LMFDB - Embedded newform 3675.2.a.cb.1.2

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			1.2
Root			$1.28734$ 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)

$f(q)$	$=$	$q-0.287336 q^{2} +1.00000 q^{3} -1.91744 q^{4} -0.287336 q^{6} +1.12562 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.287336 q^{2} +1.00000 q^{3} -1.91744 q^{4} -0.287336 q^{6} +1.12562 q^{8} +1.00000 q^{9} -3.33039 q^{11} -1.91744 q^{12} +4.54754 q^{13} +3.51145 q^{16} -5.54754 q^{17} -0.287336 q^{18} +1.65723 q^{19} +0.956942 q^{22} -7.63366 q^{23} +1.12562 q^{24} -1.30667 q^{26} +1.00000 q^{27} -0.118657 q^{29} +6.26020 q^{31} -3.26020 q^{32} -3.33039 q^{33} +1.59401 q^{34} -1.91744 q^{36} +7.75572 q^{37} -0.476183 q^{38} +4.54754 q^{39} -0.0701896 q^{41} +2.92981 q^{43} +6.38582 q^{44} +2.19342 q^{46} +6.38582 q^{47} +3.51145 q^{48} -5.54754 q^{51} -8.71963 q^{52} +0.739795 q^{53} -0.287336 q^{54} +1.65723 q^{57} +0.0340944 q^{58} +1.63010 q^{59} +7.31802 q^{61} -1.79878 q^{62} -6.08612 q^{64} +0.956942 q^{66} +3.03610 q^{67} +10.6371 q^{68} -7.63366 q^{69} -3.77048 q^{71} +1.12562 q^{72} -2.35514 q^{73} -2.22850 q^{74} -3.17764 q^{76} -1.30667 q^{78} -11.9403 q^{79} +1.00000 q^{81} +0.0201680 q^{82} -1.22411 q^{83} -0.841839 q^{86} -0.118657 q^{87} -3.74876 q^{88} +13.0001 q^{89} +14.6371 q^{92} +6.26020 q^{93} -1.83488 q^{94} -3.26020 q^{96} +3.04306 q^{97} -3.33039 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.287336	−0.203177	−0.101589	−	0.994827i	$-0.532393\pi$
$2$	−0.287336	−0.203177	−0.101589	+	0.994827i	$0.532393\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.91744	−0.958719
$5$	0	0
$5$	0	0
$6$	−0.287336	−0.117304
$7$	0	0
$7$	0	0
$8$	1.12562	0.397967
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.33039	−1.00415	−0.502076	−	0.864824i	$-0.667430\pi$
$11$	−3.33039	−1.00415	−0.502076	+	0.864824i	$0.667430\pi$
$12$	−1.91744	−0.553517
$13$	4.54754	1.26126	0.630630	−	0.776083i	$-0.282796\pi$
$13$	4.54754	1.26126	0.630630	+	0.776083i	$0.282796\pi$
$14$	0	0
$15$	0	0
$16$	3.51145	0.877861
$17$	−5.54754	−1.34548	−0.672738	−	0.739881i	$-0.734882\pi$
$17$	−5.54754	−1.34548	−0.672738	+	0.739881i	$0.734882\pi$
$18$	−0.287336	−0.0677257
$19$	1.65723	0.380195	0.190098	−	0.981765i	$-0.439120\pi$
$19$	1.65723	0.380195	0.190098	+	0.981765i	$0.439120\pi$
$20$	0	0
$21$	0	0
$22$	0.956942	0.204021
$23$	−7.63366	−1.59173	−0.795864	−	0.605476i	$-0.792983\pi$
$23$	−7.63366	−1.59173	−0.795864	+	0.605476i	$0.792983\pi$
$24$	1.12562	0.229766
$25$	0	0
$26$	−1.30667	−0.256259
$27$	1.00000	0.192450
$28$	0	0
$29$	−0.118657	−0.0220341	−0.0110170	−	0.999939i	$-0.503507\pi$
$29$	−0.118657	−0.0220341	−0.0110170	+	0.999939i	$0.503507\pi$
$30$	0	0
$31$	6.26020	1.12437	0.562183	−	0.827013i	$-0.309962\pi$
$31$	6.26020	1.12437	0.562183	+	0.827013i	$0.309962\pi$
$32$	−3.26020	−0.576328
$33$	−3.33039	−0.579747
$34$	1.59401	0.273370
$35$	0	0
$36$	−1.91744	−0.319573
$37$	7.75572	1.27503	0.637516	−	0.770437i	$-0.279962\pi$
$37$	7.75572	1.27503	0.637516	+	0.770437i	$0.279962\pi$
$38$	−0.476183	−0.0772470
$39$	4.54754	0.728189
$40$	0	0
$41$	−0.0701896	−0.0109618	−0.00548089	−	0.999985i	$-0.501745\pi$
$41$	−0.0701896	−0.0109618	−0.00548089	+	0.999985i	$0.501745\pi$
$42$	0	0
$43$	2.92981	0.446792	0.223396	−	0.974728i	$-0.428286\pi$
$43$	2.92981	0.446792	0.223396	+	0.974728i	$0.428286\pi$
$44$	6.38582	0.962699
$45$	0	0
$46$	2.19342	0.323403
$47$	6.38582	0.931468	0.465734	−	0.884925i	$-0.345790\pi$
$47$	6.38582	0.931468	0.465734	+	0.884925i	$0.345790\pi$
$48$	3.51145	0.506833
$49$	0	0
$50$	0	0
$51$	−5.54754	−0.776811
$52$	−8.71963	−1.20919
$53$	0.739795	0.101619	0.0508094	−	0.998708i	$-0.483820\pi$
$53$	0.739795	0.101619	0.0508094	+	0.998708i	$0.483820\pi$
$54$	−0.287336	−0.0391015
$55$	0	0
$56$	0	0
$57$	1.65723	0.219506
$58$	0.0340944	0.00447682
$59$	1.63010	0.212221	0.106111	−	0.994354i	$-0.466160\pi$
$59$	1.63010	0.212221	0.106111	+	0.994354i	$0.466160\pi$
$60$	0	0
$61$	7.31802	0.936977	0.468488	−	0.883470i	$-0.344799\pi$
$61$	7.31802	0.936977	0.468488	+	0.883470i	$0.344799\pi$
$62$	−1.79878	−0.228445
$63$	0	0
$64$	−6.08612	−0.760765
$65$	0	0
$66$	0.956942	0.117791
$67$	3.03610	0.370918	0.185459	−	0.982652i	$-0.440623\pi$
$67$	3.03610	0.370918	0.185459	+	0.982652i	$0.440623\pi$
$68$	10.6371	1.28993
$69$	−7.63366	−0.918984
$70$	0	0
$71$	−3.77048	−0.447474	−0.223737	−	0.974650i	$-0.571826\pi$
$71$	−3.77048	−0.447474	−0.223737	+	0.974650i	$0.571826\pi$
$72$	1.12562	0.132656
$73$	−2.35514	−0.275648	−0.137824	−	0.990457i	$-0.544011\pi$
$73$	−2.35514	−0.275648	−0.137824	+	0.990457i	$0.544011\pi$
$74$	−2.22850	−0.259058
$75$	0	0
$76$	−3.17764	−0.364501
$77$	0	0
$78$	−1.30667	−0.147951
$79$	−11.9403	−1.34339	−0.671696	−	0.740827i	$-0.734434\pi$
$79$	−11.9403	−1.34339	−0.671696	+	0.740827i	$0.734434\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0.0201680	0.00222718
$83$	−1.22411	−0.134363	−0.0671817	−	0.997741i	$-0.521401\pi$
$83$	−1.22411	−0.134363	−0.0671817	+	0.997741i	$0.521401\pi$
$84$	0	0
$85$	0	0
$86$	−0.841839	−0.0907779
$87$	−0.118657	−0.0127214
$88$	−3.74876	−0.399619
$89$	13.0001	1.37801	0.689006	−	0.724755i	$-0.258047\pi$
$89$	13.0001	1.37801	0.689006	+	0.724755i	$0.258047\pi$
$90$	0	0
$91$	0	0
$92$	14.6371	1.52602
$93$	6.26020	0.649153
$94$	−1.83488	−0.189253
$95$	0	0
$96$	−3.26020	−0.332743
$97$	3.04306	0.308976	0.154488	−	0.987995i	$-0.450627\pi$
$97$	3.04306	0.308976	0.154488	+	0.987995i	$0.450627\pi$
$98$	0	0
$99$	−3.33039	−0.334717
$100$	0	0
$101$	−16.0397	−1.59600	−0.798002	−	0.602654i	$-0.794110\pi$
$101$	−16.0397	−1.59600	−0.798002	+	0.602654i	$0.794110\pi$
$102$	1.59401	0.157830
$103$	−5.29191	−0.521428	−0.260714	−	0.965416i	$-0.583958\pi$
$103$	−5.29191	−0.521428	−0.260714	+	0.965416i	$0.583958\pi$
$104$	5.11880	0.501940
$105$	0	0
$106$	−0.212570	−0.0206466
$107$	5.16172	0.499002	0.249501	−	0.968375i	$-0.419733\pi$
$107$	5.16172	0.499002	0.249501	+	0.968375i	$0.419733\pi$
$108$	−1.91744	−0.184506
$109$	3.24087	0.310419	0.155209	−	0.987882i	$-0.450395\pi$
$109$	3.24087	0.310419	0.155209	+	0.987882i	$0.450395\pi$
$110$	0	0
$111$	7.75572	0.736141
$112$	0	0
$113$	12.6608	1.19103	0.595513	−	0.803345i	$-0.296949\pi$
$113$	12.6608	1.19103	0.595513	+	0.803345i	$0.296949\pi$
$114$	−0.476183	−0.0445986
$115$	0	0
$116$	0.227518	0.0211245
$117$	4.54754	0.420420
$118$	−0.468387	−0.0431185
$119$	0	0
$120$	0	0
$121$	0.0915259	0.00832053
$122$	−2.10273	−0.190372
$123$	−0.0701896	−0.00632879
$124$	−12.0036	−1.07795
$125$	0	0
$126$	0	0
$127$	16.5475	1.46836	0.734178	−	0.678957i	$-0.237568\pi$
$127$	16.5475	1.46836	0.734178	+	0.678957i	$0.237568\pi$
$128$	8.26917	0.730898
$129$	2.92981	0.257955
$130$	0	0
$131$	5.29785	0.462876	0.231438	−	0.972850i	$-0.425657\pi$
$131$	5.29785	0.462876	0.231438	+	0.972850i	$0.425657\pi$
$132$	6.38582	0.555815
$133$	0	0
$134$	−0.872379	−0.0753621
$135$	0	0
$136$	−6.24442	−0.535455
$137$	14.8701	1.27044	0.635221	−	0.772331i	$-0.280909\pi$
$137$	14.8701	1.27044	0.635221	+	0.772331i	$0.280909\pi$
$138$	2.19342	0.186717
$139$	9.51685	0.807209	0.403605	−	0.914934i	$-0.367757\pi$
$139$	9.51685	0.807209	0.403605	+	0.914934i	$0.367757\pi$
$140$	0	0
$141$	6.38582	0.537783
$142$	1.08339	0.0909164
$143$	−15.1451	−1.26650
$144$	3.51145	0.292620
$145$	0	0
$146$	0.676716	0.0560054
$147$	0	0
$148$	−14.8711	−1.22240
$149$	11.3700	0.931470	0.465735	−	0.884924i	$-0.345790\pi$
$149$	11.3700	0.931470	0.465735	+	0.884924i	$0.345790\pi$
$150$	0	0
$151$	8.94033	0.727554	0.363777	−	0.931486i	$-0.381487\pi$
$151$	8.94033	0.727554	0.363777	+	0.931486i	$0.381487\pi$
$152$	1.86542	0.151305
$153$	−5.54754	−0.448492
$154$	0	0
$155$	0	0
$156$	−8.71963	−0.698129
$157$	3.34632	0.267066	0.133533	−	0.991044i	$-0.457368\pi$
$157$	3.34632	0.267066	0.133533	+	0.991044i	$0.457368\pi$
$158$	3.43088	0.272946
$159$	0.739795	0.0586696
$160$	0	0
$161$	0	0
$162$	−0.287336	−0.0225752
$163$	15.7559	1.23409	0.617047	−	0.786926i	$-0.288329\pi$
$163$	15.7559	1.23409	0.617047	+	0.786926i	$0.288329\pi$
$164$	0.134584	0.0105093
$165$	0	0
$166$	0.351730	0.0272996
$167$	22.5942	1.74839	0.874194	−	0.485577i	$-0.161390\pi$
$167$	22.5942	1.74839	0.874194	+	0.485577i	$0.161390\pi$
$168$	0	0
$169$	7.68012	0.590779
$170$	0	0
$171$	1.65723	0.126732
$172$	−5.61773	−0.428348
$173$	−8.52620	−0.648235	−0.324118	−	0.946017i	$-0.605067\pi$
$173$	−8.52620	−0.648235	−0.324118	+	0.946017i	$0.605067\pi$
$174$	0.0340944	0.00258469
$175$	0	0
$176$	−11.6945	−0.881506
$177$	1.63010	0.122526
$178$	−3.73541	−0.279981
$179$	−11.7806	−0.880524	−0.440262	−	0.897869i	$-0.645115\pi$
$179$	−11.7806	−0.880524	−0.440262	+	0.897869i	$0.645115\pi$
$180$	0	0
$181$	−9.08967	−0.675630	−0.337815	−	0.941213i	$-0.609688\pi$
$181$	−9.08967	−0.675630	−0.337815	+	0.941213i	$0.609688\pi$
$182$	0	0
$183$	7.31802	0.540964
$184$	−8.59260	−0.633455
$185$	0	0
$186$	−1.79878	−0.131893
$187$	18.4755	1.35106
$188$	−12.2444	−0.893016
$189$	0	0
$190$	0	0
$191$	20.4957	1.48301	0.741507	−	0.670945i	$-0.234111\pi$
$191$	20.4957	1.48301	0.741507	+	0.670945i	$0.234111\pi$
$192$	−6.08612	−0.439228
$193$	8.54296	0.614936	0.307468	−	0.951558i	$-0.400518\pi$
$193$	8.54296	0.614936	0.307468	+	0.951558i	$0.400518\pi$
$194$	−0.874380	−0.0627768
$195$	0	0
$196$	0	0
$197$	13.8086	0.983820	0.491910	−	0.870646i	$-0.336299\pi$
$197$	13.8086	0.983820	0.491910	+	0.870646i	$0.336299\pi$
$198$	0.956942	0.0680069
$199$	−3.79099	−0.268736	−0.134368	−	0.990932i	$-0.542900\pi$
$199$	−3.79099	−0.268736	−0.134368	+	0.990932i	$0.542900\pi$
$200$	0	0
$201$	3.03610	0.214150
$202$	4.60877	0.324272
$203$	0	0
$204$	10.6371	0.744744
$205$	0	0
$206$	1.52056	0.105942
$207$	−7.63366	−0.530576
$208$	15.9684	1.10721
$209$	−5.51924	−0.381774
$210$	0	0
$211$	−0.114416	−0.00787674	−0.00393837	−	0.999992i	$-0.501254\pi$
$211$	−0.114416	−0.00787674	−0.00393837	+	0.999992i	$0.501254\pi$
$212$	−1.41851	−0.0974238
$213$	−3.77048	−0.258349
$214$	−1.48315	−0.101386
$215$	0	0
$216$	1.12562	0.0765888
$217$	0	0
$218$	−0.931218	−0.0630700
$219$	−2.35514	−0.159146
$220$	0	0
$221$	−25.2277	−1.69700
$222$	−2.22850	−0.149567
$223$	−7.86673	−0.526795	−0.263398	−	0.964687i	$-0.584843\pi$
$223$	−7.86673	−0.526795	−0.263398	+	0.964687i	$0.584843\pi$
$224$	0	0
$225$	0	0
$226$	−3.63790	−0.241989
$227$	−8.97866	−0.595935	−0.297967	−	0.954576i	$-0.596309\pi$
$227$	−8.97866	−0.595935	−0.297967	+	0.954576i	$0.596309\pi$
$228$	−3.17764	−0.210445
$229$	5.08612	0.336100	0.168050	−	0.985778i	$-0.446253\pi$
$229$	5.08612	0.336100	0.168050	+	0.985778i	$0.446253\pi$
$230$	0	0
$231$	0	0
$232$	−0.133563	−0.00876883
$233$	21.8183	1.42936	0.714681	−	0.699451i	$-0.246572\pi$
$233$	21.8183	1.42936	0.714681	+	0.699451i	$0.246572\pi$
$234$	−1.30667	−0.0854198
$235$	0	0
$236$	−3.12562	−0.203461
$237$	−11.9403	−0.775608
$238$	0	0
$239$	7.44905	0.481839	0.240920	−	0.970545i	$-0.422551\pi$
$239$	7.44905	0.481839	0.240920	+	0.970545i	$0.422551\pi$
$240$	0	0
$241$	24.1758	1.55730	0.778650	−	0.627459i	$-0.215905\pi$
$241$	24.1758	1.55730	0.778650	+	0.627459i	$0.215905\pi$
$242$	−0.0262987	−0.00169054
$243$	1.00000	0.0641500
$244$	−14.0319	−0.898297
$245$	0	0
$246$	0.0201680	0.00128586
$247$	7.53634	0.479526
$248$	7.04661	0.447460
$249$	−1.22411	−0.0775748
$250$	0	0
$251$	−6.00200	−0.378843	−0.189421	−	0.981896i	$-0.560661\pi$
$251$	−6.00200	−0.378843	−0.189421	+	0.981896i	$0.560661\pi$
$252$	0	0
$253$	25.4231	1.59834
$254$	−4.75470	−0.298336
$255$	0	0
$256$	9.79621	0.612263
$257$	−7.33395	−0.457479	−0.228740	−	0.973488i	$-0.573460\pi$
$257$	−7.33395	−0.457479	−0.228740	+	0.973488i	$0.573460\pi$
$258$	−0.841839	−0.0524106
$259$	0	0
$260$	0	0
$261$	−0.118657	−0.00734469
$262$	−1.52226	−0.0940457
$263$	9.48772	0.585038	0.292519	−	0.956260i	$-0.405507\pi$
$263$	9.48772	0.585038	0.292519	+	0.956260i	$0.405507\pi$
$264$	−3.74876	−0.230720
$265$	0	0
$266$	0	0
$267$	13.0001	0.795596
$268$	−5.82152	−0.355606
$269$	−16.0485	−0.978492	−0.489246	−	0.872146i	$-0.662728\pi$
$269$	−16.0485	−0.978492	−0.489246	+	0.872146i	$0.662728\pi$
$270$	0	0
$271$	−24.1690	−1.46816	−0.734080	−	0.679063i	$-0.762386\pi$
$271$	−24.1690	−1.46816	−0.734080	+	0.679063i	$0.762386\pi$
$272$	−19.4799	−1.18114
$273$	0	0
$274$	−4.27272	−0.258125
$275$	0	0
$276$	14.6371	0.881048
$277$	23.2177	1.39502	0.697508	−	0.716577i	$-0.254292\pi$
$277$	23.2177	1.39502	0.697508	+	0.716577i	$0.254292\pi$
$278$	−2.73453	−0.164006
$279$	6.26020	0.374789
$280$	0	0
$281$	12.4472	0.742538	0.371269	−	0.928525i	$-0.378923\pi$
$281$	12.4472	0.742538	0.371269	+	0.928525i	$0.378923\pi$
$282$	−1.83488	−0.109265
$283$	8.66775	0.515244	0.257622	−	0.966246i	$-0.417061\pi$
$283$	8.66775	0.515244	0.257622	+	0.966246i	$0.417061\pi$
$284$	7.22967	0.429002
$285$	0	0
$286$	4.35173	0.257323
$287$	0	0
$288$	−3.26020	−0.192109
$289$	13.7752	0.810306
$290$	0	0
$291$	3.04306	0.178387
$292$	4.51583	0.264269
$293$	27.0063	1.57772	0.788862	−	0.614571i	$-0.210671\pi$
$293$	27.0063	1.57772	0.788862	+	0.614571i	$0.210671\pi$
$294$	0	0
$295$	0	0
$296$	8.73000	0.507421
$297$	−3.33039	−0.193249
$298$	−3.26702	−0.189253
$299$	−34.7144	−2.00758
$300$	0	0
$301$	0	0
$302$	−2.56888	−0.147822
$303$	−16.0397	−0.921454
$304$	5.81928	0.333759
$305$	0	0
$306$	1.59401	0.0911233
$307$	−2.24681	−0.128232	−0.0641161	−	0.997942i	$-0.520423\pi$
$307$	−2.24681	−0.128232	−0.0641161	+	0.997942i	$0.520423\pi$
$308$	0	0
$309$	−5.29191	−0.301046
$310$	0	0
$311$	−28.5987	−1.62169	−0.810843	−	0.585264i	$-0.800991\pi$
$311$	−28.5987	−1.62169	−0.810843	+	0.585264i	$0.800991\pi$
$312$	5.11880	0.289795
$313$	13.3285	0.753374	0.376687	−	0.926341i	$-0.377063\pi$
$313$	13.3285	0.753374	0.376687	+	0.926341i	$0.377063\pi$
$314$	−0.961518	−0.0542616
$315$	0	0
$316$	22.8948	1.28794
$317$	29.1925	1.63962	0.819808	−	0.572638i	$-0.194080\pi$
$317$	29.1925	1.63962	0.819808	+	0.572638i	$0.194080\pi$
$318$	−0.212570	−0.0119203
$319$	0.395175	0.0221255
$320$	0	0
$321$	5.16172	0.288099
$322$	0	0
$323$	−9.19357	−0.511544
$324$	−1.91744	−0.106524
$325$	0	0
$326$	−4.52723	−0.250740
$327$	3.24087	0.179220
$328$	−0.0790069	−0.00436243
$329$	0	0
$330$	0	0
$331$	−24.8915	−1.36816	−0.684080	−	0.729407i	$-0.739796\pi$
$331$	−24.8915	−1.36816	−0.684080	+	0.729407i	$0.739796\pi$
$332$	2.34715	0.128817
$333$	7.75572	0.425011
$334$	−6.49211	−0.355232
$335$	0	0
$336$	0	0
$337$	4.72659	0.257474	0.128737	−	0.991679i	$-0.458908\pi$
$337$	4.72659	0.257474	0.128737	+	0.991679i	$0.458908\pi$
$338$	−2.20677	−0.120033
$339$	12.6608	0.687640
$340$	0	0
$341$	−20.8489	−1.12903
$342$	−0.476183	−0.0257490
$343$	0	0
$344$	3.29785	0.177808
$345$	0	0
$346$	2.44988	0.131707
$347$	−14.1583	−0.760058	−0.380029	−	0.924975i	$-0.624086\pi$
$347$	−14.1583	−0.760058	−0.380029	+	0.924975i	$0.624086\pi$
$348$	0.227518	0.0121962
$349$	−5.04930	−0.270283	−0.135141	−	0.990826i	$-0.543149\pi$
$349$	−5.04930	−0.270283	−0.135141	+	0.990826i	$0.543149\pi$
$350$	0	0
$351$	4.54754	0.242730
$352$	10.8578	0.578721
$353$	16.0985	0.856836	0.428418	−	0.903581i	$-0.359071\pi$
$353$	16.0985	0.856836	0.428418	+	0.903581i	$0.359071\pi$
$354$	−0.468387	−0.0248945
$355$	0	0
$356$	−24.9270	−1.32113
$357$	0	0
$358$	3.38499	0.178902
$359$	0.306482	0.0161755	0.00808776	−	0.999967i	$-0.497426\pi$
$359$	0.306482	0.0161755	0.00808776	+	0.999967i	$0.497426\pi$
$360$	0	0
$361$	−16.2536	−0.855451
$362$	2.61179	0.137273
$363$	0.0915259	0.00480386
$364$	0	0
$365$	0	0
$366$	−2.10273	−0.109911
$367$	−23.6497	−1.23451	−0.617253	−	0.786765i	$-0.711754\pi$
$367$	−23.6497	−1.23451	−0.617253	+	0.786765i	$0.711754\pi$
$368$	−26.8052	−1.39732
$369$	−0.0701896	−0.00365393
$370$	0	0
$371$	0	0
$372$	−12.0036	−0.622355
$373$	−18.8377	−0.975381	−0.487691	−	0.873016i	$-0.662161\pi$
$373$	−18.8377	−0.975381	−0.487691	+	0.873016i	$0.662161\pi$
$374$	−5.30867	−0.274505
$375$	0	0
$376$	7.18801	0.370694
$377$	−0.539598	−0.0277907
$378$	0	0
$379$	14.2534	0.732147	0.366074	−	0.930586i	$-0.380702\pi$
$379$	14.2534	0.732147	0.366074	+	0.930586i	$0.380702\pi$
$380$	0	0
$381$	16.5475	0.847756
$382$	−5.88914	−0.301315
$383$	−25.0564	−1.28032	−0.640161	−	0.768240i	$-0.721133\pi$
$383$	−25.0564	−1.28032	−0.640161	+	0.768240i	$0.721133\pi$
$384$	8.26917	0.421984
$385$	0	0
$386$	−2.45470	−0.124941
$387$	2.92981	0.148931
$388$	−5.83488	−0.296221
$389$	−13.4718	−0.683047	−0.341524	−	0.939873i	$-0.610943\pi$
$389$	−13.4718	−0.683047	−0.341524	+	0.939873i	$0.610943\pi$
$390$	0	0
$391$	42.3480	2.14163
$392$	0	0
$393$	5.29785	0.267241
$394$	−3.96770	−0.199890
$395$	0	0
$396$	6.38582	0.320900
$397$	−18.4484	−0.925897	−0.462948	−	0.886385i	$-0.653209\pi$
$397$	−18.4484	−0.925897	−0.462948	+	0.886385i	$0.653209\pi$
$398$	1.08929	0.0546010
$399$	0	0
$400$	0	0
$401$	25.4185	1.26934	0.634670	−	0.772783i	$-0.281136\pi$
$401$	25.4185	1.26934	0.634670	+	0.772783i	$0.281136\pi$
$402$	−0.872379	−0.0435103
$403$	28.4685	1.41812
$404$	30.7550	1.53012
$405$	0	0
$406$	0	0
$407$	−25.8296	−1.28033
$408$	−6.24442	−0.309145
$409$	18.7311	0.926195	0.463097	−	0.886307i	$-0.346738\pi$
$409$	18.7311	0.926195	0.463097	+	0.886307i	$0.346738\pi$
$410$	0	0
$411$	14.8701	0.733490
$412$	10.1469	0.499903
$413$	0	0
$414$	2.19342	0.107801
$415$	0	0
$416$	−14.8259	−0.726900
$417$	9.51685	0.466042
$418$	1.58588	0.0775677
$419$	−2.04745	−0.100024	−0.0500121	−	0.998749i	$-0.515926\pi$
$419$	−2.04745	−0.100024	−0.0500121	+	0.998749i	$0.515926\pi$
$420$	0	0
$421$	−16.0512	−0.782287	−0.391144	−	0.920330i	$-0.627920\pi$
$421$	−16.0512	−0.782287	−0.391144	+	0.920330i	$0.627920\pi$
$422$	0.0328759	0.00160037
$423$	6.38582	0.310489
$424$	0.832729	0.0404409
$425$	0	0
$426$	1.08339	0.0524906
$427$	0	0
$428$	−9.89727	−0.478403
$429$	−15.1451	−0.731212
$430$	0	0
$431$	36.6126	1.76357	0.881784	−	0.471654i	$-0.156343\pi$
$431$	36.6126	1.76357	0.881784	+	0.471654i	$0.156343\pi$
$432$	3.51145	0.168944
$433$	18.0047	0.865252	0.432626	−	0.901574i	$-0.357587\pi$
$433$	18.0047	0.865252	0.432626	+	0.901574i	$0.357587\pi$
$434$	0	0
$435$	0	0
$436$	−6.21417	−0.297605
$437$	−12.6508	−0.605168
$438$	0.676716	0.0323347
$439$	19.3746	0.924700	0.462350	−	0.886697i	$-0.347006\pi$
$439$	19.3746	0.924700	0.462350	+	0.886697i	$0.347006\pi$
$440$	0	0
$441$	0	0
$442$	7.24881	0.344791
$443$	−16.4424	−0.781203	−0.390602	−	0.920560i	$-0.627733\pi$
$443$	−16.4424	−0.781203	−0.390602	+	0.920560i	$0.627733\pi$
$444$	−14.8711	−0.705752
$445$	0	0
$446$	2.26039	0.107033
$447$	11.3700	0.537785
$448$	0	0
$449$	−32.7245	−1.54436	−0.772182	−	0.635401i	$-0.780835\pi$
$449$	−32.7245	−1.54436	−0.772182	+	0.635401i	$0.780835\pi$
$450$	0	0
$451$	0.233759	0.0110073
$452$	−24.2763	−1.14186
$453$	8.94033	0.420053
$454$	2.57989	0.121080
$455$	0	0
$456$	1.86542	0.0873561
$457$	−37.4303	−1.75092	−0.875459	−	0.483293i	$-0.839441\pi$
$457$	−37.4303	−1.75092	−0.875459	+	0.483293i	$0.839441\pi$
$458$	−1.46142	−0.0682878
$459$	−5.54754	−0.258937
$460$	0	0
$461$	−28.3604	−1.32088	−0.660438	−	0.750881i	$-0.729629\pi$
$461$	−28.3604	−1.32088	−0.660438	+	0.750881i	$0.729629\pi$
$462$	0	0
$463$	−7.20833	−0.334999	−0.167500	−	0.985872i	$-0.553569\pi$
$463$	−7.20833	−0.334999	−0.167500	+	0.985872i	$0.553569\pi$
$464$	−0.416658	−0.0193429
$465$	0	0
$466$	−6.26917	−0.290414
$467$	−12.0790	−0.558950	−0.279475	−	0.960153i	$-0.590160\pi$
$467$	−12.0790	−0.558950	−0.279475	+	0.960153i	$0.590160\pi$
$468$	−8.71963	−0.403065
$469$	0	0
$470$	0	0
$471$	3.34632	0.154190
$472$	1.83488	0.0844570
$473$	−9.75742	−0.448647
$474$	3.43088	0.157586
$475$	0	0
$476$	0	0
$477$	0.739795	0.0338729
$478$	−2.14038	−0.0978987
$479$	−20.1417	−0.920298	−0.460149	−	0.887842i	$-0.652204\pi$
$479$	−20.1417	−0.920298	−0.460149	+	0.887842i	$0.652204\pi$
$480$	0	0
$481$	35.2695	1.60815
$482$	−6.94657	−0.316408
$483$	0	0
$484$	−0.175495	−0.00797705
$485$	0	0
$486$	−0.287336	−0.0130338
$487$	−39.9396	−1.80984	−0.904919	−	0.425584i	$-0.860069\pi$
$487$	−39.9396	−1.80984	−0.904919	+	0.425584i	$0.860069\pi$
$488$	8.23731	0.372886
$489$	15.7559	0.712505
$490$	0	0
$491$	−31.5989	−1.42604	−0.713019	−	0.701145i	$-0.752673\pi$
$491$	−31.5989	−1.42604	−0.713019	+	0.701145i	$0.752673\pi$
$492$	0.134584	0.00606753
$493$	0.658255	0.0296463
$494$	−2.16546	−0.0974286
$495$	0	0
$496$	21.9824	0.987037
$497$	0	0
$498$	0.351730	0.0157614
$499$	39.7859	1.78106	0.890530	−	0.454924i	$-0.150334\pi$
$499$	39.7859	1.78106	0.890530	+	0.454924i	$0.150334\pi$
$500$	0	0
$501$	22.5942	1.00943
$502$	1.72459	0.0769722
$503$	12.8734	0.573995	0.286997	−	0.957931i	$-0.407343\pi$
$503$	12.8734	0.573995	0.286997	+	0.957931i	$0.407343\pi$
$504$	0	0
$505$	0	0
$506$	−7.30496	−0.324745
$507$	7.68012	0.341086
$508$	−31.7289	−1.40774
$509$	−33.3961	−1.48026	−0.740128	−	0.672466i	$-0.765235\pi$
$509$	−33.3961	−1.48026	−0.740128	+	0.672466i	$0.765235\pi$
$510$	0	0
$511$	0	0
$512$	−19.3531	−0.855296
$513$	1.65723	0.0731686
$514$	2.10731	0.0929493
$515$	0	0
$516$	−5.61773	−0.247307
$517$	−21.2673	−0.935335
$518$	0	0
$519$	−8.52620	−0.374259
$520$	0	0
$521$	−13.5518	−0.593714	−0.296857	−	0.954922i	$-0.595939\pi$
$521$	−13.5518	−0.593714	−0.296857	+	0.954922i	$0.595939\pi$
$522$	0.0340944	0.00149227
$523$	−20.2133	−0.883866	−0.441933	−	0.897048i	$-0.645707\pi$
$523$	−20.2133	−0.883866	−0.441933	+	0.897048i	$0.645707\pi$
$524$	−10.1583	−0.443768
$525$	0	0
$526$	−2.72616	−0.118866
$527$	−34.7287	−1.51281
$528$	−11.6945	−0.508938
$529$	35.2727	1.53360
$530$	0	0
$531$	1.63010	0.0707404
$532$	0	0
$533$	−0.319190	−0.0138257
$534$	−3.73541	−0.161647
$535$	0	0
$536$	3.41749	0.147613
$537$	−11.7806	−0.508371
$538$	4.61130	0.198807
$539$	0	0
$540$	0	0
$541$	−32.9708	−1.41752	−0.708762	−	0.705447i	$-0.750746\pi$
$541$	−32.9708	−1.41752	−0.708762	+	0.705447i	$0.750746\pi$
$542$	6.94461	0.298297
$543$	−9.08967	−0.390075
$544$	18.0861	0.775436
$545$	0	0
$546$	0	0
$547$	−24.6221	−1.05277	−0.526383	−	0.850248i	$-0.676452\pi$
$547$	−24.6221	−1.05277	−0.526383	+	0.850248i	$0.676452\pi$
$548$	−28.5126	−1.21800
$549$	7.31802	0.312326
$550$	0	0
$551$	−0.196643	−0.00837725
$552$	−8.59260	−0.365725
$553$	0	0
$554$	−6.67127	−0.283435
$555$	0	0
$556$	−18.2480	−0.773887
$557$	23.9942	1.01667	0.508334	−	0.861160i	$-0.330262\pi$
$557$	23.9942	1.01667	0.508334	+	0.861160i	$0.330262\pi$
$558$	−1.79878	−0.0761485
$559$	13.3234	0.563521
$560$	0	0
$561$	18.4755	0.780036
$562$	−3.57653	−0.150867
$563$	−0.226496	−0.00954567	−0.00477284	−	0.999989i	$-0.501519\pi$
$563$	−0.226496	−0.00954567	−0.00477284	+	0.999989i	$0.501519\pi$
$564$	−12.2444	−0.515583
$565$	0	0
$566$	−2.49056	−0.104686
$567$	0	0
$568$	−4.24413	−0.178080
$569$	−12.8055	−0.536835	−0.268417	−	0.963303i	$-0.586501\pi$
$569$	−12.8055	−0.536835	−0.268417	+	0.963303i	$0.586501\pi$
$570$	0	0
$571$	−1.56030	−0.0652965	−0.0326482	−	0.999467i	$-0.510394\pi$
$571$	−1.56030	−0.0652965	−0.0326482	+	0.999467i	$0.510394\pi$
$572$	29.0398	1.21421
$573$	20.4957	0.856219
$574$	0	0
$575$	0	0
$576$	−6.08612	−0.253588
$577$	43.6497	1.81716	0.908581	−	0.417709i	$-0.137167\pi$
$577$	43.6497	1.81716	0.908581	+	0.417709i	$0.137167\pi$
$578$	−3.95811	−0.164636
$579$	8.54296	0.355033
$580$	0	0
$581$	0	0
$582$	−0.874380	−0.0362442
$583$	−2.46381	−0.102041
$584$	−2.65099	−0.109699
$585$	0	0
$586$	−7.75987	−0.320557
$587$	13.6961	0.565297	0.282648	−	0.959224i	$-0.408787\pi$
$587$	13.6961	0.565297	0.282648	+	0.959224i	$0.408787\pi$
$588$	0	0
$589$	10.3746	0.427479
$590$	0	0
$591$	13.8086	0.568009
$592$	27.2338	1.11930
$593$	9.15490	0.375947	0.187973	−	0.982174i	$-0.439808\pi$
$593$	9.15490	0.375947	0.187973	+	0.982174i	$0.439808\pi$
$594$	0.956942	0.0392638
$595$	0	0
$596$	−21.8014	−0.893018
$597$	−3.79099	−0.155155
$598$	9.97468	0.407895
$599$	40.2736	1.64553	0.822767	−	0.568379i	$-0.192429\pi$
$599$	40.2736	1.64553	0.822767	+	0.568379i	$0.192429\pi$
$600$	0	0
$601$	8.82450	0.359959	0.179980	−	0.983670i	$-0.442397\pi$
$601$	8.82450	0.359959	0.179980	+	0.983670i	$0.442397\pi$
$602$	0	0
$603$	3.03610	0.123639
$604$	−17.1425	−0.697520
$605$	0	0
$606$	4.60877	0.187218
$607$	−28.1064	−1.14080	−0.570402	−	0.821365i	$-0.693213\pi$
$607$	−28.1064	−1.14080	−0.570402	+	0.821365i	$0.693213\pi$
$608$	−5.40292	−0.219117
$609$	0	0
$610$	0	0
$611$	29.0398	1.17482
$612$	10.6371	0.429978
$613$	−2.11339	−0.0853592	−0.0426796	−	0.999089i	$-0.513589\pi$
$613$	−2.11339	−0.0853592	−0.0426796	+	0.999089i	$0.513589\pi$
$614$	0.645589	0.0260539
$615$	0	0
$616$	0	0
$617$	18.0390	0.726221	0.363111	−	0.931746i	$-0.381715\pi$
$617$	18.0390	0.726221	0.363111	+	0.931746i	$0.381715\pi$
$618$	1.52056	0.0611657
$619$	14.6379	0.588347	0.294173	−	0.955752i	$-0.404956\pi$
$619$	14.6379	0.588347	0.294173	+	0.955752i	$0.404956\pi$
$620$	0	0
$621$	−7.63366	−0.306328
$622$	8.21744	0.329489
$623$	0	0
$624$	15.9684	0.639249
$625$	0	0
$626$	−3.82977	−0.153068
$627$	−5.51924	−0.220417
$628$	−6.41636	−0.256041
$629$	−43.0252	−1.71553
$630$	0	0
$631$	12.1251	0.482692	0.241346	−	0.970439i	$-0.422411\pi$
$631$	12.1251	0.482692	0.241346	+	0.970439i	$0.422411\pi$
$632$	−13.4403	−0.534625
$633$	−0.114416	−0.00454764
$634$	−8.38806	−0.333133
$635$	0	0
$636$	−1.41851	−0.0562477
$637$	0	0
$638$	−0.113548	−0.00449540
$639$	−3.77048	−0.149158
$640$	0	0
$641$	−13.0405	−0.515068	−0.257534	−	0.966269i	$-0.582910\pi$
$641$	−13.0405	−0.515068	−0.257534	+	0.966269i	$0.582910\pi$
$642$	−1.48315	−0.0585351
$643$	−27.0185	−1.06550	−0.532752	−	0.846271i	$-0.678842\pi$
$643$	−27.0185	−1.06550	−0.532752	+	0.846271i	$0.678842\pi$
$644$	0	0
$645$	0	0
$646$	2.64164	0.103934
$647$	34.9790	1.37516	0.687582	−	0.726106i	$-0.258672\pi$
$647$	34.9790	1.37516	0.687582	+	0.726106i	$0.258672\pi$
$648$	1.12562	0.0442185
$649$	−5.42888	−0.213102
$650$	0	0
$651$	0	0
$652$	−30.2109	−1.18315
$653$	32.3510	1.26599	0.632997	−	0.774154i	$-0.281824\pi$
$653$	32.3510	1.26599	0.632997	+	0.774154i	$0.281824\pi$
$654$	−0.931218	−0.0364135
$655$	0	0
$656$	−0.246467	−0.00962292
$657$	−2.35514	−0.0918827
$658$	0	0
$659$	8.54282	0.332781	0.166390	−	0.986060i	$-0.446789\pi$
$659$	8.54282	0.332781	0.166390	+	0.986060i	$0.446789\pi$
$660$	0	0
$661$	30.3431	1.18021	0.590104	−	0.807327i	$-0.299087\pi$
$661$	30.3431	1.18021	0.590104	+	0.807327i	$0.299087\pi$
$662$	7.15221	0.277979
$663$	−25.2277	−0.979761
$664$	−1.37788	−0.0534722
$665$	0	0
$666$	−2.22850	−0.0863525
$667$	0.905788	0.0350722
$668$	−43.3229	−1.67621
$669$	−7.86673	−0.304145
$670$	0	0
$671$	−24.3719	−0.940867
$672$	0	0
$673$	−40.5075	−1.56145	−0.780725	−	0.624875i	$-0.785150\pi$
$673$	−40.5075	−1.56145	−0.780725	+	0.624875i	$0.785150\pi$
$674$	−1.35812	−0.0523128
$675$	0	0
$676$	−14.7262	−0.566391
$677$	−23.4211	−0.900146	−0.450073	−	0.892992i	$-0.648602\pi$
$677$	−23.4211	−0.900146	−0.450073	+	0.892992i	$0.648602\pi$
$678$	−3.63790	−0.139713
$679$	0	0
$680$	0	0
$681$	−8.97866	−0.344063
$682$	5.99065	0.229394
$683$	51.0247	1.95241	0.976204	−	0.216856i	$-0.0695801\pi$
$683$	51.0247	1.95241	0.976204	+	0.216856i	$0.0695801\pi$
$684$	−3.17764	−0.121500
$685$	0	0
$686$	0	0
$687$	5.08612	0.194047
$688$	10.2879	0.392221
$689$	3.36425	0.128168
$690$	0	0
$691$	−24.6115	−0.936265	−0.468133	−	0.883658i	$-0.655073\pi$
$691$	−24.6115	−0.936265	−0.468133	+	0.883658i	$0.655073\pi$
$692$	16.3485	0.621476
$693$	0	0
$694$	4.06819	0.154426
$695$	0	0
$696$	−0.133563	−0.00506269
$697$	0.389380	0.0147488
$698$	1.45084	0.0549153
$699$	21.8183	0.825243
$700$	0	0
$701$	25.7244	0.971595	0.485798	−	0.874071i	$-0.338529\pi$
$701$	25.7244	0.971595	0.485798	+	0.874071i	$0.338529\pi$
$702$	−1.30667	−0.0493171
$703$	12.8530	0.484762
$704$	20.2692	0.763923
$705$	0	0
$706$	−4.62567	−0.174089
$707$	0	0
$708$	−3.12562	−0.117468
$709$	11.7096	0.439765	0.219882	−	0.975526i	$-0.429433\pi$
$709$	11.7096	0.439765	0.219882	+	0.975526i	$0.429433\pi$
$710$	0	0
$711$	−11.9403	−0.447797
$712$	14.6332	0.548403
$713$	−47.7883	−1.78968
$714$	0	0
$715$	0	0
$716$	22.5886	0.844176
$717$	7.44905	0.278190
$718$	−0.0880634	−0.00328650
$719$	12.3207	0.459486	0.229743	−	0.973251i	$-0.426211\pi$
$719$	12.3207	0.459486	0.229743	+	0.973251i	$0.426211\pi$
$720$	0	0
$721$	0	0
$722$	4.67023	0.173808
$723$	24.1758	0.899107
$724$	17.4289	0.647739
$725$	0	0
$726$	−0.0262987	−0.000976035 0
$727$	17.6540	0.654751	0.327376	−	0.944894i	$-0.393836\pi$
$727$	17.6540	0.654751	0.327376	+	0.944894i	$0.393836\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−16.2532	−0.601148
$732$	−14.0319	−0.518632
$733$	−10.0359	−0.370685	−0.185342	−	0.982674i	$-0.559339\pi$
$733$	−10.0359	−0.370685	−0.185342	+	0.982674i	$0.559339\pi$
$734$	6.79541	0.250823
$735$	0	0
$736$	24.8873	0.917357
$737$	−10.1114	−0.372458
$738$	0.0201680	0.000742394 0
$739$	−31.0720	−1.14300	−0.571502	−	0.820601i	$-0.693639\pi$
$739$	−31.0720	−1.14300	−0.571502	+	0.820601i	$0.693639\pi$
$740$	0	0
$741$	7.53634	0.276854
$742$	0	0
$743$	−4.04189	−0.148283	−0.0741413	−	0.997248i	$-0.523622\pi$
$743$	−4.04189	−0.148283	−0.0741413	+	0.997248i	$0.523622\pi$
$744$	7.04661	0.258341
$745$	0	0
$746$	5.41276	0.198175
$747$	−1.22411	−0.0447878
$748$	−35.4256	−1.29529
$749$	0	0
$750$	0	0
$751$	14.6945	0.536210	0.268105	−	0.963390i	$-0.413603\pi$
$751$	14.6945	0.536210	0.268105	+	0.963390i	$0.413603\pi$
$752$	22.4235	0.817700
$753$	−6.00200	−0.218725
$754$	0.155046	0.00564644
$755$	0	0
$756$	0	0
$757$	−29.6087	−1.07615	−0.538073	−	0.842898i	$-0.680847\pi$
$757$	−29.6087	−1.07615	−0.538073	+	0.842898i	$0.680847\pi$
$758$	−4.09551	−0.148756
$759$	25.4231	0.922800
$760$	0	0
$761$	14.2522	0.516643	0.258321	−	0.966059i	$-0.416831\pi$
$761$	14.2522	0.516643	0.258321	+	0.966059i	$0.416831\pi$
$762$	−4.75470	−0.172245
$763$	0	0
$764$	−39.2992	−1.42179
$765$	0	0
$766$	7.19960	0.260132
$767$	7.41296	0.267666
$768$	9.79621	0.353490
$769$	20.6367	0.744178	0.372089	−	0.928197i	$-0.378642\pi$
$769$	20.6367	0.744178	0.372089	+	0.928197i	$0.378642\pi$
$770$	0	0
$771$	−7.33395	−0.264126
$772$	−16.3806	−0.589551
$773$	42.5599	1.53077	0.765387	−	0.643571i	$-0.222548\pi$
$773$	42.5599	1.53077	0.765387	+	0.643571i	$0.222548\pi$
$774$	−0.841839	−0.0302593
$775$	0	0
$776$	3.42533	0.122962
$777$	0	0
$778$	3.87093	0.138780
$779$	−0.116321	−0.00416762
$780$	0	0
$781$	12.5572	0.449332
$782$	−12.1681	−0.435131
$783$	−0.118657	−0.00424046
$784$	0	0
$785$	0	0
$786$	−1.52226	−0.0542973
$787$	25.1991	0.898252	0.449126	−	0.893468i	$-0.351735\pi$
$787$	25.1991	0.898252	0.449126	+	0.893468i	$0.351735\pi$
$788$	−26.4771	−0.943207
$789$	9.48772	0.337772
$790$	0	0
$791$	0	0
$792$	−3.74876	−0.133206
$793$	33.2790	1.18177
$794$	5.30088	0.188121
$795$	0	0
$796$	7.26898	0.257642
$797$	−51.7211	−1.83205	−0.916027	−	0.401116i	$-0.868623\pi$
$797$	−51.7211	−1.83205	−0.916027	+	0.401116i	$0.868623\pi$
$798$	0	0
$799$	−35.4256	−1.25327
$800$	0	0
$801$	13.0001	0.459338
$802$	−7.30365	−0.257901
$803$	7.84354	0.276793
$804$	−5.82152	−0.205309
$805$	0	0
$806$	−8.18003	−0.288129
$807$	−16.0485	−0.564933
$808$	−18.0546	−0.635157
$809$	−51.7779	−1.82041	−0.910207	−	0.414153i	$-0.864078\pi$
$809$	−51.7779	−1.82041	−0.910207	+	0.414153i	$0.864078\pi$
$810$	0	0
$811$	−12.0263	−0.422299	−0.211149	−	0.977454i	$-0.567721\pi$
$811$	−12.0263	−0.422299	−0.211149	+	0.977454i	$0.567721\pi$
$812$	0	0
$813$	−24.1690	−0.847643
$814$	7.42177	0.260133
$815$	0	0
$816$	−19.4799	−0.681932
$817$	4.85538	0.169868
$818$	−5.38212	−0.188182
$819$	0	0
$820$	0	0
$821$	8.07934	0.281971	0.140985	−	0.990012i	$-0.454973\pi$
$821$	8.07934	0.281971	0.140985	+	0.990012i	$0.454973\pi$
$822$	−4.27272	−0.149028
$823$	−5.11525	−0.178306	−0.0891532	−	0.996018i	$-0.528416\pi$
$823$	−5.11525	−0.178306	−0.0891532	+	0.996018i	$0.528416\pi$
$824$	−5.95668	−0.207511
$825$	0	0
$826$	0	0
$827$	0.705254	0.0245241	0.0122620	−	0.999925i	$-0.496097\pi$
$827$	0.705254	0.0245241	0.0122620	+	0.999925i	$0.496097\pi$
$828$	14.6371	0.508673
$829$	−24.9581	−0.866830	−0.433415	−	0.901195i	$-0.642691\pi$
$829$	−24.9581	−0.866830	−0.433415	+	0.901195i	$0.642691\pi$
$830$	0	0
$831$	23.2177	0.805412
$832$	−27.6769	−0.959523
$833$	0	0
$834$	−2.73453	−0.0946891
$835$	0	0
$836$	10.5828	0.366014
$837$	6.26020	0.216384
$838$	0.588305	0.0203226
$839$	11.6389	0.401819	0.200909	−	0.979610i	$-0.435610\pi$
$839$	11.6389	0.401819	0.200909	+	0.979610i	$0.435610\pi$
$840$	0	0
$841$	−28.9859	−0.999514
$842$	4.61208	0.158943
$843$	12.4472	0.428704
$844$	0.219386	0.00755158
$845$	0	0
$846$	−1.83488	−0.0630843
$847$	0	0
$848$	2.59775	0.0892071
$849$	8.66775	0.297476
$850$	0	0
$851$	−59.2045	−2.02951
$852$	7.22967	0.247684
$853$	32.5996	1.11619	0.558094	−	0.829778i	$-0.311533\pi$
$853$	32.5996	1.11619	0.558094	+	0.829778i	$0.311533\pi$
$854$	0	0
$855$	0	0
$856$	5.81013	0.198586
$857$	31.6913	1.08255	0.541277	−	0.840845i	$-0.317941\pi$
$857$	31.6913	1.08255	0.541277	+	0.840845i	$0.317941\pi$
$858$	4.35173	0.148566
$859$	43.6912	1.49073	0.745363	−	0.666659i	$-0.232276\pi$
$859$	43.6912	1.49073	0.745363	+	0.666659i	$0.232276\pi$
$860$	0	0
$861$	0	0
$862$	−10.5201	−0.358317
$863$	−38.7985	−1.32072	−0.660358	−	0.750951i	$-0.729595\pi$
$863$	−38.7985	−1.32072	−0.660358	+	0.750951i	$0.729595\pi$
$864$	−3.26020	−0.110914
$865$	0	0
$866$	−5.17340	−0.175799
$867$	13.7752	0.467830
$868$	0	0
$869$	39.7660	1.34897
$870$	0	0
$871$	13.8068	0.467824
$872$	3.64799	0.123536
$873$	3.04306	0.102992
$874$	3.63501	0.122956
$875$	0	0
$876$	4.51583	0.152576
$877$	4.30073	0.145225	0.0726127	−	0.997360i	$-0.476866\pi$
$877$	4.30073	0.145225	0.0726127	+	0.997360i	$0.476866\pi$
$878$	−5.56702	−0.187878
$879$	27.0063	0.910899
$880$	0	0
$881$	1.29308	0.0435650	0.0217825	−	0.999763i	$-0.493066\pi$
$881$	1.29308	0.0435650	0.0217825	+	0.999763i	$0.493066\pi$
$882$	0	0
$883$	1.49533	0.0503218	0.0251609	−	0.999683i	$-0.491990\pi$
$883$	1.49533	0.0503218	0.0251609	+	0.999683i	$0.491990\pi$
$884$	48.3725	1.62694
$885$	0	0
$886$	4.72450	0.158723
$887$	−10.0917	−0.338845	−0.169423	−	0.985543i	$-0.554190\pi$
$887$	−10.0917	−0.338845	−0.169423	+	0.985543i	$0.554190\pi$
$888$	8.73000	0.292960
$889$	0	0
$890$	0	0
$891$	−3.33039	−0.111572
$892$	15.0840	0.505049
$893$	10.5828	0.354140
$894$	−3.26702	−0.109266
$895$	0	0
$896$	0	0
$897$	−34.7144	−1.15908
$898$	9.40292	0.313780
$899$	−0.742818	−0.0247744
$900$	0	0
$901$	−4.10404	−0.136726
$902$	−0.0671674	−0.00223643
$903$	0	0
$904$	14.2512	0.473989
$905$	0	0
$906$	−2.56888	−0.0853452
$907$	4.26698	0.141683	0.0708414	−	0.997488i	$-0.477432\pi$
$907$	4.26698	0.141683	0.0708414	+	0.997488i	$0.477432\pi$
$908$	17.2160	0.571334
$909$	−16.0397	−0.532002
$910$	0	0
$911$	−28.5451	−0.945742	−0.472871	−	0.881132i	$-0.656782\pi$
$911$	−28.5451	−0.945742	−0.472871	+	0.881132i	$0.656782\pi$
$912$	5.81928	0.192696
$913$	4.07677	0.134921
$914$	10.7551	0.355746
$915$	0	0
$916$	−9.75231	−0.322226
$917$	0	0
$918$	1.59401	0.0526101
$919$	−46.5643	−1.53602	−0.768008	−	0.640440i	$-0.778752\pi$
$919$	−46.5643	−1.53602	−0.768008	+	0.640440i	$0.778752\pi$
$920$	0	0
$921$	−2.24681	−0.0740349
$922$	8.14896	0.268372
$923$	−17.1464	−0.564381
$924$	0	0
$925$	0	0
$926$	2.07121	0.0680642
$927$	−5.29191	−0.173809
$928$	0.386846	0.0126989
$929$	−7.39547	−0.242638	−0.121319	−	0.992614i	$-0.538712\pi$
$929$	−7.39547	−0.242638	−0.121319	+	0.992614i	$0.538712\pi$
$930$	0	0
$931$	0	0
$932$	−41.8352	−1.37036
$933$	−28.5987	−0.936280
$934$	3.47073	0.113566
$935$	0	0
$936$	5.11880	0.167313
$937$	−44.1988	−1.44391	−0.721956	−	0.691939i	$-0.756757\pi$
$937$	−44.1988	−1.44391	−0.721956	+	0.691939i	$0.756757\pi$
$938$	0	0
$939$	13.3285	0.434960
$940$	0	0
$941$	−36.0359	−1.17474	−0.587369	−	0.809319i	$-0.699836\pi$
$941$	−36.0359	−1.17474	−0.587369	+	0.809319i	$0.699836\pi$
$942$	−0.961518	−0.0313280
$943$	0.535804	0.0174482
$944$	5.72401	0.186301
$945$	0	0
$946$	2.80366	0.0911548
$947$	−34.2341	−1.11246	−0.556229	−	0.831029i	$-0.687752\pi$
$947$	−34.2341	−1.11246	−0.556229	+	0.831029i	$0.687752\pi$
$948$	22.8948	0.743590
$949$	−10.7101	−0.347664
$950$	0	0
$951$	29.1925	0.946633
$952$	0	0
$953$	30.9689	1.00318	0.501591	−	0.865105i	$-0.332748\pi$
$953$	30.9689	1.00318	0.501591	+	0.865105i	$0.332748\pi$
$954$	−0.212570	−0.00688220
$955$	0	0
$956$	−14.2831	−0.461948
$957$	0.395175	0.0127742
$958$	5.78743	0.186983
$959$	0	0
$960$	0	0
$961$	8.19016	0.264199
$962$	−10.1342	−0.326739
$963$	5.16172	0.166334
$964$	−46.3556	−1.49301
$965$	0	0
$966$	0	0
$967$	21.0270	0.676184	0.338092	−	0.941113i	$-0.390218\pi$
$967$	21.0270	0.676184	0.338092	+	0.941113i	$0.390218\pi$
$968$	0.103023	0.00331130
$969$	−9.19357	−0.295340
$970$	0	0
$971$	−44.6937	−1.43429	−0.717144	−	0.696925i	$-0.754551\pi$
$971$	−44.6937	−1.43429	−0.717144	+	0.696925i	$0.754551\pi$
$972$	−1.91744	−0.0615019
$973$	0	0
$974$	11.4761	0.367718
$975$	0	0
$976$	25.6968	0.822536
$977$	−14.4372	−0.461886	−0.230943	−	0.972967i	$-0.574181\pi$
$977$	−14.4372	−0.461886	−0.230943	+	0.972967i	$0.574181\pi$
$978$	−4.52723	−0.144765
$979$	−43.2956	−1.38373
$980$	0	0
$981$	3.24087	0.103473
$982$	9.07949	0.289738
$983$	16.7880	0.535455	0.267727	−	0.963495i	$-0.413727\pi$
$983$	16.7880	0.535455	0.267727	+	0.963495i	$0.413727\pi$
$984$	−0.0790069	−0.00251865
$985$	0	0
$986$	−0.189140	−0.00602345
$987$	0	0
$988$	−14.4505	−0.459730
$989$	−22.3652	−0.711171
$990$	0	0
$991$	40.1078	1.27407	0.637033	−	0.770837i	$-0.280162\pi$
$991$	40.1078	1.27407	0.637033	+	0.770837i	$0.280162\pi$
$992$	−20.4095	−0.648004
$993$	−24.8915	−0.789907
$994$	0	0
$995$	0	0
$996$	2.34715	0.0743724
$997$	−51.3204	−1.62533	−0.812666	−	0.582730i	$-0.801985\pi$
$997$	−51.3204	−1.62533	−0.812666	+	0.582730i	$0.801985\pi$
$998$	−11.4319	−0.361871
$999$	7.75572	0.245380

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3675.2.a.cb.1.2		4
5.2	odd	4		735.2.d.e.589.4		8
5.3	odd	4		735.2.d.e.589.5		8
5.4	even	2		3675.2.a.bn.1.3		4
7.3	odd	6		525.2.i.h.226.3		8
7.5	odd	6		525.2.i.h.151.3		8
7.6	odd	2		3675.2.a.bz.1.2		4
15.2	even	4		2205.2.d.o.1324.5		8
15.8	even	4		2205.2.d.o.1324.4		8
35.2	odd	12		735.2.q.g.214.4		16
35.3	even	12		105.2.q.a.79.4	yes	16
35.12	even	12		105.2.q.a.4.4	✓	16
35.13	even	4		735.2.d.d.589.5		8
35.17	even	12		105.2.q.a.79.5	yes	16
35.18	odd	12		735.2.q.g.79.4		16
35.19	odd	6		525.2.i.k.151.2		8
35.23	odd	12		735.2.q.g.214.5		16
35.24	odd	6		525.2.i.k.226.2		8
35.27	even	4		735.2.d.d.589.4		8
35.32	odd	12		735.2.q.g.79.5		16
35.33	even	12		105.2.q.a.4.5	yes	16
35.34	odd	2		3675.2.a.bp.1.3		4
105.17	odd	12		315.2.bf.b.289.4		16
105.38	odd	12		315.2.bf.b.289.5		16
105.47	odd	12		315.2.bf.b.109.5		16
105.62	odd	4		2205.2.d.s.1324.5		8
105.68	odd	12		315.2.bf.b.109.4		16
105.83	odd	4		2205.2.d.s.1324.4		8
140.3	odd	12		1680.2.di.d.289.7		16
140.47	odd	12		1680.2.di.d.529.7		16
140.87	odd	12		1680.2.di.d.289.3		16
140.103	odd	12		1680.2.di.d.529.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.q.a.4.4	✓	16	35.12	even	12
105.2.q.a.4.5	yes	16	35.33	even	12
105.2.q.a.79.4	yes	16	35.3	even	12
105.2.q.a.79.5	yes	16	35.17	even	12
315.2.bf.b.109.4		16	105.68	odd	12
315.2.bf.b.109.5		16	105.47	odd	12
315.2.bf.b.289.4		16	105.17	odd	12
315.2.bf.b.289.5		16	105.38	odd	12
525.2.i.h.151.3		8	7.5	odd	6
525.2.i.h.226.3		8	7.3	odd	6
525.2.i.k.151.2		8	35.19	odd	6
525.2.i.k.226.2		8	35.24	odd	6
735.2.d.d.589.4		8	35.27	even	4
735.2.d.d.589.5		8	35.13	even	4
735.2.d.e.589.4		8	5.2	odd	4
735.2.d.e.589.5		8	5.3	odd	4
735.2.q.g.79.4		16	35.18	odd	12
735.2.q.g.79.5		16	35.32	odd	12
735.2.q.g.214.4		16	35.2	odd	12
735.2.q.g.214.5		16	35.23	odd	12
1680.2.di.d.289.3		16	140.87	odd	12
1680.2.di.d.289.7		16	140.3	odd	12
1680.2.di.d.529.3		16	140.103	odd	12
1680.2.di.d.529.7		16	140.47	odd	12
2205.2.d.o.1324.4		8	15.8	even	4
2205.2.d.o.1324.5		8	15.2	even	4
2205.2.d.s.1324.4		8	105.83	odd	4
2205.2.d.s.1324.5		8	105.62	odd	4
3675.2.a.bn.1.3		4	5.4	even	2
3675.2.a.bp.1.3		4	35.34	odd	2
3675.2.a.bz.1.2		4	7.6	odd	2
3675.2.a.cb.1.2		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)

\(f(q)\)	\(=\)	\(q-0.287336 q^{2} +1.00000 q^{3} -1.91744 q^{4} -0.287336 q^{6} +1.12562 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.287336 q^{2} +1.00000 q^{3} -1.91744 q^{4} -0.287336 q^{6} +1.12562 q^{8} +1.00000 q^{9} -3.33039 q^{11} -1.91744 q^{12} +4.54754 q^{13} +3.51145 q^{16} -5.54754 q^{17} -0.287336 q^{18} +1.65723 q^{19} +0.956942 q^{22} -7.63366 q^{23} +1.12562 q^{24} -1.30667 q^{26} +1.00000 q^{27} -0.118657 q^{29} +6.26020 q^{31} -3.26020 q^{32} -3.33039 q^{33} +1.59401 q^{34} -1.91744 q^{36} +7.75572 q^{37} -0.476183 q^{38} +4.54754 q^{39} -0.0701896 q^{41} +2.92981 q^{43} +6.38582 q^{44} +2.19342 q^{46} +6.38582 q^{47} +3.51145 q^{48} -5.54754 q^{51} -8.71963 q^{52} +0.739795 q^{53} -0.287336 q^{54} +1.65723 q^{57} +0.0340944 q^{58} +1.63010 q^{59} +7.31802 q^{61} -1.79878 q^{62} -6.08612 q^{64} +0.956942 q^{66} +3.03610 q^{67} +10.6371 q^{68} -7.63366 q^{69} -3.77048 q^{71} +1.12562 q^{72} -2.35514 q^{73} -2.22850 q^{74} -3.17764 q^{76} -1.30667 q^{78} -11.9403 q^{79} +1.00000 q^{81} +0.0201680 q^{82} -1.22411 q^{83} -0.841839 q^{86} -0.118657 q^{87} -3.74876 q^{88} +13.0001 q^{89} +14.6371 q^{92} +6.26020 q^{93} -1.83488 q^{94} -3.26020 q^{96} +3.04306 q^{97} -3.33039 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 4 q^{3} + 4 q^{4} + 2 q^{6} + 6 q^{8} + 4 q^{9}+O(q^{10}) \) 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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