LMFDB - Embedded newform 1575.2.a.x.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1575,2,Mod(1,1575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1575, 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("1575.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1575 = 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1575.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:	$12.5764383184$
Analytic rank:	$0$
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			1.3
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	1575.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.70928 q^{2} +5.34017 q^{4} +1.00000 q^{7} +9.04945 q^{8} +O(q^{10})$	$q+2.70928 q^{2} +5.34017 q^{4} +1.00000 q^{7} +9.04945 q^{8} -2.00000 q^{11} +0.921622 q^{13} +2.70928 q^{14} +13.8371 q^{16} -1.07838 q^{17} +3.07838 q^{19} -5.41855 q^{22} -2.34017 q^{23} +2.49693 q^{26} +5.34017 q^{28} +6.68035 q^{29} -7.75872 q^{31} +19.3896 q^{32} -2.92162 q^{34} +10.8371 q^{37} +8.34017 q^{38} -6.49693 q^{41} -6.52359 q^{43} -10.6803 q^{44} -6.34017 q^{46} -4.68035 q^{47} +1.00000 q^{49} +4.92162 q^{52} -3.75872 q^{53} +9.04945 q^{56} +18.0989 q^{58} -10.5236 q^{59} -4.15676 q^{61} -21.0205 q^{62} +24.8576 q^{64} +4.68035 q^{67} -5.75872 q^{68} -2.00000 q^{71} +7.07838 q^{73} +29.3607 q^{74} +16.4391 q^{76} -2.00000 q^{77} +6.15676 q^{79} -17.6020 q^{82} -6.83710 q^{83} -17.6742 q^{86} -18.0989 q^{88} -8.34017 q^{89} +0.921622 q^{91} -12.4969 q^{92} -12.6803 q^{94} -8.43907 q^{97} +2.70928 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + 5 q^{4} + 3 q^{7} + 9 q^{8}+O(q^{10}) $ 3 * q + q^2 + 5 * q^4 + 3 * q^7 + 9 * q^8	$ 3 q + q^{2} + 5 q^{4} + 3 q^{7} + 9 q^{8} - 6 q^{11} + 6 q^{13} + q^{14} + 13 q^{16} + 6 q^{19} - 2 q^{22} + 4 q^{23} - 10 q^{26} + 5 q^{28} - 2 q^{29} + 2 q^{31} + 29 q^{32} - 12 q^{34} + 4 q^{37} + 14 q^{38} - 2 q^{41} - 4 q^{43} - 10 q^{44} - 8 q^{46} + 8 q^{47} + 3 q^{49} + 18 q^{52} + 14 q^{53} + 9 q^{56} + 18 q^{58} - 16 q^{59} - 6 q^{61} - 30 q^{62} + 13 q^{64} - 8 q^{67} + 8 q^{68} - 6 q^{71} + 18 q^{73} + 44 q^{74} + 2 q^{76} - 6 q^{77} + 12 q^{79} - 34 q^{82} + 8 q^{83} + 4 q^{86} - 18 q^{88} - 14 q^{89} + 6 q^{91} - 20 q^{92} - 16 q^{94} + 22 q^{97} + q^{98}+O(q^{100}) $ 3 * q + q^2 + 5 * q^4 + 3 * q^7 + 9 * q^8 - 6 * q^11 + 6 * q^13 + q^14 + 13 * q^16 + 6 * q^19 - 2 * q^22 + 4 * q^23 - 10 * q^26 + 5 * q^28 - 2 * q^29 + 2 * q^31 + 29 * q^32 - 12 * q^34 + 4 * q^37 + 14 * q^38 - 2 * q^41 - 4 * q^43 - 10 * q^44 - 8 * q^46 + 8 * q^47 + 3 * q^49 + 18 * q^52 + 14 * q^53 + 9 * q^56 + 18 * q^58 - 16 * q^59 - 6 * q^61 - 30 * q^62 + 13 * q^64 - 8 * q^67 + 8 * q^68 - 6 * q^71 + 18 * q^73 + 44 * q^74 + 2 * q^76 - 6 * q^77 + 12 * q^79 - 34 * q^82 + 8 * q^83 + 4 * q^86 - 18 * q^88 - 14 * q^89 + 6 * q^91 - 20 * q^92 - 16 * q^94 + 22 * q^97 + q^98

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	$-0.0927213\pi$
$2$	2.70928	1.91575	0.957873	+	0.287190i	$0.0927213\pi$
$3$	0	0
$3$	0	0
$4$	5.34017	2.67009
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	9.04945	3.19946
$9$	0	0
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	0.921622	0.255612	0.127806	−	0.991799i	$-0.459207\pi$
$13$	0.921622	0.255612	0.127806	+	0.991799i	$0.459207\pi$
$14$	2.70928	0.724084
$15$	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	$0.541746\pi$
$18$	0	0
$19$	3.07838	0.706228	0.353114	−	0.935580i	$-0.385123\pi$
$19$	3.07838	0.706228	0.353114	+	0.935580i	$0.385123\pi$
$20$	0	0
$21$	0	0
$22$	−5.41855	−1.15524
$23$	−2.34017	−0.487960	−0.243980	−	0.969780i	$-0.578453\pi$
$23$	−2.34017	−0.487960	−0.243980	+	0.969780i	$0.578453\pi$
$24$	0	0
$25$	0	0
$26$	2.49693	0.489688
$27$	0	0
$28$	5.34017	1.00920
$29$	6.68035	1.24051	0.620255	−	0.784401i	$-0.287029\pi$
$29$	6.68035	1.24051	0.620255	+	0.784401i	$0.287029\pi$
$30$	0	0
$31$	−7.75872	−1.39351	−0.696754	−	0.717310i	$-0.745373\pi$
$31$	−7.75872	−1.39351	−0.696754	+	0.717310i	$0.745373\pi$
$32$	19.3896	3.42763
$33$	0	0
$34$	−2.92162	−0.501054
$35$	0	0
$36$	0	0
$37$	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	0
$40$	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	$0.610884\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	4.92162	0.682506
$53$	−3.75872	−0.516300	−0.258150	−	0.966105i	$-0.583113\pi$
$53$	−3.75872	−0.516300	−0.258150	+	0.966105i	$0.583113\pi$
$54$	0	0
$55$	0	0
$56$	9.04945	1.20928
$57$	0	0
$58$	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	$-0.585738\pi$
$61$	−4.15676	−0.532218	−0.266109	+	0.963943i	$0.585738\pi$
$62$	−21.0205	−2.66961
$63$	0	0
$64$	24.8576	3.10720
$65$	0	0
$66$	0	0
$67$	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
$69$	0	0
$70$	0	0
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	0	0
$73$	7.07838	0.828461	0.414231	−	0.910172i	$-0.364051\pi$
$73$	7.07838	0.828461	0.414231	+	0.910172i	$0.364051\pi$
$74$	29.3607	3.41311
$75$	0	0
$76$	16.4391	1.88569
$77$	−2.00000	−0.227921
$78$	0	0
$79$	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
$81$	0	0
$82$	−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
$87$	0	0
$88$	−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
$91$	0.921622	0.0966123
$92$	−12.4969	−1.30289
$93$	0	0
$94$	−12.6803	−1.30788
$95$	0	0
$96$	0	0
$97$	−8.43907	−0.856858	−0.428429	−	0.903576i	$-0.640933\pi$
$97$	−8.43907	−0.856858	−0.428429	+	0.903576i	$0.640933\pi$
$98$	2.70928	0.273678
$99$	0	0
$100$	0	0
$101$	5.81658	0.578772	0.289386	−	0.957213i	$-0.406549\pi$
$101$	5.81658	0.578772	0.289386	+	0.957213i	$0.406549\pi$
$102$	0	0
$103$	−2.15676	−0.212511	−0.106256	−	0.994339i	$-0.533886\pi$
$103$	−2.15676	−0.212511	−0.106256	+	0.994339i	$0.533886\pi$
$104$	8.34017	0.817821
$105$	0	0
$106$	−10.1834	−0.989101
$107$	16.4969	1.59482	0.797409	−	0.603439i	$-0.206203\pi$
$107$	16.4969	1.59482	0.797409	+	0.603439i	$0.206203\pi$
$108$	0	0
$109$	−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
$111$	0	0
$112$	13.8371	1.30748
$113$	5.23513	0.492480	0.246240	−	0.969209i	$-0.420805\pi$
$113$	5.23513	0.492480	0.246240	+	0.969209i	$0.420805\pi$
$114$	0	0
$115$	0	0
$116$	35.6742	3.31227
$117$	0	0
$118$	−28.5113	−2.62468
$119$	−1.07838	−0.0988547
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−11.2618	−1.01960
$123$	0	0
$124$	−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
$129$	0	0
$130$	0	0
$131$	−1.47641	−0.128995	−0.0644973	−	0.997918i	$-0.520544\pi$
$131$	−1.47641	−0.128995	−0.0644973	+	0.997918i	$0.520544\pi$
$132$	0	0
$133$	3.07838	0.266929
$134$	12.6803	1.09542
$135$	0	0
$136$	−9.75872	−0.836804
$137$	−4.43907	−0.379255	−0.189628	−	0.981856i	$-0.560728\pi$
$137$	−4.43907	−0.379255	−0.189628	+	0.981856i	$0.560728\pi$
$138$	0	0
$139$	13.6020	1.15370	0.576852	−	0.816849i	$-0.304281\pi$
$139$	13.6020	1.15370	0.576852	+	0.816849i	$0.304281\pi$
$140$	0	0
$141$	0	0
$142$	−5.41855	−0.454715
$143$	−1.84324	−0.154140
$144$	0	0
$145$	0	0
$146$	19.1773	1.58712
$147$	0	0
$148$	57.8720	4.75705
$149$	−15.6742	−1.28408	−0.642040	−	0.766671i	$-0.721912\pi$
$149$	−15.6742	−1.28408	−0.642040	+	0.766671i	$0.721912\pi$
$150$	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
$153$	0	0
$154$	−5.41855	−0.436639
$155$	0	0
$156$	0	0
$157$	4.92162	0.392788	0.196394	−	0.980525i	$-0.437077\pi$
$157$	4.92162	0.392788	0.196394	+	0.980525i	$0.437077\pi$
$158$	16.6803	1.32702
$159$	0	0
$160$	0	0
$161$	−2.34017	−0.184431
$162$	0	0
$163$	−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
$171$	0	0
$172$	−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	$0.174777\pi$
$174$	0	0
$175$	0	0
$176$	−27.6742	−2.08602
$177$	0	0
$178$	−22.5958	−1.69363
$179$	−10.0000	−0.747435	−0.373718	−	0.927543i	$-0.621917\pi$
$179$	−10.0000	−0.747435	−0.373718	+	0.927543i	$0.621917\pi$
$180$	0	0
$181$	−8.52359	−0.633553	−0.316777	−	0.948500i	$-0.602601\pi$
$181$	−8.52359	−0.633553	−0.316777	+	0.948500i	$0.602601\pi$
$182$	2.49693	0.185085
$183$	0	0
$184$	−21.1773	−1.56121
$185$	0	0
$186$	0	0
$187$	2.15676	0.157718
$188$	−24.9939	−1.82286
$189$	0	0
$190$	0	0
$191$	−15.3607	−1.11146	−0.555730	−	0.831363i	$-0.687561\pi$
$191$	−15.3607	−1.11146	−0.555730	+	0.831363i	$0.687561\pi$
$192$	0	0
$193$	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
$196$	5.34017	0.381441
$197$	11.7587	0.837774	0.418887	−	0.908038i	$-0.362420\pi$
$197$	11.7587	0.837774	0.418887	+	0.908038i	$0.362420\pi$
$198$	0	0
$199$	−22.5958	−1.60178	−0.800888	−	0.598814i	$-0.795639\pi$
$199$	−22.5958	−1.60178	−0.800888	+	0.598814i	$0.795639\pi$
$200$	0	0
$201$	0	0
$202$	15.7587	1.10878
$203$	6.68035	0.468868
$204$	0	0
$205$	0	0
$206$	−5.84324	−0.407118
$207$	0	0
$208$	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
$213$	0	0
$214$	44.6947	3.05527
$215$	0	0
$216$	0	0
$217$	−7.75872	−0.526696
$218$	−34.7792	−2.35555
$219$	0	0
$220$	0	0
$221$	−0.993857	−0.0668541
$222$	0	0
$223$	−21.6742	−1.45141	−0.725706	−	0.688005i	$-0.758487\pi$
$223$	−21.6742	−1.45141	−0.725706	+	0.688005i	$0.758487\pi$
$224$	19.3896	1.29552
$225$	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	$0.375160\pi$
$228$	0	0
$229$	12.8371	0.848300	0.424150	−	0.905592i	$-0.360573\pi$
$229$	12.8371	0.848300	0.424150	+	0.905592i	$0.360573\pi$
$230$	0	0
$231$	0	0
$232$	60.4534	3.96896
$233$	6.76487	0.443181	0.221591	−	0.975140i	$-0.428875\pi$
$233$	6.76487	0.443181	0.221591	+	0.975140i	$0.428875\pi$
$234$	0	0
$235$	0	0
$236$	−56.1978	−3.65816
$237$	0	0
$238$	−2.92162	−0.189381
$239$	23.3607	1.51108	0.755539	−	0.655104i	$-0.227375\pi$
$239$	23.3607	1.51108	0.755539	+	0.655104i	$0.227375\pi$
$240$	0	0
$241$	−14.6803	−0.945644	−0.472822	−	0.881158i	$-0.656765\pi$
$241$	−14.6803	−0.945644	−0.472822	+	0.881158i	$0.656765\pi$
$242$	−18.9649	−1.21911
$243$	0	0
$244$	−22.1978	−1.42107
$245$	0	0
$246$	0	0
$247$	2.83710	0.180520
$248$	−70.2122	−4.45848
$249$	0	0
$250$	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	$0.449370\pi$
$258$	0	0
$259$	10.8371	0.673385
$260$	0	0
$261$	0	0
$262$	−4.00000	−0.247121
$263$	−5.65983	−0.349000	−0.174500	−	0.984657i	$-0.555831\pi$
$263$	−5.65983	−0.349000	−0.174500	+	0.984657i	$0.555831\pi$
$264$	0	0
$265$	0	0
$266$	8.34017	0.511369
$267$	0	0
$268$	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	$-0.223751\pi$
$271$	25.1194	1.52590	0.762948	+	0.646460i	$0.223751\pi$
$272$	−14.9216	−0.904756
$273$	0	0
$274$	−12.0267	−0.726557
$275$	0	0
$276$	0	0
$277$	−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
$279$	0	0
$280$	0	0
$281$	20.3545	1.21425	0.607125	−	0.794606i	$-0.292323\pi$
$281$	20.3545	1.21425	0.607125	+	0.794606i	$0.292323\pi$
$282$	0	0
$283$	23.5174	1.39797	0.698984	−	0.715138i	$-0.253636\pi$
$283$	23.5174	1.39797	0.698984	+	0.715138i	$0.253636\pi$
$284$	−10.6803	−0.633762
$285$	0	0
$286$	−4.99386	−0.295293
$287$	−6.49693	−0.383502
$288$	0	0
$289$	−15.8371	−0.931594
$290$	0	0
$291$	0	0
$292$	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
$297$	0	0
$298$	−42.4657	−2.45997
$299$	−2.15676	−0.124728
$300$	0	0
$301$	−6.52359	−0.376014
$302$	15.8310	0.910969
$303$	0	0
$304$	42.5958	2.44304
$305$	0	0
$306$	0	0
$307$	−10.4703	−0.597570	−0.298785	−	0.954321i	$-0.596581\pi$
$307$	−10.4703	−0.597570	−0.298785	+	0.954321i	$0.596581\pi$
$308$	−10.6803	−0.608569
$309$	0	0
$310$	0	0
$311$	−23.8310	−1.35133	−0.675665	−	0.737209i	$-0.736143\pi$
$311$	−23.8310	−1.35133	−0.675665	+	0.737209i	$0.736143\pi$
$312$	0	0
$313$	32.7526	1.85129	0.925643	−	0.378399i	$-0.123525\pi$
$313$	32.7526	1.85129	0.925643	+	0.378399i	$0.123525\pi$
$314$	13.3340	0.752483
$315$	0	0
$316$	32.8781	1.84954
$317$	17.9155	1.00623	0.503117	−	0.864218i	$-0.332187\pi$
$317$	17.9155	1.00623	0.503117	+	0.864218i	$0.332187\pi$
$318$	0	0
$319$	−13.3607	−0.748055
$320$	0	0
$321$	0	0
$322$	−6.34017	−0.353324
$323$	−3.31965	−0.184710
$324$	0	0
$325$	0	0
$326$	−26.6681	−1.47701
$327$	0	0
$328$	−58.7936	−3.24633
$329$	−4.68035	−0.258036
$330$	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
$333$	0	0
$334$	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
$339$	0	0
$340$	0	0
$341$	15.5174	0.840317
$342$	0	0
$343$	1.00000	0.0539949
$344$	−59.0349	−3.18295
$345$	0	0
$346$	60.7936	3.26829
$347$	−16.8638	−0.905294	−0.452647	−	0.891690i	$-0.649520\pi$
$347$	−16.8638	−0.905294	−0.452647	+	0.891690i	$0.649520\pi$
$348$	0	0
$349$	9.51745	0.509457	0.254729	−	0.967013i	$-0.418014\pi$
$349$	9.51745	0.509457	0.254729	+	0.967013i	$0.418014\pi$
$350$	0	0
$351$	0	0
$352$	−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	$0.0982778\pi$
$354$	0	0
$355$	0	0
$356$	−44.5380	−2.36051
$357$	0	0
$358$	−27.0928	−1.43190
$359$	22.3135	1.17766	0.588831	−	0.808256i	$-0.299588\pi$
$359$	22.3135	1.17766	0.588831	+	0.808256i	$0.299588\pi$
$360$	0	0
$361$	−9.52359	−0.501242
$362$	−23.0928	−1.21373
$363$	0	0
$364$	4.92162	0.257963
$365$	0	0
$366$	0	0
$367$	20.3135	1.06036	0.530178	−	0.847886i	$-0.322125\pi$
$367$	20.3135	1.06036	0.530178	+	0.847886i	$0.322125\pi$
$368$	−32.3812	−1.68799
$369$	0	0
$370$	0	0
$371$	−3.75872	−0.195143
$372$	0	0
$373$	−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
$381$	0	0
$382$	−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	$0.740485\pi$
$384$	0	0
$385$	0	0
$386$	22.6681	1.15377
$387$	0	0
$388$	−45.0661	−2.28788
$389$	5.63317	0.285613	0.142806	−	0.989751i	$-0.454387\pi$
$389$	5.63317	0.285613	0.142806	+	0.989751i	$0.454387\pi$
$390$	0	0
$391$	2.52359	0.127623
$392$	9.04945	0.457066
$393$	0	0
$394$	31.8576	1.60496
$395$	0	0
$396$	0	0
$397$	37.7998	1.89712	0.948558	−	0.316604i	$-0.102543\pi$
$397$	37.7998	1.89712	0.948558	+	0.316604i	$0.102543\pi$
$398$	−61.2183	−3.06860
$399$	0	0
$400$	0	0
$401$	13.6332	0.680808	0.340404	−	0.940279i	$-0.389436\pi$
$401$	13.6332	0.680808	0.340404	+	0.940279i	$0.389436\pi$
$402$	0	0
$403$	−7.15061	−0.356197
$404$	31.0616	1.54537
$405$	0	0
$406$	18.0989	0.898233
$407$	−21.6742	−1.07435
$408$	0	0
$409$	12.3545	0.610893	0.305447	−	0.952209i	$-0.401194\pi$
$409$	12.3545	0.610893	0.305447	+	0.952209i	$0.401194\pi$
$410$	0	0
$411$	0	0
$412$	−11.5174	−0.567424
$413$	−10.5236	−0.517832
$414$	0	0
$415$	0	0
$416$	17.8699	0.876144
$417$	0	0
$418$	−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
$423$	0	0
$424$	−34.0144	−1.65188
$425$	0	0
$426$	0	0
$427$	−4.15676	−0.201159
$428$	88.0965	4.25830
$429$	0	0
$430$	0	0
$431$	10.3135	0.496784	0.248392	−	0.968660i	$-0.420098\pi$
$431$	10.3135	0.496784	0.248392	+	0.968660i	$0.420098\pi$
$432$	0	0
$433$	20.4391	0.982239	0.491120	−	0.871092i	$-0.336588\pi$
$433$	20.4391	0.982239	0.491120	+	0.871092i	$0.336588\pi$
$434$	−21.0205	−1.00902
$435$	0	0
$436$	−68.5523	−3.28306
$437$	−7.20394	−0.344611
$438$	0	0
$439$	−16.9216	−0.807625	−0.403812	−	0.914842i	$-0.632315\pi$
$439$	−16.9216	−0.807625	−0.403812	+	0.914842i	$0.632315\pi$
$440$	0	0
$441$	0	0
$442$	−2.69263	−0.128075
$443$	12.8104	0.608642	0.304321	−	0.952569i	$-0.401570\pi$
$443$	12.8104	0.608642	0.304321	+	0.952569i	$0.401570\pi$
$444$	0	0
$445$	0	0
$446$	−58.7214	−2.78054
$447$	0	0
$448$	24.8576	1.17441
$449$	14.6270	0.690292	0.345146	−	0.938549i	$-0.387829\pi$
$449$	14.6270	0.690292	0.345146	+	0.938549i	$0.387829\pi$
$450$	0	0
$451$	12.9939	0.611857
$452$	27.9565	1.31496
$453$	0	0
$454$	31.2039	1.46447
$455$	0	0
$456$	0	0
$457$	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
$459$	0	0
$460$	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	$0.414139\pi$
$468$	0	0
$469$	4.68035	0.216118
$470$	0	0
$471$	0	0
$472$	−95.2327	−4.38344
$473$	13.0472	0.599910
$474$	0	0
$475$	0	0
$476$	−5.75872	−0.263951
$477$	0	0
$478$	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
$486$	0	0
$487$	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
$489$	0	0
$490$	0	0
$491$	−2.00000	−0.0902587	−0.0451294	−	0.998981i	$-0.514370\pi$
$491$	−2.00000	−0.0902587	−0.0451294	+	0.998981i	$0.514370\pi$
$492$	0	0
$493$	−7.20394	−0.324449
$494$	7.68649	0.345831
$495$	0	0
$496$	−107.358	−4.82053
$497$	−2.00000	−0.0897123
$498$	0	0
$499$	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
$501$	0	0
$502$	−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
$507$	0	0
$508$	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
$511$	7.07838	0.313129
$512$	17.8599	0.789303
$513$	0	0
$514$	13.7587	0.606871
$515$	0	0
$516$	0	0
$517$	9.36069	0.411683
$518$	29.3607	1.29003
$519$	0	0
$520$	0	0
$521$	25.8166	1.13105	0.565523	−	0.824733i	$-0.308675\pi$
$521$	25.8166	1.13105	0.565523	+	0.824733i	$0.308675\pi$
$522$	0	0
$523$	4.00000	0.174908	0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000	0.174908	0.0874539	+	0.996169i	$0.472127\pi$
$524$	−7.88428	−0.344426
$525$	0	0
$526$	−15.3340	−0.668595
$527$	8.36683	0.364465
$528$	0	0
$529$	−17.5236	−0.761895
$530$	0	0
$531$	0	0
$532$	16.4391	0.712724
$533$	−5.98771	−0.259357
$534$	0	0
$535$	0	0
$536$	42.3545	1.82944
$537$	0	0
$538$	75.4740	3.25391
$539$	−2.00000	−0.0861461
$540$	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
$543$	0	0
$544$	−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
$549$	0	0
$550$	0	0
$551$	20.5646	0.876083
$552$	0	0
$553$	6.15676	0.261812
$554$	−76.3956	−3.24574
$555$	0	0
$556$	72.6369	3.08049
$557$	26.6491	1.12916	0.564580	−	0.825378i	$-0.309038\pi$
$557$	26.6491	1.12916	0.564580	+	0.825378i	$0.309038\pi$
$558$	0	0
$559$	−6.01229	−0.254293
$560$	0	0
$561$	0	0
$562$	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	$0.931309\pi$
$564$	0	0
$565$	0	0
$566$	63.7152	2.67815
$567$	0	0
$568$	−18.0989	−0.759413
$569$	14.3668	0.602289	0.301145	−	0.953579i	$-0.402631\pi$
$569$	14.3668	0.602289	0.301145	+	0.953579i	$0.402631\pi$
$570$	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
$573$	0	0
$574$	−17.6020	−0.734692
$575$	0	0
$576$	0	0
$577$	43.4740	1.80984	0.904922	−	0.425577i	$-0.139929\pi$
$577$	43.4740	1.80984	0.904922	+	0.425577i	$0.139929\pi$
$578$	−42.9071	−1.78470
$579$	0	0
$580$	0	0
$581$	−6.83710	−0.283651
$582$	0	0
$583$	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
$591$	0	0
$592$	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	$0.723765\pi$
$594$	0	0
$595$	0	0
$596$	−83.7030	−3.42861
$597$	0	0
$598$	−5.84324	−0.238948
$599$	−29.0349	−1.18633	−0.593167	−	0.805080i	$-0.702123\pi$
$599$	−29.0349	−1.18633	−0.593167	+	0.805080i	$0.702123\pi$
$600$	0	0
$601$	15.3607	0.626576	0.313288	−	0.949658i	$-0.398570\pi$
$601$	15.3607	0.626576	0.313288	+	0.949658i	$0.398570\pi$
$602$	−17.6742	−0.720347
$603$	0	0
$604$	31.2039	1.26967
$605$	0	0
$606$	0	0
$607$	−13.0472	−0.529569	−0.264784	−	0.964308i	$-0.585301\pi$
$607$	−13.0472	−0.529569	−0.264784	+	0.964308i	$0.585301\pi$
$608$	59.6886	2.42069
$609$	0	0
$610$	0	0
$611$	−4.31351	−0.174506
$612$	0	0
$613$	−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
$616$	−18.0989	−0.729225
$617$	22.7649	0.916479	0.458240	−	0.888829i	$-0.348480\pi$
$617$	22.7649	0.916479	0.458240	+	0.888829i	$0.348480\pi$
$618$	0	0
$619$	7.92777	0.318644	0.159322	−	0.987227i	$-0.449069\pi$
$619$	7.92777	0.318644	0.159322	+	0.987227i	$0.449069\pi$
$620$	0	0
$621$	0	0
$622$	−64.5646	−2.58881
$623$	−8.34017	−0.334142
$624$	0	0
$625$	0	0
$626$	88.7358	3.54659
$627$	0	0
$628$	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
$633$	0	0
$634$	48.5380	1.92769
$635$	0	0
$636$	0	0
$637$	0.921622	0.0365160
$638$	−36.1978	−1.43308
$639$	0	0
$640$	0	0
$641$	5.94668	0.234880	0.117440	−	0.993080i	$-0.462531\pi$
$641$	5.94668	0.234880	0.117440	+	0.993080i	$0.462531\pi$
$642$	0	0
$643$	−30.8904	−1.21820	−0.609100	−	0.793094i	$-0.708469\pi$
$643$	−30.8904	−1.21820	−0.609100	+	0.793094i	$0.708469\pi$
$644$	−12.4969	−0.492448
$645$	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	$0.376786\pi$
$648$	0	0
$649$	21.0472	0.826174
$650$	0	0
$651$	0	0
$652$	−52.5646	−2.05859
$653$	28.5548	1.11744	0.558718	−	0.829358i	$-0.311294\pi$
$653$	28.5548	1.11744	0.558718	+	0.829358i	$0.311294\pi$
$654$	0	0
$655$	0	0
$656$	−89.8987	−3.50995
$657$	0	0
$658$	−12.6803	−0.494331
$659$	27.9877	1.09025	0.545123	−	0.838356i	$-0.316483\pi$
$659$	27.9877	1.09025	0.545123	+	0.838356i	$0.316483\pi$
$660$	0	0
$661$	−22.1445	−0.861320	−0.430660	−	0.902514i	$-0.641719\pi$
$661$	−22.1445	−0.861320	−0.430660	+	0.902514i	$0.641719\pi$
$662$	−3.68649	−0.143279
$663$	0	0
$664$	−61.8720	−2.40110
$665$	0	0
$666$	0	0
$667$	−15.6332	−0.605319
$668$	102.552	3.96787
$669$	0	0
$670$	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	$0.622583\pi$
$678$	0	0
$679$	−8.43907	−0.323862
$680$	0	0
$681$	0	0
$682$	42.0410	1.60983
$683$	−11.8166	−0.452149	−0.226074	−	0.974110i	$-0.572589\pi$
$683$	−11.8166	−0.452149	−0.226074	+	0.974110i	$0.572589\pi$
$684$	0	0
$685$	0	0
$686$	2.70928	0.103441
$687$	0	0
$688$	−90.2676	−3.44142
$689$	−3.46412	−0.131973
$690$	0	0
$691$	11.7587	0.447323	0.223661	−	0.974667i	$-0.428199\pi$
$691$	11.7587	0.447323	0.223661	+	0.974667i	$0.428199\pi$
$692$	119.829	4.55520
$693$	0	0
$694$	−45.6886	−1.73431
$695$	0	0
$696$	0	0
$697$	7.00614	0.265377
$698$	25.7854	0.975991
$699$	0	0
$700$	0	0
$701$	−9.94668	−0.375681	−0.187840	−	0.982200i	$-0.560149\pi$
$701$	−9.94668	−0.375681	−0.187840	+	0.982200i	$0.560149\pi$
$702$	0	0
$703$	33.3607	1.25822
$704$	−49.7152	−1.87371
$705$	0	0
$706$	96.9914	3.65032
$707$	5.81658	0.218755
$708$	0	0
$709$	−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
$711$	0	0
$712$	−75.4740	−2.82851
$713$	18.1568	0.679976
$714$	0	0
$715$	0	0
$716$	−53.4017	−1.99572
$717$	0	0
$718$	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
$721$	−2.15676	−0.0803218
$722$	−25.8020	−0.960252
$723$	0	0
$724$	−45.5174	−1.69164
$725$	0	0
$726$	0	0
$727$	2.89043	0.107200	0.0536000	−	0.998562i	$-0.482930\pi$
$727$	2.89043	0.107200	0.0536000	+	0.998562i	$0.482930\pi$
$728$	8.34017	0.309107
$729$	0	0
$730$	0	0
$731$	7.03489	0.260195
$732$	0	0
$733$	−25.7998	−0.952936	−0.476468	−	0.879192i	$-0.658083\pi$
$733$	−25.7998	−0.952936	−0.476468	+	0.879192i	$0.658083\pi$
$734$	55.0349	2.03138
$735$	0	0
$736$	−45.3751	−1.67255
$737$	−9.36069	−0.344806
$738$	0	0
$739$	−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
$741$	0	0
$742$	−10.1834	−0.373845
$743$	9.97334	0.365886	0.182943	−	0.983123i	$-0.441438\pi$
$743$	9.97334	0.365886	0.182943	+	0.983123i	$0.441438\pi$
$744$	0	0
$745$	0	0
$746$	−43.3484	−1.58710
$747$	0	0
$748$	11.5174	0.421120
$749$	16.4969	0.602785
$750$	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
$753$	0	0
$754$	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
$759$	0	0
$760$	0	0
$761$	−2.61265	−0.0947083	−0.0473542	−	0.998878i	$-0.515079\pi$
$761$	−2.61265	−0.0947083	−0.0473542	+	0.998878i	$0.515079\pi$
$762$	0	0
$763$	−12.8371	−0.464734
$764$	−82.0288	−2.96770
$765$	0	0
$766$	−72.7091	−2.62709
$767$	−9.69878	−0.350202
$768$	0	0
$769$	−15.6742	−0.565226	−0.282613	−	0.959234i	$-0.591201\pi$
$769$	−15.6742	−0.565226	−0.282613	+	0.959234i	$0.591201\pi$
$770$	0	0
$771$	0	0
$772$	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	$0.533331\pi$
$774$	0	0
$775$	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
$783$	0	0
$784$	13.8371	0.494182
$785$	0	0
$786$	0	0
$787$	39.3484	1.40262	0.701310	−	0.712857i	$-0.252599\pi$
$787$	39.3484	1.40262	0.701310	+	0.712857i	$0.252599\pi$
$788$	62.7936	2.23693
$789$	0	0
$790$	0	0
$791$	5.23513	0.186140
$792$	0	0
$793$	−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
$801$	0	0
$802$	36.9360	1.30426
$803$	−14.1568	−0.499581
$804$	0	0
$805$	0	0
$806$	−19.3730	−0.682384
$807$	0	0
$808$	52.6369	1.85176
$809$	15.6742	0.551076	0.275538	−	0.961290i	$-0.411144\pi$
$809$	15.6742	0.551076	0.275538	+	0.961290i	$0.411144\pi$
$810$	0	0
$811$	−42.1666	−1.48067	−0.740335	−	0.672238i	$-0.765333\pi$
$811$	−42.1666	−1.48067	−0.740335	+	0.672238i	$0.765333\pi$
$812$	35.6742	1.25192
$813$	0	0
$814$	−58.7214	−2.05818
$815$	0	0
$816$	0	0
$817$	−20.0821	−0.702583
$818$	33.4719	1.17032
$819$	0	0
$820$	0	0
$821$	39.0472	1.36276	0.681378	−	0.731932i	$-0.261381\pi$
$821$	39.0472	1.36276	0.681378	+	0.731932i	$0.261381\pi$
$822$	0	0
$823$	−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
$826$	−28.5113	−0.992035
$827$	−50.2245	−1.74648	−0.873238	−	0.487294i	$-0.837984\pi$
$827$	−50.2245	−1.74648	−0.873238	+	0.487294i	$0.837984\pi$
$828$	0	0
$829$	32.8371	1.14048	0.570240	−	0.821478i	$-0.306850\pi$
$829$	32.8371	1.14048	0.570240	+	0.821478i	$0.306850\pi$
$830$	0	0
$831$	0	0
$832$	22.9093	0.794238
$833$	−1.07838	−0.0373636
$834$	0	0
$835$	0	0
$836$	−32.8781	−1.13711
$837$	0	0
$838$	−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
$843$	0	0
$844$	−73.0226	−2.51354
$845$	0	0
$846$	0	0
$847$	−7.00000	−0.240523
$848$	−52.0098	−1.78603
$849$	0	0
$850$	0	0
$851$	−25.3607	−0.869353
$852$	0	0
$853$	−39.6430	−1.35735	−0.678675	−	0.734438i	$-0.737446\pi$
$853$	−39.6430	−1.35735	−0.678675	+	0.734438i	$0.737446\pi$
$854$	−11.2618	−0.385371
$855$	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	$0.330622\pi$
$858$	0	0
$859$	−3.07838	−0.105033	−0.0525164	−	0.998620i	$-0.516724\pi$
$859$	−3.07838	−0.105033	−0.0525164	+	0.998620i	$0.516724\pi$
$860$	0	0
$861$	0	0
$862$	27.9421	0.951713
$863$	6.39350	0.217637	0.108819	−	0.994062i	$-0.465293\pi$
$863$	6.39350	0.217637	0.108819	+	0.994062i	$0.465293\pi$
$864$	0	0
$865$	0	0
$866$	55.3751	1.88172
$867$	0	0
$868$	−41.4329	−1.40633
$869$	−12.3135	−0.417707
$870$	0	0
$871$	4.31351	0.146158
$872$	−116.169	−3.93397
$873$	0	0
$874$	−19.5174	−0.660188
$875$	0	0
$876$	0	0
$877$	−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
$879$	0	0
$880$	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	$0.358467\pi$
$888$	0	0
$889$	1.84324	0.0618204
$890$	0	0
$891$	0	0
$892$	−115.744	−3.87540
$893$	−14.4079	−0.482141
$894$	0	0
$895$	0	0
$896$	28.5669	0.954353
$897$	0	0
$898$	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
$906$	0	0
$907$	−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
$909$	0	0
$910$	0	0
$911$	35.9877	1.19233	0.596163	−	0.802863i	$-0.296691\pi$
$911$	35.9877	1.19233	0.596163	+	0.802863i	$0.296691\pi$
$912$	0	0
$913$	13.6742	0.452550
$914$	38.3545	1.26866
$915$	0	0
$916$	68.5523	2.26503
$917$	−1.47641	−0.0487553
$918$	0	0
$919$	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
$921$	0	0
$922$	−0.921622	−0.0303520
$923$	−1.84324	−0.0606711
$924$	0	0
$925$	0	0
$926$	−26.6681	−0.876367
$927$	0	0
$928$	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
$931$	3.07838	0.100890
$932$	36.1256	1.18333
$933$	0	0
$934$	31.2039	1.02102
$935$	0	0
$936$	0	0
$937$	−16.1256	−0.526799	−0.263400	−	0.964687i	$-0.584844\pi$
$937$	−16.1256	−0.526799	−0.263400	+	0.964687i	$0.584844\pi$
$938$	12.6803	0.414028
$939$	0	0
$940$	0	0
$941$	−24.7070	−0.805425	−0.402713	−	0.915326i	$-0.631933\pi$
$941$	−24.7070	−0.805425	−0.402713	+	0.915326i	$0.631933\pi$
$942$	0	0
$943$	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	$-0.466123\pi$
$947$	6.53797	0.212455	0.106228	+	0.994342i	$0.466123\pi$
$948$	0	0
$949$	6.52359	0.211765
$950$	0	0
$951$	0	0
$952$	−9.75872	−0.316282
$953$	−6.11327	−0.198028	−0.0990142	−	0.995086i	$-0.531569\pi$
$953$	−6.11327	−0.198028	−0.0990142	+	0.995086i	$0.531569\pi$
$954$	0	0
$955$	0	0
$956$	124.750	4.03471
$957$	0	0
$958$	52.8781	1.70842
$959$	−4.43907	−0.143345
$960$	0	0
$961$	29.1978	0.941864
$962$	27.0595	0.872432
$963$	0	0
$964$	−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
$969$	0	0
$970$	0	0
$971$	−4.05332	−0.130077	−0.0650387	−	0.997883i	$-0.520717\pi$
$971$	−4.05332	−0.130077	−0.0650387	+	0.997883i	$0.520717\pi$
$972$	0	0
$973$	13.6020	0.436059
$974$	62.7214	2.00972
$975$	0	0
$976$	−57.5174	−1.84109
$977$	3.81205	0.121958	0.0609791	−	0.998139i	$-0.480578\pi$
$977$	3.81205	0.121958	0.0609791	+	0.998139i	$0.480578\pi$
$978$	0	0
$979$	16.6803	0.533106
$980$	0	0
$981$	0	0
$982$	−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	$0.625020\pi$
$984$	0	0
$985$	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
$993$	0	0
$994$	−5.41855	−0.171866
$995$	0	0
$996$	0	0
$997$	43.4740	1.37683	0.688417	−	0.725315i	$-0.258306\pi$
$997$	43.4740	1.37683	0.688417	+	0.725315i	$0.258306\pi$
$998$	73.7030	2.33303
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.2.a.x.1.3		3
3.2	odd	2		525.2.a.j.1.1		3
5.2	odd	4		315.2.d.e.64.6		6
5.3	odd	4		315.2.d.e.64.1		6
5.4	even	2		1575.2.a.w.1.1		3
12.11	even	2		8400.2.a.dg.1.2		3
15.2	even	4		105.2.d.b.64.1	✓	6
15.8	even	4		105.2.d.b.64.6	yes	6
15.14	odd	2		525.2.a.k.1.3		3
20.3	even	4		5040.2.t.v.1009.1		6
20.7	even	4		5040.2.t.v.1009.2		6
21.20	even	2		3675.2.a.bi.1.1		3
35.13	even	4		2205.2.d.l.1324.1		6
35.27	even	4		2205.2.d.l.1324.6		6
60.23	odd	4		1680.2.t.k.1009.3		6
60.47	odd	4		1680.2.t.k.1009.6		6
60.59	even	2		8400.2.a.dj.1.2		3
105.2	even	12		735.2.q.e.214.1		12
105.17	odd	12		735.2.q.f.79.6		12
105.23	even	12		735.2.q.e.214.6		12
105.32	even	12		735.2.q.e.79.6		12
105.38	odd	12		735.2.q.f.79.1		12
105.47	odd	12		735.2.q.f.214.1		12
105.53	even	12		735.2.q.e.79.1		12
105.62	odd	4		735.2.d.b.589.1		6
105.68	odd	12		735.2.q.f.214.6		12
105.83	odd	4		735.2.d.b.589.6		6
105.104	even	2		3675.2.a.bj.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.d.b.64.1	✓	6	15.2	even	4
105.2.d.b.64.6	yes	6	15.8	even	4
315.2.d.e.64.1		6	5.3	odd	4
315.2.d.e.64.6		6	5.2	odd	4
525.2.a.j.1.1		3	3.2	odd	2
525.2.a.k.1.3		3	15.14	odd	2
735.2.d.b.589.1		6	105.62	odd	4
735.2.d.b.589.6		6	105.83	odd	4
735.2.q.e.79.1		12	105.53	even	12
735.2.q.e.79.6		12	105.32	even	12
735.2.q.e.214.1		12	105.2	even	12
735.2.q.e.214.6		12	105.23	even	12
735.2.q.f.79.1		12	105.38	odd	12
735.2.q.f.79.6		12	105.17	odd	12
735.2.q.f.214.1		12	105.47	odd	12
735.2.q.f.214.6		12	105.68	odd	12
1575.2.a.w.1.1		3	5.4	even	2
1575.2.a.x.1.3		3	1.1	even	1	trivial
1680.2.t.k.1009.3		6	60.23	odd	4
1680.2.t.k.1009.6		6	60.47	odd	4
2205.2.d.l.1324.1		6	35.13	even	4
2205.2.d.l.1324.6		6	35.27	even	4
3675.2.a.bi.1.1		3	21.20	even	2
3675.2.a.bj.1.3		3	105.104	even	2
5040.2.t.v.1009.1		6	20.3	even	4
5040.2.t.v.1009.2		6	20.7	even	4
8400.2.a.dg.1.2		3	12.11	even	2
8400.2.a.dj.1.2		3	60.59	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:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1575.a (trivial)

\(f(q)\)	\(=\)	\(q+2.70928 q^{2} +5.34017 q^{4} +1.00000 q^{7} +9.04945 q^{8} +O(q^{10})\)	\(q+2.70928 q^{2} +5.34017 q^{4} +1.00000 q^{7} +9.04945 q^{8} -2.00000 q^{11} +0.921622 q^{13} +2.70928 q^{14} +13.8371 q^{16} -1.07838 q^{17} +3.07838 q^{19} -5.41855 q^{22} -2.34017 q^{23} +2.49693 q^{26} +5.34017 q^{28} +6.68035 q^{29} -7.75872 q^{31} +19.3896 q^{32} -2.92162 q^{34} +10.8371 q^{37} +8.34017 q^{38} -6.49693 q^{41} -6.52359 q^{43} -10.6803 q^{44} -6.34017 q^{46} -4.68035 q^{47} +1.00000 q^{49} +4.92162 q^{52} -3.75872 q^{53} +9.04945 q^{56} +18.0989 q^{58} -10.5236 q^{59} -4.15676 q^{61} -21.0205 q^{62} +24.8576 q^{64} +4.68035 q^{67} -5.75872 q^{68} -2.00000 q^{71} +7.07838 q^{73} +29.3607 q^{74} +16.4391 q^{76} -2.00000 q^{77} +6.15676 q^{79} -17.6020 q^{82} -6.83710 q^{83} -17.6742 q^{86} -18.0989 q^{88} -8.34017 q^{89} +0.921622 q^{91} -12.4969 q^{92} -12.6803 q^{94} -8.43907 q^{97} +2.70928 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + 5 q^{4} + 3 q^{7} + 9 q^{8}+O(q^{10}) \) 3 * q + q^2 + 5 * q^4 + 3 * q^7 + 9 * q^8	\( 3 q + q^{2} + 5 q^{4} + 3 q^{7} + 9 q^{8} - 6 q^{11} + 6 q^{13} + q^{14} + 13 q^{16} + 6 q^{19} - 2 q^{22} + 4 q^{23} - 10 q^{26} + 5 q^{28} - 2 q^{29} + 2 q^{31} + 29 q^{32} - 12 q^{34} + 4 q^{37} + 14 q^{38} - 2 q^{41} - 4 q^{43} - 10 q^{44} - 8 q^{46} + 8 q^{47} + 3 q^{49} + 18 q^{52} + 14 q^{53} + 9 q^{56} + 18 q^{58} - 16 q^{59} - 6 q^{61} - 30 q^{62} + 13 q^{64} - 8 q^{67} + 8 q^{68} - 6 q^{71} + 18 q^{73} + 44 q^{74} + 2 q^{76} - 6 q^{77} + 12 q^{79} - 34 q^{82} + 8 q^{83} + 4 q^{86} - 18 q^{88} - 14 q^{89} + 6 q^{91} - 20 q^{92} - 16 q^{94} + 22 q^{97} + q^{98}+O(q^{100}) \) 3 * q + q^2 + 5 * q^4 + 3 * q^7 + 9 * q^8 - 6 * q^11 + 6 * q^13 + q^14 + 13 * q^16 + 6 * q^19 - 2 * q^22 + 4 * q^23 - 10 * q^26 + 5 * q^28 - 2 * q^29 + 2 * q^31 + 29 * q^32 - 12 * q^34 + 4 * q^37 + 14 * q^38 - 2 * q^41 - 4 * q^43 - 10 * q^44 - 8 * q^46 + 8 * q^47 + 3 * q^49 + 18 * q^52 + 14 * q^53 + 9 * q^56 + 18 * q^58 - 16 * q^59 - 6 * q^61 - 30 * q^62 + 13 * q^64 - 8 * q^67 + 8 * q^68 - 6 * q^71 + 18 * q^73 + 44 * q^74 + 2 * q^76 - 6 * q^77 + 12 * q^79 - 34 * q^82 + 8 * q^83 + 4 * q^86 - 18 * q^88 - 14 * q^89 + 6 * q^91 - 20 * q^92 - 16 * q^94 + 22 * q^97 + q^98

Introduction

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