LMFDB - Embedded newform 3675.2.a.bi.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3675,2,Mod(1,3675)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3675, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3675.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$29.3450227428$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.17009$ 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)

$f(q)$	$=$	$q-2.70928 q^{2} -1.00000 q^{3} +5.34017 q^{4} +2.70928 q^{6} -9.04945 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.70928 q^{2} -1.00000 q^{3} +5.34017 q^{4} +2.70928 q^{6} -9.04945 q^{8} +1.00000 q^{9} +2.00000 q^{11} -5.34017 q^{12} -0.921622 q^{13} +13.8371 q^{16} -1.07838 q^{17} -2.70928 q^{18} -3.07838 q^{19} -5.41855 q^{22} +2.34017 q^{23} +9.04945 q^{24} +2.49693 q^{26} -1.00000 q^{27} -6.68035 q^{29} +7.75872 q^{31} -19.3896 q^{32} -2.00000 q^{33} +2.92162 q^{34} +5.34017 q^{36} +10.8371 q^{37} +8.34017 q^{38} +0.921622 q^{39} -6.49693 q^{41} -6.52359 q^{43} +10.6803 q^{44} -6.34017 q^{46} -4.68035 q^{47} -13.8371 q^{48} +1.07838 q^{51} -4.92162 q^{52} +3.75872 q^{53} +2.70928 q^{54} +3.07838 q^{57} +18.0989 q^{58} -10.5236 q^{59} +4.15676 q^{61} -21.0205 q^{62} +24.8576 q^{64} +5.41855 q^{66} +4.68035 q^{67} -5.75872 q^{68} -2.34017 q^{69} +2.00000 q^{71} -9.04945 q^{72} -7.07838 q^{73} -29.3607 q^{74} -16.4391 q^{76} -2.49693 q^{78} +6.15676 q^{79} +1.00000 q^{81} +17.6020 q^{82} -6.83710 q^{83} +17.6742 q^{86} +6.68035 q^{87} -18.0989 q^{88} -8.34017 q^{89} +12.4969 q^{92} -7.75872 q^{93} +12.6803 q^{94} +19.3896 q^{96} +8.43907 q^{97} +2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} - 3 q^{3} + 5 q^{4} + q^{6} - 9 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q - q^2 - 3 * q^3 + 5 * q^4 + q^6 - 9 * q^8 + 3 * q^9	$ 3 q - q^{2} - 3 q^{3} + 5 q^{4} + q^{6} - 9 q^{8} + 3 q^{9} + 6 q^{11} - 5 q^{12} - 6 q^{13} + 13 q^{16} - q^{18} - 6 q^{19} - 2 q^{22} - 4 q^{23} + 9 q^{24} - 10 q^{26} - 3 q^{27} + 2 q^{29} - 2 q^{31} - 29 q^{32} - 6 q^{33} + 12 q^{34} + 5 q^{36} + 4 q^{37} + 14 q^{38} + 6 q^{39} - 2 q^{41} - 4 q^{43} + 10 q^{44} - 8 q^{46} + 8 q^{47} - 13 q^{48} - 18 q^{52} - 14 q^{53} + q^{54} + 6 q^{57} + 18 q^{58} - 16 q^{59} + 6 q^{61} - 30 q^{62} + 13 q^{64} + 2 q^{66} - 8 q^{67} + 8 q^{68} + 4 q^{69} + 6 q^{71} - 9 q^{72} - 18 q^{73} - 44 q^{74} - 2 q^{76} + 10 q^{78} + 12 q^{79} + 3 q^{81} + 34 q^{82} + 8 q^{83} - 4 q^{86} - 2 q^{87} - 18 q^{88} - 14 q^{89} + 20 q^{92} + 2 q^{93} + 16 q^{94} + 29 q^{96} - 22 q^{97} + 6 q^{99}+O(q^{100}) $ 3 * q - q^2 - 3 * q^3 + 5 * q^4 + q^6 - 9 * q^8 + 3 * q^9 + 6 * q^11 - 5 * q^12 - 6 * q^13 + 13 * q^16 - q^18 - 6 * q^19 - 2 * q^22 - 4 * q^23 + 9 * q^24 - 10 * q^26 - 3 * q^27 + 2 * q^29 - 2 * q^31 - 29 * q^32 - 6 * q^33 + 12 * q^34 + 5 * q^36 + 4 * q^37 + 14 * q^38 + 6 * q^39 - 2 * q^41 - 4 * q^43 + 10 * q^44 - 8 * q^46 + 8 * q^47 - 13 * q^48 - 18 * q^52 - 14 * q^53 + q^54 + 6 * q^57 + 18 * q^58 - 16 * q^59 + 6 * q^61 - 30 * q^62 + 13 * q^64 + 2 * q^66 - 8 * q^67 + 8 * q^68 + 4 * q^69 + 6 * q^71 - 9 * q^72 - 18 * q^73 - 44 * q^74 - 2 * q^76 + 10 * q^78 + 12 * q^79 + 3 * q^81 + 34 * q^82 + 8 * q^83 - 4 * q^86 - 2 * q^87 - 18 * q^88 - 14 * q^89 + 20 * q^92 + 2 * q^93 + 16 * q^94 + 29 * q^96 - 22 * q^97 + 6 * 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 $

$2$	−2.70928	−1.91575	−0.957873	−	0.287190i	$-0.907279\pi$
$2$	−2.70928	−1.91575	−0.957873	+	0.287190i	$0.907279\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	5.34017	2.67009
$5$	0	0
$5$	0	0
$6$	2.70928	1.10606
$7$	0	0
$7$	0	0
$8$	−9.04945	−3.19946
$9$	1.00000	0.333333
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	−5.34017	−1.54158
$13$	−0.921622	−0.255612	−0.127806	−	0.991799i	$-0.540793\pi$
$13$	−0.921622	−0.255612	−0.127806	+	0.991799i	$0.540793\pi$
$14$	0	0
$15$	0	0
$16$	13.8371	3.45928
$17$	−1.07838	−0.261545	−0.130773	−	0.991412i	$-0.541746\pi$
$17$	−1.07838	−0.261545	−0.130773	+	0.991412i	$0.541746\pi$
$18$	−2.70928	−0.638582
$19$	−3.07838	−0.706228	−0.353114	−	0.935580i	$-0.614877\pi$
$19$	−3.07838	−0.706228	−0.353114	+	0.935580i	$0.614877\pi$
$20$	0	0
$21$	0	0
$22$	−5.41855	−1.15524
$23$	2.34017	0.487960	0.243980	−	0.969780i	$-0.421547\pi$
$23$	2.34017	0.487960	0.243980	+	0.969780i	$0.421547\pi$
$24$	9.04945	1.84721
$25$	0	0
$26$	2.49693	0.489688
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−6.68035	−1.24051	−0.620255	−	0.784401i	$-0.712971\pi$
$29$	−6.68035	−1.24051	−0.620255	+	0.784401i	$0.712971\pi$
$30$	0	0
$31$	7.75872	1.39351	0.696754	−	0.717310i	$-0.254627\pi$
$31$	7.75872	1.39351	0.696754	+	0.717310i	$0.254627\pi$
$32$	−19.3896	−3.42763
$33$	−2.00000	−0.348155
$34$	2.92162	0.501054
$35$	0	0
$36$	5.34017	0.890029
$37$	10.8371	1.78161	0.890804	−	0.454387i	$-0.150142\pi$
$37$	10.8371	1.78161	0.890804	+	0.454387i	$0.150142\pi$
$38$	8.34017	1.35295
$39$	0.921622	0.147578
$40$	0	0
$41$	−6.49693	−1.01465	−0.507325	−	0.861755i	$-0.669366\pi$
$41$	−6.49693	−1.01465	−0.507325	+	0.861755i	$0.669366\pi$
$42$	0	0
$43$	−6.52359	−0.994838	−0.497419	−	0.867510i	$-0.665719\pi$
$43$	−6.52359	−0.994838	−0.497419	+	0.867510i	$0.665719\pi$
$44$	10.6803	1.61012
$45$	0	0
$46$	−6.34017	−0.934808
$47$	−4.68035	−0.682699	−0.341349	−	0.939937i	$-0.610884\pi$
$47$	−4.68035	−0.682699	−0.341349	+	0.939937i	$0.610884\pi$
$48$	−13.8371	−1.99721
$49$	0	0
$50$	0	0
$51$	1.07838	0.151003
$52$	−4.92162	−0.682506
$53$	3.75872	0.516300	0.258150	−	0.966105i	$-0.416887\pi$
$53$	3.75872	0.516300	0.258150	+	0.966105i	$0.416887\pi$
$54$	2.70928	0.368686
$55$	0	0
$56$	0	0
$57$	3.07838	0.407741
$58$	18.0989	2.37650
$59$	−10.5236	−1.37005	−0.685027	−	0.728517i	$-0.740210\pi$
$59$	−10.5236	−1.37005	−0.685027	+	0.728517i	$0.740210\pi$
$60$	0	0
$61$	4.15676	0.532218	0.266109	−	0.963943i	$-0.414262\pi$
$61$	4.15676	0.532218	0.266109	+	0.963943i	$0.414262\pi$
$62$	−21.0205	−2.66961
$63$	0	0
$64$	24.8576	3.10720
$65$	0	0
$66$	5.41855	0.666977
$67$	4.68035	0.571795	0.285898	−	0.958260i	$-0.407708\pi$
$67$	4.68035	0.571795	0.285898	+	0.958260i	$0.407708\pi$
$68$	−5.75872	−0.698348
$69$	−2.34017	−0.281724
$70$	0	0
$71$	2.00000	0.237356	0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$72$	−9.04945	−1.06649
$73$	−7.07838	−0.828461	−0.414231	−	0.910172i	$-0.635949\pi$
$73$	−7.07838	−0.828461	−0.414231	+	0.910172i	$0.635949\pi$
$74$	−29.3607	−3.41311
$75$	0	0
$76$	−16.4391	−1.88569
$77$	0	0
$78$	−2.49693	−0.282721
$79$	6.15676	0.692689	0.346345	−	0.938107i	$-0.387423\pi$
$79$	6.15676	0.692689	0.346345	+	0.938107i	$0.387423\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	17.6020	1.94381
$83$	−6.83710	−0.750469	−0.375235	−	0.926930i	$-0.622438\pi$
$83$	−6.83710	−0.750469	−0.375235	+	0.926930i	$0.622438\pi$
$84$	0	0
$85$	0	0
$86$	17.6742	1.90586
$87$	6.68035	0.716208
$88$	−18.0989	−1.92935
$89$	−8.34017	−0.884057	−0.442028	−	0.897001i	$-0.645741\pi$
$89$	−8.34017	−0.884057	−0.442028	+	0.897001i	$0.645741\pi$
$90$	0	0
$91$	0	0
$92$	12.4969	1.30289
$93$	−7.75872	−0.804542
$94$	12.6803	1.30788
$95$	0	0
$96$	19.3896	1.97894
$97$	8.43907	0.856858	0.428429	−	0.903576i	$-0.359067\pi$
$97$	8.43907	0.856858	0.428429	+	0.903576i	$0.359067\pi$
$98$	0	0
$99$	2.00000	0.201008
$100$	0	0
$101$	5.81658	0.578772	0.289386	−	0.957213i	$-0.406549\pi$
$101$	5.81658	0.578772	0.289386	+	0.957213i	$0.406549\pi$
$102$	−2.92162	−0.289284
$103$	2.15676	0.212511	0.106256	−	0.994339i	$-0.466114\pi$
$103$	2.15676	0.212511	0.106256	+	0.994339i	$0.466114\pi$
$104$	8.34017	0.817821
$105$	0	0
$106$	−10.1834	−0.989101
$107$	−16.4969	−1.59482	−0.797409	−	0.603439i	$-0.793797\pi$
$107$	−16.4969	−1.59482	−0.797409	+	0.603439i	$0.793797\pi$
$108$	−5.34017	−0.513858
$109$	−12.8371	−1.22957	−0.614786	−	0.788694i	$-0.710757\pi$
$109$	−12.8371	−1.22957	−0.614786	+	0.788694i	$0.710757\pi$
$110$	0	0
$111$	−10.8371	−1.02861
$112$	0	0
$113$	−5.23513	−0.492480	−0.246240	−	0.969209i	$-0.579195\pi$
$113$	−5.23513	−0.492480	−0.246240	+	0.969209i	$0.579195\pi$
$114$	−8.34017	−0.781129
$115$	0	0
$116$	−35.6742	−3.31227
$117$	−0.921622	−0.0852040
$118$	28.5113	2.62468
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−11.2618	−1.01960
$123$	6.49693	0.585808
$124$	41.4329	3.72079
$125$	0	0
$126$	0	0
$127$	1.84324	0.163562	0.0817808	−	0.996650i	$-0.473939\pi$
$127$	1.84324	0.163562	0.0817808	+	0.996650i	$0.473939\pi$
$128$	−28.5669	−2.52498
$129$	6.52359	0.574370
$130$	0	0
$131$	−1.47641	−0.128995	−0.0644973	−	0.997918i	$-0.520544\pi$
$131$	−1.47641	−0.128995	−0.0644973	+	0.997918i	$0.520544\pi$
$132$	−10.6803	−0.929605
$133$	0	0
$134$	−12.6803	−1.09542
$135$	0	0
$136$	9.75872	0.836804
$137$	4.43907	0.379255	0.189628	−	0.981856i	$-0.439272\pi$
$137$	4.43907	0.379255	0.189628	+	0.981856i	$0.439272\pi$
$138$	6.34017	0.539711
$139$	−13.6020	−1.15370	−0.576852	−	0.816849i	$-0.695719\pi$
$139$	−13.6020	−1.15370	−0.576852	+	0.816849i	$0.695719\pi$
$140$	0	0
$141$	4.68035	0.394156
$142$	−5.41855	−0.454715
$143$	−1.84324	−0.154140
$144$	13.8371	1.15309
$145$	0	0
$146$	19.1773	1.58712
$147$	0	0
$148$	57.8720	4.75705
$149$	15.6742	1.28408	0.642040	−	0.766671i	$-0.278088\pi$
$149$	15.6742	1.28408	0.642040	+	0.766671i	$0.278088\pi$
$150$	0	0
$151$	5.84324	0.475516	0.237758	−	0.971324i	$-0.423587\pi$
$151$	5.84324	0.475516	0.237758	+	0.971324i	$0.423587\pi$
$152$	27.8576	2.25955
$153$	−1.07838	−0.0871817
$154$	0	0
$155$	0	0
$156$	4.92162	0.394045
$157$	−4.92162	−0.392788	−0.196394	−	0.980525i	$-0.562923\pi$
$157$	−4.92162	−0.392788	−0.196394	+	0.980525i	$0.562923\pi$
$158$	−16.6803	−1.32702
$159$	−3.75872	−0.298086
$160$	0	0
$161$	0	0
$162$	−2.70928	−0.212861
$163$	−9.84324	−0.770982	−0.385491	−	0.922712i	$-0.625968\pi$
$163$	−9.84324	−0.770982	−0.385491	+	0.922712i	$0.625968\pi$
$164$	−34.6947	−2.70920
$165$	0	0
$166$	18.5236	1.43771
$167$	19.2039	1.48605	0.743023	−	0.669266i	$-0.233391\pi$
$167$	19.2039	1.48605	0.743023	+	0.669266i	$0.233391\pi$
$168$	0	0
$169$	−12.1506	−0.934662
$170$	0	0
$171$	−3.07838	−0.235409
$172$	−34.8371	−2.65630
$173$	22.4391	1.70601	0.853005	−	0.521902i	$-0.174777\pi$
$173$	22.4391	1.70601	0.853005	+	0.521902i	$0.174777\pi$
$174$	−18.0989	−1.37207
$175$	0	0
$176$	27.6742	2.08602
$177$	10.5236	0.791001
$178$	22.5958	1.69363
$179$	10.0000	0.747435	0.373718	−	0.927543i	$-0.378083\pi$
$179$	10.0000	0.747435	0.373718	+	0.927543i	$0.378083\pi$
$180$	0	0
$181$	8.52359	0.633553	0.316777	−	0.948500i	$-0.397399\pi$
$181$	8.52359	0.633553	0.316777	+	0.948500i	$0.397399\pi$
$182$	0	0
$183$	−4.15676	−0.307276
$184$	−21.1773	−1.56121
$185$	0	0
$186$	21.0205	1.54130
$187$	−2.15676	−0.157718
$188$	−24.9939	−1.82286
$189$	0	0
$190$	0	0
$191$	15.3607	1.11146	0.555730	−	0.831363i	$-0.312439\pi$
$191$	15.3607	1.11146	0.555730	+	0.831363i	$0.312439\pi$
$192$	−24.8576	−1.79394
$193$	8.36683	0.602258	0.301129	−	0.953583i	$-0.402637\pi$
$193$	8.36683	0.602258	0.301129	+	0.953583i	$0.402637\pi$
$194$	−22.8638	−1.64152
$195$	0	0
$196$	0	0
$197$	−11.7587	−0.837774	−0.418887	−	0.908038i	$-0.637580\pi$
$197$	−11.7587	−0.837774	−0.418887	+	0.908038i	$0.637580\pi$
$198$	−5.41855	−0.385080
$199$	22.5958	1.60178	0.800888	−	0.598814i	$-0.204361\pi$
$199$	22.5958	1.60178	0.800888	+	0.598814i	$0.204361\pi$
$200$	0	0
$201$	−4.68035	−0.330126
$202$	−15.7587	−1.10878
$203$	0	0
$204$	5.75872	0.403191
$205$	0	0
$206$	−5.84324	−0.407118
$207$	2.34017	0.162653
$208$	−12.7526	−0.884232
$209$	−6.15676	−0.425872
$210$	0	0
$211$	−13.6742	−0.941371	−0.470685	−	0.882301i	$-0.655993\pi$
$211$	−13.6742	−0.941371	−0.470685	+	0.882301i	$0.655993\pi$
$212$	20.0722	1.37857
$213$	−2.00000	−0.137038
$214$	44.6947	3.05527
$215$	0	0
$216$	9.04945	0.615737
$217$	0	0
$218$	34.7792	2.35555
$219$	7.07838	0.478312
$220$	0	0
$221$	0.993857	0.0668541
$222$	29.3607	1.97056
$223$	21.6742	1.45141	0.725706	−	0.688005i	$-0.241513\pi$
$223$	21.6742	1.45141	0.725706	+	0.688005i	$0.241513\pi$
$224$	0	0
$225$	0	0
$226$	14.1834	0.943467
$227$	11.5174	0.764440	0.382220	−	0.924071i	$-0.375160\pi$
$227$	11.5174	0.764440	0.382220	+	0.924071i	$0.375160\pi$
$228$	16.4391	1.08870
$229$	−12.8371	−0.848300	−0.424150	−	0.905592i	$-0.639427\pi$
$229$	−12.8371	−0.848300	−0.424150	+	0.905592i	$0.639427\pi$
$230$	0	0
$231$	0	0
$232$	60.4534	3.96896
$233$	−6.76487	−0.443181	−0.221591	−	0.975140i	$-0.571125\pi$
$233$	−6.76487	−0.443181	−0.221591	+	0.975140i	$0.571125\pi$
$234$	2.49693	0.163229
$235$	0	0
$236$	−56.1978	−3.65816
$237$	−6.15676	−0.399924
$238$	0	0
$239$	−23.3607	−1.51108	−0.755539	−	0.655104i	$-0.772625\pi$
$239$	−23.3607	−1.51108	−0.755539	+	0.655104i	$0.772625\pi$
$240$	0	0
$241$	14.6803	0.945644	0.472822	−	0.881158i	$-0.343235\pi$
$241$	14.6803	0.945644	0.472822	+	0.881158i	$0.343235\pi$
$242$	18.9649	1.21911
$243$	−1.00000	−0.0641500
$244$	22.1978	1.42107
$245$	0	0
$246$	−17.6020	−1.12226
$247$	2.83710	0.180520
$248$	−70.2122	−4.45848
$249$	6.83710	0.433284
$250$	0	0
$251$	−9.16290	−0.578357	−0.289179	−	0.957275i	$-0.593382\pi$
$251$	−9.16290	−0.578357	−0.289179	+	0.957275i	$0.593382\pi$
$252$	0	0
$253$	4.68035	0.294251
$254$	−4.99386	−0.313342
$255$	0	0
$256$	27.6803	1.73002
$257$	5.07838	0.316781	0.158390	−	0.987377i	$-0.449370\pi$
$257$	5.07838	0.316781	0.158390	+	0.987377i	$0.449370\pi$
$258$	−17.6742	−1.10035
$259$	0	0
$260$	0	0
$261$	−6.68035	−0.413503
$262$	4.00000	0.247121
$263$	5.65983	0.349000	0.174500	−	0.984657i	$-0.444169\pi$
$263$	5.65983	0.349000	0.174500	+	0.984657i	$0.444169\pi$
$264$	18.0989	1.11391
$265$	0	0
$266$	0	0
$267$	8.34017	0.510410
$268$	24.9939	1.52674
$269$	27.8576	1.69851	0.849255	−	0.527984i	$-0.177052\pi$
$269$	27.8576	1.69851	0.849255	+	0.527984i	$0.177052\pi$
$270$	0	0
$271$	−25.1194	−1.52590	−0.762948	−	0.646460i	$-0.776249\pi$
$271$	−25.1194	−1.52590	−0.762948	+	0.646460i	$0.776249\pi$
$272$	−14.9216	−0.904756
$273$	0	0
$274$	−12.0267	−0.726557
$275$	0	0
$276$	−12.4969	−0.752227
$277$	−28.1978	−1.69424	−0.847121	−	0.531401i	$-0.821666\pi$
$277$	−28.1978	−1.69424	−0.847121	+	0.531401i	$0.821666\pi$
$278$	36.8515	2.21020
$279$	7.75872	0.464503
$280$	0	0
$281$	−20.3545	−1.21425	−0.607125	−	0.794606i	$-0.707677\pi$
$281$	−20.3545	−1.21425	−0.607125	+	0.794606i	$0.707677\pi$
$282$	−12.6803	−0.755104
$283$	−23.5174	−1.39797	−0.698984	−	0.715138i	$-0.746364\pi$
$283$	−23.5174	−1.39797	−0.698984	+	0.715138i	$0.746364\pi$
$284$	10.6803	0.633762
$285$	0	0
$286$	4.99386	0.295293
$287$	0	0
$288$	−19.3896	−1.14254
$289$	−15.8371	−0.931594
$290$	0	0
$291$	−8.43907	−0.494707
$292$	−37.7998	−2.21206
$293$	2.92162	0.170683	0.0853415	−	0.996352i	$-0.472802\pi$
$293$	2.92162	0.170683	0.0853415	+	0.996352i	$0.472802\pi$
$294$	0	0
$295$	0	0
$296$	−98.0698	−5.70019
$297$	−2.00000	−0.116052
$298$	−42.4657	−2.45997
$299$	−2.15676	−0.124728
$300$	0	0
$301$	0	0
$302$	−15.8310	−0.910969
$303$	−5.81658	−0.334154
$304$	−42.5958	−2.44304
$305$	0	0
$306$	2.92162	0.167018
$307$	10.4703	0.597570	0.298785	−	0.954321i	$-0.403419\pi$
$307$	10.4703	0.597570	0.298785	+	0.954321i	$0.403419\pi$
$308$	0	0
$309$	−2.15676	−0.122694
$310$	0	0
$311$	−23.8310	−1.35133	−0.675665	−	0.737209i	$-0.736143\pi$
$311$	−23.8310	−1.35133	−0.675665	+	0.737209i	$0.736143\pi$
$312$	−8.34017	−0.472169
$313$	−32.7526	−1.85129	−0.925643	−	0.378399i	$-0.876475\pi$
$313$	−32.7526	−1.85129	−0.925643	+	0.378399i	$0.876475\pi$
$314$	13.3340	0.752483
$315$	0	0
$316$	32.8781	1.84954
$317$	−17.9155	−1.00623	−0.503117	−	0.864218i	$-0.667813\pi$
$317$	−17.9155	−1.00623	−0.503117	+	0.864218i	$0.667813\pi$
$318$	10.1834	0.571058
$319$	−13.3607	−0.748055
$320$	0	0
$321$	16.4969	0.920769
$322$	0	0
$323$	3.31965	0.184710
$324$	5.34017	0.296676
$325$	0	0
$326$	26.6681	1.47701
$327$	12.8371	0.709893
$328$	58.7936	3.24633
$329$	0	0
$330$	0	0
$331$	−1.36069	−0.0747904	−0.0373952	−	0.999301i	$-0.511906\pi$
$331$	−1.36069	−0.0747904	−0.0373952	+	0.999301i	$0.511906\pi$
$332$	−36.5113	−2.00382
$333$	10.8371	0.593870
$334$	−52.0288	−2.84689
$335$	0	0
$336$	0	0
$337$	−25.3607	−1.38148	−0.690742	−	0.723101i	$-0.742716\pi$
$337$	−25.3607	−1.38148	−0.690742	+	0.723101i	$0.742716\pi$
$338$	32.9194	1.79058
$339$	5.23513	0.284333
$340$	0	0
$341$	15.5174	0.840317
$342$	8.34017	0.450985
$343$	0	0
$344$	59.0349	3.18295
$345$	0	0
$346$	−60.7936	−3.26829
$347$	16.8638	0.905294	0.452647	−	0.891690i	$-0.350480\pi$
$347$	16.8638	0.905294	0.452647	+	0.891690i	$0.350480\pi$
$348$	35.6742	1.91234
$349$	−9.51745	−0.509457	−0.254729	−	0.967013i	$-0.581986\pi$
$349$	−9.51745	−0.509457	−0.254729	+	0.967013i	$0.581986\pi$
$350$	0	0
$351$	0.921622	0.0491926
$352$	−38.7792	−2.06694
$353$	35.7998	1.90543	0.952715	−	0.303867i	$-0.0982778\pi$
$353$	35.7998	1.90543	0.952715	+	0.303867i	$0.0982778\pi$
$354$	−28.5113	−1.51536
$355$	0	0
$356$	−44.5380	−2.36051
$357$	0	0
$358$	−27.0928	−1.43190
$359$	−22.3135	−1.17766	−0.588831	−	0.808256i	$-0.700412\pi$
$359$	−22.3135	−1.17766	−0.588831	+	0.808256i	$0.700412\pi$
$360$	0	0
$361$	−9.52359	−0.501242
$362$	−23.0928	−1.21373
$363$	7.00000	0.367405
$364$	0	0
$365$	0	0
$366$	11.2618	0.588663
$367$	−20.3135	−1.06036	−0.530178	−	0.847886i	$-0.677875\pi$
$367$	−20.3135	−1.06036	−0.530178	+	0.847886i	$0.677875\pi$
$368$	32.3812	1.68799
$369$	−6.49693	−0.338217
$370$	0	0
$371$	0	0
$372$	−41.4329	−2.14820
$373$	−16.0000	−0.828449	−0.414224	−	0.910175i	$-0.635947\pi$
$373$	−16.0000	−0.828449	−0.414224	+	0.910175i	$0.635947\pi$
$374$	5.84324	0.302147
$375$	0	0
$376$	42.3545	2.18427
$377$	6.15676	0.317089
$378$	0	0
$379$	6.15676	0.316251	0.158126	−	0.987419i	$-0.449455\pi$
$379$	6.15676	0.316251	0.158126	+	0.987419i	$0.449455\pi$
$380$	0	0
$381$	−1.84324	−0.0944323
$382$	−41.6163	−2.12928
$383$	−26.8371	−1.37131	−0.685656	−	0.727926i	$-0.740485\pi$
$383$	−26.8371	−1.37131	−0.685656	+	0.727926i	$0.740485\pi$
$384$	28.5669	1.45780
$385$	0	0
$386$	−22.6681	−1.15377
$387$	−6.52359	−0.331613
$388$	45.0661	2.28788
$389$	−5.63317	−0.285613	−0.142806	−	0.989751i	$-0.545613\pi$
$389$	−5.63317	−0.285613	−0.142806	+	0.989751i	$0.545613\pi$
$390$	0	0
$391$	−2.52359	−0.127623
$392$	0	0
$393$	1.47641	0.0744750
$394$	31.8576	1.60496
$395$	0	0
$396$	10.6803	0.536708
$397$	−37.7998	−1.89712	−0.948558	−	0.316604i	$-0.897457\pi$
$397$	−37.7998	−1.89712	−0.948558	+	0.316604i	$0.897457\pi$
$398$	−61.2183	−3.06860
$399$	0	0
$400$	0	0
$401$	−13.6332	−0.680808	−0.340404	−	0.940279i	$-0.610564\pi$
$401$	−13.6332	−0.680808	−0.340404	+	0.940279i	$0.610564\pi$
$402$	12.6803	0.632438
$403$	−7.15061	−0.356197
$404$	31.0616	1.54537
$405$	0	0
$406$	0	0
$407$	21.6742	1.07435
$408$	−9.75872	−0.483129
$409$	−12.3545	−0.610893	−0.305447	−	0.952209i	$-0.598806\pi$
$409$	−12.3545	−0.610893	−0.305447	+	0.952209i	$0.598806\pi$
$410$	0	0
$411$	−4.43907	−0.218963
$412$	11.5174	0.567424
$413$	0	0
$414$	−6.34017	−0.311603
$415$	0	0
$416$	17.8699	0.876144
$417$	13.6020	0.666091
$418$	16.6803	0.815862
$419$	−28.9939	−1.41644	−0.708221	−	0.705991i	$-0.750502\pi$
$419$	−28.9939	−1.41644	−0.708221	+	0.705991i	$0.750502\pi$
$420$	0	0
$421$	−15.1629	−0.738994	−0.369497	−	0.929232i	$-0.620470\pi$
$421$	−15.1629	−0.738994	−0.369497	+	0.929232i	$0.620470\pi$
$422$	37.0472	1.80343
$423$	−4.68035	−0.227566
$424$	−34.0144	−1.65188
$425$	0	0
$426$	5.41855	0.262530
$427$	0	0
$428$	−88.0965	−4.25830
$429$	1.84324	0.0889927
$430$	0	0
$431$	−10.3135	−0.496784	−0.248392	−	0.968660i	$-0.579902\pi$
$431$	−10.3135	−0.496784	−0.248392	+	0.968660i	$0.579902\pi$
$432$	−13.8371	−0.665738
$433$	−20.4391	−0.982239	−0.491120	−	0.871092i	$-0.663412\pi$
$433$	−20.4391	−0.982239	−0.491120	+	0.871092i	$0.663412\pi$
$434$	0	0
$435$	0	0
$436$	−68.5523	−3.28306
$437$	−7.20394	−0.344611
$438$	−19.1773	−0.916326
$439$	16.9216	0.807625	0.403812	−	0.914842i	$-0.367685\pi$
$439$	16.9216	0.807625	0.403812	+	0.914842i	$0.367685\pi$
$440$	0	0
$441$	0	0
$442$	−2.69263	−0.128075
$443$	−12.8104	−0.608642	−0.304321	−	0.952569i	$-0.598430\pi$
$443$	−12.8104	−0.608642	−0.304321	+	0.952569i	$0.598430\pi$
$444$	−57.8720	−2.74648
$445$	0	0
$446$	−58.7214	−2.78054
$447$	−15.6742	−0.741364
$448$	0	0
$449$	−14.6270	−0.690292	−0.345146	−	0.938549i	$-0.612171\pi$
$449$	−14.6270	−0.690292	−0.345146	+	0.938549i	$0.612171\pi$
$450$	0	0
$451$	−12.9939	−0.611857
$452$	−27.9565	−1.31496
$453$	−5.84324	−0.274540
$454$	−31.2039	−1.46447
$455$	0	0
$456$	−27.8576	−1.30455
$457$	14.1568	0.662225	0.331113	−	0.943591i	$-0.392576\pi$
$457$	14.1568	0.662225	0.331113	+	0.943591i	$0.392576\pi$
$458$	34.7792	1.62513
$459$	1.07838	0.0503344
$460$	0	0
$461$	−0.340173	−0.0158434	−0.00792172	−	0.999969i	$-0.502522\pi$
$461$	−0.340173	−0.0158434	−0.00792172	+	0.999969i	$0.502522\pi$
$462$	0	0
$463$	−9.84324	−0.457454	−0.228727	−	0.973491i	$-0.573456\pi$
$463$	−9.84324	−0.457454	−0.228727	+	0.973491i	$0.573456\pi$
$464$	−92.4366	−4.29126
$465$	0	0
$466$	18.3279	0.849023
$467$	11.5174	0.532964	0.266482	−	0.963840i	$-0.414139\pi$
$467$	11.5174	0.532964	0.266482	+	0.963840i	$0.414139\pi$
$468$	−4.92162	−0.227502
$469$	0	0
$470$	0	0
$471$	4.92162	0.226776
$472$	95.2327	4.38344
$473$	−13.0472	−0.599910
$474$	16.6803	0.766154
$475$	0	0
$476$	0	0
$477$	3.75872	0.172100
$478$	63.2905	2.89484
$479$	19.5174	0.891775	0.445887	−	0.895089i	$-0.352888\pi$
$479$	19.5174	0.891775	0.445887	+	0.895089i	$0.352888\pi$
$480$	0	0
$481$	−9.98771	−0.455401
$482$	−39.7731	−1.81162
$483$	0	0
$484$	−37.3812	−1.69915
$485$	0	0
$486$	2.70928	0.122895
$487$	23.1506	1.04905	0.524527	−	0.851394i	$-0.324242\pi$
$487$	23.1506	1.04905	0.524527	+	0.851394i	$0.324242\pi$
$488$	−37.6163	−1.70281
$489$	9.84324	0.445127
$490$	0	0
$491$	2.00000	0.0902587	0.0451294	−	0.998981i	$-0.485630\pi$
$491$	2.00000	0.0902587	0.0451294	+	0.998981i	$0.485630\pi$
$492$	34.6947	1.56416
$493$	7.20394	0.324449
$494$	−7.68649	−0.345831
$495$	0	0
$496$	107.358	4.82053
$497$	0	0
$498$	−18.5236	−0.830062
$499$	27.2039	1.21782	0.608908	−	0.793241i	$-0.291608\pi$
$499$	27.2039	1.21782	0.608908	+	0.793241i	$0.291608\pi$
$500$	0	0
$501$	−19.2039	−0.857969
$502$	24.8248	1.10799
$503$	18.8371	0.839905	0.419952	−	0.907546i	$-0.362047\pi$
$503$	18.8371	0.839905	0.419952	+	0.907546i	$0.362047\pi$
$504$	0	0
$505$	0	0
$506$	−12.6803	−0.563710
$507$	12.1506	0.539628
$508$	9.84324	0.436723
$509$	−6.81044	−0.301867	−0.150934	−	0.988544i	$-0.548228\pi$
$509$	−6.81044	−0.301867	−0.150934	+	0.988544i	$0.548228\pi$
$510$	0	0
$511$	0	0
$512$	−17.8599	−0.789303
$513$	3.07838	0.135914
$514$	−13.7587	−0.606871
$515$	0	0
$516$	34.8371	1.53362
$517$	−9.36069	−0.411683
$518$	0	0
$519$	−22.4391	−0.984966
$520$	0	0
$521$	25.8166	1.13105	0.565523	−	0.824733i	$-0.308675\pi$
$521$	25.8166	1.13105	0.565523	+	0.824733i	$0.308675\pi$
$522$	18.0989	0.792167
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	−7.88428	−0.344426
$525$	0	0
$526$	−15.3340	−0.668595
$527$	−8.36683	−0.364465
$528$	−27.6742	−1.20437
$529$	−17.5236	−0.761895
$530$	0	0
$531$	−10.5236	−0.456685
$532$	0	0
$533$	5.98771	0.259357
$534$	−22.5958	−0.977817
$535$	0	0
$536$	−42.3545	−1.82944
$537$	−10.0000	−0.431532
$538$	−75.4740	−3.25391
$539$	0	0
$540$	0	0
$541$	25.8843	1.11285	0.556426	−	0.830897i	$-0.312172\pi$
$541$	25.8843	1.11285	0.556426	+	0.830897i	$0.312172\pi$
$542$	68.0554	2.92323
$543$	−8.52359	−0.365782
$544$	20.9093	0.896480
$545$	0	0
$546$	0	0
$547$	11.3197	0.483993	0.241997	−	0.970277i	$-0.422198\pi$
$547$	11.3197	0.483993	0.241997	+	0.970277i	$0.422198\pi$
$548$	23.7054	1.01264
$549$	4.15676	0.177406
$550$	0	0
$551$	20.5646	0.876083
$552$	21.1773	0.901365
$553$	0	0
$554$	76.3956	3.24574
$555$	0	0
$556$	−72.6369	−3.08049
$557$	−26.6491	−1.12916	−0.564580	−	0.825378i	$-0.690962\pi$
$557$	−26.6491	−1.12916	−0.564580	+	0.825378i	$0.690962\pi$
$558$	−21.0205	−0.889870
$559$	6.01229	0.254293
$560$	0	0
$561$	2.15676	0.0910583
$562$	55.1461	2.32620
$563$	−46.3545	−1.95361	−0.976806	−	0.214128i	$-0.931309\pi$
$563$	−46.3545	−1.95361	−0.976806	+	0.214128i	$0.931309\pi$
$564$	24.9939	1.05243
$565$	0	0
$566$	63.7152	2.67815
$567$	0	0
$568$	−18.0989	−0.759413
$569$	−14.3668	−0.602289	−0.301145	−	0.953579i	$-0.597369\pi$
$569$	−14.3668	−0.602289	−0.301145	+	0.953579i	$0.597369\pi$
$570$	0	0
$571$	38.7214	1.62044	0.810220	−	0.586126i	$-0.199348\pi$
$571$	38.7214	1.62044	0.810220	+	0.586126i	$0.199348\pi$
$572$	−9.84324	−0.411567
$573$	−15.3607	−0.641702
$574$	0	0
$575$	0	0
$576$	24.8576	1.03573
$577$	−43.4740	−1.80984	−0.904922	−	0.425577i	$-0.860071\pi$
$577$	−43.4740	−1.80984	−0.904922	+	0.425577i	$0.860071\pi$
$578$	42.9071	1.78470
$579$	−8.36683	−0.347714
$580$	0	0
$581$	0	0
$582$	22.8638	0.947733
$583$	7.51745	0.311341
$584$	64.0554	2.65063
$585$	0	0
$586$	−7.91548	−0.326985
$587$	−36.0288	−1.48707	−0.743533	−	0.668699i	$-0.766851\pi$
$587$	−36.0288	−1.48707	−0.743533	+	0.668699i	$0.766851\pi$
$588$	0	0
$589$	−23.8843	−0.984135
$590$	0	0
$591$	11.7587	0.483689
$592$	149.954	6.16307
$593$	−31.4863	−1.29299	−0.646493	−	0.762920i	$-0.723765\pi$
$593$	−31.4863	−1.29299	−0.646493	+	0.762920i	$0.723765\pi$
$594$	5.41855	0.222326
$595$	0	0
$596$	83.7030	3.42861
$597$	−22.5958	−0.924786
$598$	5.84324	0.238948
$599$	29.0349	1.18633	0.593167	−	0.805080i	$-0.297877\pi$
$599$	29.0349	1.18633	0.593167	+	0.805080i	$0.297877\pi$
$600$	0	0
$601$	−15.3607	−0.626576	−0.313288	−	0.949658i	$-0.601430\pi$
$601$	−15.3607	−0.626576	−0.313288	+	0.949658i	$0.601430\pi$
$602$	0	0
$603$	4.68035	0.190598
$604$	31.2039	1.26967
$605$	0	0
$606$	15.7587	0.640154
$607$	13.0472	0.529569	0.264784	−	0.964308i	$-0.414699\pi$
$607$	13.0472	0.529569	0.264784	+	0.964308i	$0.414699\pi$
$608$	59.6886	2.42069
$609$	0	0
$610$	0	0
$611$	4.31351	0.174506
$612$	−5.75872	−0.232783
$613$	−15.5174	−0.626744	−0.313372	−	0.949630i	$-0.601459\pi$
$613$	−15.5174	−0.626744	−0.313372	+	0.949630i	$0.601459\pi$
$614$	−28.3668	−1.14479
$615$	0	0
$616$	0	0
$617$	−22.7649	−0.916479	−0.458240	−	0.888829i	$-0.651520\pi$
$617$	−22.7649	−0.916479	−0.458240	+	0.888829i	$0.651520\pi$
$618$	5.84324	0.235050
$619$	−7.92777	−0.318644	−0.159322	−	0.987227i	$-0.550931\pi$
$619$	−7.92777	−0.318644	−0.159322	+	0.987227i	$0.550931\pi$
$620$	0	0
$621$	−2.34017	−0.0939079
$622$	64.5646	2.58881
$623$	0	0
$624$	12.7526	0.510512
$625$	0	0
$626$	88.7358	3.54659
$627$	6.15676	0.245877
$628$	−26.2823	−1.04878
$629$	−11.6865	−0.465971
$630$	0	0
$631$	19.2039	0.764497	0.382248	−	0.924060i	$-0.375150\pi$
$631$	19.2039	0.764497	0.382248	+	0.924060i	$0.375150\pi$
$632$	−55.7152	−2.21623
$633$	13.6742	0.543501
$634$	48.5380	1.92769
$635$	0	0
$636$	−20.0722	−0.795916
$637$	0	0
$638$	36.1978	1.43308
$639$	2.00000	0.0791188
$640$	0	0
$641$	−5.94668	−0.234880	−0.117440	−	0.993080i	$-0.537469\pi$
$641$	−5.94668	−0.234880	−0.117440	+	0.993080i	$0.537469\pi$
$642$	−44.6947	−1.76396
$643$	30.8904	1.21820	0.609100	−	0.793094i	$-0.291531\pi$
$643$	30.8904	1.21820	0.609100	+	0.793094i	$0.291531\pi$
$644$	0	0
$645$	0	0
$646$	−8.99386	−0.353859
$647$	19.2039	0.754985	0.377492	−	0.926013i	$-0.376786\pi$
$647$	19.2039	0.754985	0.377492	+	0.926013i	$0.376786\pi$
$648$	−9.04945	−0.355496
$649$	−21.0472	−0.826174
$650$	0	0
$651$	0	0
$652$	−52.5646	−2.05859
$653$	−28.5548	−1.11744	−0.558718	−	0.829358i	$-0.688706\pi$
$653$	−28.5548	−1.11744	−0.558718	+	0.829358i	$0.688706\pi$
$654$	−34.7792	−1.35998
$655$	0	0
$656$	−89.8987	−3.50995
$657$	−7.07838	−0.276154
$658$	0	0
$659$	−27.9877	−1.09025	−0.545123	−	0.838356i	$-0.683517\pi$
$659$	−27.9877	−1.09025	−0.545123	+	0.838356i	$0.683517\pi$
$660$	0	0
$661$	22.1445	0.861320	0.430660	−	0.902514i	$-0.358281\pi$
$661$	22.1445	0.861320	0.430660	+	0.902514i	$0.358281\pi$
$662$	3.68649	0.143279
$663$	−0.993857	−0.0385982
$664$	61.8720	2.40110
$665$	0	0
$666$	−29.3607	−1.13770
$667$	−15.6332	−0.605319
$668$	102.552	3.96787
$669$	−21.6742	−0.837973
$670$	0	0
$671$	8.31351	0.320940
$672$	0	0
$673$	2.21008	0.0851923	0.0425962	−	0.999092i	$-0.486437\pi$
$673$	2.21008	0.0851923	0.0425962	+	0.999092i	$0.486437\pi$
$674$	68.7091	2.64658
$675$	0	0
$676$	−64.8864	−2.49563
$677$	−19.5486	−0.751315	−0.375658	−	0.926758i	$-0.622583\pi$
$677$	−19.5486	−0.751315	−0.375658	+	0.926758i	$0.622583\pi$
$678$	−14.1834	−0.544711
$679$	0	0
$680$	0	0
$681$	−11.5174	−0.441350
$682$	−42.0410	−1.60983
$683$	11.8166	0.452149	0.226074	−	0.974110i	$-0.427411\pi$
$683$	11.8166	0.452149	0.226074	+	0.974110i	$0.427411\pi$
$684$	−16.4391	−0.628564
$685$	0	0
$686$	0	0
$687$	12.8371	0.489766
$688$	−90.2676	−3.44142
$689$	−3.46412	−0.131973
$690$	0	0
$691$	−11.7587	−0.447323	−0.223661	−	0.974667i	$-0.571801\pi$
$691$	−11.7587	−0.447323	−0.223661	+	0.974667i	$0.571801\pi$
$692$	119.829	4.55520
$693$	0	0
$694$	−45.6886	−1.73431
$695$	0	0
$696$	−60.4534	−2.29148
$697$	7.00614	0.265377
$698$	25.7854	0.975991
$699$	6.76487	0.255871
$700$	0	0
$701$	9.94668	0.375681	0.187840	−	0.982200i	$-0.439851\pi$
$701$	9.94668	0.375681	0.187840	+	0.982200i	$0.439851\pi$
$702$	−2.49693	−0.0942405
$703$	−33.3607	−1.25822
$704$	49.7152	1.87371
$705$	0	0
$706$	−96.9914	−3.65032
$707$	0	0
$708$	56.1978	2.11204
$709$	−11.0472	−0.414886	−0.207443	−	0.978247i	$-0.566514\pi$
$709$	−11.0472	−0.414886	−0.207443	+	0.978247i	$0.566514\pi$
$710$	0	0
$711$	6.15676	0.230896
$712$	75.4740	2.82851
$713$	18.1568	0.679976
$714$	0	0
$715$	0	0
$716$	53.4017	1.99572
$717$	23.3607	0.872421
$718$	60.4534	2.25610
$719$	6.15676	0.229608	0.114804	−	0.993388i	$-0.463376\pi$
$719$	6.15676	0.229608	0.114804	+	0.993388i	$0.463376\pi$
$720$	0	0
$721$	0	0
$722$	25.8020	0.960252
$723$	−14.6803	−0.545968
$724$	45.5174	1.69164
$725$	0	0
$726$	−18.9649	−0.703854
$727$	−2.89043	−0.107200	−0.0536000	−	0.998562i	$-0.517070\pi$
$727$	−2.89043	−0.107200	−0.0536000	+	0.998562i	$0.517070\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	7.03489	0.260195
$732$	−22.1978	−0.820454
$733$	25.7998	0.952936	0.476468	−	0.879192i	$-0.341917\pi$
$733$	25.7998	0.952936	0.476468	+	0.879192i	$0.341917\pi$
$734$	55.0349	2.03138
$735$	0	0
$736$	−45.3751	−1.67255
$737$	9.36069	0.344806
$738$	17.6020	0.647937
$739$	−1.04718	−0.0385212	−0.0192606	−	0.999814i	$-0.506131\pi$
$739$	−1.04718	−0.0385212	−0.0192606	+	0.999814i	$0.506131\pi$
$740$	0	0
$741$	−2.83710	−0.104224
$742$	0	0
$743$	−9.97334	−0.365886	−0.182943	−	0.983123i	$-0.558562\pi$
$743$	−9.97334	−0.365886	−0.182943	+	0.983123i	$0.558562\pi$
$744$	70.2122	2.57410
$745$	0	0
$746$	43.3484	1.58710
$747$	−6.83710	−0.250156
$748$	−11.5174	−0.421120
$749$	0	0
$750$	0	0
$751$	3.26633	0.119190	0.0595950	−	0.998223i	$-0.481019\pi$
$751$	3.26633	0.119190	0.0595950	+	0.998223i	$0.481019\pi$
$752$	−64.7624	−2.36164
$753$	9.16290	0.333915
$754$	−16.6803	−0.607462
$755$	0	0
$756$	0	0
$757$	−49.9877	−1.81683	−0.908417	−	0.418065i	$-0.862708\pi$
$757$	−49.9877	−1.81683	−0.908417	+	0.418065i	$0.862708\pi$
$758$	−16.6803	−0.605857
$759$	−4.68035	−0.169886
$760$	0	0
$761$	−2.61265	−0.0947083	−0.0473542	−	0.998878i	$-0.515079\pi$
$761$	−2.61265	−0.0947083	−0.0473542	+	0.998878i	$0.515079\pi$
$762$	4.99386	0.180908
$763$	0	0
$764$	82.0288	2.96770
$765$	0	0
$766$	72.7091	2.62709
$767$	9.69878	0.350202
$768$	−27.6803	−0.998828
$769$	15.6742	0.565226	0.282613	−	0.959234i	$-0.408799\pi$
$769$	15.6742	0.565226	0.282613	+	0.959234i	$0.408799\pi$
$770$	0	0
$771$	−5.07838	−0.182893
$772$	44.6803	1.60808
$773$	−5.81205	−0.209045	−0.104522	−	0.994523i	$-0.533331\pi$
$773$	−5.81205	−0.209045	−0.104522	+	0.994523i	$0.533331\pi$
$774$	17.6742	0.635286
$775$	0	0
$776$	−76.3689	−2.74148
$777$	0	0
$778$	15.2618	0.547162
$779$	20.0000	0.716574
$780$	0	0
$781$	4.00000	0.143131
$782$	6.83710	0.244494
$783$	6.68035	0.238736
$784$	0	0
$785$	0	0
$786$	−4.00000	−0.142675
$787$	−39.3484	−1.40262	−0.701310	−	0.712857i	$-0.747401\pi$
$787$	−39.3484	−1.40262	−0.701310	+	0.712857i	$0.747401\pi$
$788$	−62.7936	−2.23693
$789$	−5.65983	−0.201495
$790$	0	0
$791$	0	0
$792$	−18.0989	−0.643116
$793$	−3.83096	−0.136041
$794$	102.410	3.63439
$795$	0	0
$796$	120.666	4.27688
$797$	−28.2823	−1.00181	−0.500905	−	0.865502i	$-0.667000\pi$
$797$	−28.2823	−1.00181	−0.500905	+	0.865502i	$0.667000\pi$
$798$	0	0
$799$	5.04718	0.178556
$800$	0	0
$801$	−8.34017	−0.294686
$802$	36.9360	1.30426
$803$	−14.1568	−0.499581
$804$	−24.9939	−0.881465
$805$	0	0
$806$	19.3730	0.682384
$807$	−27.8576	−0.980635
$808$	−52.6369	−1.85176
$809$	−15.6742	−0.551076	−0.275538	−	0.961290i	$-0.588856\pi$
$809$	−15.6742	−0.551076	−0.275538	+	0.961290i	$0.588856\pi$
$810$	0	0
$811$	42.1666	1.48067	0.740335	−	0.672238i	$-0.234667\pi$
$811$	42.1666	1.48067	0.740335	+	0.672238i	$0.234667\pi$
$812$	0	0
$813$	25.1194	0.880976
$814$	−58.7214	−2.05818
$815$	0	0
$816$	14.9216	0.522361
$817$	20.0821	0.702583
$818$	33.4719	1.17032
$819$	0	0
$820$	0	0
$821$	−39.0472	−1.36276	−0.681378	−	0.731932i	$-0.738619\pi$
$821$	−39.0472	−1.36276	−0.681378	+	0.731932i	$0.738619\pi$
$822$	12.0267	0.419478
$823$	−36.5646	−1.27456	−0.637281	−	0.770631i	$-0.719941\pi$
$823$	−36.5646	−1.27456	−0.637281	+	0.770631i	$0.719941\pi$
$824$	−19.5174	−0.679922
$825$	0	0
$826$	0	0
$827$	50.2245	1.74648	0.873238	−	0.487294i	$-0.162016\pi$
$827$	50.2245	1.74648	0.873238	+	0.487294i	$0.162016\pi$
$828$	12.4969	0.434298
$829$	−32.8371	−1.14048	−0.570240	−	0.821478i	$-0.693150\pi$
$829$	−32.8371	−1.14048	−0.570240	+	0.821478i	$0.693150\pi$
$830$	0	0
$831$	28.1978	0.978171
$832$	−22.9093	−0.794238
$833$	0	0
$834$	−36.8515	−1.27606
$835$	0	0
$836$	−32.8781	−1.13711
$837$	−7.75872	−0.268181
$838$	78.5523	2.71355
$839$	13.3607	0.461262	0.230631	−	0.973041i	$-0.425921\pi$
$839$	13.3607	0.461262	0.230631	+	0.973041i	$0.425921\pi$
$840$	0	0
$841$	15.6270	0.538863
$842$	41.0805	1.41573
$843$	20.3545	0.701048
$844$	−73.0226	−2.51354
$845$	0	0
$846$	12.6803	0.435959
$847$	0	0
$848$	52.0098	1.78603
$849$	23.5174	0.807117
$850$	0	0
$851$	25.3607	0.869353
$852$	−10.6803	−0.365903
$853$	39.6430	1.35735	0.678675	−	0.734438i	$-0.262554\pi$
$853$	39.6430	1.35735	0.678675	+	0.734438i	$0.262554\pi$
$854$	0	0
$855$	0	0
$856$	149.288	5.10256
$857$	29.7054	1.01472	0.507359	−	0.861735i	$-0.330622\pi$
$857$	29.7054	1.01472	0.507359	+	0.861735i	$0.330622\pi$
$858$	−4.99386	−0.170487
$859$	3.07838	0.105033	0.0525164	−	0.998620i	$-0.483276\pi$
$859$	3.07838	0.105033	0.0525164	+	0.998620i	$0.483276\pi$
$860$	0	0
$861$	0	0
$862$	27.9421	0.951713
$863$	−6.39350	−0.217637	−0.108819	−	0.994062i	$-0.534707\pi$
$863$	−6.39350	−0.217637	−0.108819	+	0.994062i	$0.534707\pi$
$864$	19.3896	0.659648
$865$	0	0
$866$	55.3751	1.88172
$867$	15.8371	0.537856
$868$	0	0
$869$	12.3135	0.417707
$870$	0	0
$871$	−4.31351	−0.146158
$872$	116.169	3.93397
$873$	8.43907	0.285619
$874$	19.5174	0.660188
$875$	0	0
$876$	37.7998	1.27714
$877$	−1.21622	−0.0410689	−0.0205345	−	0.999789i	$-0.506537\pi$
$877$	−1.21622	−0.0410689	−0.0205345	+	0.999789i	$0.506537\pi$
$878$	−45.8453	−1.54721
$879$	−2.92162	−0.0985439
$880$	0	0
$881$	−15.9733	−0.538155	−0.269078	−	0.963118i	$-0.586719\pi$
$881$	−15.9733	−0.538155	−0.269078	+	0.963118i	$0.586719\pi$
$882$	0	0
$883$	11.6865	0.393282	0.196641	−	0.980476i	$-0.436997\pi$
$883$	11.6865	0.393282	0.196641	+	0.980476i	$0.436997\pi$
$884$	5.30737	0.178506
$885$	0	0
$886$	34.7070	1.16600
$887$	25.6209	0.860265	0.430132	−	0.902766i	$-0.358467\pi$
$887$	25.6209	0.860265	0.430132	+	0.902766i	$0.358467\pi$
$888$	98.0698	3.29101
$889$	0	0
$890$	0	0
$891$	2.00000	0.0670025
$892$	115.744	3.87540
$893$	14.4079	0.482141
$894$	42.4657	1.42027
$895$	0	0
$896$	0	0
$897$	2.15676	0.0720120
$898$	39.6286	1.32242
$899$	−51.8310	−1.72866
$900$	0	0
$901$	−4.05332	−0.135036
$902$	35.2039	1.17216
$903$	0	0
$904$	47.3751	1.57567
$905$	0	0
$906$	15.8310	0.525948
$907$	−57.7563	−1.91777	−0.958883	−	0.283802i	$-0.908404\pi$
$907$	−57.7563	−1.91777	−0.958883	+	0.283802i	$0.908404\pi$
$908$	61.5052	2.04112
$909$	5.81658	0.192924
$910$	0	0
$911$	−35.9877	−1.19233	−0.596163	−	0.802863i	$-0.703309\pi$
$911$	−35.9877	−1.19233	−0.596163	+	0.802863i	$0.703309\pi$
$912$	42.5958	1.41049
$913$	−13.6742	−0.452550
$914$	−38.3545	−1.26866
$915$	0	0
$916$	−68.5523	−2.26503
$917$	0	0
$918$	−2.92162	−0.0964279
$919$	46.7214	1.54120	0.770598	−	0.637321i	$-0.219958\pi$
$919$	46.7214	1.54120	0.770598	+	0.637321i	$0.219958\pi$
$920$	0	0
$921$	−10.4703	−0.345007
$922$	0.921622	0.0303520
$923$	−1.84324	−0.0606711
$924$	0	0
$925$	0	0
$926$	26.6681	0.876367
$927$	2.15676	0.0708371
$928$	129.529	4.25201
$929$	53.0493	1.74049	0.870245	−	0.492619i	$-0.163960\pi$
$929$	53.0493	1.74049	0.870245	+	0.492619i	$0.163960\pi$
$930$	0	0
$931$	0	0
$932$	−36.1256	−1.18333
$933$	23.8310	0.780191
$934$	−31.2039	−1.02102
$935$	0	0
$936$	8.34017	0.272607
$937$	16.1256	0.526799	0.263400	−	0.964687i	$-0.415156\pi$
$937$	16.1256	0.526799	0.263400	+	0.964687i	$0.415156\pi$
$938$	0	0
$939$	32.7526	1.06884
$940$	0	0
$941$	−24.7070	−0.805425	−0.402713	−	0.915326i	$-0.631933\pi$
$941$	−24.7070	−0.805425	−0.402713	+	0.915326i	$0.631933\pi$
$942$	−13.3340	−0.434446
$943$	−15.2039	−0.495108
$944$	−145.616	−4.73940
$945$	0	0
$946$	35.3484	1.14928
$947$	−6.53797	−0.212455	−0.106228	−	0.994342i	$-0.533877\pi$
$947$	−6.53797	−0.212455	−0.106228	+	0.994342i	$0.533877\pi$
$948$	−32.8781	−1.06783
$949$	6.52359	0.211765
$950$	0	0
$951$	17.9155	0.580949
$952$	0	0
$953$	6.11327	0.198028	0.0990142	−	0.995086i	$-0.468431\pi$
$953$	6.11327	0.198028	0.0990142	+	0.995086i	$0.468431\pi$
$954$	−10.1834	−0.329700
$955$	0	0
$956$	−124.750	−4.03471
$957$	13.3607	0.431890
$958$	−52.8781	−1.70842
$959$	0	0
$960$	0	0
$961$	29.1978	0.941864
$962$	27.0595	0.872432
$963$	−16.4969	−0.531606
$964$	78.3956	2.52495
$965$	0	0
$966$	0	0
$967$	−25.6209	−0.823912	−0.411956	−	0.911204i	$-0.635154\pi$
$967$	−25.6209	−0.823912	−0.411956	+	0.911204i	$0.635154\pi$
$968$	63.3461	2.03602
$969$	−3.31965	−0.106643
$970$	0	0
$971$	−4.05332	−0.130077	−0.0650387	−	0.997883i	$-0.520717\pi$
$971$	−4.05332	−0.130077	−0.0650387	+	0.997883i	$0.520717\pi$
$972$	−5.34017	−0.171286
$973$	0	0
$974$	−62.7214	−2.00972
$975$	0	0
$976$	57.5174	1.84109
$977$	−3.81205	−0.121958	−0.0609791	−	0.998139i	$-0.519422\pi$
$977$	−3.81205	−0.121958	−0.0609791	+	0.998139i	$0.519422\pi$
$978$	−26.6681	−0.852751
$979$	−16.6803	−0.533106
$980$	0	0
$981$	−12.8371	−0.409857
$982$	−5.41855	−0.172913
$983$	−24.0000	−0.765481	−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000	−0.765481	−0.382741	+	0.923856i	$0.625020\pi$
$984$	−58.7936	−1.87427
$985$	0	0
$986$	−19.5174	−0.621562
$987$	0	0
$988$	15.1506	0.482005
$989$	−15.2663	−0.485441
$990$	0	0
$991$	−42.4079	−1.34713	−0.673565	−	0.739128i	$-0.735238\pi$
$991$	−42.4079	−1.34713	−0.673565	+	0.739128i	$0.735238\pi$
$992$	−150.439	−4.77643
$993$	1.36069	0.0431803
$994$	0	0
$995$	0	0
$996$	36.5113	1.15690
$997$	−43.4740	−1.37683	−0.688417	−	0.725315i	$-0.741694\pi$
$997$	−43.4740	−1.37683	−0.688417	+	0.725315i	$0.741694\pi$
$998$	−73.7030	−2.33303
$999$	−10.8371	−0.342871

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3675.2.a.bi.1.1		3
5.2	odd	4		735.2.d.b.589.1		6
5.3	odd	4		735.2.d.b.589.6		6
5.4	even	2		3675.2.a.bj.1.3		3
7.6	odd	2		525.2.a.j.1.1		3
15.2	even	4		2205.2.d.l.1324.6		6
15.8	even	4		2205.2.d.l.1324.1		6
21.20	even	2		1575.2.a.x.1.3		3
28.27	even	2		8400.2.a.dg.1.2		3
35.2	odd	12		735.2.q.f.214.1		12
35.3	even	12		735.2.q.e.79.1		12
35.12	even	12		735.2.q.e.214.1		12
35.13	even	4		105.2.d.b.64.6	yes	6
35.17	even	12		735.2.q.e.79.6		12
35.18	odd	12		735.2.q.f.79.1		12
35.23	odd	12		735.2.q.f.214.6		12
35.27	even	4		105.2.d.b.64.1	✓	6
35.32	odd	12		735.2.q.f.79.6		12
35.33	even	12		735.2.q.e.214.6		12
35.34	odd	2		525.2.a.k.1.3		3
105.62	odd	4		315.2.d.e.64.6		6
105.83	odd	4		315.2.d.e.64.1		6
105.104	even	2		1575.2.a.w.1.1		3
140.27	odd	4		1680.2.t.k.1009.6		6
140.83	odd	4		1680.2.t.k.1009.3		6
140.139	even	2		8400.2.a.dj.1.2		3
420.83	even	4		5040.2.t.v.1009.1		6
420.167	even	4		5040.2.t.v.1009.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.d.b.64.1	✓	6	35.27	even	4
105.2.d.b.64.6	yes	6	35.13	even	4
315.2.d.e.64.1		6	105.83	odd	4
315.2.d.e.64.6		6	105.62	odd	4
525.2.a.j.1.1		3	7.6	odd	2
525.2.a.k.1.3		3	35.34	odd	2
735.2.d.b.589.1		6	5.2	odd	4
735.2.d.b.589.6		6	5.3	odd	4
735.2.q.e.79.1		12	35.3	even	12
735.2.q.e.79.6		12	35.17	even	12
735.2.q.e.214.1		12	35.12	even	12
735.2.q.e.214.6		12	35.33	even	12
735.2.q.f.79.1		12	35.18	odd	12
735.2.q.f.79.6		12	35.32	odd	12
735.2.q.f.214.1		12	35.2	odd	12
735.2.q.f.214.6		12	35.23	odd	12
1575.2.a.w.1.1		3	105.104	even	2
1575.2.a.x.1.3		3	21.20	even	2
1680.2.t.k.1009.3		6	140.83	odd	4
1680.2.t.k.1009.6		6	140.27	odd	4
2205.2.d.l.1324.1		6	15.8	even	4
2205.2.d.l.1324.6		6	15.2	even	4
3675.2.a.bi.1.1		3	1.1	even	1	trivial
3675.2.a.bj.1.3		3	5.4	even	2
5040.2.t.v.1009.1		6	420.83	even	4
5040.2.t.v.1009.2		6	420.167	even	4
8400.2.a.dg.1.2		3	28.27	even	2
8400.2.a.dj.1.2		3	140.139	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-2.70928 q^{2} -1.00000 q^{3} +5.34017 q^{4} +2.70928 q^{6} -9.04945 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.70928 q^{2} -1.00000 q^{3} +5.34017 q^{4} +2.70928 q^{6} -9.04945 q^{8} +1.00000 q^{9} +2.00000 q^{11} -5.34017 q^{12} -0.921622 q^{13} +13.8371 q^{16} -1.07838 q^{17} -2.70928 q^{18} -3.07838 q^{19} -5.41855 q^{22} +2.34017 q^{23} +9.04945 q^{24} +2.49693 q^{26} -1.00000 q^{27} -6.68035 q^{29} +7.75872 q^{31} -19.3896 q^{32} -2.00000 q^{33} +2.92162 q^{34} +5.34017 q^{36} +10.8371 q^{37} +8.34017 q^{38} +0.921622 q^{39} -6.49693 q^{41} -6.52359 q^{43} +10.6803 q^{44} -6.34017 q^{46} -4.68035 q^{47} -13.8371 q^{48} +1.07838 q^{51} -4.92162 q^{52} +3.75872 q^{53} +2.70928 q^{54} +3.07838 q^{57} +18.0989 q^{58} -10.5236 q^{59} +4.15676 q^{61} -21.0205 q^{62} +24.8576 q^{64} +5.41855 q^{66} +4.68035 q^{67} -5.75872 q^{68} -2.34017 q^{69} +2.00000 q^{71} -9.04945 q^{72} -7.07838 q^{73} -29.3607 q^{74} -16.4391 q^{76} -2.49693 q^{78} +6.15676 q^{79} +1.00000 q^{81} +17.6020 q^{82} -6.83710 q^{83} +17.6742 q^{86} +6.68035 q^{87} -18.0989 q^{88} -8.34017 q^{89} +12.4969 q^{92} -7.75872 q^{93} +12.6803 q^{94} +19.3896 q^{96} +8.43907 q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} - 3 q^{3} + 5 q^{4} + q^{6} - 9 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q - q^2 - 3 * q^3 + 5 * q^4 + q^6 - 9 * q^8 + 3 * q^9	\( 3 q - q^{2} - 3 q^{3} + 5 q^{4} + q^{6} - 9 q^{8} + 3 q^{9} + 6 q^{11} - 5 q^{12} - 6 q^{13} + 13 q^{16} - q^{18} - 6 q^{19} - 2 q^{22} - 4 q^{23} + 9 q^{24} - 10 q^{26} - 3 q^{27} + 2 q^{29} - 2 q^{31} - 29 q^{32} - 6 q^{33} + 12 q^{34} + 5 q^{36} + 4 q^{37} + 14 q^{38} + 6 q^{39} - 2 q^{41} - 4 q^{43} + 10 q^{44} - 8 q^{46} + 8 q^{47} - 13 q^{48} - 18 q^{52} - 14 q^{53} + q^{54} + 6 q^{57} + 18 q^{58} - 16 q^{59} + 6 q^{61} - 30 q^{62} + 13 q^{64} + 2 q^{66} - 8 q^{67} + 8 q^{68} + 4 q^{69} + 6 q^{71} - 9 q^{72} - 18 q^{73} - 44 q^{74} - 2 q^{76} + 10 q^{78} + 12 q^{79} + 3 q^{81} + 34 q^{82} + 8 q^{83} - 4 q^{86} - 2 q^{87} - 18 q^{88} - 14 q^{89} + 20 q^{92} + 2 q^{93} + 16 q^{94} + 29 q^{96} - 22 q^{97} + 6 q^{99}+O(q^{100}) \) 3 * q - q^2 - 3 * q^3 + 5 * q^4 + q^6 - 9 * q^8 + 3 * q^9 + 6 * q^11 - 5 * q^12 - 6 * q^13 + 13 * q^16 - q^18 - 6 * q^19 - 2 * q^22 - 4 * q^23 + 9 * q^24 - 10 * q^26 - 3 * q^27 + 2 * q^29 - 2 * q^31 - 29 * q^32 - 6 * q^33 + 12 * q^34 + 5 * q^36 + 4 * q^37 + 14 * q^38 + 6 * q^39 - 2 * q^41 - 4 * q^43 + 10 * q^44 - 8 * q^46 + 8 * q^47 - 13 * q^48 - 18 * q^52 - 14 * q^53 + q^54 + 6 * q^57 + 18 * q^58 - 16 * q^59 + 6 * q^61 - 30 * q^62 + 13 * q^64 + 2 * q^66 - 8 * q^67 + 8 * q^68 + 4 * q^69 + 6 * q^71 - 9 * q^72 - 18 * q^73 - 44 * q^74 - 2 * q^76 + 10 * q^78 + 12 * q^79 + 3 * q^81 + 34 * q^82 + 8 * q^83 - 4 * q^86 - 2 * q^87 - 18 * q^88 - 14 * q^89 + 20 * q^92 + 2 * q^93 + 16 * q^94 + 29 * q^96 - 22 * q^97 + 6 * q^99

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