LMFDB - Embedded newform 3645.2.a.d.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3645,2,Mod(1,3645)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3645, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3645.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3645 = 3^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3645.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,3,0,9,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$29.1054715368$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.3916917.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 6x^{4} + 12x^{3} + 18x^{2} - 3 $ x^6 - 3x^5 - 6x^4 + 12x^3 + 18x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-1.40798$ of defining polynomial
Character	$\chi$	$=$	3645.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.40798 q^{2} +3.79838 q^{4} -1.00000 q^{5} +3.94007 q^{7} +4.33047 q^{8} -2.40798 q^{10} +6.56863 q^{11} -1.34680 q^{13} +9.48762 q^{14} +2.83093 q^{16} -6.08941 q^{17} -0.482880 q^{19} -3.79838 q^{20} +15.8171 q^{22} +6.35832 q^{23} +1.00000 q^{25} -3.24308 q^{26} +14.9659 q^{28} +6.23816 q^{29} +3.53651 q^{31} -1.84410 q^{32} -14.6632 q^{34} -3.94007 q^{35} -3.38013 q^{37} -1.16277 q^{38} -4.33047 q^{40} -10.4476 q^{41} -6.37329 q^{43} +24.9502 q^{44} +15.3107 q^{46} +0.593703 q^{47} +8.52416 q^{49} +2.40798 q^{50} -5.11567 q^{52} +2.41105 q^{53} -6.56863 q^{55} +17.0624 q^{56} +15.0214 q^{58} +2.33092 q^{59} +1.96422 q^{61} +8.51586 q^{62} -10.1024 q^{64} +1.34680 q^{65} -8.92981 q^{67} -23.1299 q^{68} -9.48762 q^{70} +5.68055 q^{71} -6.11390 q^{73} -8.13930 q^{74} -1.83416 q^{76} +25.8809 q^{77} -3.36559 q^{79} -2.83093 q^{80} -25.1575 q^{82} +16.6900 q^{83} +6.08941 q^{85} -15.3468 q^{86} +28.4452 q^{88} +7.35902 q^{89} -5.30649 q^{91} +24.1513 q^{92} +1.42963 q^{94} +0.482880 q^{95} +2.80919 q^{97} +20.5260 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 3 q^{2} + 9 q^{4} - 6 q^{5} + 3 q^{7} + 3 q^{8} - 3 q^{10} + 9 q^{11} - 6 q^{13} + 24 q^{14} - 9 q^{16} - 15 q^{17} - 9 q^{19} - 9 q^{20} + 3 q^{22} + 6 q^{23} + 6 q^{25} + 3 q^{26} + 12 q^{28} + 30 q^{29}+ \cdots - 15 q^{97}+O(q^{100}) $ 6 * q + 3 * q^2 + 9 * q^4 - 6 * q^5 + 3 * q^7 + 3 * q^8 - 3 * q^10 + 9 * q^11 - 6 * q^13 + 24 * q^14 - 9 * q^16 - 15 * q^17 - 9 * q^19 - 9 * q^20 + 3 * q^22 + 6 * q^23 + 6 * q^25 + 3 * q^26 + 12 * q^28 + 30 * q^29 - 3 * q^31 + 9 * q^32 + 15 * q^34 - 3 * q^35 - 18 * q^37 + 18 * q^38 - 3 * q^40 + 15 * q^41 - 3 * q^43 + 36 * q^44 + 36 * q^46 + 15 * q^47 - 3 * q^49 + 3 * q^50 + 18 * q^52 + 33 * q^53 - 9 * q^55 + 24 * q^56 + 6 * q^58 + 27 * q^59 + 3 * q^61 + 6 * q^62 - 27 * q^64 + 6 * q^65 - 33 * q^67 - 12 * q^68 - 24 * q^70 + 27 * q^71 - 9 * q^73 - 3 * q^74 - 6 * q^76 + 9 * q^77 + 18 * q^79 + 9 * q^80 - 27 * q^82 + 15 * q^83 + 15 * q^85 - 15 * q^86 + 33 * q^88 - 15 * q^89 - 21 * q^91 + 9 * q^92 + 18 * q^94 + 9 * q^95 - 15 * q^97

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 $

$2$	2.40798	1.70270	0.851350	−	0.524597i	$-0.175784\pi$
$2$	2.40798	1.70270	0.851350	+	0.524597i	$0.175784\pi$
$3$	0	0
$3$	0	0
$4$	3.79838	1.89919
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.94007	1.48921	0.744604	−	0.667507i	$-0.232639\pi$
$7$	3.94007	1.48921	0.744604	+	0.667507i	$0.232639\pi$
$8$	4.33047	1.53105
$9$	0	0
$10$	−2.40798	−0.761471
$11$	6.56863	1.98052	0.990258	−	0.139244i	$-0.0444674\pi$
$11$	6.56863	1.98052	0.990258	+	0.139244i	$0.0444674\pi$
$12$	0	0
$13$	−1.34680	−0.373536	−0.186768	−	0.982404i	$-0.559801\pi$
$13$	−1.34680	−0.373536	−0.186768	+	0.982404i	$0.559801\pi$
$14$	9.48762	2.53567
$15$	0	0
$16$	2.83093	0.707734
$17$	−6.08941	−1.47690	−0.738450	−	0.674309i	$-0.764442\pi$
$17$	−6.08941	−1.47690	−0.738450	+	0.674309i	$0.764442\pi$
$18$	0	0
$19$	−0.482880	−0.110780	−0.0553902	−	0.998465i	$-0.517640\pi$
$19$	−0.482880	−0.110780	−0.0553902	+	0.998465i	$0.517640\pi$
$20$	−3.79838	−0.849344
$21$	0	0
$22$	15.8171	3.37223
$23$	6.35832	1.32580	0.662901	−	0.748707i	$-0.269325\pi$
$23$	6.35832	1.32580	0.662901	+	0.748707i	$0.269325\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−3.24308	−0.636019
$27$	0	0
$28$	14.9659	2.82829
$29$	6.23816	1.15840	0.579199	−	0.815187i	$-0.303366\pi$
$29$	6.23816	1.15840	0.579199	+	0.815187i	$0.303366\pi$
$30$	0	0
$31$	3.53651	0.635176	0.317588	−	0.948229i	$-0.397127\pi$
$31$	3.53651	0.635176	0.317588	+	0.948229i	$0.397127\pi$
$32$	−1.84410	−0.325994
$33$	0	0
$34$	−14.6632	−2.51472
$35$	−3.94007	−0.665994
$36$	0	0
$37$	−3.38013	−0.555690	−0.277845	−	0.960626i	$-0.589620\pi$
$37$	−3.38013	−0.555690	−0.277845	+	0.960626i	$0.589620\pi$
$38$	−1.16277	−0.188626
$39$	0	0
$40$	−4.33047	−0.684707
$41$	−10.4476	−1.63163	−0.815817	−	0.578310i	$-0.803713\pi$
$41$	−10.4476	−1.63163	−0.815817	+	0.578310i	$0.803713\pi$
$42$	0	0
$43$	−6.37329	−0.971917	−0.485959	−	0.873982i	$-0.661529\pi$
$43$	−6.37329	−0.971917	−0.485959	+	0.873982i	$0.661529\pi$
$44$	24.9502	3.76138
$45$	0	0
$46$	15.3107	2.25744
$47$	0.593703	0.0866005	0.0433003	−	0.999062i	$-0.486213\pi$
$47$	0.593703	0.0866005	0.0433003	+	0.999062i	$0.486213\pi$
$48$	0	0
$49$	8.52416	1.21774
$50$	2.40798	0.340540
$51$	0	0
$52$	−5.11567	−0.709415
$53$	2.41105	0.331184	0.165592	−	0.986194i	$-0.447047\pi$
$53$	2.41105	0.331184	0.165592	+	0.986194i	$0.447047\pi$
$54$	0	0
$55$	−6.56863	−0.885714
$56$	17.0624	2.28005
$57$	0	0
$58$	15.0214	1.97240
$59$	2.33092	0.303460	0.151730	−	0.988422i	$-0.451516\pi$
$59$	2.33092	0.303460	0.151730	+	0.988422i	$0.451516\pi$
$60$	0	0
$61$	1.96422	0.251492	0.125746	−	0.992062i	$-0.459868\pi$
$61$	1.96422	0.251492	0.125746	+	0.992062i	$0.459868\pi$
$62$	8.51586	1.08151
$63$	0	0
$64$	−10.1024	−1.26280
$65$	1.34680	0.167050
$66$	0	0
$67$	−8.92981	−1.09095	−0.545475	−	0.838127i	$-0.683651\pi$
$67$	−8.92981	−1.09095	−0.545475	+	0.838127i	$0.683651\pi$
$68$	−23.1299	−2.80491
$69$	0	0
$70$	−9.48762	−1.13399
$71$	5.68055	0.674158	0.337079	−	0.941476i	$-0.390561\pi$
$71$	5.68055	0.674158	0.337079	+	0.941476i	$0.390561\pi$
$72$	0	0
$73$	−6.11390	−0.715578	−0.357789	−	0.933802i	$-0.616469\pi$
$73$	−6.11390	−0.715578	−0.357789	+	0.933802i	$0.616469\pi$
$74$	−8.13930	−0.946175
$75$	0	0
$76$	−1.83416	−0.210393
$77$	25.8809	2.94940
$78$	0	0
$79$	−3.36559	−0.378658	−0.189329	−	0.981914i	$-0.560631\pi$
$79$	−3.36559	−0.378658	−0.189329	+	0.981914i	$0.560631\pi$
$80$	−2.83093	−0.316508
$81$	0	0
$82$	−25.1575	−2.77819
$83$	16.6900	1.83197	0.915985	−	0.401212i	$-0.131411\pi$
$83$	16.6900	1.83197	0.915985	+	0.401212i	$0.131411\pi$
$84$	0	0
$85$	6.08941	0.660489
$86$	−15.3468	−1.65488
$87$	0	0
$88$	28.4452	3.03227
$89$	7.35902	0.780055	0.390027	−	0.920803i	$-0.372466\pi$
$89$	7.35902	0.780055	0.390027	+	0.920803i	$0.372466\pi$
$90$	0	0
$91$	−5.30649	−0.556272
$92$	24.1513	2.51795
$93$	0	0
$94$	1.42963	0.147455
$95$	0.482880	0.0495425
$96$	0	0
$97$	2.80919	0.285230	0.142615	−	0.989778i	$-0.454449\pi$
$97$	2.80919	0.285230	0.142615	+	0.989778i	$0.454449\pi$
$98$	20.5260	2.07344
$99$	0	0
$100$	3.79838	0.379838
$101$	8.18280	0.814219	0.407110	−	0.913379i	$-0.366537\pi$
$101$	8.18280	0.814219	0.407110	+	0.913379i	$0.366537\pi$
$102$	0	0
$103$	−6.30112	−0.620868	−0.310434	−	0.950595i	$-0.600474\pi$
$103$	−6.30112	−0.620868	−0.310434	+	0.950595i	$0.600474\pi$
$104$	−5.83228	−0.571902
$105$	0	0
$106$	5.80577	0.563907
$107$	−8.81610	−0.852285	−0.426142	−	0.904656i	$-0.640128\pi$
$107$	−8.81610	−0.852285	−0.426142	+	0.904656i	$0.640128\pi$
$108$	0	0
$109$	15.3178	1.46718	0.733588	−	0.679595i	$-0.237844\pi$
$109$	15.3178	1.46718	0.733588	+	0.679595i	$0.237844\pi$
$110$	−15.8171	−1.50811
$111$	0	0
$112$	11.1541	1.05396
$113$	12.5478	1.18040	0.590199	−	0.807258i	$-0.299049\pi$
$113$	12.5478	1.18040	0.590199	+	0.807258i	$0.299049\pi$
$114$	0	0
$115$	−6.35832	−0.592916
$116$	23.6949	2.20002
$117$	0	0
$118$	5.61283	0.516702
$119$	−23.9927	−2.19941
$120$	0	0
$121$	32.1469	2.92244
$122$	4.72980	0.428216
$123$	0	0
$124$	13.4330	1.20632
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−15.6084	−1.38502	−0.692512	−	0.721406i	$-0.743496\pi$
$127$	−15.6084	−1.38502	−0.692512	+	0.721406i	$0.743496\pi$
$128$	−20.6383	−1.82418
$129$	0	0
$130$	3.24308	0.284437
$131$	−12.0675	−1.05434	−0.527171	−	0.849759i	$-0.676747\pi$
$131$	−12.0675	−1.05434	−0.527171	+	0.849759i	$0.676747\pi$
$132$	0	0
$133$	−1.90258	−0.164975
$134$	−21.5028	−1.85756
$135$	0	0
$136$	−26.3700	−2.26121
$137$	−8.53957	−0.729584	−0.364792	−	0.931089i	$-0.618860\pi$
$137$	−8.53957	−0.729584	−0.364792	+	0.931089i	$0.618860\pi$
$138$	0	0
$139$	−8.95779	−0.759790	−0.379895	−	0.925030i	$-0.624040\pi$
$139$	−8.95779	−0.759790	−0.379895	+	0.925030i	$0.624040\pi$
$140$	−14.9659	−1.26485
$141$	0	0
$142$	13.6787	1.14789
$143$	−8.84664	−0.739793
$144$	0	0
$145$	−6.23816	−0.518051
$146$	−14.7222	−1.21842
$147$	0	0
$148$	−12.8390	−1.05536
$149$	−8.90236	−0.729310	−0.364655	−	0.931143i	$-0.618813\pi$
$149$	−8.90236	−0.729310	−0.364655	+	0.931143i	$0.618813\pi$
$150$	0	0
$151$	−3.09157	−0.251589	−0.125794	−	0.992056i	$-0.540148\pi$
$151$	−3.09157	−0.251589	−0.125794	+	0.992056i	$0.540148\pi$
$152$	−2.09110	−0.169610
$153$	0	0
$154$	62.3207	5.02194
$155$	−3.53651	−0.284059
$156$	0	0
$157$	−7.16009	−0.571438	−0.285719	−	0.958313i	$-0.592232\pi$
$157$	−7.16009	−0.571438	−0.285719	+	0.958313i	$0.592232\pi$
$158$	−8.10427	−0.644741
$159$	0	0
$160$	1.84410	0.145789
$161$	25.0522	1.97439
$162$	0	0
$163$	−14.0462	−1.10018	−0.550090	−	0.835105i	$-0.685407\pi$
$163$	−14.0462	−1.10018	−0.550090	+	0.835105i	$0.685407\pi$
$164$	−39.6838	−3.09878
$165$	0	0
$166$	40.1893	3.11930
$167$	−19.8357	−1.53493	−0.767465	−	0.641091i	$-0.778482\pi$
$167$	−19.8357	−1.53493	−0.767465	+	0.641091i	$0.778482\pi$
$168$	0	0
$169$	−11.1861	−0.860471
$170$	14.6632	1.12462
$171$	0	0
$172$	−24.2082	−1.84586
$173$	−9.66735	−0.734995	−0.367498	−	0.930024i	$-0.619785\pi$
$173$	−9.66735	−0.734995	−0.367498	+	0.930024i	$0.619785\pi$
$174$	0	0
$175$	3.94007	0.297841
$176$	18.5954	1.40168
$177$	0	0
$178$	17.7204	1.32820
$179$	17.2392	1.28851	0.644257	−	0.764809i	$-0.277167\pi$
$179$	17.2392	1.28851	0.644257	+	0.764809i	$0.277167\pi$
$180$	0	0
$181$	−3.30601	−0.245734	−0.122867	−	0.992423i	$-0.539209\pi$
$181$	−3.30601	−0.245734	−0.122867	+	0.992423i	$0.539209\pi$
$182$	−12.7779	−0.947165
$183$	0	0
$184$	27.5345	2.02987
$185$	3.38013	0.248512
$186$	0	0
$187$	−39.9991	−2.92502
$188$	2.25511	0.164471
$189$	0	0
$190$	1.16277	0.0843560
$191$	11.2735	0.815719	0.407860	−	0.913045i	$-0.366275\pi$
$191$	11.2735	0.815719	0.407860	+	0.913045i	$0.366275\pi$
$192$	0	0
$193$	14.6707	1.05602	0.528010	−	0.849238i	$-0.322938\pi$
$193$	14.6707	1.05602	0.528010	+	0.849238i	$0.322938\pi$
$194$	6.76449	0.485662
$195$	0	0
$196$	32.3780	2.31272
$197$	5.79770	0.413069	0.206534	−	0.978439i	$-0.433781\pi$
$197$	5.79770	0.413069	0.206534	+	0.978439i	$0.433781\pi$
$198$	0	0
$199$	27.9170	1.97898	0.989492	−	0.144585i	$-0.0461846\pi$
$199$	27.9170	1.97898	0.989492	+	0.144585i	$0.0461846\pi$
$200$	4.33047	0.306210
$201$	0	0
$202$	19.7040	1.38637
$203$	24.5788	1.72509
$204$	0	0
$205$	10.4476	0.729689
$206$	−15.1730	−1.05715
$207$	0	0
$208$	−3.81271	−0.264364
$209$	−3.17186	−0.219402
$210$	0	0
$211$	4.34100	0.298846	0.149423	−	0.988773i	$-0.452258\pi$
$211$	4.34100	0.298846	0.149423	+	0.988773i	$0.452258\pi$
$212$	9.15810	0.628981
$213$	0	0
$214$	−21.2290	−1.45119
$215$	6.37329	0.434655
$216$	0	0
$217$	13.9341	0.945909
$218$	36.8849	2.49816
$219$	0	0
$220$	−24.9502	−1.68214
$221$	8.20123	0.551674
$222$	0	0
$223$	7.63577	0.511329	0.255665	−	0.966766i	$-0.417706\pi$
$223$	7.63577	0.511329	0.255665	+	0.966766i	$0.417706\pi$
$224$	−7.26588	−0.485472
$225$	0	0
$226$	30.2149	2.00986
$227$	−6.01960	−0.399535	−0.199767	−	0.979843i	$-0.564019\pi$
$227$	−6.01960	−0.399535	−0.199767	+	0.979843i	$0.564019\pi$
$228$	0	0
$229$	−20.5180	−1.35587	−0.677935	−	0.735122i	$-0.737125\pi$
$229$	−20.5180	−1.35587	−0.677935	+	0.735122i	$0.737125\pi$
$230$	−15.3107	−1.00956
$231$	0	0
$232$	27.0142	1.77357
$233$	12.2290	0.801146	0.400573	−	0.916265i	$-0.368811\pi$
$233$	12.2290	0.801146	0.400573	+	0.916265i	$0.368811\pi$
$234$	0	0
$235$	−0.593703	−0.0387289
$236$	8.85374	0.576329
$237$	0	0
$238$	−57.7741	−3.74494
$239$	3.96799	0.256668	0.128334	−	0.991731i	$-0.459037\pi$
$239$	3.96799	0.256668	0.128334	+	0.991731i	$0.459037\pi$
$240$	0	0
$241$	−26.7499	−1.72311	−0.861556	−	0.507663i	$-0.830510\pi$
$241$	−26.7499	−1.72311	−0.861556	+	0.507663i	$0.830510\pi$
$242$	77.4091	4.97605
$243$	0	0
$244$	7.46085	0.477632
$245$	−8.52416	−0.544589
$246$	0	0
$247$	0.650344	0.0413804
$248$	15.3147	0.972488
$249$	0	0
$250$	−2.40798	−0.152294
$251$	−25.4711	−1.60772	−0.803862	−	0.594815i	$-0.797225\pi$
$251$	−25.4711	−1.60772	−0.803862	+	0.594815i	$0.797225\pi$
$252$	0	0
$253$	41.7654	2.62577
$254$	−37.5848	−2.35828
$255$	0	0
$256$	−29.4917	−1.84323
$257$	−5.93207	−0.370032	−0.185016	−	0.982735i	$-0.559234\pi$
$257$	−5.93207	−0.370032	−0.185016	+	0.982735i	$0.559234\pi$
$258$	0	0
$259$	−13.3180	−0.827538
$260$	5.11567	0.317260
$261$	0	0
$262$	−29.0583	−1.79523
$263$	25.0690	1.54582	0.772909	−	0.634517i	$-0.218801\pi$
$263$	25.0690	1.54582	0.772909	+	0.634517i	$0.218801\pi$
$264$	0	0
$265$	−2.41105	−0.148110
$266$	−4.58139	−0.280903
$267$	0	0
$268$	−33.9188	−2.07192
$269$	−6.18572	−0.377150	−0.188575	−	0.982059i	$-0.560387\pi$
$269$	−6.18572	−0.377150	−0.188575	+	0.982059i	$0.560387\pi$
$270$	0	0
$271$	12.1585	0.738579	0.369290	−	0.929314i	$-0.379601\pi$
$271$	12.1585	0.738579	0.369290	+	0.929314i	$0.379601\pi$
$272$	−17.2387	−1.04525
$273$	0	0
$274$	−20.5631	−1.24226
$275$	6.56863	0.396103
$276$	0	0
$277$	−11.5754	−0.695497	−0.347749	−	0.937588i	$-0.613054\pi$
$277$	−11.5754	−0.695497	−0.347749	+	0.937588i	$0.613054\pi$
$278$	−21.5702	−1.29370
$279$	0	0
$280$	−17.0624	−1.01967
$281$	−14.7833	−0.881897	−0.440948	−	0.897532i	$-0.645358\pi$
$281$	−14.7833	−0.881897	−0.440948	+	0.897532i	$0.645358\pi$
$282$	0	0
$283$	−10.5967	−0.629908	−0.314954	−	0.949107i	$-0.601989\pi$
$283$	−10.5967	−0.629908	−0.314954	+	0.949107i	$0.601989\pi$
$284$	21.5769	1.28035
$285$	0	0
$286$	−21.3026	−1.25965
$287$	−41.1641	−2.42984
$288$	0	0
$289$	20.0809	1.18123
$290$	−15.0214	−0.882086
$291$	0	0
$292$	−23.2229	−1.35902
$293$	13.4017	0.782935	0.391468	−	0.920192i	$-0.371968\pi$
$293$	13.4017	0.782935	0.391468	+	0.920192i	$0.371968\pi$
$294$	0	0
$295$	−2.33092	−0.135712
$296$	−14.6376	−0.850791
$297$	0	0
$298$	−21.4367	−1.24180
$299$	−8.56339	−0.495234
$300$	0	0
$301$	−25.1112	−1.44739
$302$	−7.44445	−0.428380
$303$	0	0
$304$	−1.36700	−0.0784029
$305$	−1.96422	−0.112471
$306$	0	0
$307$	−14.3319	−0.817962	−0.408981	−	0.912543i	$-0.634116\pi$
$307$	−14.3319	−0.817962	−0.408981	+	0.912543i	$0.634116\pi$
$308$	98.3054	5.60147
$309$	0	0
$310$	−8.51586	−0.483668
$311$	6.33603	0.359283	0.179642	−	0.983732i	$-0.442506\pi$
$311$	6.33603	0.359283	0.179642	+	0.983732i	$0.442506\pi$
$312$	0	0
$313$	−11.1503	−0.630252	−0.315126	−	0.949050i	$-0.602047\pi$
$313$	−11.1503	−0.630252	−0.315126	+	0.949050i	$0.602047\pi$
$314$	−17.2414	−0.972987
$315$	0	0
$316$	−12.7838	−0.719143
$317$	9.62378	0.540525	0.270263	−	0.962787i	$-0.412889\pi$
$317$	9.62378	0.540525	0.270263	+	0.962787i	$0.412889\pi$
$318$	0	0
$319$	40.9762	2.29422
$320$	10.1024	0.564743
$321$	0	0
$322$	60.3253	3.36180
$323$	2.94046	0.163611
$324$	0	0
$325$	−1.34680	−0.0747071
$326$	−33.8229	−1.87328
$327$	0	0
$328$	−45.2428	−2.49812
$329$	2.33923	0.128966
$330$	0	0
$331$	−15.8753	−0.872584	−0.436292	−	0.899805i	$-0.643709\pi$
$331$	−15.8753	−0.872584	−0.436292	+	0.899805i	$0.643709\pi$
$332$	63.3951	3.47926
$333$	0	0
$334$	−47.7639	−2.61353
$335$	8.92981	0.487887
$336$	0	0
$337$	−11.5120	−0.627099	−0.313549	−	0.949572i	$-0.601518\pi$
$337$	−11.5120	−0.627099	−0.313549	+	0.949572i	$0.601518\pi$
$338$	−26.9360	−1.46513
$339$	0	0
$340$	23.1299	1.25440
$341$	23.2300	1.25798
$342$	0	0
$343$	6.00532	0.324257
$344$	−27.5993	−1.48806
$345$	0	0
$346$	−23.2788	−1.25148
$347$	−4.62381	−0.248219	−0.124110	−	0.992269i	$-0.539607\pi$
$347$	−4.62381	−0.248219	−0.124110	+	0.992269i	$0.539607\pi$
$348$	0	0
$349$	11.7836	0.630762	0.315381	−	0.948965i	$-0.397868\pi$
$349$	11.7836	0.630762	0.315381	+	0.948965i	$0.397868\pi$
$350$	9.48762	0.507135
$351$	0	0
$352$	−12.1132	−0.645635
$353$	−16.7720	−0.892682	−0.446341	−	0.894863i	$-0.647273\pi$
$353$	−16.7720	−0.892682	−0.446341	+	0.894863i	$0.647273\pi$
$354$	0	0
$355$	−5.68055	−0.301492
$356$	27.9524	1.48147
$357$	0	0
$358$	41.5116	2.19396
$359$	−25.6262	−1.35250	−0.676249	−	0.736673i	$-0.736396\pi$
$359$	−25.6262	−1.35250	−0.676249	+	0.736673i	$0.736396\pi$
$360$	0	0
$361$	−18.7668	−0.987728
$362$	−7.96083	−0.418412
$363$	0	0
$364$	−20.1561	−1.05647
$365$	6.11390	0.320016
$366$	0	0
$367$	−24.4708	−1.27737	−0.638684	−	0.769469i	$-0.720521\pi$
$367$	−24.4708	−1.27737	−0.638684	+	0.769469i	$0.720521\pi$
$368$	18.0000	0.938314
$369$	0	0
$370$	8.13930	0.423142
$371$	9.49972	0.493201
$372$	0	0
$373$	5.09000	0.263550	0.131775	−	0.991280i	$-0.457932\pi$
$373$	5.09000	0.263550	0.131775	+	0.991280i	$0.457932\pi$
$374$	−96.3171	−4.98044
$375$	0	0
$376$	2.57101	0.132590
$377$	−8.40156	−0.432703
$378$	0	0
$379$	−27.4745	−1.41127	−0.705636	−	0.708574i	$-0.749339\pi$
$379$	−27.4745	−1.41127	−0.705636	+	0.708574i	$0.749339\pi$
$380$	1.83416	0.0940906
$381$	0	0
$382$	27.1463	1.38893
$383$	27.5511	1.40780	0.703898	−	0.710301i	$-0.251441\pi$
$383$	27.5511	1.40780	0.703898	+	0.710301i	$0.251441\pi$
$384$	0	0
$385$	−25.8809	−1.31901
$386$	35.3268	1.79809
$387$	0	0
$388$	10.6704	0.541706
$389$	−28.1192	−1.42570	−0.712849	−	0.701317i	$-0.752596\pi$
$389$	−28.1192	−1.42570	−0.712849	+	0.701317i	$0.752596\pi$
$390$	0	0
$391$	−38.7184	−1.95807
$392$	36.9136	1.86442
$393$	0	0
$394$	13.9608	0.703333
$395$	3.36559	0.169341
$396$	0	0
$397$	20.7164	1.03973	0.519864	−	0.854249i	$-0.325983\pi$
$397$	20.7164	1.03973	0.519864	+	0.854249i	$0.325983\pi$
$398$	67.2237	3.36962
$399$	0	0
$400$	2.83093	0.141547
$401$	6.63004	0.331088	0.165544	−	0.986202i	$-0.447062\pi$
$401$	6.63004	0.331088	0.165544	+	0.986202i	$0.447062\pi$
$402$	0	0
$403$	−4.76298	−0.237261
$404$	31.0814	1.54636
$405$	0	0
$406$	59.1853	2.93732
$407$	−22.2028	−1.10055
$408$	0	0
$409$	−36.7363	−1.81649	−0.908246	−	0.418436i	$-0.862578\pi$
$409$	−36.7363	−1.81649	−0.908246	+	0.418436i	$0.862578\pi$
$410$	25.1575	1.24244
$411$	0	0
$412$	−23.9341	−1.17915
$413$	9.18401	0.451915
$414$	0	0
$415$	−16.6900	−0.819282
$416$	2.48363	0.121770
$417$	0	0
$418$	−7.63779	−0.373576
$419$	−2.06774	−0.101016	−0.0505078	−	0.998724i	$-0.516084\pi$
$419$	−2.06774	−0.101016	−0.0505078	+	0.998724i	$0.516084\pi$
$420$	0	0
$421$	17.9504	0.874850	0.437425	−	0.899255i	$-0.355891\pi$
$421$	17.9504	0.874850	0.437425	+	0.899255i	$0.355891\pi$
$422$	10.4530	0.508846
$423$	0	0
$424$	10.4410	0.507059
$425$	−6.08941	−0.295380
$426$	0	0
$427$	7.73916	0.374524
$428$	−33.4869	−1.61865
$429$	0	0
$430$	15.3468	0.740087
$431$	17.1773	0.827404	0.413702	−	0.910412i	$-0.364236\pi$
$431$	17.1773	0.827404	0.413702	+	0.910412i	$0.364236\pi$
$432$	0	0
$433$	29.3865	1.41222	0.706112	−	0.708100i	$-0.250447\pi$
$433$	29.3865	1.41222	0.706112	+	0.708100i	$0.250447\pi$
$434$	33.5531	1.61060
$435$	0	0
$436$	58.1827	2.78645
$437$	−3.07031	−0.146873
$438$	0	0
$439$	0.909475	0.0434069	0.0217034	−	0.999764i	$-0.493091\pi$
$439$	0.909475	0.0434069	0.0217034	+	0.999764i	$0.493091\pi$
$440$	−28.4452	−1.35607
$441$	0	0
$442$	19.7484	0.939337
$443$	5.88122	0.279425	0.139713	−	0.990192i	$-0.455382\pi$
$443$	5.88122	0.279425	0.139713	+	0.990192i	$0.455382\pi$
$444$	0	0
$445$	−7.35902	−0.348851
$446$	18.3868	0.870640
$447$	0	0
$448$	−39.8043	−1.88058
$449$	−15.6446	−0.738313	−0.369156	−	0.929367i	$-0.620353\pi$
$449$	−15.6446	−0.738313	−0.369156	+	0.929367i	$0.620353\pi$
$450$	0	0
$451$	−68.6261	−3.23148
$452$	47.6613	2.24180
$453$	0	0
$454$	−14.4951	−0.680288
$455$	5.30649	0.248772
$456$	0	0
$457$	3.95314	0.184920	0.0924601	−	0.995716i	$-0.470527\pi$
$457$	3.95314	0.184920	0.0924601	+	0.995716i	$0.470527\pi$
$458$	−49.4070	−2.30864
$459$	0	0
$460$	−24.1513	−1.12606
$461$	26.8327	1.24972	0.624861	−	0.780736i	$-0.285155\pi$
$461$	26.8327	1.24972	0.624861	+	0.780736i	$0.285155\pi$
$462$	0	0
$463$	−7.00138	−0.325382	−0.162691	−	0.986677i	$-0.552017\pi$
$463$	−7.00138	−0.325382	−0.162691	+	0.986677i	$0.552017\pi$
$464$	17.6598	0.819837
$465$	0	0
$466$	29.4471	1.36411
$467$	−26.2788	−1.21604	−0.608018	−	0.793923i	$-0.708035\pi$
$467$	−26.2788	−1.21604	−0.608018	+	0.793923i	$0.708035\pi$
$468$	0	0
$469$	−35.1841	−1.62465
$470$	−1.42963	−0.0659438
$471$	0	0
$472$	10.0940	0.464614
$473$	−41.8638	−1.92490
$474$	0	0
$475$	−0.482880	−0.0221561
$476$	−91.1335	−4.17710
$477$	0	0
$478$	9.55484	0.437028
$479$	−5.06329	−0.231348	−0.115674	−	0.993287i	$-0.536903\pi$
$479$	−5.06329	−0.231348	−0.115674	+	0.993287i	$0.536903\pi$
$480$	0	0
$481$	4.55237	0.207570
$482$	−64.4133	−2.93394
$483$	0	0
$484$	122.106	5.55028
$485$	−2.80919	−0.127559
$486$	0	0
$487$	−5.36202	−0.242976	−0.121488	−	0.992593i	$-0.538767\pi$
$487$	−5.36202	−0.242976	−0.121488	+	0.992593i	$0.538767\pi$
$488$	8.50599	0.385048
$489$	0	0
$490$	−20.5260	−0.927272
$491$	−21.5483	−0.972462	−0.486231	−	0.873830i	$-0.661629\pi$
$491$	−21.5483	−0.972462	−0.486231	+	0.873830i	$0.661629\pi$
$492$	0	0
$493$	−37.9867	−1.71084
$494$	1.56602	0.0704584
$495$	0	0
$496$	10.0116	0.449535
$497$	22.3818	1.00396
$498$	0	0
$499$	−6.52706	−0.292191	−0.146096	−	0.989270i	$-0.546671\pi$
$499$	−6.52706	−0.292191	−0.146096	+	0.989270i	$0.546671\pi$
$500$	−3.79838	−0.169869
$501$	0	0
$502$	−61.3341	−2.73747
$503$	−28.4359	−1.26789	−0.633947	−	0.773377i	$-0.718566\pi$
$503$	−28.4359	−1.26789	−0.633947	+	0.773377i	$0.718566\pi$
$504$	0	0
$505$	−8.18280	−0.364130
$506$	100.570	4.47090
$507$	0	0
$508$	−59.2868	−2.63042
$509$	34.3217	1.52128	0.760641	−	0.649172i	$-0.224885\pi$
$509$	34.3217	1.52128	0.760641	+	0.649172i	$0.224885\pi$
$510$	0	0
$511$	−24.0892	−1.06564
$512$	−29.7391	−1.31429
$513$	0	0
$514$	−14.2843	−0.630054
$515$	6.30112	0.277661
$516$	0	0
$517$	3.89982	0.171514
$518$	−32.0694	−1.40905
$519$	0	0
$520$	5.83228	0.255763
$521$	15.3994	0.674660	0.337330	−	0.941386i	$-0.390476\pi$
$521$	15.3994	0.674660	0.337330	+	0.941386i	$0.390476\pi$
$522$	0	0
$523$	−1.83273	−0.0801396	−0.0400698	−	0.999197i	$-0.512758\pi$
$523$	−1.83273	−0.0801396	−0.0400698	+	0.999197i	$0.512758\pi$
$524$	−45.8369	−2.00240
$525$	0	0
$526$	60.3656	2.63207
$527$	−21.5353	−0.938091
$528$	0	0
$529$	17.4282	0.757749
$530$	−5.80577	−0.252187
$531$	0	0
$532$	−7.22673	−0.313319
$533$	14.0708	0.609474
$534$	0	0
$535$	8.81610	0.381153
$536$	−38.6703	−1.67030
$537$	0	0
$538$	−14.8951	−0.642174
$539$	55.9921	2.41175
$540$	0	0
$541$	12.7445	0.547930	0.273965	−	0.961740i	$-0.411665\pi$
$541$	12.7445	0.547930	0.273965	+	0.961740i	$0.411665\pi$
$542$	29.2776	1.25758
$543$	0	0
$544$	11.2295	0.481460
$545$	−15.3178	−0.656141
$546$	0	0
$547$	19.1533	0.818934	0.409467	−	0.912325i	$-0.365715\pi$
$547$	19.1533	0.818934	0.409467	+	0.912325i	$0.365715\pi$
$548$	−32.4365	−1.38562
$549$	0	0
$550$	15.8171	0.674445
$551$	−3.01228	−0.128328
$552$	0	0
$553$	−13.2606	−0.563900
$554$	−27.8733	−1.18422
$555$	0	0
$556$	−34.0251	−1.44299
$557$	−2.77969	−0.117779	−0.0588896	−	0.998265i	$-0.518756\pi$
$557$	−2.77969	−0.117779	−0.0588896	+	0.998265i	$0.518756\pi$
$558$	0	0
$559$	8.58356	0.363046
$560$	−11.1541	−0.471346
$561$	0	0
$562$	−35.5979	−1.50161
$563$	42.8517	1.80598	0.902991	−	0.429660i	$-0.141366\pi$
$563$	42.8517	1.80598	0.902991	+	0.429660i	$0.141366\pi$
$564$	0	0
$565$	−12.5478	−0.527890
$566$	−25.5167	−1.07254
$567$	0	0
$568$	24.5995	1.03217
$569$	−14.8618	−0.623038	−0.311519	−	0.950240i	$-0.600838\pi$
$569$	−14.8618	−0.623038	−0.311519	+	0.950240i	$0.600838\pi$
$570$	0	0
$571$	2.11464	0.0884949	0.0442475	−	0.999021i	$-0.485911\pi$
$571$	2.11464	0.0884949	0.0442475	+	0.999021i	$0.485911\pi$
$572$	−33.6029	−1.40501
$573$	0	0
$574$	−99.1225	−4.13729
$575$	6.35832	0.265160
$576$	0	0
$577$	−13.2388	−0.551139	−0.275570	−	0.961281i	$-0.588866\pi$
$577$	−13.2388	−0.551139	−0.275570	+	0.961281i	$0.588866\pi$
$578$	48.3545	2.01128
$579$	0	0
$580$	−23.6949	−0.983877
$581$	65.7600	2.72818
$582$	0	0
$583$	15.8373	0.655914
$584$	−26.4761	−1.09559
$585$	0	0
$586$	32.2710	1.33310
$587$	14.3788	0.593476	0.296738	−	0.954959i	$-0.404101\pi$
$587$	14.3788	0.593476	0.296738	+	0.954959i	$0.404101\pi$
$588$	0	0
$589$	−1.70771	−0.0703650
$590$	−5.61283	−0.231076
$591$	0	0
$592$	−9.56893	−0.393281
$593$	42.8331	1.75895	0.879473	−	0.475949i	$-0.157895\pi$
$593$	42.8331	1.75895	0.879473	+	0.475949i	$0.157895\pi$
$594$	0	0
$595$	23.9927	0.983606
$596$	−33.8145	−1.38510
$597$	0	0
$598$	−20.6205	−0.843235
$599$	37.3269	1.52513	0.762567	−	0.646909i	$-0.223939\pi$
$599$	37.3269	1.52513	0.762567	+	0.646909i	$0.223939\pi$
$600$	0	0
$601$	−14.3405	−0.584960	−0.292480	−	0.956272i	$-0.594480\pi$
$601$	−14.3405	−0.584960	−0.292480	+	0.956272i	$0.594480\pi$
$602$	−60.4674	−2.46447
$603$	0	0
$604$	−11.7430	−0.477815
$605$	−32.1469	−1.30696
$606$	0	0
$607$	−2.85292	−0.115796	−0.0578981	−	0.998322i	$-0.518440\pi$
$607$	−2.85292	−0.115796	−0.0578981	+	0.998322i	$0.518440\pi$
$608$	0.890478	0.0361137
$609$	0	0
$610$	−4.72980	−0.191504
$611$	−0.799601	−0.0323484
$612$	0	0
$613$	−30.0143	−1.21227	−0.606133	−	0.795363i	$-0.707280\pi$
$613$	−30.0143	−1.21227	−0.606133	+	0.795363i	$0.707280\pi$
$614$	−34.5109	−1.39274
$615$	0	0
$616$	112.076	4.51568
$617$	−7.41672	−0.298586	−0.149293	−	0.988793i	$-0.547700\pi$
$617$	−7.41672	−0.298586	−0.149293	+	0.988793i	$0.547700\pi$
$618$	0	0
$619$	25.3423	1.01859	0.509296	−	0.860591i	$-0.329906\pi$
$619$	25.3423	1.01859	0.509296	+	0.860591i	$0.329906\pi$
$620$	−13.4330	−0.539483
$621$	0	0
$622$	15.2570	0.611752
$623$	28.9951	1.16166
$624$	0	0
$625$	1.00000	0.0400000
$626$	−26.8497	−1.07313
$627$	0	0
$628$	−27.1968	−1.08527
$629$	20.5830	0.820699
$630$	0	0
$631$	−0.549999	−0.0218951	−0.0109476	−	0.999940i	$-0.503485\pi$
$631$	−0.549999	−0.0218951	−0.0109476	+	0.999940i	$0.503485\pi$
$632$	−14.5746	−0.579745
$633$	0	0
$634$	23.1739	0.920353
$635$	15.6084	0.619402
$636$	0	0
$637$	−11.4804	−0.454868
$638$	98.6699	3.90638
$639$	0	0
$640$	20.6383	0.815799
$641$	−46.0246	−1.81786	−0.908931	−	0.416946i	$-0.863101\pi$
$641$	−46.0246	−1.81786	−0.908931	+	0.416946i	$0.863101\pi$
$642$	0	0
$643$	−43.6229	−1.72032	−0.860161	−	0.510023i	$-0.829637\pi$
$643$	−43.6229	−1.72032	−0.860161	+	0.510023i	$0.829637\pi$
$644$	95.1579	3.74975
$645$	0	0
$646$	7.08057	0.278581
$647$	−15.9348	−0.626461	−0.313230	−	0.949677i	$-0.601411\pi$
$647$	−15.9348	−0.626461	−0.313230	+	0.949677i	$0.601411\pi$
$648$	0	0
$649$	15.3110	0.601008
$650$	−3.24308	−0.127204
$651$	0	0
$652$	−53.3527	−2.08945
$653$	−29.6779	−1.16138	−0.580692	−	0.814123i	$-0.697218\pi$
$653$	−29.6779	−1.16138	−0.580692	+	0.814123i	$0.697218\pi$
$654$	0	0
$655$	12.0675	0.471516
$656$	−29.5764	−1.15476
$657$	0	0
$658$	5.63283	0.219591
$659$	23.4793	0.914622	0.457311	−	0.889307i	$-0.348813\pi$
$659$	23.4793	0.914622	0.457311	+	0.889307i	$0.348813\pi$
$660$	0	0
$661$	17.9520	0.698253	0.349126	−	0.937076i	$-0.386478\pi$
$661$	17.9520	0.698253	0.349126	+	0.937076i	$0.386478\pi$
$662$	−38.2274	−1.48575
$663$	0	0
$664$	72.2757	2.80484
$665$	1.90258	0.0737790
$666$	0	0
$667$	39.6642	1.53580
$668$	−75.3434	−2.91512
$669$	0	0
$670$	21.5028	0.830726
$671$	12.9022	0.498085
$672$	0	0
$673$	−6.87183	−0.264890	−0.132445	−	0.991190i	$-0.542283\pi$
$673$	−6.87183	−0.264890	−0.132445	+	0.991190i	$0.542283\pi$
$674$	−27.7207	−1.06776
$675$	0	0
$676$	−42.4892	−1.63420
$677$	−6.33279	−0.243389	−0.121694	−	0.992568i	$-0.538833\pi$
$677$	−6.33279	−0.243389	−0.121694	+	0.992568i	$0.538833\pi$
$678$	0	0
$679$	11.0684	0.424767
$680$	26.3700	1.01124
$681$	0	0
$682$	55.9375	2.14196
$683$	−8.46512	−0.323909	−0.161954	−	0.986798i	$-0.551780\pi$
$683$	−8.46512	−0.323909	−0.161954	+	0.986798i	$0.551780\pi$
$684$	0	0
$685$	8.53957	0.326280
$686$	14.4607	0.552112
$687$	0	0
$688$	−18.0424	−0.687859
$689$	−3.24721	−0.123709
$690$	0	0
$691$	36.4484	1.38656	0.693281	−	0.720668i	$-0.256165\pi$
$691$	36.4484	1.38656	0.693281	+	0.720668i	$0.256165\pi$
$692$	−36.7203	−1.39590
$693$	0	0
$694$	−11.1341	−0.422643
$695$	8.95779	0.339789
$696$	0	0
$697$	63.6195	2.40976
$698$	28.3747	1.07400
$699$	0	0
$700$	14.9659	0.565658
$701$	22.9932	0.868443	0.434221	−	0.900806i	$-0.357024\pi$
$701$	22.9932	0.868443	0.434221	+	0.900806i	$0.357024\pi$
$702$	0	0
$703$	1.63220	0.0615596
$704$	−66.3591	−2.50100
$705$	0	0
$706$	−40.3866	−1.51997
$707$	32.2408	1.21254
$708$	0	0
$709$	−3.28329	−0.123306	−0.0616532	−	0.998098i	$-0.519637\pi$
$709$	−3.28329	−0.123306	−0.0616532	+	0.998098i	$0.519637\pi$
$710$	−13.6787	−0.513352
$711$	0	0
$712$	31.8680	1.19430
$713$	22.4863	0.842117
$714$	0	0
$715$	8.84664	0.330846
$716$	65.4809	2.44713
$717$	0	0
$718$	−61.7074	−2.30290
$719$	−40.7581	−1.52002	−0.760010	−	0.649912i	$-0.774806\pi$
$719$	−40.7581	−1.52002	−0.760010	+	0.649912i	$0.774806\pi$
$720$	0	0
$721$	−24.8269	−0.924601
$722$	−45.1902	−1.68180
$723$	0	0
$724$	−12.5575	−0.466696
$725$	6.23816	0.231679
$726$	0	0
$727$	−18.4367	−0.683778	−0.341889	−	0.939740i	$-0.611067\pi$
$727$	−18.4367	−0.683778	−0.341889	+	0.939740i	$0.611067\pi$
$728$	−22.9796	−0.851681
$729$	0	0
$730$	14.7222	0.544892
$731$	38.8096	1.43542
$732$	0	0
$733$	−7.11572	−0.262825	−0.131413	−	0.991328i	$-0.541951\pi$
$733$	−7.11572	−0.262825	−0.131413	+	0.991328i	$0.541951\pi$
$734$	−58.9254	−2.17498
$735$	0	0
$736$	−11.7254	−0.432203
$737$	−58.6566	−2.16064
$738$	0	0
$739$	16.1585	0.594398	0.297199	−	0.954816i	$-0.403947\pi$
$739$	16.1585	0.594398	0.297199	+	0.954816i	$0.403947\pi$
$740$	12.8390	0.471972
$741$	0	0
$742$	22.8752	0.839774
$743$	−0.520135	−0.0190819	−0.00954096	−	0.999954i	$-0.503037\pi$
$743$	−0.520135	−0.0190819	−0.00954096	+	0.999954i	$0.503037\pi$
$744$	0	0
$745$	8.90236	0.326157
$746$	12.2566	0.448747
$747$	0	0
$748$	−151.932	−5.55518
$749$	−34.7361	−1.26923
$750$	0	0
$751$	−42.7669	−1.56059	−0.780293	−	0.625414i	$-0.784930\pi$
$751$	−42.7669	−1.56059	−0.780293	+	0.625414i	$0.784930\pi$
$752$	1.68073	0.0612901
$753$	0	0
$754$	−20.2308	−0.736763
$755$	3.09157	0.112514
$756$	0	0
$757$	−22.8812	−0.831630	−0.415815	−	0.909449i	$-0.636504\pi$
$757$	−22.8812	−0.831630	−0.415815	+	0.909449i	$0.636504\pi$
$758$	−66.1582	−2.40298
$759$	0	0
$760$	2.09110	0.0758521
$761$	28.8117	1.04442	0.522212	−	0.852816i	$-0.325107\pi$
$761$	28.8117	1.04442	0.522212	+	0.852816i	$0.325107\pi$
$762$	0	0
$763$	60.3531	2.18493
$764$	42.8209	1.54921
$765$	0	0
$766$	66.3426	2.39705
$767$	−3.13929	−0.113353
$768$	0	0
$769$	47.8149	1.72425	0.862125	−	0.506696i	$-0.169133\pi$
$769$	47.8149	1.72425	0.862125	+	0.506696i	$0.169133\pi$
$770$	−62.3207	−2.24588
$771$	0	0
$772$	55.7249	2.00558
$773$	17.6616	0.635243	0.317622	−	0.948218i	$-0.397116\pi$
$773$	17.6616	0.635243	0.317622	+	0.948218i	$0.397116\pi$
$774$	0	0
$775$	3.53651	0.127035
$776$	12.1651	0.436702
$777$	0	0
$778$	−67.7105	−2.42754
$779$	5.04492	0.180753
$780$	0	0
$781$	37.3134	1.33518
$782$	−93.2333	−3.33402
$783$	0	0
$784$	24.1313	0.861834
$785$	7.16009	0.255555
$786$	0	0
$787$	21.4334	0.764017	0.382009	−	0.924159i	$-0.375232\pi$
$787$	21.4334	0.764017	0.382009	+	0.924159i	$0.375232\pi$
$788$	22.0219	0.784497
$789$	0	0
$790$	8.10427	0.288337
$791$	49.4392	1.75786
$792$	0	0
$793$	−2.64541	−0.0939413
$794$	49.8848	1.77034
$795$	0	0
$796$	106.039	3.75847
$797$	50.5109	1.78919	0.894593	−	0.446882i	$-0.147466\pi$
$797$	50.5109	1.78919	0.894593	+	0.446882i	$0.147466\pi$
$798$	0	0
$799$	−3.61530	−0.127900
$800$	−1.84410	−0.0651987
$801$	0	0
$802$	15.9650	0.563744
$803$	−40.1600	−1.41721
$804$	0	0
$805$	−25.0522	−0.882975
$806$	−11.4692	−0.403984
$807$	0	0
$808$	35.4354	1.24661
$809$	−33.5209	−1.17853	−0.589266	−	0.807939i	$-0.700583\pi$
$809$	−33.5209	−1.17853	−0.589266	+	0.807939i	$0.700583\pi$
$810$	0	0
$811$	24.5151	0.860842	0.430421	−	0.902628i	$-0.358365\pi$
$811$	24.5151	0.860842	0.430421	+	0.902628i	$0.358365\pi$
$812$	93.3596	3.27628
$813$	0	0
$814$	−53.4641	−1.87391
$815$	14.0462	0.492015
$816$	0	0
$817$	3.07753	0.107669
$818$	−88.4603	−3.09294
$819$	0	0
$820$	39.6838	1.38582
$821$	−35.0050	−1.22168	−0.610842	−	0.791753i	$-0.709169\pi$
$821$	−35.0050	−1.22168	−0.610842	+	0.791753i	$0.709169\pi$
$822$	0	0
$823$	38.4754	1.34117	0.670584	−	0.741834i	$-0.266044\pi$
$823$	38.4754	1.34117	0.670584	+	0.741834i	$0.266044\pi$
$824$	−27.2868	−0.950582
$825$	0	0
$826$	22.1149	0.769477
$827$	28.7521	0.999808	0.499904	−	0.866081i	$-0.333369\pi$
$827$	28.7521	0.999808	0.499904	+	0.866081i	$0.333369\pi$
$828$	0	0
$829$	1.40464	0.0487852	0.0243926	−	0.999702i	$-0.492235\pi$
$829$	1.40464	0.0487852	0.0243926	+	0.999702i	$0.492235\pi$
$830$	−40.1893	−1.39499
$831$	0	0
$832$	13.6060	0.471702
$833$	−51.9071	−1.79848
$834$	0	0
$835$	19.8357	0.686441
$836$	−12.0479	−0.416686
$837$	0	0
$838$	−4.97907	−0.171999
$839$	−18.6650	−0.644387	−0.322193	−	0.946674i	$-0.604420\pi$
$839$	−18.6650	−0.644387	−0.322193	+	0.946674i	$0.604420\pi$
$840$	0	0
$841$	9.91463	0.341884
$842$	43.2243	1.48961
$843$	0	0
$844$	16.4888	0.567566
$845$	11.1861	0.384814
$846$	0	0
$847$	126.661	4.35212
$848$	6.82553	0.234390
$849$	0	0
$850$	−14.6632	−0.502944
$851$	−21.4920	−0.736735
$852$	0	0
$853$	21.8751	0.748991	0.374495	−	0.927229i	$-0.377816\pi$
$853$	21.8751	0.748991	0.374495	+	0.927229i	$0.377816\pi$
$854$	18.6358	0.637703
$855$	0	0
$856$	−38.1779	−1.30489
$857$	27.9593	0.955070	0.477535	−	0.878613i	$-0.341530\pi$
$857$	27.9593	0.955070	0.477535	+	0.878613i	$0.341530\pi$
$858$	0	0
$859$	32.5591	1.11090	0.555451	−	0.831549i	$-0.312546\pi$
$859$	32.5591	1.11090	0.555451	+	0.831549i	$0.312546\pi$
$860$	24.2082	0.825492
$861$	0	0
$862$	41.3627	1.40882
$863$	−34.3952	−1.17083	−0.585414	−	0.810735i	$-0.699068\pi$
$863$	−34.3952	−1.17083	−0.585414	+	0.810735i	$0.699068\pi$
$864$	0	0
$865$	9.66735	0.328700
$866$	70.7621	2.40460
$867$	0	0
$868$	52.9270	1.79646
$869$	−22.1073	−0.749938
$870$	0	0
$871$	12.0267	0.407508
$872$	66.3331	2.24632
$873$	0	0
$874$	−7.39324	−0.250080
$875$	−3.94007	−0.133199
$876$	0	0
$877$	−2.67092	−0.0901907	−0.0450953	−	0.998983i	$-0.514359\pi$
$877$	−2.67092	−0.0901907	−0.0450953	+	0.998983i	$0.514359\pi$
$878$	2.19000	0.0739089
$879$	0	0
$880$	−18.5954	−0.626849
$881$	18.5901	0.626315	0.313158	−	0.949701i	$-0.398613\pi$
$881$	18.5901	0.626315	0.313158	+	0.949701i	$0.398613\pi$
$882$	0	0
$883$	−23.5921	−0.793938	−0.396969	−	0.917832i	$-0.629938\pi$
$883$	−23.5921	−0.793938	−0.396969	+	0.917832i	$0.629938\pi$
$884$	31.1514	1.04773
$885$	0	0
$886$	14.1619	0.475778
$887$	18.6899	0.627544	0.313772	−	0.949498i	$-0.398407\pi$
$887$	18.6899	0.627544	0.313772	+	0.949498i	$0.398407\pi$
$888$	0	0
$889$	−61.4983	−2.06259
$890$	−17.7204	−0.593989
$891$	0	0
$892$	29.0036	0.971111
$893$	−0.286688	−0.00959363
$894$	0	0
$895$	−17.2392	−0.576241
$896$	−81.3163	−2.71659
$897$	0	0
$898$	−37.6719	−1.25713
$899$	22.0613	0.735786
$900$	0	0
$901$	−14.6819	−0.489125
$902$	−165.251	−5.50224
$903$	0	0
$904$	54.3378	1.80725
$905$	3.30601	0.109896
$906$	0	0
$907$	22.0745	0.732973	0.366487	−	0.930423i	$-0.380561\pi$
$907$	22.0745	0.732973	0.366487	+	0.930423i	$0.380561\pi$
$908$	−22.8647	−0.758792
$909$	0	0
$910$	12.7779	0.423585
$911$	5.69732	0.188761	0.0943803	−	0.995536i	$-0.469913\pi$
$911$	5.69732	0.188761	0.0943803	+	0.995536i	$0.469913\pi$
$912$	0	0
$913$	109.631	3.62825
$914$	9.51910	0.314864
$915$	0	0
$916$	−77.9353	−2.57505
$917$	−47.5468	−1.57013
$918$	0	0
$919$	2.18705	0.0721441	0.0360720	−	0.999349i	$-0.488515\pi$
$919$	2.18705	0.0721441	0.0360720	+	0.999349i	$0.488515\pi$
$920$	−27.5345	−0.907786
$921$	0	0
$922$	64.6126	2.12790
$923$	−7.65058	−0.251822
$924$	0	0
$925$	−3.38013	−0.111138
$926$	−16.8592	−0.554028
$927$	0	0
$928$	−11.5038	−0.377630
$929$	−15.8184	−0.518984	−0.259492	−	0.965745i	$-0.583555\pi$
$929$	−15.8184	−0.518984	−0.259492	+	0.965745i	$0.583555\pi$
$930$	0	0
$931$	−4.11615	−0.134901
$932$	46.4502	1.52153
$933$	0	0
$934$	−63.2788	−2.07055
$935$	39.9991	1.30811
$936$	0	0
$937$	19.4465	0.635291	0.317645	−	0.948210i	$-0.397108\pi$
$937$	19.4465	0.635291	0.317645	+	0.948210i	$0.397108\pi$
$938$	−84.7226	−2.76629
$939$	0	0
$940$	−2.25511	−0.0735536
$941$	53.2005	1.73429	0.867144	−	0.498058i	$-0.165953\pi$
$941$	53.2005	1.73429	0.867144	+	0.498058i	$0.165953\pi$
$942$	0	0
$943$	−66.4289	−2.16322
$944$	6.59869	0.214769
$945$	0	0
$946$	−100.807	−3.27753
$947$	−1.63964	−0.0532811	−0.0266405	−	0.999645i	$-0.508481\pi$
$947$	−1.63964	−0.0532811	−0.0266405	+	0.999645i	$0.508481\pi$
$948$	0	0
$949$	8.23421	0.267294
$950$	−1.16277	−0.0377251
$951$	0	0
$952$	−103.900	−3.36741
$953$	12.6586	0.410053	0.205026	−	0.978756i	$-0.434272\pi$
$953$	12.6586	0.410053	0.205026	+	0.978756i	$0.434272\pi$
$954$	0	0
$955$	−11.2735	−0.364801
$956$	15.0719	0.487461
$957$	0	0
$958$	−12.1923	−0.393916
$959$	−33.6465	−1.08650
$960$	0	0
$961$	−18.4931	−0.596552
$962$	10.9620	0.353430
$963$	0	0
$964$	−101.606	−3.27252
$965$	−14.6707	−0.472266
$966$	0	0
$967$	35.1289	1.12967	0.564835	−	0.825204i	$-0.308940\pi$
$967$	35.1289	1.12967	0.564835	+	0.825204i	$0.308940\pi$
$968$	139.211	4.47441
$969$	0	0
$970$	−6.76449	−0.217195
$971$	−28.8861	−0.926998	−0.463499	−	0.886097i	$-0.653406\pi$
$971$	−28.8861	−0.926998	−0.463499	+	0.886097i	$0.653406\pi$
$972$	0	0
$973$	−35.2944	−1.13148
$974$	−12.9116	−0.413716
$975$	0	0
$976$	5.56057	0.177990
$977$	52.2559	1.67181	0.835907	−	0.548871i	$-0.184942\pi$
$977$	52.2559	1.67181	0.835907	+	0.548871i	$0.184942\pi$
$978$	0	0
$979$	48.3387	1.54491
$980$	−32.3780	−1.03428
$981$	0	0
$982$	−51.8880	−1.65581
$983$	−22.9959	−0.733455	−0.366727	−	0.930328i	$-0.619522\pi$
$983$	−22.9959	−0.733455	−0.366727	+	0.930328i	$0.619522\pi$
$984$	0	0
$985$	−5.79770	−0.184730
$986$	−91.4714	−2.91304
$987$	0	0
$988$	2.47025	0.0785892
$989$	−40.5234	−1.28857
$990$	0	0
$991$	21.0308	0.668067	0.334033	−	0.942561i	$-0.391590\pi$
$991$	21.0308	0.668067	0.334033	+	0.942561i	$0.391590\pi$
$992$	−6.52167	−0.207063
$993$	0	0
$994$	53.8950	1.70944
$995$	−27.9170	−0.885029
$996$	0	0
$997$	−16.1889	−0.512709	−0.256354	−	0.966583i	$-0.582521\pi$
$997$	−16.1889	−0.512709	−0.256354	+	0.966583i	$0.582521\pi$
$998$	−15.7170	−0.497514
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3645.2.a.d.1.6	yes	6
3.2	odd	2		3645.2.a.c.1.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3645.2.a.c.1.1	✓	6	3.2	odd	2
3645.2.a.d.1.6	yes	6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3645 = 3^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3645.a (trivial)

\(f(q)\)	\(=\)	\(q+2.40798 q^{2} +3.79838 q^{4} -1.00000 q^{5} +3.94007 q^{7} +4.33047 q^{8} -2.40798 q^{10} +6.56863 q^{11} -1.34680 q^{13} +9.48762 q^{14} +2.83093 q^{16} -6.08941 q^{17} -0.482880 q^{19} -3.79838 q^{20} +15.8171 q^{22} +6.35832 q^{23} +1.00000 q^{25} -3.24308 q^{26} +14.9659 q^{28} +6.23816 q^{29} +3.53651 q^{31} -1.84410 q^{32} -14.6632 q^{34} -3.94007 q^{35} -3.38013 q^{37} -1.16277 q^{38} -4.33047 q^{40} -10.4476 q^{41} -6.37329 q^{43} +24.9502 q^{44} +15.3107 q^{46} +0.593703 q^{47} +8.52416 q^{49} +2.40798 q^{50} -5.11567 q^{52} +2.41105 q^{53} -6.56863 q^{55} +17.0624 q^{56} +15.0214 q^{58} +2.33092 q^{59} +1.96422 q^{61} +8.51586 q^{62} -10.1024 q^{64} +1.34680 q^{65} -8.92981 q^{67} -23.1299 q^{68} -9.48762 q^{70} +5.68055 q^{71} -6.11390 q^{73} -8.13930 q^{74} -1.83416 q^{76} +25.8809 q^{77} -3.36559 q^{79} -2.83093 q^{80} -25.1575 q^{82} +16.6900 q^{83} +6.08941 q^{85} -15.3468 q^{86} +28.4452 q^{88} +7.35902 q^{89} -5.30649 q^{91} +24.1513 q^{92} +1.42963 q^{94} +0.482880 q^{95} +2.80919 q^{97} +20.5260 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{2} + 9 q^{4} - 6 q^{5} + 3 q^{7} + 3 q^{8} - 3 q^{10} + 9 q^{11} - 6 q^{13} + 24 q^{14} - 9 q^{16} - 15 q^{17} - 9 q^{19} - 9 q^{20} + 3 q^{22} + 6 q^{23} + 6 q^{25} + 3 q^{26} + 12 q^{28} + 30 q^{29}+ \cdots - 15 q^{97}+O(q^{100}) \) 6 * q + 3 * q^2 + 9 * q^4 - 6 * q^5 + 3 * q^7 + 3 * q^8 - 3 * q^10 + 9 * q^11 - 6 * q^13 + 24 * q^14 - 9 * q^16 - 15 * q^17 - 9 * q^19 - 9 * q^20 + 3 * q^22 + 6 * q^23 + 6 * q^25 + 3 * q^26 + 12 * q^28 + 30 * q^29 - 3 * q^31 + 9 * q^32 + 15 * q^34 - 3 * q^35 - 18 * q^37 + 18 * q^38 - 3 * q^40 + 15 * q^41 - 3 * q^43 + 36 * q^44 + 36 * q^46 + 15 * q^47 - 3 * q^49 + 3 * q^50 + 18 * q^52 + 33 * q^53 - 9 * q^55 + 24 * q^56 + 6 * q^58 + 27 * q^59 + 3 * q^61 + 6 * q^62 - 27 * q^64 + 6 * q^65 - 33 * q^67 - 12 * q^68 - 24 * q^70 + 27 * q^71 - 9 * q^73 - 3 * q^74 - 6 * q^76 + 9 * q^77 + 18 * q^79 + 9 * q^80 - 27 * q^82 + 15 * q^83 + 15 * q^85 - 15 * q^86 + 33 * q^88 - 15 * q^89 - 21 * q^91 + 9 * q^92 + 18 * q^94 + 9 * q^95 - 15 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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