LMFDB - Embedded newform 7800.2.a.bc.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7800,2,Mod(1,7800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7800.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:	$62.2833135766$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	7800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.73205 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.73205 q^{7} +1.00000 q^{9} -1.73205 q^{11} -1.00000 q^{13} -4.46410 q^{17} +3.46410 q^{19} +1.73205 q^{21} -2.00000 q^{23} +1.00000 q^{27} -9.92820 q^{29} +1.19615 q^{31} -1.73205 q^{33} +7.46410 q^{37} -1.00000 q^{39} -9.46410 q^{41} +4.92820 q^{43} -10.6603 q^{47} -4.00000 q^{49} -4.46410 q^{51} -1.92820 q^{53} +3.46410 q^{57} -1.73205 q^{59} -5.53590 q^{61} +1.73205 q^{63} -7.73205 q^{67} -2.00000 q^{69} +6.39230 q^{71} -3.46410 q^{73} -3.00000 q^{77} +14.0000 q^{79} +1.00000 q^{81} -10.2679 q^{83} -9.92820 q^{87} +10.9282 q^{89} -1.73205 q^{91} +1.19615 q^{93} +8.92820 q^{97} -1.73205 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^9	$ 2 q + 2 q^{3} + 2 q^{9} - 2 q^{13} - 2 q^{17} - 4 q^{23} + 2 q^{27} - 6 q^{29} - 8 q^{31} + 8 q^{37} - 2 q^{39} - 12 q^{41} - 4 q^{43} - 4 q^{47} - 8 q^{49} - 2 q^{51} + 10 q^{53} - 18 q^{61} - 12 q^{67} - 4 q^{69} - 8 q^{71} - 6 q^{77} + 28 q^{79} + 2 q^{81} - 24 q^{83} - 6 q^{87} + 8 q^{89} - 8 q^{93} + 4 q^{97}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^9 - 2 * q^13 - 2 * q^17 - 4 * q^23 + 2 * q^27 - 6 * q^29 - 8 * q^31 + 8 * q^37 - 2 * q^39 - 12 * q^41 - 4 * q^43 - 4 * q^47 - 8 * q^49 - 2 * q^51 + 10 * q^53 - 18 * q^61 - 12 * q^67 - 4 * q^69 - 8 * q^71 - 6 * q^77 + 28 * q^79 + 2 * q^81 - 24 * q^83 - 6 * q^87 + 8 * q^89 - 8 * q^93 + 4 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.73205	0.654654	0.327327	−	0.944911i	$-0.393852\pi$
$7$	1.73205	0.654654	0.327327	+	0.944911i	$0.393852\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.73205	−0.522233	−0.261116	−	0.965307i	$-0.584091\pi$
$11$	−1.73205	−0.522233	−0.261116	+	0.965307i	$0.584091\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.46410	−1.08270	−0.541352	−	0.840796i	$-0.682087\pi$
$17$	−4.46410	−1.08270	−0.541352	+	0.840796i	$0.682087\pi$
$18$	0	0
$19$	3.46410	0.794719	0.397360	−	0.917663i	$-0.369927\pi$
$19$	3.46410	0.794719	0.397360	+	0.917663i	$0.369927\pi$
$20$	0	0
$21$	1.73205	0.377964
$22$	0	0
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−9.92820	−1.84362	−0.921811	−	0.387641i	$-0.873290\pi$
$29$	−9.92820	−1.84362	−0.921811	+	0.387641i	$0.873290\pi$
$30$	0	0
$31$	1.19615	0.214835	0.107418	−	0.994214i	$-0.465742\pi$
$31$	1.19615	0.214835	0.107418	+	0.994214i	$0.465742\pi$
$32$	0	0
$33$	−1.73205	−0.301511
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.46410	1.22709	0.613545	−	0.789659i	$-0.289743\pi$
$37$	7.46410	1.22709	0.613545	+	0.789659i	$0.289743\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−9.46410	−1.47804	−0.739022	−	0.673681i	$-0.764712\pi$
$41$	−9.46410	−1.47804	−0.739022	+	0.673681i	$0.764712\pi$
$42$	0	0
$43$	4.92820	0.751544	0.375772	−	0.926712i	$-0.377378\pi$
$43$	4.92820	0.751544	0.375772	+	0.926712i	$0.377378\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−10.6603	−1.55496	−0.777479	−	0.628909i	$-0.783502\pi$
$47$	−10.6603	−1.55496	−0.777479	+	0.628909i	$0.783502\pi$
$48$	0	0
$49$	−4.00000	−0.571429
$50$	0	0
$51$	−4.46410	−0.625099
$52$	0	0
$53$	−1.92820	−0.264859	−0.132430	−	0.991192i	$-0.542278\pi$
$53$	−1.92820	−0.264859	−0.132430	+	0.991192i	$0.542278\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	3.46410	0.458831
$58$	0	0
$59$	−1.73205	−0.225494	−0.112747	−	0.993624i	$-0.535965\pi$
$59$	−1.73205	−0.225494	−0.112747	+	0.993624i	$0.535965\pi$
$60$	0	0
$61$	−5.53590	−0.708799	−0.354400	−	0.935094i	$-0.615315\pi$
$61$	−5.53590	−0.708799	−0.354400	+	0.935094i	$0.615315\pi$
$62$	0	0
$63$	1.73205	0.218218
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−7.73205	−0.944620	−0.472310	−	0.881432i	$-0.656580\pi$
$67$	−7.73205	−0.944620	−0.472310	+	0.881432i	$0.656580\pi$
$68$	0	0
$69$	−2.00000	−0.240772
$70$	0	0
$71$	6.39230	0.758627	0.379314	−	0.925268i	$-0.376160\pi$
$71$	6.39230	0.758627	0.379314	+	0.925268i	$0.376160\pi$
$72$	0	0
$73$	−3.46410	−0.405442	−0.202721	−	0.979236i	$-0.564979\pi$
$73$	−3.46410	−0.405442	−0.202721	+	0.979236i	$0.564979\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−3.00000	−0.341882
$78$	0	0
$79$	14.0000	1.57512	0.787562	−	0.616236i	$-0.211343\pi$
$79$	14.0000	1.57512	0.787562	+	0.616236i	$0.211343\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−10.2679	−1.12705	−0.563527	−	0.826098i	$-0.690556\pi$
$83$	−10.2679	−1.12705	−0.563527	+	0.826098i	$0.690556\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−9.92820	−1.06442
$88$	0	0
$89$	10.9282	1.15839	0.579194	−	0.815190i	$-0.303368\pi$
$89$	10.9282	1.15839	0.579194	+	0.815190i	$0.303368\pi$
$90$	0	0
$91$	−1.73205	−0.181568
$92$	0	0
$93$	1.19615	0.124035
$94$	0	0
$95$	0	0
$96$	0	0
$97$	8.92820	0.906522	0.453261	−	0.891378i	$-0.350261\pi$
$97$	8.92820	0.906522	0.453261	+	0.891378i	$0.350261\pi$
$98$	0	0
$99$	−1.73205	−0.174078
$100$	0	0
$101$	3.00000	0.298511	0.149256	−	0.988799i	$-0.452312\pi$
$101$	3.00000	0.298511	0.149256	+	0.988799i	$0.452312\pi$
$102$	0	0
$103$	−19.3205	−1.90371	−0.951853	−	0.306554i	$-0.900824\pi$
$103$	−19.3205	−1.90371	−0.951853	+	0.306554i	$0.900824\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0.535898	0.0518073	0.0259036	−	0.999664i	$-0.491754\pi$
$107$	0.535898	0.0518073	0.0259036	+	0.999664i	$0.491754\pi$
$108$	0	0
$109$	−9.46410	−0.906497	−0.453248	−	0.891384i	$-0.649735\pi$
$109$	−9.46410	−0.906497	−0.453248	+	0.891384i	$0.649735\pi$
$110$	0	0
$111$	7.46410	0.708461
$112$	0	0
$113$	−10.0000	−0.940721	−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000	−0.940721	−0.470360	+	0.882474i	$0.655876\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−7.73205	−0.708796
$120$	0	0
$121$	−8.00000	−0.727273
$122$	0	0
$123$	−9.46410	−0.853349
$124$	0	0
$125$	0	0
$126$	0	0
$127$	12.9282	1.14719	0.573596	−	0.819138i	$-0.305548\pi$
$127$	12.9282	1.14719	0.573596	+	0.819138i	$0.305548\pi$
$128$	0	0
$129$	4.92820	0.433904
$130$	0	0
$131$	−14.5359	−1.27001	−0.635004	−	0.772509i	$-0.719001\pi$
$131$	−14.5359	−1.27001	−0.635004	+	0.772509i	$0.719001\pi$
$132$	0	0
$133$	6.00000	0.520266
$134$	0	0
$135$	0	0
$136$	0	0
$137$	10.9282	0.933659	0.466830	−	0.884347i	$-0.345396\pi$
$137$	10.9282	0.933659	0.466830	+	0.884347i	$0.345396\pi$
$138$	0	0
$139$	6.39230	0.542188	0.271094	−	0.962553i	$-0.412615\pi$
$139$	6.39230	0.542188	0.271094	+	0.962553i	$0.412615\pi$
$140$	0	0
$141$	−10.6603	−0.897755
$142$	0	0
$143$	1.73205	0.144841
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−4.00000	−0.329914
$148$	0	0
$149$	−15.8564	−1.29901	−0.649504	−	0.760358i	$-0.725023\pi$
$149$	−15.8564	−1.29901	−0.649504	+	0.760358i	$0.725023\pi$
$150$	0	0
$151$	−16.1244	−1.31218	−0.656091	−	0.754682i	$-0.727791\pi$
$151$	−16.1244	−1.31218	−0.656091	+	0.754682i	$0.727791\pi$
$152$	0	0
$153$	−4.46410	−0.360901
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−11.3923	−0.909205	−0.454602	−	0.890694i	$-0.650219\pi$
$157$	−11.3923	−0.909205	−0.454602	+	0.890694i	$0.650219\pi$
$158$	0	0
$159$	−1.92820	−0.152916
$160$	0	0
$161$	−3.46410	−0.273009
$162$	0	0
$163$	−14.3923	−1.12729	−0.563646	−	0.826016i	$-0.690602\pi$
$163$	−14.3923	−1.12729	−0.563646	+	0.826016i	$0.690602\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	17.3205	1.34030	0.670151	−	0.742225i	$-0.266230\pi$
$167$	17.3205	1.34030	0.670151	+	0.742225i	$0.266230\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	3.46410	0.264906
$172$	0	0
$173$	9.92820	0.754827	0.377414	−	0.926045i	$-0.376813\pi$
$173$	9.92820	0.754827	0.377414	+	0.926045i	$0.376813\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1.73205	−0.130189
$178$	0	0
$179$	20.3923	1.52419	0.762096	−	0.647464i	$-0.224170\pi$
$179$	20.3923	1.52419	0.762096	+	0.647464i	$0.224170\pi$
$180$	0	0
$181$	−10.4641	−0.777791	−0.388895	−	0.921282i	$-0.627143\pi$
$181$	−10.4641	−0.777791	−0.388895	+	0.921282i	$0.627143\pi$
$182$	0	0
$183$	−5.53590	−0.409225
$184$	0	0
$185$	0	0
$186$	0	0
$187$	7.73205	0.565424
$188$	0	0
$189$	1.73205	0.125988
$190$	0	0
$191$	−6.39230	−0.462531	−0.231265	−	0.972891i	$-0.574287\pi$
$191$	−6.39230	−0.462531	−0.231265	+	0.972891i	$0.574287\pi$
$192$	0	0
$193$	5.46410	0.393315	0.196657	−	0.980472i	$-0.436991\pi$
$193$	5.46410	0.393315	0.196657	+	0.980472i	$0.436991\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	16.3923	1.16790	0.583952	−	0.811788i	$-0.301506\pi$
$197$	16.3923	1.16790	0.583952	+	0.811788i	$0.301506\pi$
$198$	0	0
$199$	−13.3205	−0.944266	−0.472133	−	0.881527i	$-0.656516\pi$
$199$	−13.3205	−0.944266	−0.472133	+	0.881527i	$0.656516\pi$
$200$	0	0
$201$	−7.73205	−0.545377
$202$	0	0
$203$	−17.1962	−1.20693
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−2.00000	−0.139010
$208$	0	0
$209$	−6.00000	−0.415029
$210$	0	0
$211$	−26.0000	−1.78991	−0.894957	−	0.446153i	$-0.852794\pi$
$211$	−26.0000	−1.78991	−0.894957	+	0.446153i	$0.852794\pi$
$212$	0	0
$213$	6.39230	0.437994
$214$	0	0
$215$	0	0
$216$	0	0
$217$	2.07180	0.140643
$218$	0	0
$219$	−3.46410	−0.234082
$220$	0	0
$221$	4.46410	0.300288
$222$	0	0
$223$	20.2487	1.35595	0.677977	−	0.735083i	$-0.262857\pi$
$223$	20.2487	1.35595	0.677977	+	0.735083i	$0.262857\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5.73205	0.380450	0.190225	−	0.981741i	$-0.439078\pi$
$227$	5.73205	0.380450	0.190225	+	0.981741i	$0.439078\pi$
$228$	0	0
$229$	22.7846	1.50565	0.752825	−	0.658221i	$-0.228691\pi$
$229$	22.7846	1.50565	0.752825	+	0.658221i	$0.228691\pi$
$230$	0	0
$231$	−3.00000	−0.197386
$232$	0	0
$233$	7.85641	0.514690	0.257345	−	0.966320i	$-0.417152\pi$
$233$	7.85641	0.514690	0.257345	+	0.966320i	$0.417152\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	14.0000	0.909398
$238$	0	0
$239$	−7.19615	−0.465480	−0.232740	−	0.972539i	$-0.574769\pi$
$239$	−7.19615	−0.465480	−0.232740	+	0.972539i	$0.574769\pi$
$240$	0	0
$241$	0.392305	0.0252706	0.0126353	−	0.999920i	$-0.495978\pi$
$241$	0.392305	0.0252706	0.0126353	+	0.999920i	$0.495978\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−3.46410	−0.220416
$248$	0	0
$249$	−10.2679	−0.650705
$250$	0	0
$251$	−16.3923	−1.03467	−0.517337	−	0.855782i	$-0.673076\pi$
$251$	−16.3923	−1.03467	−0.517337	+	0.855782i	$0.673076\pi$
$252$	0	0
$253$	3.46410	0.217786
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−29.2487	−1.82448	−0.912242	−	0.409651i	$-0.865650\pi$
$257$	−29.2487	−1.82448	−0.912242	+	0.409651i	$0.865650\pi$
$258$	0	0
$259$	12.9282	0.803319
$260$	0	0
$261$	−9.92820	−0.614540
$262$	0	0
$263$	11.3205	0.698052	0.349026	−	0.937113i	$-0.386512\pi$
$263$	11.3205	0.698052	0.349026	+	0.937113i	$0.386512\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	10.9282	0.668795
$268$	0	0
$269$	−19.0000	−1.15845	−0.579225	−	0.815168i	$-0.696645\pi$
$269$	−19.0000	−1.15845	−0.579225	+	0.815168i	$0.696645\pi$
$270$	0	0
$271$	−18.2679	−1.10970	−0.554849	−	0.831951i	$-0.687224\pi$
$271$	−18.2679	−1.10970	−0.554849	+	0.831951i	$0.687224\pi$
$272$	0	0
$273$	−1.73205	−0.104828
$274$	0	0
$275$	0	0
$276$	0	0
$277$	7.07180	0.424903	0.212452	−	0.977172i	$-0.431855\pi$
$277$	7.07180	0.424903	0.212452	+	0.977172i	$0.431855\pi$
$278$	0	0
$279$	1.19615	0.0716118
$280$	0	0
$281$	−0.928203	−0.0553720	−0.0276860	−	0.999617i	$-0.508814\pi$
$281$	−0.928203	−0.0553720	−0.0276860	+	0.999617i	$0.508814\pi$
$282$	0	0
$283$	25.3205	1.50515	0.752574	−	0.658508i	$-0.228812\pi$
$283$	25.3205	1.50515	0.752574	+	0.658508i	$0.228812\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−16.3923	−0.967607
$288$	0	0
$289$	2.92820	0.172247
$290$	0	0
$291$	8.92820	0.523381
$292$	0	0
$293$	−11.8564	−0.692659	−0.346329	−	0.938113i	$-0.612572\pi$
$293$	−11.8564	−0.692659	−0.346329	+	0.938113i	$0.612572\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−1.73205	−0.100504
$298$	0	0
$299$	2.00000	0.115663
$300$	0	0
$301$	8.53590	0.492001
$302$	0	0
$303$	3.00000	0.172345
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−11.4641	−0.654291	−0.327145	−	0.944974i	$-0.606087\pi$
$307$	−11.4641	−0.654291	−0.327145	+	0.944974i	$0.606087\pi$
$308$	0	0
$309$	−19.3205	−1.09911
$310$	0	0
$311$	−0.535898	−0.0303880	−0.0151940	−	0.999885i	$-0.504837\pi$
$311$	−0.535898	−0.0303880	−0.0151940	+	0.999885i	$0.504837\pi$
$312$	0	0
$313$	−11.9282	−0.674222	−0.337111	−	0.941465i	$-0.609450\pi$
$313$	−11.9282	−0.674222	−0.337111	+	0.941465i	$0.609450\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	19.4641	1.09321	0.546606	−	0.837390i	$-0.315919\pi$
$317$	19.4641	1.09321	0.546606	+	0.837390i	$0.315919\pi$
$318$	0	0
$319$	17.1962	0.962800
$320$	0	0
$321$	0.535898	0.0299109
$322$	0	0
$323$	−15.4641	−0.860446
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−9.46410	−0.523366
$328$	0	0
$329$	−18.4641	−1.01796
$330$	0	0
$331$	−11.4641	−0.630124	−0.315062	−	0.949071i	$-0.602025\pi$
$331$	−11.4641	−0.630124	−0.315062	+	0.949071i	$0.602025\pi$
$332$	0	0
$333$	7.46410	0.409030
$334$	0	0
$335$	0	0
$336$	0	0
$337$	30.8564	1.68086	0.840428	−	0.541924i	$-0.182304\pi$
$337$	30.8564	1.68086	0.840428	+	0.541924i	$0.182304\pi$
$338$	0	0
$339$	−10.0000	−0.543125
$340$	0	0
$341$	−2.07180	−0.112194
$342$	0	0
$343$	−19.0526	−1.02874
$344$	0	0
$345$	0	0
$346$	0	0
$347$	32.2487	1.73120	0.865601	−	0.500735i	$-0.166937\pi$
$347$	32.2487	1.73120	0.865601	+	0.500735i	$0.166937\pi$
$348$	0	0
$349$	6.39230	0.342172	0.171086	−	0.985256i	$-0.445272\pi$
$349$	6.39230	0.342172	0.171086	+	0.985256i	$0.445272\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−27.3205	−1.45412	−0.727062	−	0.686572i	$-0.759115\pi$
$353$	−27.3205	−1.45412	−0.727062	+	0.686572i	$0.759115\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−7.73205	−0.409224
$358$	0	0
$359$	0.267949	0.0141418	0.00707091	−	0.999975i	$-0.497749\pi$
$359$	0.267949	0.0141418	0.00707091	+	0.999975i	$0.497749\pi$
$360$	0	0
$361$	−7.00000	−0.368421
$362$	0	0
$363$	−8.00000	−0.419891
$364$	0	0
$365$	0	0
$366$	0	0
$367$	6.92820	0.361649	0.180825	−	0.983515i	$-0.442123\pi$
$367$	6.92820	0.361649	0.180825	+	0.983515i	$0.442123\pi$
$368$	0	0
$369$	−9.46410	−0.492681
$370$	0	0
$371$	−3.33975	−0.173391
$372$	0	0
$373$	0.607695	0.0314653	0.0157326	−	0.999876i	$-0.494992\pi$
$373$	0.607695	0.0314653	0.0157326	+	0.999876i	$0.494992\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	9.92820	0.511328
$378$	0	0
$379$	15.1962	0.780574	0.390287	−	0.920693i	$-0.372376\pi$
$379$	15.1962	0.780574	0.390287	+	0.920693i	$0.372376\pi$
$380$	0	0
$381$	12.9282	0.662332
$382$	0	0
$383$	−15.4641	−0.790179	−0.395089	−	0.918643i	$-0.629286\pi$
$383$	−15.4641	−0.790179	−0.395089	+	0.918643i	$0.629286\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4.92820	0.250515
$388$	0	0
$389$	19.8564	1.00676	0.503380	−	0.864065i	$-0.332090\pi$
$389$	19.8564	1.00676	0.503380	+	0.864065i	$0.332090\pi$
$390$	0	0
$391$	8.92820	0.451519
$392$	0	0
$393$	−14.5359	−0.733239
$394$	0	0
$395$	0	0
$396$	0	0
$397$	22.0000	1.10415	0.552074	−	0.833795i	$-0.313837\pi$
$397$	22.0000	1.10415	0.552074	+	0.833795i	$0.313837\pi$
$398$	0	0
$399$	6.00000	0.300376
$400$	0	0
$401$	38.1051	1.90288	0.951439	−	0.307836i	$-0.0996049\pi$
$401$	38.1051	1.90288	0.951439	+	0.307836i	$0.0996049\pi$
$402$	0	0
$403$	−1.19615	−0.0595846
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−12.9282	−0.640827
$408$	0	0
$409$	−23.3205	−1.15312	−0.576562	−	0.817053i	$-0.695606\pi$
$409$	−23.3205	−1.15312	−0.576562	+	0.817053i	$0.695606\pi$
$410$	0	0
$411$	10.9282	0.539049
$412$	0	0
$413$	−3.00000	−0.147620
$414$	0	0
$415$	0	0
$416$	0	0
$417$	6.39230	0.313033
$418$	0	0
$419$	11.3205	0.553043	0.276522	−	0.961008i	$-0.410818\pi$
$419$	11.3205	0.553043	0.276522	+	0.961008i	$0.410818\pi$
$420$	0	0
$421$	−18.9282	−0.922504	−0.461252	−	0.887269i	$-0.652600\pi$
$421$	−18.9282	−0.922504	−0.461252	+	0.887269i	$0.652600\pi$
$422$	0	0
$423$	−10.6603	−0.518319
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−9.58846	−0.464018
$428$	0	0
$429$	1.73205	0.0836242
$430$	0	0
$431$	−18.3923	−0.885926	−0.442963	−	0.896540i	$-0.646073\pi$
$431$	−18.3923	−0.885926	−0.442963	+	0.896540i	$0.646073\pi$
$432$	0	0
$433$	12.9282	0.621290	0.310645	−	0.950526i	$-0.399455\pi$
$433$	12.9282	0.621290	0.310645	+	0.950526i	$0.399455\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−6.92820	−0.331421
$438$	0	0
$439$	−30.7846	−1.46927	−0.734635	−	0.678463i	$-0.762646\pi$
$439$	−30.7846	−1.46927	−0.734635	+	0.678463i	$0.762646\pi$
$440$	0	0
$441$	−4.00000	−0.190476
$442$	0	0
$443$	4.92820	0.234146	0.117073	−	0.993123i	$-0.462649\pi$
$443$	4.92820	0.234146	0.117073	+	0.993123i	$0.462649\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−15.8564	−0.749982
$448$	0	0
$449$	−21.0718	−0.994440	−0.497220	−	0.867625i	$-0.665646\pi$
$449$	−21.0718	−0.994440	−0.497220	+	0.867625i	$0.665646\pi$
$450$	0	0
$451$	16.3923	0.771883
$452$	0	0
$453$	−16.1244	−0.757588
$454$	0	0
$455$	0	0
$456$	0	0
$457$	23.3205	1.09089	0.545444	−	0.838147i	$-0.316361\pi$
$457$	23.3205	1.09089	0.545444	+	0.838147i	$0.316361\pi$
$458$	0	0
$459$	−4.46410	−0.208366
$460$	0	0
$461$	−20.9282	−0.974724	−0.487362	−	0.873200i	$-0.662041\pi$
$461$	−20.9282	−0.974724	−0.487362	+	0.873200i	$0.662041\pi$
$462$	0	0
$463$	9.73205	0.452287	0.226143	−	0.974094i	$-0.427388\pi$
$463$	9.73205	0.452287	0.226143	+	0.974094i	$0.427388\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1.85641	−0.0859042	−0.0429521	−	0.999077i	$-0.513676\pi$
$467$	−1.85641	−0.0859042	−0.0429521	+	0.999077i	$0.513676\pi$
$468$	0	0
$469$	−13.3923	−0.618399
$470$	0	0
$471$	−11.3923	−0.524930
$472$	0	0
$473$	−8.53590	−0.392481
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−1.92820	−0.0882864
$478$	0	0
$479$	13.0526	0.596387	0.298193	−	0.954505i	$-0.403616\pi$
$479$	13.0526	0.596387	0.298193	+	0.954505i	$0.403616\pi$
$480$	0	0
$481$	−7.46410	−0.340334
$482$	0	0
$483$	−3.46410	−0.157622
$484$	0	0
$485$	0	0
$486$	0	0
$487$	17.1962	0.779232	0.389616	−	0.920977i	$-0.372608\pi$
$487$	17.1962	0.779232	0.389616	+	0.920977i	$0.372608\pi$
$488$	0	0
$489$	−14.3923	−0.650843
$490$	0	0
$491$	−37.3205	−1.68425	−0.842125	−	0.539282i	$-0.818696\pi$
$491$	−37.3205	−1.68425	−0.842125	+	0.539282i	$0.818696\pi$
$492$	0	0
$493$	44.3205	1.99610
$494$	0	0
$495$	0	0
$496$	0	0
$497$	11.0718	0.496638
$498$	0	0
$499$	−21.5885	−0.966432	−0.483216	−	0.875501i	$-0.660531\pi$
$499$	−21.5885	−0.966432	−0.483216	+	0.875501i	$0.660531\pi$
$500$	0	0
$501$	17.3205	0.773823
$502$	0	0
$503$	−21.4641	−0.957037	−0.478518	−	0.878077i	$-0.658826\pi$
$503$	−21.4641	−0.957037	−0.478518	+	0.878077i	$0.658826\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	34.2487	1.51805	0.759024	−	0.651063i	$-0.225677\pi$
$509$	34.2487	1.51805	0.759024	+	0.651063i	$0.225677\pi$
$510$	0	0
$511$	−6.00000	−0.265424
$512$	0	0
$513$	3.46410	0.152944
$514$	0	0
$515$	0	0
$516$	0	0
$517$	18.4641	0.812050
$518$	0	0
$519$	9.92820	0.435800
$520$	0	0
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	0	0
$523$	−13.3205	−0.582465	−0.291233	−	0.956652i	$-0.594065\pi$
$523$	−13.3205	−0.582465	−0.291233	+	0.956652i	$0.594065\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−5.33975	−0.232603
$528$	0	0
$529$	−19.0000	−0.826087
$530$	0	0
$531$	−1.73205	−0.0751646
$532$	0	0
$533$	9.46410	0.409936
$534$	0	0
$535$	0	0
$536$	0	0
$537$	20.3923	0.879993
$538$	0	0
$539$	6.92820	0.298419
$540$	0	0
$541$	2.39230	0.102853	0.0514266	−	0.998677i	$-0.483623\pi$
$541$	2.39230	0.102853	0.0514266	+	0.998677i	$0.483623\pi$
$542$	0	0
$543$	−10.4641	−0.449058
$544$	0	0
$545$	0	0
$546$	0	0
$547$	2.53590	0.108427	0.0542136	−	0.998529i	$-0.482735\pi$
$547$	2.53590	0.108427	0.0542136	+	0.998529i	$0.482735\pi$
$548$	0	0
$549$	−5.53590	−0.236266
$550$	0	0
$551$	−34.3923	−1.46516
$552$	0	0
$553$	24.2487	1.03116
$554$	0	0
$555$	0	0
$556$	0	0
$557$	11.6077	0.491834	0.245917	−	0.969291i	$-0.420911\pi$
$557$	11.6077	0.491834	0.245917	+	0.969291i	$0.420911\pi$
$558$	0	0
$559$	−4.92820	−0.208441
$560$	0	0
$561$	7.73205	0.326447
$562$	0	0
$563$	−18.0000	−0.758610	−0.379305	−	0.925272i	$-0.623837\pi$
$563$	−18.0000	−0.758610	−0.379305	+	0.925272i	$0.623837\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.73205	0.0727393
$568$	0	0
$569$	−29.2487	−1.22617	−0.613085	−	0.790017i	$-0.710072\pi$
$569$	−29.2487	−1.22617	−0.613085	+	0.790017i	$0.710072\pi$
$570$	0	0
$571$	−0.143594	−0.00600920	−0.00300460	−	0.999995i	$-0.500956\pi$
$571$	−0.143594	−0.00600920	−0.00300460	+	0.999995i	$0.500956\pi$
$572$	0	0
$573$	−6.39230	−0.267042
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−36.6410	−1.52539	−0.762693	−	0.646761i	$-0.776123\pi$
$577$	−36.6410	−1.52539	−0.762693	+	0.646761i	$0.776123\pi$
$578$	0	0
$579$	5.46410	0.227080
$580$	0	0
$581$	−17.7846	−0.737830
$582$	0	0
$583$	3.33975	0.138318
$584$	0	0
$585$	0	0
$586$	0	0
$587$	7.58846	0.313209	0.156605	−	0.987661i	$-0.449945\pi$
$587$	7.58846	0.313209	0.156605	+	0.987661i	$0.449945\pi$
$588$	0	0
$589$	4.14359	0.170734
$590$	0	0
$591$	16.3923	0.674289
$592$	0	0
$593$	−0.928203	−0.0381167	−0.0190584	−	0.999818i	$-0.506067\pi$
$593$	−0.928203	−0.0381167	−0.0190584	+	0.999818i	$0.506067\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−13.3205	−0.545172
$598$	0	0
$599$	−33.1769	−1.35557	−0.677786	−	0.735259i	$-0.737060\pi$
$599$	−33.1769	−1.35557	−0.677786	+	0.735259i	$0.737060\pi$
$600$	0	0
$601$	−32.7128	−1.33438	−0.667192	−	0.744886i	$-0.732504\pi$
$601$	−32.7128	−1.33438	−0.667192	+	0.744886i	$0.732504\pi$
$602$	0	0
$603$	−7.73205	−0.314873
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−45.3205	−1.83950	−0.919751	−	0.392502i	$-0.871610\pi$
$607$	−45.3205	−1.83950	−0.919751	+	0.392502i	$0.871610\pi$
$608$	0	0
$609$	−17.1962	−0.696823
$610$	0	0
$611$	10.6603	0.431268
$612$	0	0
$613$	−24.9282	−1.00684	−0.503420	−	0.864042i	$-0.667925\pi$
$613$	−24.9282	−1.00684	−0.503420	+	0.864042i	$0.667925\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−19.8564	−0.799389	−0.399694	−	0.916648i	$-0.630884\pi$
$617$	−19.8564	−0.799389	−0.399694	+	0.916648i	$0.630884\pi$
$618$	0	0
$619$	−9.32051	−0.374623	−0.187311	−	0.982301i	$-0.559977\pi$
$619$	−9.32051	−0.374623	−0.187311	+	0.982301i	$0.559977\pi$
$620$	0	0
$621$	−2.00000	−0.0802572
$622$	0	0
$623$	18.9282	0.758342
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−6.00000	−0.239617
$628$	0	0
$629$	−33.3205	−1.32858
$630$	0	0
$631$	−8.24871	−0.328376	−0.164188	−	0.986429i	$-0.552500\pi$
$631$	−8.24871	−0.328376	−0.164188	+	0.986429i	$0.552500\pi$
$632$	0	0
$633$	−26.0000	−1.03341
$634$	0	0
$635$	0	0
$636$	0	0
$637$	4.00000	0.158486
$638$	0	0
$639$	6.39230	0.252876
$640$	0	0
$641$	28.6077	1.12994	0.564968	−	0.825113i	$-0.308888\pi$
$641$	28.6077	1.12994	0.564968	+	0.825113i	$0.308888\pi$
$642$	0	0
$643$	−13.3205	−0.525310	−0.262655	−	0.964890i	$-0.584598\pi$
$643$	−13.3205	−0.525310	−0.262655	+	0.964890i	$0.584598\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−7.71281	−0.303222	−0.151611	−	0.988440i	$-0.548446\pi$
$647$	−7.71281	−0.303222	−0.151611	+	0.988440i	$0.548446\pi$
$648$	0	0
$649$	3.00000	0.117760
$650$	0	0
$651$	2.07180	0.0812001
$652$	0	0
$653$	22.0718	0.863736	0.431868	−	0.901937i	$-0.357855\pi$
$653$	22.0718	0.863736	0.431868	+	0.901937i	$0.357855\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−3.46410	−0.135147
$658$	0	0
$659$	−25.7128	−1.00163	−0.500814	−	0.865555i	$-0.666966\pi$
$659$	−25.7128	−1.00163	−0.500814	+	0.865555i	$0.666966\pi$
$660$	0	0
$661$	−15.6077	−0.607069	−0.303534	−	0.952820i	$-0.598167\pi$
$661$	−15.6077	−0.607069	−0.303534	+	0.952820i	$0.598167\pi$
$662$	0	0
$663$	4.46410	0.173371
$664$	0	0
$665$	0	0
$666$	0	0
$667$	19.8564	0.768843
$668$	0	0
$669$	20.2487	0.782860
$670$	0	0
$671$	9.58846	0.370158
$672$	0	0
$673$	11.1436	0.429554	0.214777	−	0.976663i	$-0.431098\pi$
$673$	11.1436	0.429554	0.214777	+	0.976663i	$0.431098\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	28.6410	1.10076	0.550382	−	0.834913i	$-0.314482\pi$
$677$	28.6410	1.10076	0.550382	+	0.834913i	$0.314482\pi$
$678$	0	0
$679$	15.4641	0.593458
$680$	0	0
$681$	5.73205	0.219653
$682$	0	0
$683$	−4.94744	−0.189309	−0.0946543	−	0.995510i	$-0.530175\pi$
$683$	−4.94744	−0.189309	−0.0946543	+	0.995510i	$0.530175\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	22.7846	0.869287
$688$	0	0
$689$	1.92820	0.0734587
$690$	0	0
$691$	−13.8756	−0.527854	−0.263927	−	0.964543i	$-0.585018\pi$
$691$	−13.8756	−0.527854	−0.263927	+	0.964543i	$0.585018\pi$
$692$	0	0
$693$	−3.00000	−0.113961
$694$	0	0
$695$	0	0
$696$	0	0
$697$	42.2487	1.60028
$698$	0	0
$699$	7.85641	0.297157
$700$	0	0
$701$	28.8564	1.08989	0.544946	−	0.838471i	$-0.316550\pi$
$701$	28.8564	1.08989	0.544946	+	0.838471i	$0.316550\pi$
$702$	0	0
$703$	25.8564	0.975193
$704$	0	0
$705$	0	0
$706$	0	0
$707$	5.19615	0.195421
$708$	0	0
$709$	−10.5359	−0.395684	−0.197842	−	0.980234i	$-0.563393\pi$
$709$	−10.5359	−0.395684	−0.197842	+	0.980234i	$0.563393\pi$
$710$	0	0
$711$	14.0000	0.525041
$712$	0	0
$713$	−2.39230	−0.0895925
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−7.19615	−0.268745
$718$	0	0
$719$	−6.53590	−0.243748	−0.121874	−	0.992546i	$-0.538890\pi$
$719$	−6.53590	−0.243748	−0.121874	+	0.992546i	$0.538890\pi$
$720$	0	0
$721$	−33.4641	−1.24627
$722$	0	0
$723$	0.392305	0.0145900
$724$	0	0
$725$	0	0
$726$	0	0
$727$	26.2487	0.973511	0.486755	−	0.873538i	$-0.338180\pi$
$727$	26.2487	0.973511	0.486755	+	0.873538i	$0.338180\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−22.0000	−0.813699
$732$	0	0
$733$	24.7846	0.915440	0.457720	−	0.889096i	$-0.348666\pi$
$733$	24.7846	0.915440	0.457720	+	0.889096i	$0.348666\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	13.3923	0.493312
$738$	0	0
$739$	39.7321	1.46157	0.730784	−	0.682609i	$-0.239155\pi$
$739$	39.7321	1.46157	0.730784	+	0.682609i	$0.239155\pi$
$740$	0	0
$741$	−3.46410	−0.127257
$742$	0	0
$743$	−43.1962	−1.58471	−0.792357	−	0.610058i	$-0.791146\pi$
$743$	−43.1962	−1.58471	−0.792357	+	0.610058i	$0.791146\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−10.2679	−0.375685
$748$	0	0
$749$	0.928203	0.0339158
$750$	0	0
$751$	11.8564	0.432646	0.216323	−	0.976322i	$-0.430594\pi$
$751$	11.8564	0.432646	0.216323	+	0.976322i	$0.430594\pi$
$752$	0	0
$753$	−16.3923	−0.597369
$754$	0	0
$755$	0	0
$756$	0	0
$757$	27.5359	1.00081	0.500405	−	0.865792i	$-0.333185\pi$
$757$	27.5359	1.00081	0.500405	+	0.865792i	$0.333185\pi$
$758$	0	0
$759$	3.46410	0.125739
$760$	0	0
$761$	17.6077	0.638278	0.319139	−	0.947708i	$-0.396606\pi$
$761$	17.6077	0.638278	0.319139	+	0.947708i	$0.396606\pi$
$762$	0	0
$763$	−16.3923	−0.593441
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.73205	0.0625407
$768$	0	0
$769$	−20.7846	−0.749512	−0.374756	−	0.927123i	$-0.622274\pi$
$769$	−20.7846	−0.749512	−0.374756	+	0.927123i	$0.622274\pi$
$770$	0	0
$771$	−29.2487	−1.05337
$772$	0	0
$773$	15.8564	0.570315	0.285158	−	0.958481i	$-0.407954\pi$
$773$	15.8564	0.570315	0.285158	+	0.958481i	$0.407954\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	12.9282	0.463797
$778$	0	0
$779$	−32.7846	−1.17463
$780$	0	0
$781$	−11.0718	−0.396180
$782$	0	0
$783$	−9.92820	−0.354805
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−23.1962	−0.826854	−0.413427	−	0.910537i	$-0.635668\pi$
$787$	−23.1962	−0.826854	−0.413427	+	0.910537i	$0.635668\pi$
$788$	0	0
$789$	11.3205	0.403021
$790$	0	0
$791$	−17.3205	−0.615846
$792$	0	0
$793$	5.53590	0.196586
$794$	0	0
$795$	0	0
$796$	0	0
$797$	4.85641	0.172023	0.0860114	−	0.996294i	$-0.472588\pi$
$797$	4.85641	0.172023	0.0860114	+	0.996294i	$0.472588\pi$
$798$	0	0
$799$	47.5885	1.68356
$800$	0	0
$801$	10.9282	0.386129
$802$	0	0
$803$	6.00000	0.211735
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−19.0000	−0.668832
$808$	0	0
$809$	24.9282	0.876429	0.438214	−	0.898870i	$-0.355611\pi$
$809$	24.9282	0.876429	0.438214	+	0.898870i	$0.355611\pi$
$810$	0	0
$811$	−29.5885	−1.03899	−0.519496	−	0.854473i	$-0.673880\pi$
$811$	−29.5885	−1.03899	−0.519496	+	0.854473i	$0.673880\pi$
$812$	0	0
$813$	−18.2679	−0.640685
$814$	0	0
$815$	0	0
$816$	0	0
$817$	17.0718	0.597267
$818$	0	0
$819$	−1.73205	−0.0605228
$820$	0	0
$821$	20.5359	0.716708	0.358354	−	0.933586i	$-0.383338\pi$
$821$	20.5359	0.716708	0.358354	+	0.933586i	$0.383338\pi$
$822$	0	0
$823$	37.8564	1.31959	0.659796	−	0.751445i	$-0.270643\pi$
$823$	37.8564	1.31959	0.659796	+	0.751445i	$0.270643\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	4.12436	0.143418	0.0717089	−	0.997426i	$-0.477155\pi$
$827$	4.12436	0.143418	0.0717089	+	0.997426i	$0.477155\pi$
$828$	0	0
$829$	35.3923	1.22923	0.614613	−	0.788829i	$-0.289312\pi$
$829$	35.3923	1.22923	0.614613	+	0.788829i	$0.289312\pi$
$830$	0	0
$831$	7.07180	0.245318
$832$	0	0
$833$	17.8564	0.618688
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.19615	0.0413451
$838$	0	0
$839$	5.32051	0.183684	0.0918422	−	0.995774i	$-0.470724\pi$
$839$	5.32051	0.183684	0.0918422	+	0.995774i	$0.470724\pi$
$840$	0	0
$841$	69.5692	2.39894
$842$	0	0
$843$	−0.928203	−0.0319690
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−13.8564	−0.476112
$848$	0	0
$849$	25.3205	0.868998
$850$	0	0
$851$	−14.9282	−0.511732
$852$	0	0
$853$	7.21539	0.247050	0.123525	−	0.992341i	$-0.460580\pi$
$853$	7.21539	0.247050	0.123525	+	0.992341i	$0.460580\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	33.7128	1.15161	0.575804	−	0.817588i	$-0.304689\pi$
$857$	33.7128	1.15161	0.575804	+	0.817588i	$0.304689\pi$
$858$	0	0
$859$	45.7128	1.55970	0.779851	−	0.625966i	$-0.215295\pi$
$859$	45.7128	1.55970	0.779851	+	0.625966i	$0.215295\pi$
$860$	0	0
$861$	−16.3923	−0.558648
$862$	0	0
$863$	−49.0526	−1.66977	−0.834884	−	0.550426i	$-0.814465\pi$
$863$	−49.0526	−1.66977	−0.834884	+	0.550426i	$0.814465\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	2.92820	0.0994470
$868$	0	0
$869$	−24.2487	−0.822581
$870$	0	0
$871$	7.73205	0.261991
$872$	0	0
$873$	8.92820	0.302174
$874$	0	0
$875$	0	0
$876$	0	0
$877$	33.0718	1.11676	0.558378	−	0.829587i	$-0.311424\pi$
$877$	33.0718	1.11676	0.558378	+	0.829587i	$0.311424\pi$
$878$	0	0
$879$	−11.8564	−0.399907
$880$	0	0
$881$	6.46410	0.217781	0.108891	−	0.994054i	$-0.465270\pi$
$881$	6.46410	0.217781	0.108891	+	0.994054i	$0.465270\pi$
$882$	0	0
$883$	34.9282	1.17543	0.587714	−	0.809069i	$-0.300028\pi$
$883$	34.9282	1.17543	0.587714	+	0.809069i	$0.300028\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−38.7846	−1.30226	−0.651130	−	0.758966i	$-0.725705\pi$
$887$	−38.7846	−1.30226	−0.651130	+	0.758966i	$0.725705\pi$
$888$	0	0
$889$	22.3923	0.751014
$890$	0	0
$891$	−1.73205	−0.0580259
$892$	0	0
$893$	−36.9282	−1.23576
$894$	0	0
$895$	0	0
$896$	0	0
$897$	2.00000	0.0667781
$898$	0	0
$899$	−11.8756	−0.396075
$900$	0	0
$901$	8.60770	0.286764
$902$	0	0
$903$	8.53590	0.284057
$904$	0	0
$905$	0	0
$906$	0	0
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	0	0
$909$	3.00000	0.0995037
$910$	0	0
$911$	12.9282	0.428330	0.214165	−	0.976797i	$-0.431297\pi$
$911$	12.9282	0.428330	0.214165	+	0.976797i	$0.431297\pi$
$912$	0	0
$913$	17.7846	0.588585
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−25.1769	−0.831415
$918$	0	0
$919$	33.0333	1.08967	0.544834	−	0.838544i	$-0.316593\pi$
$919$	33.0333	1.08967	0.544834	+	0.838544i	$0.316593\pi$
$920$	0	0
$921$	−11.4641	−0.377755
$922$	0	0
$923$	−6.39230	−0.210405
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−19.3205	−0.634569
$928$	0	0
$929$	16.6410	0.545974	0.272987	−	0.962018i	$-0.411988\pi$
$929$	16.6410	0.545974	0.272987	+	0.962018i	$0.411988\pi$
$930$	0	0
$931$	−13.8564	−0.454125
$932$	0	0
$933$	−0.535898	−0.0175445
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−23.6410	−0.772318	−0.386159	−	0.922432i	$-0.626198\pi$
$937$	−23.6410	−0.772318	−0.386159	+	0.922432i	$0.626198\pi$
$938$	0	0
$939$	−11.9282	−0.389262
$940$	0	0
$941$	52.2487	1.70326	0.851630	−	0.524144i	$-0.175615\pi$
$941$	52.2487	1.70326	0.851630	+	0.524144i	$0.175615\pi$
$942$	0	0
$943$	18.9282	0.616387
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−24.6603	−0.801351	−0.400675	−	0.916220i	$-0.631224\pi$
$947$	−24.6603	−0.801351	−0.400675	+	0.916220i	$0.631224\pi$
$948$	0	0
$949$	3.46410	0.112449
$950$	0	0
$951$	19.4641	0.631167
$952$	0	0
$953$	19.2487	0.623527	0.311763	−	0.950160i	$-0.399080\pi$
$953$	19.2487	0.623527	0.311763	+	0.950160i	$0.399080\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	17.1962	0.555873
$958$	0	0
$959$	18.9282	0.611224
$960$	0	0
$961$	−29.5692	−0.953846
$962$	0	0
$963$	0.535898	0.0172691
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−25.4449	−0.818252	−0.409126	−	0.912478i	$-0.634166\pi$
$967$	−25.4449	−0.818252	−0.409126	+	0.912478i	$0.634166\pi$
$968$	0	0
$969$	−15.4641	−0.496779
$970$	0	0
$971$	35.1769	1.12888	0.564440	−	0.825474i	$-0.309092\pi$
$971$	35.1769	1.12888	0.564440	+	0.825474i	$0.309092\pi$
$972$	0	0
$973$	11.0718	0.354946
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−41.5692	−1.32992	−0.664959	−	0.746880i	$-0.731551\pi$
$977$	−41.5692	−1.32992	−0.664959	+	0.746880i	$0.731551\pi$
$978$	0	0
$979$	−18.9282	−0.604948
$980$	0	0
$981$	−9.46410	−0.302166
$982$	0	0
$983$	12.2679	0.391287	0.195643	−	0.980675i	$-0.437320\pi$
$983$	12.2679	0.391287	0.195643	+	0.980675i	$0.437320\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−18.4641	−0.587719
$988$	0	0
$989$	−9.85641	−0.313415
$990$	0	0
$991$	5.71281	0.181473	0.0907367	−	0.995875i	$-0.471078\pi$
$991$	5.71281	0.181473	0.0907367	+	0.995875i	$0.471078\pi$
$992$	0	0
$993$	−11.4641	−0.363802
$994$	0	0
$995$	0	0
$996$	0	0
$997$	44.0333	1.39455	0.697275	−	0.716804i	$-0.254396\pi$
$997$	44.0333	1.39455	0.697275	+	0.716804i	$0.254396\pi$
$998$	0	0
$999$	7.46410	0.236154

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7800.2.a.bc.1.2	yes	2
5.4	even	2		7800.2.a.y.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7800.2.a.y.1.1	✓	2	5.4	even	2
7800.2.a.bc.1.2	yes	2	1.1	even	1	trivial

Overview	Random
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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:	\( N \)	\(=\)	\( 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7800.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.73205 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.73205 q^{7} +1.00000 q^{9} -1.73205 q^{11} -1.00000 q^{13} -4.46410 q^{17} +3.46410 q^{19} +1.73205 q^{21} -2.00000 q^{23} +1.00000 q^{27} -9.92820 q^{29} +1.19615 q^{31} -1.73205 q^{33} +7.46410 q^{37} -1.00000 q^{39} -9.46410 q^{41} +4.92820 q^{43} -10.6603 q^{47} -4.00000 q^{49} -4.46410 q^{51} -1.92820 q^{53} +3.46410 q^{57} -1.73205 q^{59} -5.53590 q^{61} +1.73205 q^{63} -7.73205 q^{67} -2.00000 q^{69} +6.39230 q^{71} -3.46410 q^{73} -3.00000 q^{77} +14.0000 q^{79} +1.00000 q^{81} -10.2679 q^{83} -9.92820 q^{87} +10.9282 q^{89} -1.73205 q^{91} +1.19615 q^{93} +8.92820 q^{97} -1.73205 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^9	\( 2 q + 2 q^{3} + 2 q^{9} - 2 q^{13} - 2 q^{17} - 4 q^{23} + 2 q^{27} - 6 q^{29} - 8 q^{31} + 8 q^{37} - 2 q^{39} - 12 q^{41} - 4 q^{43} - 4 q^{47} - 8 q^{49} - 2 q^{51} + 10 q^{53} - 18 q^{61} - 12 q^{67} - 4 q^{69} - 8 q^{71} - 6 q^{77} + 28 q^{79} + 2 q^{81} - 24 q^{83} - 6 q^{87} + 8 q^{89} - 8 q^{93} + 4 q^{97}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^9 - 2 * q^13 - 2 * q^17 - 4 * q^23 + 2 * q^27 - 6 * q^29 - 8 * q^31 + 8 * q^37 - 2 * q^39 - 12 * q^41 - 4 * q^43 - 4 * q^47 - 8 * q^49 - 2 * q^51 + 10 * q^53 - 18 * q^61 - 12 * q^67 - 4 * q^69 - 8 * q^71 - 6 * q^77 + 28 * q^79 + 2 * q^81 - 24 * q^83 - 6 * q^87 + 8 * q^89 - 8 * q^93 + 4 * q^97

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