LMFDB - Embedded newform 3675.2.a.bz.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3675,2,Mod(1,3675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3675, 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("3675.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3675 = 3 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3675.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:	$29.3450227428$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.11344.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 4x + 3 $ x^4 - 2x^3 - 4x^2 + 4*x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.51658$ of defining polynomial
Character	$\chi$	$=$	3675.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.51658 q^{2} -1.00000 q^{3} +4.33317 q^{4} -2.51658 q^{6} +5.87162 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.51658 q^{2} -1.00000 q^{3} +4.33317 q^{4} -2.51658 q^{6} +5.87162 q^{8} +1.00000 q^{9} +0.978135 q^{11} -4.33317 q^{12} +5.14977 q^{13} +6.11005 q^{16} -4.14977 q^{17} +2.51658 q^{18} -2.30001 q^{19} +2.46156 q^{22} +5.07288 q^{23} -5.87162 q^{24} +12.9598 q^{26} -1.00000 q^{27} +5.92664 q^{29} +0.633188 q^{31} +3.63319 q^{32} -0.978135 q^{33} -10.4432 q^{34} +4.33317 q^{36} +9.05502 q^{37} -5.78817 q^{38} -5.14977 q^{39} +2.65505 q^{41} +0.344947 q^{43} +4.23843 q^{44} +12.7663 q^{46} -4.23843 q^{47} -6.11005 q^{48} +4.14977 q^{51} +22.3148 q^{52} +7.63319 q^{53} -2.51658 q^{54} +2.30001 q^{57} +14.9149 q^{58} +1.81659 q^{59} +0.656256 q^{61} +1.59347 q^{62} -3.07689 q^{64} -2.46156 q^{66} -9.25982 q^{67} -17.9817 q^{68} -5.07288 q^{69} -5.49351 q^{71} +5.87162 q^{72} +5.37811 q^{73} +22.7877 q^{74} -9.96636 q^{76} -12.9598 q^{78} -10.8869 q^{79} +1.00000 q^{81} +6.68165 q^{82} +6.62663 q^{83} +0.868086 q^{86} -5.92664 q^{87} +5.74324 q^{88} +16.3108 q^{89} +21.9817 q^{92} -0.633188 q^{93} -10.6663 q^{94} -3.63319 q^{96} -1.53844 q^{97} +0.978135 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} - 4 q^{3} + 4 q^{4} - 2 q^{6} + 6 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q + 2 * q^2 - 4 * q^3 + 4 * q^4 - 2 * q^6 + 6 * q^8 + 4 * q^9	$ 4 q + 2 q^{2} - 4 q^{3} + 4 q^{4} - 2 q^{6} + 6 q^{8} + 4 q^{9} - 4 q^{12} + 2 q^{13} + 2 q^{17} + 2 q^{18} - 12 q^{19} + 14 q^{22} + 10 q^{23} - 6 q^{24} + 6 q^{26} - 4 q^{27} - 6 q^{29} - 8 q^{31} + 4 q^{32} - 4 q^{34} + 4 q^{36} + 24 q^{37} + 8 q^{38} - 2 q^{39} + 4 q^{41} + 8 q^{43} + 10 q^{44} + 16 q^{46} - 10 q^{47} - 2 q^{51} + 34 q^{52} + 20 q^{53} - 2 q^{54} + 12 q^{57} + 10 q^{58} + 2 q^{59} - 8 q^{61} - 10 q^{62} - 4 q^{64} - 14 q^{66} + 6 q^{67} - 30 q^{68} - 10 q^{69} - 14 q^{71} + 6 q^{72} + 12 q^{73} + 20 q^{74} - 16 q^{76} - 6 q^{78} - 8 q^{79} + 4 q^{81} - 18 q^{82} - 6 q^{83} + 24 q^{86} + 6 q^{87} - 12 q^{88} + 8 q^{89} + 46 q^{92} + 8 q^{93} - 16 q^{94} - 4 q^{96} - 2 q^{97}+O(q^{100}) $ 4 * q + 2 * q^2 - 4 * q^3 + 4 * q^4 - 2 * q^6 + 6 * q^8 + 4 * q^9 - 4 * q^12 + 2 * q^13 + 2 * q^17 + 2 * q^18 - 12 * q^19 + 14 * q^22 + 10 * q^23 - 6 * q^24 + 6 * q^26 - 4 * q^27 - 6 * q^29 - 8 * q^31 + 4 * q^32 - 4 * q^34 + 4 * q^36 + 24 * q^37 + 8 * q^38 - 2 * q^39 + 4 * q^41 + 8 * q^43 + 10 * q^44 + 16 * q^46 - 10 * q^47 - 2 * q^51 + 34 * q^52 + 20 * q^53 - 2 * q^54 + 12 * q^57 + 10 * q^58 + 2 * q^59 - 8 * q^61 - 10 * q^62 - 4 * q^64 - 14 * q^66 + 6 * q^67 - 30 * q^68 - 10 * q^69 - 14 * q^71 + 6 * q^72 + 12 * q^73 + 20 * q^74 - 16 * q^76 - 6 * q^78 - 8 * q^79 + 4 * q^81 - 18 * q^82 - 6 * q^83 + 24 * q^86 + 6 * q^87 - 12 * q^88 + 8 * q^89 + 46 * q^92 + 8 * q^93 - 16 * q^94 - 4 * q^96 - 2 * 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 $

$2$	2.51658	1.77949	0.889745	−	0.456457i	$-0.150882\pi$
$2$	2.51658	1.77949	0.889745	+	0.456457i	$0.150882\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	4.33317	2.16659
$5$	0	0
$5$	0	0
$6$	−2.51658	−1.02739
$7$	0	0
$7$	0	0
$8$	5.87162	2.07593
$9$	1.00000	0.333333
$10$	0	0
$11$	0.978135	0.294919	0.147459	−	0.989068i	$-0.452890\pi$
$11$	0.978135	0.294919	0.147459	+	0.989068i	$0.452890\pi$
$12$	−4.33317	−1.25088
$13$	5.14977	1.42829	0.714144	−	0.699998i	$-0.246816\pi$
$13$	5.14977	1.42829	0.714144	+	0.699998i	$0.246816\pi$
$14$	0	0
$15$	0	0
$16$	6.11005	1.52751
$17$	−4.14977	−1.00647	−0.503233	−	0.864151i	$-0.667856\pi$
$17$	−4.14977	−1.00647	−0.503233	+	0.864151i	$0.667856\pi$
$18$	2.51658	0.593164
$19$	−2.30001	−0.527659	−0.263830	−	0.964569i	$-0.584986\pi$
$19$	−2.30001	−0.527659	−0.263830	+	0.964569i	$0.584986\pi$
$20$	0	0
$21$	0	0
$22$	2.46156	0.524805
$23$	5.07288	1.05777	0.528884	−	0.848694i	$-0.322611\pi$
$23$	5.07288	1.05777	0.528884	+	0.848694i	$0.322611\pi$
$24$	−5.87162	−1.19854
$25$	0	0
$26$	12.9598	2.54163
$27$	−1.00000	−0.192450
$28$	0	0
$29$	5.92664	1.10055	0.550275	−	0.834983i	$-0.314523\pi$
$29$	5.92664	1.10055	0.550275	+	0.834983i	$0.314523\pi$
$30$	0	0
$31$	0.633188	0.113724	0.0568620	−	0.998382i	$-0.481890\pi$
$31$	0.633188	0.113724	0.0568620	+	0.998382i	$0.481890\pi$
$32$	3.63319	0.642263
$33$	−0.978135	−0.170271
$34$	−10.4432	−1.79100
$35$	0	0
$36$	4.33317	0.722196
$37$	9.05502	1.48864	0.744318	−	0.667825i	$-0.232775\pi$
$37$	9.05502	1.48864	0.744318	+	0.667825i	$0.232775\pi$
$38$	−5.78817	−0.938965
$39$	−5.14977	−0.824623
$40$	0	0
$41$	2.65505	0.414650	0.207325	−	0.978272i	$-0.433524\pi$
$41$	2.65505	0.414650	0.207325	+	0.978272i	$0.433524\pi$
$42$	0	0
$43$	0.344947	0.0526039	0.0263020	−	0.999654i	$-0.491627\pi$
$43$	0.344947	0.0526039	0.0263020	+	0.999654i	$0.491627\pi$
$44$	4.23843	0.638967
$45$	0	0
$46$	12.7663	1.88229
$47$	−4.23843	−0.618239	−0.309119	−	0.951023i	$-0.600034\pi$
$47$	−4.23843	−0.618239	−0.309119	+	0.951023i	$0.600034\pi$
$48$	−6.11005	−0.881910
$49$	0	0
$50$	0	0
$51$	4.14977	0.581084
$52$	22.3148	3.09451
$53$	7.63319	1.04850	0.524250	−	0.851565i	$-0.324346\pi$
$53$	7.63319	1.04850	0.524250	+	0.851565i	$0.324346\pi$
$54$	−2.51658	−0.342463
$55$	0	0
$56$	0	0
$57$	2.30001	0.304644
$58$	14.9149	1.95842
$59$	1.81659	0.236500	0.118250	−	0.992984i	$-0.462272\pi$
$59$	1.81659	0.236500	0.118250	+	0.992984i	$0.462272\pi$
$60$	0	0
$61$	0.656256	0.0840250	0.0420125	−	0.999117i	$-0.486623\pi$
$61$	0.656256	0.0840250	0.0420125	+	0.999117i	$0.486623\pi$
$62$	1.59347	0.202371
$63$	0	0
$64$	−3.07689	−0.384611
$65$	0	0
$66$	−2.46156	−0.302997
$67$	−9.25982	−1.13127	−0.565633	−	0.824657i	$-0.691368\pi$
$67$	−9.25982	−1.13127	−0.565633	+	0.824657i	$0.691368\pi$
$68$	−17.9817	−2.18060
$69$	−5.07288	−0.610703
$70$	0	0
$71$	−5.49351	−0.651960	−0.325980	−	0.945377i	$-0.605694\pi$
$71$	−5.49351	−0.651960	−0.325980	+	0.945377i	$0.605694\pi$
$72$	5.87162	0.691977
$73$	5.37811	0.629460	0.314730	−	0.949181i	$-0.398086\pi$
$73$	5.37811	0.629460	0.314730	+	0.949181i	$0.398086\pi$
$74$	22.7877	2.64902
$75$	0	0
$76$	−9.96636	−1.14322
$77$	0	0
$78$	−12.9598	−1.46741
$79$	−10.8869	−1.22487	−0.612437	−	0.790519i	$-0.709811\pi$
$79$	−10.8869	−1.22487	−0.612437	+	0.790519i	$0.709811\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	6.68165	0.737865
$83$	6.62663	0.727367	0.363683	−	0.931523i	$-0.381519\pi$
$83$	6.62663	0.727367	0.363683	+	0.931523i	$0.381519\pi$
$84$	0	0
$85$	0	0
$86$	0.868086	0.0936082
$87$	−5.92664	−0.635403
$88$	5.74324	0.612231
$89$	16.3108	1.72894	0.864472	−	0.502680i	$-0.167653\pi$
$89$	16.3108	1.72894	0.864472	+	0.502680i	$0.167653\pi$
$90$	0	0
$91$	0	0
$92$	21.9817	2.29175
$93$	−0.633188	−0.0656586
$94$	−10.6663	−1.10015
$95$	0	0
$96$	−3.63319	−0.370811
$97$	−1.53844	−0.156205	−0.0781027	−	0.996945i	$-0.524886\pi$
$97$	−1.53844	−0.156205	−0.0781027	+	0.996945i	$0.524886\pi$
$98$	0	0
$99$	0.978135	0.0983063
$100$	0	0
$101$	−5.51610	−0.548873	−0.274436	−	0.961605i	$-0.588491\pi$
$101$	−5.51610	−0.548873	−0.274436	+	0.961605i	$0.588491\pi$
$102$	10.4432	1.03403
$103$	16.5213	1.62789	0.813947	−	0.580939i	$-0.197315\pi$
$103$	16.5213	1.62789	0.813947	+	0.580939i	$0.197315\pi$
$104$	30.2375	2.96503
$105$	0	0
$106$	19.2095	1.86579
$107$	−2.38820	−0.230876	−0.115438	−	0.993315i	$-0.536827\pi$
$107$	−2.38820	−0.230876	−0.115438	+	0.993315i	$0.536827\pi$
$108$	−4.33317	−0.416960
$109$	−18.1096	−1.73458	−0.867291	−	0.497801i	$-0.834141\pi$
$109$	−18.1096	−1.73458	−0.867291	+	0.497801i	$0.834141\pi$
$110$	0	0
$111$	−9.05502	−0.859465
$112$	0	0
$113$	4.04373	0.380402	0.190201	−	0.981745i	$-0.439086\pi$
$113$	4.04373	0.380402	0.190201	+	0.981745i	$0.439086\pi$
$114$	5.78817	0.542112
$115$	0	0
$116$	25.6812	2.38444
$117$	5.14977	0.476096
$118$	4.57160	0.420850
$119$	0	0
$120$	0	0
$121$	−10.0433	−0.913023
$122$	1.65152	0.149522
$123$	−2.65505	−0.239398
$124$	2.74372	0.246393
$125$	0	0
$126$	0	0
$127$	6.85023	0.607860	0.303930	−	0.952694i	$-0.401701\pi$
$127$	6.85023	0.607860	0.303930	+	0.952694i	$0.401701\pi$
$128$	−15.0096	−1.32667
$129$	−0.344947	−0.0303709
$130$	0	0
$131$	−4.02540	−0.351701	−0.175850	−	0.984417i	$-0.556267\pi$
$131$	−4.02540	−0.351701	−0.175850	+	0.984417i	$0.556267\pi$
$132$	−4.23843	−0.368908
$133$	0	0
$134$	−23.3031	−2.01308
$135$	0	0
$136$	−24.3659	−2.08935
$137$	11.2319	0.959603	0.479802	−	0.877377i	$-0.340709\pi$
$137$	11.2319	0.959603	0.479802	+	0.877377i	$0.340709\pi$
$138$	−12.7663	−1.08674
$139$	−4.98991	−0.423238	−0.211619	−	0.977352i	$-0.567874\pi$
$139$	−4.98991	−0.423238	−0.211619	+	0.977352i	$0.567874\pi$
$140$	0	0
$141$	4.23843	0.356940
$142$	−13.8249	−1.16016
$143$	5.03717	0.421229
$144$	6.11005	0.509171
$145$	0	0
$146$	13.5344	1.12012
$147$	0	0
$148$	39.2370	3.22526
$149$	−14.4942	−1.18741	−0.593707	−	0.804681i	$-0.702336\pi$
$149$	−14.4942	−1.18741	−0.593707	+	0.804681i	$0.702336\pi$
$150$	0	0
$151$	7.88692	0.641829	0.320914	−	0.947108i	$-0.396010\pi$
$151$	7.88692	0.641829	0.320914	+	0.947108i	$0.396010\pi$
$152$	−13.5048	−1.09538
$153$	−4.14977	−0.335489
$154$	0	0
$155$	0	0
$156$	−22.3148	−1.78662
$157$	6.55630	0.523250	0.261625	−	0.965170i	$-0.415742\pi$
$157$	6.55630	0.523250	0.261625	+	0.965170i	$0.415742\pi$
$158$	−27.3978	−2.17965
$159$	−7.63319	−0.605351
$160$	0	0
$161$	0	0
$162$	2.51658	0.197721
$163$	−12.2558	−0.959949	−0.479974	−	0.877282i	$-0.659354\pi$
$163$	−12.2558	−0.959949	−0.479974	+	0.877282i	$0.659354\pi$
$164$	11.5048	0.898374
$165$	0	0
$166$	16.6764	1.29434
$167$	−2.13239	−0.165009	−0.0825047	−	0.996591i	$-0.526292\pi$
$167$	−2.13239	−0.165009	−0.0825047	+	0.996591i	$0.526292\pi$
$168$	0	0
$169$	13.5201	1.04001
$170$	0	0
$171$	−2.30001	−0.175886
$172$	1.49472	0.113971
$173$	11.5485	0.878019	0.439009	−	0.898482i	$-0.355329\pi$
$173$	11.5485	0.878019	0.439009	+	0.898482i	$0.355329\pi$
$174$	−14.9149	−1.13069
$175$	0	0
$176$	5.97645	0.450492
$177$	−1.81659	−0.136544
$178$	41.0475	3.07664
$179$	8.89956	0.665185	0.332592	−	0.943071i	$-0.392077\pi$
$179$	8.89956	0.665185	0.332592	+	0.943071i	$0.392077\pi$
$180$	0	0
$181$	−3.17940	−0.236323	−0.118161	−	0.992994i	$-0.537700\pi$
$181$	−3.17940	−0.236323	−0.118161	+	0.992994i	$0.537700\pi$
$182$	0	0
$183$	−0.656256	−0.0485119
$184$	29.7860	2.19585
$185$	0	0
$186$	−1.59347	−0.116839
$187$	−4.05903	−0.296826
$188$	−18.3659	−1.33947
$189$	0	0
$190$	0	0
$191$	−0.622618	−0.0450511	−0.0225255	−	0.999746i	$-0.507171\pi$
$191$	−0.622618	−0.0450511	−0.0225255	+	0.999746i	$0.507171\pi$
$192$	3.07689	0.222055
$193$	−15.1877	−1.09323	−0.546616	−	0.837383i	$-0.684084\pi$
$193$	−15.1877	−1.09323	−0.546616	+	0.837383i	$0.684084\pi$
$194$	−3.87162	−0.277966
$195$	0	0
$196$	0	0
$197$	−23.9410	−1.70573	−0.852863	−	0.522136i	$-0.825136\pi$
$197$	−23.9410	−1.70573	−0.852863	+	0.522136i	$0.825136\pi$
$198$	2.46156	0.174935
$199$	13.9532	0.989119	0.494560	−	0.869144i	$-0.335329\pi$
$199$	13.9532	0.989119	0.494560	+	0.869144i	$0.335329\pi$
$200$	0	0
$201$	9.25982	0.653137
$202$	−13.8817	−0.976714
$203$	0	0
$204$	17.9817	1.25897
$205$	0	0
$206$	41.5772	2.89682
$207$	5.07288	0.352589
$208$	31.4653	2.18173
$209$	−2.24973	−0.155617
$210$	0	0
$211$	4.82315	0.332040	0.166020	−	0.986122i	$-0.446908\pi$
$211$	4.82315	0.332040	0.166020	+	0.986122i	$0.446908\pi$
$212$	33.0759	2.27166
$213$	5.49351	0.376409
$214$	−6.01009	−0.410841
$215$	0	0
$216$	−5.87162	−0.399513
$217$	0	0
$218$	−45.5742	−3.08667
$219$	−5.37811	−0.363419
$220$	0	0
$221$	−21.3703	−1.43752
$222$	−22.7877	−1.52941
$223$	−15.8227	−1.05956	−0.529782	−	0.848134i	$-0.677726\pi$
$223$	−15.8227	−1.05956	−0.529782	+	0.848134i	$0.677726\pi$
$224$	0	0
$225$	0	0
$226$	10.1764	0.676922
$227$	21.6983	1.44017	0.720084	−	0.693887i	$-0.244103\pi$
$227$	21.6983	1.44017	0.720084	+	0.693887i	$0.244103\pi$
$228$	9.96636	0.660038
$229$	−2.07689	−0.137245	−0.0686223	−	0.997643i	$-0.521860\pi$
$229$	−2.07689	−0.137245	−0.0686223	+	0.997643i	$0.521860\pi$
$230$	0	0
$231$	0	0
$232$	34.7990	2.28467
$233$	6.75902	0.442798	0.221399	−	0.975183i	$-0.428938\pi$
$233$	6.75902	0.442798	0.221399	+	0.975183i	$0.428938\pi$
$234$	12.9598	0.847209
$235$	0	0
$236$	7.87162	0.512399
$237$	10.8869	0.707182
$238$	0	0
$239$	−2.90478	−0.187894	−0.0939472	−	0.995577i	$-0.529949\pi$
$239$	−2.90478	−0.187894	−0.0939472	+	0.995577i	$0.529949\pi$
$240$	0	0
$241$	−8.89749	−0.573138	−0.286569	−	0.958060i	$-0.592515\pi$
$241$	−8.89749	−0.573138	−0.286569	+	0.958060i	$0.592515\pi$
$242$	−25.2746	−1.62472
$243$	−1.00000	−0.0641500
$244$	2.84367	0.182047
$245$	0	0
$246$	−6.68165	−0.426007
$247$	−11.8445	−0.753650
$248$	3.71784	0.236083
$249$	−6.62663	−0.419946
$250$	0	0
$251$	−21.1747	−1.33653	−0.668267	−	0.743921i	$-0.732964\pi$
$251$	−21.1747	−1.33653	−0.668267	+	0.743921i	$0.732964\pi$
$252$	0	0
$253$	4.96196	0.311956
$254$	17.2392	1.08168
$255$	0	0
$256$	−31.6191	−1.97619
$257$	−6.23442	−0.388892	−0.194446	−	0.980913i	$-0.562291\pi$
$257$	−6.23442	−0.388892	−0.194446	+	0.980913i	$0.562291\pi$
$258$	−0.868086	−0.0540447
$259$	0	0
$260$	0	0
$261$	5.92664	0.366850
$262$	−10.1302	−0.625848
$263$	28.0480	1.72951	0.864756	−	0.502191i	$-0.167473\pi$
$263$	28.0480	1.72951	0.864756	+	0.502191i	$0.167473\pi$
$264$	−5.74324	−0.353472
$265$	0	0
$266$	0	0
$267$	−16.3108	−0.998207
$268$	−40.1244	−2.45099
$269$	7.41830	0.452302	0.226151	−	0.974092i	$-0.427386\pi$
$269$	7.41830	0.452302	0.226151	+	0.974092i	$0.427386\pi$
$270$	0	0
$271$	−31.2116	−1.89597	−0.947985	−	0.318316i	$-0.896883\pi$
$271$	−31.2116	−1.89597	−0.947985	+	0.318316i	$0.896883\pi$
$272$	−25.3553	−1.53739
$273$	0	0
$274$	28.2659	1.70761
$275$	0	0
$276$	−21.9817	−1.32314
$277$	12.4526	0.748204	0.374102	−	0.927388i	$-0.377951\pi$
$277$	12.4526	0.748204	0.374102	+	0.927388i	$0.377951\pi$
$278$	−12.5575	−0.753149
$279$	0.633188	0.0379080
$280$	0	0
$281$	−0.0409225	−0.00244123	−0.00122061	−	0.999999i	$-0.500389\pi$
$281$	−0.0409225	−0.00244123	−0.00122061	+	0.999999i	$0.500389\pi$
$282$	10.6663	0.635172
$283$	−10.8420	−0.644489	−0.322245	−	0.946656i	$-0.604437\pi$
$283$	−10.8420	−0.644489	−0.322245	+	0.946656i	$0.604437\pi$
$284$	−23.8043	−1.41253
$285$	0	0
$286$	12.6764	0.749574
$287$	0	0
$288$	3.63319	0.214088
$289$	0.220576	0.0129750
$290$	0	0
$291$	1.53844	0.0901852
$292$	23.3043	1.36378
$293$	−29.6455	−1.73191	−0.865953	−	0.500125i	$-0.833287\pi$
$293$	−29.6455	−1.73191	−0.865953	+	0.500125i	$0.833287\pi$
$294$	0	0
$295$	0	0
$296$	53.1676	3.09031
$297$	−0.978135	−0.0567572
$298$	−36.4759	−2.11299
$299$	26.1242	1.51080
$300$	0	0
$301$	0	0
$302$	19.8481	1.14213
$303$	5.51610	0.316892
$304$	−14.0532	−0.806006
$305$	0	0
$306$	−10.4432	−0.596999
$307$	−31.6055	−1.80382	−0.901910	−	0.431923i	$-0.857835\pi$
$307$	−31.6055	−1.80382	−0.901910	+	0.431923i	$0.857835\pi$
$308$	0	0
$309$	−16.5213	−0.939865
$310$	0	0
$311$	22.1703	1.25716	0.628581	−	0.777744i	$-0.283636\pi$
$311$	22.1703	1.25716	0.628581	+	0.777744i	$0.283636\pi$
$312$	−30.2375	−1.71186
$313$	−6.88572	−0.389204	−0.194602	−	0.980882i	$-0.562342\pi$
$313$	−6.88572	−0.389204	−0.194602	+	0.980882i	$0.562342\pi$
$314$	16.4995	0.931118
$315$	0	0
$316$	−47.1749	−2.65380
$317$	−12.8387	−0.721094	−0.360547	−	0.932741i	$-0.617410\pi$
$317$	−12.8387	−0.721094	−0.360547	+	0.932741i	$0.617410\pi$
$318$	−19.2095	−1.07722
$319$	5.79706	0.324573
$320$	0	0
$321$	2.38820	0.133296
$322$	0	0
$323$	9.54453	0.531072
$324$	4.33317	0.240732
$325$	0	0
$326$	−30.8427	−1.70822
$327$	18.1096	1.00146
$328$	15.5895	0.860784
$329$	0	0
$330$	0	0
$331$	−8.53357	−0.469047	−0.234524	−	0.972110i	$-0.575353\pi$
$331$	−8.53357	−0.469047	−0.234524	+	0.972110i	$0.575353\pi$
$332$	28.7143	1.57590
$333$	9.05502	0.496212
$334$	−5.36633	−0.293633
$335$	0	0
$336$	0	0
$337$	29.1131	1.58589	0.792946	−	0.609292i	$-0.208546\pi$
$337$	29.1131	1.58589	0.792946	+	0.609292i	$0.208546\pi$
$338$	34.0244	1.85069
$339$	−4.04373	−0.219625
$340$	0	0
$341$	0.619344	0.0335393
$342$	−5.78817	−0.312988
$343$	0	0
$344$	2.02540	0.109202
$345$	0	0
$346$	29.0628	1.56243
$347$	13.4427	0.721644	0.360822	−	0.932635i	$-0.382496\pi$
$347$	13.4427	0.721644	0.360822	+	0.932635i	$0.382496\pi$
$348$	−25.6812	−1.37666
$349$	−24.7397	−1.32429	−0.662144	−	0.749377i	$-0.730353\pi$
$349$	−24.7397	−1.32429	−0.662144	+	0.749377i	$0.730353\pi$
$350$	0	0
$351$	−5.14977	−0.274874
$352$	3.55375	0.189415
$353$	−16.7550	−0.891779	−0.445890	−	0.895088i	$-0.647113\pi$
$353$	−16.7550	−0.891779	−0.445890	+	0.895088i	$0.647113\pi$
$354$	−4.57160	−0.242978
$355$	0	0
$356$	70.6777	3.74591
$357$	0	0
$358$	22.3965	1.18369
$359$	−32.4924	−1.71488	−0.857442	−	0.514581i	$-0.827948\pi$
$359$	−32.4924	−1.71488	−0.857442	+	0.514581i	$0.827948\pi$
$360$	0	0
$361$	−13.7099	−0.721575
$362$	−8.00120	−0.420534
$363$	10.0433	0.527134
$364$	0	0
$365$	0	0
$366$	−1.65152	−0.0863264
$367$	−23.9619	−1.25080	−0.625400	−	0.780304i	$-0.715064\pi$
$367$	−23.9619	−1.25080	−0.625400	+	0.780304i	$0.715064\pi$
$368$	30.9955	1.61575
$369$	2.65505	0.138217
$370$	0	0
$371$	0	0
$372$	−2.74372	−0.142255
$373$	11.0754	0.573464	0.286732	−	0.958011i	$-0.407431\pi$
$373$	11.0754	0.573464	0.286732	+	0.958011i	$0.407431\pi$
$374$	−10.2149	−0.528199
$375$	0	0
$376$	−24.8864	−1.28342
$377$	30.5208	1.57190
$378$	0	0
$379$	−32.7423	−1.68186	−0.840929	−	0.541145i	$-0.817991\pi$
$379$	−32.7423	−1.68186	−0.840929	+	0.541145i	$0.817991\pi$
$380$	0	0
$381$	−6.85023	−0.350948
$382$	−1.56687	−0.0801680
$383$	−23.2523	−1.18814	−0.594069	−	0.804414i	$-0.702479\pi$
$383$	−23.2523	−1.18814	−0.594069	+	0.804414i	$0.702479\pi$
$384$	15.0096	0.765956
$385$	0	0
$386$	−38.2210	−1.94540
$387$	0.344947	0.0175346
$388$	−6.66635	−0.338433
$389$	−37.6262	−1.90772	−0.953861	−	0.300249i	$-0.902930\pi$
$389$	−37.6262	−1.90772	−0.953861	+	0.300249i	$0.902930\pi$
$390$	0	0
$391$	−21.0513	−1.06461
$392$	0	0
$393$	4.02540	0.203054
$394$	−60.2494	−3.03532
$395$	0	0
$396$	4.23843	0.212989
$397$	−0.0575740	−0.00288956	−0.00144478	−	0.999999i	$-0.500460\pi$
$397$	−0.0575740	−0.00288956	−0.00144478	+	0.999999i	$0.500460\pi$
$398$	35.1144	1.76013
$399$	0	0
$400$	0	0
$401$	−9.07593	−0.453230	−0.226615	−	0.973984i	$-0.572766\pi$
$401$	−9.07593	−0.453230	−0.226615	+	0.973984i	$0.572766\pi$
$402$	23.3031	1.16225
$403$	3.26077	0.162431
$404$	−23.9022	−1.18918
$405$	0	0
$406$	0	0
$407$	8.85704	0.439027
$408$	24.3659	1.20629
$409$	16.6120	0.821413	0.410706	−	0.911768i	$-0.365282\pi$
$409$	16.6120	0.821413	0.410706	+	0.911768i	$0.365282\pi$
$410$	0	0
$411$	−11.2319	−0.554027
$412$	71.5897	3.52697
$413$	0	0
$414$	12.7663	0.627430
$415$	0	0
$416$	18.7101	0.917337
$417$	4.98991	0.244357
$418$	−5.66161	−0.276919
$419$	−29.8759	−1.45953	−0.729766	−	0.683697i	$-0.760371\pi$
$419$	−29.8759	−1.45953	−0.729766	+	0.683697i	$0.760371\pi$
$420$	0	0
$421$	−19.3201	−0.941602	−0.470801	−	0.882239i	$-0.656035\pi$
$421$	−19.3201	−0.941602	−0.470801	+	0.882239i	$0.656035\pi$
$422$	12.1379	0.590861
$423$	−4.23843	−0.206080
$424$	44.8192	2.17661
$425$	0	0
$426$	13.8249	0.669817
$427$	0	0
$428$	−10.3485	−0.500213
$429$	−5.03717	−0.243197
$430$	0	0
$431$	−21.9962	−1.05952	−0.529761	−	0.848147i	$-0.677718\pi$
$431$	−21.9962	−1.05952	−0.529761	+	0.848147i	$0.677718\pi$
$432$	−6.11005	−0.293970
$433$	−2.72706	−0.131054	−0.0655272	−	0.997851i	$-0.520873\pi$
$433$	−2.72706	−0.131054	−0.0655272	+	0.997851i	$0.520873\pi$
$434$	0	0
$435$	0	0
$436$	−78.4719	−3.75812
$437$	−11.6677	−0.558142
$438$	−13.5344	−0.646700
$439$	−7.54366	−0.360039	−0.180020	−	0.983663i	$-0.557616\pi$
$439$	−7.54366	−0.360039	−0.180020	+	0.983663i	$0.557616\pi$
$440$	0	0
$441$	0	0
$442$	−53.7802	−2.55806
$443$	−10.4383	−0.495941	−0.247970	−	0.968768i	$-0.579764\pi$
$443$	−10.4383	−0.495941	−0.247970	+	0.968768i	$0.579764\pi$
$444$	−39.2370	−1.86211
$445$	0	0
$446$	−39.8190	−1.88549
$447$	14.4942	0.685554
$448$	0	0
$449$	−1.73107	−0.0816944	−0.0408472	−	0.999165i	$-0.513006\pi$
$449$	−1.73107	−0.0816944	−0.0408472	+	0.999165i	$0.513006\pi$
$450$	0	0
$451$	2.59700	0.122288
$452$	17.5222	0.824174
$453$	−7.88692	−0.370560
$454$	54.6055	2.56276
$455$	0	0
$456$	13.5048	0.632421
$457$	30.8614	1.44364	0.721819	−	0.692082i	$-0.243306\pi$
$457$	30.8614	1.44364	0.721819	+	0.692082i	$0.243306\pi$
$458$	−5.22666	−0.244226
$459$	4.14977	0.193695
$460$	0	0
$461$	10.7294	0.499718	0.249859	−	0.968282i	$-0.419616\pi$
$461$	10.7294	0.499718	0.249859	+	0.968282i	$0.419616\pi$
$462$	0	0
$463$	11.1060	0.516141	0.258071	−	0.966126i	$-0.416913\pi$
$463$	11.1060	0.516141	0.258071	+	0.966126i	$0.416913\pi$
$464$	36.2121	1.68110
$465$	0	0
$466$	17.0096	0.787955
$467$	27.5895	1.27669	0.638344	−	0.769751i	$-0.279620\pi$
$467$	27.5895	1.27669	0.638344	+	0.769751i	$0.279620\pi$
$468$	22.3148	1.03150
$469$	0	0
$470$	0	0
$471$	−6.55630	−0.302098
$472$	10.6663	0.490958
$473$	0.337405	0.0155139
$474$	27.3978	1.25842
$475$	0	0
$476$	0	0
$477$	7.63319	0.349500
$478$	−7.31011	−0.334356
$479$	−10.0174	−0.457706	−0.228853	−	0.973461i	$-0.573497\pi$
$479$	−10.0174	−0.457706	−0.228853	+	0.973461i	$0.573497\pi$
$480$	0	0
$481$	46.6313	2.12620
$482$	−22.3913	−1.01989
$483$	0	0
$484$	−43.5192	−1.97814
$485$	0	0
$486$	−2.51658	−0.114154
$487$	−30.7341	−1.39270	−0.696348	−	0.717704i	$-0.745193\pi$
$487$	−30.7341	−1.39270	−0.696348	+	0.717704i	$0.745193\pi$
$488$	3.85329	0.174430
$489$	12.2558	0.554227
$490$	0	0
$491$	4.14054	0.186860	0.0934301	−	0.995626i	$-0.470217\pi$
$491$	4.14054	0.186860	0.0934301	+	0.995626i	$0.470217\pi$
$492$	−11.5048	−0.518677
$493$	−24.5942	−1.10767
$494$	−29.8077	−1.34111
$495$	0	0
$496$	3.86881	0.173715
$497$	0	0
$498$	−16.6764	−0.747289
$499$	−1.54828	−0.0693105	−0.0346552	−	0.999399i	$-0.511033\pi$
$499$	−1.54828	−0.0693105	−0.0346552	+	0.999399i	$0.511033\pi$
$500$	0	0
$501$	2.13239	0.0952682
$502$	−53.2878	−2.37835
$503$	15.1658	0.676210	0.338105	−	0.941108i	$-0.390214\pi$
$503$	15.1658	0.676210	0.338105	+	0.941108i	$0.390214\pi$
$504$	0	0
$505$	0	0
$506$	12.4872	0.555123
$507$	−13.5201	−0.600449
$508$	29.6832	1.31698
$509$	−20.4655	−0.907116	−0.453558	−	0.891227i	$-0.649846\pi$
$509$	−20.4655	−0.907116	−0.453558	+	0.891227i	$0.649846\pi$
$510$	0	0
$511$	0	0
$512$	−49.5528	−2.18994
$513$	2.30001	0.101548
$514$	−15.6894	−0.692030
$515$	0	0
$516$	−1.49472	−0.0658012
$517$	−4.14576	−0.182330
$518$	0	0
$519$	−11.5485	−0.506924
$520$	0	0
$521$	2.74674	0.120337	0.0601685	−	0.998188i	$-0.480836\pi$
$521$	2.74674	0.120337	0.0601685	+	0.998188i	$0.480836\pi$
$522$	14.9149	0.652806
$523$	39.8669	1.74326	0.871629	−	0.490166i	$-0.163064\pi$
$523$	39.8669	1.74326	0.871629	+	0.490166i	$0.163064\pi$
$524$	−17.4427	−0.761990
$525$	0	0
$526$	70.5850	3.07765
$527$	−2.62758	−0.114459
$528$	−5.97645	−0.260092
$529$	2.73410	0.118874
$530$	0	0
$531$	1.81659	0.0788335
$532$	0	0
$533$	13.6729	0.592239
$534$	−41.0475	−1.77630
$535$	0	0
$536$	−54.3701	−2.34843
$537$	−8.89956	−0.384045
$538$	18.6688	0.804867
$539$	0	0
$540$	0	0
$541$	26.4986	1.13927	0.569633	−	0.821899i	$-0.307085\pi$
$541$	26.4986	1.13927	0.569633	+	0.821899i	$0.307085\pi$
$542$	−78.5465	−3.37386
$543$	3.17940	0.136441
$544$	−15.0769	−0.646416
$545$	0	0
$546$	0	0
$547$	12.9090	0.551950	0.275975	−	0.961165i	$-0.410999\pi$
$547$	12.9090	0.551950	0.275975	+	0.961165i	$0.410999\pi$
$548$	48.6696	2.07906
$549$	0.656256	0.0280083
$550$	0	0
$551$	−13.6314	−0.580716
$552$	−29.7860	−1.26778
$553$	0	0
$554$	31.3379	1.33142
$555$	0	0
$556$	−21.6221	−0.916983
$557$	7.18509	0.304442	0.152221	−	0.988346i	$-0.451357\pi$
$557$	7.18509	0.304442	0.152221	+	0.988346i	$0.451357\pi$
$558$	1.59347	0.0674569
$559$	1.77640	0.0751336
$560$	0	0
$561$	4.05903	0.171373
$562$	−0.102985	−0.00434415
$563$	2.38700	0.100600	0.0502999	−	0.998734i	$-0.483982\pi$
$563$	2.38700	0.100600	0.0502999	+	0.998734i	$0.483982\pi$
$564$	18.3659	0.773342
$565$	0	0
$566$	−27.2847	−1.14686
$567$	0	0
$568$	−32.2558	−1.35342
$569$	29.8542	1.25155	0.625776	−	0.780003i	$-0.284783\pi$
$569$	29.8542	1.25155	0.625776	+	0.780003i	$0.284783\pi$
$570$	0	0
$571$	−19.4634	−0.814518	−0.407259	−	0.913313i	$-0.633515\pi$
$571$	−19.4634	−0.814518	−0.407259	+	0.913313i	$0.633515\pi$
$572$	21.8269	0.912630
$573$	0.622618	0.0260103
$574$	0	0
$575$	0	0
$576$	−3.07689	−0.128204
$577$	3.96187	0.164935	0.0824675	−	0.996594i	$-0.473720\pi$
$577$	3.96187	0.164935	0.0824675	+	0.996594i	$0.473720\pi$
$578$	0.555096	0.0230890
$579$	15.1877	0.631178
$580$	0	0
$581$	0	0
$582$	3.87162	0.160484
$583$	7.46629	0.309222
$584$	31.5782	1.30671
$585$	0	0
$586$	−74.6052	−3.08191
$587$	−13.9419	−0.575446	−0.287723	−	0.957714i	$-0.592898\pi$
$587$	−13.9419	−0.575446	−0.287723	+	0.957714i	$0.592898\pi$
$588$	0	0
$589$	−1.45634	−0.0600075
$590$	0	0
$591$	23.9410	0.984801
$592$	55.3266	2.27391
$593$	38.4973	1.58089	0.790447	−	0.612530i	$-0.209848\pi$
$593$	38.4973	1.58089	0.790447	+	0.612530i	$0.209848\pi$
$594$	−2.46156	−0.100999
$595$	0	0
$596$	−62.8061	−2.57264
$597$	−13.9532	−0.571068
$598$	65.7435	2.68845
$599$	17.4997	0.715019	0.357509	−	0.933910i	$-0.383626\pi$
$599$	17.4997	0.715019	0.357509	+	0.933910i	$0.383626\pi$
$600$	0	0
$601$	34.5192	1.40807	0.704033	−	0.710167i	$-0.251381\pi$
$601$	34.5192	1.40807	0.704033	+	0.710167i	$0.251381\pi$
$602$	0	0
$603$	−9.25982	−0.377089
$604$	34.1754	1.39058
$605$	0	0
$606$	13.8817	0.563906
$607$	−10.9156	−0.443050	−0.221525	−	0.975155i	$-0.571103\pi$
$607$	−10.9156	−0.443050	−0.221525	+	0.975155i	$0.571103\pi$
$608$	−8.35639	−0.338896
$609$	0	0
$610$	0	0
$611$	−21.8269	−0.883023
$612$	−17.9817	−0.726866
$613$	26.1173	1.05487	0.527435	−	0.849596i	$-0.323154\pi$
$613$	26.1173	1.05487	0.527435	+	0.849596i	$0.323154\pi$
$614$	−79.5377	−3.20988
$615$	0	0
$616$	0	0
$617$	−11.6689	−0.469772	−0.234886	−	0.972023i	$-0.575472\pi$
$617$	−11.6689	−0.469772	−0.234886	+	0.972023i	$0.575472\pi$
$618$	−41.5772	−1.67248
$619$	−0.823632	−0.0331046	−0.0165523	−	0.999863i	$-0.505269\pi$
$619$	−0.823632	−0.0331046	−0.0165523	+	0.999863i	$0.505269\pi$
$620$	0	0
$621$	−5.07288	−0.203568
$622$	55.7933	2.23711
$623$	0	0
$624$	−31.4653	−1.25962
$625$	0	0
$626$	−17.3285	−0.692585
$627$	2.24973	0.0898454
$628$	28.4096	1.13367
$629$	−37.5763	−1.49826
$630$	0	0
$631$	−20.5920	−0.819755	−0.409877	−	0.912141i	$-0.634429\pi$
$631$	−20.5920	−0.819755	−0.409877	+	0.912141i	$0.634429\pi$
$632$	−63.9239	−2.54275
$633$	−4.82315	−0.191703
$634$	−32.3096	−1.28318
$635$	0	0
$636$	−33.0759	−1.31155
$637$	0	0
$638$	14.5888	0.577575
$639$	−5.49351	−0.217320
$640$	0	0
$641$	29.6741	1.17206	0.586029	−	0.810290i	$-0.300690\pi$
$641$	29.6741	1.17206	0.586029	+	0.810290i	$0.300690\pi$
$642$	6.01009	0.237199
$643$	−11.1286	−0.438870	−0.219435	−	0.975627i	$-0.570421\pi$
$643$	−11.1286	−0.438870	−0.219435	+	0.975627i	$0.570421\pi$
$644$	0	0
$645$	0	0
$646$	24.0196	0.945037
$647$	10.9234	0.429442	0.214721	−	0.976675i	$-0.431116\pi$
$647$	10.9234	0.429442	0.214721	+	0.976675i	$0.431116\pi$
$648$	5.87162	0.230659
$649$	1.77687	0.0697484
$650$	0	0
$651$	0	0
$652$	−53.1065	−2.07981
$653$	7.17076	0.280614	0.140307	−	0.990108i	$-0.455191\pi$
$653$	7.17076	0.280614	0.140307	+	0.990108i	$0.455191\pi$
$654$	45.5742	1.78209
$655$	0	0
$656$	16.2225	0.633382
$657$	5.37811	0.209820
$658$	0	0
$659$	14.1232	0.550161	0.275080	−	0.961421i	$-0.411296\pi$
$659$	14.1232	0.550161	0.275080	+	0.961421i	$0.411296\pi$
$660$	0	0
$661$	28.9217	1.12492	0.562461	−	0.826824i	$-0.309854\pi$
$661$	28.9217	1.12492	0.562461	+	0.826824i	$0.309854\pi$
$662$	−21.4754	−0.834665
$663$	21.3703	0.829955
$664$	38.9090	1.50996
$665$	0	0
$666$	22.7877	0.883005
$667$	30.0651	1.16413
$668$	−9.24002	−0.357507
$669$	15.8227	0.611740
$670$	0	0
$671$	0.641907	0.0247806
$672$	0	0
$673$	14.4081	0.555392	0.277696	−	0.960669i	$-0.410429\pi$
$673$	14.4081	0.555392	0.277696	+	0.960669i	$0.410429\pi$
$674$	73.2654	2.82208
$675$	0	0
$676$	58.5850	2.25327
$677$	30.1366	1.15825	0.579123	−	0.815240i	$-0.303395\pi$
$677$	30.1366	1.15825	0.579123	+	0.815240i	$0.303395\pi$
$678$	−10.1764	−0.390821
$679$	0	0
$680$	0	0
$681$	−21.6983	−0.831481
$682$	1.55863	0.0596830
$683$	15.5168	0.593735	0.296868	−	0.954919i	$-0.404058\pi$
$683$	15.5168	0.593735	0.296868	+	0.954919i	$0.404058\pi$
$684$	−9.96636	−0.381073
$685$	0	0
$686$	0	0
$687$	2.07689	0.0792383
$688$	2.10764	0.0803531
$689$	39.3092	1.49756
$690$	0	0
$691$	45.7835	1.74168	0.870842	−	0.491562i	$-0.163574\pi$
$691$	45.7835	1.74168	0.870842	+	0.491562i	$0.163574\pi$
$692$	50.0418	1.90230
$693$	0	0
$694$	33.8297	1.28416
$695$	0	0
$696$	−34.7990	−1.31905
$697$	−11.0179	−0.417331
$698$	−62.2595	−2.35656
$699$	−6.75902	−0.255650
$700$	0	0
$701$	24.0419	0.908050	0.454025	−	0.890989i	$-0.349988\pi$
$701$	24.0419	0.908050	0.454025	+	0.890989i	$0.349988\pi$
$702$	−12.9598	−0.489136
$703$	−20.8267	−0.785493
$704$	−3.00961	−0.113429
$705$	0	0
$706$	−42.1653	−1.58691
$707$	0	0
$708$	−7.87162	−0.295834
$709$	18.3971	0.690917	0.345459	−	0.938434i	$-0.387723\pi$
$709$	18.3971	0.690917	0.345459	+	0.938434i	$0.387723\pi$
$710$	0	0
$711$	−10.8869	−0.408292
$712$	95.7710	3.58917
$713$	3.21209	0.120294
$714$	0	0
$715$	0	0
$716$	38.5634	1.44118
$717$	2.90478	0.108481
$718$	−81.7697	−3.05162
$719$	−16.2455	−0.605855	−0.302927	−	0.953014i	$-0.597964\pi$
$719$	−16.2455	−0.605855	−0.302927	+	0.953014i	$0.597964\pi$
$720$	0	0
$721$	0	0
$722$	−34.5021	−1.28404
$723$	8.89749	0.330901
$724$	−13.7769	−0.512014
$725$	0	0
$726$	25.2746	0.938030
$727$	−42.6977	−1.58357	−0.791785	−	0.610800i	$-0.790848\pi$
$727$	−42.6977	−1.58357	−0.791785	+	0.610800i	$0.790848\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−1.43145	−0.0529441
$732$	−2.84367	−0.105105
$733$	−46.7120	−1.72535	−0.862674	−	0.505760i	$-0.831212\pi$
$733$	−46.7120	−1.72535	−0.862674	+	0.505760i	$0.831212\pi$
$734$	−60.3020	−2.22579
$735$	0	0
$736$	18.4307	0.679366
$737$	−9.05735	−0.333632
$738$	6.68165	0.245955
$739$	7.04821	0.259272	0.129636	−	0.991562i	$-0.458619\pi$
$739$	7.04821	0.259272	0.129636	+	0.991562i	$0.458619\pi$
$740$	0	0
$741$	11.8445	0.435120
$742$	0	0
$743$	−8.55510	−0.313856	−0.156928	−	0.987610i	$-0.550159\pi$
$743$	−8.55510	−0.313856	−0.156928	+	0.987610i	$0.550159\pi$
$744$	−3.71784	−0.136303
$745$	0	0
$746$	27.8722	1.02047
$747$	6.62663	0.242456
$748$	−17.5885	−0.643099
$749$	0	0
$750$	0	0
$751$	−2.97645	−0.108612	−0.0543062	−	0.998524i	$-0.517295\pi$
$751$	−2.97645	−0.108612	−0.0543062	+	0.998524i	$0.517295\pi$
$752$	−25.8970	−0.944367
$753$	21.1747	0.771648
$754$	76.8081	2.79719
$755$	0	0
$756$	0	0
$757$	43.6750	1.58740	0.793698	−	0.608313i	$-0.208153\pi$
$757$	43.6750	1.58740	0.793698	+	0.608313i	$0.208153\pi$
$758$	−82.3986	−2.99285
$759$	−4.96196	−0.180108
$760$	0	0
$761$	26.7256	0.968803	0.484402	−	0.874846i	$-0.339037\pi$
$761$	26.7256	0.968803	0.484402	+	0.874846i	$0.339037\pi$
$762$	−17.2392	−0.624509
$763$	0	0
$764$	−2.69791	−0.0976071
$765$	0	0
$766$	−58.5163	−2.11428
$767$	9.35504	0.337791
$768$	31.6191	1.14096
$769$	−4.04661	−0.145925	−0.0729623	−	0.997335i	$-0.523245\pi$
$769$	−4.04661	−0.145925	−0.0729623	+	0.997335i	$0.523245\pi$
$770$	0	0
$771$	6.23442	0.224527
$772$	−65.8108	−2.36858
$773$	6.31105	0.226993	0.113496	−	0.993538i	$-0.463795\pi$
$773$	6.31105	0.226993	0.113496	+	0.993538i	$0.463795\pi$
$774$	0.868086	0.0312027
$775$	0	0
$776$	−9.03316	−0.324272
$777$	0	0
$778$	−94.6892	−3.39477
$779$	−6.10666	−0.218794
$780$	0	0
$781$	−5.37340	−0.192275
$782$	−52.9772	−1.89446
$783$	−5.92664	−0.211801
$784$	0	0
$785$	0	0
$786$	10.1302	0.361333
$787$	29.9755	1.06851	0.534256	−	0.845323i	$-0.320592\pi$
$787$	29.9755	1.06851	0.534256	+	0.845323i	$0.320592\pi$
$788$	−103.740	−3.69560
$789$	−28.0480	−0.998535
$790$	0	0
$791$	0	0
$792$	5.74324	0.204077
$793$	3.37957	0.120012
$794$	−0.144890	−0.00514194
$795$	0	0
$796$	60.4618	2.14301
$797$	0.676527	0.0239638	0.0119819	−	0.999928i	$-0.496186\pi$
$797$	0.676527	0.0239638	0.0119819	+	0.999928i	$0.496186\pi$
$798$	0	0
$799$	17.5885	0.622236
$800$	0	0
$801$	16.3108	0.576315
$802$	−22.8403	−0.806519
$803$	5.26052	0.185640
$804$	40.1244	1.41508
$805$	0	0
$806$	8.20600	0.289044
$807$	−7.41830	−0.261137
$808$	−32.3884	−1.13942
$809$	−50.1223	−1.76221	−0.881104	−	0.472923i	$-0.843199\pi$
$809$	−50.1223	−1.76221	−0.881104	+	0.472923i	$0.843199\pi$
$810$	0	0
$811$	−36.4884	−1.28128	−0.640641	−	0.767841i	$-0.721331\pi$
$811$	−36.4884	−1.28128	−0.640641	+	0.767841i	$0.721331\pi$
$812$	0	0
$813$	31.2116	1.09464
$814$	22.2894	0.781245
$815$	0	0
$816$	25.3553	0.887613
$817$	−0.793383	−0.0277570
$818$	41.8055	1.46170
$819$	0	0
$820$	0	0
$821$	38.7308	1.35172	0.675858	−	0.737032i	$-0.263773\pi$
$821$	38.7308	1.35172	0.675858	+	0.737032i	$0.263773\pi$
$822$	−28.2659	−0.985886
$823$	20.9812	0.731358	0.365679	−	0.930741i	$-0.380837\pi$
$823$	20.9812	0.731358	0.365679	+	0.930741i	$0.380837\pi$
$824$	97.0069	3.37939
$825$	0	0
$826$	0	0
$827$	37.8114	1.31483	0.657416	−	0.753528i	$-0.271650\pi$
$827$	37.8114	1.31483	0.657416	+	0.753528i	$0.271650\pi$
$828$	21.9817	0.763916
$829$	−53.3181	−1.85182	−0.925908	−	0.377750i	$-0.876698\pi$
$829$	−53.3181	−1.85182	−0.925908	+	0.377750i	$0.876698\pi$
$830$	0	0
$831$	−12.4526	−0.431976
$832$	−15.8453	−0.549336
$833$	0	0
$834$	12.5575	0.434831
$835$	0	0
$836$	−9.74845	−0.337157
$837$	−0.633188	−0.0218862
$838$	−75.1850	−2.59722
$839$	52.6452	1.81752	0.908758	−	0.417324i	$-0.137032\pi$
$839$	52.6452	1.81752	0.908758	+	0.417324i	$0.137032\pi$
$840$	0	0
$841$	6.12510	0.211210
$842$	−48.6205	−1.67557
$843$	0.0409225	0.00140944
$844$	20.8996	0.719393
$845$	0	0
$846$	−10.6663	−0.366717
$847$	0	0
$848$	46.6392	1.60160
$849$	10.8420	0.372096
$850$	0	0
$851$	45.9350	1.57463
$852$	23.8043	0.815523
$853$	−5.01225	−0.171616	−0.0858081	−	0.996312i	$-0.527347\pi$
$853$	−5.01225	−0.171616	−0.0858081	+	0.996312i	$0.527347\pi$
$854$	0	0
$855$	0	0
$856$	−14.0226	−0.479282
$857$	−38.4212	−1.31244	−0.656222	−	0.754568i	$-0.727846\pi$
$857$	−38.4212	−1.31244	−0.656222	+	0.754568i	$0.727846\pi$
$858$	−12.6764	−0.432767
$859$	23.3418	0.796413	0.398207	−	0.917296i	$-0.369633\pi$
$859$	23.3418	0.796413	0.398207	+	0.917296i	$0.369633\pi$
$860$	0	0
$861$	0	0
$862$	−55.3553	−1.88541
$863$	−54.3757	−1.85097	−0.925486	−	0.378783i	$-0.876343\pi$
$863$	−54.3757	−1.85097	−0.925486	+	0.378783i	$0.876343\pi$
$864$	−3.63319	−0.123604
$865$	0	0
$866$	−6.86287	−0.233210
$867$	−0.220576	−0.00749114
$868$	0	0
$869$	−10.6489	−0.361239
$870$	0	0
$871$	−47.6859	−1.61578
$872$	−106.332	−3.60087
$873$	−1.53844	−0.0520685
$874$	−29.3627	−0.993208
$875$	0	0
$876$	−23.3043	−0.787378
$877$	28.4557	0.960881	0.480441	−	0.877027i	$-0.340477\pi$
$877$	28.4557	0.960881	0.480441	+	0.877027i	$0.340477\pi$
$878$	−18.9842	−0.640686
$879$	29.6455	0.999917
$880$	0	0
$881$	−6.50466	−0.219148	−0.109574	−	0.993979i	$-0.534949\pi$
$881$	−6.50466	−0.219148	−0.109574	+	0.993979i	$0.534949\pi$
$882$	0	0
$883$	−34.7640	−1.16990	−0.584951	−	0.811069i	$-0.698886\pi$
$883$	−34.7640	−1.16990	−0.584951	+	0.811069i	$0.698886\pi$
$884$	−92.6014	−3.11452
$885$	0	0
$886$	−26.2689	−0.882522
$887$	−29.3541	−0.985614	−0.492807	−	0.870139i	$-0.664029\pi$
$887$	−29.3541	−0.985614	−0.492807	+	0.870139i	$0.664029\pi$
$888$	−53.1676	−1.78419
$889$	0	0
$890$	0	0
$891$	0.978135	0.0327688
$892$	−68.5624	−2.29564
$893$	9.74845	0.326219
$894$	36.4759	1.21994
$895$	0	0
$896$	0	0
$897$	−26.1242	−0.872260
$898$	−4.35639	−0.145374
$899$	3.75268	0.125159
$900$	0	0
$901$	−31.6760	−1.05528
$902$	6.53556	0.217610
$903$	0	0
$904$	23.7432	0.789688
$905$	0	0
$906$	−19.8481	−0.659408
$907$	−36.2871	−1.20489	−0.602447	−	0.798159i	$-0.705808\pi$
$907$	−36.2871	−1.20489	−0.602447	+	0.798159i	$0.705808\pi$
$908$	94.0225	3.12025
$909$	−5.51610	−0.182958
$910$	0	0
$911$	51.6732	1.71201	0.856004	−	0.516968i	$-0.172940\pi$
$911$	51.6732	1.71201	0.856004	+	0.516968i	$0.172940\pi$
$912$	14.0532	0.465348
$913$	6.48174	0.214514
$914$	77.6653	2.56894
$915$	0	0
$916$	−8.99952	−0.297353
$917$	0	0
$918$	10.4432	0.344678
$919$	−41.0377	−1.35371	−0.676854	−	0.736117i	$-0.736657\pi$
$919$	−41.0377	−1.35371	−0.676854	+	0.736117i	$0.736657\pi$
$920$	0	0
$921$	31.6055	1.04144
$922$	27.0014	0.889243
$923$	−28.2903	−0.931187
$924$	0	0
$925$	0	0
$926$	27.9492	0.918469
$927$	16.5213	0.542631
$928$	21.5326	0.706843
$929$	−2.98520	−0.0979412	−0.0489706	−	0.998800i	$-0.515594\pi$
$929$	−2.98520	−0.0979412	−0.0489706	+	0.998800i	$0.515594\pi$
$930$	0	0
$931$	0	0
$932$	29.2880	0.959361
$933$	−22.1703	−0.725823
$934$	69.4311	2.27185
$935$	0	0
$936$	30.2375	0.988343
$937$	−26.1169	−0.853201	−0.426601	−	0.904440i	$-0.640289\pi$
$937$	−26.1169	−0.853201	−0.426601	+	0.904440i	$0.640289\pi$
$938$	0	0
$939$	6.88572	0.224707
$940$	0	0
$941$	10.2116	0.332889	0.166444	−	0.986051i	$-0.446771\pi$
$941$	10.2116	0.332889	0.166444	+	0.986051i	$0.446771\pi$
$942$	−16.4995	−0.537581
$943$	13.4688	0.438603
$944$	11.0995	0.361257
$945$	0	0
$946$	0.849106	0.0276068
$947$	−46.5444	−1.51249	−0.756245	−	0.654289i	$-0.772968\pi$
$947$	−46.5444	−1.51249	−0.756245	+	0.654289i	$0.772968\pi$
$948$	47.1749	1.53217
$949$	27.6960	0.899050
$950$	0	0
$951$	12.8387	0.416324
$952$	0	0
$953$	−30.6348	−0.992358	−0.496179	−	0.868220i	$-0.665264\pi$
$953$	−30.6348	−0.992358	−0.496179	+	0.868220i	$0.665264\pi$
$954$	19.2095	0.621931
$955$	0	0
$956$	−12.5869	−0.407090
$957$	−5.79706	−0.187392
$958$	−25.2095	−0.814483
$959$	0	0
$960$	0	0
$961$	−30.5991	−0.987067
$962$	117.351	3.78356
$963$	−2.38820	−0.0769586
$964$	−38.5544	−1.24175
$965$	0	0
$966$	0	0
$967$	−57.4401	−1.84715	−0.923575	−	0.383419i	$-0.874747\pi$
$967$	−57.4401	−1.84715	−0.923575	+	0.383419i	$0.874747\pi$
$968$	−58.9701	−1.89537
$969$	−9.54453	−0.306614
$970$	0	0
$971$	48.1816	1.54622	0.773110	−	0.634272i	$-0.218700\pi$
$971$	48.1816	1.54622	0.773110	+	0.634272i	$0.218700\pi$
$972$	−4.33317	−0.138987
$973$	0	0
$974$	−77.3449	−2.47829
$975$	0	0
$976$	4.00976	0.128349
$977$	13.7523	0.439977	0.219988	−	0.975503i	$-0.429398\pi$
$977$	13.7523	0.439977	0.219988	+	0.975503i	$0.429398\pi$
$978$	30.8427	0.986241
$979$	15.9542	0.509898
$980$	0	0
$981$	−18.1096	−0.578194
$982$	10.4200	0.332516
$983$	38.1944	1.21821	0.609106	−	0.793089i	$-0.291528\pi$
$983$	38.1944	1.21821	0.609106	+	0.793089i	$0.291528\pi$
$984$	−15.5895	−0.496974
$985$	0	0
$986$	−61.8933	−1.97108
$987$	0	0
$988$	−51.3245	−1.63285
$989$	1.74987	0.0556428
$990$	0	0
$991$	39.5200	1.25539	0.627697	−	0.778458i	$-0.283998\pi$
$991$	39.5200	1.25539	0.627697	+	0.778458i	$0.283998\pi$
$992$	2.30049	0.0730407
$993$	8.53357	0.270805
$994$	0	0
$995$	0	0
$996$	−28.7143	−0.909849
$997$	−11.5290	−0.365126	−0.182563	−	0.983194i	$-0.558439\pi$
$997$	−11.5290	−0.365126	−0.182563	+	0.983194i	$0.558439\pi$
$998$	−3.89637	−0.123337
$999$	−9.05502	−0.286488

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3675.2.a.bz.1.4		4
5.2	odd	4		735.2.d.d.589.8		8
5.3	odd	4		735.2.d.d.589.1		8
5.4	even	2		3675.2.a.bp.1.1		4
7.2	even	3		525.2.i.h.151.1		8
7.4	even	3		525.2.i.h.226.1		8
7.6	odd	2		3675.2.a.cb.1.4		4
15.2	even	4		2205.2.d.s.1324.1		8
15.8	even	4		2205.2.d.s.1324.8		8
35.2	odd	12		105.2.q.a.4.8	yes	16
35.3	even	12		735.2.q.g.79.8		16
35.4	even	6		525.2.i.k.226.4		8
35.9	even	6		525.2.i.k.151.4		8
35.12	even	12		735.2.q.g.214.8		16
35.13	even	4		735.2.d.e.589.1		8
35.17	even	12		735.2.q.g.79.1		16
35.18	odd	12		105.2.q.a.79.8	yes	16
35.23	odd	12		105.2.q.a.4.1	✓	16
35.27	even	4		735.2.d.e.589.8		8
35.32	odd	12		105.2.q.a.79.1	yes	16
35.33	even	12		735.2.q.g.214.1		16
35.34	odd	2		3675.2.a.bn.1.1		4
105.2	even	12		315.2.bf.b.109.1		16
105.23	even	12		315.2.bf.b.109.8		16
105.32	even	12		315.2.bf.b.289.8		16
105.53	even	12		315.2.bf.b.289.1		16
105.62	odd	4		2205.2.d.o.1324.1		8
105.83	odd	4		2205.2.d.o.1324.8		8
140.23	even	12		1680.2.di.d.529.4		16
140.67	even	12		1680.2.di.d.289.4		16
140.107	even	12		1680.2.di.d.529.5		16
140.123	even	12		1680.2.di.d.289.5		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.q.a.4.1	✓	16	35.23	odd	12
105.2.q.a.4.8	yes	16	35.2	odd	12
105.2.q.a.79.1	yes	16	35.32	odd	12
105.2.q.a.79.8	yes	16	35.18	odd	12
315.2.bf.b.109.1		16	105.2	even	12
315.2.bf.b.109.8		16	105.23	even	12
315.2.bf.b.289.1		16	105.53	even	12
315.2.bf.b.289.8		16	105.32	even	12
525.2.i.h.151.1		8	7.2	even	3
525.2.i.h.226.1		8	7.4	even	3
525.2.i.k.151.4		8	35.9	even	6
525.2.i.k.226.4		8	35.4	even	6
735.2.d.d.589.1		8	5.3	odd	4
735.2.d.d.589.8		8	5.2	odd	4
735.2.d.e.589.1		8	35.13	even	4
735.2.d.e.589.8		8	35.27	even	4
735.2.q.g.79.1		16	35.17	even	12
735.2.q.g.79.8		16	35.3	even	12
735.2.q.g.214.1		16	35.33	even	12
735.2.q.g.214.8		16	35.12	even	12
1680.2.di.d.289.4		16	140.67	even	12
1680.2.di.d.289.5		16	140.123	even	12
1680.2.di.d.529.4		16	140.23	even	12
1680.2.di.d.529.5		16	140.107	even	12
2205.2.d.o.1324.1		8	105.62	odd	4
2205.2.d.o.1324.8		8	105.83	odd	4
2205.2.d.s.1324.1		8	15.2	even	4
2205.2.d.s.1324.8		8	15.8	even	4
3675.2.a.bn.1.1		4	35.34	odd	2
3675.2.a.bp.1.1		4	5.4	even	2
3675.2.a.bz.1.4		4	1.1	even	1	trivial
3675.2.a.cb.1.4		4	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3675.a (trivial)

\(f(q)\)	\(=\)	\(q+2.51658 q^{2} -1.00000 q^{3} +4.33317 q^{4} -2.51658 q^{6} +5.87162 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.51658 q^{2} -1.00000 q^{3} +4.33317 q^{4} -2.51658 q^{6} +5.87162 q^{8} +1.00000 q^{9} +0.978135 q^{11} -4.33317 q^{12} +5.14977 q^{13} +6.11005 q^{16} -4.14977 q^{17} +2.51658 q^{18} -2.30001 q^{19} +2.46156 q^{22} +5.07288 q^{23} -5.87162 q^{24} +12.9598 q^{26} -1.00000 q^{27} +5.92664 q^{29} +0.633188 q^{31} +3.63319 q^{32} -0.978135 q^{33} -10.4432 q^{34} +4.33317 q^{36} +9.05502 q^{37} -5.78817 q^{38} -5.14977 q^{39} +2.65505 q^{41} +0.344947 q^{43} +4.23843 q^{44} +12.7663 q^{46} -4.23843 q^{47} -6.11005 q^{48} +4.14977 q^{51} +22.3148 q^{52} +7.63319 q^{53} -2.51658 q^{54} +2.30001 q^{57} +14.9149 q^{58} +1.81659 q^{59} +0.656256 q^{61} +1.59347 q^{62} -3.07689 q^{64} -2.46156 q^{66} -9.25982 q^{67} -17.9817 q^{68} -5.07288 q^{69} -5.49351 q^{71} +5.87162 q^{72} +5.37811 q^{73} +22.7877 q^{74} -9.96636 q^{76} -12.9598 q^{78} -10.8869 q^{79} +1.00000 q^{81} +6.68165 q^{82} +6.62663 q^{83} +0.868086 q^{86} -5.92664 q^{87} +5.74324 q^{88} +16.3108 q^{89} +21.9817 q^{92} -0.633188 q^{93} -10.6663 q^{94} -3.63319 q^{96} -1.53844 q^{97} +0.978135 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} - 4 q^{3} + 4 q^{4} - 2 q^{6} + 6 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q + 2 * q^2 - 4 * q^3 + 4 * q^4 - 2 * q^6 + 6 * q^8 + 4 * q^9	\( 4 q + 2 q^{2} - 4 q^{3} + 4 q^{4} - 2 q^{6} + 6 q^{8} + 4 q^{9} - 4 q^{12} + 2 q^{13} + 2 q^{17} + 2 q^{18} - 12 q^{19} + 14 q^{22} + 10 q^{23} - 6 q^{24} + 6 q^{26} - 4 q^{27} - 6 q^{29} - 8 q^{31} + 4 q^{32} - 4 q^{34} + 4 q^{36} + 24 q^{37} + 8 q^{38} - 2 q^{39} + 4 q^{41} + 8 q^{43} + 10 q^{44} + 16 q^{46} - 10 q^{47} - 2 q^{51} + 34 q^{52} + 20 q^{53} - 2 q^{54} + 12 q^{57} + 10 q^{58} + 2 q^{59} - 8 q^{61} - 10 q^{62} - 4 q^{64} - 14 q^{66} + 6 q^{67} - 30 q^{68} - 10 q^{69} - 14 q^{71} + 6 q^{72} + 12 q^{73} + 20 q^{74} - 16 q^{76} - 6 q^{78} - 8 q^{79} + 4 q^{81} - 18 q^{82} - 6 q^{83} + 24 q^{86} + 6 q^{87} - 12 q^{88} + 8 q^{89} + 46 q^{92} + 8 q^{93} - 16 q^{94} - 4 q^{96} - 2 q^{97}+O(q^{100}) \) 4 * q + 2 * q^2 - 4 * q^3 + 4 * q^4 - 2 * q^6 + 6 * q^8 + 4 * q^9 - 4 * q^12 + 2 * q^13 + 2 * q^17 + 2 * q^18 - 12 * q^19 + 14 * q^22 + 10 * q^23 - 6 * q^24 + 6 * q^26 - 4 * q^27 - 6 * q^29 - 8 * q^31 + 4 * q^32 - 4 * q^34 + 4 * q^36 + 24 * q^37 + 8 * q^38 - 2 * q^39 + 4 * q^41 + 8 * q^43 + 10 * q^44 + 16 * q^46 - 10 * q^47 - 2 * q^51 + 34 * q^52 + 20 * q^53 - 2 * q^54 + 12 * q^57 + 10 * q^58 + 2 * q^59 - 8 * q^61 - 10 * q^62 - 4 * q^64 - 14 * q^66 + 6 * q^67 - 30 * q^68 - 10 * q^69 - 14 * q^71 + 6 * q^72 + 12 * q^73 + 20 * q^74 - 16 * q^76 - 6 * q^78 - 8 * q^79 + 4 * q^81 - 18 * q^82 - 6 * q^83 + 24 * q^86 + 6 * q^87 - 12 * q^88 + 8 * q^89 + 46 * q^92 + 8 * q^93 - 16 * q^94 - 4 * q^96 - 2 * q^97

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