LMFDB - Embedded newform 2925.2.a.ba.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2925,2,Mod(1,2925)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2925, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2925.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2925 = 3^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2925.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:	$23.3562425912$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	2925.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205 q^{2} +1.00000 q^{4} -1.00000 q^{7} +1.73205 q^{8} +O(q^{10})$	$q-1.73205 q^{2} +1.00000 q^{4} -1.00000 q^{7} +1.73205 q^{8} -1.73205 q^{11} +1.00000 q^{13} +1.73205 q^{14} -5.00000 q^{16} +1.73205 q^{17} -4.00000 q^{19} +3.00000 q^{22} +3.46410 q^{23} -1.73205 q^{26} -1.00000 q^{28} +8.66025 q^{29} -7.00000 q^{31} +5.19615 q^{32} -3.00000 q^{34} +2.00000 q^{37} +6.92820 q^{38} -6.92820 q^{41} +2.00000 q^{43} -1.73205 q^{44} -6.00000 q^{46} -5.19615 q^{47} -6.00000 q^{49} +1.00000 q^{52} +5.19615 q^{53} -1.73205 q^{56} -15.0000 q^{58} +1.73205 q^{59} -1.00000 q^{61} +12.1244 q^{62} +1.00000 q^{64} +5.00000 q^{67} +1.73205 q^{68} -3.46410 q^{71} +2.00000 q^{73} -3.46410 q^{74} -4.00000 q^{76} +1.73205 q^{77} -10.0000 q^{79} +12.0000 q^{82} +5.19615 q^{83} -3.46410 q^{86} -3.00000 q^{88} +13.8564 q^{89} -1.00000 q^{91} +3.46410 q^{92} +9.00000 q^{94} -10.0000 q^{97} +10.3923 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} - 2 q^{7}+O(q^{10}) $ 2 * q + 2 * q^4 - 2 * q^7	$ 2 q + 2 q^{4} - 2 q^{7} + 2 q^{13} - 10 q^{16} - 8 q^{19} + 6 q^{22} - 2 q^{28} - 14 q^{31} - 6 q^{34} + 4 q^{37} + 4 q^{43} - 12 q^{46} - 12 q^{49} + 2 q^{52} - 30 q^{58} - 2 q^{61} + 2 q^{64} + 10 q^{67} + 4 q^{73} - 8 q^{76} - 20 q^{79} + 24 q^{82} - 6 q^{88} - 2 q^{91} + 18 q^{94} - 20 q^{97}+O(q^{100}) $ 2 * q + 2 * q^4 - 2 * q^7 + 2 * q^13 - 10 * q^16 - 8 * q^19 + 6 * q^22 - 2 * q^28 - 14 * q^31 - 6 * q^34 + 4 * q^37 + 4 * q^43 - 12 * q^46 - 12 * q^49 + 2 * q^52 - 30 * q^58 - 2 * q^61 + 2 * q^64 + 10 * q^67 + 4 * q^73 - 8 * q^76 - 20 * q^79 + 24 * q^82 - 6 * q^88 - 2 * q^91 + 18 * q^94 - 20 * 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 $

$2$	−1.73205	−1.22474	−0.612372	−	0.790569i	$-0.709785\pi$
$2$	−1.73205	−1.22474	−0.612372	+	0.790569i	$0.709785\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	1.73205	0.612372
$9$	0	0
$10$	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
$14$	1.73205	0.462910
$15$	0	0
$16$	−5.00000	−1.25000
$17$	1.73205	0.420084	0.210042	−	0.977692i	$-0.432640\pi$
$17$	1.73205	0.420084	0.210042	+	0.977692i	$0.432640\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	0	0
$22$	3.00000	0.639602
$23$	3.46410	0.722315	0.361158	−	0.932505i	$-0.382382\pi$
$23$	3.46410	0.722315	0.361158	+	0.932505i	$0.382382\pi$
$24$	0	0
$25$	0	0
$26$	−1.73205	−0.339683
$27$	0	0
$28$	−1.00000	−0.188982
$29$	8.66025	1.60817	0.804084	−	0.594515i	$-0.202656\pi$
$29$	8.66025	1.60817	0.804084	+	0.594515i	$0.202656\pi$
$30$	0	0
$31$	−7.00000	−1.25724	−0.628619	−	0.777714i	$-0.716379\pi$
$31$	−7.00000	−1.25724	−0.628619	+	0.777714i	$0.716379\pi$
$32$	5.19615	0.918559
$33$	0	0
$34$	−3.00000	−0.514496
$35$	0	0
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	6.92820	1.12390
$39$	0	0
$40$	0	0
$41$	−6.92820	−1.08200	−0.541002	−	0.841021i	$-0.681955\pi$
$41$	−6.92820	−1.08200	−0.541002	+	0.841021i	$0.681955\pi$
$42$	0	0
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	−1.73205	−0.261116
$45$	0	0
$46$	−6.00000	−0.884652
$47$	−5.19615	−0.757937	−0.378968	−	0.925410i	$-0.623721\pi$
$47$	−5.19615	−0.757937	−0.378968	+	0.925410i	$0.623721\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	0	0
$52$	1.00000	0.138675
$53$	5.19615	0.713746	0.356873	−	0.934153i	$-0.383843\pi$
$53$	5.19615	0.713746	0.356873	+	0.934153i	$0.383843\pi$
$54$	0	0
$55$	0	0
$56$	−1.73205	−0.231455
$57$	0	0
$58$	−15.0000	−1.96960
$59$	1.73205	0.225494	0.112747	−	0.993624i	$-0.464035\pi$
$59$	1.73205	0.225494	0.112747	+	0.993624i	$0.464035\pi$
$60$	0	0
$61$	−1.00000	−0.128037	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000	−0.128037	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	12.1244	1.53979
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	5.00000	0.610847	0.305424	−	0.952217i	$-0.401202\pi$
$67$	5.00000	0.610847	0.305424	+	0.952217i	$0.401202\pi$
$68$	1.73205	0.210042
$69$	0	0
$70$	0	0
$71$	−3.46410	−0.411113	−0.205557	−	0.978645i	$-0.565900\pi$
$71$	−3.46410	−0.411113	−0.205557	+	0.978645i	$0.565900\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	−3.46410	−0.402694
$75$	0	0
$76$	−4.00000	−0.458831
$77$	1.73205	0.197386
$78$	0	0
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	0	0
$81$	0	0
$82$	12.0000	1.32518
$83$	5.19615	0.570352	0.285176	−	0.958475i	$-0.407948\pi$
$83$	5.19615	0.570352	0.285176	+	0.958475i	$0.407948\pi$
$84$	0	0
$85$	0	0
$86$	−3.46410	−0.373544
$87$	0	0
$88$	−3.00000	−0.319801
$89$	13.8564	1.46878	0.734388	−	0.678730i	$-0.237469\pi$
$89$	13.8564	1.46878	0.734388	+	0.678730i	$0.237469\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	3.46410	0.361158
$93$	0	0
$94$	9.00000	0.928279
$95$	0	0
$96$	0	0
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	10.3923	1.04978
$99$	0	0
$100$	0	0
$101$	−1.73205	−0.172345	−0.0861727	−	0.996280i	$-0.527464\pi$
$101$	−1.73205	−0.172345	−0.0861727	+	0.996280i	$0.527464\pi$
$102$	0	0
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	1.73205	0.169842
$105$	0	0
$106$	−9.00000	−0.874157
$107$	3.46410	0.334887	0.167444	−	0.985882i	$-0.446449\pi$
$107$	3.46410	0.334887	0.167444	+	0.985882i	$0.446449\pi$
$108$	0	0
$109$	8.00000	0.766261	0.383131	−	0.923694i	$-0.374846\pi$
$109$	8.00000	0.766261	0.383131	+	0.923694i	$0.374846\pi$
$110$	0	0
$111$	0	0
$112$	5.00000	0.472456
$113$	6.92820	0.651751	0.325875	−	0.945413i	$-0.394341\pi$
$113$	6.92820	0.651751	0.325875	+	0.945413i	$0.394341\pi$
$114$	0	0
$115$	0	0
$116$	8.66025	0.804084
$117$	0	0
$118$	−3.00000	−0.276172
$119$	−1.73205	−0.158777
$120$	0	0
$121$	−8.00000	−0.727273
$122$	1.73205	0.156813
$123$	0	0
$124$	−7.00000	−0.628619
$125$	0	0
$126$	0	0
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	−12.1244	−1.07165
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	−8.66025	−0.748132
$135$	0	0
$136$	3.00000	0.257248
$137$	−13.8564	−1.18383	−0.591916	−	0.805999i	$-0.701628\pi$
$137$	−13.8564	−1.18383	−0.591916	+	0.805999i	$0.701628\pi$
$138$	0	0
$139$	−10.0000	−0.848189	−0.424094	−	0.905618i	$-0.639408\pi$
$139$	−10.0000	−0.848189	−0.424094	+	0.905618i	$0.639408\pi$
$140$	0	0
$141$	0	0
$142$	6.00000	0.503509
$143$	−1.73205	−0.144841
$144$	0	0
$145$	0	0
$146$	−3.46410	−0.286691
$147$	0	0
$148$	2.00000	0.164399
$149$	−3.46410	−0.283790	−0.141895	−	0.989882i	$-0.545320\pi$
$149$	−3.46410	−0.283790	−0.141895	+	0.989882i	$0.545320\pi$
$150$	0	0
$151$	−19.0000	−1.54620	−0.773099	−	0.634285i	$-0.781294\pi$
$151$	−19.0000	−1.54620	−0.773099	+	0.634285i	$0.781294\pi$
$152$	−6.92820	−0.561951
$153$	0	0
$154$	−3.00000	−0.241747
$155$	0	0
$156$	0	0
$157$	23.0000	1.83560	0.917800	−	0.397043i	$-0.129964\pi$
$157$	23.0000	1.83560	0.917800	+	0.397043i	$0.129964\pi$
$158$	17.3205	1.37795
$159$	0	0
$160$	0	0
$161$	−3.46410	−0.273009
$162$	0	0
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	−6.92820	−0.541002
$165$	0	0
$166$	−9.00000	−0.698535
$167$	3.46410	0.268060	0.134030	−	0.990977i	$-0.457208\pi$
$167$	3.46410	0.268060	0.134030	+	0.990977i	$0.457208\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	2.00000	0.152499
$173$	−8.66025	−0.658427	−0.329213	−	0.944256i	$-0.606784\pi$
$173$	−8.66025	−0.658427	−0.329213	+	0.944256i	$0.606784\pi$
$174$	0	0
$175$	0	0
$176$	8.66025	0.652791
$177$	0	0
$178$	−24.0000	−1.79888
$179$	13.8564	1.03568	0.517838	−	0.855479i	$-0.326737\pi$
$179$	13.8564	1.03568	0.517838	+	0.855479i	$0.326737\pi$
$180$	0	0
$181$	−1.00000	−0.0743294	−0.0371647	−	0.999309i	$-0.511833\pi$
$181$	−1.00000	−0.0743294	−0.0371647	+	0.999309i	$0.511833\pi$
$182$	1.73205	0.128388
$183$	0	0
$184$	6.00000	0.442326
$185$	0	0
$186$	0	0
$187$	−3.00000	−0.219382
$188$	−5.19615	−0.378968
$189$	0	0
$190$	0	0
$191$	−24.2487	−1.75458	−0.877288	−	0.479965i	$-0.840649\pi$
$191$	−24.2487	−1.75458	−0.877288	+	0.479965i	$0.840649\pi$
$192$	0	0
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	17.3205	1.24354
$195$	0	0
$196$	−6.00000	−0.428571
$197$	−6.92820	−0.493614	−0.246807	−	0.969065i	$-0.579381\pi$
$197$	−6.92820	−0.493614	−0.246807	+	0.969065i	$0.579381\pi$
$198$	0	0
$199$	2.00000	0.141776	0.0708881	−	0.997484i	$-0.477417\pi$
$199$	2.00000	0.141776	0.0708881	+	0.997484i	$0.477417\pi$
$200$	0	0
$201$	0	0
$202$	3.00000	0.211079
$203$	−8.66025	−0.607831
$204$	0	0
$205$	0	0
$206$	27.7128	1.93084
$207$	0	0
$208$	−5.00000	−0.346688
$209$	6.92820	0.479234
$210$	0	0
$211$	−10.0000	−0.688428	−0.344214	−	0.938891i	$-0.611855\pi$
$211$	−10.0000	−0.688428	−0.344214	+	0.938891i	$0.611855\pi$
$212$	5.19615	0.356873
$213$	0	0
$214$	−6.00000	−0.410152
$215$	0	0
$216$	0	0
$217$	7.00000	0.475191
$218$	−13.8564	−0.938474
$219$	0	0
$220$	0	0
$221$	1.73205	0.116510
$222$	0	0
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	−5.19615	−0.347183
$225$	0	0
$226$	−12.0000	−0.798228
$227$	−19.0526	−1.26456	−0.632281	−	0.774739i	$-0.717881\pi$
$227$	−19.0526	−1.26456	−0.632281	+	0.774739i	$0.717881\pi$
$228$	0	0
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	0	0
$231$	0	0
$232$	15.0000	0.984798
$233$	−20.7846	−1.36165	−0.680823	−	0.732448i	$-0.738378\pi$
$233$	−20.7846	−1.36165	−0.680823	+	0.732448i	$0.738378\pi$
$234$	0	0
$235$	0	0
$236$	1.73205	0.112747
$237$	0	0
$238$	3.00000	0.194461
$239$	−25.9808	−1.68056	−0.840278	−	0.542156i	$-0.817608\pi$
$239$	−25.9808	−1.68056	−0.840278	+	0.542156i	$0.817608\pi$
$240$	0	0
$241$	8.00000	0.515325	0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000	0.515325	0.257663	+	0.966235i	$0.417048\pi$
$242$	13.8564	0.890724
$243$	0	0
$244$	−1.00000	−0.0640184
$245$	0	0
$246$	0	0
$247$	−4.00000	−0.254514
$248$	−12.1244	−0.769897
$249$	0	0
$250$	0	0
$251$	6.92820	0.437304	0.218652	−	0.975803i	$-0.429834\pi$
$251$	6.92820	0.437304	0.218652	+	0.975803i	$0.429834\pi$
$252$	0	0
$253$	−6.00000	−0.377217
$254$	−3.46410	−0.217357
$255$	0	0
$256$	19.0000	1.18750
$257$	−12.1244	−0.756297	−0.378148	−	0.925745i	$-0.623439\pi$
$257$	−12.1244	−0.756297	−0.378148	+	0.925745i	$0.623439\pi$
$258$	0	0
$259$	−2.00000	−0.124274
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−27.7128	−1.70885	−0.854423	−	0.519579i	$-0.826089\pi$
$263$	−27.7128	−1.70885	−0.854423	+	0.519579i	$0.826089\pi$
$264$	0	0
$265$	0	0
$266$	−6.92820	−0.424795
$267$	0	0
$268$	5.00000	0.305424
$269$	−29.4449	−1.79529	−0.897643	−	0.440724i	$-0.854722\pi$
$269$	−29.4449	−1.79529	−0.897643	+	0.440724i	$0.854722\pi$
$270$	0	0
$271$	11.0000	0.668202	0.334101	−	0.942537i	$-0.391567\pi$
$271$	11.0000	0.668202	0.334101	+	0.942537i	$0.391567\pi$
$272$	−8.66025	−0.525105
$273$	0	0
$274$	24.0000	1.44989
$275$	0	0
$276$	0	0
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	17.3205	1.03882
$279$	0	0
$280$	0	0
$281$	10.3923	0.619953	0.309976	−	0.950744i	$-0.399679\pi$
$281$	10.3923	0.619953	0.309976	+	0.950744i	$0.399679\pi$
$282$	0	0
$283$	14.0000	0.832214	0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000	0.832214	0.416107	+	0.909316i	$0.363394\pi$
$284$	−3.46410	−0.205557
$285$	0	0
$286$	3.00000	0.177394
$287$	6.92820	0.408959
$288$	0	0
$289$	−14.0000	−0.823529
$290$	0	0
$291$	0	0
$292$	2.00000	0.117041
$293$	−3.46410	−0.202375	−0.101187	−	0.994867i	$-0.532264\pi$
$293$	−3.46410	−0.202375	−0.101187	+	0.994867i	$0.532264\pi$
$294$	0	0
$295$	0	0
$296$	3.46410	0.201347
$297$	0	0
$298$	6.00000	0.347571
$299$	3.46410	0.200334
$300$	0	0
$301$	−2.00000	−0.115278
$302$	32.9090	1.89370
$303$	0	0
$304$	20.0000	1.14708
$305$	0	0
$306$	0	0
$307$	−28.0000	−1.59804	−0.799022	−	0.601302i	$-0.794649\pi$
$307$	−28.0000	−1.59804	−0.799022	+	0.601302i	$0.794649\pi$
$308$	1.73205	0.0986928
$309$	0	0
$310$	0	0
$311$	10.3923	0.589294	0.294647	−	0.955606i	$-0.404798\pi$
$311$	10.3923	0.589294	0.294647	+	0.955606i	$0.404798\pi$
$312$	0	0
$313$	23.0000	1.30004	0.650018	−	0.759918i	$-0.274761\pi$
$313$	23.0000	1.30004	0.650018	+	0.759918i	$0.274761\pi$
$314$	−39.8372	−2.24814
$315$	0	0
$316$	−10.0000	−0.562544
$317$	−31.1769	−1.75107	−0.875535	−	0.483155i	$-0.839491\pi$
$317$	−31.1769	−1.75107	−0.875535	+	0.483155i	$0.839491\pi$
$318$	0	0
$319$	−15.0000	−0.839839
$320$	0	0
$321$	0	0
$322$	6.00000	0.334367
$323$	−6.92820	−0.385496
$324$	0	0
$325$	0	0
$326$	6.92820	0.383718
$327$	0	0
$328$	−12.0000	−0.662589
$329$	5.19615	0.286473
$330$	0	0
$331$	−16.0000	−0.879440	−0.439720	−	0.898135i	$-0.644922\pi$
$331$	−16.0000	−0.879440	−0.439720	+	0.898135i	$0.644922\pi$
$332$	5.19615	0.285176
$333$	0	0
$334$	−6.00000	−0.328305
$335$	0	0
$336$	0	0
$337$	−25.0000	−1.36184	−0.680918	−	0.732359i	$-0.738419\pi$
$337$	−25.0000	−1.36184	−0.680918	+	0.732359i	$0.738419\pi$
$338$	−1.73205	−0.0942111
$339$	0	0
$340$	0	0
$341$	12.1244	0.656571
$342$	0	0
$343$	13.0000	0.701934
$344$	3.46410	0.186772
$345$	0	0
$346$	15.0000	0.806405
$347$	−24.2487	−1.30174	−0.650870	−	0.759190i	$-0.725596\pi$
$347$	−24.2487	−1.30174	−0.650870	+	0.759190i	$0.725596\pi$
$348$	0	0
$349$	−34.0000	−1.81998	−0.909989	−	0.414632i	$-0.863910\pi$
$349$	−34.0000	−1.81998	−0.909989	+	0.414632i	$0.863910\pi$
$350$	0	0
$351$	0	0
$352$	−9.00000	−0.479702
$353$	13.8564	0.737502	0.368751	−	0.929528i	$-0.379785\pi$
$353$	13.8564	0.737502	0.368751	+	0.929528i	$0.379785\pi$
$354$	0	0
$355$	0	0
$356$	13.8564	0.734388
$357$	0	0
$358$	−24.0000	−1.26844
$359$	−5.19615	−0.274242	−0.137121	−	0.990554i	$-0.543785\pi$
$359$	−5.19615	−0.274242	−0.137121	+	0.990554i	$0.543785\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	1.73205	0.0910346
$363$	0	0
$364$	−1.00000	−0.0524142
$365$	0	0
$366$	0	0
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	−17.3205	−0.902894
$369$	0	0
$370$	0	0
$371$	−5.19615	−0.269771
$372$	0	0
$373$	5.00000	0.258890	0.129445	−	0.991587i	$-0.458680\pi$
$373$	5.00000	0.258890	0.129445	+	0.991587i	$0.458680\pi$
$374$	5.19615	0.268687
$375$	0	0
$376$	−9.00000	−0.464140
$377$	8.66025	0.446026
$378$	0	0
$379$	−25.0000	−1.28416	−0.642082	−	0.766636i	$-0.721929\pi$
$379$	−25.0000	−1.28416	−0.642082	+	0.766636i	$0.721929\pi$
$380$	0	0
$381$	0	0
$382$	42.0000	2.14891
$383$	−24.2487	−1.23905	−0.619526	−	0.784976i	$-0.712675\pi$
$383$	−24.2487	−1.23905	−0.619526	+	0.784976i	$0.712675\pi$
$384$	0	0
$385$	0	0
$386$	6.92820	0.352636
$387$	0	0
$388$	−10.0000	−0.507673
$389$	20.7846	1.05382	0.526911	−	0.849921i	$-0.323350\pi$
$389$	20.7846	1.05382	0.526911	+	0.849921i	$0.323350\pi$
$390$	0	0
$391$	6.00000	0.303433
$392$	−10.3923	−0.524891
$393$	0	0
$394$	12.0000	0.604551
$395$	0	0
$396$	0	0
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	−3.46410	−0.173640
$399$	0	0
$400$	0	0
$401$	3.46410	0.172989	0.0864945	−	0.996252i	$-0.472434\pi$
$401$	3.46410	0.172989	0.0864945	+	0.996252i	$0.472434\pi$
$402$	0	0
$403$	−7.00000	−0.348695
$404$	−1.73205	−0.0861727
$405$	0	0
$406$	15.0000	0.744438
$407$	−3.46410	−0.171709
$408$	0	0
$409$	8.00000	0.395575	0.197787	−	0.980245i	$-0.436624\pi$
$409$	8.00000	0.395575	0.197787	+	0.980245i	$0.436624\pi$
$410$	0	0
$411$	0	0
$412$	−16.0000	−0.788263
$413$	−1.73205	−0.0852286
$414$	0	0
$415$	0	0
$416$	5.19615	0.254762
$417$	0	0
$418$	−12.0000	−0.586939
$419$	34.6410	1.69232	0.846162	−	0.532925i	$-0.178907\pi$
$419$	34.6410	1.69232	0.846162	+	0.532925i	$0.178907\pi$
$420$	0	0
$421$	−4.00000	−0.194948	−0.0974740	−	0.995238i	$-0.531076\pi$
$421$	−4.00000	−0.194948	−0.0974740	+	0.995238i	$0.531076\pi$
$422$	17.3205	0.843149
$423$	0	0
$424$	9.00000	0.437079
$425$	0	0
$426$	0	0
$427$	1.00000	0.0483934
$428$	3.46410	0.167444
$429$	0	0
$430$	0	0
$431$	3.46410	0.166860	0.0834300	−	0.996514i	$-0.473413\pi$
$431$	3.46410	0.166860	0.0834300	+	0.996514i	$0.473413\pi$
$432$	0	0
$433$	38.0000	1.82616	0.913082	−	0.407777i	$-0.133696\pi$
$433$	38.0000	1.82616	0.913082	+	0.407777i	$0.133696\pi$
$434$	−12.1244	−0.581988
$435$	0	0
$436$	8.00000	0.383131
$437$	−13.8564	−0.662842
$438$	0	0
$439$	2.00000	0.0954548	0.0477274	−	0.998860i	$-0.484802\pi$
$439$	2.00000	0.0954548	0.0477274	+	0.998860i	$0.484802\pi$
$440$	0	0
$441$	0	0
$442$	−3.00000	−0.142695
$443$	−10.3923	−0.493753	−0.246877	−	0.969047i	$-0.579404\pi$
$443$	−10.3923	−0.493753	−0.246877	+	0.969047i	$0.579404\pi$
$444$	0	0
$445$	0	0
$446$	−13.8564	−0.656120
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	−34.6410	−1.63481	−0.817405	−	0.576063i	$-0.804588\pi$
$449$	−34.6410	−1.63481	−0.817405	+	0.576063i	$0.804588\pi$
$450$	0	0
$451$	12.0000	0.565058
$452$	6.92820	0.325875
$453$	0	0
$454$	33.0000	1.54877
$455$	0	0
$456$	0	0
$457$	−28.0000	−1.30978	−0.654892	−	0.755722i	$-0.727286\pi$
$457$	−28.0000	−1.30978	−0.654892	+	0.755722i	$0.727286\pi$
$458$	−3.46410	−0.161867
$459$	0	0
$460$	0	0
$461$	−3.46410	−0.161339	−0.0806696	−	0.996741i	$-0.525706\pi$
$461$	−3.46410	−0.161339	−0.0806696	+	0.996741i	$0.525706\pi$
$462$	0	0
$463$	−31.0000	−1.44069	−0.720346	−	0.693615i	$-0.756017\pi$
$463$	−31.0000	−1.44069	−0.720346	+	0.693615i	$0.756017\pi$
$464$	−43.3013	−2.01021
$465$	0	0
$466$	36.0000	1.66767
$467$	20.7846	0.961797	0.480899	−	0.876776i	$-0.340311\pi$
$467$	20.7846	0.961797	0.480899	+	0.876776i	$0.340311\pi$
$468$	0	0
$469$	−5.00000	−0.230879
$470$	0	0
$471$	0	0
$472$	3.00000	0.138086
$473$	−3.46410	−0.159280
$474$	0	0
$475$	0	0
$476$	−1.73205	−0.0793884
$477$	0	0
$478$	45.0000	2.05825
$479$	39.8372	1.82021	0.910103	−	0.414381i	$-0.136002\pi$
$479$	39.8372	1.82021	0.910103	+	0.414381i	$0.136002\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	−13.8564	−0.631142
$483$	0	0
$484$	−8.00000	−0.363636
$485$	0	0
$486$	0	0
$487$	−7.00000	−0.317200	−0.158600	−	0.987343i	$-0.550698\pi$
$487$	−7.00000	−0.317200	−0.158600	+	0.987343i	$0.550698\pi$
$488$	−1.73205	−0.0784063
$489$	0	0
$490$	0	0
$491$	24.2487	1.09433	0.547165	−	0.837025i	$-0.315707\pi$
$491$	24.2487	1.09433	0.547165	+	0.837025i	$0.315707\pi$
$492$	0	0
$493$	15.0000	0.675566
$494$	6.92820	0.311715
$495$	0	0
$496$	35.0000	1.57155
$497$	3.46410	0.155386
$498$	0	0
$499$	11.0000	0.492428	0.246214	−	0.969216i	$-0.420813\pi$
$499$	11.0000	0.492428	0.246214	+	0.969216i	$0.420813\pi$
$500$	0	0
$501$	0	0
$502$	−12.0000	−0.535586
$503$	27.7128	1.23565	0.617827	−	0.786314i	$-0.288013\pi$
$503$	27.7128	1.23565	0.617827	+	0.786314i	$0.288013\pi$
$504$	0	0
$505$	0	0
$506$	10.3923	0.461994
$507$	0	0
$508$	2.00000	0.0887357
$509$	−27.7128	−1.22835	−0.614174	−	0.789170i	$-0.710511\pi$
$509$	−27.7128	−1.22835	−0.614174	+	0.789170i	$0.710511\pi$
$510$	0	0
$511$	−2.00000	−0.0884748
$512$	−8.66025	−0.382733
$513$	0	0
$514$	21.0000	0.926270
$515$	0	0
$516$	0	0
$517$	9.00000	0.395820
$518$	3.46410	0.152204
$519$	0	0
$520$	0	0
$521$	−20.7846	−0.910590	−0.455295	−	0.890341i	$-0.650466\pi$
$521$	−20.7846	−0.910590	−0.455295	+	0.890341i	$0.650466\pi$
$522$	0	0
$523$	14.0000	0.612177	0.306089	−	0.952003i	$-0.400980\pi$
$523$	14.0000	0.612177	0.306089	+	0.952003i	$0.400980\pi$
$524$	0	0
$525$	0	0
$526$	48.0000	2.09290
$527$	−12.1244	−0.528145
$528$	0	0
$529$	−11.0000	−0.478261
$530$	0	0
$531$	0	0
$532$	4.00000	0.173422
$533$	−6.92820	−0.300094
$534$	0	0
$535$	0	0
$536$	8.66025	0.374066
$537$	0	0
$538$	51.0000	2.19877
$539$	10.3923	0.447628
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	−19.0526	−0.818377
$543$	0	0
$544$	9.00000	0.385872
$545$	0	0
$546$	0	0
$547$	8.00000	0.342055	0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000	0.342055	0.171028	+	0.985266i	$0.445291\pi$
$548$	−13.8564	−0.591916
$549$	0	0
$550$	0	0
$551$	−34.6410	−1.47576
$552$	0	0
$553$	10.0000	0.425243
$554$	17.3205	0.735878
$555$	0	0
$556$	−10.0000	−0.424094
$557$	34.6410	1.46779	0.733893	−	0.679265i	$-0.237701\pi$
$557$	34.6410	1.46779	0.733893	+	0.679265i	$0.237701\pi$
$558$	0	0
$559$	2.00000	0.0845910
$560$	0	0
$561$	0	0
$562$	−18.0000	−0.759284
$563$	17.3205	0.729972	0.364986	−	0.931013i	$-0.381074\pi$
$563$	17.3205	0.729972	0.364986	+	0.931013i	$0.381074\pi$
$564$	0	0
$565$	0	0
$566$	−24.2487	−1.01925
$567$	0	0
$568$	−6.00000	−0.251754
$569$	39.8372	1.67006	0.835030	−	0.550204i	$-0.185450\pi$
$569$	39.8372	1.67006	0.835030	+	0.550204i	$0.185450\pi$
$570$	0	0
$571$	−22.0000	−0.920671	−0.460336	−	0.887745i	$-0.652271\pi$
$571$	−22.0000	−0.920671	−0.460336	+	0.887745i	$0.652271\pi$
$572$	−1.73205	−0.0724207
$573$	0	0
$574$	−12.0000	−0.500870
$575$	0	0
$576$	0	0
$577$	−34.0000	−1.41544	−0.707719	−	0.706494i	$-0.750276\pi$
$577$	−34.0000	−1.41544	−0.707719	+	0.706494i	$0.750276\pi$
$578$	24.2487	1.00861
$579$	0	0
$580$	0	0
$581$	−5.19615	−0.215573
$582$	0	0
$583$	−9.00000	−0.372742
$584$	3.46410	0.143346
$585$	0	0
$586$	6.00000	0.247858
$587$	−43.3013	−1.78723	−0.893617	−	0.448830i	$-0.851841\pi$
$587$	−43.3013	−1.78723	−0.893617	+	0.448830i	$0.851841\pi$
$588$	0	0
$589$	28.0000	1.15372
$590$	0	0
$591$	0	0
$592$	−10.0000	−0.410997
$593$	31.1769	1.28028	0.640141	−	0.768257i	$-0.278876\pi$
$593$	31.1769	1.28028	0.640141	+	0.768257i	$0.278876\pi$
$594$	0	0
$595$	0	0
$596$	−3.46410	−0.141895
$597$	0	0
$598$	−6.00000	−0.245358
$599$	−20.7846	−0.849236	−0.424618	−	0.905373i	$-0.639592\pi$
$599$	−20.7846	−0.849236	−0.424618	+	0.905373i	$0.639592\pi$
$600$	0	0
$601$	35.0000	1.42768	0.713840	−	0.700309i	$-0.246954\pi$
$601$	35.0000	1.42768	0.713840	+	0.700309i	$0.246954\pi$
$602$	3.46410	0.141186
$603$	0	0
$604$	−19.0000	−0.773099
$605$	0	0
$606$	0	0
$607$	14.0000	0.568242	0.284121	−	0.958788i	$-0.408298\pi$
$607$	14.0000	0.568242	0.284121	+	0.958788i	$0.408298\pi$
$608$	−20.7846	−0.842927
$609$	0	0
$610$	0	0
$611$	−5.19615	−0.210214
$612$	0	0
$613$	−22.0000	−0.888572	−0.444286	−	0.895885i	$-0.646543\pi$
$613$	−22.0000	−0.888572	−0.444286	+	0.895885i	$0.646543\pi$
$614$	48.4974	1.95720
$615$	0	0
$616$	3.00000	0.120873
$617$	38.1051	1.53405	0.767027	−	0.641615i	$-0.221735\pi$
$617$	38.1051	1.53405	0.767027	+	0.641615i	$0.221735\pi$
$618$	0	0
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	0	0
$621$	0	0
$622$	−18.0000	−0.721734
$623$	−13.8564	−0.555145
$624$	0	0
$625$	0	0
$626$	−39.8372	−1.59221
$627$	0	0
$628$	23.0000	0.917800
$629$	3.46410	0.138123
$630$	0	0
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	−17.3205	−0.688973
$633$	0	0
$634$	54.0000	2.14461
$635$	0	0
$636$	0	0
$637$	−6.00000	−0.237729
$638$	25.9808	1.02859
$639$	0	0
$640$	0	0
$641$	−19.0526	−0.752531	−0.376265	−	0.926512i	$-0.622792\pi$
$641$	−19.0526	−0.752531	−0.376265	+	0.926512i	$0.622792\pi$
$642$	0	0
$643$	−4.00000	−0.157745	−0.0788723	−	0.996885i	$-0.525132\pi$
$643$	−4.00000	−0.157745	−0.0788723	+	0.996885i	$0.525132\pi$
$644$	−3.46410	−0.136505
$645$	0	0
$646$	12.0000	0.472134
$647$	34.6410	1.36188	0.680939	−	0.732340i	$-0.261572\pi$
$647$	34.6410	1.36188	0.680939	+	0.732340i	$0.261572\pi$
$648$	0	0
$649$	−3.00000	−0.117760
$650$	0	0
$651$	0	0
$652$	−4.00000	−0.156652
$653$	−22.5167	−0.881145	−0.440573	−	0.897717i	$-0.645225\pi$
$653$	−22.5167	−0.881145	−0.440573	+	0.897717i	$0.645225\pi$
$654$	0	0
$655$	0	0
$656$	34.6410	1.35250
$657$	0	0
$658$	−9.00000	−0.350857
$659$	−45.0333	−1.75425	−0.877125	−	0.480263i	$-0.840541\pi$
$659$	−45.0333	−1.75425	−0.877125	+	0.480263i	$0.840541\pi$
$660$	0	0
$661$	32.0000	1.24466	0.622328	−	0.782757i	$-0.286187\pi$
$661$	32.0000	1.24466	0.622328	+	0.782757i	$0.286187\pi$
$662$	27.7128	1.07709
$663$	0	0
$664$	9.00000	0.349268
$665$	0	0
$666$	0	0
$667$	30.0000	1.16160
$668$	3.46410	0.134030
$669$	0	0
$670$	0	0
$671$	1.73205	0.0668651
$672$	0	0
$673$	47.0000	1.81172	0.905858	−	0.423581i	$-0.139227\pi$
$673$	47.0000	1.81172	0.905858	+	0.423581i	$0.139227\pi$
$674$	43.3013	1.66790
$675$	0	0
$676$	1.00000	0.0384615
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	10.0000	0.383765
$680$	0	0
$681$	0	0
$682$	−21.0000	−0.804132
$683$	32.9090	1.25923	0.629613	−	0.776909i	$-0.283213\pi$
$683$	32.9090	1.25923	0.629613	+	0.776909i	$0.283213\pi$
$684$	0	0
$685$	0	0
$686$	−22.5167	−0.859690
$687$	0	0
$688$	−10.0000	−0.381246
$689$	5.19615	0.197958
$690$	0	0
$691$	−1.00000	−0.0380418	−0.0190209	−	0.999819i	$-0.506055\pi$
$691$	−1.00000	−0.0380418	−0.0190209	+	0.999819i	$0.506055\pi$
$692$	−8.66025	−0.329213
$693$	0	0
$694$	42.0000	1.59430
$695$	0	0
$696$	0	0
$697$	−12.0000	−0.454532
$698$	58.8897	2.22901
$699$	0	0
$700$	0	0
$701$	−5.19615	−0.196256	−0.0981280	−	0.995174i	$-0.531285\pi$
$701$	−5.19615	−0.196256	−0.0981280	+	0.995174i	$0.531285\pi$
$702$	0	0
$703$	−8.00000	−0.301726
$704$	−1.73205	−0.0652791
$705$	0	0
$706$	−24.0000	−0.903252
$707$	1.73205	0.0651405
$708$	0	0
$709$	−28.0000	−1.05156	−0.525781	−	0.850620i	$-0.676227\pi$
$709$	−28.0000	−1.05156	−0.525781	+	0.850620i	$0.676227\pi$
$710$	0	0
$711$	0	0
$712$	24.0000	0.899438
$713$	−24.2487	−0.908121
$714$	0	0
$715$	0	0
$716$	13.8564	0.517838
$717$	0	0
$718$	9.00000	0.335877
$719$	6.92820	0.258378	0.129189	−	0.991620i	$-0.458763\pi$
$719$	6.92820	0.258378	0.129189	+	0.991620i	$0.458763\pi$
$720$	0	0
$721$	16.0000	0.595871
$722$	5.19615	0.193381
$723$	0	0
$724$	−1.00000	−0.0371647
$725$	0	0
$726$	0	0
$727$	−28.0000	−1.03846	−0.519231	−	0.854634i	$-0.673782\pi$
$727$	−28.0000	−1.03846	−0.519231	+	0.854634i	$0.673782\pi$
$728$	−1.73205	−0.0641941
$729$	0	0
$730$	0	0
$731$	3.46410	0.128124
$732$	0	0
$733$	20.0000	0.738717	0.369358	−	0.929287i	$-0.379577\pi$
$733$	20.0000	0.738717	0.369358	+	0.929287i	$0.379577\pi$
$734$	−13.8564	−0.511449
$735$	0	0
$736$	18.0000	0.663489
$737$	−8.66025	−0.319005
$738$	0	0
$739$	−13.0000	−0.478213	−0.239106	−	0.970993i	$-0.576854\pi$
$739$	−13.0000	−0.478213	−0.239106	+	0.970993i	$0.576854\pi$
$740$	0	0
$741$	0	0
$742$	9.00000	0.330400
$743$	50.2295	1.84274	0.921370	−	0.388686i	$-0.127071\pi$
$743$	50.2295	1.84274	0.921370	+	0.388686i	$0.127071\pi$
$744$	0	0
$745$	0	0
$746$	−8.66025	−0.317074
$747$	0	0
$748$	−3.00000	−0.109691
$749$	−3.46410	−0.126576
$750$	0	0
$751$	−22.0000	−0.802791	−0.401396	−	0.915905i	$-0.631475\pi$
$751$	−22.0000	−0.802791	−0.401396	+	0.915905i	$0.631475\pi$
$752$	25.9808	0.947421
$753$	0	0
$754$	−15.0000	−0.546268
$755$	0	0
$756$	0	0
$757$	5.00000	0.181728	0.0908640	−	0.995863i	$-0.471037\pi$
$757$	5.00000	0.181728	0.0908640	+	0.995863i	$0.471037\pi$
$758$	43.3013	1.57277
$759$	0	0
$760$	0	0
$761$	17.3205	0.627868	0.313934	−	0.949445i	$-0.398353\pi$
$761$	17.3205	0.627868	0.313934	+	0.949445i	$0.398353\pi$
$762$	0	0
$763$	−8.00000	−0.289619
$764$	−24.2487	−0.877288
$765$	0	0
$766$	42.0000	1.51752
$767$	1.73205	0.0625407
$768$	0	0
$769$	32.0000	1.15395	0.576975	−	0.816762i	$-0.304233\pi$
$769$	32.0000	1.15395	0.576975	+	0.816762i	$0.304233\pi$
$770$	0	0
$771$	0	0
$772$	−4.00000	−0.143963
$773$	38.1051	1.37055	0.685273	−	0.728286i	$-0.259683\pi$
$773$	38.1051	1.37055	0.685273	+	0.728286i	$0.259683\pi$
$774$	0	0
$775$	0	0
$776$	−17.3205	−0.621770
$777$	0	0
$778$	−36.0000	−1.29066
$779$	27.7128	0.992915
$780$	0	0
$781$	6.00000	0.214697
$782$	−10.3923	−0.371628
$783$	0	0
$784$	30.0000	1.07143
$785$	0	0
$786$	0	0
$787$	23.0000	0.819861	0.409931	−	0.912117i	$-0.365553\pi$
$787$	23.0000	0.819861	0.409931	+	0.912117i	$0.365553\pi$
$788$	−6.92820	−0.246807
$789$	0	0
$790$	0	0
$791$	−6.92820	−0.246339
$792$	0	0
$793$	−1.00000	−0.0355110
$794$	38.1051	1.35230
$795$	0	0
$796$	2.00000	0.0708881
$797$	−8.66025	−0.306762	−0.153381	−	0.988167i	$-0.549016\pi$
$797$	−8.66025	−0.306762	−0.153381	+	0.988167i	$0.549016\pi$
$798$	0	0
$799$	−9.00000	−0.318397
$800$	0	0
$801$	0	0
$802$	−6.00000	−0.211867
$803$	−3.46410	−0.122245
$804$	0	0
$805$	0	0
$806$	12.1244	0.427062
$807$	0	0
$808$	−3.00000	−0.105540
$809$	34.6410	1.21791	0.608957	−	0.793204i	$-0.291588\pi$
$809$	34.6410	1.21791	0.608957	+	0.793204i	$0.291588\pi$
$810$	0	0
$811$	−43.0000	−1.50993	−0.754967	−	0.655763i	$-0.772347\pi$
$811$	−43.0000	−1.50993	−0.754967	+	0.655763i	$0.772347\pi$
$812$	−8.66025	−0.303915
$813$	0	0
$814$	6.00000	0.210300
$815$	0	0
$816$	0	0
$817$	−8.00000	−0.279885
$818$	−13.8564	−0.484478
$819$	0	0
$820$	0	0
$821$	−17.3205	−0.604490	−0.302245	−	0.953230i	$-0.597736\pi$
$821$	−17.3205	−0.604490	−0.302245	+	0.953230i	$0.597736\pi$
$822$	0	0
$823$	32.0000	1.11545	0.557725	−	0.830026i	$-0.311674\pi$
$823$	32.0000	1.11545	0.557725	+	0.830026i	$0.311674\pi$
$824$	−27.7128	−0.965422
$825$	0	0
$826$	3.00000	0.104383
$827$	25.9808	0.903440	0.451720	−	0.892160i	$-0.350811\pi$
$827$	25.9808	0.903440	0.451720	+	0.892160i	$0.350811\pi$
$828$	0	0
$829$	23.0000	0.798823	0.399412	−	0.916772i	$-0.369214\pi$
$829$	23.0000	0.798823	0.399412	+	0.916772i	$0.369214\pi$
$830$	0	0
$831$	0	0
$832$	1.00000	0.0346688
$833$	−10.3923	−0.360072
$834$	0	0
$835$	0	0
$836$	6.92820	0.239617
$837$	0	0
$838$	−60.0000	−2.07267
$839$	17.3205	0.597970	0.298985	−	0.954258i	$-0.403352\pi$
$839$	17.3205	0.597970	0.298985	+	0.954258i	$0.403352\pi$
$840$	0	0
$841$	46.0000	1.58621
$842$	6.92820	0.238762
$843$	0	0
$844$	−10.0000	−0.344214
$845$	0	0
$846$	0	0
$847$	8.00000	0.274883
$848$	−25.9808	−0.892183
$849$	0	0
$850$	0	0
$851$	6.92820	0.237496
$852$	0	0
$853$	44.0000	1.50653	0.753266	−	0.657716i	$-0.228477\pi$
$853$	44.0000	1.50653	0.753266	+	0.657716i	$0.228477\pi$
$854$	−1.73205	−0.0592696
$855$	0	0
$856$	6.00000	0.205076
$857$	41.5692	1.41998	0.709989	−	0.704213i	$-0.248700\pi$
$857$	41.5692	1.41998	0.709989	+	0.704213i	$0.248700\pi$
$858$	0	0
$859$	−34.0000	−1.16007	−0.580033	−	0.814593i	$-0.696960\pi$
$859$	−34.0000	−1.16007	−0.580033	+	0.814593i	$0.696960\pi$
$860$	0	0
$861$	0	0
$862$	−6.00000	−0.204361
$863$	46.7654	1.59191	0.795956	−	0.605355i	$-0.206969\pi$
$863$	46.7654	1.59191	0.795956	+	0.605355i	$0.206969\pi$
$864$	0	0
$865$	0	0
$866$	−65.8179	−2.23658
$867$	0	0
$868$	7.00000	0.237595
$869$	17.3205	0.587558
$870$	0	0
$871$	5.00000	0.169419
$872$	13.8564	0.469237
$873$	0	0
$874$	24.0000	0.811812
$875$	0	0
$876$	0	0
$877$	−28.0000	−0.945493	−0.472746	−	0.881199i	$-0.656737\pi$
$877$	−28.0000	−0.945493	−0.472746	+	0.881199i	$0.656737\pi$
$878$	−3.46410	−0.116908
$879$	0	0
$880$	0	0
$881$	39.8372	1.34215	0.671074	−	0.741390i	$-0.265833\pi$
$881$	39.8372	1.34215	0.671074	+	0.741390i	$0.265833\pi$
$882$	0	0
$883$	8.00000	0.269221	0.134611	−	0.990899i	$-0.457022\pi$
$883$	8.00000	0.269221	0.134611	+	0.990899i	$0.457022\pi$
$884$	1.73205	0.0582552
$885$	0	0
$886$	18.0000	0.604722
$887$	24.2487	0.814192	0.407096	−	0.913385i	$-0.366541\pi$
$887$	24.2487	0.814192	0.407096	+	0.913385i	$0.366541\pi$
$888$	0	0
$889$	−2.00000	−0.0670778
$890$	0	0
$891$	0	0
$892$	8.00000	0.267860
$893$	20.7846	0.695530
$894$	0	0
$895$	0	0
$896$	12.1244	0.405046
$897$	0	0
$898$	60.0000	2.00223
$899$	−60.6218	−2.02185
$900$	0	0
$901$	9.00000	0.299833
$902$	−20.7846	−0.692052
$903$	0	0
$904$	12.0000	0.399114
$905$	0	0
$906$	0	0
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	−19.0526	−0.632281
$909$	0	0
$910$	0	0
$911$	31.1769	1.03294	0.516469	−	0.856306i	$-0.327246\pi$
$911$	31.1769	1.03294	0.516469	+	0.856306i	$0.327246\pi$
$912$	0	0
$913$	−9.00000	−0.297857
$914$	48.4974	1.60415
$915$	0	0
$916$	2.00000	0.0660819
$917$	0	0
$918$	0	0
$919$	−34.0000	−1.12156	−0.560778	−	0.827966i	$-0.689498\pi$
$919$	−34.0000	−1.12156	−0.560778	+	0.827966i	$0.689498\pi$
$920$	0	0
$921$	0	0
$922$	6.00000	0.197599
$923$	−3.46410	−0.114022
$924$	0	0
$925$	0	0
$926$	53.6936	1.76448
$927$	0	0
$928$	45.0000	1.47720
$929$	17.3205	0.568267	0.284134	−	0.958785i	$-0.408294\pi$
$929$	17.3205	0.568267	0.284134	+	0.958785i	$0.408294\pi$
$930$	0	0
$931$	24.0000	0.786568
$932$	−20.7846	−0.680823
$933$	0	0
$934$	−36.0000	−1.17796
$935$	0	0
$936$	0	0
$937$	−7.00000	−0.228680	−0.114340	−	0.993442i	$-0.536475\pi$
$937$	−7.00000	−0.228680	−0.114340	+	0.993442i	$0.536475\pi$
$938$	8.66025	0.282767
$939$	0	0
$940$	0	0
$941$	−51.9615	−1.69390	−0.846949	−	0.531675i	$-0.821563\pi$
$941$	−51.9615	−1.69390	−0.846949	+	0.531675i	$0.821563\pi$
$942$	0	0
$943$	−24.0000	−0.781548
$944$	−8.66025	−0.281867
$945$	0	0
$946$	6.00000	0.195077
$947$	8.66025	0.281420	0.140710	−	0.990051i	$-0.455061\pi$
$947$	8.66025	0.281420	0.140710	+	0.990051i	$0.455061\pi$
$948$	0	0
$949$	2.00000	0.0649227
$950$	0	0
$951$	0	0
$952$	−3.00000	−0.0972306
$953$	−8.66025	−0.280533	−0.140267	−	0.990114i	$-0.544796\pi$
$953$	−8.66025	−0.280533	−0.140267	+	0.990114i	$0.544796\pi$
$954$	0	0
$955$	0	0
$956$	−25.9808	−0.840278
$957$	0	0
$958$	−69.0000	−2.22929
$959$	13.8564	0.447447
$960$	0	0
$961$	18.0000	0.580645
$962$	−3.46410	−0.111687
$963$	0	0
$964$	8.00000	0.257663
$965$	0	0
$966$	0	0
$967$	29.0000	0.932577	0.466289	−	0.884633i	$-0.345591\pi$
$967$	29.0000	0.932577	0.466289	+	0.884633i	$0.345591\pi$
$968$	−13.8564	−0.445362
$969$	0	0
$970$	0	0
$971$	−45.0333	−1.44519	−0.722594	−	0.691273i	$-0.757050\pi$
$971$	−45.0333	−1.44519	−0.722594	+	0.691273i	$0.757050\pi$
$972$	0	0
$973$	10.0000	0.320585
$974$	12.1244	0.388489
$975$	0	0
$976$	5.00000	0.160046
$977$	−6.92820	−0.221653	−0.110826	−	0.993840i	$-0.535350\pi$
$977$	−6.92820	−0.221653	−0.110826	+	0.993840i	$0.535350\pi$
$978$	0	0
$979$	−24.0000	−0.767043
$980$	0	0
$981$	0	0
$982$	−42.0000	−1.34027
$983$	25.9808	0.828658	0.414329	−	0.910127i	$-0.364016\pi$
$983$	25.9808	0.828658	0.414329	+	0.910127i	$0.364016\pi$
$984$	0	0
$985$	0	0
$986$	−25.9808	−0.827396
$987$	0	0
$988$	−4.00000	−0.127257
$989$	6.92820	0.220304
$990$	0	0
$991$	−58.0000	−1.84243	−0.921215	−	0.389053i	$-0.872802\pi$
$991$	−58.0000	−1.84243	−0.921215	+	0.389053i	$0.872802\pi$
$992$	−36.3731	−1.15485
$993$	0	0
$994$	−6.00000	−0.190308
$995$	0	0
$996$	0	0
$997$	−55.0000	−1.74187	−0.870934	−	0.491400i	$-0.836485\pi$
$997$	−55.0000	−1.74187	−0.870934	+	0.491400i	$0.836485\pi$
$998$	−19.0526	−0.603098
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2925.2.a.ba.1.1	✓	2
3.2	odd	2	inner	2925.2.a.ba.1.2	yes	2
5.2	odd	4		2925.2.c.t.2224.1		4
5.3	odd	4		2925.2.c.t.2224.4		4
5.4	even	2		2925.2.a.bb.1.2	yes	2
15.2	even	4		2925.2.c.t.2224.3		4
15.8	even	4		2925.2.c.t.2224.2		4
15.14	odd	2		2925.2.a.bb.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2925.2.a.ba.1.1	✓	2	1.1	even	1	trivial
2925.2.a.ba.1.2	yes	2	3.2	odd	2	inner
2925.2.a.bb.1.1	yes	2	15.14	odd	2
2925.2.a.bb.1.2	yes	2	5.4	even	2
2925.2.c.t.2224.1		4	5.2	odd	4
2925.2.c.t.2224.2		4	15.8	even	4
2925.2.c.t.2224.3		4	15.2	even	4
2925.2.c.t.2224.4		4	5.3	odd	4

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:	\( N \)	\(=\)	\( 2925 = 3^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2925.a (trivial)

\(f(q)\)	\(=\)	\(q-1.73205 q^{2} +1.00000 q^{4} -1.00000 q^{7} +1.73205 q^{8} +O(q^{10})\)	\(q-1.73205 q^{2} +1.00000 q^{4} -1.00000 q^{7} +1.73205 q^{8} -1.73205 q^{11} +1.00000 q^{13} +1.73205 q^{14} -5.00000 q^{16} +1.73205 q^{17} -4.00000 q^{19} +3.00000 q^{22} +3.46410 q^{23} -1.73205 q^{26} -1.00000 q^{28} +8.66025 q^{29} -7.00000 q^{31} +5.19615 q^{32} -3.00000 q^{34} +2.00000 q^{37} +6.92820 q^{38} -6.92820 q^{41} +2.00000 q^{43} -1.73205 q^{44} -6.00000 q^{46} -5.19615 q^{47} -6.00000 q^{49} +1.00000 q^{52} +5.19615 q^{53} -1.73205 q^{56} -15.0000 q^{58} +1.73205 q^{59} -1.00000 q^{61} +12.1244 q^{62} +1.00000 q^{64} +5.00000 q^{67} +1.73205 q^{68} -3.46410 q^{71} +2.00000 q^{73} -3.46410 q^{74} -4.00000 q^{76} +1.73205 q^{77} -10.0000 q^{79} +12.0000 q^{82} +5.19615 q^{83} -3.46410 q^{86} -3.00000 q^{88} +13.8564 q^{89} -1.00000 q^{91} +3.46410 q^{92} +9.00000 q^{94} -10.0000 q^{97} +10.3923 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} - 2 q^{7}+O(q^{10}) \) 2 * q + 2 * q^4 - 2 * q^7	\( 2 q + 2 q^{4} - 2 q^{7} + 2 q^{13} - 10 q^{16} - 8 q^{19} + 6 q^{22} - 2 q^{28} - 14 q^{31} - 6 q^{34} + 4 q^{37} + 4 q^{43} - 12 q^{46} - 12 q^{49} + 2 q^{52} - 30 q^{58} - 2 q^{61} + 2 q^{64} + 10 q^{67} + 4 q^{73} - 8 q^{76} - 20 q^{79} + 24 q^{82} - 6 q^{88} - 2 q^{91} + 18 q^{94} - 20 q^{97}+O(q^{100}) \) 2 * q + 2 * q^4 - 2 * q^7 + 2 * q^13 - 10 * q^16 - 8 * q^19 + 6 * q^22 - 2 * q^28 - 14 * q^31 - 6 * q^34 + 4 * q^37 + 4 * q^43 - 12 * q^46 - 12 * q^49 + 2 * q^52 - 30 * q^58 - 2 * q^61 + 2 * q^64 + 10 * q^67 + 4 * q^73 - 8 * q^76 - 20 * q^79 + 24 * q^82 - 6 * q^88 - 2 * q^91 + 18 * q^94 - 20 * q^97

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