LMFDB - Embedded newform 3675.2.a.bx.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3675,2,Mod(1,3675)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [4,1,4,7,0,1,0,3,4,0,8,7,7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$29.3450227428$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.88404.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 7x^{2} + 5x + 4 $ x^4 - x^3 - 7x^2 + 5x + 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 525)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.70285$ of defining polynomial
Character	$\chi$	$=$	3675.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.70285 q^{2} +1.00000 q^{3} +5.30542 q^{4} +2.70285 q^{6} +8.93406 q^{8} +1.00000 q^{9} +2.82550 q^{11} +5.30542 q^{12} +4.47992 q^{13} +13.5366 q^{16} -4.23121 q^{17} +2.70285 q^{18} -2.10029 q^{19} +7.63692 q^{22} -6.61084 q^{23} +8.93406 q^{24} +12.1086 q^{26} +1.00000 q^{27} +4.00000 q^{29} -9.33150 q^{31} +18.7194 q^{32} +2.82550 q^{33} -11.4363 q^{34} +5.30542 q^{36} +4.20513 q^{37} -5.67677 q^{38} +4.47992 q^{39} -2.37963 q^{41} -3.13092 q^{43} +14.9905 q^{44} -17.8681 q^{46} -7.78534 q^{47} +13.5366 q^{48} -4.23121 q^{51} +23.7678 q^{52} +2.58021 q^{53} +2.70285 q^{54} -2.10029 q^{57} +10.8114 q^{58} +3.78534 q^{59} +5.00000 q^{61} -25.2217 q^{62} +23.5225 q^{64} +7.63692 q^{66} -3.30542 q^{67} -22.4483 q^{68} -6.61084 q^{69} -11.4223 q^{71} +8.93406 q^{72} -2.65100 q^{73} +11.3658 q^{74} -11.1429 q^{76} +12.1086 q^{78} +12.6369 q^{79} +1.00000 q^{81} -6.43179 q^{82} +3.02608 q^{83} -8.46242 q^{86} +4.00000 q^{87} +25.2432 q^{88} +8.23121 q^{89} -35.0733 q^{92} -9.33150 q^{93} -21.0426 q^{94} +18.7194 q^{96} +15.0165 q^{97} +2.82550 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{2} + 4 q^{3} + 7 q^{4} + q^{6} + 3 q^{8} + 4 q^{9} + 8 q^{11} + 7 q^{12} + 7 q^{13} + 17 q^{16} + 6 q^{17} + q^{18} - 3 q^{19} - 12 q^{22} + 2 q^{23} + 3 q^{24} + 19 q^{26} + 4 q^{27} + 16 q^{29}+ \cdots + 8 q^{99}+O(q^{100}) $ 4 * q + q^2 + 4 * q^3 + 7 * q^4 + q^6 + 3 * q^8 + 4 * q^9 + 8 * q^11 + 7 * q^12 + 7 * q^13 + 17 * q^16 + 6 * q^17 + q^18 - 3 * q^19 - 12 * q^22 + 2 * q^23 + 3 * q^24 + 19 * q^26 + 4 * q^27 + 16 * q^29 - 9 * q^31 + 17 * q^32 + 8 * q^33 - 14 * q^34 + 7 * q^36 + 8 * q^37 - 27 * q^38 + 7 * q^39 - 4 * q^41 + 5 * q^43 + 26 * q^44 - 6 * q^46 - 6 * q^47 + 17 * q^48 + 6 * q^51 + 35 * q^52 - 6 * q^53 + q^54 - 3 * q^57 + 4 * q^58 - 10 * q^59 + 20 * q^61 - 44 * q^62 + 21 * q^64 - 12 * q^66 + q^67 - 8 * q^68 + 2 * q^69 + 22 * q^71 + 3 * q^72 - 4 * q^73 - 21 * q^74 + 23 * q^76 + 19 * q^78 + 8 * q^79 + 4 * q^81 + 8 * q^82 - 2 * q^83 + 12 * q^86 + 16 * q^87 + 28 * q^88 + 10 * q^89 - 66 * q^92 - 9 * q^93 - 22 * q^94 + 17 * q^96 + 12 * q^97 + 8 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.70285	1.91121	0.955603	−	0.294657i	$-0.0952054\pi$
$2$	2.70285	1.91121	0.955603	+	0.294657i	$0.0952054\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	5.30542	2.65271
$5$	0	0
$5$	0	0
$6$	2.70285	1.10344
$7$	0	0
$7$	0	0
$8$	8.93406	3.15867
$9$	1.00000	0.333333
$10$	0	0
$11$	2.82550	0.851921	0.425960	−	0.904742i	$-0.359936\pi$
$11$	2.82550	0.851921	0.425960	+	0.904742i	$0.359936\pi$
$12$	5.30542	1.53154
$13$	4.47992	1.24251	0.621253	−	0.783610i	$-0.286624\pi$
$13$	4.47992	1.24251	0.621253	+	0.783610i	$0.286624\pi$
$14$	0	0
$15$	0	0
$16$	13.5366	3.38416
$17$	−4.23121	−1.02622	−0.513109	−	0.858323i	$-0.671506\pi$
$17$	−4.23121	−1.02622	−0.513109	+	0.858323i	$0.671506\pi$
$18$	2.70285	0.637069
$19$	−2.10029	−0.481839	−0.240920	−	0.970545i	$-0.577449\pi$
$19$	−2.10029	−0.481839	−0.240920	+	0.970545i	$0.577449\pi$
$20$	0	0
$21$	0	0
$22$	7.63692	1.62820
$23$	−6.61084	−1.37845	−0.689227	−	0.724545i	$-0.742050\pi$
$23$	−6.61084	−1.37845	−0.689227	+	0.724545i	$0.742050\pi$
$24$	8.93406	1.82366
$25$	0	0
$26$	12.1086	2.37468
$27$	1.00000	0.192450
$28$	0	0
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	0	0
$31$	−9.33150	−1.67599	−0.837993	−	0.545681i	$-0.816271\pi$
$31$	−9.33150	−1.67599	−0.837993	+	0.545681i	$0.816271\pi$
$32$	18.7194	3.30915
$33$	2.82550	0.491857
$34$	−11.4363	−1.96132
$35$	0	0
$36$	5.30542	0.884236
$37$	4.20513	0.691319	0.345659	−	0.938360i	$-0.387655\pi$
$37$	4.20513	0.691319	0.345659	+	0.938360i	$0.387655\pi$
$38$	−5.67677	−0.920894
$39$	4.47992	0.717361
$40$	0	0
$41$	−2.37963	−0.371635	−0.185818	−	0.982584i	$-0.559493\pi$
$41$	−2.37963	−0.371635	−0.185818	+	0.982584i	$0.559493\pi$
$42$	0	0
$43$	−3.13092	−0.477461	−0.238730	−	0.971086i	$-0.576731\pi$
$43$	−3.13092	−0.477461	−0.238730	+	0.971086i	$0.576731\pi$
$44$	14.9905	2.25990
$45$	0	0
$46$	−17.8681	−2.63451
$47$	−7.78534	−1.13561	−0.567804	−	0.823164i	$-0.692207\pi$
$47$	−7.78534	−1.13561	−0.567804	+	0.823164i	$0.692207\pi$
$48$	13.5366	1.95384
$49$	0	0
$50$	0	0
$51$	−4.23121	−0.592488
$52$	23.7678	3.29601
$53$	2.58021	0.354419	0.177209	−	0.984173i	$-0.443293\pi$
$53$	2.58021	0.354419	0.177209	+	0.984173i	$0.443293\pi$
$54$	2.70285	0.367812
$55$	0	0
$56$	0	0
$57$	−2.10029	−0.278190
$58$	10.8114	1.41961
$59$	3.78534	0.492809	0.246404	−	0.969167i	$-0.420751\pi$
$59$	3.78534	0.492809	0.246404	+	0.969167i	$0.420751\pi$
$60$	0	0
$61$	5.00000	0.640184	0.320092	−	0.947386i	$-0.396286\pi$
$61$	5.00000	0.640184	0.320092	+	0.947386i	$0.396286\pi$
$62$	−25.2217	−3.20316
$63$	0	0
$64$	23.5225	2.94032
$65$	0	0
$66$	7.63692	0.940039
$67$	−3.30542	−0.403821	−0.201911	−	0.979404i	$-0.564715\pi$
$67$	−3.30542	−0.403821	−0.201911	+	0.979404i	$0.564715\pi$
$68$	−22.4483	−2.72226
$69$	−6.61084	−0.795851
$70$	0	0
$71$	−11.4223	−1.35557	−0.677786	−	0.735259i	$-0.737060\pi$
$71$	−11.4223	−1.35557	−0.677786	+	0.735259i	$0.737060\pi$
$72$	8.93406	1.05289
$73$	−2.65100	−0.310276	−0.155138	−	0.987893i	$-0.549582\pi$
$73$	−2.65100	−0.310276	−0.155138	+	0.987893i	$0.549582\pi$
$74$	11.3658	1.32125
$75$	0	0
$76$	−11.1429	−1.27818
$77$	0	0
$78$	12.1086	1.37102
$79$	12.6369	1.42176	0.710882	−	0.703311i	$-0.248296\pi$
$79$	12.6369	1.42176	0.710882	+	0.703311i	$0.248296\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−6.43179	−0.710272
$83$	3.02608	0.332155	0.166078	−	0.986113i	$-0.446890\pi$
$83$	3.02608	0.332155	0.166078	+	0.986113i	$0.446890\pi$
$84$	0	0
$85$	0	0
$86$	−8.46242	−0.912526
$87$	4.00000	0.428845
$88$	25.2432	2.69093
$89$	8.23121	0.872506	0.436253	−	0.899824i	$-0.356305\pi$
$89$	8.23121	0.872506	0.436253	+	0.899824i	$0.356305\pi$
$90$	0	0
$91$	0	0
$92$	−35.0733	−3.65664
$93$	−9.33150	−0.967631
$94$	−21.0426	−2.17038
$95$	0	0
$96$	18.7194	1.91054
$97$	15.0165	1.52470	0.762350	−	0.647166i	$-0.224046\pi$
$97$	15.0165	1.52470	0.762350	+	0.647166i	$0.224046\pi$
$98$	0	0
$99$	2.82550	0.283974
$100$	0	0
$101$	2.23121	0.222014	0.111007	−	0.993820i	$-0.464592\pi$
$101$	2.23121	0.222014	0.111007	+	0.993820i	$0.464592\pi$
$102$	−11.4363	−1.13237
$103$	−7.84205	−0.772700	−0.386350	−	0.922352i	$-0.626264\pi$
$103$	−7.84205	−0.772700	−0.386350	+	0.922352i	$0.626264\pi$
$104$	40.0239	3.92466
$105$	0	0
$106$	6.97392	0.677367
$107$	−15.6369	−1.51168	−0.755839	−	0.654758i	$-0.772771\pi$
$107$	−15.6369	−1.51168	−0.755839	+	0.654758i	$0.772771\pi$
$108$	5.30542	0.510514
$109$	3.07421	0.294456	0.147228	−	0.989103i	$-0.452965\pi$
$109$	3.07421	0.294456	0.147228	+	0.989103i	$0.452965\pi$
$110$	0	0
$111$	4.20513	0.399133
$112$	0	0
$113$	0.348998	0.0328310	0.0164155	−	0.999865i	$-0.494775\pi$
$113$	0.348998	0.0328310	0.0164155	+	0.999865i	$0.494775\pi$
$114$	−5.67677	−0.531679
$115$	0	0
$116$	21.2217	1.97038
$117$	4.47992	0.414169
$118$	10.2312	0.941859
$119$	0	0
$120$	0	0
$121$	−3.01654	−0.274231
$122$	13.5143	1.22352
$123$	−2.37963	−0.214564
$124$	−49.5075	−4.44590
$125$	0	0
$126$	0	0
$127$	−2.27137	−0.201552	−0.100776	−	0.994909i	$-0.532133\pi$
$127$	−2.27137	−0.201552	−0.100776	+	0.994909i	$0.532133\pi$
$128$	26.1392	2.31040
$129$	−3.13092	−0.275662
$130$	0	0
$131$	−19.2217	−1.67941	−0.839703	−	0.543046i	$-0.817271\pi$
$131$	−19.2217	−1.67941	−0.839703	+	0.543046i	$0.817271\pi$
$132$	14.9905	1.30475
$133$	0	0
$134$	−8.93406	−0.771785
$135$	0	0
$136$	−37.8019	−3.24148
$137$	8.26184	0.705857	0.352928	−	0.935650i	$-0.385186\pi$
$137$	8.26184	0.705857	0.352928	+	0.935650i	$0.385186\pi$
$138$	−17.8681	−1.52104
$139$	−0.985914	−0.0836241	−0.0418121	−	0.999125i	$-0.513313\pi$
$139$	−0.985914	−0.0836241	−0.0418121	+	0.999125i	$0.513313\pi$
$140$	0	0
$141$	−7.78534	−0.655644
$142$	−30.8727	−2.59078
$143$	12.6580	1.05852
$144$	13.5366	1.12805
$145$	0	0
$146$	−7.16527	−0.593002
$147$	0	0
$148$	22.3100	1.83387
$149$	7.82095	0.640717	0.320359	−	0.947296i	$-0.396197\pi$
$149$	7.82095	0.640717	0.320359	+	0.947296i	$0.396197\pi$
$150$	0	0
$151$	−3.13433	−0.255068	−0.127534	−	0.991834i	$-0.540706\pi$
$151$	−3.13433	−0.255068	−0.127534	+	0.991834i	$0.540706\pi$
$152$	−18.7641	−1.52197
$153$	−4.23121	−0.342073
$154$	0	0
$155$	0	0
$156$	23.7678	1.90295
$157$	−23.4790	−1.87383	−0.936913	−	0.349564i	$-0.886330\pi$
$157$	−23.4790	−1.87383	−0.936913	+	0.349564i	$0.886330\pi$
$158$	34.1557	2.71728
$159$	2.58021	0.204624
$160$	0	0
$161$	0	0
$162$	2.70285	0.212356
$163$	12.4363	0.974089	0.487045	−	0.873377i	$-0.338075\pi$
$163$	12.4363	0.974089	0.487045	+	0.873377i	$0.338075\pi$
$164$	−12.6249	−0.985841
$165$	0	0
$166$	8.17905	0.634817
$167$	18.4483	1.42757	0.713787	−	0.700362i	$-0.246978\pi$
$167$	18.4483	1.42757	0.713787	+	0.700362i	$0.246978\pi$
$168$	0	0
$169$	7.06966	0.543820
$170$	0	0
$171$	−2.10029	−0.160613
$172$	−16.6108	−1.26656
$173$	7.65100	0.581695	0.290847	−	0.956769i	$-0.406063\pi$
$173$	7.65100	0.581695	0.290847	+	0.956769i	$0.406063\pi$
$174$	10.8114	0.819611
$175$	0	0
$176$	38.2478	2.88303
$177$	3.78534	0.284523
$178$	22.2478	1.66754
$179$	−2.41026	−0.180151	−0.0900756	−	0.995935i	$-0.528711\pi$
$179$	−2.41026	−0.180151	−0.0900756	+	0.995935i	$0.528711\pi$
$180$	0	0
$181$	13.6970	1.01809	0.509046	−	0.860739i	$-0.329998\pi$
$181$	13.6970	1.01809	0.509046	+	0.860739i	$0.329998\pi$
$182$	0	0
$183$	5.00000	0.369611
$184$	−59.0616	−4.35408
$185$	0	0
$186$	−25.2217	−1.84934
$187$	−11.9553	−0.874257
$188$	−41.3045	−3.01244
$189$	0	0
$190$	0	0
$191$	−12.1343	−0.878010	−0.439005	−	0.898485i	$-0.644669\pi$
$191$	−12.1343	−0.878010	−0.439005	+	0.898485i	$0.644669\pi$
$192$	23.5225	1.69759
$193$	−11.9258	−0.858437	−0.429219	−	0.903201i	$-0.641211\pi$
$193$	−11.9258	−0.858437	−0.429219	+	0.903201i	$0.641211\pi$
$194$	40.5875	2.91401
$195$	0	0
$196$	0	0
$197$	5.50258	0.392043	0.196021	−	0.980600i	$-0.437198\pi$
$197$	5.50258	0.392043	0.196021	+	0.980600i	$0.437198\pi$
$198$	7.63692	0.542732
$199$	11.3281	0.803027	0.401513	−	0.915853i	$-0.368484\pi$
$199$	11.3281	0.803027	0.401513	+	0.915853i	$0.368484\pi$
$200$	0	0
$201$	−3.30542	−0.233146
$202$	6.03063	0.424314
$203$	0	0
$204$	−22.4483	−1.57170
$205$	0	0
$206$	−21.1959	−1.47679
$207$	−6.61084	−0.459485
$208$	60.6430	4.20483
$209$	−5.93437	−0.410489
$210$	0	0
$211$	8.82095	0.607259	0.303630	−	0.952790i	$-0.401801\pi$
$211$	8.82095	0.607259	0.303630	+	0.952790i	$0.401801\pi$
$212$	13.6891	0.940170
$213$	−11.4223	−0.782640
$214$	−42.2643	−2.88913
$215$	0	0
$216$	8.93406	0.607886
$217$	0	0
$218$	8.30914	0.562766
$219$	−2.65100	−0.179138
$220$	0	0
$221$	−18.9555	−1.27508
$222$	11.3658	0.762826
$223$	26.4885	1.77380	0.886900	−	0.461961i	$-0.152854\pi$
$223$	26.4885	1.77380	0.886900	+	0.461961i	$0.152854\pi$
$224$	0	0
$225$	0	0
$226$	0.943291	0.0627468
$227$	−17.7712	−1.17952	−0.589760	−	0.807579i	$-0.700778\pi$
$227$	−17.7712	−1.17952	−0.589760	+	0.807579i	$0.700778\pi$
$228$	−11.1429	−0.737957
$229$	−18.6449	−1.23209	−0.616044	−	0.787712i	$-0.711266\pi$
$229$	−18.6449	−1.23209	−0.616044	+	0.787712i	$0.711266\pi$
$230$	0	0
$231$	0	0
$232$	35.7362	2.34620
$233$	9.76879	0.639975	0.319987	−	0.947422i	$-0.396321\pi$
$233$	9.76879	0.639975	0.319987	+	0.947422i	$0.396321\pi$
$234$	12.1086	0.791561
$235$	0	0
$236$	20.0828	1.30728
$237$	12.6369	0.820856
$238$	0	0
$239$	−6.20058	−0.401082	−0.200541	−	0.979685i	$-0.564270\pi$
$239$	−6.20058	−0.401082	−0.200541	+	0.979685i	$0.564270\pi$
$240$	0	0
$241$	20.2382	1.30366	0.651829	−	0.758366i	$-0.274002\pi$
$241$	20.2382	1.30366	0.651829	+	0.758366i	$0.274002\pi$
$242$	−8.15328	−0.524113
$243$	1.00000	0.0641500
$244$	26.5271	1.69822
$245$	0	0
$246$	−6.43179	−0.410076
$247$	−9.40912	−0.598688
$248$	−83.3682	−5.29388
$249$	3.02608	0.191770
$250$	0	0
$251$	6.21466	0.392266	0.196133	−	0.980577i	$-0.437162\pi$
$251$	6.21466	0.392266	0.196133	+	0.980577i	$0.437162\pi$
$252$	0	0
$253$	−18.6789	−1.17433
$254$	−6.13919	−0.385207
$255$	0	0
$256$	23.6053	1.47533
$257$	5.65346	0.352653	0.176327	−	0.984332i	$-0.443579\pi$
$257$	5.65346	0.352653	0.176327	+	0.984332i	$0.443579\pi$
$258$	−8.46242	−0.526847
$259$	0	0
$260$	0	0
$261$	4.00000	0.247594
$262$	−51.9534	−3.20969
$263$	−25.0120	−1.54231	−0.771153	−	0.636650i	$-0.780320\pi$
$263$	−25.0120	−1.54231	−0.771153	+	0.636650i	$0.780320\pi$
$264$	25.2432	1.55361
$265$	0	0
$266$	0	0
$267$	8.23121	0.503742
$268$	−17.5366	−1.07122
$269$	7.88221	0.480587	0.240293	−	0.970700i	$-0.422756\pi$
$269$	7.88221	0.480587	0.240293	+	0.970700i	$0.422756\pi$
$270$	0	0
$271$	−4.74271	−0.288099	−0.144050	−	0.989570i	$-0.546012\pi$
$271$	−4.74271	−0.288099	−0.144050	+	0.989570i	$0.546012\pi$
$272$	−57.2763	−3.47289
$273$	0	0
$274$	22.3305	1.34904
$275$	0	0
$276$	−35.0733	−2.11116
$277$	−7.73721	−0.464884	−0.232442	−	0.972610i	$-0.574672\pi$
$277$	−7.73721	−0.464884	−0.232442	+	0.972610i	$0.574672\pi$
$278$	−2.66478	−0.159823
$279$	−9.33150	−0.558662
$280$	0	0
$281$	2.11779	0.126337	0.0631684	−	0.998003i	$-0.479879\pi$
$281$	2.11779	0.126337	0.0631684	+	0.998003i	$0.479879\pi$
$282$	−21.0426	−1.25307
$283$	4.39774	0.261419	0.130709	−	0.991421i	$-0.458275\pi$
$283$	4.39774	0.261419	0.130709	+	0.991421i	$0.458275\pi$
$284$	−60.5998	−3.59594
$285$	0	0
$286$	34.2128	2.02304
$287$	0	0
$288$	18.7194	1.10305
$289$	0.903125	0.0531250
$290$	0	0
$291$	15.0165	0.880285
$292$	−14.0647	−0.823073
$293$	−14.2006	−0.829607	−0.414803	−	0.909911i	$-0.636150\pi$
$293$	−14.2006	−0.829607	−0.414803	+	0.909911i	$0.636150\pi$
$294$	0	0
$295$	0	0
$296$	37.5689	2.18365
$297$	2.82550	0.163952
$298$	21.1389	1.22454
$299$	−29.6160	−1.71274
$300$	0	0
$301$	0	0
$302$	−8.47165	−0.487488
$303$	2.23121	0.128180
$304$	−28.4308	−1.63062
$305$	0	0
$306$	−11.4363	−0.653772
$307$	−13.9423	−0.795731	−0.397866	−	0.917444i	$-0.630249\pi$
$307$	−13.9423	−0.795731	−0.397866	+	0.917444i	$0.630249\pi$
$308$	0	0
$309$	−7.84205	−0.446118
$310$	0	0
$311$	3.79942	0.215445	0.107723	−	0.994181i	$-0.465644\pi$
$311$	3.79942	0.215445	0.107723	+	0.994181i	$0.465644\pi$
$312$	40.0239	2.26590
$313$	27.8976	1.57687	0.788433	−	0.615120i	$-0.210893\pi$
$313$	27.8976	1.57687	0.788433	+	0.615120i	$0.210893\pi$
$314$	−63.4602	−3.58127
$315$	0	0
$316$	67.0441	3.77153
$317$	−34.4955	−1.93746	−0.968730	−	0.248116i	$-0.920189\pi$
$317$	−34.4955	−1.93746	−0.968730	+	0.248116i	$0.920189\pi$
$318$	6.97392	0.391078
$319$	11.3020	0.632791
$320$	0	0
$321$	−15.6369	−0.872768
$322$	0	0
$323$	8.88676	0.494473
$324$	5.30542	0.294745
$325$	0	0
$326$	33.6136	1.86169
$327$	3.07421	0.170004
$328$	−21.2597	−1.17387
$329$	0	0
$330$	0	0
$331$	−29.8942	−1.64313	−0.821567	−	0.570112i	$-0.806900\pi$
$331$	−29.8942	−1.64313	−0.821567	+	0.570112i	$0.806900\pi$
$332$	16.0546	0.881112
$333$	4.20513	0.230440
$334$	49.8631	2.72839
$335$	0	0
$336$	0	0
$337$	−19.3480	−1.05395	−0.526977	−	0.849879i	$-0.676675\pi$
$337$	−19.3480	−1.05395	−0.526977	+	0.849879i	$0.676675\pi$
$338$	19.1083	1.03935
$339$	0.348998	0.0189550
$340$	0	0
$341$	−26.3662	−1.42781
$342$	−5.67677	−0.306965
$343$	0	0
$344$	−27.9718	−1.50814
$345$	0	0
$346$	20.6795	1.11174
$347$	−26.2999	−1.41185	−0.705927	−	0.708285i	$-0.749469\pi$
$347$	−26.2999	−1.41185	−0.705927	+	0.708285i	$0.749469\pi$
$348$	21.2217	1.13760
$349$	14.9083	0.798022	0.399011	−	0.916946i	$-0.369353\pi$
$349$	14.9083	0.798022	0.399011	+	0.916946i	$0.369353\pi$
$350$	0	0
$351$	4.47992	0.239120
$352$	52.8917	2.81914
$353$	−15.8210	−0.842064	−0.421032	−	0.907046i	$-0.638332\pi$
$353$	−15.8210	−0.842064	−0.421032	+	0.907046i	$0.638332\pi$
$354$	10.2312	0.543783
$355$	0	0
$356$	43.6700	2.31451
$357$	0	0
$358$	−6.51458	−0.344306
$359$	16.6771	0.880183	0.440091	−	0.897953i	$-0.354946\pi$
$359$	16.6771	0.880183	0.440091	+	0.897953i	$0.354946\pi$
$360$	0	0
$361$	−14.5888	−0.767831
$362$	37.0211	1.94579
$363$	−3.01654	−0.158328
$364$	0	0
$365$	0	0
$366$	13.5143	0.706402
$367$	28.1876	1.47138	0.735691	−	0.677317i	$-0.236858\pi$
$367$	28.1876	1.47138	0.735691	+	0.677317i	$0.236858\pi$
$368$	−89.4884	−4.66491
$369$	−2.37963	−0.123878
$370$	0	0
$371$	0	0
$372$	−49.5075	−2.56684
$373$	−23.9990	−1.24262	−0.621312	−	0.783564i	$-0.713400\pi$
$373$	−23.9990	−1.24262	−0.621312	+	0.783564i	$0.713400\pi$
$374$	−32.3134	−1.67089
$375$	0	0
$376$	−69.5547	−3.58701
$377$	17.9197	0.922910
$378$	0	0
$379$	13.8550	0.711683	0.355842	−	0.934546i	$-0.384194\pi$
$379$	13.8550	0.711683	0.355842	+	0.934546i	$0.384194\pi$
$380$	0	0
$381$	−2.27137	−0.116366
$382$	−32.7973	−1.67806
$383$	−30.7764	−1.57260	−0.786301	−	0.617844i	$-0.788006\pi$
$383$	−30.7764	−1.57260	−0.786301	+	0.617844i	$0.788006\pi$
$384$	26.1392	1.33391
$385$	0	0
$386$	−32.2337	−1.64065
$387$	−3.13092	−0.159154
$388$	79.6690	4.04458
$389$	38.4127	1.94760	0.973801	−	0.227402i	$-0.0730231\pi$
$389$	38.4127	1.94760	0.973801	+	0.227402i	$0.0730231\pi$
$390$	0	0
$391$	27.9718	1.41460
$392$	0	0
$393$	−19.2217	−0.969605
$394$	14.8727	0.749275
$395$	0	0
$396$	14.9905	0.753299
$397$	−33.4615	−1.67938	−0.839691	−	0.543064i	$-0.817264\pi$
$397$	−33.4615	−1.67938	−0.839691	+	0.543064i	$0.817264\pi$
$398$	30.6182	1.53475
$399$	0	0
$400$	0	0
$401$	14.8420	0.741176	0.370588	−	0.928797i	$-0.379156\pi$
$401$	14.8420	0.741176	0.370588	+	0.928797i	$0.379156\pi$
$402$	−8.93406	−0.445591
$403$	−41.8043	−2.08242
$404$	11.8375	0.588937
$405$	0	0
$406$	0	0
$407$	11.8816	0.588949
$408$	−37.8019	−1.87147
$409$	−8.44492	−0.417574	−0.208787	−	0.977961i	$-0.566952\pi$
$409$	−8.44492	−0.417574	−0.208787	+	0.977961i	$0.566952\pi$
$410$	0	0
$411$	8.26184	0.407526
$412$	−41.6053	−2.04975
$413$	0	0
$414$	−17.8681	−0.878170
$415$	0	0
$416$	83.8614	4.11164
$417$	−0.985914	−0.0482804
$418$	−16.0397	−0.784529
$419$	18.8427	0.920524	0.460262	−	0.887783i	$-0.347755\pi$
$419$	18.8427	0.920524	0.460262	+	0.887783i	$0.347755\pi$
$420$	0	0
$421$	−9.09687	−0.443355	−0.221677	−	0.975120i	$-0.571153\pi$
$421$	−9.09687	−0.443355	−0.221677	+	0.975120i	$0.571153\pi$
$422$	23.8417	1.16060
$423$	−7.78534	−0.378536
$424$	23.0517	1.11949
$425$	0	0
$426$	−30.8727	−1.49579
$427$	0	0
$428$	−82.9604	−4.01004
$429$	12.6580	0.611135
$430$	0	0
$431$	−4.95983	−0.238907	−0.119453	−	0.992840i	$-0.538114\pi$
$431$	−4.95983	−0.238907	−0.119453	+	0.992840i	$0.538114\pi$
$432$	13.5366	0.651281
$433$	−2.61696	−0.125763	−0.0628815	−	0.998021i	$-0.520029\pi$
$433$	−2.61696	−0.125763	−0.0628815	+	0.998021i	$0.520029\pi$
$434$	0	0
$435$	0	0
$436$	16.3100	0.781106
$437$	13.8847	0.664194
$438$	−7.16527	−0.342370
$439$	26.5050	1.26502	0.632508	−	0.774554i	$-0.282025\pi$
$439$	26.5050	1.26502	0.632508	+	0.774554i	$0.282025\pi$
$440$	0	0
$441$	0	0
$442$	−51.2338	−2.43695
$443$	−39.4903	−1.87624	−0.938121	−	0.346307i	$-0.887436\pi$
$443$	−39.4903	−1.87624	−0.938121	+	0.346307i	$0.887436\pi$
$444$	22.3100	1.05878
$445$	0	0
$446$	71.5945	3.39010
$447$	7.82095	0.369918
$448$	0	0
$449$	21.3045	1.00542	0.502710	−	0.864455i	$-0.332336\pi$
$449$	21.3045	1.00542	0.502710	+	0.864455i	$0.332336\pi$
$450$	0	0
$451$	−6.72364	−0.316604
$452$	1.85158	0.0870910
$453$	−3.13433	−0.147264
$454$	−48.0331	−2.25430
$455$	0	0
$456$	−18.7641	−0.878710
$457$	−27.8801	−1.30418	−0.652088	−	0.758143i	$-0.726107\pi$
$457$	−27.8801	−1.30418	−0.652088	+	0.758143i	$0.726107\pi$
$458$	−50.3944	−2.35478
$459$	−4.23121	−0.197496
$460$	0	0
$461$	−22.5237	−1.04903	−0.524516	−	0.851401i	$-0.675754\pi$
$461$	−22.5237	−1.04903	−0.524516	+	0.851401i	$0.675754\pi$
$462$	0	0
$463$	1.69301	0.0786809	0.0393405	−	0.999226i	$-0.487474\pi$
$463$	1.69301	0.0786809	0.0393405	+	0.999226i	$0.487474\pi$
$464$	54.1465	2.51369
$465$	0	0
$466$	26.4036	1.22312
$467$	−0.0803308	−0.00371727	−0.00185863	−	0.999998i	$-0.500592\pi$
$467$	−0.0803308	−0.00371727	−0.00185863	+	0.999998i	$0.500592\pi$
$468$	23.7678	1.09867
$469$	0	0
$470$	0	0
$471$	−23.4790	−1.08185
$472$	33.8184	1.55662
$473$	−8.84642	−0.406759
$474$	34.1557	1.56882
$475$	0	0
$476$	0	0
$477$	2.58021	0.118140
$478$	−16.7593	−0.766551
$479$	38.7783	1.77182	0.885912	−	0.463853i	$-0.153534\pi$
$479$	38.7783	1.77182	0.885912	+	0.463853i	$0.153534\pi$
$480$	0	0
$481$	18.8386	0.858968
$482$	54.7009	2.49156
$483$	0	0
$484$	−16.0040	−0.727456
$485$	0	0
$486$	2.70285	0.122604
$487$	27.1309	1.22942	0.614710	−	0.788753i	$-0.289273\pi$
$487$	27.1309	1.22942	0.614710	+	0.788753i	$0.289273\pi$
$488$	44.6703	2.02213
$489$	12.4363	0.562391
$490$	0	0
$491$	16.7243	0.754755	0.377378	−	0.926059i	$-0.376826\pi$
$491$	16.7243	0.754755	0.377378	+	0.926059i	$0.376826\pi$
$492$	−12.6249	−0.569175
$493$	−16.9248	−0.762256
$494$	−25.4315	−1.14422
$495$	0	0
$496$	−126.317	−5.67180
$497$	0	0
$498$	8.17905	0.366512
$499$	19.7844	0.885670	0.442835	−	0.896603i	$-0.353973\pi$
$499$	19.7844	0.885670	0.442835	+	0.896603i	$0.353973\pi$
$500$	0	0
$501$	18.4483	0.824211
$502$	16.7973	0.749701
$503$	−6.75015	−0.300975	−0.150487	−	0.988612i	$-0.548084\pi$
$503$	−6.75015	−0.300975	−0.150487	+	0.988612i	$0.548084\pi$
$504$	0	0
$505$	0	0
$506$	−50.4864	−2.24439
$507$	7.06966	0.313975
$508$	−12.0506	−0.534658
$509$	40.3471	1.78835	0.894177	−	0.447715i	$-0.147762\pi$
$509$	40.3471	1.78835	0.894177	+	0.447715i	$0.147762\pi$
$510$	0	0
$511$	0	0
$512$	11.5234	0.509266
$513$	−2.10029	−0.0927300
$514$	15.2805	0.673993
$515$	0	0
$516$	−16.6108	−0.731251
$517$	−21.9975	−0.967448
$518$	0	0
$519$	7.65100	0.335842
$520$	0	0
$521$	−37.6038	−1.64745	−0.823725	−	0.566989i	$-0.808108\pi$
$521$	−37.6038	−1.64745	−0.823725	+	0.566989i	$0.808108\pi$
$522$	10.8114	0.473203
$523$	39.3124	1.71901	0.859506	−	0.511125i	$-0.170771\pi$
$523$	39.3124	1.71901	0.859506	+	0.511125i	$0.170771\pi$
$524$	−101.979	−4.45497
$525$	0	0
$526$	−67.6038	−2.94766
$527$	39.4835	1.71993
$528$	38.2478	1.66452
$529$	20.7032	0.900137
$530$	0	0
$531$	3.78534	0.164270
$532$	0	0
$533$	−10.6605	−0.461759
$534$	22.2478	0.962754
$535$	0	0
$536$	−29.5308	−1.27554
$537$	−2.41026	−0.104010
$538$	21.3045	0.918501
$539$	0	0
$540$	0	0
$541$	21.5722	0.927463	0.463732	−	0.885976i	$-0.346510\pi$
$541$	21.5722	0.927463	0.463732	+	0.885976i	$0.346510\pi$
$542$	−12.8189	−0.550617
$543$	13.6970	0.587796
$544$	−79.2057	−3.39592
$545$	0	0
$546$	0	0
$547$	21.3707	0.913745	0.456873	−	0.889532i	$-0.348969\pi$
$547$	21.3707	0.913745	0.456873	+	0.889532i	$0.348969\pi$
$548$	43.8325	1.87243
$549$	5.00000	0.213395
$550$	0	0
$551$	−8.40116	−0.357901
$552$	−59.0616	−2.51383
$553$	0	0
$554$	−20.9125	−0.888488
$555$	0	0
$556$	−5.23069	−0.221830
$557$	−14.2618	−0.604293	−0.302147	−	0.953261i	$-0.597703\pi$
$557$	−14.2618	−0.604293	−0.302147	+	0.953261i	$0.597703\pi$
$558$	−25.2217	−1.06772
$559$	−14.0263	−0.593248
$560$	0	0
$561$	−11.9553	−0.504752
$562$	5.72408	0.241456
$563$	−2.36308	−0.0995921	−0.0497961	−	0.998759i	$-0.515857\pi$
$563$	−2.36308	−0.0995921	−0.0497961	+	0.998759i	$0.515857\pi$
$564$	−41.3045	−1.73923
$565$	0	0
$566$	11.8865	0.499625
$567$	0	0
$568$	−102.047	−4.28180
$569$	−12.7458	−0.534331	−0.267166	−	0.963651i	$-0.586087\pi$
$569$	−12.7458	−0.534331	−0.267166	+	0.963651i	$0.586087\pi$
$570$	0	0
$571$	0.361514	0.0151289	0.00756444	−	0.999971i	$-0.497592\pi$
$571$	0.361514	0.0151289	0.00756444	+	0.999971i	$0.497592\pi$
$572$	67.1560	2.80794
$573$	−12.1343	−0.506919
$574$	0	0
$575$	0	0
$576$	23.5225	0.980106
$577$	9.59334	0.399376	0.199688	−	0.979860i	$-0.436007\pi$
$577$	9.59334	0.399376	0.199688	+	0.979860i	$0.436007\pi$
$578$	2.44102	0.101533
$579$	−11.9258	−0.495619
$580$	0	0
$581$	0	0
$582$	40.5875	1.68241
$583$	7.29038	0.301937
$584$	−23.6842	−0.980060
$585$	0	0
$586$	−38.3821	−1.58555
$587$	19.2217	0.793363	0.396682	−	0.917956i	$-0.370162\pi$
$587$	19.2217	0.793363	0.396682	+	0.917956i	$0.370162\pi$
$588$	0	0
$589$	19.5988	0.807556
$590$	0	0
$591$	5.50258	0.226346
$592$	56.9233	2.33953
$593$	11.7994	0.484544	0.242272	−	0.970208i	$-0.422107\pi$
$593$	11.7994	0.484544	0.242272	+	0.970208i	$0.422107\pi$
$594$	7.63692	0.313346
$595$	0	0
$596$	41.4934	1.69964
$597$	11.3281	0.463628
$598$	−80.0477	−3.27339
$599$	40.3962	1.65054	0.825271	−	0.564736i	$-0.191022\pi$
$599$	40.3962	1.65054	0.825271	+	0.564736i	$0.191022\pi$
$600$	0	0
$601$	20.9073	0.852828	0.426414	−	0.904528i	$-0.359777\pi$
$601$	20.9073	0.852828	0.426414	+	0.904528i	$0.359777\pi$
$602$	0	0
$603$	−3.30542	−0.134607
$604$	−16.6290	−0.676622
$605$	0	0
$606$	6.03063	0.244978
$607$	−4.20759	−0.170781	−0.0853903	−	0.996348i	$-0.527214\pi$
$607$	−4.20759	−0.170781	−0.0853903	+	0.996348i	$0.527214\pi$
$608$	−39.3161	−1.59448
$609$	0	0
$610$	0	0
$611$	−34.8777	−1.41100
$612$	−22.4483	−0.907420
$613$	16.4268	0.663472	0.331736	−	0.943372i	$-0.392366\pi$
$613$	16.4268	0.663472	0.331736	+	0.943372i	$0.392366\pi$
$614$	−37.6841	−1.52081
$615$	0	0
$616$	0	0
$617$	21.9037	0.881811	0.440906	−	0.897553i	$-0.354657\pi$
$617$	21.9037	0.881811	0.440906	+	0.897553i	$0.354657\pi$
$618$	−21.1959	−0.852624
$619$	−0.319504	−0.0128420	−0.00642098	−	0.999979i	$-0.502044\pi$
$619$	−0.319504	−0.0128420	−0.00642098	+	0.999979i	$0.502044\pi$
$620$	0	0
$621$	−6.61084	−0.265284
$622$	10.2693	0.411761
$623$	0	0
$624$	60.6430	2.42766
$625$	0	0
$626$	75.4032	3.01372
$627$	−5.93437	−0.236996
$628$	−124.566	−4.97071
$629$	−17.7928	−0.709445
$630$	0	0
$631$	17.3090	0.689061	0.344530	−	0.938775i	$-0.388038\pi$
$631$	17.3090	0.689061	0.344530	+	0.938775i	$0.388038\pi$
$632$	112.899	4.49088
$633$	8.82095	0.350601
$634$	−93.2363	−3.70289
$635$	0	0
$636$	13.6891	0.542807
$637$	0	0
$638$	30.5477	1.20939
$639$	−11.4223	−0.451857
$640$	0	0
$641$	48.2858	1.90718	0.953588	−	0.301115i	$-0.0973589\pi$
$641$	48.2858	1.90718	0.953588	+	0.301115i	$0.0973589\pi$
$642$	−42.2643	−1.66804
$643$	−12.6460	−0.498710	−0.249355	−	0.968412i	$-0.580219\pi$
$643$	−12.6460	−0.498710	−0.249355	+	0.968412i	$0.580219\pi$
$644$	0	0
$645$	0	0
$646$	24.0196	0.945039
$647$	15.8534	0.623262	0.311631	−	0.950203i	$-0.399125\pi$
$647$	15.8534	0.623262	0.311631	+	0.950203i	$0.399125\pi$
$648$	8.93406	0.350963
$649$	10.6955	0.419834
$650$	0	0
$651$	0	0
$652$	65.9800	2.58398
$653$	1.15795	0.0453143	0.0226571	−	0.999743i	$-0.492787\pi$
$653$	1.15795	0.0453143	0.0226571	+	0.999743i	$0.492787\pi$
$654$	8.30914	0.324913
$655$	0	0
$656$	−32.2121	−1.25767
$657$	−2.65100	−0.103425
$658$	0	0
$659$	−34.5617	−1.34633	−0.673167	−	0.739490i	$-0.735067\pi$
$659$	−34.5617	−1.34633	−0.673167	+	0.739490i	$0.735067\pi$
$660$	0	0
$661$	1.28030	0.0497977	0.0248989	−	0.999690i	$-0.492074\pi$
$661$	1.28030	0.0497977	0.0248989	+	0.999690i	$0.492074\pi$
$662$	−80.7997	−3.14037
$663$	−18.9555	−0.736169
$664$	27.0352	1.04917
$665$	0	0
$666$	11.3658	0.440418
$667$	−26.4433	−1.02389
$668$	97.8761	3.78694
$669$	26.4885	1.02410
$670$	0	0
$671$	14.1275	0.545386
$672$	0	0
$673$	45.9058	1.76954	0.884769	−	0.466031i	$-0.154316\pi$
$673$	45.9058	1.76954	0.884769	+	0.466031i	$0.154316\pi$
$674$	−52.2949	−2.01433
$675$	0	0
$676$	37.5075	1.44260
$677$	−19.7100	−0.757516	−0.378758	−	0.925496i	$-0.623649\pi$
$677$	−19.7100	−0.757516	−0.378758	+	0.925496i	$0.623649\pi$
$678$	0.943291	0.0362269
$679$	0	0
$680$	0	0
$681$	−17.7712	−0.680996
$682$	−71.2639	−2.72883
$683$	1.44317	0.0552212	0.0276106	−	0.999619i	$-0.491210\pi$
$683$	1.44317	0.0552212	0.0276106	+	0.999619i	$0.491210\pi$
$684$	−11.1429	−0.426060
$685$	0	0
$686$	0	0
$687$	−18.6449	−0.711347
$688$	−42.3821	−1.61580
$689$	11.5591	0.440367
$690$	0	0
$691$	30.9221	1.17633	0.588167	−	0.808740i	$-0.299850\pi$
$691$	30.9221	1.17633	0.588167	+	0.808740i	$0.299850\pi$
$692$	40.5918	1.54307
$693$	0	0
$694$	−71.0848	−2.69834
$695$	0	0
$696$	35.7362	1.35458
$697$	10.0687	0.381379
$698$	40.2949	1.52519
$699$	9.76879	0.369490
$700$	0	0
$701$	50.6746	1.91395	0.956976	−	0.290168i	$-0.0937111\pi$
$701$	50.6746	1.91395	0.956976	+	0.290168i	$0.0937111\pi$
$702$	12.1086	0.457008
$703$	−8.83199	−0.333105
$704$	66.4630	2.50492
$705$	0	0
$706$	−42.7617	−1.60936
$707$	0	0
$708$	20.0828	0.754757
$709$	−20.1770	−0.757762	−0.378881	−	0.925445i	$-0.623691\pi$
$709$	−20.1770	−0.757762	−0.378881	+	0.925445i	$0.623691\pi$
$710$	0	0
$711$	12.6369	0.473921
$712$	73.5381	2.75596
$713$	61.6890	2.31027
$714$	0	0
$715$	0	0
$716$	−12.7874	−0.477889
$717$	−6.20058	−0.231565
$718$	45.0757	1.68221
$719$	−45.3278	−1.69044	−0.845221	−	0.534416i	$-0.820532\pi$
$719$	−45.3278	−1.69044	−0.845221	+	0.534416i	$0.820532\pi$
$720$	0	0
$721$	0	0
$722$	−39.4314	−1.46748
$723$	20.2382	0.752667
$724$	72.6685	2.70070
$725$	0	0
$726$	−8.15328	−0.302597
$727$	16.9117	0.627220	0.313610	−	0.949552i	$-0.398461\pi$
$727$	16.9117	0.627220	0.313610	+	0.949552i	$0.398461\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	13.2476	0.489979
$732$	26.5271	0.980470
$733$	7.16558	0.264667	0.132333	−	0.991205i	$-0.457753\pi$
$733$	7.16558	0.264667	0.132333	+	0.991205i	$0.457753\pi$
$734$	76.1870	2.81211
$735$	0	0
$736$	−123.751	−4.56152
$737$	−9.33946	−0.344024
$738$	−6.43179	−0.236757
$739$	−20.3848	−0.749867	−0.374933	−	0.927052i	$-0.622334\pi$
$739$	−20.3848	−0.749867	−0.374933	+	0.927052i	$0.622334\pi$
$740$	0	0
$741$	−9.40912	−0.345653
$742$	0	0
$743$	−16.9267	−0.620982	−0.310491	−	0.950576i	$-0.600493\pi$
$743$	−16.9267	−0.620982	−0.310491	+	0.950576i	$0.600493\pi$
$744$	−83.3682	−3.05643
$745$	0	0
$746$	−64.8659	−2.37491
$747$	3.02608	0.110718
$748$	−63.4278	−2.31915
$749$	0	0
$750$	0	0
$751$	39.0877	1.42633	0.713165	−	0.700996i	$-0.247261\pi$
$751$	39.0877	1.42633	0.713165	+	0.700996i	$0.247261\pi$
$752$	−105.387	−3.84308
$753$	6.21466	0.226475
$754$	48.4342	1.76387
$755$	0	0
$756$	0	0
$757$	22.6675	0.823866	0.411933	−	0.911214i	$-0.364854\pi$
$757$	22.6675	0.823866	0.411933	+	0.911214i	$0.364854\pi$
$758$	37.4480	1.36017
$759$	−18.6789	−0.678002
$760$	0	0
$761$	−10.3275	−0.374370	−0.187185	−	0.982325i	$-0.559936\pi$
$761$	−10.3275	−0.374370	−0.187185	+	0.982325i	$0.559936\pi$
$762$	−6.13919	−0.222399
$763$	0	0
$764$	−64.3777	−2.32910
$765$	0	0
$766$	−83.1841	−3.00557
$767$	16.9580	0.612318
$768$	23.6053	0.851784
$769$	25.8785	0.933204	0.466602	−	0.884467i	$-0.345478\pi$
$769$	25.8785	0.933204	0.466602	+	0.884467i	$0.345478\pi$
$770$	0	0
$771$	5.65346	0.203604
$772$	−63.2713	−2.27718
$773$	9.35416	0.336446	0.168223	−	0.985749i	$-0.446197\pi$
$773$	9.35416	0.336446	0.168223	+	0.985749i	$0.446197\pi$
$774$	−8.46242	−0.304175
$775$	0	0
$776$	134.159	4.81602
$777$	0	0
$778$	103.824	3.72227
$779$	4.99791	0.179069
$780$	0	0
$781$	−32.2736	−1.15484
$782$	75.6038	2.70358
$783$	4.00000	0.142948
$784$	0	0
$785$	0	0
$786$	−51.9534	−1.85312
$787$	−2.80188	−0.0998762	−0.0499381	−	0.998752i	$-0.515902\pi$
$787$	−2.80188	−0.0998762	−0.0499381	+	0.998752i	$0.515902\pi$
$788$	29.1935	1.03998
$789$	−25.0120	−0.890451
$790$	0	0
$791$	0	0
$792$	25.2432	0.896978
$793$	22.3996	0.795433
$794$	−90.4414	−3.20965
$795$	0	0
$796$	60.1002	2.13020
$797$	13.0211	0.461231	0.230615	−	0.973045i	$-0.425926\pi$
$797$	13.0211	0.461231	0.230615	+	0.973045i	$0.425926\pi$
$798$	0	0
$799$	32.9414	1.16538
$800$	0	0
$801$	8.23121	0.290835
$802$	40.1159	1.41654
$803$	−7.49041	−0.264331
$804$	−17.5366	−0.618469
$805$	0	0
$806$	−112.991	−3.97994
$807$	7.88221	0.277467
$808$	19.9338	0.701267
$809$	21.0211	0.739062	0.369531	−	0.929218i	$-0.379518\pi$
$809$	21.0211	0.739062	0.369531	+	0.929218i	$0.379518\pi$
$810$	0	0
$811$	46.2591	1.62438	0.812189	−	0.583395i	$-0.198276\pi$
$811$	46.2591	1.62438	0.812189	+	0.583395i	$0.198276\pi$
$812$	0	0
$813$	−4.74271	−0.166334
$814$	32.1142	1.12560
$815$	0	0
$816$	−57.2763	−2.00507
$817$	6.57584	0.230059
$818$	−22.8254	−0.798070
$819$	0	0
$820$	0	0
$821$	16.9383	0.591151	0.295575	−	0.955319i	$-0.404489\pi$
$821$	16.9383	0.591151	0.295575	+	0.955319i	$0.404489\pi$
$822$	22.3305	0.778867
$823$	−18.5332	−0.646027	−0.323014	−	0.946394i	$-0.604696\pi$
$823$	−18.5332	−0.646027	−0.323014	+	0.946394i	$0.604696\pi$
$824$	−70.0613	−2.44070
$825$	0	0
$826$	0	0
$827$	32.5496	1.13186	0.565930	−	0.824453i	$-0.308517\pi$
$827$	32.5496	1.13186	0.565930	+	0.824453i	$0.308517\pi$
$828$	−35.0733	−1.21888
$829$	10.9553	0.380493	0.190246	−	0.981736i	$-0.439071\pi$
$829$	10.9553	0.380493	0.190246	+	0.981736i	$0.439071\pi$
$830$	0	0
$831$	−7.73721	−0.268401
$832$	105.379	3.65336
$833$	0	0
$834$	−2.66478	−0.0922738
$835$	0	0
$836$	−31.4843	−1.08891
$837$	−9.33150	−0.322544
$838$	50.9290	1.75931
$839$	−1.24074	−0.0428352	−0.0214176	−	0.999771i	$-0.506818\pi$
$839$	−1.24074	−0.0428352	−0.0214176	+	0.999771i	$0.506818\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	−24.5875	−0.847342
$843$	2.11779	0.0729405
$844$	46.7988	1.61088
$845$	0	0
$846$	−21.0426	−0.723460
$847$	0	0
$848$	34.9273	1.19941
$849$	4.39774	0.150930
$850$	0	0
$851$	−27.7994	−0.952952
$852$	−60.5998	−2.07612
$853$	−44.8500	−1.53564	−0.767818	−	0.640669i	$-0.778657\pi$
$853$	−44.8500	−1.53564	−0.767818	+	0.640669i	$0.778657\pi$
$854$	0	0
$855$	0	0
$856$	−139.701	−4.77489
$857$	−38.6108	−1.31892	−0.659461	−	0.751739i	$-0.729215\pi$
$857$	−38.6108	−1.31892	−0.659461	+	0.751739i	$0.729215\pi$
$858$	34.2128	1.16800
$859$	−35.1083	−1.19788	−0.598939	−	0.800795i	$-0.704411\pi$
$859$	−35.1083	−1.19788	−0.598939	+	0.800795i	$0.704411\pi$
$860$	0	0
$861$	0	0
$862$	−13.4057	−0.456600
$863$	4.94575	0.168355	0.0841776	−	0.996451i	$-0.473174\pi$
$863$	4.94575	0.168355	0.0841776	+	0.996451i	$0.473174\pi$
$864$	18.7194	0.636847
$865$	0	0
$866$	−7.07325	−0.240359
$867$	0.903125	0.0306717
$868$	0	0
$869$	35.7056	1.21123
$870$	0	0
$871$	−14.8080	−0.501750
$872$	27.4652	0.930088
$873$	15.0165	0.508233
$874$	37.5282	1.26941
$875$	0	0
$876$	−14.0647	−0.475201
$877$	−10.7123	−0.361727	−0.180864	−	0.983508i	$-0.557889\pi$
$877$	−10.7123	−0.361727	−0.180864	+	0.983508i	$0.557889\pi$
$878$	71.6392	2.41771
$879$	−14.2006	−0.478974
$880$	0	0
$881$	−40.2527	−1.35615	−0.678075	−	0.734993i	$-0.737185\pi$
$881$	−40.2527	−1.35615	−0.678075	+	0.734993i	$0.737185\pi$
$882$	0	0
$883$	−26.2228	−0.882468	−0.441234	−	0.897392i	$-0.645459\pi$
$883$	−26.2228	−0.882468	−0.441234	+	0.897392i	$0.645459\pi$
$884$	−100.567	−3.38242
$885$	0	0
$886$	−106.737	−3.58589
$887$	−49.3964	−1.65857	−0.829284	−	0.558828i	$-0.811251\pi$
$887$	−49.3964	−1.65857	−0.829284	+	0.558828i	$0.811251\pi$
$888$	37.5689	1.26073
$889$	0	0
$890$	0	0
$891$	2.82550	0.0946578
$892$	140.533	4.70538
$893$	16.3515	0.547181
$894$	21.1389	0.706990
$895$	0	0
$896$	0	0
$897$	−29.6160	−0.988850
$898$	57.5828	1.92156
$899$	−37.3260	−1.24489
$900$	0	0
$901$	−10.9174	−0.363711
$902$	−18.1730	−0.605095
$903$	0	0
$904$	3.11797	0.103702
$905$	0	0
$906$	−8.47165	−0.281452
$907$	56.1629	1.86486	0.932429	−	0.361354i	$-0.117685\pi$
$907$	56.1629	1.86486	0.932429	+	0.361354i	$0.117685\pi$
$908$	−94.2839	−3.12892
$909$	2.23121	0.0740045
$910$	0	0
$911$	−35.9981	−1.19267	−0.596335	−	0.802736i	$-0.703377\pi$
$911$	−35.9981	−1.19267	−0.596335	+	0.802736i	$0.703377\pi$
$912$	−28.4308	−0.941439
$913$	8.55019	0.282970
$914$	−75.3559	−2.49255
$915$	0	0
$916$	−98.9189	−3.26837
$917$	0	0
$918$	−11.4363	−0.377455
$919$	−7.54801	−0.248986	−0.124493	−	0.992221i	$-0.539730\pi$
$919$	−7.54801	−0.248986	−0.124493	+	0.992221i	$0.539730\pi$
$920$	0	0
$921$	−13.9423	−0.459416
$922$	−60.8782	−2.00492
$923$	−51.1707	−1.68431
$924$	0	0
$925$	0	0
$926$	4.57596	0.150375
$927$	−7.84205	−0.257567
$928$	74.8776	2.45798
$929$	−5.19104	−0.170313	−0.0851563	−	0.996368i	$-0.527139\pi$
$929$	−5.19104	−0.170313	−0.0851563	+	0.996368i	$0.527139\pi$
$930$	0	0
$931$	0	0
$932$	51.8275	1.69767
$933$	3.79942	0.124387
$934$	−0.217122	−0.00710446
$935$	0	0
$936$	40.0239	1.30822
$937$	5.23690	0.171082	0.0855410	−	0.996335i	$-0.472738\pi$
$937$	5.23690	0.171082	0.0855410	+	0.996335i	$0.472738\pi$
$938$	0	0
$939$	27.8976	0.910404
$940$	0	0
$941$	12.9439	0.421959	0.210980	−	0.977490i	$-0.432335\pi$
$941$	12.9439	0.421959	0.210980	+	0.977490i	$0.432335\pi$
$942$	−63.4602	−2.06764
$943$	15.7313	0.512283
$944$	51.2407	1.66774
$945$	0	0
$946$	−23.9106	−0.777400
$947$	−23.0143	−0.747863	−0.373932	−	0.927456i	$-0.621991\pi$
$947$	−23.0143	−0.747863	−0.373932	+	0.927456i	$0.621991\pi$
$948$	67.0441	2.17749
$949$	−11.8763	−0.385520
$950$	0	0
$951$	−34.4955	−1.11859
$952$	0	0
$953$	−15.1560	−0.490952	−0.245476	−	0.969403i	$-0.578944\pi$
$953$	−15.1560	−0.490952	−0.245476	+	0.969403i	$0.578944\pi$
$954$	6.97392	0.225789
$955$	0	0
$956$	−32.8967	−1.06395
$957$	11.3020	0.365342
$958$	104.812	3.38632
$959$	0	0
$960$	0	0
$961$	56.0768	1.80893
$962$	50.9181	1.64166
$963$	−15.6369	−0.503893
$964$	107.372	3.45823
$965$	0	0
$966$	0	0
$967$	−45.5145	−1.46365	−0.731824	−	0.681494i	$-0.761331\pi$
$967$	−45.5145	−1.46365	−0.731824	+	0.681494i	$0.761331\pi$
$968$	−26.9500	−0.866206
$969$	8.88676	0.285484
$970$	0	0
$971$	7.60192	0.243957	0.121979	−	0.992533i	$-0.461076\pi$
$971$	7.60192	0.243957	0.121979	+	0.992533i	$0.461076\pi$
$972$	5.30542	0.170171
$973$	0	0
$974$	73.3309	2.34967
$975$	0	0
$976$	67.6831	2.16648
$977$	−34.6936	−1.10995	−0.554974	−	0.831868i	$-0.687272\pi$
$977$	−34.6936	−1.10995	−0.554974	+	0.831868i	$0.687272\pi$
$978$	33.6136	1.07484
$979$	23.2573	0.743306
$980$	0	0
$981$	3.07421	0.0981520
$982$	45.2032	1.44249
$983$	−15.5779	−0.496859	−0.248429	−	0.968650i	$-0.579914\pi$
$983$	−15.5779	−0.496859	−0.248429	+	0.968650i	$0.579914\pi$
$984$	−21.2597	−0.677736
$985$	0	0
$986$	−45.7454	−1.45683
$987$	0	0
$988$	−49.9193	−1.58815
$989$	20.6980	0.658158
$990$	0	0
$991$	−45.9589	−1.45993	−0.729966	−	0.683484i	$-0.760464\pi$
$991$	−45.9589	−1.45993	−0.729966	+	0.683484i	$0.760464\pi$
$992$	−174.680	−5.54610
$993$	−29.8942	−0.948664
$994$	0	0
$995$	0	0
$996$	16.0546	0.508710
$997$	−41.0687	−1.30066	−0.650329	−	0.759652i	$-0.725369\pi$
$997$	−41.0687	−1.30066	−0.650329	+	0.759652i	$0.725369\pi$
$998$	53.4743	1.69270
$999$	4.20513	0.133044

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3675.2.a.bx.1.4		4
5.4	even	2		3675.2.a.bq.1.1		4
7.3	odd	6		525.2.i.i.226.1	yes	8
7.5	odd	6		525.2.i.i.151.1	✓	8
7.6	odd	2		3675.2.a.bw.1.4		4
35.3	even	12		525.2.r.h.499.8		16
35.12	even	12		525.2.r.h.424.8		16
35.17	even	12		525.2.r.h.499.1		16
35.19	odd	6		525.2.i.j.151.4	yes	8
35.24	odd	6		525.2.i.j.226.4	yes	8
35.33	even	12		525.2.r.h.424.1		16
35.34	odd	2		3675.2.a.br.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.2.i.i.151.1	✓	8	7.5	odd	6
525.2.i.i.226.1	yes	8	7.3	odd	6
525.2.i.j.151.4	yes	8	35.19	odd	6
525.2.i.j.226.4	yes	8	35.24	odd	6
525.2.r.h.424.1		16	35.33	even	12
525.2.r.h.424.8		16	35.12	even	12
525.2.r.h.499.1		16	35.17	even	12
525.2.r.h.499.8		16	35.3	even	12
3675.2.a.bq.1.1		4	5.4	even	2
3675.2.a.br.1.1		4	35.34	odd	2
3675.2.a.bw.1.4		4	7.6	odd	2
3675.2.a.bx.1.4		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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+2.70285 q^{2} +1.00000 q^{3} +5.30542 q^{4} +2.70285 q^{6} +8.93406 q^{8} +1.00000 q^{9} +2.82550 q^{11} +5.30542 q^{12} +4.47992 q^{13} +13.5366 q^{16} -4.23121 q^{17} +2.70285 q^{18} -2.10029 q^{19} +7.63692 q^{22} -6.61084 q^{23} +8.93406 q^{24} +12.1086 q^{26} +1.00000 q^{27} +4.00000 q^{29} -9.33150 q^{31} +18.7194 q^{32} +2.82550 q^{33} -11.4363 q^{34} +5.30542 q^{36} +4.20513 q^{37} -5.67677 q^{38} +4.47992 q^{39} -2.37963 q^{41} -3.13092 q^{43} +14.9905 q^{44} -17.8681 q^{46} -7.78534 q^{47} +13.5366 q^{48} -4.23121 q^{51} +23.7678 q^{52} +2.58021 q^{53} +2.70285 q^{54} -2.10029 q^{57} +10.8114 q^{58} +3.78534 q^{59} +5.00000 q^{61} -25.2217 q^{62} +23.5225 q^{64} +7.63692 q^{66} -3.30542 q^{67} -22.4483 q^{68} -6.61084 q^{69} -11.4223 q^{71} +8.93406 q^{72} -2.65100 q^{73} +11.3658 q^{74} -11.1429 q^{76} +12.1086 q^{78} +12.6369 q^{79} +1.00000 q^{81} -6.43179 q^{82} +3.02608 q^{83} -8.46242 q^{86} +4.00000 q^{87} +25.2432 q^{88} +8.23121 q^{89} -35.0733 q^{92} -9.33150 q^{93} -21.0426 q^{94} +18.7194 q^{96} +15.0165 q^{97} +2.82550 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{2} + 4 q^{3} + 7 q^{4} + q^{6} + 3 q^{8} + 4 q^{9} + 8 q^{11} + 7 q^{12} + 7 q^{13} + 17 q^{16} + 6 q^{17} + q^{18} - 3 q^{19} - 12 q^{22} + 2 q^{23} + 3 q^{24} + 19 q^{26} + 4 q^{27} + 16 q^{29}+ \cdots + 8 q^{99}+O(q^{100}) \) 4 * q + q^2 + 4 * q^3 + 7 * q^4 + q^6 + 3 * q^8 + 4 * q^9 + 8 * q^11 + 7 * q^12 + 7 * q^13 + 17 * q^16 + 6 * q^17 + q^18 - 3 * q^19 - 12 * q^22 + 2 * q^23 + 3 * q^24 + 19 * q^26 + 4 * q^27 + 16 * q^29 - 9 * q^31 + 17 * q^32 + 8 * q^33 - 14 * q^34 + 7 * q^36 + 8 * q^37 - 27 * q^38 + 7 * q^39 - 4 * q^41 + 5 * q^43 + 26 * q^44 - 6 * q^46 - 6 * q^47 + 17 * q^48 + 6 * q^51 + 35 * q^52 - 6 * q^53 + q^54 - 3 * q^57 + 4 * q^58 - 10 * q^59 + 20 * q^61 - 44 * q^62 + 21 * q^64 - 12 * q^66 + q^67 - 8 * q^68 + 2 * q^69 + 22 * q^71 + 3 * q^72 - 4 * q^73 - 21 * q^74 + 23 * q^76 + 19 * q^78 + 8 * q^79 + 4 * q^81 + 8 * q^82 - 2 * q^83 + 12 * q^86 + 16 * q^87 + 28 * q^88 + 10 * q^89 - 66 * q^92 - 9 * q^93 - 22 * q^94 + 17 * q^96 + 12 * q^97 + 8 * q^99

Introduction

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