LMFDB - Embedded newform 2925.2.a.bq.1.2

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:	$0$
Dimension:	$6$
Coefficient field:	6.6.1509051136.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 13x^{4} + 43x^{2} - 19 $ x^6 - 13x^4 + 43x^2 - 19
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.16355$ 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)

$f(q)$	$=$	$q-2.16355 q^{2} +2.68093 q^{4} -3.26817 q^{7} -1.47322 q^{8} +O(q^{10})$	$q-2.16355 q^{2} +2.68093 q^{4} -3.26817 q^{7} -1.47322 q^{8} -2.85387 q^{11} +1.00000 q^{13} +7.07083 q^{14} -2.17448 q^{16} -6.38050 q^{17} +3.68093 q^{19} +6.17448 q^{22} -5.01742 q^{23} -2.16355 q^{26} -8.76172 q^{28} -2.05341 q^{29} +5.58724 q^{31} +7.65103 q^{32} +13.8045 q^{34} -10.2173 q^{37} -7.96385 q^{38} -5.01742 q^{41} -0.319072 q^{43} -7.65103 q^{44} +10.8554 q^{46} +2.16355 q^{47} +3.68093 q^{49} +2.68093 q^{52} -2.96401 q^{53} +4.81472 q^{56} +4.44265 q^{58} +6.49064 q^{59} +12.1745 q^{61} -12.0882 q^{62} -12.2044 q^{64} +7.58724 q^{67} -17.1057 q^{68} -16.3978 q^{71} +5.04278 q^{73} +22.1055 q^{74} +9.86830 q^{76} +9.32694 q^{77} +0.957217 q^{79} +10.8554 q^{82} -10.5975 q^{83} +0.690328 q^{86} +4.20438 q^{88} +7.05326 q^{89} -3.26817 q^{91} -13.4513 q^{92} -4.68093 q^{94} +10.5363 q^{97} -7.96385 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 14 q^{4} + 6 q^{7}+O(q^{10}) $ 6 * q + 14 * q^4 + 6 * q^7	$ 6 q + 14 q^{4} + 6 q^{7} + 6 q^{13} + 34 q^{16} + 20 q^{19} - 10 q^{22} + 18 q^{28} + 10 q^{31} + 6 q^{34} - 8 q^{37} - 4 q^{43} + 16 q^{46} + 20 q^{49} + 14 q^{52} - 46 q^{58} + 26 q^{61} + 70 q^{64} + 22 q^{67} + 24 q^{73} + 100 q^{76} + 12 q^{79} + 16 q^{82} - 118 q^{88} + 6 q^{91} - 26 q^{94} + 12 q^{97}+O(q^{100}) $ 6 * q + 14 * q^4 + 6 * q^7 + 6 * q^13 + 34 * q^16 + 20 * q^19 - 10 * q^22 + 18 * q^28 + 10 * q^31 + 6 * q^34 - 8 * q^37 - 4 * q^43 + 16 * q^46 + 20 * q^49 + 14 * q^52 - 46 * q^58 + 26 * q^61 + 70 * q^64 + 22 * q^67 + 24 * q^73 + 100 * q^76 + 12 * q^79 + 16 * q^82 - 118 * q^88 + 6 * q^91 - 26 * q^94 + 12 * 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 $

$2$	−2.16355	−1.52986	−0.764929	−	0.644115i	$-0.777226\pi$
$2$	−2.16355	−1.52986	−0.764929	+	0.644115i	$0.777226\pi$
$3$	0	0
$3$	0	0
$4$	2.68093	1.34046
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.26817	−1.23525	−0.617626	−	0.786472i	$-0.711905\pi$
$7$	−3.26817	−1.23525	−0.617626	+	0.786472i	$0.711905\pi$
$8$	−1.47322	−0.520861
$9$	0	0
$10$	0	0
$11$	−2.85387	−0.860475	−0.430238	−	0.902716i	$-0.641570\pi$
$11$	−2.85387	−0.860475	−0.430238	+	0.902716i	$0.641570\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	7.07083	1.88976
$15$	0	0
$16$	−2.17448	−0.543621
$17$	−6.38050	−1.54750	−0.773750	−	0.633491i	$-0.781621\pi$
$17$	−6.38050	−1.54750	−0.773750	+	0.633491i	$0.781621\pi$
$18$	0	0
$19$	3.68093	0.844463	0.422231	−	0.906488i	$-0.361247\pi$
$19$	3.68093	0.844463	0.422231	+	0.906488i	$0.361247\pi$
$20$	0	0
$21$	0	0
$22$	6.17448	1.31640
$23$	−5.01742	−1.04620	−0.523102	−	0.852270i	$-0.675225\pi$
$23$	−5.01742	−1.04620	−0.523102	+	0.852270i	$0.675225\pi$
$24$	0	0
$25$	0	0
$26$	−2.16355	−0.424306
$27$	0	0
$28$	−8.76172	−1.65581
$29$	−2.05341	−0.381309	−0.190655	−	0.981657i	$-0.561061\pi$
$29$	−2.05341	−0.381309	−0.190655	+	0.981657i	$0.561061\pi$
$30$	0	0
$31$	5.58724	1.00350	0.501749	−	0.865013i	$-0.332690\pi$
$31$	5.58724	1.00350	0.501749	+	0.865013i	$0.332690\pi$
$32$	7.65103	1.35252
$33$	0	0
$34$	13.8045	2.36745
$35$	0	0
$36$	0	0
$37$	−10.2173	−1.67971	−0.839854	−	0.542812i	$-0.817359\pi$
$37$	−10.2173	−1.67971	−0.839854	+	0.542812i	$0.817359\pi$
$38$	−7.96385	−1.29191
$39$	0	0
$40$	0	0
$41$	−5.01742	−0.783589	−0.391795	−	0.920053i	$-0.628146\pi$
$41$	−5.01742	−0.783589	−0.391795	+	0.920053i	$0.628146\pi$
$42$	0	0
$43$	−0.319072	−0.0486581	−0.0243290	−	0.999704i	$-0.507745\pi$
$43$	−0.319072	−0.0486581	−0.0243290	+	0.999704i	$0.507745\pi$
$44$	−7.65103	−1.15344
$45$	0	0
$46$	10.8554	1.60054
$47$	2.16355	0.315585	0.157793	−	0.987472i	$-0.449562\pi$
$47$	2.16355	0.315585	0.157793	+	0.987472i	$0.449562\pi$
$48$	0	0
$49$	3.68093	0.525847
$50$	0	0
$51$	0	0
$52$	2.68093	0.371778
$53$	−2.96401	−0.407137	−0.203569	−	0.979061i	$-0.565254\pi$
$53$	−2.96401	−0.407137	−0.203569	+	0.979061i	$0.565254\pi$
$54$	0	0
$55$	0	0
$56$	4.81472	0.643394
$57$	0	0
$58$	4.44265	0.583349
$59$	6.49064	0.845009	0.422504	−	0.906361i	$-0.361151\pi$
$59$	6.49064	0.845009	0.422504	+	0.906361i	$0.361151\pi$
$60$	0	0
$61$	12.1745	1.55878	0.779391	−	0.626537i	$-0.215528\pi$
$61$	12.1745	1.55878	0.779391	+	0.626537i	$0.215528\pi$
$62$	−12.0882	−1.53521
$63$	0	0
$64$	−12.2044	−1.52555
$65$	0	0
$66$	0	0
$67$	7.58724	0.926929	0.463465	−	0.886115i	$-0.346606\pi$
$67$	7.58724	0.926929	0.463465	+	0.886115i	$0.346606\pi$
$68$	−17.1057	−2.07437
$69$	0	0
$70$	0	0
$71$	−16.3978	−1.94606	−0.973029	−	0.230685i	$-0.925903\pi$
$71$	−16.3978	−1.94606	−0.973029	+	0.230685i	$0.925903\pi$
$72$	0	0
$73$	5.04278	0.590213	0.295107	−	0.955464i	$-0.404645\pi$
$73$	5.04278	0.590213	0.295107	+	0.955464i	$0.404645\pi$
$74$	22.1055	2.56971
$75$	0	0
$76$	9.86830	1.13197
$77$	9.32694	1.06290
$78$	0	0
$79$	0.957217	0.107695	0.0538477	−	0.998549i	$-0.482851\pi$
$79$	0.957217	0.107695	0.0538477	+	0.998549i	$0.482851\pi$
$80$	0	0
$81$	0	0
$82$	10.8554	1.19878
$83$	−10.5975	−1.16322	−0.581611	−	0.813467i	$-0.697577\pi$
$83$	−10.5975	−1.16322	−0.581611	+	0.813467i	$0.697577\pi$
$84$	0	0
$85$	0	0
$86$	0.690328	0.0744399
$87$	0	0
$88$	4.20438	0.448188
$89$	7.05326	0.747644	0.373822	−	0.927500i	$-0.378047\pi$
$89$	7.05326	0.747644	0.373822	+	0.927500i	$0.378047\pi$
$90$	0	0
$91$	−3.26817	−0.342597
$92$	−13.4513	−1.40240
$93$	0	0
$94$	−4.68093	−0.482801
$95$	0	0
$96$	0	0
$97$	10.5363	1.06980	0.534902	−	0.844914i	$-0.320349\pi$
$97$	10.5363	1.06980	0.534902	+	0.844914i	$0.320349\pi$
$98$	−7.96385	−0.804471
$99$	0	0
$100$	0	0
$101$	−7.29110	−0.725491	−0.362746	−	0.931888i	$-0.618161\pi$
$101$	−7.29110	−0.725491	−0.362746	+	0.931888i	$0.618161\pi$
$102$	0	0
$103$	−5.68093	−0.559758	−0.279879	−	0.960035i	$-0.590294\pi$
$103$	−5.68093	−0.559758	−0.279879	+	0.960035i	$0.590294\pi$
$104$	−1.47322	−0.144461
$105$	0	0
$106$	6.41276	0.622862
$107$	5.70775	0.551789	0.275894	−	0.961188i	$-0.411026\pi$
$107$	5.70775	0.551789	0.275894	+	0.961188i	$0.411026\pi$
$108$	0	0
$109$	4.95722	0.474815	0.237408	−	0.971410i	$-0.423702\pi$
$109$	4.95722	0.474815	0.237408	+	0.971410i	$0.423702\pi$
$110$	0	0
$111$	0	0
$112$	7.10658	0.671508
$113$	17.0881	1.60751	0.803756	−	0.594958i	$-0.202832\pi$
$113$	17.0881	1.60751	0.803756	+	0.594958i	$0.202832\pi$
$114$	0	0
$115$	0	0
$116$	−5.50505	−0.511131
$117$	0	0
$118$	−14.0428	−1.29274
$119$	20.8526	1.91155
$120$	0	0
$121$	−2.85541	−0.259583
$122$	−26.3400	−2.38472
$123$	0	0
$124$	14.9790	1.34515
$125$	0	0
$126$	0	0
$127$	−11.5791	−1.02748	−0.513740	−	0.857946i	$-0.671741\pi$
$127$	−11.5791	−1.02748	−0.513740	+	0.857946i	$0.671741\pi$
$128$	11.1027	0.981346
$129$	0	0
$130$	0	0
$131$	9.12424	0.797189	0.398594	−	0.917127i	$-0.369498\pi$
$131$	9.12424	0.797189	0.398594	+	0.917127i	$0.369498\pi$
$132$	0	0
$133$	−12.0299	−1.04312
$134$	−16.4153	−1.41807
$135$	0	0
$136$	9.39987	0.806032
$137$	17.0881	1.45993	0.729967	−	0.683482i	$-0.239535\pi$
$137$	17.0881	1.45993	0.729967	+	0.683482i	$0.239535\pi$
$138$	0	0
$139$	−18.4345	−1.56360	−0.781798	−	0.623531i	$-0.785697\pi$
$139$	−18.4345	−1.56360	−0.781798	+	0.623531i	$0.785697\pi$
$140$	0	0
$141$	0	0
$142$	35.4773	2.97719
$143$	−2.85387	−0.238653
$144$	0	0
$145$	0	0
$146$	−10.9103	−0.902942
$147$	0	0
$148$	−27.3917	−2.25159
$149$	−2.94643	−0.241381	−0.120691	−	0.992690i	$-0.538511\pi$
$149$	−2.94643	−0.241381	−0.120691	+	0.992690i	$0.538511\pi$
$150$	0	0
$151$	2.54446	0.207065	0.103533	−	0.994626i	$-0.466985\pi$
$151$	2.54446	0.207065	0.103533	+	0.994626i	$0.466985\pi$
$152$	−5.42281	−0.439848
$153$	0	0
$154$	−20.1793	−1.62609
$155$	0	0
$156$	0	0
$157$	−8.04278	−0.641884	−0.320942	−	0.947099i	$-0.603999\pi$
$157$	−8.04278	−0.641884	−0.320942	+	0.947099i	$0.603999\pi$
$158$	−2.07098	−0.164759
$159$	0	0
$160$	0	0
$161$	16.3978	1.29233
$162$	0	0
$163$	13.3917	1.04892	0.524461	−	0.851434i	$-0.324267\pi$
$163$	13.3917	1.04892	0.524461	+	0.851434i	$0.324267\pi$
$164$	−13.4513	−1.05037
$165$	0	0
$166$	22.9281	1.77956
$167$	17.9987	1.39278	0.696390	−	0.717663i	$-0.254788\pi$
$167$	17.9987	1.39278	0.696390	+	0.717663i	$0.254788\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	−0.855410	−0.0652244
$173$	7.29110	0.554332	0.277166	−	0.960822i	$-0.410605\pi$
$173$	7.29110	0.554332	0.277166	+	0.960822i	$0.410605\pi$
$174$	0	0
$175$	0	0
$176$	6.20570	0.467772
$177$	0	0
$178$	−15.2600	−1.14379
$179$	−7.96385	−0.595246	−0.297623	−	0.954683i	$-0.596194\pi$
$179$	−7.96385	−0.595246	−0.297623	+	0.954683i	$0.596194\pi$
$180$	0	0
$181$	−17.3028	−1.28611	−0.643055	−	0.765820i	$-0.722333\pi$
$181$	−17.3028	−1.28611	−0.643055	+	0.765820i	$0.722333\pi$
$182$	7.07083	0.524125
$183$	0	0
$184$	7.39175	0.544927
$185$	0	0
$186$	0	0
$187$	18.2091	1.33158
$188$	5.80031	0.423031
$189$	0	0
$190$	0	0
$191$	−11.3804	−0.823453	−0.411727	−	0.911307i	$-0.635074\pi$
$191$	−11.3804	−0.823453	−0.411727	+	0.911307i	$0.635074\pi$
$192$	0	0
$193$	−20.1155	−1.44794	−0.723971	−	0.689830i	$-0.757685\pi$
$193$	−20.1155	−1.44794	−0.723971	+	0.689830i	$0.757685\pi$
$194$	−22.7958	−1.63665
$195$	0	0
$196$	9.86830	0.704879
$197$	−15.0523	−1.07243	−0.536214	−	0.844082i	$-0.680146\pi$
$197$	−15.0523	−1.07243	−0.536214	+	0.844082i	$0.680146\pi$
$198$	0	0
$199$	−10.4345	−0.739684	−0.369842	−	0.929095i	$-0.620588\pi$
$199$	−10.4345	−0.739684	−0.369842	+	0.929095i	$0.620588\pi$
$200$	0	0
$201$	0	0
$202$	15.7746	1.10990
$203$	6.71090	0.471013
$204$	0	0
$205$	0	0
$206$	12.2909	0.856351
$207$	0	0
$208$	−2.17448	−0.150773
$209$	−10.5049	−0.726639
$210$	0	0
$211$	16.8554	1.16037	0.580187	−	0.814483i	$-0.302979\pi$
$211$	16.8554	1.16037	0.580187	+	0.814483i	$0.302979\pi$
$212$	−7.94628	−0.545753
$213$	0	0
$214$	−12.3490	−0.844158
$215$	0	0
$216$	0	0
$217$	−18.2600	−1.23957
$218$	−10.7252	−0.726400
$219$	0	0
$220$	0	0
$221$	−6.38050	−0.429199
$222$	0	0
$223$	21.3917	1.43250	0.716249	−	0.697845i	$-0.245858\pi$
$223$	21.3917	1.43250	0.716249	+	0.697845i	$0.245858\pi$
$224$	−25.0048	−1.67071
$225$	0	0
$226$	−36.9709	−2.45927
$227$	8.78188	0.582874	0.291437	−	0.956590i	$-0.405867\pi$
$227$	8.78188	0.582874	0.291437	+	0.956590i	$0.405867\pi$
$228$	0	0
$229$	21.7964	1.44035	0.720173	−	0.693795i	$-0.244062\pi$
$229$	21.7964	1.44035	0.720173	+	0.693795i	$0.244062\pi$
$230$	0	0
$231$	0	0
$232$	3.02512	0.198609
$233$	−15.7074	−1.02903	−0.514514	−	0.857482i	$-0.672028\pi$
$233$	−15.7074	−1.02903	−0.514514	+	0.857482i	$0.672028\pi$
$234$	0	0
$235$	0	0
$236$	17.4009	1.13270
$237$	0	0
$238$	−45.1155	−2.92440
$239$	24.9594	1.61449	0.807244	−	0.590217i	$-0.200958\pi$
$239$	24.9594	1.61449	0.807244	+	0.590217i	$0.200958\pi$
$240$	0	0
$241$	−9.39175	−0.604976	−0.302488	−	0.953153i	$-0.597817\pi$
$241$	−9.39175	−0.604976	−0.302488	+	0.953153i	$0.597817\pi$
$242$	6.17781	0.397125
$243$	0	0
$244$	32.6389	2.08949
$245$	0	0
$246$	0	0
$247$	3.68093	0.234212
$248$	−8.23122	−0.522683
$249$	0	0
$250$	0	0
$251$	12.1058	0.764113	0.382056	−	0.924139i	$-0.375216\pi$
$251$	12.1058	0.764113	0.382056	+	0.924139i	$0.375216\pi$
$252$	0	0
$253$	14.3191	0.900232
$254$	25.0519	1.57190
$255$	0	0
$256$	0.387635	0.0242272
$257$	11.6182	0.724723	0.362361	−	0.932038i	$-0.381971\pi$
$257$	11.6182	0.724723	0.362361	+	0.932038i	$0.381971\pi$
$258$	0	0
$259$	33.3917	2.07486
$260$	0	0
$261$	0	0
$262$	−19.7407	−1.21959
$263$	26.2123	1.61632	0.808161	−	0.588962i	$-0.200463\pi$
$263$	26.2123	1.61632	0.808161	+	0.588962i	$0.200463\pi$
$264$	0	0
$265$	0	0
$266$	26.0272	1.59583
$267$	0	0
$268$	20.3408	1.24251
$269$	22.3433	1.36230	0.681149	−	0.732145i	$-0.261481\pi$
$269$	22.3433	1.36230	0.681149	+	0.732145i	$0.261481\pi$
$270$	0	0
$271$	23.8045	1.44602	0.723010	−	0.690837i	$-0.242758\pi$
$271$	23.8045	1.44602	0.723010	+	0.690837i	$0.242758\pi$
$272$	13.8743	0.841253
$273$	0	0
$274$	−36.9709	−2.23349
$275$	0	0
$276$	0	0
$277$	−32.8853	−1.97589	−0.987943	−	0.154817i	$-0.950521\pi$
$277$	−32.8853	−1.97589	−0.987943	+	0.154817i	$0.950521\pi$
$278$	39.8839	2.39208
$279$	0	0
$280$	0	0
$281$	−11.3804	−0.678895	−0.339447	−	0.940625i	$-0.610240\pi$
$281$	−11.3804	−0.678895	−0.339447	+	0.940625i	$0.610240\pi$
$282$	0	0
$283$	−3.91443	−0.232689	−0.116344	−	0.993209i	$-0.537118\pi$
$283$	−3.91443	−0.232689	−0.116344	+	0.993209i	$0.537118\pi$
$284$	−43.9612	−2.60862
$285$	0	0
$286$	6.17448	0.365105
$287$	16.3978	0.967930
$288$	0	0
$289$	23.7108	1.39475
$290$	0	0
$291$	0	0
$292$	13.5193	0.791159
$293$	34.2113	1.99865	0.999324	−	0.0367711i	$-0.0117072\pi$
$293$	34.2113	1.99865	0.999324	+	0.0367711i	$0.0117072\pi$
$294$	0	0
$295$	0	0
$296$	15.0523	0.874894
$297$	0	0
$298$	6.37475	0.369279
$299$	−5.01742	−0.290165
$300$	0	0
$301$	1.04278	0.0601050
$302$	−5.50505	−0.316780
$303$	0	0
$304$	−8.00411	−0.459067
$305$	0	0
$306$	0	0
$307$	25.9281	1.47979	0.739897	−	0.672720i	$-0.234874\pi$
$307$	25.9281	1.47979	0.739897	+	0.672720i	$0.234874\pi$
$308$	25.0048	1.42478
$309$	0	0
$310$	0	0
$311$	20.0345	1.13605	0.568027	−	0.823010i	$-0.307707\pi$
$311$	20.0345	1.13605	0.568027	+	0.823010i	$0.307707\pi$
$312$	0	0
$313$	−13.7536	−0.777400	−0.388700	−	0.921364i	$-0.627076\pi$
$313$	−13.7536	−0.777400	−0.388700	+	0.921364i	$0.627076\pi$
$314$	17.4009	0.981991
$315$	0	0
$316$	2.56623	0.144362
$317$	17.9987	1.01091	0.505454	−	0.862854i	$-0.331325\pi$
$317$	17.9987	1.01091	0.505454	+	0.862854i	$0.331325\pi$
$318$	0	0
$319$	5.86018	0.328107
$320$	0	0
$321$	0	0
$322$	−35.4773	−1.97707
$323$	−23.4862	−1.30681
$324$	0	0
$325$	0	0
$326$	−28.9737	−1.60470
$327$	0	0
$328$	7.39175	0.408141
$329$	−7.07083	−0.389828
$330$	0	0
$331$	8.31907	0.457258	0.228629	−	0.973514i	$-0.426576\pi$
$331$	8.31907	0.457258	0.228629	+	0.973514i	$0.426576\pi$
$332$	−28.4110	−1.55926
$333$	0	0
$334$	−38.9410	−2.13076
$335$	0	0
$336$	0	0
$337$	27.0137	1.47153	0.735764	−	0.677238i	$-0.236823\pi$
$337$	27.0137	1.47153	0.735764	+	0.677238i	$0.236823\pi$
$338$	−2.16355	−0.117681
$339$	0	0
$340$	0	0
$341$	−15.9453	−0.863485
$342$	0	0
$343$	10.8473	0.585699
$344$	0.470063	0.0253441
$345$	0	0
$346$	−15.7746	−0.848048
$347$	−24.1765	−1.29786	−0.648931	−	0.760847i	$-0.724784\pi$
$347$	−24.1765	−1.29786	−0.648931	+	0.760847i	$0.724784\pi$
$348$	0	0
$349$	−6.21727	−0.332803	−0.166401	−	0.986058i	$-0.553215\pi$
$349$	−6.21727	−0.332803	−0.166401	+	0.986058i	$0.553215\pi$
$350$	0	0
$351$	0	0
$352$	−21.8351	−1.16381
$353$	30.9800	1.64890	0.824448	−	0.565937i	$-0.191485\pi$
$353$	30.9800	1.64890	0.824448	+	0.565937i	$0.191485\pi$
$354$	0	0
$355$	0	0
$356$	18.9093	1.00219
$357$	0	0
$358$	17.2302	0.910642
$359$	3.50906	0.185201	0.0926006	−	0.995703i	$-0.470482\pi$
$359$	3.50906	0.185201	0.0926006	+	0.995703i	$0.470482\pi$
$360$	0	0
$361$	−5.45077	−0.286883
$362$	37.4355	1.96756
$363$	0	0
$364$	−8.76172	−0.459239
$365$	0	0
$366$	0	0
$367$	5.71082	0.298102	0.149051	−	0.988829i	$-0.452378\pi$
$367$	5.71082	0.298102	0.149051	+	0.988829i	$0.452378\pi$
$368$	10.9103	0.568738
$369$	0	0
$370$	0	0
$371$	9.68687	0.502917
$372$	0	0
$373$	26.4936	1.37178	0.685892	−	0.727703i	$-0.259412\pi$
$373$	26.4936	1.37178	0.685892	+	0.727703i	$0.259412\pi$
$374$	−39.3963	−2.03713
$375$	0	0
$376$	−3.18737	−0.164376
$377$	−2.05341	−0.105756
$378$	0	0
$379$	22.3110	1.14604	0.573018	−	0.819543i	$-0.305772\pi$
$379$	22.3110	1.14604	0.573018	+	0.819543i	$0.305772\pi$
$380$	0	0
$381$	0	0
$382$	24.6219	1.25977
$383$	26.4326	1.35064	0.675322	−	0.737523i	$-0.264005\pi$
$383$	26.4326	1.35064	0.675322	+	0.737523i	$0.264005\pi$
$384$	0	0
$385$	0	0
$386$	43.5207	2.21515
$387$	0	0
$388$	28.2472	1.43403
$389$	−2.98158	−0.151172	−0.0755859	−	0.997139i	$-0.524083\pi$
$389$	−2.98158	−0.151172	−0.0755859	+	0.997139i	$0.524083\pi$
$390$	0	0
$391$	32.0137	1.61900
$392$	−5.42281	−0.273893
$393$	0	0
$394$	32.5662	1.64066
$395$	0	0
$396$	0	0
$397$	−21.8982	−1.09904	−0.549519	−	0.835481i	$-0.685189\pi$
$397$	−21.8982	−1.09904	−0.549519	+	0.835481i	$0.685189\pi$
$398$	22.5756	1.13161
$399$	0	0
$400$	0	0
$401$	−19.1591	−0.956759	−0.478379	−	0.878153i	$-0.658776\pi$
$401$	−19.1591	−0.956759	−0.478379	+	0.878153i	$0.658776\pi$
$402$	0	0
$403$	5.58724	0.278320
$404$	−19.5469	−0.972495
$405$	0	0
$406$	−14.5193	−0.720582
$407$	29.1588	1.44535
$408$	0	0
$409$	20.0137	0.989611	0.494806	−	0.869004i	$-0.335239\pi$
$409$	20.0137	0.989611	0.494806	+	0.869004i	$0.335239\pi$
$410$	0	0
$411$	0	0
$412$	−15.2302	−0.750336
$413$	−21.2125	−1.04380
$414$	0	0
$415$	0	0
$416$	7.65103	0.375122
$417$	0	0
$418$	22.7278	1.11165
$419$	−19.3793	−0.946743	−0.473371	−	0.880863i	$-0.656963\pi$
$419$	−19.3793	−0.946743	−0.473371	+	0.880863i	$0.656963\pi$
$420$	0	0
$421$	−34.3490	−1.67407	−0.837033	−	0.547152i	$-0.815712\pi$
$421$	−34.3490	−1.67407	−0.837033	+	0.547152i	$0.815712\pi$
$422$	−36.4674	−1.77521
$423$	0	0
$424$	4.36662	0.212062
$425$	0	0
$426$	0	0
$427$	−39.7883	−1.92549
$428$	15.3021	0.739653
$429$	0	0
$430$	0	0
$431$	12.3261	0.593727	0.296863	−	0.954920i	$-0.404059\pi$
$431$	12.3261	0.593727	0.296863	+	0.954920i	$0.404059\pi$
$432$	0	0
$433$	30.3327	1.45770	0.728849	−	0.684675i	$-0.240056\pi$
$433$	30.3327	1.45770	0.728849	+	0.684675i	$0.240056\pi$
$434$	39.5064	1.89637
$435$	0	0
$436$	13.2899	0.636473
$437$	−18.4688	−0.883480
$438$	0	0
$439$	14.1317	0.674469	0.337235	−	0.941421i	$-0.390508\pi$
$439$	14.1317	0.674469	0.337235	+	0.941421i	$0.390508\pi$
$440$	0	0
$441$	0	0
$442$	13.8045	0.656613
$443$	14.8320	0.704689	0.352345	−	0.935870i	$-0.385385\pi$
$443$	14.8320	0.704689	0.352345	+	0.935870i	$0.385385\pi$
$444$	0	0
$445$	0	0
$446$	−46.2820	−2.19152
$447$	0	0
$448$	39.8860	1.88443
$449$	−36.9375	−1.74319	−0.871594	−	0.490228i	$-0.836914\pi$
$449$	−36.9375	−1.74319	−0.871594	+	0.490228i	$0.836914\pi$
$450$	0	0
$451$	14.3191	0.674259
$452$	45.8119	2.15481
$453$	0	0
$454$	−19.0000	−0.891714
$455$	0	0
$456$	0	0
$457$	−7.78273	−0.364061	−0.182030	−	0.983293i	$-0.558267\pi$
$457$	−7.78273	−0.364061	−0.182030	+	0.983293i	$0.558267\pi$
$458$	−47.1575	−2.20352
$459$	0	0
$460$	0	0
$461$	−24.1765	−1.12601	−0.563006	−	0.826453i	$-0.690355\pi$
$461$	−24.1765	−1.12601	−0.563006	+	0.826453i	$0.690355\pi$
$462$	0	0
$463$	5.13647	0.238712	0.119356	−	0.992852i	$-0.461917\pi$
$463$	5.13647	0.238712	0.119356	+	0.992852i	$0.461917\pi$
$464$	4.46511	0.207288
$465$	0	0
$466$	33.9838	1.57427
$467$	8.87445	0.410660	0.205330	−	0.978693i	$-0.434173\pi$
$467$	8.87445	0.410660	0.205330	+	0.978693i	$0.434173\pi$
$468$	0	0
$469$	−24.7964	−1.14499
$470$	0	0
$471$	0	0
$472$	−9.56212	−0.440132
$473$	0.910592	0.0418691
$474$	0	0
$475$	0	0
$476$	55.9042	2.56237
$477$	0	0
$478$	−54.0008	−2.46994
$479$	−42.7027	−1.95113	−0.975567	−	0.219700i	$-0.929492\pi$
$479$	−42.7027	−1.95113	−0.975567	+	0.219700i	$0.929492\pi$
$480$	0	0
$481$	−10.2173	−0.465867
$482$	20.3195	0.925527
$483$	0	0
$484$	−7.65515	−0.347961
$485$	0	0
$486$	0	0
$487$	30.1073	1.36429	0.682147	−	0.731215i	$-0.261046\pi$
$487$	30.1073	1.36429	0.682147	+	0.731215i	$0.261046\pi$
$488$	−17.9357	−0.811909
$489$	0	0
$490$	0	0
$491$	−31.4500	−1.41932	−0.709660	−	0.704544i	$-0.751151\pi$
$491$	−31.4500	−1.41932	−0.709660	+	0.704544i	$0.751151\pi$
$492$	0	0
$493$	13.1018	0.590076
$494$	−7.96385	−0.358311
$495$	0	0
$496$	−12.1494	−0.545522
$497$	53.5907	2.40387
$498$	0	0
$499$	22.0937	0.989049	0.494525	−	0.869164i	$-0.335342\pi$
$499$	22.0937	0.989049	0.494525	+	0.869164i	$0.335342\pi$
$500$	0	0
$501$	0	0
$502$	−26.1915	−1.16898
$503$	3.67190	0.163722	0.0818610	−	0.996644i	$-0.473914\pi$
$503$	3.67190	0.163722	0.0818610	+	0.996644i	$0.473914\pi$
$504$	0	0
$505$	0	0
$506$	−30.9800	−1.37723
$507$	0	0
$508$	−31.0428	−1.37730
$509$	43.5558	1.93058	0.965289	−	0.261183i	$-0.0841125\pi$
$509$	43.5558	1.93058	0.965289	+	0.261183i	$0.0841125\pi$
$510$	0	0
$511$	−16.4807	−0.729062
$512$	−23.0440	−1.01841
$513$	0	0
$514$	−25.1365	−1.10872
$515$	0	0
$516$	0	0
$517$	−6.17448	−0.271553
$518$	−72.2446	−3.17424
$519$	0	0
$520$	0	0
$521$	10.0348	0.439634	0.219817	−	0.975541i	$-0.429454\pi$
$521$	10.0348	0.439634	0.219817	+	0.975541i	$0.429454\pi$
$522$	0	0
$523$	21.5234	0.941155	0.470577	−	0.882359i	$-0.344046\pi$
$523$	21.5234	0.941155	0.470577	+	0.882359i	$0.344046\pi$
$524$	24.4614	1.06860
$525$	0	0
$526$	−56.7116	−2.47274
$527$	−35.6494	−1.55291
$528$	0	0
$529$	2.17448	0.0945427
$530$	0	0
$531$	0	0
$532$	−32.2513	−1.39827
$533$	−5.01742	−0.217328
$534$	0	0
$535$	0	0
$536$	−11.1777	−0.482801
$537$	0	0
$538$	−48.3408	−2.08412
$539$	−10.5049	−0.452478
$540$	0	0
$541$	−3.97011	−0.170688	−0.0853441	−	0.996352i	$-0.527199\pi$
$541$	−3.97011	−0.170688	−0.0853441	+	0.996352i	$0.527199\pi$
$542$	−51.5021	−2.21221
$543$	0	0
$544$	−48.8174	−2.09303
$545$	0	0
$546$	0	0
$547$	−43.3917	−1.85530	−0.927649	−	0.373454i	$-0.878173\pi$
$547$	−43.3917	−1.85530	−0.927649	+	0.373454i	$0.878173\pi$
$548$	45.8119	1.95699
$549$	0	0
$550$	0	0
$551$	−7.55846	−0.322001
$552$	0	0
$553$	−3.12835	−0.133031
$554$	71.1488	3.02282
$555$	0	0
$556$	−49.4216	−2.09594
$557$	−10.6900	−0.452951	−0.226475	−	0.974017i	$-0.572720\pi$
$557$	−10.6900	−0.452951	−0.226475	+	0.974017i	$0.572720\pi$
$558$	0	0
$559$	−0.319072	−0.0134953
$560$	0	0
$561$	0	0
$562$	24.6219	1.03861
$563$	3.19623	0.134705	0.0673526	−	0.997729i	$-0.478545\pi$
$563$	3.19623	0.134705	0.0673526	+	0.997729i	$0.478545\pi$
$564$	0	0
$565$	0	0
$566$	8.46906	0.355981
$567$	0	0
$568$	24.1575	1.01363
$569$	16.8503	0.706400	0.353200	−	0.935548i	$-0.385094\pi$
$569$	16.8503	0.706400	0.353200	+	0.935548i	$0.385094\pi$
$570$	0	0
$571$	−2.13170	−0.0892089	−0.0446044	−	0.999005i	$-0.514203\pi$
$571$	−2.13170	−0.0892089	−0.0446044	+	0.999005i	$0.514203\pi$
$572$	−7.65103	−0.319905
$573$	0	0
$574$	−35.4773	−1.48079
$575$	0	0
$576$	0	0
$577$	−10.1018	−0.420544	−0.210272	−	0.977643i	$-0.567435\pi$
$577$	−10.1018	−0.420544	−0.210272	+	0.977643i	$0.567435\pi$
$578$	−51.2994	−2.13377
$579$	0	0
$580$	0	0
$581$	34.6343	1.43687
$582$	0	0
$583$	8.45889	0.350332
$584$	−7.42912	−0.307419
$585$	0	0
$586$	−74.0178	−3.05765
$587$	−24.2691	−1.00169	−0.500846	−	0.865537i	$-0.666978\pi$
$587$	−24.2691	−1.00169	−0.500846	+	0.865537i	$0.666978\pi$
$588$	0	0
$589$	20.5662	0.847417
$590$	0	0
$591$	0	0
$592$	22.2173	0.913124
$593$	−17.3435	−0.712212	−0.356106	−	0.934446i	$-0.615896\pi$
$593$	−17.3435	−0.712212	−0.356106	+	0.934446i	$0.615896\pi$
$594$	0	0
$595$	0	0
$596$	−7.89918	−0.323563
$597$	0	0
$598$	10.8554	0.443911
$599$	−20.9803	−0.857230	−0.428615	−	0.903487i	$-0.640998\pi$
$599$	−20.9803	−0.857230	−0.428615	+	0.903487i	$0.640998\pi$
$600$	0	0
$601$	−31.7536	−1.29526	−0.647628	−	0.761956i	$-0.724239\pi$
$601$	−31.7536	−1.29526	−0.647628	+	0.761956i	$0.724239\pi$
$602$	−2.25611	−0.0919521
$603$	0	0
$604$	6.82151	0.277563
$605$	0	0
$606$	0	0
$607$	2.73995	0.111211	0.0556056	−	0.998453i	$-0.482291\pi$
$607$	2.73995	0.111211	0.0556056	+	0.998453i	$0.482291\pi$
$608$	28.1629	1.14216
$609$	0	0
$610$	0	0
$611$	2.16355	0.0875277
$612$	0	0
$613$	−23.5072	−0.949447	−0.474724	−	0.880135i	$-0.657452\pi$
$613$	−23.5072	−0.949447	−0.474724	+	0.880135i	$0.657452\pi$
$614$	−56.0966	−2.26387
$615$	0	0
$616$	−13.7406	−0.553625
$617$	−12.9813	−0.522606	−0.261303	−	0.965257i	$-0.584152\pi$
$617$	−12.9813	−0.522606	−0.261303	+	0.965257i	$0.584152\pi$
$618$	0	0
$619$	28.2173	1.13415	0.567074	−	0.823667i	$-0.308075\pi$
$619$	28.2173	1.13415	0.567074	+	0.823667i	$0.308075\pi$
$620$	0	0
$621$	0	0
$622$	−43.3456	−1.73800
$623$	−23.0512	−0.923529
$624$	0	0
$625$	0	0
$626$	29.7565	1.18931
$627$	0	0
$628$	−21.5621	−0.860422
$629$	65.1913	2.59935
$630$	0	0
$631$	−8.65180	−0.344423	−0.172211	−	0.985060i	$-0.555091\pi$
$631$	−8.65180	−0.344423	−0.172211	+	0.985060i	$0.555091\pi$
$632$	−1.41019	−0.0560943
$633$	0	0
$634$	−38.9410	−1.54654
$635$	0	0
$636$	0	0
$637$	3.68093	0.145844
$638$	−12.6788	−0.501957
$639$	0	0
$640$	0	0
$641$	3.17866	0.125550	0.0627748	−	0.998028i	$-0.480005\pi$
$641$	3.17866	0.125550	0.0627748	+	0.998028i	$0.480005\pi$
$642$	0	0
$643$	4.52269	0.178357	0.0891787	−	0.996016i	$-0.471576\pi$
$643$	4.52269	0.178357	0.0891787	+	0.996016i	$0.471576\pi$
$644$	43.9612	1.73232
$645$	0	0
$646$	50.8134	1.99923
$647$	6.61273	0.259973	0.129987	−	0.991516i	$-0.458507\pi$
$647$	6.61273	0.259973	0.129987	+	0.991516i	$0.458507\pi$
$648$	0	0
$649$	−18.5234	−0.727109
$650$	0	0
$651$	0	0
$652$	35.9023	1.40604
$653$	−45.1041	−1.76506	−0.882529	−	0.470258i	$-0.844161\pi$
$653$	−45.1041	−1.76506	−0.882529	+	0.470258i	$0.844161\pi$
$654$	0	0
$655$	0	0
$656$	10.9103	0.425975
$657$	0	0
$658$	15.2981	0.596381
$659$	42.6157	1.66007	0.830036	−	0.557709i	$-0.188320\pi$
$659$	42.6157	1.66007	0.830036	+	0.557709i	$0.188320\pi$
$660$	0	0
$661$	−8.42088	−0.327534	−0.163767	−	0.986499i	$-0.552365\pi$
$661$	−8.42088	−0.327534	−0.163767	+	0.986499i	$0.552365\pi$
$662$	−17.9987	−0.699539
$663$	0	0
$664$	15.6124	0.605877
$665$	0	0
$666$	0	0
$667$	10.3028	0.398927
$668$	48.2532	1.86697
$669$	0	0
$670$	0	0
$671$	−34.7444	−1.34129
$672$	0	0
$673$	27.2173	1.04915	0.524574	−	0.851365i	$-0.324225\pi$
$673$	27.2173	1.04915	0.524574	+	0.851365i	$0.324225\pi$
$674$	−58.4453	−2.25123
$675$	0	0
$676$	2.68093	0.103113
$677$	15.9277	0.612151	0.306076	−	0.952007i	$-0.400984\pi$
$677$	15.9277	0.612151	0.306076	+	0.952007i	$0.400984\pi$
$678$	0	0
$679$	−34.4345	−1.32148
$680$	0	0
$681$	0	0
$682$	34.4983	1.32101
$683$	25.3943	0.971686	0.485843	−	0.874046i	$-0.338513\pi$
$683$	25.3943	0.971686	0.485843	+	0.874046i	$0.338513\pi$
$684$	0	0
$685$	0	0
$686$	−23.4686	−0.896035
$687$	0	0
$688$	0.693817	0.0264515
$689$	−2.96401	−0.112920
$690$	0	0
$691$	30.6898	1.16750	0.583748	−	0.811935i	$-0.301586\pi$
$691$	30.6898	1.16750	0.583748	+	0.811935i	$0.301586\pi$
$692$	19.5469	0.743062
$693$	0	0
$694$	52.3069	1.98554
$695$	0	0
$696$	0	0
$697$	32.0137	1.21260
$698$	13.4513	0.509140
$699$	0	0
$700$	0	0
$701$	10.7427	0.405747	0.202874	−	0.979205i	$-0.434972\pi$
$701$	10.7427	0.405747	0.202874	+	0.979205i	$0.434972\pi$
$702$	0	0
$703$	−37.6090	−1.41845
$704$	34.8297	1.31270
$705$	0	0
$706$	−67.0265	−2.52258
$707$	23.8285	0.896164
$708$	0	0
$709$	−25.2899	−0.949784	−0.474892	−	0.880044i	$-0.657513\pi$
$709$	−25.2899	−0.949784	−0.474892	+	0.880044i	$0.657513\pi$
$710$	0	0
$711$	0	0
$712$	−10.3910	−0.389419
$713$	−28.0335	−1.04986
$714$	0	0
$715$	0	0
$716$	−21.3505	−0.797906
$717$	0	0
$718$	−7.59201	−0.283331
$719$	42.1400	1.57156	0.785779	−	0.618507i	$-0.212262\pi$
$719$	42.1400	1.57156	0.785779	+	0.618507i	$0.212262\pi$
$720$	0	0
$721$	18.5662	0.691443
$722$	11.7930	0.438890
$723$	0	0
$724$	−46.3876	−1.72398
$725$	0	0
$726$	0	0
$727$	−9.68093	−0.359046	−0.179523	−	0.983754i	$-0.557455\pi$
$727$	−9.68093	−0.359046	−0.179523	+	0.983754i	$0.557455\pi$
$728$	4.81472	0.178446
$729$	0	0
$730$	0	0
$731$	2.03584	0.0752984
$732$	0	0
$733$	−24.9015	−0.919760	−0.459880	−	0.887981i	$-0.652107\pi$
$733$	−24.9015	−0.919760	−0.459880	+	0.887981i	$0.652107\pi$
$734$	−12.3556	−0.456054
$735$	0	0
$736$	−38.3884	−1.41502
$737$	−21.6530	−0.797599
$738$	0	0
$739$	−24.3707	−0.896492	−0.448246	−	0.893910i	$-0.647951\pi$
$739$	−24.3707	−0.896492	−0.448246	+	0.893910i	$0.647951\pi$
$740$	0	0
$741$	0	0
$742$	−20.9580	−0.769392
$743$	−16.7401	−0.614136	−0.307068	−	0.951688i	$-0.599348\pi$
$743$	−16.7401	−0.614136	−0.307068	+	0.951688i	$0.599348\pi$
$744$	0	0
$745$	0	0
$746$	−57.3200	−2.09863
$747$	0	0
$748$	48.8174	1.78494
$749$	−18.6539	−0.681598
$750$	0	0
$751$	−6.13170	−0.223749	−0.111874	−	0.993722i	$-0.535685\pi$
$751$	−6.13170	−0.223749	−0.111874	+	0.993722i	$0.535685\pi$
$752$	−4.70459	−0.171559
$753$	0	0
$754$	4.44265	0.161792
$755$	0	0
$756$	0	0
$757$	−23.9871	−0.871826	−0.435913	−	0.899989i	$-0.643575\pi$
$757$	−23.9871	−0.871826	−0.435913	+	0.899989i	$0.643575\pi$
$758$	−48.2708	−1.75327
$759$	0	0
$760$	0	0
$761$	−20.6897	−0.750002	−0.375001	−	0.927024i	$-0.622358\pi$
$761$	−20.6897	−0.750002	−0.375001	+	0.927024i	$0.622358\pi$
$762$	0	0
$763$	−16.2010	−0.586516
$764$	−30.5099	−1.10381
$765$	0	0
$766$	−57.1881	−2.06629
$767$	6.49064	0.234363
$768$	0	0
$769$	−17.1745	−0.619328	−0.309664	−	0.950846i	$-0.600217\pi$
$769$	−17.1745	−0.619328	−0.309664	+	0.950846i	$0.600217\pi$
$770$	0	0
$771$	0	0
$772$	−53.9281	−1.94091
$773$	−24.3968	−0.877491	−0.438745	−	0.898611i	$-0.644577\pi$
$773$	−24.3968	−0.877491	−0.438745	+	0.898611i	$0.644577\pi$
$774$	0	0
$775$	0	0
$776$	−15.5223	−0.557219
$777$	0	0
$778$	6.45077	0.231271
$779$	−18.4688	−0.661712
$780$	0	0
$781$	46.7971	1.67453
$782$	−69.2630	−2.47684
$783$	0	0
$784$	−8.00411	−0.285861
$785$	0	0
$786$	0	0
$787$	−12.7617	−0.454906	−0.227453	−	0.973789i	$-0.573040\pi$
$787$	−12.7617	−0.454906	−0.227453	+	0.973789i	$0.573040\pi$
$788$	−40.3540	−1.43755
$789$	0	0
$790$	0	0
$791$	−55.8468	−1.98568
$792$	0	0
$793$	12.1745	0.432329
$794$	47.3777	1.68137
$795$	0	0
$796$	−27.9742	−0.991520
$797$	2.01827	0.0714909	0.0357454	−	0.999361i	$-0.488619\pi$
$797$	2.01827	0.0714909	0.0357454	+	0.999361i	$0.488619\pi$
$798$	0	0
$799$	−13.8045	−0.488368
$800$	0	0
$801$	0	0
$802$	41.4515	1.46370
$803$	−14.3915	−0.507864
$804$	0	0
$805$	0	0
$806$	−12.0882	−0.425790
$807$	0	0
$808$	10.7414	0.377880
$809$	−53.0855	−1.86639	−0.933193	−	0.359376i	$-0.882990\pi$
$809$	−53.0855	−1.86639	−0.933193	+	0.359376i	$0.882990\pi$
$810$	0	0
$811$	−13.0808	−0.459329	−0.229664	−	0.973270i	$-0.573763\pi$
$811$	−13.0808	−0.459329	−0.229664	+	0.973270i	$0.573763\pi$
$812$	17.9914	0.631376
$813$	0	0
$814$	−63.0863	−2.21117
$815$	0	0
$816$	0	0
$817$	−1.17448	−0.0410899
$818$	−43.3004	−1.51396
$819$	0	0
$820$	0	0
$821$	−0.910592	−0.0317799	−0.0158899	−	0.999874i	$-0.505058\pi$
$821$	−0.910592	−0.0317799	−0.0158899	+	0.999874i	$0.505058\pi$
$822$	0	0
$823$	−52.1616	−1.81824	−0.909119	−	0.416536i	$-0.863244\pi$
$823$	−52.1616	−1.81824	−0.909119	+	0.416536i	$0.863244\pi$
$824$	8.36924	0.291556
$825$	0	0
$826$	45.8942	1.59686
$827$	−47.5406	−1.65315	−0.826574	−	0.562828i	$-0.809713\pi$
$827$	−47.5406	−1.65315	−0.826574	+	0.562828i	$0.809713\pi$
$828$	0	0
$829$	−19.0435	−0.661410	−0.330705	−	0.943734i	$-0.607286\pi$
$829$	−19.0435	−0.661410	−0.330705	+	0.943734i	$0.607286\pi$
$830$	0	0
$831$	0	0
$832$	−12.2044	−0.423111
$833$	−23.4862	−0.813748
$834$	0	0
$835$	0	0
$836$	−28.1629	−0.974033
$837$	0	0
$838$	41.9281	1.44838
$839$	39.2288	1.35433	0.677163	−	0.735833i	$-0.263209\pi$
$839$	39.2288	1.35433	0.677163	+	0.735833i	$0.263209\pi$
$840$	0	0
$841$	−24.7835	−0.854603
$842$	74.3155	2.56108
$843$	0	0
$844$	45.1881	1.55544
$845$	0	0
$846$	0	0
$847$	9.33196	0.320650
$848$	6.44518	0.221328
$849$	0	0
$850$	0	0
$851$	51.2643	1.75732
$852$	0	0
$853$	3.82887	0.131098	0.0655490	−	0.997849i	$-0.479120\pi$
$853$	3.82887	0.131098	0.0655490	+	0.997849i	$0.479120\pi$
$854$	86.0837	2.94572
$855$	0	0
$856$	−8.40875	−0.287405
$857$	14.1768	0.484270	0.242135	−	0.970243i	$-0.422152\pi$
$857$	14.1768	0.484270	0.242135	+	0.970243i	$0.422152\pi$
$858$	0	0
$859$	37.1881	1.26884	0.634421	−	0.772987i	$-0.281238\pi$
$859$	37.1881	1.26884	0.634421	+	0.772987i	$0.281238\pi$
$860$	0	0
$861$	0	0
$862$	−26.6680	−0.908317
$863$	−4.41965	−0.150447	−0.0752234	−	0.997167i	$-0.523967\pi$
$863$	−4.41965	−0.150447	−0.0752234	+	0.997167i	$0.523967\pi$
$864$	0	0
$865$	0	0
$866$	−65.6262	−2.23007
$867$	0	0
$868$	−48.9539	−1.66160
$869$	−2.73178	−0.0926692
$870$	0	0
$871$	7.58724	0.257084
$872$	−7.30306	−0.247313
$873$	0	0
$874$	39.9580	1.35160
$875$	0	0
$876$	0	0
$877$	−40.3327	−1.36194	−0.680970	−	0.732312i	$-0.738441\pi$
$877$	−40.3327	−1.36194	−0.680970	+	0.732312i	$0.738441\pi$
$878$	−30.5746	−1.03184
$879$	0	0
$880$	0	0
$881$	−22.9985	−0.774840	−0.387420	−	0.921903i	$-0.626634\pi$
$881$	−22.9985	−0.774840	−0.387420	+	0.921903i	$0.626634\pi$
$882$	0	0
$883$	15.6946	0.528165	0.264082	−	0.964500i	$-0.414931\pi$
$883$	15.6946	0.528165	0.264082	+	0.964500i	$0.414931\pi$
$884$	−17.1057	−0.575326
$885$	0	0
$886$	−32.0897	−1.07807
$887$	25.3074	0.849738	0.424869	−	0.905255i	$-0.360320\pi$
$887$	25.3074	0.849738	0.424869	+	0.905255i	$0.360320\pi$
$888$	0	0
$889$	37.8425	1.26920
$890$	0	0
$891$	0	0
$892$	57.3497	1.92021
$893$	7.96385	0.266500
$894$	0	0
$895$	0	0
$896$	−36.2854	−1.21221
$897$	0	0
$898$	79.9160	2.66683
$899$	−11.4729	−0.382643
$900$	0	0
$901$	18.9118	0.630045
$902$	−30.9800	−1.03152
$903$	0	0
$904$	−25.1745	−0.837291
$905$	0	0
$906$	0	0
$907$	−10.3490	−0.343632	−0.171816	−	0.985129i	$-0.554963\pi$
$907$	−10.3490	−0.343632	−0.171816	+	0.985129i	$0.554963\pi$
$908$	23.5436	0.781322
$909$	0	0
$910$	0	0
$911$	41.0148	1.35888	0.679440	−	0.733731i	$-0.262223\pi$
$911$	41.0148	1.35888	0.679440	+	0.733731i	$0.262223\pi$
$912$	0	0
$913$	30.2438	1.00092
$914$	16.8383	0.556961
$915$	0	0
$916$	58.4345	1.93073
$917$	−29.8196	−0.984729
$918$	0	0
$919$	23.3456	0.770101	0.385050	−	0.922896i	$-0.374184\pi$
$919$	23.3456	0.770101	0.385050	+	0.922896i	$0.374184\pi$
$920$	0	0
$921$	0	0
$922$	52.3069	1.72264
$923$	−16.3978	−0.539739
$924$	0	0
$925$	0	0
$926$	−11.1130	−0.365195
$927$	0	0
$928$	−15.7107	−0.515730
$929$	32.8307	1.07714	0.538570	−	0.842581i	$-0.318965\pi$
$929$	32.8307	1.07714	0.538570	+	0.842581i	$0.318965\pi$
$930$	0	0
$931$	13.5492	0.444058
$932$	−42.1105	−1.37938
$933$	0	0
$934$	−19.2003	−0.628252
$935$	0	0
$936$	0	0
$937$	−7.39099	−0.241453	−0.120727	−	0.992686i	$-0.538522\pi$
$937$	−7.39099	−0.241453	−0.120727	+	0.992686i	$0.538522\pi$
$938$	53.6481	1.75167
$939$	0	0
$940$	0	0
$941$	38.0684	1.24099	0.620497	−	0.784209i	$-0.286931\pi$
$941$	38.0684	1.24099	0.620497	+	0.784209i	$0.286931\pi$
$942$	0	0
$943$	25.1745	0.819794
$944$	−14.1138	−0.459364
$945$	0	0
$946$	−1.97011	−0.0640537
$947$	6.02057	0.195642	0.0978212	−	0.995204i	$-0.468813\pi$
$947$	6.02057	0.195642	0.0978212	+	0.995204i	$0.468813\pi$
$948$	0	0
$949$	5.04278	0.163696
$950$	0	0
$951$	0	0
$952$	−30.7204	−0.995653
$953$	−48.7760	−1.58001	−0.790004	−	0.613102i	$-0.789922\pi$
$953$	−48.7760	−1.58001	−0.790004	+	0.613102i	$0.789922\pi$
$954$	0	0
$955$	0	0
$956$	66.9143	2.16416
$957$	0	0
$958$	92.3892	2.98496
$959$	−55.8468	−1.80339
$960$	0	0
$961$	0.217265	0.00700856
$962$	22.1055	0.712710
$963$	0	0
$964$	−25.1786	−0.810948
$965$	0	0
$966$	0	0
$967$	−3.19625	−0.102785	−0.0513923	−	0.998679i	$-0.516366\pi$
$967$	−3.19625	−0.102785	−0.0513923	+	0.998679i	$0.516366\pi$
$968$	4.20664	0.135207
$969$	0	0
$970$	0	0
$971$	34.3613	1.10271	0.551354	−	0.834272i	$-0.314112\pi$
$971$	34.3613	1.10271	0.551354	+	0.834272i	$0.314112\pi$
$972$	0	0
$973$	60.2472	1.93144
$974$	−65.1386	−2.08717
$975$	0	0
$976$	−26.4732	−0.847386
$977$	−30.1045	−0.963129	−0.481564	−	0.876411i	$-0.659931\pi$
$977$	−30.1045	−0.963129	−0.481564	+	0.876411i	$0.659931\pi$
$978$	0	0
$979$	−20.1291	−0.643329
$980$	0	0
$981$	0	0
$982$	68.0435	2.17136
$983$	39.7914	1.26915	0.634574	−	0.772862i	$-0.281176\pi$
$983$	39.7914	1.26915	0.634574	+	0.772862i	$0.281176\pi$
$984$	0	0
$985$	0	0
$986$	−28.3464	−0.902732
$987$	0	0
$988$	9.86830	0.313952
$989$	1.60092	0.0509063
$990$	0	0
$991$	46.3028	1.47086	0.735429	−	0.677602i	$-0.236981\pi$
$991$	46.3028	1.47086	0.735429	+	0.677602i	$0.236981\pi$
$992$	42.7481	1.35725
$993$	0	0
$994$	−115.946	−3.67758
$995$	0	0
$996$	0	0
$997$	−57.7673	−1.82951	−0.914754	−	0.404012i	$-0.867615\pi$
$997$	−57.7673	−1.82951	−0.914754	+	0.404012i	$0.867615\pi$
$998$	−47.8007	−1.51310
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2925.2.a.bq.1.2	yes	6
3.2	odd	2	inner	2925.2.a.bq.1.5	yes	6
5.2	odd	4		2925.2.c.z.2224.3		12
5.3	odd	4		2925.2.c.z.2224.10		12
5.4	even	2		2925.2.a.bp.1.5	yes	6
15.2	even	4		2925.2.c.z.2224.9		12
15.8	even	4		2925.2.c.z.2224.4		12
15.14	odd	2		2925.2.a.bp.1.2	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2925.2.a.bp.1.2	✓	6	15.14	odd	2
2925.2.a.bp.1.5	yes	6	5.4	even	2
2925.2.a.bq.1.2	yes	6	1.1	even	1	trivial
2925.2.a.bq.1.5	yes	6	3.2	odd	2	inner
2925.2.c.z.2224.3		12	5.2	odd	4
2925.2.c.z.2224.4		12	15.8	even	4
2925.2.c.z.2224.9		12	15.2	even	4
2925.2.c.z.2224.10		12	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.16355 q^{2} +2.68093 q^{4} -3.26817 q^{7} -1.47322 q^{8} +O(q^{10})\)	\(q-2.16355 q^{2} +2.68093 q^{4} -3.26817 q^{7} -1.47322 q^{8} -2.85387 q^{11} +1.00000 q^{13} +7.07083 q^{14} -2.17448 q^{16} -6.38050 q^{17} +3.68093 q^{19} +6.17448 q^{22} -5.01742 q^{23} -2.16355 q^{26} -8.76172 q^{28} -2.05341 q^{29} +5.58724 q^{31} +7.65103 q^{32} +13.8045 q^{34} -10.2173 q^{37} -7.96385 q^{38} -5.01742 q^{41} -0.319072 q^{43} -7.65103 q^{44} +10.8554 q^{46} +2.16355 q^{47} +3.68093 q^{49} +2.68093 q^{52} -2.96401 q^{53} +4.81472 q^{56} +4.44265 q^{58} +6.49064 q^{59} +12.1745 q^{61} -12.0882 q^{62} -12.2044 q^{64} +7.58724 q^{67} -17.1057 q^{68} -16.3978 q^{71} +5.04278 q^{73} +22.1055 q^{74} +9.86830 q^{76} +9.32694 q^{77} +0.957217 q^{79} +10.8554 q^{82} -10.5975 q^{83} +0.690328 q^{86} +4.20438 q^{88} +7.05326 q^{89} -3.26817 q^{91} -13.4513 q^{92} -4.68093 q^{94} +10.5363 q^{97} -7.96385 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 14 q^{4} + 6 q^{7}+O(q^{10}) \) 6 * q + 14 * q^4 + 6 * q^7	\( 6 q + 14 q^{4} + 6 q^{7} + 6 q^{13} + 34 q^{16} + 20 q^{19} - 10 q^{22} + 18 q^{28} + 10 q^{31} + 6 q^{34} - 8 q^{37} - 4 q^{43} + 16 q^{46} + 20 q^{49} + 14 q^{52} - 46 q^{58} + 26 q^{61} + 70 q^{64} + 22 q^{67} + 24 q^{73} + 100 q^{76} + 12 q^{79} + 16 q^{82} - 118 q^{88} + 6 q^{91} - 26 q^{94} + 12 q^{97}+O(q^{100}) \) 6 * q + 14 * q^4 + 6 * q^7 + 6 * q^13 + 34 * q^16 + 20 * q^19 - 10 * q^22 + 18 * q^28 + 10 * q^31 + 6 * q^34 - 8 * q^37 - 4 * q^43 + 16 * q^46 + 20 * q^49 + 14 * q^52 - 46 * q^58 + 26 * q^61 + 70 * q^64 + 22 * q^67 + 24 * q^73 + 100 * q^76 + 12 * q^79 + 16 * q^82 - 118 * q^88 + 6 * q^91 - 26 * q^94 + 12 * q^97

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