LMFDB - Embedded newform 1904.2.a.m.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1904,2,Mod(1,1904)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1904.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1904, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1904 = 2^{4} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1904.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,-1,0,3] 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:	$15.2035165449$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 952)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.254102$ of defining polynomial
Character	$\chi$	$=$	1904.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.93543 q^{3} +1.25410 q^{5} -1.00000 q^{7} +5.61676 q^{9} -2.50820 q^{11} -3.68133 q^{15} -1.00000 q^{17} +3.87086 q^{19} +2.93543 q^{21} +4.37907 q^{23} -3.42723 q^{25} -7.68133 q^{27} -7.87086 q^{29} -1.25410 q^{31} +7.36266 q^{33} -1.25410 q^{35} +3.87086 q^{37} +9.28169 q^{41} +1.41082 q^{43} +7.04399 q^{45} +7.74173 q^{47} +1.00000 q^{49} +2.93543 q^{51} +1.38324 q^{53} -3.14554 q^{55} -11.3627 q^{57} -8.34625 q^{59} -11.6608 q^{61} -5.61676 q^{63} -10.9630 q^{67} -12.8545 q^{69} -0.475390 q^{71} +0.427229 q^{73} +10.0604 q^{75} +2.50820 q^{77} -8.00000 q^{79} +5.69774 q^{81} +1.83805 q^{83} -1.25410 q^{85} +23.1044 q^{87} -2.37907 q^{89} +3.68133 q^{93} +4.85446 q^{95} -1.12497 q^{97} -14.0880 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{3} + 3 q^{5} - 3 q^{7} + 2 q^{9} - 6 q^{11} - 4 q^{15} - 3 q^{17} - 4 q^{19} + q^{21} - 4 q^{23} - 4 q^{25} - 16 q^{27} - 8 q^{29} - 3 q^{31} + 8 q^{33} - 3 q^{35} - 4 q^{37} + 9 q^{41} + q^{43}+ \cdots + 13 q^{97}+O(q^{100}) $ 3 * q - q^3 + 3 * q^5 - 3 * q^7 + 2 * q^9 - 6 * q^11 - 4 * q^15 - 3 * q^17 - 4 * q^19 + q^21 - 4 * q^23 - 4 * q^25 - 16 * q^27 - 8 * q^29 - 3 * q^31 + 8 * q^33 - 3 * q^35 - 4 * q^37 + 9 * q^41 + q^43 - 8 * q^47 + 3 * q^49 + q^51 + 19 * q^53 - 22 * q^55 - 20 * q^57 - 14 * q^59 + q^61 - 2 * q^63 - 7 * q^67 - 26 * q^69 - 6 * q^71 - 5 * q^73 + 6 * q^75 + 6 * q^77 - 24 * q^79 + 7 * q^81 - 4 * q^83 - 3 * q^85 + 24 * q^87 + 10 * q^89 + 4 * q^93 + 2 * q^95 + 13 * 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 $

$2$	0	0
$2$	0	0
$3$	−2.93543	−1.69477	−0.847386	−	0.530977i	$-0.821825\pi$
$3$	−2.93543	−1.69477	−0.847386	+	0.530977i	$0.821825\pi$
$4$	0	0
$5$	1.25410	0.560851	0.280426	−	0.959876i	$-0.409524\pi$
$5$	1.25410	0.560851	0.280426	+	0.959876i	$0.409524\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	5.61676	1.87225
$10$	0	0
$11$	−2.50820	−0.756252	−0.378126	−	0.925754i	$-0.623431\pi$
$11$	−2.50820	−0.756252	−0.378126	+	0.925754i	$0.623431\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	−3.68133	−0.950515
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	3.87086	0.888037	0.444019	−	0.896018i	$-0.353552\pi$
$19$	3.87086	0.888037	0.444019	+	0.896018i	$0.353552\pi$
$20$	0	0
$21$	2.93543	0.640564
$22$	0	0
$23$	4.37907	0.913099	0.456549	−	0.889698i	$-0.349085\pi$
$23$	4.37907	0.913099	0.456549	+	0.889698i	$0.349085\pi$
$24$	0	0
$25$	−3.42723	−0.685446
$26$	0	0
$27$	−7.68133	−1.47827
$28$	0	0
$29$	−7.87086	−1.46158	−0.730791	−	0.682601i	$-0.760849\pi$
$29$	−7.87086	−1.46158	−0.730791	+	0.682601i	$0.760849\pi$
$30$	0	0
$31$	−1.25410	−0.225243	−0.112622	−	0.993638i	$-0.535925\pi$
$31$	−1.25410	−0.225243	−0.112622	+	0.993638i	$0.535925\pi$
$32$	0	0
$33$	7.36266	1.28167
$34$	0	0
$35$	−1.25410	−0.211982
$36$	0	0
$37$	3.87086	0.636366	0.318183	−	0.948029i	$-0.396927\pi$
$37$	3.87086	0.636366	0.318183	+	0.948029i	$0.396927\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	9.28169	1.44956	0.724778	−	0.688982i	$-0.241942\pi$
$41$	9.28169	1.44956	0.724778	+	0.688982i	$0.241942\pi$
$42$	0	0
$43$	1.41082	0.215148	0.107574	−	0.994197i	$-0.465692\pi$
$43$	1.41082	0.215148	0.107574	+	0.994197i	$0.465692\pi$
$44$	0	0
$45$	7.04399	1.05006
$46$	0	0
$47$	7.74173	1.12925	0.564624	−	0.825348i	$-0.309021\pi$
$47$	7.74173	1.12925	0.564624	+	0.825348i	$0.309021\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	2.93543	0.411043
$52$	0	0
$53$	1.38324	0.190002	0.0950011	−	0.995477i	$-0.469715\pi$
$53$	1.38324	0.190002	0.0950011	+	0.995477i	$0.469715\pi$
$54$	0	0
$55$	−3.14554	−0.424145
$56$	0	0
$57$	−11.3627	−1.50502
$58$	0	0
$59$	−8.34625	−1.08659	−0.543295	−	0.839542i	$-0.682823\pi$
$59$	−8.34625	−1.08659	−0.543295	+	0.839542i	$0.682823\pi$
$60$	0	0
$61$	−11.6608	−1.49301	−0.746503	−	0.665382i	$-0.768269\pi$
$61$	−11.6608	−1.49301	−0.746503	+	0.665382i	$0.768269\pi$
$62$	0	0
$63$	−5.61676	−0.707646
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−10.9630	−1.33935	−0.669673	−	0.742656i	$-0.733566\pi$
$67$	−10.9630	−1.33935	−0.669673	+	0.742656i	$0.733566\pi$
$68$	0	0
$69$	−12.8545	−1.54749
$70$	0	0
$71$	−0.475390	−0.0564184	−0.0282092	−	0.999602i	$-0.508980\pi$
$71$	−0.475390	−0.0564184	−0.0282092	+	0.999602i	$0.508980\pi$
$72$	0	0
$73$	0.427229	0.0500034	0.0250017	−	0.999687i	$-0.492041\pi$
$73$	0.427229	0.0500034	0.0250017	+	0.999687i	$0.492041\pi$
$74$	0	0
$75$	10.0604	1.16167
$76$	0	0
$77$	2.50820	0.285836
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	5.69774	0.633082
$82$	0	0
$83$	1.83805	0.201752	0.100876	−	0.994899i	$-0.467835\pi$
$83$	1.83805	0.201752	0.100876	+	0.994899i	$0.467835\pi$
$84$	0	0
$85$	−1.25410	−0.136026
$86$	0	0
$87$	23.1044	2.47705
$88$	0	0
$89$	−2.37907	−0.252181	−0.126090	−	0.992019i	$-0.540243\pi$
$89$	−2.37907	−0.252181	−0.126090	+	0.992019i	$0.540243\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	3.68133	0.381736
$94$	0	0
$95$	4.85446	0.498057
$96$	0	0
$97$	−1.12497	−0.114223	−0.0571115	−	0.998368i	$-0.518189\pi$
$97$	−1.12497	−0.114223	−0.0571115	+	0.998368i	$0.518189\pi$
$98$	0	0
$99$	−14.0880	−1.41590
$100$	0	0
$101$	−4.47539	−0.445318	−0.222659	−	0.974896i	$-0.571474\pi$
$101$	−4.47539	−0.445318	−0.222659	+	0.974896i	$0.571474\pi$
$102$	0	0
$103$	10.2499	1.00996	0.504978	−	0.863132i	$-0.331501\pi$
$103$	10.2499	1.00996	0.504978	+	0.863132i	$0.331501\pi$
$104$	0	0
$105$	3.68133	0.359261
$106$	0	0
$107$	−15.7417	−1.52181	−0.760905	−	0.648863i	$-0.775245\pi$
$107$	−15.7417	−1.52181	−0.760905	+	0.648863i	$0.775245\pi$
$108$	0	0
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	0	0
$111$	−11.3627	−1.07850
$112$	0	0
$113$	−16.0880	−1.51343	−0.756715	−	0.653745i	$-0.773197\pi$
$113$	−16.0880	−1.51343	−0.756715	+	0.653745i	$0.773197\pi$
$114$	0	0
$115$	5.49180	0.512113
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.00000	0.0916698
$120$	0	0
$121$	−4.70892	−0.428083
$122$	0	0
$123$	−27.2458	−2.45667
$124$	0	0
$125$	−10.5686	−0.945285
$126$	0	0
$127$	−3.50403	−0.310933	−0.155466	−	0.987841i	$-0.549688\pi$
$127$	−3.50403	−0.310933	−0.155466	+	0.987841i	$0.549688\pi$
$128$	0	0
$129$	−4.14137	−0.364628
$130$	0	0
$131$	−17.1044	−1.49442	−0.747209	−	0.664589i	$-0.768606\pi$
$131$	−17.1044	−1.49442	−0.747209	+	0.664589i	$0.768606\pi$
$132$	0	0
$133$	−3.87086	−0.335647
$134$	0	0
$135$	−9.63317	−0.829091
$136$	0	0
$137$	17.0234	1.45441	0.727204	−	0.686421i	$-0.240819\pi$
$137$	17.0234	1.45441	0.727204	+	0.686421i	$0.240819\pi$
$138$	0	0
$139$	5.12497	0.434694	0.217347	−	0.976094i	$-0.430260\pi$
$139$	5.12497	0.434694	0.217347	+	0.976094i	$0.430260\pi$
$140$	0	0
$141$	−22.7253	−1.91382
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−9.87086	−0.819731
$146$	0	0
$147$	−2.93543	−0.242110
$148$	0	0
$149$	−1.70191	−0.139426	−0.0697128	−	0.997567i	$-0.522208\pi$
$149$	−1.70191	−0.139426	−0.0697128	+	0.997567i	$0.522208\pi$
$150$	0	0
$151$	−0.298094	−0.0242585	−0.0121293	−	0.999926i	$-0.503861\pi$
$151$	−0.298094	−0.0242585	−0.0121293	+	0.999926i	$0.503861\pi$
$152$	0	0
$153$	−5.61676	−0.454088
$154$	0	0
$155$	−1.57277	−0.126328
$156$	0	0
$157$	20.3379	1.62314	0.811571	−	0.584253i	$-0.198613\pi$
$157$	20.3379	1.62314	0.811571	+	0.584253i	$0.198613\pi$
$158$	0	0
$159$	−4.06040	−0.322011
$160$	0	0
$161$	−4.37907	−0.345119
$162$	0	0
$163$	−10.3463	−0.810381	−0.405191	−	0.914232i	$-0.632795\pi$
$163$	−10.3463	−0.810381	−0.405191	+	0.914232i	$0.632795\pi$
$164$	0	0
$165$	9.23353	0.718829
$166$	0	0
$167$	−17.3145	−1.33984	−0.669918	−	0.742435i	$-0.733671\pi$
$167$	−17.3145	−1.33984	−0.669918	+	0.742435i	$0.733671\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	21.7417	1.66263
$172$	0	0
$173$	2.04816	0.155719	0.0778594	−	0.996964i	$-0.475191\pi$
$173$	2.04816	0.155719	0.0778594	+	0.996964i	$0.475191\pi$
$174$	0	0
$175$	3.42723	0.259074
$176$	0	0
$177$	24.4999	1.84152
$178$	0	0
$179$	−17.9107	−1.33871	−0.669354	−	0.742944i	$-0.733429\pi$
$179$	−17.9107	−1.33871	−0.669354	+	0.742944i	$0.733429\pi$
$180$	0	0
$181$	−17.3955	−1.29300	−0.646498	−	0.762916i	$-0.723767\pi$
$181$	−17.3955	−1.29300	−0.646498	+	0.762916i	$0.723767\pi$
$182$	0	0
$183$	34.2294	2.53031
$184$	0	0
$185$	4.85446	0.356907
$186$	0	0
$187$	2.50820	0.183418
$188$	0	0
$189$	7.68133	0.558735
$190$	0	0
$191$	−5.25410	−0.380173	−0.190087	−	0.981767i	$-0.560877\pi$
$191$	−5.25410	−0.380173	−0.190087	+	0.981767i	$0.560877\pi$
$192$	0	0
$193$	−23.1372	−1.66545	−0.832726	−	0.553685i	$-0.813221\pi$
$193$	−23.1372	−1.66545	−0.832726	+	0.553685i	$0.813221\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−4.03281	−0.287326	−0.143663	−	0.989627i	$-0.545888\pi$
$197$	−4.03281	−0.287326	−0.143663	+	0.989627i	$0.545888\pi$
$198$	0	0
$199$	−1.31450	−0.0931825	−0.0465912	−	0.998914i	$-0.514836\pi$
$199$	−1.31450	−0.0931825	−0.0465912	+	0.998914i	$0.514836\pi$
$200$	0	0
$201$	32.1812	2.26989
$202$	0	0
$203$	7.87086	0.552426
$204$	0	0
$205$	11.6402	0.812985
$206$	0	0
$207$	24.5962	1.70955
$208$	0	0
$209$	−9.70892	−0.671580
$210$	0	0
$211$	10.8873	0.749511	0.374755	−	0.927124i	$-0.377727\pi$
$211$	10.8873	0.749511	0.374755	+	0.927124i	$0.377727\pi$
$212$	0	0
$213$	1.39547	0.0956164
$214$	0	0
$215$	1.76931	0.120666
$216$	0	0
$217$	1.25410	0.0851340
$218$	0	0
$219$	−1.25410	−0.0847443
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−3.45898	−0.231631	−0.115815	−	0.993271i	$-0.536948\pi$
$223$	−3.45898	−0.231631	−0.115815	+	0.993271i	$0.536948\pi$
$224$	0	0
$225$	−19.2499	−1.28333
$226$	0	0
$227$	23.8503	1.58300	0.791500	−	0.611170i	$-0.209301\pi$
$227$	23.8503	1.58300	0.791500	+	0.611170i	$0.209301\pi$
$228$	0	0
$229$	5.49180	0.362908	0.181454	−	0.983399i	$-0.441920\pi$
$229$	5.49180	0.362908	0.181454	+	0.983399i	$0.441920\pi$
$230$	0	0
$231$	−7.36266	−0.484428
$232$	0	0
$233$	−18.9753	−1.24311	−0.621555	−	0.783370i	$-0.713499\pi$
$233$	−18.9753	−1.24311	−0.621555	+	0.783370i	$0.713499\pi$
$234$	0	0
$235$	9.70892	0.633340
$236$	0	0
$237$	23.4835	1.52541
$238$	0	0
$239$	30.4272	1.96817	0.984087	−	0.177688i	$-0.0568617\pi$
$239$	30.4272	1.96817	0.984087	+	0.177688i	$0.0568617\pi$
$240$	0	0
$241$	−10.8339	−0.697872	−0.348936	−	0.937147i	$-0.613457\pi$
$241$	−10.8339	−0.697872	−0.348936	+	0.937147i	$0.613457\pi$
$242$	0	0
$243$	6.31867	0.405343
$244$	0	0
$245$	1.25410	0.0801216
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−5.39547	−0.341924
$250$	0	0
$251$	−10.1619	−0.641417	−0.320708	−	0.947178i	$-0.603921\pi$
$251$	−10.1619	−0.641417	−0.320708	+	0.947178i	$0.603921\pi$
$252$	0	0
$253$	−10.9836	−0.690533
$254$	0	0
$255$	3.68133	0.230534
$256$	0	0
$257$	10.6373	0.663539	0.331769	−	0.943360i	$-0.392354\pi$
$257$	10.6373	0.663539	0.331769	+	0.943360i	$0.392354\pi$
$258$	0	0
$259$	−3.87086	−0.240524
$260$	0	0
$261$	−44.2088	−2.73645
$262$	0	0
$263$	−5.96719	−0.367952	−0.183976	−	0.982931i	$-0.558897\pi$
$263$	−5.96719	−0.367952	−0.183976	+	0.982931i	$0.558897\pi$
$264$	0	0
$265$	1.73472	0.106563
$266$	0	0
$267$	6.98359	0.427389
$268$	0	0
$269$	26.1536	1.59461	0.797307	−	0.603574i	$-0.206257\pi$
$269$	26.1536	1.59461	0.797307	+	0.603574i	$0.206257\pi$
$270$	0	0
$271$	−16.1208	−0.979269	−0.489634	−	0.871928i	$-0.662870\pi$
$271$	−16.1208	−0.979269	−0.489634	+	0.871928i	$0.662870\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	8.59619	0.518370
$276$	0	0
$277$	−26.7581	−1.60774	−0.803870	−	0.594805i	$-0.797229\pi$
$277$	−26.7581	−1.60774	−0.803870	+	0.594805i	$0.797229\pi$
$278$	0	0
$279$	−7.04399	−0.421713
$280$	0	0
$281$	28.0674	1.67436	0.837181	−	0.546927i	$-0.184202\pi$
$281$	28.0674	1.67436	0.837181	+	0.546927i	$0.184202\pi$
$282$	0	0
$283$	30.1002	1.78927	0.894636	−	0.446795i	$-0.147435\pi$
$283$	30.1002	1.78927	0.894636	+	0.446795i	$0.147435\pi$
$284$	0	0
$285$	−14.2499	−0.844093
$286$	0	0
$287$	−9.28169	−0.547881
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	3.30226	0.193582
$292$	0	0
$293$	−24.4999	−1.43130	−0.715649	−	0.698460i	$-0.753869\pi$
$293$	−24.4999	−1.43130	−0.715649	+	0.698460i	$0.753869\pi$
$294$	0	0
$295$	−10.4671	−0.609415
$296$	0	0
$297$	19.2663	1.11795
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−1.41082	−0.0813184
$302$	0	0
$303$	13.1372	0.754713
$304$	0	0
$305$	−14.6238	−0.837355
$306$	0	0
$307$	−2.21712	−0.126538	−0.0632688	−	0.997997i	$-0.520153\pi$
$307$	−2.21712	−0.126538	−0.0632688	+	0.997997i	$0.520153\pi$
$308$	0	0
$309$	−30.0880	−1.71165
$310$	0	0
$311$	−13.2130	−0.749238	−0.374619	−	0.927179i	$-0.622226\pi$
$311$	−13.2130	−0.749238	−0.374619	+	0.927179i	$0.622226\pi$
$312$	0	0
$313$	33.3421	1.88461	0.942303	−	0.334761i	$-0.108656\pi$
$313$	33.3421	1.88461	0.942303	+	0.334761i	$0.108656\pi$
$314$	0	0
$315$	−7.04399	−0.396884
$316$	0	0
$317$	28.8461	1.62016	0.810080	−	0.586320i	$-0.199424\pi$
$317$	28.8461	1.62016	0.810080	+	0.586320i	$0.199424\pi$
$318$	0	0
$319$	19.7417	1.10532
$320$	0	0
$321$	46.2088	2.57912
$322$	0	0
$323$	−3.87086	−0.215381
$324$	0	0
$325$	0	0
$326$	0	0
$327$	41.0961	2.27262
$328$	0	0
$329$	−7.74173	−0.426815
$330$	0	0
$331$	−7.00417	−0.384984	−0.192492	−	0.981299i	$-0.561657\pi$
$331$	−7.00417	−0.384984	−0.192492	+	0.981299i	$0.561657\pi$
$332$	0	0
$333$	21.7417	1.19144
$334$	0	0
$335$	−13.7487	−0.751174
$336$	0	0
$337$	28.8461	1.57135	0.785674	−	0.618641i	$-0.212316\pi$
$337$	28.8461	1.57135	0.785674	+	0.618641i	$0.212316\pi$
$338$	0	0
$339$	47.2252	2.56492
$340$	0	0
$341$	3.14554	0.170341
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−16.1208	−0.867915
$346$	0	0
$347$	16.1208	0.865410	0.432705	−	0.901536i	$-0.357559\pi$
$347$	16.1208	0.865410	0.432705	+	0.901536i	$0.357559\pi$
$348$	0	0
$349$	−19.9037	−1.06542	−0.532710	−	0.846298i	$-0.678826\pi$
$349$	−19.9037	−1.06542	−0.532710	+	0.846298i	$0.678826\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	11.1127	0.591471	0.295735	−	0.955270i	$-0.404435\pi$
$353$	11.1127	0.591471	0.295735	+	0.955270i	$0.404435\pi$
$354$	0	0
$355$	−0.596187	−0.0316423
$356$	0	0
$357$	−2.93543	−0.155360
$358$	0	0
$359$	6.96302	0.367494	0.183747	−	0.982974i	$-0.441177\pi$
$359$	6.96302	0.367494	0.183747	+	0.982974i	$0.441177\pi$
$360$	0	0
$361$	−4.01641	−0.211390
$362$	0	0
$363$	13.8227	0.725504
$364$	0	0
$365$	0.535789	0.0280445
$366$	0	0
$367$	−3.60558	−0.188210	−0.0941050	−	0.995562i	$-0.529999\pi$
$367$	−3.60558	−0.188210	−0.0941050	+	0.995562i	$0.529999\pi$
$368$	0	0
$369$	52.1330	2.71394
$370$	0	0
$371$	−1.38324	−0.0718141
$372$	0	0
$373$	−18.2981	−0.947439	−0.473720	−	0.880676i	$-0.657089\pi$
$373$	−18.2981	−0.947439	−0.473720	+	0.880676i	$0.657089\pi$
$374$	0	0
$375$	31.0234	1.60204
$376$	0	0
$377$	0	0
$378$	0	0
$379$	1.23353	0.0633620	0.0316810	−	0.999498i	$-0.489914\pi$
$379$	1.23353	0.0633620	0.0316810	+	0.999498i	$0.489914\pi$
$380$	0	0
$381$	10.2859	0.526960
$382$	0	0
$383$	−5.55742	−0.283971	−0.141986	−	0.989869i	$-0.545349\pi$
$383$	−5.55742	−0.283971	−0.141986	+	0.989869i	$0.545349\pi$
$384$	0	0
$385$	3.14554	0.160312
$386$	0	0
$387$	7.92425	0.402812
$388$	0	0
$389$	5.38324	0.272941	0.136470	−	0.990644i	$-0.456424\pi$
$389$	5.38324	0.272941	0.136470	+	0.990644i	$0.456424\pi$
$390$	0	0
$391$	−4.37907	−0.221459
$392$	0	0
$393$	50.2088	2.53270
$394$	0	0
$395$	−10.0328	−0.504806
$396$	0	0
$397$	−27.1801	−1.36413	−0.682066	−	0.731291i	$-0.738918\pi$
$397$	−27.1801	−1.36413	−0.682066	+	0.731291i	$0.738918\pi$
$398$	0	0
$399$	11.3627	0.568845
$400$	0	0
$401$	4.44258	0.221852	0.110926	−	0.993829i	$-0.464618\pi$
$401$	4.44258	0.221852	0.110926	+	0.993829i	$0.464618\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	7.14554	0.355065
$406$	0	0
$407$	−9.70892	−0.481253
$408$	0	0
$409$	9.26634	0.458191	0.229095	−	0.973404i	$-0.426423\pi$
$409$	9.26634	0.458191	0.229095	+	0.973404i	$0.426423\pi$
$410$	0	0
$411$	−49.9711	−2.46489
$412$	0	0
$413$	8.34625	0.410692
$414$	0	0
$415$	2.30510	0.113153
$416$	0	0
$417$	−15.0440	−0.736707
$418$	0	0
$419$	−8.36683	−0.408746	−0.204373	−	0.978893i	$-0.565516\pi$
$419$	−8.36683	−0.408746	−0.204373	+	0.978893i	$0.565516\pi$
$420$	0	0
$421$	−25.3473	−1.23535	−0.617676	−	0.786432i	$-0.711926\pi$
$421$	−25.3473	−1.23535	−0.617676	+	0.786432i	$0.711926\pi$
$422$	0	0
$423$	43.4835	2.11424
$424$	0	0
$425$	3.42723	0.166245
$426$	0	0
$427$	11.6608	0.564303
$428$	0	0
$429$	0	0
$430$	0	0
$431$	14.1291	0.680577	0.340288	−	0.940321i	$-0.389475\pi$
$431$	14.1291	0.680577	0.340288	+	0.940321i	$0.389475\pi$
$432$	0	0
$433$	2.25827	0.108526	0.0542628	−	0.998527i	$-0.482719\pi$
$433$	2.25827	0.108526	0.0542628	+	0.998527i	$0.482719\pi$
$434$	0	0
$435$	28.9753	1.38926
$436$	0	0
$437$	16.9508	0.810866
$438$	0	0
$439$	−35.8779	−1.71236	−0.856179	−	0.516680i	$-0.827168\pi$
$439$	−35.8779	−1.71236	−0.856179	+	0.516680i	$0.827168\pi$
$440$	0	0
$441$	5.61676	0.267465
$442$	0	0
$443$	−19.4835	−0.925687	−0.462844	−	0.886440i	$-0.653171\pi$
$443$	−19.4835	−0.925687	−0.462844	+	0.886440i	$0.653171\pi$
$444$	0	0
$445$	−2.98359	−0.141436
$446$	0	0
$447$	4.99583	0.236295
$448$	0	0
$449$	4.79095	0.226099	0.113049	−	0.993589i	$-0.463938\pi$
$449$	4.79095	0.226099	0.113049	+	0.993589i	$0.463938\pi$
$450$	0	0
$451$	−23.2804	−1.09623
$452$	0	0
$453$	0.875034	0.0411127
$454$	0	0
$455$	0	0
$456$	0	0
$457$	33.1494	1.55067	0.775333	−	0.631553i	$-0.217582\pi$
$457$	33.1494	1.55067	0.775333	+	0.631553i	$0.217582\pi$
$458$	0	0
$459$	7.68133	0.358534
$460$	0	0
$461$	33.3955	1.55538	0.777691	−	0.628647i	$-0.216391\pi$
$461$	33.3955	1.55538	0.777691	+	0.628647i	$0.216391\pi$
$462$	0	0
$463$	−11.2817	−0.524304	−0.262152	−	0.965027i	$-0.584432\pi$
$463$	−11.2817	−0.524304	−0.262152	+	0.965027i	$0.584432\pi$
$464$	0	0
$465$	4.61676	0.214097
$466$	0	0
$467$	7.45065	0.344775	0.172387	−	0.985029i	$-0.444852\pi$
$467$	7.45065	0.344775	0.172387	+	0.985029i	$0.444852\pi$
$468$	0	0
$469$	10.9630	0.506225
$470$	0	0
$471$	−59.7006	−2.75086
$472$	0	0
$473$	−3.53863	−0.162706
$474$	0	0
$475$	−13.2663	−0.608701
$476$	0	0
$477$	7.76931	0.355732
$478$	0	0
$479$	−20.8943	−0.954684	−0.477342	−	0.878718i	$-0.658400\pi$
$479$	−20.8943	−0.954684	−0.477342	+	0.878718i	$0.658400\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	12.8545	0.584898
$484$	0	0
$485$	−1.41082	−0.0640621
$486$	0	0
$487$	3.90368	0.176893	0.0884463	−	0.996081i	$-0.471810\pi$
$487$	3.90368	0.176893	0.0884463	+	0.996081i	$0.471810\pi$
$488$	0	0
$489$	30.3707	1.37341
$490$	0	0
$491$	−12.9958	−0.586494	−0.293247	−	0.956037i	$-0.594736\pi$
$491$	−12.9958	−0.586494	−0.293247	+	0.956037i	$0.594736\pi$
$492$	0	0
$493$	7.87086	0.354486
$494$	0	0
$495$	−17.6678	−0.794107
$496$	0	0
$497$	0.475390	0.0213241
$498$	0	0
$499$	24.3379	1.08951	0.544757	−	0.838594i	$-0.316622\pi$
$499$	24.3379	1.08951	0.544757	+	0.838594i	$0.316622\pi$
$500$	0	0
$501$	50.8255	2.27072
$502$	0	0
$503$	−38.0122	−1.69488	−0.847441	−	0.530890i	$-0.821858\pi$
$503$	−38.0122	−1.69488	−0.847441	+	0.530890i	$0.821858\pi$
$504$	0	0
$505$	−5.61259	−0.249757
$506$	0	0
$507$	38.1606	1.69477
$508$	0	0
$509$	41.3132	1.83117	0.915587	−	0.402120i	$-0.131726\pi$
$509$	41.3132	1.83117	0.915587	+	0.402120i	$0.131726\pi$
$510$	0	0
$511$	−0.427229	−0.0188995
$512$	0	0
$513$	−29.7334	−1.31276
$514$	0	0
$515$	12.8545	0.566435
$516$	0	0
$517$	−19.4178	−0.853995
$518$	0	0
$519$	−6.01224	−0.263908
$520$	0	0
$521$	40.9271	1.79305	0.896524	−	0.442995i	$-0.146084\pi$
$521$	40.9271	1.79305	0.896524	+	0.442995i	$0.146084\pi$
$522$	0	0
$523$	−20.2088	−0.883668	−0.441834	−	0.897097i	$-0.645672\pi$
$523$	−20.2088	−0.883668	−0.441834	+	0.897097i	$0.645672\pi$
$524$	0	0
$525$	−10.0604	−0.439072
$526$	0	0
$527$	1.25410	0.0546295
$528$	0	0
$529$	−3.82376	−0.166251
$530$	0	0
$531$	−46.8789	−2.03437
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−19.7417	−0.853509
$536$	0	0
$537$	52.5756	2.26881
$538$	0	0
$539$	−2.50820	−0.108036
$540$	0	0
$541$	−28.0328	−1.20522	−0.602612	−	0.798034i	$-0.705873\pi$
$541$	−28.0328	−1.20522	−0.602612	+	0.798034i	$0.705873\pi$
$542$	0	0
$543$	51.0632	2.19133
$544$	0	0
$545$	−17.5574	−0.752077
$546$	0	0
$547$	10.3051	0.440614	0.220307	−	0.975431i	$-0.429294\pi$
$547$	10.3051	0.440614	0.220307	+	0.975431i	$0.429294\pi$
$548$	0	0
$549$	−65.4957	−2.79529
$550$	0	0
$551$	−30.4671	−1.29794
$552$	0	0
$553$	8.00000	0.340195
$554$	0	0
$555$	−14.2499	−0.604876
$556$	0	0
$557$	6.32390	0.267952	0.133976	−	0.990985i	$-0.457225\pi$
$557$	6.32390	0.267952	0.133976	+	0.990985i	$0.457225\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−7.36266	−0.310852
$562$	0	0
$563$	11.1784	0.471112	0.235556	−	0.971861i	$-0.424309\pi$
$563$	11.1784	0.471112	0.235556	+	0.971861i	$0.424309\pi$
$564$	0	0
$565$	−20.1760	−0.848809
$566$	0	0
$567$	−5.69774	−0.239282
$568$	0	0
$569$	−21.8779	−0.917168	−0.458584	−	0.888651i	$-0.651643\pi$
$569$	−21.8779	−0.917168	−0.458584	+	0.888651i	$0.651643\pi$
$570$	0	0
$571$	−32.9096	−1.37723	−0.688613	−	0.725129i	$-0.741780\pi$
$571$	−32.9096	−1.37723	−0.688613	+	0.725129i	$0.741780\pi$
$572$	0	0
$573$	15.4231	0.644308
$574$	0	0
$575$	−15.0081	−0.625880
$576$	0	0
$577$	−36.5327	−1.52088	−0.760438	−	0.649411i	$-0.775016\pi$
$577$	−36.5327	−1.52088	−0.760438	+	0.649411i	$0.775016\pi$
$578$	0	0
$579$	67.9177	2.82256
$580$	0	0
$581$	−1.83805	−0.0762552
$582$	0	0
$583$	−3.46944	−0.143690
$584$	0	0
$585$	0	0
$586$	0	0
$587$	34.4999	1.42396	0.711981	−	0.702199i	$-0.247798\pi$
$587$	34.4999	1.42396	0.711981	+	0.702199i	$0.247798\pi$
$588$	0	0
$589$	−4.85446	−0.200024
$590$	0	0
$591$	11.8381	0.486952
$592$	0	0
$593$	17.8381	0.732521	0.366260	−	0.930512i	$-0.380638\pi$
$593$	17.8381	0.732521	0.366260	+	0.930512i	$0.380638\pi$
$594$	0	0
$595$	1.25410	0.0514132
$596$	0	0
$597$	3.85863	0.157923
$598$	0	0
$599$	−39.6196	−1.61881	−0.809407	−	0.587249i	$-0.800211\pi$
$599$	−39.6196	−1.61881	−0.809407	+	0.587249i	$0.800211\pi$
$600$	0	0
$601$	−2.25827	−0.0921168	−0.0460584	−	0.998939i	$-0.514666\pi$
$601$	−2.25827	−0.0921168	−0.0460584	+	0.998939i	$0.514666\pi$
$602$	0	0
$603$	−61.5767	−2.50760
$604$	0	0
$605$	−5.90546	−0.240091
$606$	0	0
$607$	−40.0039	−1.62371	−0.811854	−	0.583860i	$-0.801542\pi$
$607$	−40.0039	−1.62371	−0.811854	+	0.583860i	$0.801542\pi$
$608$	0	0
$609$	−23.1044	−0.936237
$610$	0	0
$611$	0	0
$612$	0	0
$613$	1.88310	0.0760578	0.0380289	−	0.999277i	$-0.487892\pi$
$613$	1.88310	0.0760578	0.0380289	+	0.999277i	$0.487892\pi$
$614$	0	0
$615$	−34.1690	−1.37783
$616$	0	0
$617$	−6.85446	−0.275950	−0.137975	−	0.990436i	$-0.544059\pi$
$617$	−6.85446	−0.275950	−0.137975	+	0.990436i	$0.544059\pi$
$618$	0	0
$619$	−16.3463	−0.657011	−0.328506	−	0.944502i	$-0.606545\pi$
$619$	−16.3463	−0.657011	−0.328506	+	0.944502i	$0.606545\pi$
$620$	0	0
$621$	−33.6371	−1.34981
$622$	0	0
$623$	2.37907	0.0953153
$624$	0	0
$625$	3.88204	0.155282
$626$	0	0
$627$	28.4999	1.13817
$628$	0	0
$629$	−3.87086	−0.154341
$630$	0	0
$631$	−2.27051	−0.0903875	−0.0451938	−	0.998978i	$-0.514391\pi$
$631$	−2.27051	−0.0903875	−0.0451938	+	0.998978i	$0.514391\pi$
$632$	0	0
$633$	−31.9588	−1.27025
$634$	0	0
$635$	−4.39442	−0.174387
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−2.67015	−0.105630
$640$	0	0
$641$	−33.4423	−1.32089	−0.660446	−	0.750874i	$-0.729633\pi$
$641$	−33.4423	−1.32089	−0.660446	+	0.750874i	$0.729633\pi$
$642$	0	0
$643$	12.7100	0.501232	0.250616	−	0.968087i	$-0.419367\pi$
$643$	12.7100	0.501232	0.250616	+	0.968087i	$0.419367\pi$
$644$	0	0
$645$	−5.19370	−0.204502
$646$	0	0
$647$	−29.9917	−1.17909	−0.589547	−	0.807734i	$-0.700694\pi$
$647$	−29.9917	−1.17909	−0.589547	+	0.807734i	$0.700694\pi$
$648$	0	0
$649$	20.9341	0.821735
$650$	0	0
$651$	−3.68133	−0.144283
$652$	0	0
$653$	7.01641	0.274573	0.137287	−	0.990531i	$-0.456162\pi$
$653$	7.01641	0.274573	0.137287	+	0.990531i	$0.456162\pi$
$654$	0	0
$655$	−21.4506	−0.838146
$656$	0	0
$657$	2.39964	0.0936190
$658$	0	0
$659$	26.9875	1.05128	0.525642	−	0.850706i	$-0.323825\pi$
$659$	26.9875	1.05128	0.525642	+	0.850706i	$0.323825\pi$
$660$	0	0
$661$	−4.16195	−0.161881	−0.0809405	−	0.996719i	$-0.525792\pi$
$661$	−4.16195	−0.161881	−0.0809405	+	0.996719i	$0.525792\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−4.85446	−0.188248
$666$	0	0
$667$	−34.4671	−1.33457
$668$	0	0
$669$	10.1536	0.392561
$670$	0	0
$671$	29.2475	1.12909
$672$	0	0
$673$	−18.1760	−0.700632	−0.350316	−	0.936632i	$-0.613926\pi$
$673$	−18.1760	−0.700632	−0.350316	+	0.936632i	$0.613926\pi$
$674$	0	0
$675$	26.3257	1.01328
$676$	0	0
$677$	4.57171	0.175705	0.0878526	−	0.996133i	$-0.472000\pi$
$677$	4.57171	0.175705	0.0878526	+	0.996133i	$0.472000\pi$
$678$	0	0
$679$	1.12497	0.0431722
$680$	0	0
$681$	−70.0109	−2.68282
$682$	0	0
$683$	6.15361	0.235461	0.117731	−	0.993046i	$-0.462438\pi$
$683$	6.15361	0.235461	0.117731	+	0.993046i	$0.462438\pi$
$684$	0	0
$685$	21.3491	0.815707
$686$	0	0
$687$	−16.1208	−0.615047
$688$	0	0
$689$	0	0
$690$	0	0
$691$	29.2405	1.11236	0.556181	−	0.831061i	$-0.312266\pi$
$691$	29.2405	1.11236	0.556181	+	0.831061i	$0.312266\pi$
$692$	0	0
$693$	14.0880	0.535158
$694$	0	0
$695$	6.42723	0.243799
$696$	0	0
$697$	−9.28169	−0.351569
$698$	0	0
$699$	55.7006	2.10679
$700$	0	0
$701$	−14.4342	−0.545174	−0.272587	−	0.962131i	$-0.587879\pi$
$701$	−14.4342	−0.545174	−0.272587	+	0.962131i	$0.587879\pi$
$702$	0	0
$703$	14.9836	0.565117
$704$	0	0
$705$	−28.4999	−1.07337
$706$	0	0
$707$	4.47539	0.168314
$708$	0	0
$709$	−15.3955	−0.578189	−0.289095	−	0.957301i	$-0.593354\pi$
$709$	−15.3955	−0.578189	−0.289095	+	0.957301i	$0.593354\pi$
$710$	0	0
$711$	−44.9341	−1.68516
$712$	0	0
$713$	−5.49180	−0.205669
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−89.3171	−3.33561
$718$	0	0
$719$	−3.02342	−0.112754	−0.0563772	−	0.998410i	$-0.517955\pi$
$719$	−3.02342	−0.112754	−0.0563772	+	0.998410i	$0.517955\pi$
$720$	0	0
$721$	−10.2499	−0.381727
$722$	0	0
$723$	31.8021	1.18273
$724$	0	0
$725$	26.9753	1.00184
$726$	0	0
$727$	−2.12914	−0.0789653	−0.0394826	−	0.999220i	$-0.512571\pi$
$727$	−2.12914	−0.0789653	−0.0394826	+	0.999220i	$0.512571\pi$
$728$	0	0
$729$	−35.6412	−1.32005
$730$	0	0
$731$	−1.41082	−0.0521811
$732$	0	0
$733$	11.3627	0.419689	0.209845	−	0.977735i	$-0.432704\pi$
$733$	11.3627	0.419689	0.209845	+	0.977735i	$0.432704\pi$
$734$	0	0
$735$	−3.68133	−0.135788
$736$	0	0
$737$	27.4975	1.01288
$738$	0	0
$739$	46.6995	1.71787	0.858935	−	0.512085i	$-0.171127\pi$
$739$	46.6995	1.71787	0.858935	+	0.512085i	$0.171127\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	44.0245	1.61510	0.807551	−	0.589798i	$-0.200793\pi$
$743$	44.0245	1.61510	0.807551	+	0.589798i	$0.200793\pi$
$744$	0	0
$745$	−2.13436	−0.0781970
$746$	0	0
$747$	10.3239	0.377732
$748$	0	0
$749$	15.7417	0.575190
$750$	0	0
$751$	−49.6678	−1.81240	−0.906201	−	0.422847i	$-0.861031\pi$
$751$	−49.6678	−1.81240	−0.906201	+	0.422847i	$0.861031\pi$
$752$	0	0
$753$	29.8297	1.08706
$754$	0	0
$755$	−0.373840	−0.0136054
$756$	0	0
$757$	−40.7651	−1.48163	−0.740817	−	0.671707i	$-0.765562\pi$
$757$	−40.7651	−1.48163	−0.740817	+	0.671707i	$0.765562\pi$
$758$	0	0
$759$	32.2416	1.17030
$760$	0	0
$761$	26.6618	0.966490	0.483245	−	0.875485i	$-0.339458\pi$
$761$	26.6618	0.966490	0.483245	+	0.875485i	$0.339458\pi$
$762$	0	0
$763$	14.0000	0.506834
$764$	0	0
$765$	−7.04399	−0.254676
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−23.9260	−0.862795	−0.431397	−	0.902162i	$-0.641979\pi$
$769$	−23.9260	−0.862795	−0.431397	+	0.902162i	$0.641979\pi$
$770$	0	0
$771$	−31.2252	−1.12455
$772$	0	0
$773$	34.7253	1.24898	0.624492	−	0.781032i	$-0.285306\pi$
$773$	34.7253	1.24898	0.624492	+	0.781032i	$0.285306\pi$
$774$	0	0
$775$	4.29809	0.154392
$776$	0	0
$777$	11.3627	0.407633
$778$	0	0
$779$	35.9282	1.28726
$780$	0	0
$781$	1.19237	0.0426665
$782$	0	0
$783$	60.4587	2.16062
$784$	0	0
$785$	25.5058	0.910342
$786$	0	0
$787$	−33.5386	−1.19552	−0.597762	−	0.801674i	$-0.703943\pi$
$787$	−33.5386	−1.19552	−0.597762	+	0.801674i	$0.703943\pi$
$788$	0	0
$789$	17.5163	0.623596
$790$	0	0
$791$	16.0880	0.572023
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−5.09215	−0.180600
$796$	0	0
$797$	−3.13509	−0.111050	−0.0555252	−	0.998457i	$-0.517683\pi$
$797$	−3.13509	−0.111050	−0.0555252	+	0.998457i	$0.517683\pi$
$798$	0	0
$799$	−7.74173	−0.273883
$800$	0	0
$801$	−13.3627	−0.472146
$802$	0	0
$803$	−1.07158	−0.0378151
$804$	0	0
$805$	−5.49180	−0.193560
$806$	0	0
$807$	−76.7722	−2.70251
$808$	0	0
$809$	34.4999	1.21295	0.606475	−	0.795102i	$-0.292583\pi$
$809$	34.4999	1.21295	0.606475	+	0.795102i	$0.292583\pi$
$810$	0	0
$811$	−2.08097	−0.0730729	−0.0365364	−	0.999332i	$-0.511633\pi$
$811$	−2.08097	−0.0730729	−0.0365364	+	0.999332i	$0.511633\pi$
$812$	0	0
$813$	47.3215	1.65964
$814$	0	0
$815$	−12.9753	−0.454503
$816$	0	0
$817$	5.46110	0.191060
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−8.64568	−0.301736	−0.150868	−	0.988554i	$-0.548207\pi$
$821$	−8.64568	−0.301736	−0.150868	+	0.988554i	$0.548207\pi$
$822$	0	0
$823$	−7.52461	−0.262291	−0.131146	−	0.991363i	$-0.541866\pi$
$823$	−7.52461	−0.262291	−0.131146	+	0.991363i	$0.541866\pi$
$824$	0	0
$825$	−25.2335	−0.878519
$826$	0	0
$827$	−40.8545	−1.42065	−0.710324	−	0.703875i	$-0.751452\pi$
$827$	−40.8545	−1.42065	−0.710324	+	0.703875i	$0.751452\pi$
$828$	0	0
$829$	−38.9836	−1.35396	−0.676978	−	0.736003i	$-0.736711\pi$
$829$	−38.9836	−1.35396	−0.676978	+	0.736003i	$0.736711\pi$
$830$	0	0
$831$	78.5467	2.72475
$832$	0	0
$833$	−1.00000	−0.0346479
$834$	0	0
$835$	−21.7141	−0.751449
$836$	0	0
$837$	9.63317	0.332971
$838$	0	0
$839$	36.7581	1.26903	0.634516	−	0.772910i	$-0.281200\pi$
$839$	36.7581	1.26903	0.634516	+	0.772910i	$0.281200\pi$
$840$	0	0
$841$	32.9505	1.13622
$842$	0	0
$843$	−82.3900	−2.83766
$844$	0	0
$845$	−16.3033	−0.560851
$846$	0	0
$847$	4.70892	0.161800
$848$	0	0
$849$	−88.3572	−3.03241
$850$	0	0
$851$	16.9508	0.581065
$852$	0	0
$853$	20.2968	0.694948	0.347474	−	0.937690i	$-0.387039\pi$
$853$	20.2968	0.694948	0.347474	+	0.937690i	$0.387039\pi$
$854$	0	0
$855$	27.2663	0.932489
$856$	0	0
$857$	−20.8255	−0.711387	−0.355694	−	0.934603i	$-0.615755\pi$
$857$	−20.8255	−0.711387	−0.355694	+	0.934603i	$0.615755\pi$
$858$	0	0
$859$	35.7641	1.22025	0.610127	−	0.792303i	$-0.291118\pi$
$859$	35.7641	1.22025	0.610127	+	0.792303i	$0.291118\pi$
$860$	0	0
$861$	27.2458	0.928533
$862$	0	0
$863$	16.5616	0.563763	0.281882	−	0.959449i	$-0.409041\pi$
$863$	16.5616	0.563763	0.281882	+	0.959449i	$0.409041\pi$
$864$	0	0
$865$	2.56860	0.0873351
$866$	0	0
$867$	−2.93543	−0.0996925
$868$	0	0
$869$	20.0656	0.680680
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−6.31867	−0.213855
$874$	0	0
$875$	10.5686	0.357284
$876$	0	0
$877$	52.2744	1.76518	0.882591	−	0.470142i	$-0.155797\pi$
$877$	52.2744	1.76518	0.882591	+	0.470142i	$0.155797\pi$
$878$	0	0
$879$	71.9177	2.42572
$880$	0	0
$881$	7.85029	0.264483	0.132241	−	0.991218i	$-0.457783\pi$
$881$	7.85029	0.264483	0.132241	+	0.991218i	$0.457783\pi$
$882$	0	0
$883$	36.9135	1.24224	0.621120	−	0.783716i	$-0.286678\pi$
$883$	36.9135	1.24224	0.621120	+	0.783716i	$0.286678\pi$
$884$	0	0
$885$	30.7253	1.03282
$886$	0	0
$887$	−49.8196	−1.67278	−0.836389	−	0.548136i	$-0.815338\pi$
$887$	−49.8196	−1.67278	−0.836389	+	0.548136i	$0.815338\pi$
$888$	0	0
$889$	3.50403	0.117522
$890$	0	0
$891$	−14.2911	−0.478769
$892$	0	0
$893$	29.9672	1.00281
$894$	0	0
$895$	−22.4618	−0.750816
$896$	0	0
$897$	0	0
$898$	0	0
$899$	9.87086	0.329212
$900$	0	0
$901$	−1.38324	−0.0460823
$902$	0	0
$903$	4.14137	0.137816
$904$	0	0
$905$	−21.8157	−0.725178
$906$	0	0
$907$	−8.70297	−0.288977	−0.144489	−	0.989506i	$-0.546154\pi$
$907$	−8.70297	−0.288977	−0.144489	+	0.989506i	$0.546154\pi$
$908$	0	0
$909$	−25.1372	−0.833748
$910$	0	0
$911$	56.8378	1.88312	0.941560	−	0.336846i	$-0.109360\pi$
$911$	56.8378	1.88312	0.941560	+	0.336846i	$0.109360\pi$
$912$	0	0
$913$	−4.61021	−0.152576
$914$	0	0
$915$	42.9271	1.41913
$916$	0	0
$917$	17.1044	0.564837
$918$	0	0
$919$	23.3473	0.770157	0.385078	−	0.922884i	$-0.374174\pi$
$919$	23.3473	0.770157	0.385078	+	0.922884i	$0.374174\pi$
$920$	0	0
$921$	6.50820	0.214453
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−13.2663	−0.436195
$926$	0	0
$927$	57.5714	1.89089
$928$	0	0
$929$	16.8667	0.553378	0.276689	−	0.960959i	$-0.410763\pi$
$929$	16.8667	0.553378	0.276689	+	0.960959i	$0.410763\pi$
$930$	0	0
$931$	3.87086	0.126862
$932$	0	0
$933$	38.7857	1.26979
$934$	0	0
$935$	3.14554	0.102870
$936$	0	0
$937$	1.55530	0.0508096	0.0254048	−	0.999677i	$-0.491913\pi$
$937$	1.55530	0.0508096	0.0254048	+	0.999677i	$0.491913\pi$
$938$	0	0
$939$	−97.8734	−3.19398
$940$	0	0
$941$	40.3226	1.31448	0.657239	−	0.753682i	$-0.271724\pi$
$941$	40.3226	1.31448	0.657239	+	0.753682i	$0.271724\pi$
$942$	0	0
$943$	40.6451	1.32359
$944$	0	0
$945$	9.63317	0.313367
$946$	0	0
$947$	12.7337	0.413788	0.206894	−	0.978363i	$-0.433664\pi$
$947$	12.7337	0.413788	0.206894	+	0.978363i	$0.433664\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−84.6758	−2.74580
$952$	0	0
$953$	−2.80941	−0.0910056	−0.0455028	−	0.998964i	$-0.514489\pi$
$953$	−2.80941	−0.0910056	−0.0455028	+	0.998964i	$0.514489\pi$
$954$	0	0
$955$	−6.58918	−0.213221
$956$	0	0
$957$	−57.9505	−1.87327
$958$	0	0
$959$	−17.0234	−0.549715
$960$	0	0
$961$	−29.4272	−0.949265
$962$	0	0
$963$	−88.4176	−2.84922
$964$	0	0
$965$	−29.0164	−0.934071
$966$	0	0
$967$	42.4106	1.36383	0.681916	−	0.731431i	$-0.261147\pi$
$967$	42.4106	1.36383	0.681916	+	0.731431i	$0.261147\pi$
$968$	0	0
$969$	11.3627	0.365021
$970$	0	0
$971$	11.8709	0.380954	0.190477	−	0.981692i	$-0.438997\pi$
$971$	11.8709	0.380954	0.190477	+	0.981692i	$0.438997\pi$
$972$	0	0
$973$	−5.12497	−0.164299
$974$	0	0
$975$	0	0
$976$	0	0
$977$	29.5837	0.946466	0.473233	−	0.880937i	$-0.343087\pi$
$977$	29.5837	0.946466	0.473233	+	0.880937i	$0.343087\pi$
$978$	0	0
$979$	5.96719	0.190712
$980$	0	0
$981$	−78.6347	−2.51061
$982$	0	0
$983$	−33.8196	−1.07868	−0.539339	−	0.842089i	$-0.681326\pi$
$983$	−33.8196	−1.07868	−0.539339	+	0.842089i	$0.681326\pi$
$984$	0	0
$985$	−5.05756	−0.161147
$986$	0	0
$987$	22.7253	0.723355
$988$	0	0
$989$	6.17809	0.196452
$990$	0	0
$991$	−48.1208	−1.52861	−0.764304	−	0.644856i	$-0.776917\pi$
$991$	−48.1208	−1.52861	−0.764304	+	0.644856i	$0.776917\pi$
$992$	0	0
$993$	20.5603	0.652460
$994$	0	0
$995$	−1.64852	−0.0522615
$996$	0	0
$997$	56.2403	1.78115	0.890574	−	0.454839i	$-0.150303\pi$
$997$	56.2403	1.78115	0.890574	+	0.454839i	$0.150303\pi$
$998$	0	0
$999$	−29.7334	−0.940723

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1904.2.a.m.1.1		3
4.3	odd	2		952.2.a.f.1.3	✓	3
8.3	odd	2		7616.2.a.bc.1.1		3
8.5	even	2		7616.2.a.be.1.3		3
12.11	even	2		8568.2.a.x.1.2		3
28.27	even	2		6664.2.a.j.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
952.2.a.f.1.3	✓	3	4.3	odd	2
1904.2.a.m.1.1		3	1.1	even	1	trivial
6664.2.a.j.1.1		3	28.27	even	2
7616.2.a.bc.1.1		3	8.3	odd	2
7616.2.a.be.1.3		3	8.5	even	2
8568.2.a.x.1.2		3	12.11	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1904 = 2^{4} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1904.a (trivial)

\(f(q)\)	\(=\)	\(q-2.93543 q^{3} +1.25410 q^{5} -1.00000 q^{7} +5.61676 q^{9} -2.50820 q^{11} -3.68133 q^{15} -1.00000 q^{17} +3.87086 q^{19} +2.93543 q^{21} +4.37907 q^{23} -3.42723 q^{25} -7.68133 q^{27} -7.87086 q^{29} -1.25410 q^{31} +7.36266 q^{33} -1.25410 q^{35} +3.87086 q^{37} +9.28169 q^{41} +1.41082 q^{43} +7.04399 q^{45} +7.74173 q^{47} +1.00000 q^{49} +2.93543 q^{51} +1.38324 q^{53} -3.14554 q^{55} -11.3627 q^{57} -8.34625 q^{59} -11.6608 q^{61} -5.61676 q^{63} -10.9630 q^{67} -12.8545 q^{69} -0.475390 q^{71} +0.427229 q^{73} +10.0604 q^{75} +2.50820 q^{77} -8.00000 q^{79} +5.69774 q^{81} +1.83805 q^{83} -1.25410 q^{85} +23.1044 q^{87} -2.37907 q^{89} +3.68133 q^{93} +4.85446 q^{95} -1.12497 q^{97} -14.0880 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{3} + 3 q^{5} - 3 q^{7} + 2 q^{9} - 6 q^{11} - 4 q^{15} - 3 q^{17} - 4 q^{19} + q^{21} - 4 q^{23} - 4 q^{25} - 16 q^{27} - 8 q^{29} - 3 q^{31} + 8 q^{33} - 3 q^{35} - 4 q^{37} + 9 q^{41} + q^{43}+ \cdots + 13 q^{97}+O(q^{100}) \) 3 * q - q^3 + 3 * q^5 - 3 * q^7 + 2 * q^9 - 6 * q^11 - 4 * q^15 - 3 * q^17 - 4 * q^19 + q^21 - 4 * q^23 - 4 * q^25 - 16 * q^27 - 8 * q^29 - 3 * q^31 + 8 * q^33 - 3 * q^35 - 4 * q^37 + 9 * q^41 + q^43 - 8 * q^47 + 3 * q^49 + q^51 + 19 * q^53 - 22 * q^55 - 20 * q^57 - 14 * q^59 + q^61 - 2 * q^63 - 7 * q^67 - 26 * q^69 - 6 * q^71 - 5 * q^73 + 6 * q^75 + 6 * q^77 - 24 * q^79 + 7 * q^81 - 4 * q^83 - 3 * q^85 + 24 * q^87 + 10 * q^89 + 4 * q^93 + 2 * q^95 + 13 * q^97

Introduction

L-functions

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