LMFDB - Embedded newform 3675.2.a.cb.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	$-0.642928\pi$
$13$	−3.13023	−0.868170	−0.434085	+	0.900872i	$0.642928\pi$
$14$	0	0
$15$	0	0
$16$	−4.96962	−1.24241
$17$	2.13023	0.516657	0.258329	−	0.966057i	$-0.416828\pi$
$17$	2.13023	0.516657	0.258329	+	0.966057i	$0.416828\pi$
$18$	−1.78165	−0.419940
$19$	7.73760	1.77513	0.887563	−	0.460686i	$-0.152397\pi$
$19$	7.73760	1.77513	0.887563	+	0.460686i	$0.152397\pi$
$20$	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	$-0.584215\pi$
$31$	−2.91188	−0.522990	−0.261495	+	0.965205i	$0.584215\pi$
$32$	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	$-0.714471\pi$
$41$	−7.99038	−1.24789	−0.623944	+	0.781469i	$0.714471\pi$
$42$	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	$-0.556966\pi$
$47$	−2.44075	−0.356021	−0.178010	+	0.984029i	$0.556966\pi$
$48$	−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	$-0.561632\pi$
$59$	−2.95594	−0.384831	−0.192415	+	0.981314i	$0.561632\pi$
$60$	0	0
$61$	10.8946	1.39491	0.697454	−	0.716629i	$-0.254316\pi$
$61$	10.8946	1.39491	0.697454	+	0.716629i	$0.254316\pi$
$62$	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	$-0.333125\pi$
$73$	8.55369	1.00113	0.500567	+	0.865698i	$0.333125\pi$
$74$	−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	$-0.340532\pi$
$83$	8.75128	0.960577	0.480289	+	0.877110i	$0.340532\pi$
$84$	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	$-0.510439\pi$
$89$	−0.618661	−0.0655779	−0.0327890	+	0.999462i	$0.510439\pi$
$90$	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	$-0.495203\pi$
$97$	0.296842	0.0301397	0.0150699	+	0.999886i	$0.495203\pi$
$98$	0	0
$99$	−2.07850	−0.208897
$100$	0	0
$101$	−8.25879	−0.821780	−0.410890	−	0.911685i	$-0.634782\pi$
$101$	−8.25879	−0.821780	−0.410890	+	0.911685i	$0.634782\pi$
$102$	−3.79533	−0.375794
$103$	17.0866	1.68359	0.841797	−	0.539794i	$-0.181498\pi$
$103$	17.0866	1.68359	0.841797	+	0.539794i	$0.181498\pi$
$104$	−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	$-0.574967\pi$
$131$	−5.34150	−0.466689	−0.233345	+	0.972394i	$0.574967\pi$
$132$	−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	$-0.503283\pi$
$139$	−0.243164	−0.0206249	−0.0103125	+	0.999947i	$0.503283\pi$
$140$	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	$-0.649163\pi$
$157$	−11.3182	−0.903291	−0.451645	+	0.892198i	$0.649163\pi$
$158$	−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	$-0.455693\pi$
$167$	3.58600	0.277493	0.138747	+	0.990328i	$0.455693\pi$
$168$	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	$-0.701165\pi$
$173$	−15.5400	−1.18148	−0.590742	+	0.806860i	$0.701165\pi$
$174$	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	$-0.440662\pi$
$181$	4.98692	0.370675	0.185337	+	0.982675i	$0.440662\pi$
$182$	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	$-0.210971\pi$
$199$	22.2401	1.57656	0.788281	+	0.615315i	$0.210971\pi$
$200$	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	$-0.181487\pi$
$223$	25.1420	1.68363	0.841815	+	0.539765i	$0.181487\pi$
$224$	0	0
$225$	0	0
$226$	−18.0962	−1.20374
$227$	−23.6702	−1.57105	−0.785524	−	0.618831i	$-0.787607\pi$
$227$	−23.6702	−1.57105	−0.785524	+	0.618831i	$0.787607\pi$
$228$	9.08617	0.601747
$229$	−0.406316	−0.0268501	−0.0134251	−	0.999910i	$-0.504273\pi$
$229$	−0.406316	−0.0268501	−0.0134251	+	0.999910i	$0.504273\pi$
$230$	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	$-0.452596\pi$
$241$	4.60676	0.296748	0.148374	+	0.988931i	$0.452596\pi$
$242$	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	$-0.496870\pi$
$251$	0.311597	0.0196678	0.00983390	+	0.999952i	$0.496870\pi$
$252$	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	$-0.475134\pi$
$257$	2.50211	0.156077	0.0780387	+	0.996950i	$0.475134\pi$
$258$	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	$-0.619457\pi$
$269$	−12.0233	−0.733074	−0.366537	+	0.930404i	$0.619457\pi$
$270$	0	0
$271$	5.46935	0.332239	0.166120	−	0.986106i	$-0.446876\pi$
$271$	5.46935	0.332239	0.166120	+	0.986106i	$0.446876\pi$
$272$	−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	$-0.475240\pi$
$283$	2.61444	0.155412	0.0777062	+	0.996976i	$0.475240\pi$
$284$	−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	$-0.621535\pi$
$293$	−12.7559	−0.745210	−0.372605	+	0.927990i	$0.621535\pi$
$294$	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	$-0.622855\pi$
$307$	−13.1919	−0.752900	−0.376450	+	0.926437i	$0.622855\pi$
$308$	0	0
$309$	17.0866	0.972024
$310$	0	0
$311$	14.2823	0.809873	0.404937	−	0.914345i	$-0.367294\pi$
$311$	14.2823	0.809873	0.404937	+	0.914345i	$0.367294\pi$
$312$	−4.60498	−0.260706
$313$	4.77143	0.269697	0.134849	−	0.990866i	$-0.456945\pi$
$313$	4.77143	0.269697	0.134849	+	0.990866i	$0.456945\pi$
$314$	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	$-0.717816\pi$
$349$	−23.6180	−1.26424	−0.632122	+	0.774869i	$0.717816\pi$
$350$	0	0
$351$	−3.13023	−0.167079
$352$	−12.2878	−0.654943
$353$	5.77759	0.307510	0.153755	−	0.988109i	$-0.450863\pi$
$353$	5.77759	0.307510	0.153755	+	0.988109i	$0.450863\pi$
$354$	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	$-0.357808\pi$
$367$	16.5519	0.864002	0.432001	+	0.901873i	$0.357808\pi$
$368$	−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	$-0.736262\pi$
$383$	−26.4568	−1.35188	−0.675940	+	0.736957i	$0.736262\pi$
$384$	−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	$-0.376111\pi$
$397$	15.1212	0.758912	0.379456	+	0.925210i	$0.376111\pi$
$398$	−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	$-0.416177\pi$
$409$	10.5287	0.520611	0.260306	+	0.965526i	$0.416177\pi$
$410$	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	$-0.605412\pi$
$419$	−13.3110	−0.650283	−0.325142	+	0.945665i	$0.605412\pi$
$420$	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	$-0.655112\pi$
$433$	−19.4869	−0.936482	−0.468241	+	0.883601i	$0.655112\pi$
$434$	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	$-0.604658\pi$
$439$	−13.5310	−0.645799	−0.322899	+	0.946433i	$0.604658\pi$
$440$	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	$-0.447714\pi$
$461$	7.02196	0.327045	0.163523	+	0.986540i	$0.447714\pi$
$462$	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	$-0.685229\pi$
$467$	−23.7549	−1.09925	−0.549623	+	0.835413i	$0.685229\pi$
$468$	−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	$-0.452231\pi$
$479$	6.54423	0.299013	0.149507	+	0.988761i	$0.452231\pi$
$480$	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	$-0.287076\pi$
$503$	27.8165	1.24028	0.620139	+	0.784492i	$0.287076\pi$
$504$	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	$-0.288531\pi$
$509$	27.8199	1.23309	0.616547	+	0.787318i	$0.288531\pi$
$510$	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	$-0.762296\pi$
$521$	−33.5026	−1.46777	−0.733887	+	0.679272i	$0.762296\pi$
$522$	7.15099	0.312990
$523$	−12.7958	−0.559522	−0.279761	−	0.960070i	$-0.590255\pi$
$523$	−12.7958	−0.559522	−0.279761	+	0.960070i	$0.590255\pi$
$524$	−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	$-0.571529\pi$
$563$	−10.5744	−0.445660	−0.222830	+	0.974857i	$0.571529\pi$
$564$	−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	$-0.477134\pi$
$577$	3.44809	0.143546	0.0717730	+	0.997421i	$0.477134\pi$
$578$	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	$-0.570188\pi$
$587$	−10.5983	−0.437441	−0.218720	+	0.975788i	$0.570188\pi$
$588$	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	$-0.498468\pi$
$593$	0.234412	0.00962614	0.00481307	+	0.999988i	$0.498468\pi$
$594$	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	$-0.492495\pi$
$601$	1.15592	0.0471508	0.0235754	+	0.999722i	$0.492495\pi$
$602$	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	$-0.656127\pi$
$607$	−23.2111	−0.942110	−0.471055	+	0.882104i	$0.656127\pi$
$608$	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	$-0.301197\pi$
$619$	29.0962	1.16948	0.584738	+	0.811222i	$0.301197\pi$
$620$	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	$-0.631157\pi$
$643$	−20.3104	−0.800963	−0.400481	+	0.916305i	$0.631157\pi$
$644$	0	0
$645$	0	0
$646$	−29.3668	−1.15542
$647$	22.4329	0.881929	0.440964	−	0.897525i	$-0.354636\pi$
$647$	22.4329	0.881929	0.440964	+	0.897525i	$0.354636\pi$
$648$	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	$-0.518426\pi$
$661$	−2.97492	−0.115711	−0.0578554	+	0.998325i	$0.518426\pi$
$662$	−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	$-0.455841\pi$
$677$	7.19609	0.276568	0.138284	+	0.990393i	$0.455841\pi$
$678$	−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	$-0.521443\pi$
$691$	−3.53896	−0.134628	−0.0673142	+	0.997732i	$0.521443\pi$
$692$	−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	$-0.591973\pi$
$719$	−15.2807	−0.569876	−0.284938	+	0.958546i	$0.591973\pi$
$720$	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	$-0.615766\pi$
$727$	−19.1829	−0.711453	−0.355726	+	0.934590i	$0.615766\pi$
$728$	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	$-0.856281\pi$
$733$	−48.7218	−1.79958	−0.899790	+	0.436323i	$0.856281\pi$
$734$	−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	$-0.357272\pi$
$761$	23.9183	0.867039	0.433519	+	0.901144i	$0.357272\pi$
$762$	−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	$-0.582034\pi$
$769$	−14.1358	−0.509750	−0.254875	+	0.966974i	$0.582034\pi$
$770$	0	0
$771$	2.50211	0.0901113
$772$	29.0496	1.04552
$773$	−6.75972	−0.243130	−0.121565	−	0.992583i	$-0.538791\pi$
$773$	−6.75972	−0.243130	−0.121565	+	0.992583i	$0.538791\pi$
$774$	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	$-0.588357\pi$
$787$	−15.3751	−0.548062	−0.274031	+	0.961721i	$0.588357\pi$
$788$	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	$-0.446371\pi$
$797$	9.46785	0.335369	0.167684	+	0.985841i	$0.446371\pi$
$798$	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	$-0.640691\pi$
$811$	−24.3625	−0.855485	−0.427742	+	0.903901i	$0.640691\pi$
$812$	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	$-0.631151\pi$
$829$	−23.0606	−0.800927	−0.400463	+	0.916313i	$0.631151\pi$
$830$	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	$-0.293007\pi$
$839$	35.0723	1.21083	0.605415	+	0.795910i	$0.293007\pi$
$840$	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	$-0.432398\pi$
$853$	12.3125	0.421571	0.210785	+	0.977532i	$0.432398\pi$
$854$	0	0
$855$	0	0
$856$	9.28360	0.317307
$857$	−33.2498	−1.13579	−0.567896	−	0.823101i	$-0.692242\pi$
$857$	−33.2498	−1.13579	−0.567896	+	0.823101i	$0.692242\pi$
$858$	−11.5917	−0.395736
$859$	3.01395	0.102834	0.0514172	−	0.998677i	$-0.483626\pi$
$859$	3.01395	0.102834	0.0514172	+	0.998677i	$0.483626\pi$
$860$	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	$-0.818712\pi$
$881$	−49.9929	−1.68431	−0.842153	+	0.539239i	$0.818712\pi$
$882$	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	$-0.444688\pi$
$887$	10.2985	0.345790	0.172895	+	0.984940i	$0.444688\pi$
$888$	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	$-0.634044\pi$
$929$	−24.9185	−0.817549	−0.408775	+	0.912635i	$0.634044\pi$
$930$	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	$-0.856758\pi$
$937$	−55.1260	−1.80089	−0.900444	+	0.434973i	$0.856758\pi$
$938$	0	0
$939$	4.77143	0.155710
$940$	0	0
$941$	−13.6447	−0.444803	−0.222402	−	0.974955i	$-0.571390\pi$
$941$	−13.6447	−0.444803	−0.222402	+	0.974955i	$0.571390\pi$
$942$	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	$-0.500388\pi$
$971$	−0.0759319	−0.00243677	−0.00121839	+	0.999999i	$0.500388\pi$
$972$	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	$-0.594378\pi$
$983$	−18.3208	−0.584343	−0.292171	+	0.956366i	$0.594378\pi$
$984$	−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	$-0.283401\pi$
$997$	39.7315	1.25831	0.629154	+	0.777281i	$0.283401\pi$
$998$	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.cb.1.1		4
5.2	odd	4		735.2.d.e.589.2		8
5.3	odd	4		735.2.d.e.589.7		8
5.4	even	2		3675.2.a.bn.1.4		4
7.3	odd	6		525.2.i.h.226.4		8
7.5	odd	6		525.2.i.h.151.4		8
7.6	odd	2		3675.2.a.bz.1.1		4
15.2	even	4		2205.2.d.o.1324.7		8
15.8	even	4		2205.2.d.o.1324.2		8
35.2	odd	12		735.2.q.g.214.2		16
35.3	even	12		105.2.q.a.79.2	yes	16
35.12	even	12		105.2.q.a.4.2	✓	16
35.13	even	4		735.2.d.d.589.7		8
35.17	even	12		105.2.q.a.79.7	yes	16
35.18	odd	12		735.2.q.g.79.2		16
35.19	odd	6		525.2.i.k.151.1		8
35.23	odd	12		735.2.q.g.214.7		16
35.24	odd	6		525.2.i.k.226.1		8
35.27	even	4		735.2.d.d.589.2		8
35.32	odd	12		735.2.q.g.79.7		16
35.33	even	12		105.2.q.a.4.7	yes	16
35.34	odd	2		3675.2.a.bp.1.4		4
105.17	odd	12		315.2.bf.b.289.2		16
105.38	odd	12		315.2.bf.b.289.7		16
105.47	odd	12		315.2.bf.b.109.7		16
105.62	odd	4		2205.2.d.s.1324.7		8
105.68	odd	12		315.2.bf.b.109.2		16
105.83	odd	4		2205.2.d.s.1324.2		8
140.3	odd	12		1680.2.di.d.289.6		16
140.47	odd	12		1680.2.di.d.529.6		16
140.87	odd	12		1680.2.di.d.289.2		16
140.103	odd	12		1680.2.di.d.529.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.q.a.4.2	✓	16	35.12	even	12
105.2.q.a.4.7	yes	16	35.33	even	12
105.2.q.a.79.2	yes	16	35.3	even	12
105.2.q.a.79.7	yes	16	35.17	even	12
315.2.bf.b.109.2		16	105.68	odd	12
315.2.bf.b.109.7		16	105.47	odd	12
315.2.bf.b.289.2		16	105.17	odd	12
315.2.bf.b.289.7		16	105.38	odd	12
525.2.i.h.151.4		8	7.5	odd	6
525.2.i.h.226.4		8	7.3	odd	6
525.2.i.k.151.1		8	35.19	odd	6
525.2.i.k.226.1		8	35.24	odd	6
735.2.d.d.589.2		8	35.27	even	4
735.2.d.d.589.7		8	35.13	even	4
735.2.d.e.589.2		8	5.2	odd	4
735.2.d.e.589.7		8	5.3	odd	4
735.2.q.g.79.2		16	35.18	odd	12
735.2.q.g.79.7		16	35.32	odd	12
735.2.q.g.214.2		16	35.2	odd	12
735.2.q.g.214.7		16	35.23	odd	12
1680.2.di.d.289.2		16	140.87	odd	12
1680.2.di.d.289.6		16	140.3	odd	12
1680.2.di.d.529.2		16	140.103	odd	12
1680.2.di.d.529.6		16	140.47	odd	12
2205.2.d.o.1324.2		8	15.8	even	4
2205.2.d.o.1324.7		8	15.2	even	4
2205.2.d.s.1324.2		8	105.83	odd	4
2205.2.d.s.1324.7		8	105.62	odd	4
3675.2.a.bn.1.4		4	5.4	even	2
3675.2.a.bp.1.4		4	35.34	odd	2
3675.2.a.bz.1.1		4	7.6	odd	2
3675.2.a.cb.1.1		4	1.1	even	1	trivial

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more