.755276\pi$
$569$	−817.915	−1.43746	−0.718730	+	0.695289i	$0.755276\pi$
$570$	0	0
$571$	− 470.642i	− 0.824242i	−0.911129	−	0.412121i	$-0.864788\pi$
$571$	− 470.642i	− 0.824242i	0.911129	−	0.412121i	$-0.135212\pi$
$572$	0	0
$573$	− 286.728i	− 0.500397i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 1051.17i	− 1.82178i	−0.412649	−	0.910890i	$-0.635396\pi$
$577$	− 1051.17i	− 1.82178i	0.412649	−	0.910890i	$-0.364604\pi$
$578$	0	0
$579$	− 208.138i	− 0.359478i
$580$	0	0
$581$	−198.577	−0.341786
$582$	0	0
$583$	−1120.38	−1.92175
$584$	0	0
$585$	0	0
$586$	0	0
$587$	267.708	0.456062	0.228031	−	0.973654i	$-0.426771\pi$
$587$	267.708	0.456062	0.228031	+	0.973654i	$0.426771\pi$
$588$	0	0
$589$	1030.32	1.74927
$590$	0	0
$591$	38.4195i	0.0650076i
$592$	0	0
$593$	− 168.440i	− 0.284047i	−0.989863	−	0.142023i	$-0.954639\pi$
$593$	− 168.440i	− 0.284047i	0.989863	−	0.142023i	$-0.0453608\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	152.997i	0.256276i
$598$	0	0
$599$	626.715i	1.04627i	0.852250	+	0.523135i	$0.175237\pi$
$599$	626.715i	1.04627i	−0.852250	+	0.523135i	$0.824763\pi$
$600$	0	0
$601$	−359.252	−0.597758	−0.298879	−	0.954291i	$-0.596613\pi$
$601$	−359.252	−0.597758	−0.298879	+	0.954291i	$0.596613\pi$
$602$	0	0
$603$	37.4590	0.0621210
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−679.728	−1.11982	−0.559908	−	0.828555i	$-0.689164\pi$
$607$	−679.728	−1.11982	−0.559908	+	0.828555i	$0.689164\pi$
$608$	0	0
$609$	32.1703	0.0528248
$610$	0	0
$611$	− 343.862i	− 0.562786i
$612$	0	0
$613$	748.958i	1.22179i	0.791711	+	0.610896i	$0.209191\pi$
$613$	748.958i	1.22179i	−0.791711	+	0.610896i	$0.790809\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	947.876i	1.53627i	0.640290	+	0.768133i	$0.278814\pi$
$617$	947.876i	1.53627i	−0.640290	+	0.768133i	$0.721186\pi$
$618$	0	0
$619$	− 1074.32i	− 1.73558i	−0.496934	−	0.867788i	$-0.665541\pi$
$619$	− 1074.32i	− 1.73558i	0.496934	−	0.867788i	$-0.334459\pi$
$620$	0	0
$621$	−389.390	−0.627037
$622$	0	0
$623$	57.1347	0.0917089
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−421.771	−0.672681
$628$	0	0
$629$	−134.059	−0.213130
$630$	0	0
$631$	1061.67i	1.68253i	0.540627	+	0.841263i	$0.318187\pi$
$631$	1061.67i	1.68253i	−0.540627	+	0.841263i	$0.681813\pi$
$632$	0	0
$633$	− 42.3344i	− 0.0668789i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	257.128i	0.403655i
$638$	0	0
$639$	− 630.354i	− 0.986470i
$640$	0	0
$641$	497.404	0.775981	0.387991	−	0.921663i	$-0.373169\pi$
$641$	497.404	0.775981	0.387991	+	0.921663i	$0.373169\pi$
$642$	0	0
$643$	−604.397	−0.939964	−0.469982	−	0.882676i	$-0.655740\pi$
$643$	−604.397	−0.939964	−0.469982	+	0.882676i	$0.655740\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	693.151	1.07133	0.535665	−	0.844430i	$-0.320061\pi$
$647$	693.151	1.07133	0.535665	+	0.844430i	$0.320061\pi$
$648$	0	0
$649$	220.093	0.339126
$650$	0	0
$651$	52.6687i	0.0809044i
$652$	0	0
$653$	948.853i	1.45307i	0.687131	+	0.726534i	$0.258870\pi$
$653$	948.853i	1.45307i	−0.687131	+	0.726534i	$0.741130\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	528.440i	0.804322i
$658$	0	0
$659$	889.260i	1.34941i	0.738088	+	0.674704i	$0.235729\pi$
$659$	889.260i	1.34941i	−0.738088	+	0.674704i	$0.764271\pi$
$660$	0	0
$661$	664.354	1.00507	0.502537	−	0.864555i	$-0.332400\pi$
$661$	664.354	1.00507	0.502537	+	0.864555i	$0.332400\pi$
$662$	0	0
$663$	46.4922	0.0701240
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−402.683	−0.603722
$668$	0	0
$669$	−250.906	−0.375046
$670$	0	0
$671$	− 13.6254i	− 0.0203062i
$672$	0	0
$673$	429.437i	0.638093i	0.947739	+	0.319046i	$0.103363\pi$
$673$	429.437i	0.638093i	−0.947739	+	0.319046i	$0.896637\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	336.918i	0.497663i	0.968547	+	0.248832i	$0.0800466\pi$
$677$	336.918i	0.497663i	−0.968547	+	0.248832i	$0.919953\pi$
$678$	0	0
$679$	164.807i	0.242719i
$680$	0	0
$681$	−287.023	−0.421473
$682$	0	0
$683$	818.200	1.19795	0.598975	−	0.800767i	$-0.295575\pi$
$683$	818.200	1.19795	0.598975	+	0.800767i	$0.295575\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	201.961	0.293976
$688$	0	0
$689$	531.554	0.771486
$690$	0	0
$691$	− 356.689i	− 0.516192i	−0.966119	−	0.258096i	$-0.916905\pi$
$691$	− 356.689i	− 0.516192i	0.966119	−	0.258096i	$-0.0830951\pi$
$692$	0	0
$693$	105.443i	0.152154i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	403.724i	0.579232i
$698$	0	0
$699$	13.8684i	0.0198404i
$700$	0	0
$701$	−50.0851	−0.0714481	−0.0357241	−	0.999362i	$-0.511374\pi$
$701$	−50.0851	−0.0714481	−0.0357241	+	0.999362i	$0.511374\pi$
$702$	0	0
$703$	−576.997	−0.820764
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−61.5279	−0.0870267
$708$	0	0
$709$	−698.604	−0.985337	−0.492668	−	0.870217i	$-0.663978\pi$
$709$	−698.604	−0.985337	−0.492668	+	0.870217i	$0.663978\pi$
$710$	0	0
$711$	− 930.269i	− 1.30840i
$712$	0	0
$713$	− 659.266i	− 0.924637i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	267.882i	0.373616i
$718$	0	0
$719$	246.387i	0.342680i	0.985212	+	0.171340i	$0.0548097\pi$
$719$	246.387i	0.342680i	−0.985212	+	0.171340i	$0.945190\pi$
$720$	0	0
$721$	−243.141	−0.337227
$722$	0	0
$723$	60.7415	0.0840131
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−256.134	−0.352316	−0.176158	−	0.984362i	$-0.556367\pi$
$727$	−256.134	−0.352316	−0.176158	+	0.984362i	$0.556367\pi$
$728$	0	0
$729$	−87.9381	−0.120628
$730$	0	0
$731$	− 435.850i	− 0.596238i
$732$	0	0
$733$	− 897.574i	− 1.22452i	−0.790656	−	0.612261i	$-0.790260\pi$
$733$	− 897.574i	− 1.22452i	0.790656	−	0.612261i	$-0.209740\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 57.2322i	− 0.0776557i
$738$	0	0
$739$	268.486i	0.363310i	0.983362	+	0.181655i	$0.0581454\pi$
$739$	268.486i	0.363310i	−0.983362	+	0.181655i	$0.941855\pi$
$740$	0	0
$741$	200.105	0.270048
$742$	0	0
$743$	612.049	0.823753	0.411877	−	0.911240i	$-0.364873\pi$
$743$	612.049	0.823753	0.411877	+	0.911240i	$0.364873\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−1200.42	−1.60698
$748$	0	0
$749$	18.4195	0.0245921
$750$	0	0
$751$	142.485i	0.189726i	0.995490	+	0.0948632i	$0.0302414\pi$
$751$	142.485i	0.189726i	−0.995490	+	0.0948632i	$0.969759\pi$
$752$	0	0
$753$	− 349.220i	− 0.463771i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 636.342i	− 0.840610i	−0.907383	−	0.420305i	$-0.861923\pi$
$757$	− 636.342i	− 0.840610i	0.907383	−	0.420305i	$-0.138077\pi$
$758$	0	0
$759$	269.876i	0.355568i
$760$	0	0
$761$	591.666	0.777484	0.388742	−	0.921347i	$-0.372910\pi$
$761$	591.666	0.777484	0.388742	+	0.921347i	$0.372910\pi$
$762$	0	0
$763$	−216.138	−0.283274
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−104.421	−0.136142
$768$	0	0
$769$	−0.216701	−0.000281796 0	−0.000140898	−	1.00000i	$-0.500045\pi$
$769$	−0.216701	−0.000281796 0	−0.000140898		1.00000i	$0.500045\pi$
$770$	0	0
$771$	247.141i	0.320546i
$772$	0	0
$773$	− 762.807i	− 0.986813i	−0.869799	−	0.493407i	$-0.835751\pi$
$773$	− 762.807i	− 0.986813i	0.869799	−	0.493407i	$-0.164249\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 29.4953i	− 0.0379605i
$778$	0	0
$779$	1737.65i	2.23062i
$780$	0	0
$781$	−963.096	−1.23316
$782$	0	0
$783$	428.709	0.547521
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1311.80	1.66684	0.833419	−	0.552642i	$-0.186380\pi$
$787$	1311.80	1.66684	0.833419	+	0.552642i	$0.186380\pi$
$788$	0	0
$789$	−306.249	−0.388149
$790$	0	0
$791$	40.8591i	0.0516550i
$792$	0	0
$793$	6.46446i	0.00815190i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 1341.85i	− 1.68363i	−0.539769	−	0.841813i	$-0.681489\pi$
$797$	− 1341.85i	− 1.68363i	0.539769	−	0.841813i	$-0.318511\pi$
$798$	0	0
$799$	− 440.859i	− 0.551764i
$800$	0	0
$801$	345.384	0.431191
$802$	0	0
$803$	807.384	1.00546
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−16.5836	−0.0205497
$808$	0	0
$809$	53.8948	0.0666190	0.0333095	−	0.999445i	$-0.489395\pi$
$809$	53.8948	0.0666190	0.0333095	+	0.999445i	$0.489395\pi$
$810$	0	0
$811$	20.9644i	0.0258500i	0.999916	+	0.0129250i	$0.00411427\pi$
$811$	20.9644i	0.0258500i	−0.999916	+	0.0129250i	$0.995886\pi$
$812$	0	0
$813$	73.5078i	0.0904155i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 1875.92i	− 2.29611i
$818$	0	0
$819$	− 50.0263i	− 0.0610822i
$820$	0	0
$821$	363.240	0.442436	0.221218	−	0.975224i	$-0.428997\pi$
$821$	363.240	0.442436	0.221218	+	0.975224i	$0.428997\pi$
$822$	0	0
$823$	1487.42	1.80732	0.903658	−	0.428256i	$-0.140872\pi$
$823$	1487.42	1.80732	0.903658	+	0.428256i	$0.140872\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	948.981	1.14750	0.573749	−	0.819031i	$-0.305489\pi$
$827$	948.981	1.14750	0.573749	+	0.819031i	$0.305489\pi$
$828$	0	0
$829$	1114.06	1.34386	0.671930	−	0.740614i	$-0.265465\pi$
$829$	1114.06	1.34386	0.671930	+	0.740614i	$0.265465\pi$
$830$	0	0
$831$	− 631.457i	− 0.759876i
$832$	0	0
$833$	329.659i	0.395750i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	701.876i	0.838562i
$838$	0	0
$839$	145.272i	0.173149i	0.996245	+	0.0865747i	$0.0275921\pi$
$839$	145.272i	0.173149i	−0.996245	+	0.0865747i	$0.972408\pi$
$840$	0	0
$841$	−397.656	−0.472837
$842$	0	0
$843$	−604.807	−0.717445
$844$	0	0
$845$	0	0
$846$	0	0
$847$	11.5379	0.0136221
$848$	0	0
$849$	259.862	0.306080
$850$	0	0
$851$	369.200i	0.433842i
$852$	0	0
$853$	127.423i	0.149382i	0.997207	+	0.0746909i	$0.0237970\pi$
$853$	127.423i	0.149382i	−0.997207	+	0.0746909i	$0.976203\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	576.715i	0.672946i	0.941693	+	0.336473i	$0.109234\pi$
$857$	576.715i	0.672946i	−0.941693	+	0.336473i	$0.890766\pi$
$858$	0	0
$859$	− 513.155i	− 0.597386i	−0.954349	−	0.298693i	$-0.903449\pi$
$859$	− 513.155i	− 0.597386i	0.954349	−	0.298693i	$-0.0965507\pi$
$860$	0	0
$861$	−88.8266	−0.103167
$862$	0	0
$863$	774.488	0.897437	0.448719	−	0.893673i	$-0.351881\pi$
$863$	774.488	0.897437	0.448719	+	0.893673i	$0.351881\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−297.617	−0.343272
$868$	0	0
$869$	−1421.33	−1.63559
$870$	0	0
$871$	27.1533i	0.0311749i
$872$	0	0
$873$	996.269i	1.14120i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1335.13i	1.52239i	0.648524	+	0.761194i	$0.275387\pi$
$877$	1335.13i	1.52239i	−0.648524	+	0.761194i	$0.724613\pi$
$878$	0	0
$879$	257.686i	0.293158i
$880$	0	0
$881$	29.7570	0.0337764	0.0168882	−	0.999857i	$-0.494624\pi$
$881$	29.7570	0.0337764	0.0168882	+	0.999857i	$0.494624\pi$
$882$	0	0
$883$	105.544	0.119529	0.0597645	−	0.998213i	$-0.480965\pi$
$883$	105.544	0.119529	0.0597645	+	0.998213i	$0.480965\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−60.2268	−0.0678994	−0.0339497	−	0.999424i	$-0.510809\pi$
$887$	−60.2268	−0.0678994	−0.0339497	+	0.999424i	$0.510809\pi$
$888$	0	0
$889$	115.862	0.130329
$890$	0	0
$891$	480.426i	0.539198i
$892$	0	0
$893$	− 1897.48i	− 2.12484i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 128.040i	− 0.142743i
$898$	0	0
$899$	725.836i	0.807381i
$900$	0	0
$901$	681.495	0.756377
$902$	0	0
$903$	95.8948	0.106196
$904$	0	0
$905$	0	0
$906$	0	0
$907$	155.473	0.171414	0.0857072	−	0.996320i	$-0.472685\pi$
$907$	155.473	0.171414	0.0857072	+	0.996320i	$0.472685\pi$
$908$	0	0
$909$	−371.941	−0.409176
$910$	0	0
$911$	− 686.630i	− 0.753710i	−0.926272	−	0.376855i	$-0.877005\pi$
$911$	− 686.630i	− 0.753710i	0.926272	−	0.376855i	$-0.122995\pi$
$912$	0	0
$913$	1834.07i	2.00884i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 99.8823i	− 0.108923i
$918$	0	0
$919$	− 1339.16i	− 1.45719i	−0.684943	−	0.728597i	$-0.740173\pi$
$919$	− 1339.16i	− 1.45719i	0.684943	−	0.728597i	$-0.259827\pi$
$920$	0	0
$921$	283.757	0.308097
$922$	0	0
$923$	456.932	0.495051
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−1469.81	−1.58555
$928$	0	0
$929$	636.302	0.684932	0.342466	−	0.939530i	$-0.388738\pi$
$929$	636.302	0.684932	0.342466	+	0.939530i	$0.388738\pi$
$930$	0	0
$931$	1418.87i	1.52403i
$932$	0	0
$933$	434.715i	0.465933i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	310.210i	0.331068i	0.986204	+	0.165534i	$0.0529347\pi$
$937$	310.210i	0.331068i	−0.986204	+	0.165534i	$0.947065\pi$
$938$	0	0
$939$	− 517.358i	− 0.550967i
$940$	0	0
$941$	−1322.66	−1.40559	−0.702793	−	0.711394i	$-0.748064\pi$
$941$	−1322.66	−1.40559	−0.702793	+	0.711394i	$0.748064\pi$
$942$	0	0
$943$	1111.86	1.17907
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−724.883	−0.765452	−0.382726	−	0.923862i	$-0.625015\pi$
$947$	−724.883	−0.765452	−0.382726	+	0.923862i	$0.625015\pi$
$948$	0	0
$949$	−383.056	−0.403641
$950$	0	0
$951$	433.961i	0.456321i
$952$	0	0
$953$	− 298.223i	− 0.312931i	−0.987683	−	0.156465i	$-0.949990\pi$
$953$	− 298.223i	− 0.312931i	0.987683	−	0.156465i	$-0.0500099\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 297.127i	− 0.310478i
$958$	0	0
$959$	− 237.528i	− 0.247683i
$960$	0	0
$961$	−227.328	−0.236554
$962$	0	0
$963$	111.348	0.115626
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−249.236	−0.257742	−0.128871	−	0.991661i	$-0.541135\pi$
$967$	−249.236	−0.257742	−0.128871	+	0.991661i	$0.541135\pi$
$968$	0	0
$969$	256.551	0.264759
$970$	0	0
$971$	339.246i	0.349378i	0.984624	+	0.174689i	$0.0558920\pi$
$971$	339.246i	0.349378i	−0.984624	+	0.174689i	$0.944108\pi$
$972$	0	0
$973$	− 8.17029i	− 0.00839701i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	40.8854i	0.0418479i	0.999781	+	0.0209240i	$0.00666079\pi$
$977$	40.8854i	0.0418479i	−0.999781	+	0.0209240i	$0.993339\pi$
$978$	0	0
$979$	− 527.700i	− 0.539019i
$980$	0	0
$981$	−1306.57	−1.33188
$982$	0	0
$983$	−5.56423	−0.00566045	−0.00283023	−	0.999996i	$-0.500901\pi$
$983$	−5.56423	−0.00566045	−0.00283023	+	0.999996i	$0.500901\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	96.9969	0.0982745
$988$	0	0
$989$	−1200.34	−1.21369
$990$	0	0
$991$	1562.45i	1.57664i	0.615267	+	0.788319i	$0.289048\pi$
$991$	1562.45i	1.57664i	−0.615267	+	0.788319i	$0.710952\pi$
$992$	0	0
$993$	779.777i	0.785274i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 704.224i	− 0.706343i	−0.935559	−	0.353172i	$-0.885103\pi$
$997$	− 704.224i	− 0.706343i	0.935559	−	0.353172i	$-0.114897\pi$
$998$	0	0
$999$	− 393.062i	− 0.393455i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.3.h.k.1599.2		4
4.3	odd	2		1600.3.h.f.1599.3		4
5.2	odd	4		320.3.b.d.191.3		4
5.3	odd	4		1600.3.b.u.1151.2		4
5.4	even	2		1600.3.h.f.1599.4		4
8.3	odd	2		800.3.h.h.799.2		4
8.5	even	2		800.3.h.e.799.3		4
15.2	even	4		2880.3.e.h.2431.3		4
20.3	even	4		1600.3.b.u.1151.3		4
20.7	even	4		320.3.b.d.191.2		4
20.19	odd	2	inner	1600.3.h.k.1599.1		4
40.3	even	4		800.3.b.g.351.2		4
40.13	odd	4		800.3.b.g.351.3		4
40.19	odd	2		800.3.h.e.799.4		4
40.27	even	4		160.3.b.b.31.3	yes	4
40.29	even	2		800.3.h.h.799.1		4
40.37	odd	4		160.3.b.b.31.2	✓	4
60.47	odd	4		2880.3.e.h.2431.4		4
80.27	even	4		1280.3.g.c.1151.1		4
80.37	odd	4		1280.3.g.b.1151.3		4
80.67	even	4		1280.3.g.b.1151.4		4
80.77	odd	4		1280.3.g.c.1151.2		4
120.77	even	4		1440.3.e.a.991.1		4
120.107	odd	4		1440.3.e.a.991.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.3.b.b.31.2	✓	4	40.37	odd	4
160.3.b.b.31.3	yes	4	40.27	even	4
320.3.b.d.191.2		4	20.7	even	4
320.3.b.d.191.3		4	5.2	odd	4
800.3.b.g.351.2		4	40.3	even	4
800.3.b.g.351.3		4	40.13	odd	4
800.3.h.e.799.3		4	8.5	even	2
800.3.h.e.799.4		4	40.19	odd	2
800.3.h.h.799.1		4	40.29	even	2
800.3.h.h.799.2		4	8.3	odd	2
1280.3.g.b.1151.3		4	80.37	odd	4
1280.3.g.b.1151.4		4	80.67	even	4
1280.3.g.c.1151.1		4	80.27	even	4
1280.3.g.c.1151.2		4	80.77	odd	4
1440.3.e.a.991.1		4	120.77	even	4
1440.3.e.a.991.2		4	120.107	odd	4
1600.3.b.u.1151.2		4	5.3	odd	4
1600.3.b.u.1151.3		4	20.3	even	4
1600.3.h.f.1599.3		4	4.3	odd	2
1600.3.h.f.1599.4		4	5.4	even	2
1600.3.h.k.1599.1		4	20.19	odd	2	inner
1600.3.h.k.1599.2		4	1.1	even	1	trivial
2880.3.e.h.2431.3		4	15.2	even	4
2880.3.e.h.2431.4		4	60.47	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1600.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.23607 q^{3} -1.23607 q^{7} -7.47214 q^{9} +O(q^{10})\)	\(q-1.23607 q^{3} -1.23607 q^{7} -7.47214 q^{9} +11.4164i q^{11} -5.41641i q^{13} -6.94427i q^{17} -29.8885i q^{19} +1.52786 q^{21} -19.1246 q^{23} +20.3607 q^{27} +21.0557 q^{29} +34.4721i q^{31} -14.1115i q^{33} -19.3050i q^{37} +6.69505i q^{39} -58.1378 q^{41} +62.7639 q^{43} +63.4853 q^{47} -47.4721 q^{49} +8.58359i q^{51} +98.1378i q^{53} +36.9443i q^{57} -19.2786i q^{59} -1.19350 q^{61} +9.23607 q^{63} -5.01316 q^{67} +23.6393 q^{69} +84.3607i q^{71} -70.7214i q^{73} -14.1115i q^{77} +124.498i q^{79} +42.0820 q^{81} +160.652 q^{83} -26.0263 q^{87} -46.2229 q^{89} +6.69505i q^{91} -42.6099i q^{93} -133.331i q^{97} -85.3050i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 4 q^{7} - 12 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 4 * q^7 - 12 * q^9	\( 4 q + 4 q^{3} + 4 q^{7} - 12 q^{9} + 24 q^{21} + 4 q^{23} - 8 q^{27} + 120 q^{29} + 260 q^{43} + 84 q^{47} - 172 q^{49} + 192 q^{61} + 28 q^{63} + 132 q^{67} + 184 q^{69} - 100 q^{81} + 580 q^{83} + 200 q^{87} - 328 q^{89}+O(q^{100}) \) 4 * q + 4 * q^3 + 4 * q^7 - 12 * q^9 + 24 * q^21 + 4 * q^23 - 8 * q^27 + 120 * q^29 + 260 * q^43 + 84 * q^47 - 172 * q^49 + 192 * q^61 + 28 * q^63 + 132 * q^67 + 184 * q^69 - 100 * q^81 + 580 * q^83 + 200 * q^87 - 328 * q^89

Introduction

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