LMFDB - Embedded newform 3675.2.a.bx.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:	4.4.88404.1
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			1.2
Root			$-0.494173$ 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)

$f(q)$	$=$	$q-0.494173 q^{2} +1.00000 q^{3} -1.75579 q^{4} -0.494173 q^{6} +1.85601 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.494173 q^{2} +1.00000 q^{3} -1.75579 q^{4} -0.494173 q^{6} +1.85601 q^{8} +1.00000 q^{9} +5.33853 q^{11} -1.75579 q^{12} -5.09433 q^{13} +2.59439 q^{16} -0.350186 q^{17} -0.494173 q^{18} -2.76745 q^{19} -2.63816 q^{22} +7.51159 q^{23} +1.85601 q^{24} +2.51748 q^{26} +1.00000 q^{27} +4.00000 q^{29} -6.11763 q^{31} -4.99411 q^{32} +5.33853 q^{33} +0.173053 q^{34} -1.75579 q^{36} -3.52324 q^{37} +1.36760 q^{38} -5.09433 q^{39} +7.86177 q^{41} +1.41726 q^{43} -9.37336 q^{44} -3.71203 q^{46} +8.85012 q^{47} +2.59439 q^{48} -0.350186 q^{51} +8.94458 q^{52} -6.32688 q^{53} -0.494173 q^{54} -2.76745 q^{57} -1.97669 q^{58} -12.8501 q^{59} +5.00000 q^{61} +3.02317 q^{62} -2.72083 q^{64} -2.63816 q^{66} +3.75579 q^{67} +0.614854 q^{68} +7.51159 q^{69} +15.4883 q^{71} +1.85601 q^{72} -7.67707 q^{73} +1.74109 q^{74} +4.85906 q^{76} +2.51748 q^{78} +2.36184 q^{79} +1.00000 q^{81} -3.88508 q^{82} +6.87342 q^{83} -0.700372 q^{86} +4.00000 q^{87} +9.90838 q^{88} +4.35019 q^{89} -13.1888 q^{92} -6.11763 q^{93} -4.37349 q^{94} -4.99411 q^{96} -5.49993 q^{97} +5.33853 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{2} + 4 q^{3} + 7 q^{4} + q^{6} + 3 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q + q^2 + 4 * q^3 + 7 * q^4 + q^6 + 3 * q^8 + 4 * q^9	$ 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}+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

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 $

$2$	−0.494173	−0.349433	−0.174717	−	0.984619i	$-0.555901\pi$
$2$	−0.494173	−0.349433	−0.174717	+	0.984619i	$0.555901\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.75579	−0.877896
$5$	0	0
$5$	0	0
$6$	−0.494173	−0.201745
$7$	0	0
$7$	0	0
$8$	1.85601	0.656200
$9$	1.00000	0.333333
$10$	0	0
$11$	5.33853	1.60963	0.804814	−	0.593527i	$-0.202265\pi$
$11$	5.33853	1.60963	0.804814	+	0.593527i	$0.202265\pi$
$12$	−1.75579	−0.506854
$13$	−5.09433	−1.41291	−0.706456	−	0.707757i	$-0.749707\pi$
$13$	−5.09433	−1.41291	−0.706456	+	0.707757i	$0.749707\pi$
$14$	0	0
$15$	0	0
$16$	2.59439	0.648598
$17$	−0.350186	−0.0849326	−0.0424663	−	0.999098i	$-0.513522\pi$
$17$	−0.350186	−0.0849326	−0.0424663	+	0.999098i	$0.513522\pi$
$18$	−0.494173	−0.116478
$19$	−2.76745	−0.634896	−0.317448	−	0.948276i	$-0.602826\pi$
$19$	−2.76745	−0.634896	−0.317448	+	0.948276i	$0.602826\pi$
$20$	0	0
$21$	0	0
$22$	−2.63816	−0.562458
$23$	7.51159	1.56627	0.783137	−	0.621849i	$-0.213618\pi$
$23$	7.51159	1.56627	0.783137	+	0.621849i	$0.213618\pi$
$24$	1.85601	0.378857
$25$	0	0
$26$	2.51748	0.493718
$27$	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
$31$	−6.11763	−1.09876	−0.549380	−	0.835573i	$-0.685136\pi$
$31$	−6.11763	−1.09876	−0.549380	+	0.835573i	$0.685136\pi$
$32$	−4.99411	−0.882841
$33$	5.33853	0.929319
$34$	0.173053	0.0296783
$35$	0	0
$36$	−1.75579	−0.292632
$37$	−3.52324	−0.579217	−0.289608	−	0.957145i	$-0.593525\pi$
$37$	−3.52324	−0.579217	−0.289608	+	0.957145i	$0.593525\pi$
$38$	1.36760	0.221854
$39$	−5.09433	−0.815745
$40$	0	0
$41$	7.86177	1.22780	0.613901	−	0.789383i	$-0.289599\pi$
$41$	7.86177	1.22780	0.613901	+	0.789383i	$0.289599\pi$
$42$	0	0
$43$	1.41726	0.216130	0.108065	−	0.994144i	$-0.465535\pi$
$43$	1.41726	0.216130	0.108065	+	0.994144i	$0.465535\pi$
$44$	−9.37336	−1.41309
$45$	0	0
$46$	−3.71203	−0.547308
$47$	8.85012	1.29092	0.645461	−	0.763793i	$-0.276665\pi$
$47$	8.85012	1.29092	0.645461	+	0.763793i	$0.276665\pi$
$48$	2.59439	0.374468
$49$	0	0
$50$	0	0
$51$	−0.350186	−0.0490359
$52$	8.94458	1.24039
$53$	−6.32688	−0.869064	−0.434532	−	0.900656i	$-0.643086\pi$
$53$	−6.32688	−0.869064	−0.434532	+	0.900656i	$0.643086\pi$
$54$	−0.494173	−0.0672485
$55$	0	0
$56$	0	0
$57$	−2.76745	−0.366557
$58$	−1.97669	−0.259553
$59$	−12.8501	−1.67294	−0.836471	−	0.548011i	$-0.815385\pi$
$59$	−12.8501	−1.67294	−0.836471	+	0.548011i	$0.815385\pi$
$60$	0	0
$61$	5.00000	0.640184	0.320092	−	0.947386i	$-0.396286\pi$
$61$	5.00000	0.640184	0.320092	+	0.947386i	$0.396286\pi$
$62$	3.02317	0.383943
$63$	0	0
$64$	−2.72083	−0.340104
$65$	0	0
$66$	−2.63816	−0.324735
$67$	3.75579	0.458843	0.229422	−	0.973327i	$-0.426317\pi$
$67$	3.75579	0.458843	0.229422	+	0.973327i	$0.426317\pi$
$68$	0.614854	0.0745620
$69$	7.51159	0.904289
$70$	0	0
$71$	15.4883	1.83812	0.919060	−	0.394117i	$-0.128950\pi$
$71$	15.4883	1.83812	0.919060	+	0.394117i	$0.128950\pi$
$72$	1.85601	0.218733
$73$	−7.67707	−0.898533	−0.449266	−	0.893398i	$-0.648315\pi$
$73$	−7.67707	−0.898533	−0.449266	+	0.893398i	$0.648315\pi$
$74$	1.74109	0.202398
$75$	0	0
$76$	4.85906	0.557373
$77$	0	0
$78$	2.51748	0.285048
$79$	2.36184	0.265728	0.132864	−	0.991134i	$-0.457583\pi$
$79$	2.36184	0.265728	0.132864	+	0.991134i	$0.457583\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−3.88508	−0.429035
$83$	6.87342	0.754456	0.377228	−	0.926120i	$-0.376877\pi$
$83$	6.87342	0.754456	0.377228	+	0.926120i	$0.376877\pi$
$84$	0	0
$85$	0	0
$86$	−0.700372	−0.0755231
$87$	4.00000	0.428845
$88$	9.90838	1.05624
$89$	4.35019	0.461119	0.230559	−	0.973058i	$-0.425944\pi$
$89$	4.35019	0.461119	0.230559	+	0.973058i	$0.425944\pi$
$90$	0	0
$91$	0	0
$92$	−13.1888	−1.37503
$93$	−6.11763	−0.634369
$94$	−4.37349	−0.451091
$95$	0	0
$96$	−4.99411	−0.509709
$97$	−5.49993	−0.558433	−0.279217	−	0.960228i	$-0.590075\pi$
$97$	−5.49993	−0.558433	−0.279217	+	0.960228i	$0.590075\pi$
$98$	0	0
$99$	5.33853	0.536543
$100$	0	0
$101$	−1.64981	−0.164163	−0.0820813	−	0.996626i	$-0.526157\pi$
$101$	−1.64981	−0.164163	−0.0820813	+	0.996626i	$0.526157\pi$
$102$	0.173053	0.0171348
$103$	10.1614	1.00123	0.500616	−	0.865669i	$-0.333107\pi$
$103$	10.1614	1.00123	0.500616	+	0.865669i	$0.333107\pi$
$104$	−9.45513	−0.927152
$105$	0	0
$106$	3.12658	0.303680
$107$	−5.36184	−0.518349	−0.259174	−	0.965831i	$-0.583450\pi$
$107$	−5.36184	−0.518349	−0.259174	+	0.965831i	$0.583450\pi$
$108$	−1.75579	−0.168951
$109$	−0.105979	−0.0101509	−0.00507546	−	0.999987i	$-0.501616\pi$
$109$	−0.105979	−0.0101509	−0.00507546	+	0.999987i	$0.501616\pi$
$110$	0	0
$111$	−3.52324	−0.334411
$112$	0	0
$113$	−4.67707	−0.439981	−0.219991	−	0.975502i	$-0.570603\pi$
$113$	−4.67707	−0.439981	−0.219991	+	0.975502i	$0.570603\pi$
$114$	1.36760	0.128087
$115$	0	0
$116$	−7.02317	−0.652085
$117$	−5.09433	−0.470971
$118$	6.35019	0.584582
$119$	0	0
$120$	0	0
$121$	17.4999	1.59090
$122$	−2.47087	−0.223702
$123$	7.86177	0.708872
$124$	10.7413	0.964597
$125$	0	0
$126$	0	0
$127$	−17.5388	−1.55632	−0.778160	−	0.628066i	$-0.783847\pi$
$127$	−17.5388	−1.55632	−0.778160	+	0.628066i	$0.783847\pi$
$128$	11.3328	1.00169
$129$	1.41726	0.124783
$130$	0	0
$131$	9.02317	0.788358	0.394179	−	0.919034i	$-0.371029\pi$
$131$	9.02317	0.788358	0.394179	+	0.919034i	$0.371029\pi$
$132$	−9.37336	−0.815846
$133$	0	0
$134$	−1.85601	−0.160335
$135$	0	0
$136$	−0.649950	−0.0557327
$137$	−0.834520	−0.0712978	−0.0356489	−	0.999364i	$-0.511350\pi$
$137$	−0.834520	−0.0712978	−0.0356489	+	0.999364i	$0.511350\pi$
$138$	−3.71203	−0.315989
$139$	14.3152	1.21420	0.607101	−	0.794625i	$-0.292332\pi$
$139$	14.3152	1.21420	0.607101	+	0.794625i	$0.292332\pi$
$140$	0	0
$141$	8.85012	0.745314
$142$	−7.65389	−0.642301
$143$	−27.1962	−2.27426
$144$	2.59439	0.216199
$145$	0	0
$146$	3.79380	0.313977
$147$	0	0
$148$	6.18608	0.508492
$149$	19.3967	1.58904	0.794518	−	0.607240i	$-0.207723\pi$
$149$	19.3967	1.58904	0.794518	+	0.607240i	$0.207723\pi$
$150$	0	0
$151$	18.5272	1.50772	0.753860	−	0.657035i	$-0.228189\pi$
$151$	18.5272	1.50772	0.753860	+	0.657035i	$0.228189\pi$
$152$	−5.13641	−0.416618
$153$	−0.350186	−0.0283109
$154$	0	0
$155$	0	0
$156$	8.94458	0.716139
$157$	4.79956	0.383047	0.191523	−	0.981488i	$-0.438657\pi$
$157$	4.79956	0.383047	0.191523	+	0.981488i	$0.438657\pi$
$158$	−1.16716	−0.0928541
$159$	−6.32688	−0.501754
$160$	0	0
$161$	0	0
$162$	−0.494173	−0.0388259
$163$	0.826947	0.0647715	0.0323858	−	0.999475i	$-0.489689\pi$
$163$	0.826947	0.0647715	0.0323858	+	0.999475i	$0.489689\pi$
$164$	−13.8036	−1.07788
$165$	0	0
$166$	−3.39666	−0.263632
$167$	−4.61485	−0.357108	−0.178554	−	0.983930i	$-0.557142\pi$
$167$	−4.61485	−0.357108	−0.178554	+	0.983930i	$0.557142\pi$
$168$	0	0
$169$	12.9522	0.996319
$170$	0	0
$171$	−2.76745	−0.211632
$172$	−2.48841	−0.189740
$173$	12.6771	0.963819	0.481910	−	0.876221i	$-0.339943\pi$
$173$	12.6771	0.963819	0.481910	+	0.876221i	$0.339943\pi$
$174$	−1.97669	−0.149853
$175$	0	0
$176$	13.8503	1.04400
$177$	−12.8501	−0.965874
$178$	−2.14975	−0.161130
$179$	13.0465	0.975139	0.487570	−	0.873084i	$-0.337883\pi$
$179$	13.0465	0.975139	0.487570	+	0.873084i	$0.337883\pi$
$180$	0	0
$181$	−15.0594	−1.11935	−0.559677	−	0.828711i	$-0.689075\pi$
$181$	−15.0594	−1.11935	−0.559677	+	0.828711i	$0.689075\pi$
$182$	0	0
$183$	5.00000	0.369611
$184$	13.9416	1.02779
$185$	0	0
$186$	3.02317	0.221670
$187$	−1.86948	−0.136710
$188$	−15.5390	−1.13330
$189$	0	0
$190$	0	0
$191$	9.52718	0.689363	0.344681	−	0.938720i	$-0.387987\pi$
$191$	9.52718	0.689363	0.344681	+	0.938720i	$0.387987\pi$
$192$	−2.72083	−0.196359
$193$	−15.1060	−1.08735	−0.543676	−	0.839295i	$-0.682968\pi$
$193$	−15.1060	−1.08735	−0.543676	+	0.839295i	$0.682968\pi$
$194$	2.71792	0.195135
$195$	0	0
$196$	0	0
$197$	16.8890	1.20329	0.601647	−	0.798762i	$-0.294512\pi$
$197$	16.8890	1.20329	0.601647	+	0.798762i	$0.294512\pi$
$198$	−2.63816	−0.187486
$199$	25.2276	1.78833	0.894167	−	0.447734i	$-0.147769\pi$
$199$	25.2276	1.78833	0.894167	+	0.447734i	$0.147769\pi$
$200$	0	0
$201$	3.75579	0.264913
$202$	0.815294	0.0573639
$203$	0	0
$204$	0.614854	0.0430484
$205$	0	0
$206$	−5.02149	−0.349864
$207$	7.51159	0.522091
$208$	−13.2167	−0.916412
$209$	−14.7741	−1.02195
$210$	0	0
$211$	20.3967	1.40416	0.702082	−	0.712096i	$-0.252254\pi$
$211$	20.3967	1.40416	0.702082	+	0.712096i	$0.252254\pi$
$212$	11.1087	0.762948
$213$	15.4883	1.06124
$214$	2.64968	0.181128
$215$	0	0
$216$	1.85601	0.126286
$217$	0	0
$218$	0.0523719	0.00354707
$219$	−7.67707	−0.518768
$220$	0	0
$221$	1.78396	0.120002
$222$	1.74109	0.116854
$223$	22.5738	1.51165	0.755827	−	0.654772i	$-0.227235\pi$
$223$	22.5738	1.51165	0.755827	+	0.654772i	$0.227235\pi$
$224$	0	0
$225$	0	0
$226$	2.31128	0.153744
$227$	14.1653	0.940187	0.470093	−	0.882617i	$-0.344220\pi$
$227$	14.1653	0.940187	0.470093	+	0.882617i	$0.344220\pi$
$228$	4.85906	0.321799
$229$	17.8062	1.17667	0.588334	−	0.808618i	$-0.299784\pi$
$229$	17.8062	1.17667	0.588334	+	0.808618i	$0.299784\pi$
$230$	0	0
$231$	0	0
$232$	7.42405	0.487413
$233$	13.6498	0.894229	0.447115	−	0.894477i	$-0.352452\pi$
$233$	13.6498	0.894229	0.447115	+	0.894477i	$0.352452\pi$
$234$	2.51748	0.164573
$235$	0	0
$236$	22.5621	1.46867
$237$	2.36184	0.153418
$238$	0	0
$239$	−7.53489	−0.487392	−0.243696	−	0.969852i	$-0.578360\pi$
$239$	−7.53489	−0.487392	−0.243696	+	0.969852i	$0.578360\pi$
$240$	0	0
$241$	−28.5231	−1.83733	−0.918667	−	0.395032i	$-0.870733\pi$
$241$	−28.5231	−1.83733	−0.918667	+	0.395032i	$0.870733\pi$
$242$	−8.64800	−0.555915
$243$	1.00000	0.0641500
$244$	−8.77896	−0.562016
$245$	0	0
$246$	−3.88508	−0.247704
$247$	14.0983	0.897051
$248$	−11.3544	−0.721005
$249$	6.87342	0.435586
$250$	0	0
$251$	22.8501	1.44229	0.721143	−	0.692786i	$-0.243617\pi$
$251$	22.8501	1.44229	0.721143	+	0.692786i	$0.243617\pi$
$252$	0	0
$253$	40.1008	2.52112
$254$	8.66723	0.543830
$255$	0	0
$256$	−0.158689	−0.00991808
$257$	−25.1381	−1.56807	−0.784036	−	0.620716i	$-0.786842\pi$
$257$	−25.1381	−1.56807	−0.784036	+	0.620716i	$0.786842\pi$
$258$	−0.700372	−0.0436033
$259$	0	0
$260$	0	0
$261$	4.00000	0.247594
$262$	−4.45901	−0.275479
$263$	−13.5582	−0.836034	−0.418017	−	0.908439i	$-0.637275\pi$
$263$	−13.5582	−0.836034	−0.418017	+	0.908439i	$0.637275\pi$
$264$	9.90838	0.609819
$265$	0	0
$266$	0	0
$267$	4.35019	0.266227
$268$	−6.59439	−0.402817
$269$	9.02725	0.550401	0.275201	−	0.961387i	$-0.411256\pi$
$269$	9.02725	0.550401	0.275201	+	0.961387i	$0.411256\pi$
$270$	0	0
$271$	−4.77639	−0.290145	−0.145072	−	0.989421i	$-0.546342\pi$
$271$	−4.77639	−0.290145	−0.145072	+	0.989421i	$0.546342\pi$
$272$	−0.908520	−0.0550871
$273$	0	0
$274$	0.412397	0.0249138
$275$	0	0
$276$	−13.1888	−0.793872
$277$	1.87071	0.112400	0.0562002	−	0.998420i	$-0.482101\pi$
$277$	1.87071	0.112400	0.0562002	+	0.998420i	$0.482101\pi$
$278$	−7.07420	−0.424283
$279$	−6.11763	−0.366253
$280$	0	0
$281$	0.972748	0.0580293	0.0290147	−	0.999579i	$-0.490763\pi$
$281$	0.972748	0.0580293	0.0290147	+	0.999579i	$0.490763\pi$
$282$	−4.37349	−0.260438
$283$	24.1797	1.43733	0.718667	−	0.695354i	$-0.244752\pi$
$283$	24.1797	1.43733	0.718667	+	0.695354i	$0.244752\pi$
$284$	−27.1942	−1.61368
$285$	0	0
$286$	13.4396	0.794703
$287$	0	0
$288$	−4.99411	−0.294280
$289$	−16.8774	−0.992786
$290$	0	0
$291$	−5.49993	−0.322412
$292$	13.4793	0.788818
$293$	−15.5349	−0.907558	−0.453779	−	0.891114i	$-0.649924\pi$
$293$	−15.5349	−0.907558	−0.453779	+	0.891114i	$0.649924\pi$
$294$	0	0
$295$	0	0
$296$	−6.53918	−0.380082
$297$	5.33853	0.309773
$298$	−9.58531	−0.555262
$299$	−38.2665	−2.21301
$300$	0	0
$301$	0	0
$302$	−9.15564	−0.526848
$303$	−1.64981	−0.0947793
$304$	−7.17984	−0.411792
$305$	0	0
$306$	0.173053	0.00989276
$307$	3.39395	0.193703	0.0968516	−	0.995299i	$-0.469123\pi$
$307$	3.39395	0.193703	0.0968516	+	0.995299i	$0.469123\pi$
$308$	0	0
$309$	10.1614	0.578062
$310$	0	0
$311$	2.46511	0.139783	0.0698917	−	0.997555i	$-0.477735\pi$
$311$	2.46511	0.139783	0.0698917	+	0.997555i	$0.477735\pi$
$312$	−9.45513	−0.535291
$313$	0.475526	0.0268783	0.0134392	−	0.999910i	$-0.495722\pi$
$313$	0.475526	0.0268783	0.0134392	+	0.999910i	$0.495722\pi$
$314$	−2.37181	−0.133849
$315$	0	0
$316$	−4.14690	−0.233281
$317$	14.2995	0.803139	0.401570	−	0.915828i	$-0.368465\pi$
$317$	14.2995	0.803139	0.401570	+	0.915828i	$0.368465\pi$
$318$	3.12658	0.175330
$319$	21.3541	1.19560
$320$	0	0
$321$	−5.36184	−0.299269
$322$	0	0
$323$	0.969121	0.0539233
$324$	−1.75579	−0.0975440
$325$	0	0
$326$	−0.408655	−0.0226333
$327$	−0.105979	−0.00586064
$328$	14.5915	0.805683
$329$	0	0
$330$	0	0
$331$	−19.5855	−1.07651	−0.538257	−	0.842781i	$-0.680917\pi$
$331$	−19.5855	−1.07651	−0.538257	+	0.842781i	$0.680917\pi$
$332$	−12.0683	−0.662334
$333$	−3.52324	−0.193072
$334$	2.28054	0.124785
$335$	0	0
$336$	0	0
$337$	4.38230	0.238719	0.119360	−	0.992851i	$-0.461916\pi$
$337$	4.38230	0.238719	0.119360	+	0.992851i	$0.461916\pi$
$338$	−6.40061	−0.348147
$339$	−4.67707	−0.254023
$340$	0	0
$341$	−32.6592	−1.76859
$342$	1.36760	0.0739512
$343$	0	0
$344$	2.63045	0.141824
$345$	0	0
$346$	−6.26467	−0.336791
$347$	−9.59710	−0.515200	−0.257600	−	0.966252i	$-0.582932\pi$
$347$	−9.59710	−0.515200	−0.257600	+	0.966252i	$0.582932\pi$
$348$	−7.02317	−0.376481
$349$	19.9007	1.06526	0.532629	−	0.846349i	$-0.321204\pi$
$349$	19.9007	1.06526	0.532629	+	0.846349i	$0.321204\pi$
$350$	0	0
$351$	−5.09433	−0.271915
$352$	−26.6612	−1.42105
$353$	−27.3967	−1.45818	−0.729089	−	0.684419i	$-0.760056\pi$
$353$	−27.3967	−1.45818	−0.729089	+	0.684419i	$0.760056\pi$
$354$	6.35019	0.337509
$355$	0	0
$356$	−7.63802	−0.404815
$357$	0	0
$358$	−6.44722	−0.340746
$359$	25.5505	1.34850	0.674252	−	0.738502i	$-0.264466\pi$
$359$	25.5505	1.34850	0.674252	+	0.738502i	$0.264466\pi$
$360$	0	0
$361$	−11.3412	−0.596908
$362$	7.44194	0.391140
$363$	17.4999	0.918508
$364$	0	0
$365$	0	0
$366$	−2.47087	−0.129154
$367$	22.2715	1.16256	0.581280	−	0.813703i	$-0.302552\pi$
$367$	22.2715	1.16256	0.581280	+	0.813703i	$0.302552\pi$
$368$	19.4880	1.01588
$369$	7.86177	0.409267
$370$	0	0
$371$	0	0
$372$	10.7413	0.556910
$373$	−5.29477	−0.274153	−0.137076	−	0.990560i	$-0.543771\pi$
$373$	−5.29477	−0.274153	−0.137076	+	0.990560i	$0.543771\pi$
$374$	0.923847	0.0477710
$375$	0	0
$376$	16.4259	0.847103
$377$	−20.3773	−1.04948
$378$	0	0
$379$	3.10203	0.159341	0.0796704	−	0.996821i	$-0.474613\pi$
$379$	3.10203	0.159341	0.0796704	+	0.996821i	$0.474613\pi$
$380$	0	0
$381$	−17.5388	−0.898542
$382$	−4.70808	−0.240886
$383$	−21.6127	−1.10436	−0.552179	−	0.833726i	$-0.686203\pi$
$383$	−21.6127	−1.10436	−0.552179	+	0.833726i	$0.686203\pi$
$384$	11.3328	0.578323
$385$	0	0
$386$	7.46497	0.379957
$387$	1.41726	0.0720434
$388$	9.65674	0.490247
$389$	−12.8616	−0.652111	−0.326055	−	0.945351i	$-0.605720\pi$
$389$	−12.8616	−0.652111	−0.326055	+	0.945351i	$0.605720\pi$
$390$	0	0
$391$	−2.63045	−0.133028
$392$	0	0
$393$	9.02317	0.455159
$394$	−8.34611	−0.420471
$395$	0	0
$396$	−9.37336	−0.471029
$397$	−6.99514	−0.351076	−0.175538	−	0.984473i	$-0.556166\pi$
$397$	−6.99514	−0.351076	−0.175538	+	0.984473i	$0.556166\pi$
$398$	−12.4668	−0.624904
$399$	0	0
$400$	0	0
$401$	−3.16140	−0.157873	−0.0789364	−	0.996880i	$-0.525152\pi$
$401$	−3.16140	−0.157873	−0.0789364	+	0.996880i	$0.525152\pi$
$402$	−1.85601	−0.0925695
$403$	31.1652	1.55245
$404$	2.89673	0.144118
$405$	0	0
$406$	0	0
$407$	−18.8089	−0.932324
$408$	−0.649950	−0.0321773
$409$	−2.49507	−0.123373	−0.0616866	−	0.998096i	$-0.519648\pi$
$409$	−2.49507	−0.123373	−0.0616866	+	0.998096i	$0.519648\pi$
$410$	0	0
$411$	−0.834520	−0.0411638
$412$	−17.8413	−0.878978
$413$	0	0
$414$	−3.71203	−0.182436
$415$	0	0
$416$	25.4416	1.24738
$417$	14.3152	0.701020
$418$	7.30097	0.357102
$419$	32.6748	1.59627	0.798134	−	0.602480i	$-0.205821\pi$
$419$	32.6748	1.59627	0.798134	+	0.602480i	$0.205821\pi$
$420$	0	0
$421$	−26.8774	−1.30992	−0.654961	−	0.755662i	$-0.727315\pi$
$421$	−26.8774	−1.30992	−0.654961	+	0.755662i	$0.727315\pi$
$422$	−10.0795	−0.490662
$423$	8.85012	0.430307
$424$	−11.7428	−0.570279
$425$	0	0
$426$	−7.65389	−0.370832
$427$	0	0
$428$	9.41428	0.455056
$429$	−27.1962	−1.31305
$430$	0	0
$431$	14.1887	0.683443	0.341722	−	0.939801i	$-0.388990\pi$
$431$	14.1887	0.683443	0.341722	+	0.939801i	$0.388990\pi$
$432$	2.59439	0.124823
$433$	−29.9717	−1.44035	−0.720174	−	0.693794i	$-0.755938\pi$
$433$	−29.9717	−1.44035	−0.720174	+	0.693794i	$0.755938\pi$
$434$	0	0
$435$	0	0
$436$	0.186077	0.00891146
$437$	−20.7879	−0.994420
$438$	3.79380	0.181275
$439$	2.07386	0.0989802	0.0494901	−	0.998775i	$-0.484240\pi$
$439$	2.07386	0.0989802	0.0494901	+	0.998775i	$0.484240\pi$
$440$	0	0
$441$	0	0
$442$	−0.881586	−0.0419328
$443$	32.0775	1.52405	0.762025	−	0.647548i	$-0.224205\pi$
$443$	32.0775	1.52405	0.762025	+	0.647548i	$0.224205\pi$
$444$	6.18608	0.293578
$445$	0	0
$446$	−11.1554	−0.528222
$447$	19.3967	0.917431
$448$	0	0
$449$	−4.46103	−0.210529	−0.105264	−	0.994444i	$-0.533569\pi$
$449$	−4.46103	−0.210529	−0.105264	+	0.994444i	$0.533569\pi$
$450$	0	0
$451$	41.9703	1.97631
$452$	8.21196	0.386258
$453$	18.5272	0.870483
$454$	−7.00014	−0.328533
$455$	0	0
$456$	−5.13641	−0.240535
$457$	−2.27022	−0.106197	−0.0530983	−	0.998589i	$-0.516910\pi$
$457$	−2.27022	−0.106197	−0.0530983	+	0.998589i	$0.516910\pi$
$458$	−8.79936	−0.411167
$459$	−0.350186	−0.0163453
$460$	0	0
$461$	−4.33096	−0.201713	−0.100856	−	0.994901i	$-0.532158\pi$
$461$	−4.33096	−0.201713	−0.100856	+	0.994901i	$0.532158\pi$
$462$	0	0
$463$	−41.7856	−1.94194	−0.970971	−	0.239196i	$-0.923116\pi$
$463$	−41.7856	−1.94194	−0.970971	+	0.239196i	$0.923116\pi$
$464$	10.3776	0.481767
$465$	0	0
$466$	−6.74537	−0.312473
$467$	−38.3773	−1.77589	−0.887945	−	0.459950i	$-0.847867\pi$
$467$	−38.3773	−1.77589	−0.887945	+	0.459950i	$0.847867\pi$
$468$	8.94458	0.413463
$469$	0	0
$470$	0	0
$471$	4.79956	0.221152
$472$	−23.8500	−1.09778
$473$	7.56609	0.347889
$474$	−1.16716	−0.0536093
$475$	0	0
$476$	0	0
$477$	−6.32688	−0.289688
$478$	3.72354	0.170311
$479$	−38.0386	−1.73803	−0.869015	−	0.494786i	$-0.835246\pi$
$479$	−38.0386	−1.73803	−0.869015	+	0.494786i	$0.835246\pi$
$480$	0	0
$481$	17.9485	0.818382
$482$	14.0954	0.642026
$483$	0	0
$484$	−30.7263	−1.39665
$485$	0	0
$486$	−0.494173	−0.0224162
$487$	22.5827	1.02332	0.511661	−	0.859188i	$-0.329030\pi$
$487$	22.5827	1.02332	0.511661	+	0.859188i	$0.329030\pi$
$488$	9.28006	0.420089
$489$	0.826947	0.0373959
$490$	0	0
$491$	−0.134148	−0.00605400	−0.00302700	−	0.999995i	$-0.500964\pi$
$491$	−0.134148	−0.00605400	−0.00302700	+	0.999995i	$0.500964\pi$
$492$	−13.8036	−0.622316
$493$	−1.40074	−0.0630863
$494$	−6.96699	−0.313460
$495$	0	0
$496$	−15.8715	−0.712653
$497$	0	0
$498$	−3.39666	−0.152208
$499$	−15.5554	−0.696353	−0.348177	−	0.937429i	$-0.613199\pi$
$499$	−15.5554	−0.696353	−0.348177	+	0.937429i	$0.613199\pi$
$500$	0	0
$501$	−4.61485	−0.206176
$502$	−11.2919	−0.503983
$503$	−4.39272	−0.195862	−0.0979308	−	0.995193i	$-0.531222\pi$
$503$	−4.39272	−0.195862	−0.0979308	+	0.995193i	$0.531222\pi$
$504$	0	0
$505$	0	0
$506$	−19.8168	−0.880963
$507$	12.9522	0.575225
$508$	30.7946	1.36629
$509$	−2.08753	−0.0925283	−0.0462642	−	0.998929i	$-0.514732\pi$
$509$	−2.08753	−0.0925283	−0.0462642	+	0.998929i	$0.514732\pi$
$510$	0	0
$511$	0	0
$512$	−22.5871	−0.998219
$513$	−2.76745	−0.122186
$514$	12.4226	0.547936
$515$	0	0
$516$	−2.48841	−0.109546
$517$	47.2466	2.07791
$518$	0	0
$519$	12.6771	0.556461
$520$	0	0
$521$	36.7001	1.60786	0.803930	−	0.594724i	$-0.202738\pi$
$521$	36.7001	1.60786	0.803930	+	0.594724i	$0.202738\pi$
$522$	−1.97669	−0.0865175
$523$	−12.6291	−0.552231	−0.276116	−	0.961124i	$-0.589047\pi$
$523$	−12.6291	−0.552231	−0.276116	+	0.961124i	$0.589047\pi$
$524$	−15.8428	−0.692097
$525$	0	0
$526$	6.70010	0.292138
$527$	2.14231	0.0933205
$528$	13.8503	0.602755
$529$	33.4239	1.45321
$530$	0	0
$531$	−12.8501	−0.557648
$532$	0	0
$533$	−40.0504	−1.73478
$534$	−2.14975	−0.0930286
$535$	0	0
$536$	6.97080	0.301093
$537$	13.0465	0.562997
$538$	−4.46103	−0.192329
$539$	0	0
$540$	0	0
$541$	38.8412	1.66991	0.834956	−	0.550316i	$-0.185493\pi$
$541$	38.8412	1.66991	0.834956	+	0.550316i	$0.185493\pi$
$542$	2.36036	0.101386
$543$	−15.0594	−0.646259
$544$	1.74887	0.0749820
$545$	0	0
$546$	0	0
$547$	18.6010	0.795323	0.397662	−	0.917532i	$-0.369822\pi$
$547$	18.6010	0.795323	0.397662	+	0.917532i	$0.369822\pi$
$548$	1.46524	0.0625921
$549$	5.00000	0.213395
$550$	0	0
$551$	−11.0698	−0.471589
$552$	13.9416	0.593394
$553$	0	0
$554$	−0.924457	−0.0392764
$555$	0	0
$556$	−25.1346	−1.06594
$557$	−5.16548	−0.218868	−0.109434	−	0.993994i	$-0.534904\pi$
$557$	−5.16548	−0.218868	−0.109434	+	0.993994i	$0.534904\pi$
$558$	3.02317	0.127981
$559$	−7.21998	−0.305373
$560$	0	0
$561$	−1.86948	−0.0789295
$562$	−0.480706	−0.0202774
$563$	−12.6382	−0.532635	−0.266317	−	0.963885i	$-0.585807\pi$
$563$	−12.6382	−0.532635	−0.266317	+	0.963885i	$0.585807\pi$
$564$	−15.5390	−0.654309
$565$	0	0
$566$	−11.9490	−0.502253
$567$	0	0
$568$	28.7464	1.20617
$569$	−8.79741	−0.368807	−0.184403	−	0.982851i	$-0.559035\pi$
$569$	−8.79741	−0.368807	−0.184403	+	0.982851i	$0.559035\pi$
$570$	0	0
$571$	−39.9033	−1.66990	−0.834950	−	0.550327i	$-0.814503\pi$
$571$	−39.9033	−1.66990	−0.834950	+	0.550327i	$0.814503\pi$
$572$	47.7509	1.99657
$573$	9.52718	0.398004
$574$	0	0
$575$	0	0
$576$	−2.72083	−0.113368
$577$	−2.71689	−0.113106	−0.0565528	−	0.998400i	$-0.518011\pi$
$577$	−2.71689	−0.113106	−0.0565528	+	0.998400i	$0.518011\pi$
$578$	8.34035	0.346913
$579$	−15.1060	−0.627783
$580$	0	0
$581$	0	0
$582$	2.71792	0.112661
$583$	−33.7763	−1.39887
$584$	−14.2487	−0.589617
$585$	0	0
$586$	7.67693	0.317131
$587$	−9.02317	−0.372426	−0.186213	−	0.982509i	$-0.559621\pi$
$587$	−9.02317	−0.372426	−0.186213	+	0.982509i	$0.559621\pi$
$588$	0	0
$589$	16.9302	0.697597
$590$	0	0
$591$	16.8890	0.694722
$592$	−9.14067	−0.375679
$593$	10.4651	0.429750	0.214875	−	0.976642i	$-0.431066\pi$
$593$	10.4651	0.429750	0.214875	+	0.976642i	$0.431066\pi$
$594$	−2.63816	−0.108245
$595$	0	0
$596$	−34.0565	−1.39501
$597$	25.2276	1.03250
$598$	18.9103	0.773298
$599$	9.63830	0.393810	0.196905	−	0.980423i	$-0.436911\pi$
$599$	9.63830	0.393810	0.196905	+	0.980423i	$0.436911\pi$
$600$	0	0
$601$	7.19544	0.293508	0.146754	−	0.989173i	$-0.453117\pi$
$601$	7.19544	0.293508	0.146754	+	0.989173i	$0.453117\pi$
$602$	0	0
$603$	3.75579	0.152948
$604$	−32.5299	−1.32362
$605$	0	0
$606$	0.815294	0.0331191
$607$	39.3384	1.59670	0.798348	−	0.602196i	$-0.205708\pi$
$607$	39.3384	1.59670	0.798348	+	0.602196i	$0.205708\pi$
$608$	13.8209	0.560512
$609$	0	0
$610$	0	0
$611$	−45.0854	−1.82396
$612$	0.614854	0.0248540
$613$	−19.5464	−0.789472	−0.394736	−	0.918795i	$-0.629164\pi$
$613$	−19.5464	−0.789472	−0.394736	+	0.918795i	$0.629164\pi$
$614$	−1.67720	−0.0676863
$615$	0	0
$616$	0	0
$617$	35.9588	1.44765	0.723824	−	0.689985i	$-0.242383\pi$
$617$	35.9588	1.44765	0.723824	+	0.689985i	$0.242383\pi$
$618$	−5.02149	−0.201994
$619$	−8.55943	−0.344033	−0.172016	−	0.985094i	$-0.555028\pi$
$619$	−8.55943	−0.344033	−0.172016	+	0.985094i	$0.555028\pi$
$620$	0	0
$621$	7.51159	0.301430
$622$	−1.21819	−0.0488450
$623$	0	0
$624$	−13.2167	−0.529091
$625$	0	0
$626$	−0.234992	−0.00939219
$627$	−14.7741	−0.590021
$628$	−8.42703	−0.336275
$629$	1.23379	0.0491944
$630$	0	0
$631$	−17.5192	−0.697427	−0.348713	−	0.937229i	$-0.613381\pi$
$631$	−17.5192	−0.697427	−0.348713	+	0.937229i	$0.613381\pi$
$632$	4.38360	0.174370
$633$	20.3967	0.810695
$634$	−7.06643	−0.280644
$635$	0	0
$636$	11.1087	0.440488
$637$	0	0
$638$	−10.5526	−0.417783
$639$	15.4883	0.612707
$640$	0	0
$641$	16.2819	0.643095	0.321548	−	0.946893i	$-0.395797\pi$
$641$	16.2819	0.643095	0.321548	+	0.946893i	$0.395797\pi$
$642$	2.64968	0.104574
$643$	15.7544	0.621294	0.310647	−	0.950525i	$-0.399454\pi$
$643$	15.7544	0.621294	0.310647	+	0.950525i	$0.399454\pi$
$644$	0	0
$645$	0	0
$646$	−0.478914	−0.0188426
$647$	−45.4394	−1.78641	−0.893203	−	0.449653i	$-0.851548\pi$
$647$	−45.4394	−1.78641	−0.893203	+	0.449653i	$0.851548\pi$
$648$	1.85601	0.0729111
$649$	−68.6008	−2.69282
$650$	0	0
$651$	0	0
$652$	−1.45195	−0.0568627
$653$	19.1614	0.749844	0.374922	−	0.927056i	$-0.377670\pi$
$653$	19.1614	0.749844	0.374922	+	0.927056i	$0.377670\pi$
$654$	0.0523719	0.00204790
$655$	0	0
$656$	20.3965	0.796351
$657$	−7.67707	−0.299511
$658$	0	0
$659$	−8.76258	−0.341342	−0.170671	−	0.985328i	$-0.554593\pi$
$659$	−8.76258	−0.341342	−0.170671	+	0.985328i	$0.554593\pi$
$660$	0	0
$661$	9.07602	0.353016	0.176508	−	0.984299i	$-0.443520\pi$
$661$	9.07602	0.353016	0.176508	+	0.984299i	$0.443520\pi$
$662$	9.67861	0.376170
$663$	1.78396	0.0692833
$664$	12.7572	0.495074
$665$	0	0
$666$	1.74109	0.0674659
$667$	30.0463	1.16340
$668$	8.10273	0.313504
$669$	22.5738	0.872753
$670$	0	0
$671$	26.6927	1.03046
$672$	0	0
$673$	−18.3460	−0.707185	−0.353593	−	0.935400i	$-0.615040\pi$
$673$	−18.3460	−0.707185	−0.353593	+	0.935400i	$0.615040\pi$
$674$	−2.16562	−0.0834164
$675$	0	0
$676$	−22.7413	−0.874665
$677$	1.79593	0.0690233	0.0345116	−	0.999404i	$-0.489012\pi$
$677$	1.79593	0.0690233	0.0345116	+	0.999404i	$0.489012\pi$
$678$	2.31128	0.0887642
$679$	0	0
$680$	0	0
$681$	14.1653	0.542817
$682$	16.1393	0.618006
$683$	−44.3929	−1.69865	−0.849324	−	0.527873i	$-0.822990\pi$
$683$	−44.3929	−1.69865	−0.849324	+	0.527873i	$0.822990\pi$
$684$	4.85906	0.185791
$685$	0	0
$686$	0	0
$687$	17.8062	0.679349
$688$	3.67693	0.140182
$689$	32.2312	1.22791
$690$	0	0
$691$	−43.1925	−1.64312	−0.821559	−	0.570123i	$-0.806896\pi$
$691$	−43.1925	−1.64312	−0.821559	+	0.570123i	$0.806896\pi$
$692$	−22.2583	−0.846134
$693$	0	0
$694$	4.74263	0.180028
$695$	0	0
$696$	7.42405	0.281408
$697$	−2.75308	−0.104280
$698$	−9.83438	−0.372237
$699$	13.6498	0.516283
$700$	0	0
$701$	−9.69616	−0.366219	−0.183109	−	0.983093i	$-0.558616\pi$
$701$	−9.69616	−0.366219	−0.183109	+	0.983093i	$0.558616\pi$
$702$	2.51748	0.0950162
$703$	9.75037	0.367742
$704$	−14.5253	−0.547441
$705$	0	0
$706$	13.5387	0.509536
$707$	0	0
$708$	22.5621	0.847937
$709$	18.1537	0.681776	0.340888	−	0.940104i	$-0.389272\pi$
$709$	18.1537	0.681776	0.340888	+	0.940104i	$0.389272\pi$
$710$	0	0
$711$	2.36184	0.0885759
$712$	8.07400	0.302586
$713$	−45.9531	−1.72096
$714$	0	0
$715$	0	0
$716$	−22.9069	−0.856071
$717$	−7.53489	−0.281396
$718$	−12.6264	−0.471212
$719$	35.1808	1.31202	0.656011	−	0.754751i	$-0.272242\pi$
$719$	35.1808	1.31202	0.656011	+	0.754751i	$0.272242\pi$
$720$	0	0
$721$	0	0
$722$	5.60454	0.208579
$723$	−28.5231	−1.06079
$724$	26.4411	0.982677
$725$	0	0
$726$	−8.64800	−0.320957
$727$	4.79075	0.177679	0.0888396	−	0.996046i	$-0.471684\pi$
$727$	4.79075	0.177679	0.0888396	+	0.996046i	$0.471684\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−0.496305	−0.0183565
$732$	−8.77896	−0.324480
$733$	12.1243	0.447821	0.223910	−	0.974610i	$-0.428118\pi$
$733$	12.1243	0.447821	0.223910	+	0.974610i	$0.428118\pi$
$734$	−11.0060	−0.406237
$735$	0	0
$736$	−37.5136	−1.38277
$737$	20.0504	0.738567
$738$	−3.88508	−0.143012
$739$	−32.9163	−1.21084	−0.605422	−	0.795904i	$-0.706996\pi$
$739$	−32.9163	−1.21084	−0.605422	+	0.795904i	$0.706996\pi$
$740$	0	0
$741$	14.0983	0.517913
$742$	0	0
$743$	−38.8112	−1.42385	−0.711923	−	0.702258i	$-0.752175\pi$
$743$	−38.8112	−1.42385	−0.711923	+	0.702258i	$0.752175\pi$
$744$	−11.3544	−0.416273
$745$	0	0
$746$	2.61653	0.0957980
$747$	6.87342	0.251485
$748$	3.28242	0.120017
$749$	0	0
$750$	0	0
$751$	−30.0682	−1.09720	−0.548602	−	0.836084i	$-0.684840\pi$
$751$	−30.0682	−1.09720	−0.548602	+	0.836084i	$0.684840\pi$
$752$	22.9607	0.837290
$753$	22.8501	0.832705
$754$	10.0699	0.366725
$755$	0	0
$756$	0	0
$757$	7.17713	0.260857	0.130429	−	0.991458i	$-0.458365\pi$
$757$	7.17713	0.260857	0.130429	+	0.991458i	$0.458365\pi$
$758$	−1.53294	−0.0556790
$759$	40.1008	1.45557
$760$	0	0
$761$	7.60862	0.275812	0.137906	−	0.990445i	$-0.455963\pi$
$761$	7.60862	0.275812	0.137906	+	0.990445i	$0.455963\pi$
$762$	8.66723	0.313980
$763$	0	0
$764$	−16.7278	−0.605189
$765$	0	0
$766$	10.6804	0.385899
$767$	65.4627	2.36372
$768$	−0.158689	−0.00572621
$769$	−50.2712	−1.81283	−0.906413	−	0.422393i	$-0.861190\pi$
$769$	−50.2712	−1.81283	−0.906413	+	0.422393i	$0.861190\pi$
$770$	0	0
$771$	−25.1381	−0.905326
$772$	26.5230	0.954582
$773$	27.1010	0.974755	0.487377	−	0.873192i	$-0.337954\pi$
$773$	27.1010	0.974755	0.487377	+	0.873192i	$0.337954\pi$
$774$	−0.700372	−0.0251744
$775$	0	0
$776$	−10.2079	−0.366444
$777$	0	0
$778$	6.35588	0.227869
$779$	−21.7570	−0.779526
$780$	0	0
$781$	82.6847	2.95869
$782$	1.29990	0.0464843
$783$	4.00000	0.142948
$784$	0	0
$785$	0	0
$786$	−4.45901	−0.159048
$787$	34.3501	1.22445	0.612224	−	0.790685i	$-0.290275\pi$
$787$	34.3501	1.22445	0.612224	+	0.790685i	$0.290275\pi$
$788$	−29.6536	−1.05637
$789$	−13.5582	−0.482685
$790$	0	0
$791$	0	0
$792$	9.90838	0.352079
$793$	−25.4716	−0.904524
$794$	3.45681	0.122678
$795$	0	0
$796$	−44.2944	−1.56997
$797$	−16.5581	−0.586517	−0.293258	−	0.956033i	$-0.594740\pi$
$797$	−16.5581	−0.586517	−0.293258	+	0.956033i	$0.594740\pi$
$798$	0	0
$799$	−3.09919	−0.109641
$800$	0	0
$801$	4.35019	0.153706
$802$	1.56228	0.0551660
$803$	−40.9843	−1.44630
$804$	−6.59439	−0.232566
$805$	0	0
$806$	−15.4010	−0.542478
$807$	9.02725	0.317774
$808$	−3.06208	−0.107723
$809$	−8.55806	−0.300885	−0.150443	−	0.988619i	$-0.548070\pi$
$809$	−8.55806	−0.300885	−0.150443	+	0.988619i	$0.548070\pi$
$810$	0	0
$811$	−21.4277	−0.752429	−0.376215	−	0.926533i	$-0.622774\pi$
$811$	−21.4277	−0.752429	−0.376215	+	0.926533i	$0.622774\pi$
$812$	0	0
$813$	−4.77639	−0.167515
$814$	9.29487	0.325785
$815$	0	0
$816$	−0.908520	−0.0318046
$817$	−3.92219	−0.137220
$818$	1.23300	0.0431107
$819$	0	0
$820$	0	0
$821$	−15.1202	−0.527699	−0.263849	−	0.964564i	$-0.584992\pi$
$821$	−15.1202	−0.527699	−0.263849	+	0.964564i	$0.584992\pi$
$822$	0.412397	0.0143840
$823$	−24.7043	−0.861138	−0.430569	−	0.902558i	$-0.641687\pi$
$823$	−24.7043	−0.861138	−0.430569	+	0.902558i	$0.641687\pi$
$824$	18.8597	0.657008
$825$	0	0
$826$	0	0
$827$	28.8578	1.00348	0.501742	−	0.865017i	$-0.332693\pi$
$827$	28.8578	1.00348	0.501742	+	0.865017i	$0.332693\pi$
$828$	−13.1888	−0.458342
$829$	0.869480	0.0301983	0.0150991	−	0.999886i	$-0.495194\pi$
$829$	0.869480	0.0301983	0.0150991	+	0.999886i	$0.495194\pi$
$830$	0	0
$831$	1.87071	0.0648944
$832$	13.8608	0.480537
$833$	0	0
$834$	−7.07420	−0.244960
$835$	0	0
$836$	25.9403	0.897163
$837$	−6.11763	−0.211456
$838$	−16.1470	−0.557789
$839$	−21.7235	−0.749980	−0.374990	−	0.927029i	$-0.622354\pi$
$839$	−21.7235	−0.749980	−0.374990	+	0.927029i	$0.622354\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	13.2821	0.457731
$843$	0.972748	0.0335032
$844$	−35.8123	−1.23271
$845$	0	0
$846$	−4.37349	−0.150364
$847$	0	0
$848$	−16.4144	−0.563673
$849$	24.1797	0.829845
$850$	0	0
$851$	−26.4651	−0.907212
$852$	−27.1942	−0.931658
$853$	−0.670546	−0.0229591	−0.0114795	−	0.999934i	$-0.503654\pi$
$853$	−0.670546	−0.0229591	−0.0114795	+	0.999934i	$0.503654\pi$
$854$	0	0
$855$	0	0
$856$	−9.95164	−0.340140
$857$	−24.4884	−0.836508	−0.418254	−	0.908330i	$-0.637358\pi$
$857$	−24.4884	−0.836508	−0.418254	+	0.908330i	$0.637358\pi$
$858$	13.4396	0.458822
$859$	−9.59939	−0.327527	−0.163764	−	0.986500i	$-0.552363\pi$
$859$	−9.59939	−0.327527	−0.163764	+	0.986500i	$0.552363\pi$
$860$	0	0
$861$	0	0
$862$	−7.01165	−0.238818
$863$	−29.5039	−1.00432	−0.502162	−	0.864774i	$-0.667462\pi$
$863$	−29.5039	−1.00432	−0.502162	+	0.864774i	$0.667462\pi$
$864$	−4.99411	−0.169903
$865$	0	0
$866$	14.8112	0.503306
$867$	−16.8774	−0.573186
$868$	0	0
$869$	12.6088	0.427723
$870$	0	0
$871$	−19.1332	−0.648305
$872$	−0.196698	−0.00666103
$873$	−5.49993	−0.186144
$874$	10.2728	0.347484
$875$	0	0
$876$	13.4793	0.455425
$877$	−5.30765	−0.179227	−0.0896134	−	0.995977i	$-0.528563\pi$
$877$	−5.30765	−0.179227	−0.0896134	+	0.995977i	$0.528563\pi$
$878$	−1.02485	−0.0345870
$879$	−15.5349	−0.523979
$880$	0	0
$881$	−49.2817	−1.66034	−0.830172	−	0.557507i	$-0.811758\pi$
$881$	−49.2817	−1.66034	−0.830172	+	0.557507i	$0.811758\pi$
$882$	0	0
$883$	−6.02862	−0.202879	−0.101440	−	0.994842i	$-0.532345\pi$
$883$	−6.02862	−0.202879	−0.101440	+	0.994842i	$0.532345\pi$
$884$	−3.13227	−0.105350
$885$	0	0
$886$	−15.8519	−0.532554
$887$	−7.98486	−0.268105	−0.134053	−	0.990974i	$-0.542799\pi$
$887$	−7.98486	−0.268105	−0.134053	+	0.990974i	$0.542799\pi$
$888$	−6.53918	−0.219440
$889$	0	0
$890$	0	0
$891$	5.33853	0.178848
$892$	−39.6349	−1.32707
$893$	−24.4922	−0.819601
$894$	−9.58531	−0.320581
$895$	0	0
$896$	0	0
$897$	−38.2665	−1.27768
$898$	2.20452	0.0735658
$899$	−24.4705	−0.816138
$900$	0	0
$901$	2.21558	0.0738118
$902$	−20.7406	−0.690587
$903$	0	0
$904$	−8.68069	−0.288716
$905$	0	0
$906$	−9.15564	−0.304176
$907$	2.53108	0.0840432	0.0420216	−	0.999117i	$-0.486620\pi$
$907$	2.53108	0.0840432	0.0420216	+	0.999117i	$0.486620\pi$
$908$	−24.8714	−0.825387
$909$	−1.64981	−0.0547209
$910$	0	0
$911$	1.41047	0.0467309	0.0233655	−	0.999727i	$-0.492562\pi$
$911$	1.41047	0.0467309	0.0233655	+	0.999727i	$0.492562\pi$
$912$	−7.17984	−0.237748
$913$	36.6940	1.21439
$914$	1.12188	0.0371086
$915$	0	0
$916$	−31.2640	−1.03299
$917$	0	0
$918$	0.173053	0.00571159
$919$	46.6836	1.53995	0.769975	−	0.638074i	$-0.220269\pi$
$919$	46.6836	1.53995	0.769975	+	0.638074i	$0.220269\pi$
$920$	0	0
$921$	3.39395	0.111835
$922$	2.14025	0.0704852
$923$	−78.9023	−2.59710
$924$	0	0
$925$	0	0
$926$	20.6493	0.678579
$927$	10.1614	0.333744
$928$	−19.9764	−0.655758
$929$	17.8385	0.585261	0.292631	−	0.956226i	$-0.405469\pi$
$929$	17.8385	0.585261	0.292631	+	0.956226i	$0.405469\pi$
$930$	0	0
$931$	0	0
$932$	−23.9662	−0.785040
$933$	2.46511	0.0807040
$934$	18.9650	0.620555
$935$	0	0
$936$	−9.45513	−0.309051
$937$	0.343849	0.0112330	0.00561652	−	0.999984i	$-0.498212\pi$
$937$	0.343849	0.0112330	0.00561652	+	0.999984i	$0.498212\pi$
$938$	0	0
$939$	0.475526	0.0155182
$940$	0	0
$941$	46.1475	1.50436	0.752182	−	0.658955i	$-0.229001\pi$
$941$	46.1475	1.50436	0.752182	+	0.658955i	$0.229001\pi$
$942$	−2.37181	−0.0772779
$943$	59.0544	1.92307
$944$	−33.3383	−1.08507
$945$	0	0
$946$	−3.73896	−0.121564
$947$	−27.6618	−0.898887	−0.449444	−	0.893309i	$-0.648378\pi$
$947$	−27.6618	−0.898887	−0.449444	+	0.893309i	$0.648378\pi$
$948$	−4.14690	−0.134685
$949$	39.1095	1.26955
$950$	0	0
$951$	14.2995	0.463693
$952$	0	0
$953$	4.24907	0.137641	0.0688204	−	0.997629i	$-0.478076\pi$
$953$	4.24907	0.137641	0.0688204	+	0.997629i	$0.478076\pi$
$954$	3.12658	0.101227
$955$	0	0
$956$	13.2297	0.427879
$957$	21.3541	0.690281
$958$	18.7977	0.607325
$959$	0	0
$960$	0	0
$961$	6.42542	0.207272
$962$	−8.86968	−0.285970
$963$	−5.36184	−0.172783
$964$	50.0807	1.61299
$965$	0	0
$966$	0	0
$967$	59.6146	1.91708	0.958538	−	0.284965i	$-0.0919820\pi$
$967$	59.6146	1.91708	0.958538	+	0.284965i	$0.0919820\pi$
$968$	32.4801	1.04395
$969$	0.969121	0.0311327
$970$	0	0
$971$	0.951234	0.0305266	0.0152633	−	0.999884i	$-0.495141\pi$
$971$	0.951234	0.0305266	0.0152633	+	0.999884i	$0.495141\pi$
$972$	−1.75579	−0.0563171
$973$	0	0
$974$	−11.1598	−0.357583
$975$	0	0
$976$	12.9720	0.415223
$977$	−23.0506	−0.737453	−0.368726	−	0.929538i	$-0.620206\pi$
$977$	−23.0506	−0.737453	−0.368726	+	0.929538i	$0.620206\pi$
$978$	−0.408655	−0.0130674
$979$	23.2236	0.742230
$980$	0	0
$981$	−0.105979	−0.00338364
$982$	0.0662922	0.00211547
$983$	−31.8348	−1.01537	−0.507687	−	0.861542i	$-0.669499\pi$
$983$	−31.8348	−1.01537	−0.507687	+	0.861542i	$0.669499\pi$
$984$	14.5915	0.465162
$985$	0	0
$986$	0.692210	0.0220445
$987$	0	0
$988$	−24.7536	−0.787518
$989$	10.6459	0.338519
$990$	0	0
$991$	−8.10611	−0.257499	−0.128750	−	0.991677i	$-0.541096\pi$
$991$	−8.10611	−0.257499	−0.128750	+	0.991677i	$0.541096\pi$
$992$	30.5521	0.970030
$993$	−19.5855	−0.621525
$994$	0	0
$995$	0	0
$996$	−12.0683	−0.382399
$997$	−28.2469	−0.894589	−0.447294	−	0.894387i	$-0.647612\pi$
$997$	−28.2469	−0.894589	−0.447294	+	0.894387i	$0.647612\pi$
$998$	7.68704	0.243329
$999$	−3.52324	−0.111470

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3675.2.a.bx.1.2		4
5.4	even	2		3675.2.a.bq.1.3		4
7.3	odd	6		525.2.i.i.226.3	yes	8
7.5	odd	6		525.2.i.i.151.3	✓	8
7.6	odd	2		3675.2.a.bw.1.2		4
35.3	even	12		525.2.r.h.499.4		16
35.12	even	12		525.2.r.h.424.4		16
35.17	even	12		525.2.r.h.499.5		16
35.19	odd	6		525.2.i.j.151.2	yes	8
35.24	odd	6		525.2.i.j.226.2	yes	8
35.33	even	12		525.2.r.h.424.5		16
35.34	odd	2		3675.2.a.br.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.2.i.i.151.3	✓	8	7.5	odd	6
525.2.i.i.226.3	yes	8	7.3	odd	6
525.2.i.j.151.2	yes	8	35.19	odd	6
525.2.i.j.226.2	yes	8	35.24	odd	6
525.2.r.h.424.4		16	35.12	even	12
525.2.r.h.424.5		16	35.33	even	12
525.2.r.h.499.4		16	35.3	even	12
525.2.r.h.499.5		16	35.17	even	12
3675.2.a.bq.1.3		4	5.4	even	2
3675.2.a.br.1.3		4	35.34	odd	2
3675.2.a.bw.1.2		4	7.6	odd	2
3675.2.a.bx.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)

\(f(q)\)	\(=\)	\(q-0.494173 q^{2} +1.00000 q^{3} -1.75579 q^{4} -0.494173 q^{6} +1.85601 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.494173 q^{2} +1.00000 q^{3} -1.75579 q^{4} -0.494173 q^{6} +1.85601 q^{8} +1.00000 q^{9} +5.33853 q^{11} -1.75579 q^{12} -5.09433 q^{13} +2.59439 q^{16} -0.350186 q^{17} -0.494173 q^{18} -2.76745 q^{19} -2.63816 q^{22} +7.51159 q^{23} +1.85601 q^{24} +2.51748 q^{26} +1.00000 q^{27} +4.00000 q^{29} -6.11763 q^{31} -4.99411 q^{32} +5.33853 q^{33} +0.173053 q^{34} -1.75579 q^{36} -3.52324 q^{37} +1.36760 q^{38} -5.09433 q^{39} +7.86177 q^{41} +1.41726 q^{43} -9.37336 q^{44} -3.71203 q^{46} +8.85012 q^{47} +2.59439 q^{48} -0.350186 q^{51} +8.94458 q^{52} -6.32688 q^{53} -0.494173 q^{54} -2.76745 q^{57} -1.97669 q^{58} -12.8501 q^{59} +5.00000 q^{61} +3.02317 q^{62} -2.72083 q^{64} -2.63816 q^{66} +3.75579 q^{67} +0.614854 q^{68} +7.51159 q^{69} +15.4883 q^{71} +1.85601 q^{72} -7.67707 q^{73} +1.74109 q^{74} +4.85906 q^{76} +2.51748 q^{78} +2.36184 q^{79} +1.00000 q^{81} -3.88508 q^{82} +6.87342 q^{83} -0.700372 q^{86} +4.00000 q^{87} +9.90838 q^{88} +4.35019 q^{89} -13.1888 q^{92} -6.11763 q^{93} -4.37349 q^{94} -4.99411 q^{96} -5.49993 q^{97} +5.33853 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{2} + 4 q^{3} + 7 q^{4} + q^{6} + 3 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q + q^2 + 4 * q^3 + 7 * q^4 + q^6 + 3 * q^8 + 4 * q^9	\( 4 q + q^{2} + 4 q^{3} + 7 q^{4} + q^{6} + 3 q^{8} + 4 q^{9} + 8 q^{11} + 7 q^{12} + 7 q^{13} + 17 q^{16} + 6 q^{17} + q^{18} - 3 q^{19} - 12 q^{22} + 2 q^{23} + 3 q^{24} + 19 q^{26} + 4 q^{27} + 16 q^{29} - 9 q^{31} + 17 q^{32} + 8 q^{33} - 14 q^{34} + 7 q^{36} + 8 q^{37} - 27 q^{38} + 7 q^{39} - 4 q^{41} + 5 q^{43} + 26 q^{44} - 6 q^{46} - 6 q^{47} + 17 q^{48} + 6 q^{51} + 35 q^{52} - 6 q^{53} + q^{54} - 3 q^{57} + 4 q^{58} - 10 q^{59} + 20 q^{61} - 44 q^{62} + 21 q^{64} - 12 q^{66} + q^{67} - 8 q^{68} + 2 q^{69} + 22 q^{71} + 3 q^{72} - 4 q^{73} - 21 q^{74} + 23 q^{76} + 19 q^{78} + 8 q^{79} + 4 q^{81} + 8 q^{82} - 2 q^{83} + 12 q^{86} + 16 q^{87} + 28 q^{88} + 10 q^{89} - 66 q^{92} - 9 q^{93} - 22 q^{94} + 17 q^{96} + 12 q^{97} + 8 q^{99}+O(q^{100}) \) 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

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