LMFDB - Embedded newform 9200.2.a.bn.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9200,2,Mod(1,9200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9200, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("9200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9200 = 2^{4} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9200.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:	$73.4623698596$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 4600)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	9200.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.23607 q^{3} +4.23607 q^{7} +7.47214 q^{9} -1.00000 q^{11} -2.23607 q^{13} +6.47214 q^{17} -1.00000 q^{19} -13.7082 q^{21} +1.00000 q^{23} -14.4721 q^{27} -1.76393 q^{29} -0.472136 q^{31} +3.23607 q^{33} +11.2361 q^{37} +7.23607 q^{39} -5.94427 q^{41} -2.52786 q^{43} -11.7082 q^{47} +10.9443 q^{49} -20.9443 q^{51} +1.23607 q^{53} +3.23607 q^{57} -3.23607 q^{59} -7.23607 q^{61} +31.6525 q^{63} -12.9443 q^{67} -3.23607 q^{69} -10.0000 q^{71} -9.47214 q^{73} -4.23607 q^{77} -15.1803 q^{79} +24.4164 q^{81} -9.00000 q^{83} +5.70820 q^{87} +2.00000 q^{89} -9.47214 q^{91} +1.52786 q^{93} +6.18034 q^{97} -7.47214 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 4 q^{7} + 6 q^{9} - 2 q^{11} + 4 q^{17} - 2 q^{19} - 14 q^{21} + 2 q^{23} - 20 q^{27} - 8 q^{29} + 8 q^{31} + 2 q^{33} + 18 q^{37} + 10 q^{39} + 6 q^{41} - 14 q^{43} - 10 q^{47} + 4 q^{49}+ \cdots - 6 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 4 * q^7 + 6 * q^9 - 2 * q^11 + 4 * q^17 - 2 * q^19 - 14 * q^21 + 2 * q^23 - 20 * q^27 - 8 * q^29 + 8 * q^31 + 2 * q^33 + 18 * q^37 + 10 * q^39 + 6 * q^41 - 14 * q^43 - 10 * q^47 + 4 * q^49 - 24 * q^51 - 2 * q^53 + 2 * q^57 - 2 * q^59 - 10 * q^61 + 32 * q^63 - 8 * q^67 - 2 * q^69 - 20 * q^71 - 10 * q^73 - 4 * q^77 - 8 * q^79 + 22 * q^81 - 18 * q^83 - 2 * q^87 + 4 * q^89 - 10 * q^91 + 12 * q^93 - 10 * q^97 - 6 * q^99

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
$3$	−3.23607	−1.86834	−0.934172	−	0.356822i	$-0.883860\pi$
$3$	−3.23607	−1.86834	−0.934172	+	0.356822i	$0.883860\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.23607	1.60108	0.800542	−	0.599277i	$-0.204545\pi$
$7$	4.23607	1.60108	0.800542	+	0.599277i	$0.204545\pi$
$8$	0	0
$9$	7.47214	2.49071
$10$	0	0
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	0	0
$13$	−2.23607	−0.620174	−0.310087	−	0.950708i	$-0.600358\pi$
$13$	−2.23607	−0.620174	−0.310087	+	0.950708i	$0.600358\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.47214	1.56972	0.784862	−	0.619671i	$-0.212734\pi$
$17$	6.47214	1.56972	0.784862	+	0.619671i	$0.212734\pi$
$18$	0	0
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	0	0
$21$	−13.7082	−2.99138
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−14.4721	−2.78516
$28$	0	0
$29$	−1.76393	−0.327554	−0.163777	−	0.986497i	$-0.552368\pi$
$29$	−1.76393	−0.327554	−0.163777	+	0.986497i	$0.552368\pi$
$30$	0	0
$31$	−0.472136	−0.0847981	−0.0423991	−	0.999101i	$-0.513500\pi$
$31$	−0.472136	−0.0847981	−0.0423991	+	0.999101i	$0.513500\pi$
$32$	0	0
$33$	3.23607	0.563327
$34$	0	0
$35$	0	0
$36$	0	0
$37$	11.2361	1.84720	0.923599	−	0.383360i	$-0.125233\pi$
$37$	11.2361	1.84720	0.923599	+	0.383360i	$0.125233\pi$
$38$	0	0
$39$	7.23607	1.15870
$40$	0	0
$41$	−5.94427	−0.928339	−0.464170	−	0.885746i	$-0.653647\pi$
$41$	−5.94427	−0.928339	−0.464170	+	0.885746i	$0.653647\pi$
$42$	0	0
$43$	−2.52786	−0.385496	−0.192748	−	0.981248i	$-0.561740\pi$
$43$	−2.52786	−0.385496	−0.192748	+	0.981248i	$0.561740\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−11.7082	−1.70782	−0.853909	−	0.520423i	$-0.825774\pi$
$47$	−11.7082	−1.70782	−0.853909	+	0.520423i	$0.825774\pi$
$48$	0	0
$49$	10.9443	1.56347
$50$	0	0
$51$	−20.9443	−2.93278
$52$	0	0
$53$	1.23607	0.169787	0.0848935	−	0.996390i	$-0.472945\pi$
$53$	1.23607	0.169787	0.0848935	+	0.996390i	$0.472945\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	3.23607	0.428628
$58$	0	0
$59$	−3.23607	−0.421300	−0.210650	−	0.977562i	$-0.567558\pi$
$59$	−3.23607	−0.421300	−0.210650	+	0.977562i	$0.567558\pi$
$60$	0	0
$61$	−7.23607	−0.926484	−0.463242	−	0.886232i	$-0.653314\pi$
$61$	−7.23607	−0.926484	−0.463242	+	0.886232i	$0.653314\pi$
$62$	0	0
$63$	31.6525	3.98784
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−12.9443	−1.58139	−0.790697	−	0.612207i	$-0.790282\pi$
$67$	−12.9443	−1.58139	−0.790697	+	0.612207i	$0.790282\pi$
$68$	0	0
$69$	−3.23607	−0.389577
$70$	0	0
$71$	−10.0000	−1.18678	−0.593391	−	0.804914i	$-0.702211\pi$
$71$	−10.0000	−1.18678	−0.593391	+	0.804914i	$0.702211\pi$
$72$	0	0
$73$	−9.47214	−1.10863	−0.554315	−	0.832307i	$-0.687020\pi$
$73$	−9.47214	−1.10863	−0.554315	+	0.832307i	$0.687020\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−4.23607	−0.482745
$78$	0	0
$79$	−15.1803	−1.70792	−0.853961	−	0.520337i	$-0.825806\pi$
$79$	−15.1803	−1.70792	−0.853961	+	0.520337i	$0.825806\pi$
$80$	0	0
$81$	24.4164	2.71293
$82$	0	0
$83$	−9.00000	−0.987878	−0.493939	−	0.869496i	$-0.664443\pi$
$83$	−9.00000	−0.987878	−0.493939	+	0.869496i	$0.664443\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	5.70820	0.611984
$88$	0	0
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	−9.47214	−0.992950
$92$	0	0
$93$	1.52786	0.158432
$94$	0	0
$95$	0	0
$96$	0	0
$97$	6.18034	0.627518	0.313759	−	0.949503i	$-0.398412\pi$
$97$	6.18034	0.627518	0.313759	+	0.949503i	$0.398412\pi$
$98$	0	0
$99$	−7.47214	−0.750978
$100$	0	0
$101$	−18.9443	−1.88503	−0.942513	−	0.334170i	$-0.891544\pi$
$101$	−18.9443	−1.88503	−0.942513	+	0.334170i	$0.891544\pi$
$102$	0	0
$103$	2.23607	0.220326	0.110163	−	0.993914i	$-0.464863\pi$
$103$	2.23607	0.220326	0.110163	+	0.993914i	$0.464863\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	19.4164	1.87705	0.938527	−	0.345204i	$-0.112190\pi$
$107$	19.4164	1.87705	0.938527	+	0.345204i	$0.112190\pi$
$108$	0	0
$109$	−12.7639	−1.22256	−0.611281	−	0.791413i	$-0.709346\pi$
$109$	−12.7639	−1.22256	−0.611281	+	0.791413i	$0.709346\pi$
$110$	0	0
$111$	−36.3607	−3.45120
$112$	0	0
$113$	6.76393	0.636297	0.318149	−	0.948041i	$-0.396939\pi$
$113$	6.76393	0.636297	0.318149	+	0.948041i	$0.396939\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−16.7082	−1.54467
$118$	0	0
$119$	27.4164	2.51326
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	19.2361	1.73446
$124$	0	0
$125$	0	0
$126$	0	0
$127$	4.18034	0.370945	0.185473	−	0.982649i	$-0.440618\pi$
$127$	4.18034	0.370945	0.185473	+	0.982649i	$0.440618\pi$
$128$	0	0
$129$	8.18034	0.720239
$130$	0	0
$131$	14.9443	1.30569	0.652844	−	0.757493i	$-0.273576\pi$
$131$	14.9443	1.30569	0.652844	+	0.757493i	$0.273576\pi$
$132$	0	0
$133$	−4.23607	−0.367314
$134$	0	0
$135$	0	0
$136$	0	0
$137$	11.5279	0.984892	0.492446	−	0.870343i	$-0.336103\pi$
$137$	11.5279	0.984892	0.492446	+	0.870343i	$0.336103\pi$
$138$	0	0
$139$	−20.0000	−1.69638	−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000	−1.69638	−0.848189	+	0.529694i	$0.822307\pi$
$140$	0	0
$141$	37.8885	3.19079
$142$	0	0
$143$	2.23607	0.186989
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−35.4164	−2.92110
$148$	0	0
$149$	10.9443	0.896590	0.448295	−	0.893886i	$-0.352031\pi$
$149$	10.9443	0.896590	0.448295	+	0.893886i	$0.352031\pi$
$150$	0	0
$151$	−11.5279	−0.938124	−0.469062	−	0.883165i	$-0.655408\pi$
$151$	−11.5279	−0.938124	−0.469062	+	0.883165i	$0.655408\pi$
$152$	0	0
$153$	48.3607	3.90973
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0.291796	0.0232879	0.0116439	−	0.999932i	$-0.496294\pi$
$157$	0.291796	0.0232879	0.0116439	+	0.999932i	$0.496294\pi$
$158$	0	0
$159$	−4.00000	−0.317221
$160$	0	0
$161$	4.23607	0.333849
$162$	0	0
$163$	11.7082	0.917057	0.458529	−	0.888680i	$-0.348377\pi$
$163$	11.7082	0.917057	0.458529	+	0.888680i	$0.348377\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	5.41641	0.419134	0.209567	−	0.977794i	$-0.432795\pi$
$167$	5.41641	0.419134	0.209567	+	0.977794i	$0.432795\pi$
$168$	0	0
$169$	−8.00000	−0.615385
$170$	0	0
$171$	−7.47214	−0.571409
$172$	0	0
$173$	14.2361	1.08235	0.541174	−	0.840911i	$-0.317980\pi$
$173$	14.2361	1.08235	0.541174	+	0.840911i	$0.317980\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	10.4721	0.787134
$178$	0	0
$179$	15.7082	1.17409	0.587043	−	0.809556i	$-0.300292\pi$
$179$	15.7082	1.17409	0.587043	+	0.809556i	$0.300292\pi$
$180$	0	0
$181$	−0.944272	−0.0701872	−0.0350936	−	0.999384i	$-0.511173\pi$
$181$	−0.944272	−0.0701872	−0.0350936	+	0.999384i	$0.511173\pi$
$182$	0	0
$183$	23.4164	1.73099
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−6.47214	−0.473289
$188$	0	0
$189$	−61.3050	−4.45928
$190$	0	0
$191$	19.1803	1.38784	0.693920	−	0.720052i	$-0.255882\pi$
$191$	19.1803	1.38784	0.693920	+	0.720052i	$0.255882\pi$
$192$	0	0
$193$	2.94427	0.211933	0.105967	−	0.994370i	$-0.466206\pi$
$193$	2.94427	0.211933	0.105967	+	0.994370i	$0.466206\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−8.23607	−0.586796	−0.293398	−	0.955990i	$-0.594786\pi$
$197$	−8.23607	−0.586796	−0.293398	+	0.955990i	$0.594786\pi$
$198$	0	0
$199$	−12.7082	−0.900861	−0.450430	−	0.892812i	$-0.648729\pi$
$199$	−12.7082	−0.900861	−0.450430	+	0.892812i	$0.648729\pi$
$200$	0	0
$201$	41.8885	2.95459
$202$	0	0
$203$	−7.47214	−0.524441
$204$	0	0
$205$	0	0
$206$	0	0
$207$	7.47214	0.519349
$208$	0	0
$209$	1.00000	0.0691714
$210$	0	0
$211$	14.9443	1.02881	0.514403	−	0.857549i	$-0.328014\pi$
$211$	14.9443	1.02881	0.514403	+	0.857549i	$0.328014\pi$
$212$	0	0
$213$	32.3607	2.21732
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−2.00000	−0.135769
$218$	0	0
$219$	30.6525	2.07130
$220$	0	0
$221$	−14.4721	−0.973501
$222$	0	0
$223$	−13.2361	−0.886353	−0.443176	−	0.896434i	$-0.646148\pi$
$223$	−13.2361	−0.886353	−0.443176	+	0.896434i	$0.646148\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−12.9443	−0.859142	−0.429571	−	0.903033i	$-0.641335\pi$
$227$	−12.9443	−0.859142	−0.429571	+	0.903033i	$0.641335\pi$
$228$	0	0
$229$	6.94427	0.458890	0.229445	−	0.973322i	$-0.426309\pi$
$229$	6.94427	0.458890	0.229445	+	0.973322i	$0.426309\pi$
$230$	0	0
$231$	13.7082	0.901934
$232$	0	0
$233$	4.52786	0.296630	0.148315	−	0.988940i	$-0.452615\pi$
$233$	4.52786	0.296630	0.148315	+	0.988940i	$0.452615\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	49.1246	3.19099
$238$	0	0
$239$	−6.76393	−0.437522	−0.218761	−	0.975778i	$-0.570202\pi$
$239$	−6.76393	−0.437522	−0.218761	+	0.975778i	$0.570202\pi$
$240$	0	0
$241$	21.2361	1.36794	0.683968	−	0.729512i	$-0.260253\pi$
$241$	21.2361	1.36794	0.683968	+	0.729512i	$0.260253\pi$
$242$	0	0
$243$	−35.5967	−2.28353
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.23607	0.142278
$248$	0	0
$249$	29.1246	1.84570
$250$	0	0
$251$	5.52786	0.348916	0.174458	−	0.984665i	$-0.444183\pi$
$251$	5.52786	0.348916	0.174458	+	0.984665i	$0.444183\pi$
$252$	0	0
$253$	−1.00000	−0.0628695
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−14.0000	−0.873296	−0.436648	−	0.899632i	$-0.643834\pi$
$257$	−14.0000	−0.873296	−0.436648	+	0.899632i	$0.643834\pi$
$258$	0	0
$259$	47.5967	2.95752
$260$	0	0
$261$	−13.1803	−0.815843
$262$	0	0
$263$	−8.00000	−0.493301	−0.246651	−	0.969104i	$-0.579330\pi$
$263$	−8.00000	−0.493301	−0.246651	+	0.969104i	$0.579330\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−6.47214	−0.396088
$268$	0	0
$269$	−25.1803	−1.53527	−0.767636	−	0.640886i	$-0.778567\pi$
$269$	−25.1803	−1.53527	−0.767636	+	0.640886i	$0.778567\pi$
$270$	0	0
$271$	32.0689	1.94805	0.974023	−	0.226449i	$-0.0727117\pi$
$271$	32.0689	1.94805	0.974023	+	0.226449i	$0.0727117\pi$
$272$	0	0
$273$	30.6525	1.85517
$274$	0	0
$275$	0	0
$276$	0	0
$277$	14.2361	0.855362	0.427681	−	0.903930i	$-0.359331\pi$
$277$	14.2361	0.855362	0.427681	+	0.903930i	$0.359331\pi$
$278$	0	0
$279$	−3.52786	−0.211208
$280$	0	0
$281$	−22.1803	−1.32317	−0.661584	−	0.749871i	$-0.730116\pi$
$281$	−22.1803	−1.32317	−0.661584	+	0.749871i	$0.730116\pi$
$282$	0	0
$283$	−28.3607	−1.68587	−0.842934	−	0.538017i	$-0.819173\pi$
$283$	−28.3607	−1.68587	−0.842934	+	0.538017i	$0.819173\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−25.1803	−1.48635
$288$	0	0
$289$	24.8885	1.46403
$290$	0	0
$291$	−20.0000	−1.17242
$292$	0	0
$293$	−14.4721	−0.845471	−0.422736	−	0.906253i	$-0.638930\pi$
$293$	−14.4721	−0.845471	−0.422736	+	0.906253i	$0.638930\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	14.4721	0.839759
$298$	0	0
$299$	−2.23607	−0.129315
$300$	0	0
$301$	−10.7082	−0.617211
$302$	0	0
$303$	61.3050	3.52188
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−24.0000	−1.36975	−0.684876	−	0.728659i	$-0.740144\pi$
$307$	−24.0000	−1.36975	−0.684876	+	0.728659i	$0.740144\pi$
$308$	0	0
$309$	−7.23607	−0.411646
$310$	0	0
$311$	3.23607	0.183501	0.0917503	−	0.995782i	$-0.470754\pi$
$311$	3.23607	0.183501	0.0917503	+	0.995782i	$0.470754\pi$
$312$	0	0
$313$	−10.6525	−0.602114	−0.301057	−	0.953606i	$-0.597339\pi$
$313$	−10.6525	−0.602114	−0.301057	+	0.953606i	$0.597339\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−22.1246	−1.24264	−0.621321	−	0.783556i	$-0.713404\pi$
$317$	−22.1246	−1.24264	−0.621321	+	0.783556i	$0.713404\pi$
$318$	0	0
$319$	1.76393	0.0987612
$320$	0	0
$321$	−62.8328	−3.50699
$322$	0	0
$323$	−6.47214	−0.360119
$324$	0	0
$325$	0	0
$326$	0	0
$327$	41.3050	2.28417
$328$	0	0
$329$	−49.5967	−2.73436
$330$	0	0
$331$	−30.9443	−1.70085	−0.850426	−	0.526095i	$-0.823655\pi$
$331$	−30.9443	−1.70085	−0.850426	+	0.526095i	$0.823655\pi$
$332$	0	0
$333$	83.9574	4.60084
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−24.6525	−1.34291	−0.671453	−	0.741047i	$-0.734329\pi$
$337$	−24.6525	−1.34291	−0.671453	+	0.741047i	$0.734329\pi$
$338$	0	0
$339$	−21.8885	−1.18882
$340$	0	0
$341$	0.472136	0.0255676
$342$	0	0
$343$	16.7082	0.902158
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−3.05573	−0.164040	−0.0820200	−	0.996631i	$-0.526137\pi$
$347$	−3.05573	−0.164040	−0.0820200	+	0.996631i	$0.526137\pi$
$348$	0	0
$349$	−5.18034	−0.277297	−0.138649	−	0.990342i	$-0.544276\pi$
$349$	−5.18034	−0.277297	−0.138649	+	0.990342i	$0.544276\pi$
$350$	0	0
$351$	32.3607	1.72729
$352$	0	0
$353$	−4.88854	−0.260191	−0.130095	−	0.991501i	$-0.541528\pi$
$353$	−4.88854	−0.260191	−0.130095	+	0.991501i	$0.541528\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−88.7214	−4.69563
$358$	0	0
$359$	−19.7639	−1.04310	−0.521550	−	0.853221i	$-0.674646\pi$
$359$	−19.7639	−1.04310	−0.521550	+	0.853221i	$0.674646\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	32.3607	1.69850
$364$	0	0
$365$	0	0
$366$	0	0
$367$	23.6525	1.23465	0.617325	−	0.786709i	$-0.288217\pi$
$367$	23.6525	1.23465	0.617325	+	0.786709i	$0.288217\pi$
$368$	0	0
$369$	−44.4164	−2.31223
$370$	0	0
$371$	5.23607	0.271843
$372$	0	0
$373$	−12.0000	−0.621336	−0.310668	−	0.950518i	$-0.600553\pi$
$373$	−12.0000	−0.621336	−0.310668	+	0.950518i	$0.600553\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.94427	0.203140
$378$	0	0
$379$	−3.05573	−0.156962	−0.0784811	−	0.996916i	$-0.525007\pi$
$379$	−3.05573	−0.156962	−0.0784811	+	0.996916i	$0.525007\pi$
$380$	0	0
$381$	−13.5279	−0.693053
$382$	0	0
$383$	0.236068	0.0120625	0.00603126	−	0.999982i	$-0.498080\pi$
$383$	0.236068	0.0120625	0.00603126	+	0.999982i	$0.498080\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−18.8885	−0.960159
$388$	0	0
$389$	6.18034	0.313356	0.156678	−	0.987650i	$-0.449922\pi$
$389$	6.18034	0.313356	0.156678	+	0.987650i	$0.449922\pi$
$390$	0	0
$391$	6.47214	0.327310
$392$	0	0
$393$	−48.3607	−2.43947
$394$	0	0
$395$	0	0
$396$	0	0
$397$	22.3607	1.12225	0.561125	−	0.827731i	$-0.310369\pi$
$397$	22.3607	1.12225	0.561125	+	0.827731i	$0.310369\pi$
$398$	0	0
$399$	13.7082	0.686269
$400$	0	0
$401$	−13.1246	−0.655412	−0.327706	−	0.944780i	$-0.606276\pi$
$401$	−13.1246	−0.655412	−0.327706	+	0.944780i	$0.606276\pi$
$402$	0	0
$403$	1.05573	0.0525896
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−11.2361	−0.556951
$408$	0	0
$409$	−26.4164	−1.30621	−0.653104	−	0.757269i	$-0.726533\pi$
$409$	−26.4164	−1.30621	−0.653104	+	0.757269i	$0.726533\pi$
$410$	0	0
$411$	−37.3050	−1.84012
$412$	0	0
$413$	−13.7082	−0.674537
$414$	0	0
$415$	0	0
$416$	0	0
$417$	64.7214	3.16942
$418$	0	0
$419$	36.7771	1.79668	0.898339	−	0.439303i	$-0.144774\pi$
$419$	36.7771	1.79668	0.898339	+	0.439303i	$0.144774\pi$
$420$	0	0
$421$	15.8885	0.774360	0.387180	−	0.922004i	$-0.373449\pi$
$421$	15.8885	0.774360	0.387180	+	0.922004i	$0.373449\pi$
$422$	0	0
$423$	−87.4853	−4.25368
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−30.6525	−1.48338
$428$	0	0
$429$	−7.23607	−0.349361
$430$	0	0
$431$	−12.9443	−0.623504	−0.311752	−	0.950164i	$-0.600916\pi$
$431$	−12.9443	−0.623504	−0.311752	+	0.950164i	$0.600916\pi$
$432$	0	0
$433$	−29.1246	−1.39964	−0.699820	−	0.714319i	$-0.746736\pi$
$433$	−29.1246	−1.39964	−0.699820	+	0.714319i	$0.746736\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.00000	−0.0478365
$438$	0	0
$439$	−12.3607	−0.589943	−0.294972	−	0.955506i	$-0.595310\pi$
$439$	−12.3607	−0.589943	−0.294972	+	0.955506i	$0.595310\pi$
$440$	0	0
$441$	81.7771	3.89415
$442$	0	0
$443$	10.0000	0.475114	0.237557	−	0.971374i	$-0.423653\pi$
$443$	10.0000	0.475114	0.237557	+	0.971374i	$0.423653\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−35.4164	−1.67514
$448$	0	0
$449$	−36.8328	−1.73825	−0.869124	−	0.494594i	$-0.835317\pi$
$449$	−36.8328	−1.73825	−0.869124	+	0.494594i	$0.835317\pi$
$450$	0	0
$451$	5.94427	0.279905
$452$	0	0
$453$	37.3050	1.75274
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−20.6525	−0.966082	−0.483041	−	0.875598i	$-0.660468\pi$
$457$	−20.6525	−0.966082	−0.483041	+	0.875598i	$0.660468\pi$
$458$	0	0
$459$	−93.6656	−4.37194
$460$	0	0
$461$	33.6525	1.56735	0.783676	−	0.621170i	$-0.213342\pi$
$461$	33.6525	1.56735	0.783676	+	0.621170i	$0.213342\pi$
$462$	0	0
$463$	26.4721	1.23026	0.615132	−	0.788424i	$-0.289103\pi$
$463$	26.4721	1.23026	0.615132	+	0.788424i	$0.289103\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	17.0000	0.786666	0.393333	−	0.919396i	$-0.371322\pi$
$467$	17.0000	0.786666	0.393333	+	0.919396i	$0.371322\pi$
$468$	0	0
$469$	−54.8328	−2.53194
$470$	0	0
$471$	−0.944272	−0.0435098
$472$	0	0
$473$	2.52786	0.116231
$474$	0	0
$475$	0	0
$476$	0	0
$477$	9.23607	0.422891
$478$	0	0
$479$	12.1246	0.553988	0.276994	−	0.960872i	$-0.410662\pi$
$479$	12.1246	0.553988	0.276994	+	0.960872i	$0.410662\pi$
$480$	0	0
$481$	−25.1246	−1.14558
$482$	0	0
$483$	−13.7082	−0.623745
$484$	0	0
$485$	0	0
$486$	0	0
$487$	39.1246	1.77291	0.886453	−	0.462819i	$-0.153162\pi$
$487$	39.1246	1.77291	0.886453	+	0.462819i	$0.153162\pi$
$488$	0	0
$489$	−37.8885	−1.71338
$490$	0	0
$491$	3.23607	0.146042	0.0730209	−	0.997330i	$-0.476736\pi$
$491$	3.23607	0.146042	0.0730209	+	0.997330i	$0.476736\pi$
$492$	0	0
$493$	−11.4164	−0.514169
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−42.3607	−1.90014
$498$	0	0
$499$	42.5410	1.90440	0.952199	−	0.305479i	$-0.0988166\pi$
$499$	42.5410	1.90440	0.952199	+	0.305479i	$0.0988166\pi$
$500$	0	0
$501$	−17.5279	−0.783087
$502$	0	0
$503$	−16.5967	−0.740012	−0.370006	−	0.929029i	$-0.620644\pi$
$503$	−16.5967	−0.740012	−0.370006	+	0.929029i	$0.620644\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	25.8885	1.14975
$508$	0	0
$509$	30.3607	1.34571	0.672857	−	0.739773i	$-0.265067\pi$
$509$	30.3607	1.34571	0.672857	+	0.739773i	$0.265067\pi$
$510$	0	0
$511$	−40.1246	−1.77501
$512$	0	0
$513$	14.4721	0.638960
$514$	0	0
$515$	0	0
$516$	0	0
$517$	11.7082	0.514926
$518$	0	0
$519$	−46.0689	−2.02220
$520$	0	0
$521$	9.34752	0.409522	0.204761	−	0.978812i	$-0.434358\pi$
$521$	9.34752	0.409522	0.204761	+	0.978812i	$0.434358\pi$
$522$	0	0
$523$	18.8885	0.825938	0.412969	−	0.910745i	$-0.364492\pi$
$523$	18.8885	0.825938	0.412969	+	0.910745i	$0.364492\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3.05573	−0.133110
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−24.1803	−1.04934
$532$	0	0
$533$	13.2918	0.575732
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−50.8328	−2.19360
$538$	0	0
$539$	−10.9443	−0.471403
$540$	0	0
$541$	8.70820	0.374395	0.187197	−	0.982322i	$-0.440060\pi$
$541$	8.70820	0.374395	0.187197	+	0.982322i	$0.440060\pi$
$542$	0	0
$543$	3.05573	0.131134
$544$	0	0
$545$	0	0
$546$	0	0
$547$	15.7082	0.671634	0.335817	−	0.941927i	$-0.390988\pi$
$547$	15.7082	0.671634	0.335817	+	0.941927i	$0.390988\pi$
$548$	0	0
$549$	−54.0689	−2.30760
$550$	0	0
$551$	1.76393	0.0751460
$552$	0	0
$553$	−64.3050	−2.73452
$554$	0	0
$555$	0	0
$556$	0	0
$557$	36.1803	1.53301	0.766505	−	0.642238i	$-0.221994\pi$
$557$	36.1803	1.53301	0.766505	+	0.642238i	$0.221994\pi$
$558$	0	0
$559$	5.65248	0.239074
$560$	0	0
$561$	20.9443	0.884268
$562$	0	0
$563$	39.3607	1.65885	0.829427	−	0.558614i	$-0.188667\pi$
$563$	39.3607	1.65885	0.829427	+	0.558614i	$0.188667\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	103.430	4.34363
$568$	0	0
$569$	12.5836	0.527532	0.263766	−	0.964587i	$-0.415035\pi$
$569$	12.5836	0.527532	0.263766	+	0.964587i	$0.415035\pi$
$570$	0	0
$571$	−24.0000	−1.00437	−0.502184	−	0.864761i	$-0.667470\pi$
$571$	−24.0000	−1.00437	−0.502184	+	0.864761i	$0.667470\pi$
$572$	0	0
$573$	−62.0689	−2.59296
$574$	0	0
$575$	0	0
$576$	0	0
$577$	9.00000	0.374675	0.187337	−	0.982296i	$-0.440014\pi$
$577$	9.00000	0.374675	0.187337	+	0.982296i	$0.440014\pi$
$578$	0	0
$579$	−9.52786	−0.395965
$580$	0	0
$581$	−38.1246	−1.58168
$582$	0	0
$583$	−1.23607	−0.0511927
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−27.5967	−1.13904	−0.569520	−	0.821978i	$-0.692871\pi$
$587$	−27.5967	−1.13904	−0.569520	+	0.821978i	$0.692871\pi$
$588$	0	0
$589$	0.472136	0.0194540
$590$	0	0
$591$	26.6525	1.09634
$592$	0	0
$593$	44.8885	1.84335	0.921676	−	0.387961i	$-0.126820\pi$
$593$	44.8885	1.84335	0.921676	+	0.387961i	$0.126820\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	41.1246	1.68312
$598$	0	0
$599$	−16.8328	−0.687770	−0.343885	−	0.939012i	$-0.611743\pi$
$599$	−16.8328	−0.687770	−0.343885	+	0.939012i	$0.611743\pi$
$600$	0	0
$601$	−14.9443	−0.609590	−0.304795	−	0.952418i	$-0.598588\pi$
$601$	−14.9443	−0.609590	−0.304795	+	0.952418i	$0.598588\pi$
$602$	0	0
$603$	−96.7214	−3.93880
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−23.1246	−0.938599	−0.469300	−	0.883039i	$-0.655494\pi$
$607$	−23.1246	−0.938599	−0.469300	+	0.883039i	$0.655494\pi$
$608$	0	0
$609$	24.1803	0.979837
$610$	0	0
$611$	26.1803	1.05914
$612$	0	0
$613$	−18.0689	−0.729795	−0.364898	−	0.931048i	$-0.618896\pi$
$613$	−18.0689	−0.729795	−0.364898	+	0.931048i	$0.618896\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	25.3050	1.01874	0.509369	−	0.860548i	$-0.329879\pi$
$617$	25.3050	1.01874	0.509369	+	0.860548i	$0.329879\pi$
$618$	0	0
$619$	9.52786	0.382957	0.191479	−	0.981497i	$-0.438672\pi$
$619$	9.52786	0.382957	0.191479	+	0.981497i	$0.438672\pi$
$620$	0	0
$621$	−14.4721	−0.580747
$622$	0	0
$623$	8.47214	0.339429
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−3.23607	−0.129236
$628$	0	0
$629$	72.7214	2.89959
$630$	0	0
$631$	24.1246	0.960386	0.480193	−	0.877163i	$-0.340567\pi$
$631$	24.1246	0.960386	0.480193	+	0.877163i	$0.340567\pi$
$632$	0	0
$633$	−48.3607	−1.92216
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−24.4721	−0.969621
$638$	0	0
$639$	−74.7214	−2.95593
$640$	0	0
$641$	27.7771	1.09713	0.548564	−	0.836108i	$-0.315175\pi$
$641$	27.7771	1.09713	0.548564	+	0.836108i	$0.315175\pi$
$642$	0	0
$643$	−1.94427	−0.0766746	−0.0383373	−	0.999265i	$-0.512206\pi$
$643$	−1.94427	−0.0766746	−0.0383373	+	0.999265i	$0.512206\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−10.1803	−0.400230	−0.200115	−	0.979772i	$-0.564132\pi$
$647$	−10.1803	−0.400230	−0.200115	+	0.979772i	$0.564132\pi$
$648$	0	0
$649$	3.23607	0.127027
$650$	0	0
$651$	6.47214	0.253663
$652$	0	0
$653$	−39.5410	−1.54736	−0.773680	−	0.633577i	$-0.781586\pi$
$653$	−39.5410	−1.54736	−0.773680	+	0.633577i	$0.781586\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−70.7771	−2.76128
$658$	0	0
$659$	−13.4721	−0.524800	−0.262400	−	0.964959i	$-0.584514\pi$
$659$	−13.4721	−0.524800	−0.262400	+	0.964959i	$0.584514\pi$
$660$	0	0
$661$	−21.8197	−0.848686	−0.424343	−	0.905501i	$-0.639495\pi$
$661$	−21.8197	−0.848686	−0.424343	+	0.905501i	$0.639495\pi$
$662$	0	0
$663$	46.8328	1.81884
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.76393	−0.0682997
$668$	0	0
$669$	42.8328	1.65601
$670$	0	0
$671$	7.23607	0.279345
$672$	0	0
$673$	48.3050	1.86202	0.931010	−	0.364995i	$-0.118929\pi$
$673$	48.3050	1.86202	0.931010	+	0.364995i	$0.118929\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−17.7082	−0.680582	−0.340291	−	0.940320i	$-0.610526\pi$
$677$	−17.7082	−0.680582	−0.340291	+	0.940320i	$0.610526\pi$
$678$	0	0
$679$	26.1803	1.00471
$680$	0	0
$681$	41.8885	1.60517
$682$	0	0
$683$	−41.7771	−1.59856	−0.799278	−	0.600962i	$-0.794784\pi$
$683$	−41.7771	−1.59856	−0.799278	+	0.600962i	$0.794784\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−22.4721	−0.857365
$688$	0	0
$689$	−2.76393	−0.105297
$690$	0	0
$691$	42.4721	1.61572	0.807858	−	0.589377i	$-0.200627\pi$
$691$	42.4721	1.61572	0.807858	+	0.589377i	$0.200627\pi$
$692$	0	0
$693$	−31.6525	−1.20238
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−38.4721	−1.45724
$698$	0	0
$699$	−14.6525	−0.554208
$700$	0	0
$701$	−37.5279	−1.41741	−0.708704	−	0.705506i	$-0.750720\pi$
$701$	−37.5279	−1.41741	−0.708704	+	0.705506i	$0.750720\pi$
$702$	0	0
$703$	−11.2361	−0.423776
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−80.2492	−3.01808
$708$	0	0
$709$	9.41641	0.353641	0.176820	−	0.984243i	$-0.443419\pi$
$709$	9.41641	0.353641	0.176820	+	0.984243i	$0.443419\pi$
$710$	0	0
$711$	−113.430	−4.25394
$712$	0	0
$713$	−0.472136	−0.0176816
$714$	0	0
$715$	0	0
$716$	0	0
$717$	21.8885	0.817443
$718$	0	0
$719$	−37.7082	−1.40628	−0.703139	−	0.711052i	$-0.748219\pi$
$719$	−37.7082	−1.40628	−0.703139	+	0.711052i	$0.748219\pi$
$720$	0	0
$721$	9.47214	0.352761
$722$	0	0
$723$	−68.7214	−2.55577
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−44.7214	−1.65862	−0.829312	−	0.558786i	$-0.811267\pi$
$727$	−44.7214	−1.65862	−0.829312	+	0.558786i	$0.811267\pi$
$728$	0	0
$729$	41.9443	1.55349
$730$	0	0
$731$	−16.3607	−0.605122
$732$	0	0
$733$	−39.5279	−1.45999	−0.729997	−	0.683450i	$-0.760479\pi$
$733$	−39.5279	−1.45999	−0.729997	+	0.683450i	$0.760479\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	12.9443	0.476808
$738$	0	0
$739$	−9.05573	−0.333120	−0.166560	−	0.986031i	$-0.553266\pi$
$739$	−9.05573	−0.333120	−0.166560	+	0.986031i	$0.553266\pi$
$740$	0	0
$741$	−7.23607	−0.265824
$742$	0	0
$743$	−39.1803	−1.43739	−0.718694	−	0.695327i	$-0.755260\pi$
$743$	−39.1803	−1.43739	−0.718694	+	0.695327i	$0.755260\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−67.2492	−2.46052
$748$	0	0
$749$	82.2492	3.00532
$750$	0	0
$751$	22.5967	0.824567	0.412284	−	0.911056i	$-0.364731\pi$
$751$	22.5967	0.824567	0.412284	+	0.911056i	$0.364731\pi$
$752$	0	0
$753$	−17.8885	−0.651895
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−16.7639	−0.609295	−0.304648	−	0.952465i	$-0.598539\pi$
$757$	−16.7639	−0.609295	−0.304648	+	0.952465i	$0.598539\pi$
$758$	0	0
$759$	3.23607	0.117462
$760$	0	0
$761$	−52.8885	−1.91721	−0.958604	−	0.284742i	$-0.908092\pi$
$761$	−52.8885	−1.91721	−0.958604	+	0.284742i	$0.908092\pi$
$762$	0	0
$763$	−54.0689	−1.95743
$764$	0	0
$765$	0	0
$766$	0	0
$767$	7.23607	0.261279
$768$	0	0
$769$	−46.0689	−1.66129	−0.830643	−	0.556805i	$-0.812027\pi$
$769$	−46.0689	−1.66129	−0.830643	+	0.556805i	$0.812027\pi$
$770$	0	0
$771$	45.3050	1.63162
$772$	0	0
$773$	−4.76393	−0.171347	−0.0856734	−	0.996323i	$-0.527304\pi$
$773$	−4.76393	−0.171347	−0.0856734	+	0.996323i	$0.527304\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−154.026	−5.52566
$778$	0	0
$779$	5.94427	0.212976
$780$	0	0
$781$	10.0000	0.357828
$782$	0	0
$783$	25.5279	0.912291
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−39.7214	−1.41591	−0.707957	−	0.706256i	$-0.750383\pi$
$787$	−39.7214	−1.41591	−0.707957	+	0.706256i	$0.750383\pi$
$788$	0	0
$789$	25.8885	0.921657
$790$	0	0
$791$	28.6525	1.01876
$792$	0	0
$793$	16.1803	0.574581
$794$	0	0
$795$	0	0
$796$	0	0
$797$	15.2361	0.539689	0.269845	−	0.962904i	$-0.413028\pi$
$797$	15.2361	0.539689	0.269845	+	0.962904i	$0.413028\pi$
$798$	0	0
$799$	−75.7771	−2.68080
$800$	0	0
$801$	14.9443	0.528030
$802$	0	0
$803$	9.47214	0.334264
$804$	0	0
$805$	0	0
$806$	0	0
$807$	81.4853	2.86842
$808$	0	0
$809$	−9.11146	−0.320342	−0.160171	−	0.987089i	$-0.551205\pi$
$809$	−9.11146	−0.320342	−0.160171	+	0.987089i	$0.551205\pi$
$810$	0	0
$811$	−15.4164	−0.541343	−0.270672	−	0.962672i	$-0.587246\pi$
$811$	−15.4164	−0.541343	−0.270672	+	0.962672i	$0.587246\pi$
$812$	0	0
$813$	−103.777	−3.63962
$814$	0	0
$815$	0	0
$816$	0	0
$817$	2.52786	0.0884388
$818$	0	0
$819$	−70.7771	−2.47315
$820$	0	0
$821$	−22.2361	−0.776044	−0.388022	−	0.921650i	$-0.626842\pi$
$821$	−22.2361	−0.776044	−0.388022	+	0.921650i	$0.626842\pi$
$822$	0	0
$823$	−8.94427	−0.311778	−0.155889	−	0.987775i	$-0.549824\pi$
$823$	−8.94427	−0.311778	−0.155889	+	0.987775i	$0.549824\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−3.00000	−0.104320	−0.0521601	−	0.998639i	$-0.516611\pi$
$827$	−3.00000	−0.104320	−0.0521601	+	0.998639i	$0.516611\pi$
$828$	0	0
$829$	−4.23607	−0.147125	−0.0735624	−	0.997291i	$-0.523437\pi$
$829$	−4.23607	−0.147125	−0.0735624	+	0.997291i	$0.523437\pi$
$830$	0	0
$831$	−46.0689	−1.59811
$832$	0	0
$833$	70.8328	2.45421
$834$	0	0
$835$	0	0
$836$	0	0
$837$	6.83282	0.236177
$838$	0	0
$839$	−8.59675	−0.296793	−0.148396	−	0.988928i	$-0.547411\pi$
$839$	−8.59675	−0.296793	−0.148396	+	0.988928i	$0.547411\pi$
$840$	0	0
$841$	−25.8885	−0.892708
$842$	0	0
$843$	71.7771	2.47213
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−42.3607	−1.45553
$848$	0	0
$849$	91.7771	3.14978
$850$	0	0
$851$	11.2361	0.385167
$852$	0	0
$853$	28.2361	0.966785	0.483392	−	0.875404i	$-0.339404\pi$
$853$	28.2361	0.966785	0.483392	+	0.875404i	$0.339404\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−35.3050	−1.20599	−0.602997	−	0.797743i	$-0.706027\pi$
$857$	−35.3050	−1.20599	−0.602997	+	0.797743i	$0.706027\pi$
$858$	0	0
$859$	21.7082	0.740674	0.370337	−	0.928897i	$-0.379242\pi$
$859$	21.7082	0.740674	0.370337	+	0.928897i	$0.379242\pi$
$860$	0	0
$861$	81.4853	2.77701
$862$	0	0
$863$	34.5410	1.17579	0.587895	−	0.808937i	$-0.299957\pi$
$863$	34.5410	1.17579	0.587895	+	0.808937i	$0.299957\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−80.5410	−2.73532
$868$	0	0
$869$	15.1803	0.514958
$870$	0	0
$871$	28.9443	0.980739
$872$	0	0
$873$	46.1803	1.56297
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−2.58359	−0.0872417	−0.0436209	−	0.999048i	$-0.513889\pi$
$877$	−2.58359	−0.0872417	−0.0436209	+	0.999048i	$0.513889\pi$
$878$	0	0
$879$	46.8328	1.57963
$880$	0	0
$881$	37.7082	1.27042	0.635211	−	0.772339i	$-0.280913\pi$
$881$	37.7082	1.27042	0.635211	+	0.772339i	$0.280913\pi$
$882$	0	0
$883$	3.05573	0.102833	0.0514167	−	0.998677i	$-0.483626\pi$
$883$	3.05573	0.102833	0.0514167	+	0.998677i	$0.483626\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	12.3607	0.415031	0.207516	−	0.978232i	$-0.433462\pi$
$887$	12.3607	0.415031	0.207516	+	0.978232i	$0.433462\pi$
$888$	0	0
$889$	17.7082	0.593914
$890$	0	0
$891$	−24.4164	−0.817980
$892$	0	0
$893$	11.7082	0.391800
$894$	0	0
$895$	0	0
$896$	0	0
$897$	7.23607	0.241605
$898$	0	0
$899$	0.832816	0.0277760
$900$	0	0
$901$	8.00000	0.266519
$902$	0	0
$903$	34.6525	1.15316
$904$	0	0
$905$	0	0
$906$	0	0
$907$	5.58359	0.185400	0.0927001	−	0.995694i	$-0.470450\pi$
$907$	5.58359	0.185400	0.0927001	+	0.995694i	$0.470450\pi$
$908$	0	0
$909$	−141.554	−4.69506
$910$	0	0
$911$	31.0689	1.02936	0.514679	−	0.857383i	$-0.327911\pi$
$911$	31.0689	1.02936	0.514679	+	0.857383i	$0.327911\pi$
$912$	0	0
$913$	9.00000	0.297857
$914$	0	0
$915$	0	0
$916$	0	0
$917$	63.3050	2.09051
$918$	0	0
$919$	−43.7771	−1.44407	−0.722036	−	0.691855i	$-0.756794\pi$
$919$	−43.7771	−1.44407	−0.722036	+	0.691855i	$0.756794\pi$
$920$	0	0
$921$	77.6656	2.55917
$922$	0	0
$923$	22.3607	0.736011
$924$	0	0
$925$	0	0
$926$	0	0
$927$	16.7082	0.548769
$928$	0	0
$929$	31.0000	1.01708	0.508539	−	0.861039i	$-0.330186\pi$
$929$	31.0000	1.01708	0.508539	+	0.861039i	$0.330186\pi$
$930$	0	0
$931$	−10.9443	−0.358684
$932$	0	0
$933$	−10.4721	−0.342842
$934$	0	0
$935$	0	0
$936$	0	0
$937$	7.12461	0.232751	0.116375	−	0.993205i	$-0.462872\pi$
$937$	7.12461	0.232751	0.116375	+	0.993205i	$0.462872\pi$
$938$	0	0
$939$	34.4721	1.12496
$940$	0	0
$941$	−13.2361	−0.431483	−0.215742	−	0.976450i	$-0.569217\pi$
$941$	−13.2361	−0.431483	−0.215742	+	0.976450i	$0.569217\pi$
$942$	0	0
$943$	−5.94427	−0.193572
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−45.1935	−1.46859	−0.734296	−	0.678830i	$-0.762487\pi$
$947$	−45.1935	−1.46859	−0.734296	+	0.678830i	$0.762487\pi$
$948$	0	0
$949$	21.1803	0.687543
$950$	0	0
$951$	71.5967	2.32168
$952$	0	0
$953$	5.34752	0.173223	0.0866116	−	0.996242i	$-0.472396\pi$
$953$	5.34752	0.173223	0.0866116	+	0.996242i	$0.472396\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−5.70820	−0.184520
$958$	0	0
$959$	48.8328	1.57689
$960$	0	0
$961$	−30.7771	−0.992809
$962$	0	0
$963$	145.082	4.67520
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−5.88854	−0.189363	−0.0946814	−	0.995508i	$-0.530183\pi$
$967$	−5.88854	−0.189363	−0.0946814	+	0.995508i	$0.530183\pi$
$968$	0	0
$969$	20.9443	0.672827
$970$	0	0
$971$	−37.2492	−1.19538	−0.597692	−	0.801726i	$-0.703916\pi$
$971$	−37.2492	−1.19538	−0.597692	+	0.801726i	$0.703916\pi$
$972$	0	0
$973$	−84.7214	−2.71604
$974$	0	0
$975$	0	0
$976$	0	0
$977$	22.5410	0.721151	0.360576	−	0.932730i	$-0.382580\pi$
$977$	22.5410	0.721151	0.360576	+	0.932730i	$0.382580\pi$
$978$	0	0
$979$	−2.00000	−0.0639203
$980$	0	0
$981$	−95.3738	−3.04505
$982$	0	0
$983$	−27.6525	−0.881977	−0.440989	−	0.897513i	$-0.645372\pi$
$983$	−27.6525	−0.881977	−0.440989	+	0.897513i	$0.645372\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	160.498	5.10872
$988$	0	0
$989$	−2.52786	−0.0803814
$990$	0	0
$991$	41.5967	1.32136	0.660682	−	0.750666i	$-0.270267\pi$
$991$	41.5967	1.32136	0.660682	+	0.750666i	$0.270267\pi$
$992$	0	0
$993$	100.138	3.17778
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−2.12461	−0.0672871	−0.0336436	−	0.999434i	$-0.510711\pi$
$997$	−2.12461	−0.0672871	−0.0336436	+	0.999434i	$0.510711\pi$
$998$	0	0
$999$	−162.610	−5.14475

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9200.2.a.bn.1.1		2
4.3	odd	2		4600.2.a.u.1.2	yes	2
5.4	even	2		9200.2.a.bz.1.2		2
20.3	even	4		4600.2.e.l.4049.4		4
20.7	even	4		4600.2.e.l.4049.1		4
20.19	odd	2		4600.2.a.q.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4600.2.a.q.1.1	✓	2	20.19	odd	2
4600.2.a.u.1.2	yes	2	4.3	odd	2
4600.2.e.l.4049.1		4	20.7	even	4
4600.2.e.l.4049.4		4	20.3	even	4
9200.2.a.bn.1.1		2	1.1	even	1	trivial
9200.2.a.bz.1.2		2	5.4	even	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:	\( N \)	\(=\)	\( 9200 = 2^{4} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9200.a (trivial)

\(f(q)\)	\(=\)	\(q-3.23607 q^{3} +4.23607 q^{7} +7.47214 q^{9} -1.00000 q^{11} -2.23607 q^{13} +6.47214 q^{17} -1.00000 q^{19} -13.7082 q^{21} +1.00000 q^{23} -14.4721 q^{27} -1.76393 q^{29} -0.472136 q^{31} +3.23607 q^{33} +11.2361 q^{37} +7.23607 q^{39} -5.94427 q^{41} -2.52786 q^{43} -11.7082 q^{47} +10.9443 q^{49} -20.9443 q^{51} +1.23607 q^{53} +3.23607 q^{57} -3.23607 q^{59} -7.23607 q^{61} +31.6525 q^{63} -12.9443 q^{67} -3.23607 q^{69} -10.0000 q^{71} -9.47214 q^{73} -4.23607 q^{77} -15.1803 q^{79} +24.4164 q^{81} -9.00000 q^{83} +5.70820 q^{87} +2.00000 q^{89} -9.47214 q^{91} +1.52786 q^{93} +6.18034 q^{97} -7.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 4 q^{7} + 6 q^{9} - 2 q^{11} + 4 q^{17} - 2 q^{19} - 14 q^{21} + 2 q^{23} - 20 q^{27} - 8 q^{29} + 8 q^{31} + 2 q^{33} + 18 q^{37} + 10 q^{39} + 6 q^{41} - 14 q^{43} - 10 q^{47} + 4 q^{49}+ \cdots - 6 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 4 * q^7 + 6 * q^9 - 2 * q^11 + 4 * q^17 - 2 * q^19 - 14 * q^21 + 2 * q^23 - 20 * q^27 - 8 * q^29 + 8 * q^31 + 2 * q^33 + 18 * q^37 + 10 * q^39 + 6 * q^41 - 14 * q^43 - 10 * q^47 + 4 * q^49 - 24 * q^51 - 2 * q^53 + 2 * q^57 - 2 * q^59 - 10 * q^61 + 32 * q^63 - 8 * q^67 - 2 * q^69 - 20 * q^71 - 10 * q^73 - 4 * q^77 - 8 * q^79 + 22 * q^81 - 18 * q^83 - 2 * q^87 + 4 * q^89 - 10 * q^91 + 12 * q^93 - 10 * q^97 - 6 * q^99

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