LMFDB - Embedded newform 3675.2.a.bz.1.1

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			1.1
Root			$2.78165$ 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)

$f(q)$	$=$	$q-1.78165 q^{2} -1.00000 q^{3} +1.17429 q^{4} +1.78165 q^{6} +1.47113 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.78165 q^{2} -1.00000 q^{3} +1.17429 q^{4} +1.78165 q^{6} +1.47113 q^{8} +1.00000 q^{9} -2.07850 q^{11} -1.17429 q^{12} +3.13023 q^{13} -4.96962 q^{16} -2.13023 q^{17} -1.78165 q^{18} -7.73760 q^{19} +3.70316 q^{22} +5.53655 q^{23} -1.47113 q^{24} -5.57699 q^{26} -1.00000 q^{27} -4.01368 q^{29} +2.91188 q^{31} +5.91188 q^{32} +2.07850 q^{33} +3.79533 q^{34} +1.17429 q^{36} +3.51519 q^{37} +13.7857 q^{38} -3.13023 q^{39} +7.99038 q^{41} -4.99038 q^{43} -2.44075 q^{44} -9.86421 q^{46} +2.44075 q^{47} +4.96962 q^{48} +2.13023 q^{51} +3.67580 q^{52} +9.91188 q^{53} +1.78165 q^{54} +7.73760 q^{57} +7.15099 q^{58} +2.95594 q^{59} -10.8946 q^{61} -5.18797 q^{62} -0.593684 q^{64} -3.70316 q^{66} +3.83939 q^{67} -2.50151 q^{68} -5.53655 q^{69} -15.0248 q^{71} +1.47113 q^{72} -8.55369 q^{73} -6.26285 q^{74} -9.08617 q^{76} +5.57699 q^{78} +8.11354 q^{79} +1.00000 q^{81} -14.2361 q^{82} -8.75128 q^{83} +8.89113 q^{86} +4.01368 q^{87} -3.05774 q^{88} +0.618661 q^{89} +6.50151 q^{92} -2.91188 q^{93} -4.34858 q^{94} -5.91188 q^{96} -0.296842 q^{97} -2.07850 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.78165	−1.25982	−0.629910	−	0.776668i	$-0.716908\pi$
$2$	−1.78165	−1.25982	−0.629910	+	0.776668i	$0.716908\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.17429	0.587145
$5$	0	0
$5$	0	0
$6$	1.78165	0.727357
$7$	0	0
$7$	0	0
$8$	1.47113	0.520123
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.07850	−0.626690	−0.313345	−	0.949639i	$-0.601450\pi$
$11$	−2.07850	−0.626690	−0.313345	+	0.949639i	$0.601450\pi$
$12$	−1.17429	−0.338988
$13$	3.13023	0.868170	0.434085	−	0.900872i	$-0.357072\pi$
$13$	3.13023	0.868170	0.434085	+	0.900872i	$0.357072\pi$
$14$	0	0
$15$	0	0
$16$	−4.96962	−1.24241
$17$	−2.13023	−0.516657	−0.258329	−	0.966057i	$-0.583172\pi$
$17$	−2.13023	−0.516657	−0.258329	+	0.966057i	$0.583172\pi$
$18$	−1.78165	−0.419940
$19$	−7.73760	−1.77513	−0.887563	−	0.460686i	$-0.847603\pi$
$19$	−7.73760	−1.77513	−0.887563	+	0.460686i	$0.847603\pi$
$20$	0	0
$21$	0	0
$22$	3.70316	0.789516
$23$	5.53655	1.15445	0.577225	−	0.816585i	$-0.304136\pi$
$23$	5.53655	1.15445	0.577225	+	0.816585i	$0.304136\pi$
$24$	−1.47113	−0.300293
$25$	0	0
$26$	−5.57699	−1.09374
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−4.01368	−0.745322	−0.372661	−	0.927968i	$-0.621555\pi$
$29$	−4.01368	−0.745322	−0.372661	+	0.927968i	$0.621555\pi$
$30$	0	0
$31$	2.91188	0.522990	0.261495	−	0.965205i	$-0.415785\pi$
$31$	2.91188	0.522990	0.261495	+	0.965205i	$0.415785\pi$
$32$	5.91188	1.04508
$33$	2.07850	0.361820
$34$	3.79533	0.650894
$35$	0	0
$36$	1.17429	0.195715
$37$	3.51519	0.577893	0.288947	−	0.957345i	$-0.406695\pi$
$37$	3.51519	0.577893	0.288947	+	0.957345i	$0.406695\pi$
$38$	13.7857	2.23634
$39$	−3.13023	−0.501238
$40$	0	0
$41$	7.99038	1.24789	0.623944	−	0.781469i	$-0.285529\pi$
$41$	7.99038	1.24789	0.623944	+	0.781469i	$0.285529\pi$
$42$	0	0
$43$	−4.99038	−0.761026	−0.380513	−	0.924776i	$-0.624253\pi$
$43$	−4.99038	−0.761026	−0.380513	+	0.924776i	$0.624253\pi$
$44$	−2.44075	−0.367958
$45$	0	0
$46$	−9.86421	−1.45440
$47$	2.44075	0.356021	0.178010	−	0.984029i	$-0.443034\pi$
$47$	2.44075	0.356021	0.178010	+	0.984029i	$0.443034\pi$
$48$	4.96962	0.717303
$49$	0	0
$50$	0	0
$51$	2.13023	0.298292
$52$	3.67580	0.509741
$53$	9.91188	1.36150	0.680751	−	0.732515i	$-0.261654\pi$
$53$	9.91188	1.36150	0.680751	+	0.732515i	$0.261654\pi$
$54$	1.78165	0.242452
$55$	0	0
$56$	0	0
$57$	7.73760	1.02487
$58$	7.15099	0.938971
$59$	2.95594	0.384831	0.192415	−	0.981314i	$-0.438368\pi$
$59$	2.95594	0.384831	0.192415	+	0.981314i	$0.438368\pi$
$60$	0	0
$61$	−10.8946	−1.39491	−0.697454	−	0.716629i	$-0.745684\pi$
$61$	−10.8946	−1.39491	−0.697454	+	0.716629i	$0.745684\pi$
$62$	−5.18797	−0.658873
$63$	0	0
$64$	−0.593684	−0.0742104
$65$	0	0
$66$	−3.70316	−0.455827
$67$	3.83939	0.469056	0.234528	−	0.972109i	$-0.424645\pi$
$67$	3.83939	0.469056	0.234528	+	0.972109i	$0.424645\pi$
$68$	−2.50151	−0.303352
$69$	−5.53655	−0.666522
$70$	0	0
$71$	−15.0248	−1.78312	−0.891559	−	0.452905i	$-0.850388\pi$
$71$	−15.0248	−1.78312	−0.891559	+	0.452905i	$0.850388\pi$
$72$	1.47113	0.173374
$73$	−8.55369	−1.00113	−0.500567	−	0.865698i	$-0.666875\pi$
$73$	−8.55369	−1.00113	−0.500567	+	0.865698i	$0.666875\pi$
$74$	−6.26285	−0.728041
$75$	0	0
$76$	−9.08617	−1.04226
$77$	0	0
$78$	5.57699	0.631470
$79$	8.11354	0.912844	0.456422	−	0.889763i	$-0.349131\pi$
$79$	8.11354	0.912844	0.456422	+	0.889763i	$0.349131\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−14.2361	−1.57211
$83$	−8.75128	−0.960577	−0.480289	−	0.877110i	$-0.659468\pi$
$83$	−8.75128	−0.960577	−0.480289	+	0.877110i	$0.659468\pi$
$84$	0	0
$85$	0	0
$86$	8.89113	0.958755
$87$	4.01368	0.430312
$88$	−3.05774	−0.325956
$89$	0.618661	0.0655779	0.0327890	−	0.999462i	$-0.489561\pi$
$89$	0.618661	0.0655779	0.0327890	+	0.999462i	$0.489561\pi$
$90$	0	0
$91$	0	0
$92$	6.50151	0.677829
$93$	−2.91188	−0.301948
$94$	−4.34858	−0.448522
$95$	0	0
$96$	−5.91188	−0.603379
$97$	−0.296842	−0.0301397	−0.0150699	−	0.999886i	$-0.504797\pi$
$97$	−0.296842	−0.0301397	−0.0150699	+	0.999886i	$0.504797\pi$
$98$	0	0
$99$	−2.07850	−0.208897
$100$	0	0
$101$	8.25879	0.821780	0.410890	−	0.911685i	$-0.365218\pi$
$101$	8.25879	0.821780	0.410890	+	0.911685i	$0.365218\pi$
$102$	−3.79533	−0.375794
$103$	−17.0866	−1.68359	−0.841797	−	0.539794i	$-0.818502\pi$
$103$	−17.0866	−1.68359	−0.841797	+	0.539794i	$0.818502\pi$
$104$	4.60498	0.451555
$105$	0	0
$106$	−17.6595	−1.71525
$107$	6.31052	0.610061	0.305031	−	0.952343i	$-0.401333\pi$
$107$	6.31052	0.610061	0.305031	+	0.952343i	$0.401333\pi$
$108$	−1.17429	−0.112996
$109$	2.44676	0.234357	0.117178	−	0.993111i	$-0.462615\pi$
$109$	2.44676	0.234357	0.117178	+	0.993111i	$0.462615\pi$
$110$	0	0
$111$	−3.51519	−0.333647
$112$	0	0
$113$	10.1570	0.955489	0.477745	−	0.878499i	$-0.341454\pi$
$113$	10.1570	0.955489	0.477745	+	0.878499i	$0.341454\pi$
$114$	−13.7857	−1.29115
$115$	0	0
$116$	−4.71322	−0.437612
$117$	3.13023	0.289390
$118$	−5.26647	−0.484817
$119$	0	0
$120$	0	0
$121$	−6.67986	−0.607260
$122$	19.4104	1.75733
$123$	−7.99038	−0.720468
$124$	3.41939	0.307071
$125$	0	0
$126$	0	0
$127$	8.86977	0.787065	0.393532	−	0.919311i	$-0.371253\pi$
$127$	8.86977	0.787065	0.393532	+	0.919311i	$0.371253\pi$
$128$	−10.7660	−0.951592
$129$	4.99038	0.439378
$130$	0	0
$131$	5.34150	0.466689	0.233345	−	0.972394i	$-0.425033\pi$
$131$	5.34150	0.466689	0.233345	+	0.972394i	$0.425033\pi$
$132$	2.44075	0.212440
$133$	0	0
$134$	−6.84047	−0.590926
$135$	0	0
$136$	−3.13385	−0.268725
$137$	−13.1039	−1.11954	−0.559772	−	0.828647i	$-0.689111\pi$
$137$	−13.1039	−1.11954	−0.559772	+	0.828647i	$0.689111\pi$
$138$	9.86421	0.839697
$139$	0.243164	0.0206249	0.0103125	−	0.999947i	$-0.496717\pi$
$139$	0.243164	0.0206249	0.0103125	+	0.999947i	$0.496717\pi$
$140$	0	0
$141$	−2.44075	−0.205549
$142$	26.7690	2.24641
$143$	−6.50617	−0.544073
$144$	−4.96962	−0.414135
$145$	0	0
$146$	15.2397	1.26125
$147$	0	0
$148$	4.12785	0.339307
$149$	2.33728	0.191478	0.0957388	−	0.995406i	$-0.469479\pi$
$149$	2.33728	0.191478	0.0957388	+	0.995406i	$0.469479\pi$
$150$	0	0
$151$	−11.1135	−0.904407	−0.452203	−	0.891915i	$-0.649362\pi$
$151$	−11.1135	−0.904407	−0.452203	+	0.891915i	$0.649362\pi$
$152$	−11.3830	−0.923284
$153$	−2.13023	−0.172219
$154$	0	0
$155$	0	0
$156$	−3.67580	−0.294299
$157$	11.3182	0.903291	0.451645	−	0.892198i	$-0.350837\pi$
$157$	11.3182	0.903291	0.451645	+	0.892198i	$0.350837\pi$
$158$	−14.4555	−1.15002
$159$	−9.91188	−0.786064
$160$	0	0
$161$	0	0
$162$	−1.78165	−0.139980
$163$	−2.10347	−0.164757	−0.0823783	−	0.996601i	$-0.526252\pi$
$163$	−2.10347	−0.164757	−0.0823783	+	0.996601i	$0.526252\pi$
$164$	9.38302	0.732690
$165$	0	0
$166$	15.5917	1.21015
$167$	−3.58600	−0.277493	−0.138747	−	0.990328i	$-0.544307\pi$
$167$	−3.58600	−0.277493	−0.138747	+	0.990328i	$0.544307\pi$
$168$	0	0
$169$	−3.20165	−0.246281
$170$	0	0
$171$	−7.73760	−0.591709
$172$	−5.86015	−0.446832
$173$	15.5400	1.18148	0.590742	−	0.806860i	$-0.298835\pi$
$173$	15.5400	1.18148	0.590742	+	0.806860i	$0.298835\pi$
$174$	−7.15099	−0.542115
$175$	0	0
$176$	10.3293	0.778603
$177$	−2.95594	−0.222182
$178$	−1.10224	−0.0826163
$179$	15.7357	1.17614	0.588069	−	0.808811i	$-0.299888\pi$
$179$	15.7357	1.17614	0.588069	+	0.808811i	$0.299888\pi$
$180$	0	0
$181$	−4.98692	−0.370675	−0.185337	−	0.982675i	$-0.559338\pi$
$181$	−4.98692	−0.370675	−0.185337	+	0.982675i	$0.559338\pi$
$182$	0	0
$183$	10.8946	0.805351
$184$	8.14499	0.600456
$185$	0	0
$186$	5.18797	0.380400
$187$	4.42768	0.323784
$188$	2.86615	0.209036
$189$	0	0
$190$	0	0
$191$	11.8084	0.854427	0.427213	−	0.904151i	$-0.359495\pi$
$191$	11.8084	0.854427	0.427213	+	0.904151i	$0.359495\pi$
$192$	0.593684	0.0428454
$193$	24.7380	1.78068	0.890342	−	0.455293i	$-0.150466\pi$
$193$	24.7380	1.78068	0.890342	+	0.455293i	$0.150466\pi$
$194$	0.528869	0.0379706
$195$	0	0
$196$	0	0
$197$	19.5526	1.39307	0.696533	−	0.717525i	$-0.254725\pi$
$197$	19.5526	1.39307	0.696533	+	0.717525i	$0.254725\pi$
$198$	3.70316	0.263172
$199$	−22.2401	−1.57656	−0.788281	−	0.615315i	$-0.789029\pi$
$199$	−22.2401	−1.57656	−0.788281	+	0.615315i	$0.789029\pi$
$200$	0	0
$201$	−3.83939	−0.270810
$202$	−14.7143	−1.03529
$203$	0	0
$204$	2.50151	0.175141
$205$	0	0
$206$	30.4424	2.12102
$207$	5.53655	0.384817
$208$	−15.5561	−1.07862
$209$	16.0826	1.11245
$210$	0	0
$211$	23.6191	1.62601	0.813003	−	0.582259i	$-0.197831\pi$
$211$	23.6191	1.62601	0.813003	+	0.582259i	$0.197831\pi$
$212$	11.6394	0.799398
$213$	15.0248	1.02948
$214$	−11.2432	−0.768567
$215$	0	0
$216$	−1.47113	−0.100098
$217$	0	0
$218$	−4.35927	−0.295247
$219$	8.55369	0.578005
$220$	0	0
$221$	−6.66812	−0.448546
$222$	6.26285	0.420335
$223$	−25.1420	−1.68363	−0.841815	−	0.539765i	$-0.818513\pi$
$223$	−25.1420	−1.68363	−0.841815	+	0.539765i	$0.818513\pi$
$224$	0	0
$225$	0	0
$226$	−18.0962	−1.20374
$227$	23.6702	1.57105	0.785524	−	0.618831i	$-0.212393\pi$
$227$	23.6702	1.57105	0.785524	+	0.618831i	$0.212393\pi$
$228$	9.08617	0.601747
$229$	0.406316	0.0268501	0.0134251	−	0.999910i	$-0.495727\pi$
$229$	0.406316	0.0268501	0.0134251	+	0.999910i	$0.495727\pi$
$230$	0	0
$231$	0	0
$232$	−5.90465	−0.387659
$233$	−7.16527	−0.469413	−0.234706	−	0.972066i	$-0.575413\pi$
$233$	−7.16527	−0.469413	−0.234706	+	0.972066i	$0.575413\pi$
$234$	−5.57699	−0.364579
$235$	0	0
$236$	3.47113	0.225951
$237$	−8.11354	−0.527031
$238$	0	0
$239$	10.0922	0.652809	0.326404	−	0.945230i	$-0.394163\pi$
$239$	10.0922	0.652809	0.326404	+	0.945230i	$0.394163\pi$
$240$	0	0
$241$	−4.60676	−0.296748	−0.148374	−	0.988931i	$-0.547404\pi$
$241$	−4.60676	−0.296748	−0.148374	+	0.988931i	$0.547404\pi$
$242$	11.9012	0.765038
$243$	−1.00000	−0.0641500
$244$	−12.7934	−0.819013
$245$	0	0
$246$	14.2361	0.907660
$247$	−24.2205	−1.54111
$248$	4.28376	0.272019
$249$	8.75128	0.554590
$250$	0	0
$251$	−0.311597	−0.0196678	−0.00983390	−	0.999952i	$-0.503130\pi$
$251$	−0.311597	−0.0196678	−0.00983390	+	0.999952i	$0.503130\pi$
$252$	0	0
$253$	−11.5077	−0.723482
$254$	−15.8029	−0.991559
$255$	0	0
$256$	20.3687	1.27304
$257$	−2.50211	−0.156077	−0.0780387	−	0.996950i	$-0.524866\pi$
$257$	−2.50211	−0.156077	−0.0780387	+	0.996950i	$0.524866\pi$
$258$	−8.89113	−0.553537
$259$	0	0
$260$	0	0
$261$	−4.01368	−0.248441
$262$	−9.51671	−0.587944
$263$	−4.62511	−0.285196	−0.142598	−	0.989781i	$-0.545546\pi$
$263$	−4.62511	−0.285196	−0.142598	+	0.989781i	$0.545546\pi$
$264$	3.05774	0.188191
$265$	0	0
$266$	0	0
$267$	−0.618661	−0.0378614
$268$	4.50856	0.275404
$269$	12.0233	0.733074	0.366537	−	0.930404i	$-0.380543\pi$
$269$	12.0233	0.733074	0.366537	+	0.930404i	$0.380543\pi$
$270$	0	0
$271$	−5.46935	−0.332239	−0.166120	−	0.986106i	$-0.553124\pi$
$271$	−5.46935	−0.332239	−0.166120	+	0.986106i	$0.553124\pi$
$272$	10.5864	0.641898
$273$	0	0
$274$	23.3466	1.41042
$275$	0	0
$276$	−6.50151	−0.391345
$277$	29.8099	1.79111	0.895553	−	0.444956i	$-0.146781\pi$
$277$	29.8099	1.79111	0.895553	+	0.444956i	$0.146781\pi$
$278$	−0.433235	−0.0259837
$279$	2.91188	0.174330
$280$	0	0
$281$	7.78511	0.464421	0.232210	−	0.972666i	$-0.425404\pi$
$281$	7.78511	0.464421	0.232210	+	0.972666i	$0.425404\pi$
$282$	4.34858	0.258954
$283$	−2.61444	−0.155412	−0.0777062	−	0.996976i	$-0.524760\pi$
$283$	−2.61444	−0.155412	−0.0777062	+	0.996976i	$0.524760\pi$
$284$	−17.6435	−1.04695
$285$	0	0
$286$	11.5917	0.685434
$287$	0	0
$288$	5.91188	0.348361
$289$	−12.4621	−0.733066
$290$	0	0
$291$	0.296842	0.0174012
$292$	−10.0445	−0.587810
$293$	12.7559	0.745210	0.372605	−	0.927990i	$-0.378465\pi$
$293$	12.7559	0.745210	0.372605	+	0.927990i	$0.378465\pi$
$294$	0	0
$295$	0	0
$296$	5.17130	0.300576
$297$	2.07850	0.120607
$298$	−4.16423	−0.241227
$299$	17.3307	1.00226
$300$	0	0
$301$	0	0
$302$	19.8005	1.13939
$303$	−8.25879	−0.474455
$304$	38.4529	2.20543
$305$	0	0
$306$	3.79533	0.216965
$307$	13.1919	0.752900	0.376450	−	0.926437i	$-0.377145\pi$
$307$	13.1919	0.752900	0.376450	+	0.926437i	$0.377145\pi$
$308$	0	0
$309$	17.0866	0.972024
$310$	0	0
$311$	−14.2823	−0.809873	−0.404937	−	0.914345i	$-0.632706\pi$
$311$	−14.2823	−0.809873	−0.404937	+	0.914345i	$0.632706\pi$
$312$	−4.60498	−0.260706
$313$	−4.77143	−0.269697	−0.134849	−	0.990866i	$-0.543055\pi$
$313$	−4.77143	−0.269697	−0.134849	+	0.990866i	$0.543055\pi$
$314$	−20.1651	−1.13798
$315$	0	0
$316$	9.52764	0.535971
$317$	18.8048	1.05618	0.528091	−	0.849188i	$-0.322908\pi$
$317$	18.8048	1.05618	0.528091	+	0.849188i	$0.322908\pi$
$318$	17.6595	0.990298
$319$	8.34242	0.467086
$320$	0	0
$321$	−6.31052	−0.352219
$322$	0	0
$323$	16.4829	0.917131
$324$	1.17429	0.0652383
$325$	0	0
$326$	3.74766	0.207564
$327$	−2.44676	−0.135306
$328$	11.7549	0.649055
$329$	0	0
$330$	0	0
$331$	17.7742	0.976956	0.488478	−	0.872576i	$-0.337552\pi$
$331$	17.7742	0.976956	0.488478	+	0.872576i	$0.337552\pi$
$332$	−10.2765	−0.563998
$333$	3.51519	0.192631
$334$	6.38902	0.349592
$335$	0	0
$336$	0	0
$337$	−3.86675	−0.210635	−0.105318	−	0.994439i	$-0.533586\pi$
$337$	−3.86675	−0.210635	−0.105318	+	0.994439i	$0.533586\pi$
$338$	5.70423	0.310269
$339$	−10.1570	−0.551652
$340$	0	0
$341$	−6.05234	−0.327753
$342$	13.7857	0.745446
$343$	0	0
$344$	−7.34150	−0.395827
$345$	0	0
$346$	−27.6869	−1.48846
$347$	−10.2725	−0.551455	−0.275727	−	0.961236i	$-0.588919\pi$
$347$	−10.2725	−0.551455	−0.275727	+	0.961236i	$0.588919\pi$
$348$	4.71322	0.252655
$349$	23.6180	1.26424	0.632122	−	0.774869i	$-0.282184\pi$
$349$	23.6180	1.26424	0.632122	+	0.774869i	$0.282184\pi$
$350$	0	0
$351$	−3.13023	−0.167079
$352$	−12.2878	−0.654943
$353$	−5.77759	−0.307510	−0.153755	−	0.988109i	$-0.549137\pi$
$353$	−5.77759	−0.307510	−0.153755	+	0.988109i	$0.549137\pi$
$354$	5.26647	0.279909
$355$	0	0
$356$	0.726487	0.0385037
$357$	0	0
$358$	−28.0355	−1.48172
$359$	31.3054	1.65224	0.826118	−	0.563497i	$-0.190544\pi$
$359$	31.3054	1.65224	0.826118	+	0.563497i	$0.190544\pi$
$360$	0	0
$361$	40.8704	2.15107
$362$	8.88497	0.466983
$363$	6.67986	0.350602
$364$	0	0
$365$	0	0
$366$	−19.4104	−1.01460
$367$	−16.5519	−0.864002	−0.432001	−	0.901873i	$-0.642192\pi$
$367$	−16.5519	−0.864002	−0.432001	+	0.901873i	$0.642192\pi$
$368$	−27.5146	−1.43430
$369$	7.99038	0.415963
$370$	0	0
$371$	0	0
$372$	−3.41939	−0.177287
$373$	32.1426	1.66428	0.832140	−	0.554566i	$-0.187116\pi$
$373$	32.1426	1.66428	0.832140	+	0.554566i	$0.187116\pi$
$374$	−7.88858	−0.407909
$375$	0	0
$376$	3.59067	0.185175
$377$	−12.5638	−0.647066
$378$	0	0
$379$	−4.98800	−0.256216	−0.128108	−	0.991760i	$-0.540890\pi$
$379$	−4.98800	−0.256216	−0.128108	+	0.991760i	$0.540890\pi$
$380$	0	0
$381$	−8.86977	−0.454412
$382$	−21.0385	−1.07642
$383$	26.4568	1.35188	0.675940	−	0.736957i	$-0.263738\pi$
$383$	26.4568	1.35188	0.675940	+	0.736957i	$0.263738\pi$
$384$	10.7660	0.549402
$385$	0	0
$386$	−44.0746	−2.24334
$387$	−4.99038	−0.253675
$388$	−0.348578	−0.0176964
$389$	−12.7716	−0.647545	−0.323773	−	0.946135i	$-0.604951\pi$
$389$	−12.7716	−0.647545	−0.323773	+	0.946135i	$0.604951\pi$
$390$	0	0
$391$	−11.7941	−0.596455
$392$	0	0
$393$	−5.34150	−0.269443
$394$	−34.8360	−1.75501
$395$	0	0
$396$	−2.44075	−0.122653
$397$	−15.1212	−0.758912	−0.379456	−	0.925210i	$-0.623889\pi$
$397$	−15.1212	−0.758912	−0.379456	+	0.925210i	$0.623889\pi$
$398$	39.6242	1.98618
$399$	0	0
$400$	0	0
$401$	12.3606	0.617258	0.308629	−	0.951182i	$-0.400130\pi$
$401$	12.3606	0.617258	0.308629	+	0.951182i	$0.400130\pi$
$402$	6.84047	0.341171
$403$	9.11487	0.454044
$404$	9.69820	0.482504
$405$	0	0
$406$	0	0
$407$	−7.30630	−0.362160
$408$	3.13385	0.155149
$409$	−10.5287	−0.520611	−0.260306	−	0.965526i	$-0.583823\pi$
$409$	−10.5287	−0.520611	−0.260306	+	0.965526i	$0.583823\pi$
$410$	0	0
$411$	13.1039	0.646369
$412$	−20.0646	−0.988513
$413$	0	0
$414$	−9.86421	−0.484799
$415$	0	0
$416$	18.5056	0.907310
$417$	−0.243164	−0.0119078
$418$	−28.6535	−1.40149
$419$	13.3110	0.650283	0.325142	−	0.945665i	$-0.394588\pi$
$419$	13.3110	0.650283	0.325142	+	0.945665i	$0.394588\pi$
$420$	0	0
$421$	19.1520	0.933413	0.466707	−	0.884412i	$-0.345440\pi$
$421$	19.1520	0.933413	0.466707	+	0.884412i	$0.345440\pi$
$422$	−42.0811	−2.04847
$423$	2.44075	0.118674
$424$	14.5817	0.708149
$425$	0	0
$426$	−26.7690	−1.29696
$427$	0	0
$428$	7.41038	0.358194
$429$	6.50617	0.314121
$430$	0	0
$431$	10.8964	0.524860	0.262430	−	0.964951i	$-0.415476\pi$
$431$	10.8964	0.524860	0.262430	+	0.964951i	$0.415476\pi$
$432$	4.96962	0.239101
$433$	19.4869	0.936482	0.468241	−	0.883601i	$-0.344888\pi$
$433$	19.4869	0.936482	0.468241	+	0.883601i	$0.344888\pi$
$434$	0	0
$435$	0	0
$436$	2.87320	0.137601
$437$	−42.8396	−2.04929
$438$	−15.2397	−0.728181
$439$	13.5310	0.645799	0.322899	−	0.946433i	$-0.395342\pi$
$439$	13.5310	0.645799	0.322899	+	0.946433i	$0.395342\pi$
$440$	0	0
$441$	0	0
$442$	11.8803	0.565087
$443$	28.8663	1.37148	0.685740	−	0.727846i	$-0.259479\pi$
$443$	28.8663	1.37148	0.685740	+	0.727846i	$0.259479\pi$
$444$	−4.12785	−0.195899
$445$	0	0
$446$	44.7943	2.12107
$447$	−2.33728	−0.110550
$448$	0	0
$449$	23.4298	1.10572	0.552860	−	0.833274i	$-0.313536\pi$
$449$	23.4298	1.10572	0.552860	+	0.833274i	$0.313536\pi$
$450$	0	0
$451$	−16.6080	−0.782039
$452$	11.9272	0.561010
$453$	11.1135	0.522159
$454$	−42.1722	−1.97924
$455$	0	0
$456$	11.3830	0.533059
$457$	30.2876	1.41679	0.708396	−	0.705815i	$-0.249419\pi$
$457$	30.2876	1.41679	0.708396	+	0.705815i	$0.249419\pi$
$458$	−0.723915	−0.0338263
$459$	2.13023	0.0994307
$460$	0	0
$461$	−7.02196	−0.327045	−0.163523	−	0.986540i	$-0.552286\pi$
$461$	−7.02196	−0.327045	−0.163523	+	0.986540i	$0.552286\pi$
$462$	0	0
$463$	2.97324	0.138178	0.0690891	−	0.997610i	$-0.477991\pi$
$463$	2.97324	0.138178	0.0690891	+	0.997610i	$0.477991\pi$
$464$	19.9465	0.925992
$465$	0	0
$466$	12.7660	0.591375
$467$	23.7549	1.09925	0.549623	−	0.835413i	$-0.314771\pi$
$467$	23.7549	1.09925	0.549623	+	0.835413i	$0.314771\pi$
$468$	3.67580	0.169914
$469$	0	0
$470$	0	0
$471$	−11.3182	−0.521515
$472$	4.34858	0.200160
$473$	10.3725	0.476927
$474$	14.4555	0.663964
$475$	0	0
$476$	0	0
$477$	9.91188	0.453834
$478$	−17.9808	−0.822421
$479$	−6.54423	−0.299013	−0.149507	−	0.988761i	$-0.547769\pi$
$479$	−6.54423	−0.299013	−0.149507	+	0.988761i	$0.547769\pi$
$480$	0	0
$481$	11.0034	0.501710
$482$	8.20765	0.373848
$483$	0	0
$484$	−7.84408	−0.356549
$485$	0	0
$486$	1.78165	0.0808174
$487$	16.3269	0.739844	0.369922	−	0.929063i	$-0.379384\pi$
$487$	16.3269	0.739844	0.369922	+	0.929063i	$0.379384\pi$
$488$	−16.0274	−0.725525
$489$	2.10347	0.0951223
$490$	0	0
$491$	24.9009	1.12376	0.561882	−	0.827218i	$-0.310078\pi$
$491$	24.9009	1.12376	0.561882	+	0.827218i	$0.310078\pi$
$492$	−9.38302	−0.423019
$493$	8.55007	0.385076
$494$	43.1525	1.94152
$495$	0	0
$496$	−14.4710	−0.649766
$497$	0	0
$498$	−15.5917	−0.698683
$499$	−12.2039	−0.546323	−0.273161	−	0.961968i	$-0.588069\pi$
$499$	−12.2039	−0.546323	−0.273161	+	0.961968i	$0.588069\pi$
$500$	0	0
$501$	3.58600	0.160211
$502$	0.555157	0.0247779
$503$	−27.8165	−1.24028	−0.620139	−	0.784492i	$-0.712924\pi$
$503$	−27.8165	−1.24028	−0.620139	+	0.784492i	$0.712924\pi$
$504$	0	0
$505$	0	0
$506$	20.5027	0.911457
$507$	3.20165	0.142190
$508$	10.4157	0.462121
$509$	−27.8199	−1.23309	−0.616547	−	0.787318i	$-0.711469\pi$
$509$	−27.8199	−1.23309	−0.616547	+	0.787318i	$0.711469\pi$
$510$	0	0
$511$	0	0
$512$	−14.7579	−0.652214
$513$	7.73760	0.341623
$514$	4.45789	0.196629
$515$	0	0
$516$	5.86015	0.257979
$517$	−5.07310	−0.223115
$518$	0	0
$519$	−15.5400	−0.682131
$520$	0	0
$521$	33.5026	1.46777	0.733887	−	0.679272i	$-0.237704\pi$
$521$	33.5026	1.46777	0.733887	+	0.679272i	$0.237704\pi$
$522$	7.15099	0.312990
$523$	12.7958	0.559522	0.279761	−	0.960070i	$-0.409745\pi$
$523$	12.7958	0.559522	0.279761	+	0.960070i	$0.409745\pi$
$524$	6.27247	0.274014
$525$	0	0
$526$	8.24034	0.359296
$527$	−6.20299	−0.270206
$528$	−10.3293	−0.449527
$529$	7.65336	0.332755
$530$	0	0
$531$	2.95594	0.128277
$532$	0	0
$533$	25.0117	1.08338
$534$	1.10224	0.0476986
$535$	0	0
$536$	5.64825	0.243967
$537$	−15.7357	−0.679044
$538$	−21.4214	−0.923540
$539$	0	0
$540$	0	0
$541$	25.2566	1.08587	0.542933	−	0.839776i	$-0.317314\pi$
$541$	25.2566	1.08587	0.542933	+	0.839776i	$0.317314\pi$
$542$	9.74448	0.418561
$543$	4.98692	0.214009
$544$	−12.5937	−0.539950
$545$	0	0
$546$	0	0
$547$	−38.8743	−1.66214	−0.831072	−	0.556165i	$-0.812272\pi$
$547$	−38.8743	−1.66214	−0.831072	+	0.556165i	$0.812272\pi$
$548$	−15.3878	−0.657334
$549$	−10.8946	−0.464970
$550$	0	0
$551$	31.0562	1.32304
$552$	−8.14499	−0.346674
$553$	0	0
$554$	−53.1110	−2.25647
$555$	0	0
$556$	0.285545	0.0121098
$557$	−1.36378	−0.0577850	−0.0288925	−	0.999583i	$-0.509198\pi$
$557$	−1.36378	−0.0577850	−0.0288925	+	0.999583i	$0.509198\pi$
$558$	−5.18797	−0.219624
$559$	−15.6210	−0.660700
$560$	0	0
$561$	−4.42768	−0.186937
$562$	−13.8704	−0.585086
$563$	10.5744	0.445660	0.222830	−	0.974857i	$-0.428471\pi$
$563$	10.5744	0.445660	0.222830	+	0.974857i	$0.428471\pi$
$564$	−2.86615	−0.120687
$565$	0	0
$566$	4.65803	0.195791
$567$	0	0
$568$	−22.1035	−0.927441
$569$	37.9865	1.59248	0.796238	−	0.604983i	$-0.206820\pi$
$569$	37.9865	1.59248	0.796238	+	0.604983i	$0.206820\pi$
$570$	0	0
$571$	−15.6910	−0.656648	−0.328324	−	0.944565i	$-0.606484\pi$
$571$	−15.6910	−0.656648	−0.328324	+	0.944565i	$0.606484\pi$
$572$	−7.64013	−0.319450
$573$	−11.8084	−0.493304
$574$	0	0
$575$	0	0
$576$	−0.593684	−0.0247368
$577$	−3.44809	−0.143546	−0.0717730	−	0.997421i	$-0.522866\pi$
$577$	−3.44809	−0.143546	−0.0717730	+	0.997421i	$0.522866\pi$
$578$	22.2032	0.923530
$579$	−24.7380	−1.02808
$580$	0	0
$581$	0	0
$582$	−0.528869	−0.0219223
$583$	−20.6018	−0.853240
$584$	−12.5836	−0.520713
$585$	0	0
$586$	−22.7267	−0.938830
$587$	10.5983	0.437441	0.218720	−	0.975788i	$-0.429812\pi$
$587$	10.5983	0.437441	0.218720	+	0.975788i	$0.429812\pi$
$588$	0	0
$589$	−22.5310	−0.928373
$590$	0	0
$591$	−19.5526	−0.804287
$592$	−17.4692	−0.717978
$593$	−0.234412	−0.00962614	−0.00481307	−	0.999988i	$-0.501532\pi$
$593$	−0.234412	−0.00962614	−0.00481307	+	0.999988i	$0.501532\pi$
$594$	−3.70316	−0.151942
$595$	0	0
$596$	2.74464	0.112425
$597$	22.2401	0.910229
$598$	−30.8773	−1.26267
$599$	−25.8290	−1.05535	−0.527673	−	0.849448i	$-0.676935\pi$
$599$	−25.8290	−1.05535	−0.527673	+	0.849448i	$0.676935\pi$
$600$	0	0
$601$	−1.15592	−0.0471508	−0.0235754	−	0.999722i	$-0.507505\pi$
$601$	−1.15592	−0.0471508	−0.0235754	+	0.999722i	$0.507505\pi$
$602$	0	0
$603$	3.83939	0.156352
$604$	−13.0505	−0.531017
$605$	0	0
$606$	14.7143	0.597727
$607$	23.2111	0.942110	0.471055	−	0.882104i	$-0.343873\pi$
$607$	23.2111	0.942110	0.471055	+	0.882104i	$0.343873\pi$
$608$	−45.7438	−1.85516
$609$	0	0
$610$	0	0
$611$	7.64013	0.309086
$612$	−2.50151	−0.101117
$613$	6.33144	0.255724	0.127862	−	0.991792i	$-0.459188\pi$
$613$	6.33144	0.255724	0.127862	+	0.991792i	$0.459188\pi$
$614$	−23.5033	−0.948518
$615$	0	0
$616$	0	0
$617$	−25.9546	−1.04489	−0.522447	−	0.852672i	$-0.674981\pi$
$617$	−25.9546	−1.04489	−0.522447	+	0.852672i	$0.674981\pi$
$618$	−30.4424	−1.22457
$619$	−29.0962	−1.16948	−0.584738	−	0.811222i	$-0.698803\pi$
$619$	−29.0962	−1.16948	−0.584738	+	0.811222i	$0.698803\pi$
$620$	0	0
$621$	−5.53655	−0.222174
$622$	25.4460	1.02029
$623$	0	0
$624$	15.5561	0.622741
$625$	0	0
$626$	8.50104	0.339770
$627$	−16.0826	−0.642275
$628$	13.2908	0.530362
$629$	−7.48816	−0.298573
$630$	0	0
$631$	−37.3609	−1.48731	−0.743657	−	0.668561i	$-0.766911\pi$
$631$	−37.3609	−1.48731	−0.743657	+	0.668561i	$0.766911\pi$
$632$	11.9361	0.474791
$633$	−23.6191	−0.938775
$634$	−33.5036	−1.33060
$635$	0	0
$636$	−11.6394	−0.461533
$637$	0	0
$638$	−14.8633	−0.588443
$639$	−15.0248	−0.594373
$640$	0	0
$641$	−27.8535	−1.10015	−0.550074	−	0.835116i	$-0.685400\pi$
$641$	−27.8535	−1.10015	−0.550074	+	0.835116i	$0.685400\pi$
$642$	11.2432	0.443732
$643$	20.3104	0.800963	0.400481	−	0.916305i	$-0.368843\pi$
$643$	20.3104	0.800963	0.400481	+	0.916305i	$0.368843\pi$
$644$	0	0
$645$	0	0
$646$	−29.3668	−1.15542
$647$	−22.4329	−0.881929	−0.440964	−	0.897525i	$-0.645364\pi$
$647$	−22.4329	−0.881929	−0.440964	+	0.897525i	$0.645364\pi$
$648$	1.47113	0.0577915
$649$	−6.14391	−0.241170
$650$	0	0
$651$	0	0
$652$	−2.47008	−0.0967360
$653$	−19.8051	−0.775035	−0.387517	−	0.921862i	$-0.626667\pi$
$653$	−19.8051	−0.775035	−0.387517	+	0.921862i	$0.626667\pi$
$654$	4.35927	0.170461
$655$	0	0
$656$	−39.7092	−1.55038
$657$	−8.55369	−0.333711
$658$	0	0
$659$	38.3567	1.49416	0.747082	−	0.664731i	$-0.231454\pi$
$659$	38.3567	1.49416	0.747082	+	0.664731i	$0.231454\pi$
$660$	0	0
$661$	2.97492	0.115711	0.0578554	−	0.998325i	$-0.481574\pi$
$661$	2.97492	0.115711	0.0578554	+	0.998325i	$0.481574\pi$
$662$	−31.6674	−1.23079
$663$	6.66812	0.258968
$664$	−12.8743	−0.499619
$665$	0	0
$666$	−6.26285	−0.242680
$667$	−22.2219	−0.860437
$668$	−4.21101	−0.162929
$669$	25.1420	0.972045
$670$	0	0
$671$	22.6443	0.874175
$672$	0	0
$673$	22.8397	0.880407	0.440203	−	0.897898i	$-0.354906\pi$
$673$	22.8397	0.880407	0.440203	+	0.897898i	$0.354906\pi$
$674$	6.88921	0.265363
$675$	0	0
$676$	−3.75966	−0.144602
$677$	−7.19609	−0.276568	−0.138284	−	0.990393i	$-0.544159\pi$
$677$	−7.19609	−0.276568	−0.138284	+	0.990393i	$0.544159\pi$
$678$	18.0962	0.694982
$679$	0	0
$680$	0	0
$681$	−23.6702	−0.907045
$682$	10.7832	0.412909
$683$	4.55442	0.174270	0.0871351	−	0.996197i	$-0.472229\pi$
$683$	4.55442	0.174270	0.0871351	+	0.996197i	$0.472229\pi$
$684$	−9.08617	−0.347419
$685$	0	0
$686$	0	0
$687$	−0.406316	−0.0155019
$688$	24.8003	0.945503
$689$	31.0265	1.18202
$690$	0	0
$691$	3.53896	0.134628	0.0673142	−	0.997732i	$-0.478557\pi$
$691$	3.53896	0.134628	0.0673142	+	0.997732i	$0.478557\pi$
$692$	18.2485	0.693702
$693$	0	0
$694$	18.3020	0.694734
$695$	0	0
$696$	5.90465	0.223815
$697$	−17.0214	−0.644730
$698$	−42.0791	−1.59272
$699$	7.16527	0.271015
$700$	0	0
$701$	−16.8111	−0.634948	−0.317474	−	0.948267i	$-0.602835\pi$
$701$	−16.8111	−0.634948	−0.317474	+	0.948267i	$0.602835\pi$
$702$	5.57699	0.210490
$703$	−27.1991	−1.02583
$704$	1.23397	0.0465069
$705$	0	0
$706$	10.2937	0.387407
$707$	0	0
$708$	−3.47113	−0.130453
$709$	31.8176	1.19494	0.597468	−	0.801893i	$-0.296174\pi$
$709$	31.8176	1.19494	0.597468	+	0.801893i	$0.296174\pi$
$710$	0	0
$711$	8.11354	0.304281
$712$	0.910131	0.0341086
$713$	16.1218	0.603766
$714$	0	0
$715$	0	0
$716$	18.4782	0.690563
$717$	−10.0922	−0.376899
$718$	−55.7754	−2.08152
$719$	15.2807	0.569876	0.284938	−	0.958546i	$-0.408027\pi$
$719$	15.2807	0.569876	0.284938	+	0.958546i	$0.408027\pi$
$720$	0	0
$721$	0	0
$722$	−72.8169	−2.70996
$723$	4.60676	0.171327
$724$	−5.85609	−0.217640
$725$	0	0
$726$	−11.9012	−0.441695
$727$	19.1829	0.711453	0.355726	−	0.934590i	$-0.384234\pi$
$727$	19.1829	0.711453	0.355726	+	0.934590i	$0.384234\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	10.6307	0.393189
$732$	12.7934	0.472857
$733$	48.7218	1.79958	0.899790	−	0.436323i	$-0.143719\pi$
$733$	48.7218	1.79958	0.899790	+	0.436323i	$0.143719\pi$
$734$	29.4898	1.08849
$735$	0	0
$736$	32.7314	1.20650
$737$	−7.98016	−0.293953
$738$	−14.2361	−0.524038
$739$	−9.48405	−0.348876	−0.174438	−	0.984668i	$-0.555811\pi$
$739$	−9.48405	−0.348876	−0.174438	+	0.984668i	$0.555811\pi$
$740$	0	0
$741$	24.2205	0.889761
$742$	0	0
$743$	−30.2032	−1.10805	−0.554023	−	0.832501i	$-0.686908\pi$
$743$	−30.2032	−1.10805	−0.554023	+	0.832501i	$0.686908\pi$
$744$	−4.28376	−0.157050
$745$	0	0
$746$	−57.2669	−2.09669
$747$	−8.75128	−0.320192
$748$	5.19937	0.190108
$749$	0	0
$750$	0	0
$751$	−7.32934	−0.267451	−0.133726	−	0.991018i	$-0.542694\pi$
$751$	−7.32934	−0.267451	−0.133726	+	0.991018i	$0.542694\pi$
$752$	−12.1296	−0.442322
$753$	0.311597	0.0113552
$754$	22.3842	0.815186
$755$	0	0
$756$	0	0
$757$	14.1603	0.514666	0.257333	−	0.966323i	$-0.417156\pi$
$757$	14.1603	0.514666	0.257333	+	0.966323i	$0.417156\pi$
$758$	8.88688	0.322786
$759$	11.5077	0.417703
$760$	0	0
$761$	−23.9183	−0.867039	−0.433519	−	0.901144i	$-0.642728\pi$
$761$	−23.9183	−0.867039	−0.433519	+	0.901144i	$0.642728\pi$
$762$	15.8029	0.572477
$763$	0	0
$764$	13.8665	0.501672
$765$	0	0
$766$	−47.1369	−1.70312
$767$	9.25278	0.334099
$768$	−20.3687	−0.734992
$769$	14.1358	0.509750	0.254875	−	0.966974i	$-0.417966\pi$
$769$	14.1358	0.509750	0.254875	+	0.966974i	$0.417966\pi$
$770$	0	0
$771$	2.50211	0.0901113
$772$	29.0496	1.04552
$773$	6.75972	0.243130	0.121565	−	0.992583i	$-0.461209\pi$
$773$	6.75972	0.243130	0.121565	+	0.992583i	$0.461209\pi$
$774$	8.89113	0.319585
$775$	0	0
$776$	−0.436693	−0.0156764
$777$	0	0
$778$	22.7545	0.815790
$779$	−61.8263	−2.21516
$780$	0	0
$781$	31.2290	1.11746
$782$	21.0131	0.751425
$783$	4.01368	0.143437
$784$	0	0
$785$	0	0
$786$	9.51671	0.339450
$787$	15.3751	0.548062	0.274031	−	0.961721i	$-0.411643\pi$
$787$	15.3751	0.548062	0.274031	+	0.961721i	$0.411643\pi$
$788$	22.9604	0.817931
$789$	4.62511	0.164658
$790$	0	0
$791$	0	0
$792$	−3.05774	−0.108652
$793$	−34.1026	−1.21102
$794$	26.9408	0.956092
$795$	0	0
$796$	−26.1164	−0.925670
$797$	−9.46785	−0.335369	−0.167684	−	0.985841i	$-0.553629\pi$
$797$	−9.46785	−0.335369	−0.167684	+	0.985841i	$0.553629\pi$
$798$	0	0
$799$	−5.19937	−0.183941
$800$	0	0
$801$	0.618661	0.0218593
$802$	−22.0223	−0.777634
$803$	17.7788	0.627400
$804$	−4.50856	−0.159004
$805$	0	0
$806$	−16.2395	−0.572014
$807$	−12.0233	−0.423240
$808$	12.1498	0.427427
$809$	5.63745	0.198202	0.0991011	−	0.995077i	$-0.468403\pi$
$809$	5.63745	0.198202	0.0991011	+	0.995077i	$0.468403\pi$
$810$	0	0
$811$	24.3625	0.855485	0.427742	−	0.903901i	$-0.359309\pi$
$811$	24.3625	0.855485	0.427742	+	0.903901i	$0.359309\pi$
$812$	0	0
$813$	5.46935	0.191818
$814$	13.0173	0.456256
$815$	0	0
$816$	−10.5864	−0.370600
$817$	38.6135	1.35092
$818$	18.7585	0.655876
$819$	0	0
$820$	0	0
$821$	−31.7462	−1.10795	−0.553974	−	0.832534i	$-0.686889\pi$
$821$	−31.7462	−1.10795	−0.553974	+	0.832534i	$0.686889\pi$
$822$	−23.3466	−0.814307
$823$	−3.97563	−0.138582	−0.0692908	−	0.997597i	$-0.522074\pi$
$823$	−3.97563	−0.138582	−0.0692908	+	0.997597i	$0.522074\pi$
$824$	−25.1366	−0.875677
$825$	0	0
$826$	0	0
$827$	6.80348	0.236580	0.118290	−	0.992979i	$-0.462259\pi$
$827$	6.80348	0.236580	0.118290	+	0.992979i	$0.462259\pi$
$828$	6.50151	0.225943
$829$	23.0606	0.800927	0.400463	−	0.916313i	$-0.368849\pi$
$829$	23.0606	0.800927	0.400463	+	0.916313i	$0.368849\pi$
$830$	0	0
$831$	−29.8099	−1.03410
$832$	−1.85837	−0.0644273
$833$	0	0
$834$	0.433235	0.0150017
$835$	0	0
$836$	18.8856	0.653171
$837$	−2.91188	−0.100649
$838$	−23.7155	−0.819239
$839$	−35.0723	−1.21083	−0.605415	−	0.795910i	$-0.706993\pi$
$839$	−35.0723	−1.21083	−0.605415	+	0.795910i	$0.706993\pi$
$840$	0	0
$841$	−12.8904	−0.444495
$842$	−34.1223	−1.17593
$843$	−7.78511	−0.268133
$844$	27.7357	0.954701
$845$	0	0
$846$	−4.34858	−0.149507
$847$	0	0
$848$	−49.2583	−1.69154
$849$	2.61444	0.0897274
$850$	0	0
$851$	19.4620	0.667149
$852$	17.6435	0.604456
$853$	−12.3125	−0.421571	−0.210785	−	0.977532i	$-0.567602\pi$
$853$	−12.3125	−0.421571	−0.210785	+	0.977532i	$0.567602\pi$
$854$	0	0
$855$	0	0
$856$	9.28360	0.317307
$857$	33.2498	1.13579	0.567896	−	0.823101i	$-0.307758\pi$
$857$	33.2498	1.13579	0.567896	+	0.823101i	$0.307758\pi$
$858$	−11.5917	−0.395736
$859$	−3.01395	−0.102834	−0.0514172	−	0.998677i	$-0.516374\pi$
$859$	−3.01395	−0.102834	−0.0514172	+	0.998677i	$0.516374\pi$
$860$	0	0
$861$	0	0
$862$	−19.4136	−0.661228
$863$	−22.2360	−0.756921	−0.378460	−	0.925617i	$-0.623546\pi$
$863$	−22.2360	−0.756921	−0.378460	+	0.925617i	$0.623546\pi$
$864$	−5.91188	−0.201126
$865$	0	0
$866$	−34.7190	−1.17980
$867$	12.4621	0.423236
$868$	0	0
$869$	−16.8639	−0.572070
$870$	0	0
$871$	12.0182	0.407221
$872$	3.59950	0.121894
$873$	−0.296842	−0.0100466
$874$	76.3253	2.58174
$875$	0	0
$876$	10.0445	0.339372
$877$	−14.3221	−0.483623	−0.241812	−	0.970323i	$-0.577742\pi$
$877$	−14.3221	−0.483623	−0.241812	+	0.970323i	$0.577742\pi$
$878$	−24.1075	−0.813590
$879$	−12.7559	−0.430247
$880$	0	0
$881$	49.9929	1.68431	0.842153	−	0.539239i	$-0.181288\pi$
$881$	49.9929	1.68431	0.842153	+	0.539239i	$0.181288\pi$
$882$	0	0
$883$	44.3095	1.49113	0.745566	−	0.666432i	$-0.232179\pi$
$883$	44.3095	1.49113	0.745566	+	0.666432i	$0.232179\pi$
$884$	−7.83030	−0.263361
$885$	0	0
$886$	−51.4298	−1.72782
$887$	−10.2985	−0.345790	−0.172895	−	0.984940i	$-0.555312\pi$
$887$	−10.2985	−0.345790	−0.172895	+	0.984940i	$0.555312\pi$
$888$	−5.17130	−0.173538
$889$	0	0
$890$	0	0
$891$	−2.07850	−0.0696322
$892$	−29.5239	−0.988535
$893$	−18.8856	−0.631981
$894$	4.16423	0.139273
$895$	0	0
$896$	0	0
$897$	−17.3307	−0.578654
$898$	−41.7438	−1.39301
$899$	−11.6874	−0.389796
$900$	0	0
$901$	−21.1146	−0.703430
$902$	29.5896	0.985227
$903$	0	0
$904$	14.9423	0.496972
$905$	0	0
$906$	−19.8005	−0.657827
$907$	29.4280	0.977139	0.488570	−	0.872525i	$-0.337519\pi$
$907$	29.4280	0.977139	0.488570	+	0.872525i	$0.337519\pi$
$908$	27.7957	0.922433
$909$	8.25879	0.273927
$910$	0	0
$911$	−24.8078	−0.821918	−0.410959	−	0.911654i	$-0.634806\pi$
$911$	−24.8078	−0.821918	−0.410959	+	0.911654i	$0.634806\pi$
$912$	−38.4529	−1.27330
$913$	18.1895	0.601984
$914$	−53.9619	−1.78490
$915$	0	0
$916$	0.477133	0.0157649
$917$	0	0
$918$	−3.79533	−0.125265
$919$	13.0093	0.429138	0.214569	−	0.976709i	$-0.431165\pi$
$919$	13.0093	0.429138	0.214569	+	0.976709i	$0.431165\pi$
$920$	0	0
$921$	−13.1919	−0.434687
$922$	12.5107	0.412018
$923$	−47.0312	−1.54805
$924$	0	0
$925$	0	0
$926$	−5.29729	−0.174080
$927$	−17.0866	−0.561198
$928$	−23.7284	−0.778924
$929$	24.9185	0.817549	0.408775	−	0.912635i	$-0.365956\pi$
$929$	24.9185	0.817549	0.408775	+	0.912635i	$0.365956\pi$
$930$	0	0
$931$	0	0
$932$	−8.41410	−0.275613
$933$	14.2823	0.467580
$934$	−42.3230	−1.38485
$935$	0	0
$936$	4.60498	0.150518
$937$	55.1260	1.80089	0.900444	−	0.434973i	$-0.143242\pi$
$937$	55.1260	1.80089	0.900444	+	0.434973i	$0.143242\pi$
$938$	0	0
$939$	4.77143	0.155710
$940$	0	0
$941$	13.6447	0.444803	0.222402	−	0.974955i	$-0.428610\pi$
$941$	13.6447	0.444803	0.222402	+	0.974955i	$0.428610\pi$
$942$	20.1651	0.657015
$943$	44.2391	1.44062
$944$	−14.6899	−0.478116
$945$	0	0
$946$	−18.4802	−0.600842
$947$	0.893089	0.0290215	0.0145107	−	0.999895i	$-0.495381\pi$
$947$	0.893089	0.0290215	0.0145107	+	0.999895i	$0.495381\pi$
$948$	−9.52764	−0.309443
$949$	−26.7750	−0.869154
$950$	0	0
$951$	−18.8048	−0.609787
$952$	0	0
$953$	−34.5636	−1.11963	−0.559813	−	0.828619i	$-0.689127\pi$
$953$	−34.5636	−1.11963	−0.559813	+	0.828619i	$0.689127\pi$
$954$	−17.6595	−0.571749
$955$	0	0
$956$	11.8511	0.383293
$957$	−8.34242	−0.269672
$958$	11.6595	0.376703
$959$	0	0
$960$	0	0
$961$	−22.5209	−0.726481
$962$	−19.6042	−0.632064
$963$	6.31052	0.203354
$964$	−5.40967	−0.174234
$965$	0	0
$966$	0	0
$967$	−22.1811	−0.713296	−0.356648	−	0.934239i	$-0.616080\pi$
$967$	−22.1811	−0.713296	−0.356648	+	0.934239i	$0.616080\pi$
$968$	−9.82694	−0.315850
$969$	−16.4829	−0.529506
$970$	0	0
$971$	0.0759319	0.00243677	0.00121839	−	0.999999i	$-0.499612\pi$
$971$	0.0759319	0.00243677	0.00121839	+	0.999999i	$0.499612\pi$
$972$	−1.17429	−0.0376653
$973$	0	0
$974$	−29.0889	−0.932069
$975$	0	0
$976$	54.1420	1.73304
$977$	43.2114	1.38246	0.691228	−	0.722636i	$-0.257070\pi$
$977$	43.2114	1.38246	0.691228	+	0.722636i	$0.257070\pi$
$978$	−3.74766	−0.119837
$979$	−1.28588	−0.0410970
$980$	0	0
$981$	2.44676	0.0781189
$982$	−44.3648	−1.41574
$983$	18.3208	0.584343	0.292171	−	0.956366i	$-0.405622\pi$
$983$	18.3208	0.584343	0.292171	+	0.956366i	$0.405622\pi$
$984$	−11.7549	−0.374732
$985$	0	0
$986$	−15.2333	−0.485126
$987$	0	0
$988$	−28.4418	−0.904855
$989$	−27.6295	−0.878566
$990$	0	0
$991$	−31.5776	−1.00310	−0.501548	−	0.865130i	$-0.667236\pi$
$991$	−31.5776	−1.00310	−0.501548	+	0.865130i	$0.667236\pi$
$992$	17.2147	0.546568
$993$	−17.7742	−0.564046
$994$	0	0
$995$	0	0
$996$	10.2765	0.325624
$997$	−39.7315	−1.25831	−0.629154	−	0.777281i	$-0.716599\pi$
$997$	−39.7315	−1.25831	−0.629154	+	0.777281i	$0.716599\pi$
$998$	21.7432	0.688268
$999$	−3.51519	−0.111216

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3675.2.a.bz.1.1		4
5.2	odd	4		735.2.d.d.589.2		8
5.3	odd	4		735.2.d.d.589.7		8
5.4	even	2		3675.2.a.bp.1.4		4
7.2	even	3		525.2.i.h.151.4		8
7.4	even	3		525.2.i.h.226.4		8
7.6	odd	2		3675.2.a.cb.1.1		4
15.2	even	4		2205.2.d.s.1324.7		8
15.8	even	4		2205.2.d.s.1324.2		8
35.2	odd	12		105.2.q.a.4.2	✓	16
35.3	even	12		735.2.q.g.79.2		16
35.4	even	6		525.2.i.k.226.1		8
35.9	even	6		525.2.i.k.151.1		8
35.12	even	12		735.2.q.g.214.2		16
35.13	even	4		735.2.d.e.589.7		8
35.17	even	12		735.2.q.g.79.7		16
35.18	odd	12		105.2.q.a.79.2	yes	16
35.23	odd	12		105.2.q.a.4.7	yes	16
35.27	even	4		735.2.d.e.589.2		8
35.32	odd	12		105.2.q.a.79.7	yes	16
35.33	even	12		735.2.q.g.214.7		16
35.34	odd	2		3675.2.a.bn.1.4		4
105.2	even	12		315.2.bf.b.109.7		16
105.23	even	12		315.2.bf.b.109.2		16
105.32	even	12		315.2.bf.b.289.2		16
105.53	even	12		315.2.bf.b.289.7		16
105.62	odd	4		2205.2.d.o.1324.7		8
105.83	odd	4		2205.2.d.o.1324.2		8
140.23	even	12		1680.2.di.d.529.2		16
140.67	even	12		1680.2.di.d.289.2		16
140.107	even	12		1680.2.di.d.529.6		16
140.123	even	12		1680.2.di.d.289.6		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.q.a.4.2	✓	16	35.2	odd	12
105.2.q.a.4.7	yes	16	35.23	odd	12
105.2.q.a.79.2	yes	16	35.18	odd	12
105.2.q.a.79.7	yes	16	35.32	odd	12
315.2.bf.b.109.2		16	105.23	even	12
315.2.bf.b.109.7		16	105.2	even	12
315.2.bf.b.289.2		16	105.32	even	12
315.2.bf.b.289.7		16	105.53	even	12
525.2.i.h.151.4		8	7.2	even	3
525.2.i.h.226.4		8	7.4	even	3
525.2.i.k.151.1		8	35.9	even	6
525.2.i.k.226.1		8	35.4	even	6
735.2.d.d.589.2		8	5.2	odd	4
735.2.d.d.589.7		8	5.3	odd	4
735.2.d.e.589.2		8	35.27	even	4
735.2.d.e.589.7		8	35.13	even	4
735.2.q.g.79.2		16	35.3	even	12
735.2.q.g.79.7		16	35.17	even	12
735.2.q.g.214.2		16	35.12	even	12
735.2.q.g.214.7		16	35.33	even	12
1680.2.di.d.289.2		16	140.67	even	12
1680.2.di.d.289.6		16	140.123	even	12
1680.2.di.d.529.2		16	140.23	even	12
1680.2.di.d.529.6		16	140.107	even	12
2205.2.d.o.1324.2		8	105.83	odd	4
2205.2.d.o.1324.7		8	105.62	odd	4
2205.2.d.s.1324.2		8	15.8	even	4
2205.2.d.s.1324.7		8	15.2	even	4
3675.2.a.bn.1.4		4	35.34	odd	2
3675.2.a.bp.1.4		4	5.4	even	2
3675.2.a.bz.1.1		4	1.1	even	1	trivial
3675.2.a.cb.1.1		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)

\(f(q)\)	\(=\)	\(q-1.78165 q^{2} -1.00000 q^{3} +1.17429 q^{4} +1.78165 q^{6} +1.47113 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.78165 q^{2} -1.00000 q^{3} +1.17429 q^{4} +1.78165 q^{6} +1.47113 q^{8} +1.00000 q^{9} -2.07850 q^{11} -1.17429 q^{12} +3.13023 q^{13} -4.96962 q^{16} -2.13023 q^{17} -1.78165 q^{18} -7.73760 q^{19} +3.70316 q^{22} +5.53655 q^{23} -1.47113 q^{24} -5.57699 q^{26} -1.00000 q^{27} -4.01368 q^{29} +2.91188 q^{31} +5.91188 q^{32} +2.07850 q^{33} +3.79533 q^{34} +1.17429 q^{36} +3.51519 q^{37} +13.7857 q^{38} -3.13023 q^{39} +7.99038 q^{41} -4.99038 q^{43} -2.44075 q^{44} -9.86421 q^{46} +2.44075 q^{47} +4.96962 q^{48} +2.13023 q^{51} +3.67580 q^{52} +9.91188 q^{53} +1.78165 q^{54} +7.73760 q^{57} +7.15099 q^{58} +2.95594 q^{59} -10.8946 q^{61} -5.18797 q^{62} -0.593684 q^{64} -3.70316 q^{66} +3.83939 q^{67} -2.50151 q^{68} -5.53655 q^{69} -15.0248 q^{71} +1.47113 q^{72} -8.55369 q^{73} -6.26285 q^{74} -9.08617 q^{76} +5.57699 q^{78} +8.11354 q^{79} +1.00000 q^{81} -14.2361 q^{82} -8.75128 q^{83} +8.89113 q^{86} +4.01368 q^{87} -3.05774 q^{88} +0.618661 q^{89} +6.50151 q^{92} -2.91188 q^{93} -4.34858 q^{94} -5.91188 q^{96} -0.296842 q^{97} -2.07850 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} - 4 q^{3} + 4 q^{4} - 2 q^{6} + 6 q^{8} + 4 q^{9}+O(q^{10}) \) 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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