LMFDB - Embedded newform 3800.2.d.o.3649.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3800,2,Mod(3649,3800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))

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

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

chi := DirichletCharacter("3800.3649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3800 = 2^{3} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3800.d (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$30.3431527681$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.419904.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 6x^{4} + 9x^{2} + 1 $ x^6 + 6x^4 + 9x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3649.6
Root			$1.87939i$ of defining polynomial
Character	$\chi$	$=$	3800.3649
Dual form			3800.2.d.o.3649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.87939i q^{3} +1.18479i q^{7} -0.532089 q^{9} +O(q^{10})$	$q+1.87939i q^{3} +1.18479i q^{7} -0.532089 q^{9} -2.18479 q^{11} +1.71688i q^{13} -0.120615i q^{17} -1.00000 q^{19} -2.22668 q^{21} -7.98545i q^{23} +4.63816i q^{27} -3.24897 q^{29} -8.41147 q^{31} -4.10607i q^{33} +3.33275i q^{37} -3.22668 q^{39} -8.98545 q^{41} -4.06418i q^{43} -1.71688i q^{47} +5.59627 q^{49} +0.226682 q^{51} +6.51754i q^{53} -1.87939i q^{57} -10.2121 q^{59} +6.53983 q^{61} -0.630415i q^{63} -2.18479i q^{67} +15.0077 q^{69} -9.12836 q^{71} -0.773318i q^{73} -2.58853i q^{77} -1.63816 q^{79} -10.3131 q^{81} -2.44831i q^{83} -6.10607i q^{87} -2.83750 q^{89} -2.03415 q^{91} -15.8084i q^{93} -2.19934i q^{97} +1.16250 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{9}+O(q^{10}) $ 6 * q + 6 * q^9	$ 6 q + 6 q^{9} - 6 q^{11} - 6 q^{19} + 6 q^{29} - 30 q^{31} - 6 q^{39} - 18 q^{41} + 6 q^{49} - 12 q^{51} - 12 q^{59} - 18 q^{61} + 42 q^{69} - 18 q^{71} + 24 q^{79} - 18 q^{81} - 12 q^{89} - 54 q^{91} + 12 q^{99}+O(q^{100}) $ 6 * q + 6 * q^9 - 6 * q^11 - 6 * q^19 + 6 * q^29 - 30 * q^31 - 6 * q^39 - 18 * q^41 + 6 * q^49 - 12 * q^51 - 12 * q^59 - 18 * q^61 + 42 * q^69 - 18 * q^71 + 24 * q^79 - 18 * q^81 - 12 * q^89 - 54 * q^91 + 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times$.

$n$	$401$	$951$	$1901$	$1977$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	1.87939i	1.08506i	0.840035	+	0.542532i	$0.182534\pi$
$3$	1.87939i	1.08506i	−0.840035	+	0.542532i	$0.817466\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.18479i	0.447809i	0.974611	+	0.223905i	$0.0718805\pi$
$7$	1.18479i	0.447809i	−0.974611	+	0.223905i	$0.928120\pi$
$8$	0	0
$9$	−0.532089	−0.177363
$10$	0	0
$11$	−2.18479	−0.658740	−0.329370	−	0.944201i	$-0.606836\pi$
$11$	−2.18479	−0.658740	−0.329370	+	0.944201i	$0.606836\pi$
$12$	0	0
$13$	1.71688i	0.476177i	0.971243	+	0.238089i	$0.0765209\pi$
$13$	1.71688i	0.476177i	−0.971243	+	0.238089i	$0.923479\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 0.120615i	− 0.0292534i	−0.999893	−	0.0146267i	$-0.995344\pi$
$17$	− 0.120615i	− 0.0292534i	0.999893	−	0.0146267i	$-0.00465599\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−2.22668	−0.485902
$22$	0	0
$23$	− 7.98545i	− 1.66508i	−0.553964	−	0.832541i	$-0.686885\pi$
$23$	− 7.98545i	− 1.66508i	0.553964	−	0.832541i	$-0.313115\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	4.63816i	0.892613i
$28$	0	0
$29$	−3.24897	−0.603319	−0.301659	−	0.953416i	$-0.597541\pi$
$29$	−3.24897	−0.603319	−0.301659	+	0.953416i	$0.597541\pi$
$30$	0	0
$31$	−8.41147	−1.51075	−0.755373	−	0.655295i	$-0.772544\pi$
$31$	−8.41147	−1.51075	−0.755373	+	0.655295i	$0.772544\pi$
$32$	0	0
$33$	− 4.10607i	− 0.714774i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.33275i	0.547900i	0.961744	+	0.273950i	$0.0883304\pi$
$37$	3.33275i	0.547900i	−0.961744	+	0.273950i	$0.911670\pi$
$38$	0	0
$39$	−3.22668	−0.516683
$40$	0	0
$41$	−8.98545	−1.40329	−0.701646	−	0.712526i	$-0.747551\pi$
$41$	−8.98545	−1.40329	−0.701646	+	0.712526i	$0.747551\pi$
$42$	0	0
$43$	− 4.06418i	− 0.619781i	−0.950772	−	0.309891i	$-0.899708\pi$
$43$	− 4.06418i	− 0.619781i	0.950772	−	0.309891i	$-0.100292\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 1.71688i	− 0.250433i	−0.992129	−	0.125216i	$-0.960037\pi$
$47$	− 1.71688i	− 0.250433i	0.992129	−	0.125216i	$-0.0399625\pi$
$48$	0	0
$49$	5.59627	0.799467
$50$	0	0
$51$	0.226682	0.0317418
$52$	0	0
$53$	6.51754i	0.895253i	0.894221	+	0.447627i	$0.147731\pi$
$53$	6.51754i	0.895253i	−0.894221	+	0.447627i	$0.852269\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 1.87939i	− 0.248931i
$58$	0	0
$59$	−10.2121	−1.32951	−0.664753	−	0.747063i	$-0.731463\pi$
$59$	−10.2121	−1.32951	−0.664753	+	0.747063i	$0.731463\pi$
$60$	0	0
$61$	6.53983	0.837339	0.418670	−	0.908139i	$-0.362497\pi$
$61$	6.53983	0.837339	0.418670	+	0.908139i	$0.362497\pi$
$62$	0	0
$63$	− 0.630415i	− 0.0794248i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 2.18479i	− 0.266915i	−0.991055	−	0.133457i	$-0.957392\pi$
$67$	− 2.18479i	− 0.266915i	0.991055	−	0.133457i	$-0.0426079\pi$
$68$	0	0
$69$	15.0077	1.80672
$70$	0	0
$71$	−9.12836	−1.08334	−0.541668	−	0.840592i	$-0.682207\pi$
$71$	−9.12836	−1.08334	−0.541668	+	0.840592i	$0.682207\pi$
$72$	0	0
$73$	− 0.773318i	− 0.0905101i	−0.998975	−	0.0452550i	$-0.985590\pi$
$73$	− 0.773318i	− 0.0905101i	0.998975	−	0.0452550i	$-0.0144100\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 2.58853i	− 0.294990i
$78$	0	0
$79$	−1.63816	−0.184307	−0.0921535	−	0.995745i	$-0.529375\pi$
$79$	−1.63816	−0.184307	−0.0921535	+	0.995745i	$0.529375\pi$
$80$	0	0
$81$	−10.3131	−1.14591
$82$	0	0
$83$	− 2.44831i	− 0.268737i	−0.990931	−	0.134369i	$-0.957099\pi$
$83$	− 2.44831i	− 0.268737i	0.990931	−	0.134369i	$-0.0429006\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 6.10607i	− 0.654639i
$88$	0	0
$89$	−2.83750	−0.300774	−0.150387	−	0.988627i	$-0.548052\pi$
$89$	−2.83750	−0.300774	−0.150387	+	0.988627i	$0.548052\pi$
$90$	0	0
$91$	−2.03415	−0.213237
$92$	0	0
$93$	− 15.8084i	− 1.63925i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 2.19934i	− 0.223309i	−0.993747	−	0.111655i	$-0.964385\pi$
$97$	− 2.19934i	− 0.223309i	0.993747	−	0.111655i	$-0.0356150\pi$
$98$	0	0
$99$	1.16250	0.116836
$100$	0	0
$101$	−10.3327	−1.02815	−0.514073	−	0.857746i	$-0.671864\pi$
$101$	−10.3327	−1.02815	−0.514073	+	0.857746i	$0.671864\pi$
$102$	0	0
$103$	− 17.5253i	− 1.72682i	−0.504505	−	0.863409i	$-0.668325\pi$
$103$	− 17.5253i	− 1.72682i	0.504505	−	0.863409i	$-0.331675\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.16250i	0.305731i	0.988247	+	0.152865i	$0.0488501\pi$
$107$	3.16250i	0.305731i	−0.988247	+	0.152865i	$0.951150\pi$
$108$	0	0
$109$	8.77332	0.840331	0.420166	−	0.907447i	$-0.361972\pi$
$109$	8.77332	0.840331	0.420166	+	0.907447i	$0.361972\pi$
$110$	0	0
$111$	−6.26352	−0.594507
$112$	0	0
$113$	15.0128i	1.41228i	0.708070	+	0.706142i	$0.249566\pi$
$113$	15.0128i	1.41228i	−0.708070	+	0.706142i	$0.750434\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 0.913534i	− 0.0844562i
$118$	0	0
$119$	0.142903	0.0130999
$120$	0	0
$121$	−6.22668	−0.566062
$122$	0	0
$123$	− 16.8871i	− 1.52266i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 6.35504i	− 0.563918i	−0.959426	−	0.281959i	$-0.909016\pi$
$127$	− 6.35504i	− 0.563918i	0.959426	−	0.281959i	$-0.0909843\pi$
$128$	0	0
$129$	7.63816	0.672502
$130$	0	0
$131$	8.41921	0.735590	0.367795	−	0.929907i	$-0.380113\pi$
$131$	8.41921	0.735590	0.367795	+	0.929907i	$0.380113\pi$
$132$	0	0
$133$	− 1.18479i	− 0.102735i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 0.644963i	− 0.0551029i	−0.999620	−	0.0275514i	$-0.991229\pi$
$137$	− 0.644963i	− 0.0551029i	0.999620	−	0.0275514i	$-0.00877101\pi$
$138$	0	0
$139$	−10.0351	−0.851165	−0.425582	−	0.904920i	$-0.639931\pi$
$139$	−10.0351	−0.851165	−0.425582	+	0.904920i	$0.639931\pi$
$140$	0	0
$141$	3.22668	0.271736
$142$	0	0
$143$	− 3.75103i	− 0.313677i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	10.5175i	0.867472i
$148$	0	0
$149$	4.24628	0.347869	0.173934	−	0.984757i	$-0.444352\pi$
$149$	4.24628	0.347869	0.173934	+	0.984757i	$0.444352\pi$
$150$	0	0
$151$	−17.3378	−1.41093	−0.705465	−	0.708745i	$-0.749262\pi$
$151$	−17.3378	−1.41093	−0.705465	+	0.708745i	$0.749262\pi$
$152$	0	0
$153$	0.0641778i	0.00518847i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2.27126i	0.181266i	0.995884	+	0.0906331i	$0.0288891\pi$
$157$	2.27126i	0.181266i	−0.995884	+	0.0906331i	$0.971111\pi$
$158$	0	0
$159$	−12.2490	−0.971407
$160$	0	0
$161$	9.46110	0.745639
$162$	0	0
$163$	0.0418891i	0.00328100i	0.999999	+	0.00164050i	$0.000522188\pi$
$163$	0.0418891i	0.00328100i	−0.999999	+	0.00164050i	$0.999478\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 0.246282i	− 0.0190579i	−0.999955	−	0.00952893i	$-0.996967\pi$
$167$	− 0.246282i	− 0.0190579i	0.999955	−	0.00952893i	$-0.00303320\pi$
$168$	0	0
$169$	10.0523	0.773255
$170$	0	0
$171$	0.532089	0.0406899
$172$	0	0
$173$	− 9.50980i	− 0.723017i	−0.932369	−	0.361508i	$-0.882262\pi$
$173$	− 9.50980i	− 0.723017i	0.932369	−	0.361508i	$-0.117738\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 19.1925i	− 1.44260i
$178$	0	0
$179$	2.10338	0.157214	0.0786069	−	0.996906i	$-0.474953\pi$
$179$	2.10338	0.157214	0.0786069	+	0.996906i	$0.474953\pi$
$180$	0	0
$181$	−7.86753	−0.584789	−0.292394	−	0.956298i	$-0.594452\pi$
$181$	−7.86753	−0.584789	−0.292394	+	0.956298i	$0.594452\pi$
$182$	0	0
$183$	12.2909i	0.908566i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0.263518i	0.0192704i
$188$	0	0
$189$	−5.49525	−0.399721
$190$	0	0
$191$	−20.0719	−1.45235	−0.726177	−	0.687508i	$-0.758704\pi$
$191$	−20.0719	−1.45235	−0.726177	+	0.687508i	$0.758704\pi$
$192$	0	0
$193$	15.6800i	1.12867i	0.825544	+	0.564337i	$0.190868\pi$
$193$	15.6800i	1.12867i	−0.825544	+	0.564337i	$0.809132\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 9.65776i	− 0.688087i	−0.938954	−	0.344043i	$-0.888203\pi$
$197$	− 9.65776i	− 0.688087i	0.938954	−	0.344043i	$-0.111797\pi$
$198$	0	0
$199$	28.1712	1.99700	0.998501	−	0.0547346i	$-0.0174313\pi$
$199$	28.1712	1.99700	0.998501	+	0.0547346i	$0.0174313\pi$
$200$	0	0
$201$	4.10607	0.289620
$202$	0	0
$203$	− 3.84936i	− 0.270172i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	4.24897i	0.295324i
$208$	0	0
$209$	2.18479	0.151125
$210$	0	0
$211$	−2.17705	−0.149874	−0.0749372	−	0.997188i	$-0.523876\pi$
$211$	−2.17705	−0.149874	−0.0749372	+	0.997188i	$0.523876\pi$
$212$	0	0
$213$	− 17.1557i	− 1.17549i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 9.96585i	− 0.676526i
$218$	0	0
$219$	1.45336	0.0982092
$220$	0	0
$221$	0.207081	0.0139298
$222$	0	0
$223$	12.2371i	0.819458i	0.912207	+	0.409729i	$0.134377\pi$
$223$	12.2371i	0.819458i	−0.912207	+	0.409729i	$0.865623\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 24.5476i	− 1.62928i	−0.579967	−	0.814640i	$-0.696935\pi$
$227$	− 24.5476i	− 1.62928i	0.579967	−	0.814640i	$-0.303065\pi$
$228$	0	0
$229$	16.1061	1.06432	0.532159	−	0.846644i	$-0.321381\pi$
$229$	16.1061	1.06432	0.532159	+	0.846644i	$0.321381\pi$
$230$	0	0
$231$	4.86484	0.320083
$232$	0	0
$233$	− 7.46110i	− 0.488793i	−0.969675	−	0.244397i	$-0.921410\pi$
$233$	− 7.46110i	− 0.488793i	0.969675	−	0.244397i	$-0.0785899\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 3.07873i	− 0.199985i
$238$	0	0
$239$	−14.1702	−0.916597	−0.458298	−	0.888798i	$-0.651541\pi$
$239$	−14.1702	−0.916597	−0.458298	+	0.888798i	$0.651541\pi$
$240$	0	0
$241$	−4.40373	−0.283669	−0.141835	−	0.989890i	$-0.545300\pi$
$241$	−4.40373	−0.283669	−0.141835	+	0.989890i	$0.545300\pi$
$242$	0	0
$243$	− 5.46791i	− 0.350767i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 1.71688i	− 0.109243i
$248$	0	0
$249$	4.60132	0.291597
$250$	0	0
$251$	13.7520	0.868016	0.434008	−	0.900909i	$-0.357099\pi$
$251$	13.7520	0.868016	0.434008	+	0.900909i	$0.357099\pi$
$252$	0	0
$253$	17.4466i	1.09686i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 21.0719i	− 1.31443i	−0.753703	−	0.657215i	$-0.771734\pi$
$257$	− 21.0719i	− 1.31443i	0.753703	−	0.657215i	$-0.228266\pi$
$258$	0	0
$259$	−3.94862	−0.245355
$260$	0	0
$261$	1.72874	0.107006
$262$	0	0
$263$	4.66819i	0.287853i	0.989588	+	0.143926i	$0.0459728\pi$
$263$	4.66819i	0.287853i	−0.989588	+	0.143926i	$0.954027\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 5.33275i	− 0.326359i
$268$	0	0
$269$	−7.08647	−0.432069	−0.216035	−	0.976386i	$-0.569312\pi$
$269$	−7.08647	−0.432069	−0.216035	+	0.976386i	$0.569312\pi$
$270$	0	0
$271$	6.28405	0.381729	0.190864	−	0.981616i	$-0.438871\pi$
$271$	6.28405	0.381729	0.190864	+	0.981616i	$0.438871\pi$
$272$	0	0
$273$	− 3.82295i	− 0.231375i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 17.8675i	− 1.07356i	−0.843724	−	0.536778i	$-0.819641\pi$
$277$	− 17.8675i	− 1.07356i	0.843724	−	0.536778i	$-0.180359\pi$
$278$	0	0
$279$	4.47565	0.267950
$280$	0	0
$281$	−8.36959	−0.499288	−0.249644	−	0.968338i	$-0.580314\pi$
$281$	−8.36959	−0.499288	−0.249644	+	0.968338i	$0.580314\pi$
$282$	0	0
$283$	7.19759i	0.427852i	0.976850	+	0.213926i	$0.0686252\pi$
$283$	7.19759i	0.427852i	−0.976850	+	0.213926i	$0.931375\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 10.6459i	− 0.628407i
$288$	0	0
$289$	16.9855	0.999144
$290$	0	0
$291$	4.13341	0.242305
$292$	0	0
$293$	4.39424i	0.256714i	0.991728	+	0.128357i	$0.0409704\pi$
$293$	4.39424i	0.256714i	−0.991728	+	0.128357i	$0.959030\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 10.1334i	− 0.588000i
$298$	0	0
$299$	13.7101	0.792874
$300$	0	0
$301$	4.81521	0.277544
$302$	0	0
$303$	− 19.4192i	− 1.11560i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	1.19160i	0.0680082i	0.999422	+	0.0340041i	$0.0108259\pi$
$307$	1.19160i	0.0680082i	−0.999422	+	0.0340041i	$0.989174\pi$
$308$	0	0
$309$	32.9368	1.87371
$310$	0	0
$311$	−12.5844	−0.713596	−0.356798	−	0.934182i	$-0.616132\pi$
$311$	−12.5844	−0.713596	−0.356798	+	0.934182i	$0.616132\pi$
$312$	0	0
$313$	4.34224i	0.245438i	0.992441	+	0.122719i	$0.0391614\pi$
$313$	4.34224i	0.245438i	−0.992441	+	0.122719i	$0.960839\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 14.7050i	− 0.825916i	−0.910750	−	0.412958i	$-0.864496\pi$
$317$	− 14.7050i	− 0.825916i	0.910750	−	0.412958i	$-0.135504\pi$
$318$	0	0
$319$	7.09833	0.397430
$320$	0	0
$321$	−5.94356	−0.331737
$322$	0	0
$323$	0.120615i	0.00671118i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	16.4884i	0.911813i
$328$	0	0
$329$	2.03415	0.112146
$330$	0	0
$331$	−32.1215	−1.76556	−0.882780	−	0.469787i	$-0.844331\pi$
$331$	−32.1215	−1.76556	−0.882780	+	0.469787i	$0.844331\pi$
$332$	0	0
$333$	− 1.77332i	− 0.0971772i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 9.27900i	− 0.505459i	−0.967537	−	0.252730i	$-0.918672\pi$
$337$	− 9.27900i	− 0.505459i	0.967537	−	0.252730i	$-0.0813283\pi$
$338$	0	0
$339$	−28.2148	−1.53242
$340$	0	0
$341$	18.3773	0.995188
$342$	0	0
$343$	14.9240i	0.805818i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	33.3928i	1.79262i	0.443428	+	0.896310i	$0.353762\pi$
$347$	33.3928i	1.79262i	−0.443428	+	0.896310i	$0.646238\pi$
$348$	0	0
$349$	−4.31820	−0.231148	−0.115574	−	0.993299i	$-0.536871\pi$
$349$	−4.31820	−0.231148	−0.115574	+	0.993299i	$0.536871\pi$
$350$	0	0
$351$	−7.96316	−0.425042
$352$	0	0
$353$	− 9.82026i	− 0.522680i	−0.965247	−	0.261340i	$-0.915836\pi$
$353$	− 9.82026i	− 0.522680i	0.965247	−	0.261340i	$-0.0841643\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0.268571i	0.0142143i
$358$	0	0
$359$	−14.7365	−0.777762	−0.388881	−	0.921288i	$-0.627138\pi$
$359$	−14.7365	−0.777762	−0.388881	+	0.921288i	$0.627138\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	− 11.7023i	− 0.614213i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	18.9017i	0.986659i	0.869842	+	0.493330i	$0.164220\pi$
$367$	18.9017i	0.986659i	−0.869842	+	0.493330i	$0.835780\pi$
$368$	0	0
$369$	4.78106	0.248892
$370$	0	0
$371$	−7.72193	−0.400903
$372$	0	0
$373$	23.7033i	1.22731i	0.789575	+	0.613654i	$0.210301\pi$
$373$	23.7033i	1.22731i	−0.789575	+	0.613654i	$0.789699\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 5.57810i	− 0.287287i
$378$	0	0
$379$	−15.1061	−0.775947	−0.387973	−	0.921671i	$-0.626825\pi$
$379$	−15.1061	−0.775947	−0.387973	+	0.921671i	$0.626825\pi$
$380$	0	0
$381$	11.9436	0.611887
$382$	0	0
$383$	− 35.5357i	− 1.81579i	−0.419198	−	0.907895i	$-0.637689\pi$
$383$	− 35.5357i	− 1.81579i	0.419198	−	0.907895i	$-0.362311\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2.16250i	0.109926i
$388$	0	0
$389$	−0.0641778	−0.00325394	−0.00162697	−	0.999999i	$-0.500518\pi$
$389$	−0.0641778	−0.00325394	−0.00162697	+	0.999999i	$0.500518\pi$
$390$	0	0
$391$	−0.963163	−0.0487093
$392$	0	0
$393$	15.8229i	0.798162i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 31.9195i	− 1.60199i	−0.598668	−	0.800997i	$-0.704303\pi$
$397$	− 31.9195i	− 1.60199i	0.598668	−	0.800997i	$-0.295697\pi$
$398$	0	0
$399$	2.22668	0.111474
$400$	0	0
$401$	25.7374	1.28527	0.642633	−	0.766174i	$-0.277842\pi$
$401$	25.7374	1.28527	0.642633	+	0.766174i	$0.277842\pi$
$402$	0	0
$403$	− 14.4415i	− 0.719383i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 7.28136i	− 0.360924i
$408$	0	0
$409$	−29.5631	−1.46180	−0.730899	−	0.682485i	$-0.760899\pi$
$409$	−29.5631	−1.46180	−0.730899	+	0.682485i	$0.760899\pi$
$410$	0	0
$411$	1.21213	0.0597901
$412$	0	0
$413$	− 12.0993i	− 0.595366i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 18.8598i	− 0.923568i
$418$	0	0
$419$	16.8452	0.822944	0.411472	−	0.911422i	$-0.365015\pi$
$419$	16.8452	0.822944	0.411472	+	0.911422i	$0.365015\pi$
$420$	0	0
$421$	−1.76146	−0.0858483	−0.0429241	−	0.999078i	$-0.513667\pi$
$421$	−1.76146	−0.0858483	−0.0429241	+	0.999078i	$0.513667\pi$
$422$	0	0
$423$	0.913534i	0.0444175i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	7.74834i	0.374969i
$428$	0	0
$429$	7.04963	0.340359
$430$	0	0
$431$	−13.4766	−0.649144	−0.324572	−	0.945861i	$-0.605220\pi$
$431$	−13.4766	−0.649144	−0.324572	+	0.945861i	$0.605220\pi$
$432$	0	0
$433$	21.4037i	1.02860i	0.857611	+	0.514299i	$0.171948\pi$
$433$	21.4037i	1.02860i	−0.857611	+	0.514299i	$0.828052\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7.98545i	0.381996i
$438$	0	0
$439$	16.9786	0.810347	0.405173	−	0.914240i	$-0.367211\pi$
$439$	16.9786	0.810347	0.405173	+	0.914240i	$0.367211\pi$
$440$	0	0
$441$	−2.97771	−0.141796
$442$	0	0
$443$	26.5921i	1.26343i	0.775200	+	0.631716i	$0.217649\pi$
$443$	26.5921i	1.26343i	−0.775200	+	0.631716i	$0.782351\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	7.98040i	0.377460i
$448$	0	0
$449$	−25.7425	−1.21486	−0.607431	−	0.794372i	$-0.707800\pi$
$449$	−25.7425	−1.21486	−0.607431	+	0.794372i	$0.707800\pi$
$450$	0	0
$451$	19.6313	0.924404
$452$	0	0
$453$	− 32.5844i	− 1.53095i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	32.4715i	1.51895i	0.650534	+	0.759477i	$0.274545\pi$
$457$	32.4715i	1.51895i	−0.650534	+	0.759477i	$0.725455\pi$
$458$	0	0
$459$	0.559430	0.0261120
$460$	0	0
$461$	11.4115	0.531485	0.265743	−	0.964044i	$-0.414383\pi$
$461$	11.4115	0.531485	0.265743	+	0.964044i	$0.414383\pi$
$462$	0	0
$463$	8.59863i	0.399612i	0.979835	+	0.199806i	$0.0640312\pi$
$463$	8.59863i	0.399612i	−0.979835	+	0.199806i	$0.935969\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	26.4056i	1.22191i	0.791667	+	0.610953i	$0.209213\pi$
$467$	26.4056i	1.22191i	−0.791667	+	0.610953i	$0.790787\pi$
$468$	0	0
$469$	2.58853	0.119527
$470$	0	0
$471$	−4.26857	−0.196685
$472$	0	0
$473$	8.87939i	0.408275i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 3.46791i	− 0.158785i
$478$	0	0
$479$	19.2472	0.879428	0.439714	−	0.898138i	$-0.355080\pi$
$479$	19.2472	0.879428	0.439714	+	0.898138i	$0.355080\pi$
$480$	0	0
$481$	−5.72193	−0.260898
$482$	0	0
$483$	17.7811i	0.809066i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 9.48339i	− 0.429734i	−0.976643	−	0.214867i	$-0.931068\pi$
$487$	− 9.48339i	− 0.429734i	0.976643	−	0.214867i	$-0.0689317\pi$
$488$	0	0
$489$	−0.0787257	−0.00356010
$490$	0	0
$491$	−18.2618	−0.824142	−0.412071	−	0.911152i	$-0.635194\pi$
$491$	−18.2618	−0.824142	−0.412071	+	0.911152i	$0.635194\pi$
$492$	0	0
$493$	0.391874i	0.0176491i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 10.8152i	− 0.485128i
$498$	0	0
$499$	10.9828	0.491656	0.245828	−	0.969313i	$-0.420940\pi$
$499$	10.9828	0.491656	0.245828	+	0.969313i	$0.420940\pi$
$500$	0	0
$501$	0.462859	0.0206790
$502$	0	0
$503$	− 18.9590i	− 0.845342i	−0.906283	−	0.422671i	$-0.861093\pi$
$503$	− 18.9590i	− 0.845342i	0.906283	−	0.422671i	$-0.138907\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	18.8922i	0.839031i
$508$	0	0
$509$	−12.6705	−0.561612	−0.280806	−	0.959765i	$-0.590602\pi$
$509$	−12.6705	−0.561612	−0.280806	+	0.959765i	$0.590602\pi$
$510$	0	0
$511$	0.916222	0.0405313
$512$	0	0
$513$	− 4.63816i	− 0.204780i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	3.75103i	0.164970i
$518$	0	0
$519$	17.8726	0.784519
$520$	0	0
$521$	41.3756	1.81270	0.906348	−	0.422531i	$-0.138858\pi$
$521$	41.3756	1.81270	0.906348	+	0.422531i	$0.138858\pi$
$522$	0	0
$523$	35.2104i	1.53964i	0.638260	+	0.769821i	$0.279655\pi$
$523$	35.2104i	1.53964i	−0.638260	+	0.769821i	$0.720345\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.01455i	0.0441944i
$528$	0	0
$529$	−40.7674	−1.77250
$530$	0	0
$531$	5.43376	0.235805
$532$	0	0
$533$	− 15.4270i	− 0.668216i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	3.95306i	0.170587i
$538$	0	0
$539$	−12.2267	−0.526640
$540$	0	0
$541$	25.8307	1.11055	0.555274	−	0.831667i	$-0.312613\pi$
$541$	25.8307	1.11055	0.555274	+	0.831667i	$0.312613\pi$
$542$	0	0
$543$	− 14.7861i	− 0.634533i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	28.6878i	1.22660i	0.789850	+	0.613301i	$0.210159\pi$
$547$	28.6878i	1.22660i	−0.789850	+	0.613301i	$0.789841\pi$
$548$	0	0
$549$	−3.47977	−0.148513
$550$	0	0
$551$	3.24897	0.138411
$552$	0	0
$553$	− 1.94087i	− 0.0825344i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	7.50980i	0.318200i	0.987262	+	0.159100i	$0.0508593\pi$
$557$	7.50980i	0.318200i	−0.987262	+	0.159100i	$0.949141\pi$
$558$	0	0
$559$	6.97771	0.295126
$560$	0	0
$561$	−0.495252	−0.0209096
$562$	0	0
$563$	29.6287i	1.24870i	0.781145	+	0.624350i	$0.214636\pi$
$563$	29.6287i	1.24870i	−0.781145	+	0.624350i	$0.785364\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 12.2189i	− 0.513147i
$568$	0	0
$569$	−9.34461	−0.391746	−0.195873	−	0.980629i	$-0.562754\pi$
$569$	−9.34461	−0.391746	−0.195873	+	0.980629i	$0.562754\pi$
$570$	0	0
$571$	27.1239	1.13510	0.567550	−	0.823339i	$-0.307891\pi$
$571$	27.1239	1.13510	0.567550	+	0.823339i	$0.307891\pi$
$572$	0	0
$573$	− 37.7229i	− 1.57590i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	11.7273i	0.488214i	0.969748	+	0.244107i	$0.0784949\pi$
$577$	11.7273i	0.488214i	−0.969748	+	0.244107i	$0.921505\pi$
$578$	0	0
$579$	−29.4688	−1.22468
$580$	0	0
$581$	2.90074	0.120343
$582$	0	0
$583$	− 14.2395i	− 0.589739i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	7.37195i	0.304273i	0.988359	+	0.152136i	$0.0486153\pi$
$587$	7.37195i	0.304273i	−0.988359	+	0.152136i	$0.951385\pi$
$588$	0	0
$589$	8.41147	0.346589
$590$	0	0
$591$	18.1506	0.746618
$592$	0	0
$593$	22.2831i	0.915058i	0.889195	+	0.457529i	$0.151265\pi$
$593$	22.2831i	0.915058i	−0.889195	+	0.457529i	$0.848735\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	52.9445i	2.16687i
$598$	0	0
$599$	−28.7766	−1.17578	−0.587890	−	0.808941i	$-0.700041\pi$
$599$	−28.7766	−1.17578	−0.587890	+	0.808941i	$0.700041\pi$
$600$	0	0
$601$	−36.2719	−1.47956	−0.739780	−	0.672849i	$-0.765071\pi$
$601$	−36.2719	−1.47956	−0.739780	+	0.672849i	$0.765071\pi$
$602$	0	0
$603$	1.16250i	0.0473408i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 11.2867i	− 0.458115i	−0.973413	−	0.229057i	$-0.926436\pi$
$607$	− 11.2867i	− 0.458115i	0.973413	−	0.229057i	$-0.0735643\pi$
$608$	0	0
$609$	7.23442	0.293154
$610$	0	0
$611$	2.94768	0.119250
$612$	0	0
$613$	− 1.45067i	− 0.0585922i	−0.999571	−	0.0292961i	$-0.990673\pi$
$613$	− 1.45067i	− 0.0585922i	0.999571	−	0.0292961i	$-0.00932657\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 1.65951i	− 0.0668094i	−0.999442	−	0.0334047i	$-0.989365\pi$
$617$	− 1.65951i	− 0.0668094i	0.999442	−	0.0334047i	$-0.0106350\pi$
$618$	0	0
$619$	−9.66456	−0.388452	−0.194226	−	0.980957i	$-0.562219\pi$
$619$	−9.66456	−0.388452	−0.194226	+	0.980957i	$0.562219\pi$
$620$	0	0
$621$	37.0378	1.48627
$622$	0	0
$623$	− 3.36184i	− 0.134689i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	4.10607i	0.163981i
$628$	0	0
$629$	0.401979	0.0160279
$630$	0	0
$631$	−16.0205	−0.637767	−0.318884	−	0.947794i	$-0.603308\pi$
$631$	−16.0205	−0.637767	−0.318884	+	0.947794i	$0.603308\pi$
$632$	0	0
$633$	− 4.09152i	− 0.162623i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	9.60813i	0.380688i
$638$	0	0
$639$	4.85710	0.192144
$640$	0	0
$641$	−39.8093	−1.57237	−0.786187	−	0.617989i	$-0.787948\pi$
$641$	−39.8093	−1.57237	−0.786187	+	0.617989i	$0.787948\pi$
$642$	0	0
$643$	− 0.295912i	− 0.0116696i	−0.999983	−	0.00583481i	$-0.998143\pi$
$643$	− 0.295912i	− 0.0116696i	0.999983	−	0.00583481i	$-0.00185729\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	44.3651i	1.74417i	0.489351	+	0.872087i	$0.337234\pi$
$647$	44.3651i	1.74417i	−0.489351	+	0.872087i	$0.662766\pi$
$648$	0	0
$649$	22.3114	0.875799
$650$	0	0
$651$	18.7297	0.734074
$652$	0	0
$653$	38.7110i	1.51488i	0.652905	+	0.757439i	$0.273550\pi$
$653$	38.7110i	1.51488i	−0.652905	+	0.757439i	$0.726450\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0.411474i	0.0160531i
$658$	0	0
$659$	9.17562	0.357431	0.178716	−	0.983901i	$-0.442806\pi$
$659$	9.17562	0.357431	0.178716	+	0.983901i	$0.442806\pi$
$660$	0	0
$661$	−5.04727	−0.196316	−0.0981579	−	0.995171i	$-0.531295\pi$
$661$	−5.04727	−0.196316	−0.0981579	+	0.995171i	$0.531295\pi$
$662$	0	0
$663$	0.389185i	0.0151147i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	25.9445i	1.00457i
$668$	0	0
$669$	−22.9982	−0.889164
$670$	0	0
$671$	−14.2882	−0.551589
$672$	0	0
$673$	− 6.24216i	− 0.240618i	−0.992737	−	0.120309i	$-0.961612\pi$
$673$	− 6.24216i	− 0.240618i	0.992737	−	0.120309i	$-0.0383885\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	13.2044i	0.507486i	0.967272	+	0.253743i	$0.0816618\pi$
$677$	13.2044i	0.507486i	−0.967272	+	0.253743i	$0.918338\pi$
$678$	0	0
$679$	2.60576	0.100000
$680$	0	0
$681$	46.1343	1.76787
$682$	0	0
$683$	41.3988i	1.58408i	0.610469	+	0.792040i	$0.290981\pi$
$683$	41.3988i	1.58408i	−0.610469	+	0.792040i	$0.709019\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	30.2695i	1.15485i
$688$	0	0
$689$	−11.1898	−0.426299
$690$	0	0
$691$	−18.1985	−0.692304	−0.346152	−	0.938178i	$-0.612512\pi$
$691$	−18.1985	−0.692304	−0.346152	+	0.938178i	$0.612512\pi$
$692$	0	0
$693$	1.37733i	0.0523203i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1.08378i	0.0410510i
$698$	0	0
$699$	14.0223	0.530372
$700$	0	0
$701$	−23.7383	−0.896585	−0.448293	−	0.893887i	$-0.647968\pi$
$701$	−23.7383	−0.896585	−0.448293	+	0.893887i	$0.647968\pi$
$702$	0	0
$703$	− 3.33275i	− 0.125697i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 12.2422i	− 0.460414i
$708$	0	0
$709$	11.9426	0.448515	0.224257	−	0.974530i	$-0.428004\pi$
$709$	11.9426	0.448515	0.224257	+	0.974530i	$0.428004\pi$
$710$	0	0
$711$	0.871644	0.0326892
$712$	0	0
$713$	67.1694i	2.51551i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 26.6313i	− 0.994566i
$718$	0	0
$719$	−13.1352	−0.489859	−0.244929	−	0.969541i	$-0.578765\pi$
$719$	−13.1352	−0.489859	−0.244929	+	0.969541i	$0.578765\pi$
$720$	0	0
$721$	20.7638	0.773285
$722$	0	0
$723$	− 8.27631i	− 0.307799i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	13.5253i	0.501625i	0.968036	+	0.250812i	$0.0806977\pi$
$727$	13.5253i	0.501625i	−0.968036	+	0.250812i	$0.919302\pi$
$728$	0	0
$729$	−20.6631	−0.765301
$730$	0	0
$731$	−0.490200	−0.0181307
$732$	0	0
$733$	− 49.7256i	− 1.83666i	−0.395821	−	0.918328i	$-0.629540\pi$
$733$	− 49.7256i	− 1.83666i	0.395821	−	0.918328i	$-0.370460\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.77332i	0.175827i
$738$	0	0
$739$	−39.7282	−1.46143	−0.730714	−	0.682684i	$-0.760812\pi$
$739$	−39.7282	−1.46143	−0.730714	+	0.682684i	$0.760812\pi$
$740$	0	0
$741$	3.22668	0.118535
$742$	0	0
$743$	− 38.6955i	− 1.41960i	−0.704403	−	0.709801i	$-0.748785\pi$
$743$	− 38.6955i	− 1.41960i	0.704403	−	0.709801i	$-0.251215\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	1.30272i	0.0476640i
$748$	0	0
$749$	−3.74691	−0.136909
$750$	0	0
$751$	−53.1353	−1.93893	−0.969467	−	0.245222i	$-0.921139\pi$
$751$	−53.1353	−1.93893	−0.969467	+	0.245222i	$0.921139\pi$
$752$	0	0
$753$	25.8452i	0.941853i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	30.3259i	1.10222i	0.834434	+	0.551108i	$0.185795\pi$
$757$	30.3259i	1.10222i	−0.834434	+	0.551108i	$0.814205\pi$
$758$	0	0
$759$	−32.7888	−1.19016
$760$	0	0
$761$	−37.4935	−1.35914	−0.679569	−	0.733611i	$-0.737833\pi$
$761$	−37.4935	−1.35914	−0.679569	+	0.733611i	$0.737833\pi$
$762$	0	0
$763$	10.3946i	0.376308i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 17.5330i	− 0.633081i
$768$	0	0
$769$	−9.36009	−0.337533	−0.168767	−	0.985656i	$-0.553978\pi$
$769$	−9.36009	−0.337533	−0.168767	+	0.985656i	$0.553978\pi$
$770$	0	0
$771$	39.6023	1.42624
$772$	0	0
$773$	− 35.8881i	− 1.29080i	−0.763843	−	0.645402i	$-0.776690\pi$
$773$	− 35.8881i	− 1.29080i	0.763843	−	0.645402i	$-0.223310\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 7.42097i	− 0.266226i
$778$	0	0
$779$	8.98545	0.321937
$780$	0	0
$781$	19.9436	0.713637
$782$	0	0
$783$	− 15.0692i	− 0.538530i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	0.781059i	0.0278418i	0.999903	+	0.0139209i	$0.00443130\pi$
$787$	0.781059i	0.0278418i	−0.999903	+	0.0139209i	$0.995569\pi$
$788$	0	0
$789$	−8.77332	−0.312338
$790$	0	0
$791$	−17.7870	−0.632435
$792$	0	0
$793$	11.2281i	0.398722i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0.0240434i	0 0.000851661i	1.00000		0.000425830i	$0.000135546\pi$
$797$	0.0240434i	0 0.000851661i	−1.00000		0.000425830i	$0.999864\pi$
$798$	0	0
$799$	−0.207081	−0.00732601
$800$	0	0
$801$	1.50980	0.0533462
$802$	0	0
$803$	1.68954i	0.0596226i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 13.3182i	− 0.468823i
$808$	0	0
$809$	−39.3919	−1.38494	−0.692472	−	0.721444i	$-0.743479\pi$
$809$	−39.3919	−1.38494	−0.692472	+	0.721444i	$0.743479\pi$
$810$	0	0
$811$	11.4216	0.401066	0.200533	−	0.979687i	$-0.435733\pi$
$811$	11.4216	0.401066	0.200533	+	0.979687i	$0.435733\pi$
$812$	0	0
$813$	11.8102i	0.414200i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	4.06418i	0.142188i
$818$	0	0
$819$	1.08235	0.0378203
$820$	0	0
$821$	53.9778	1.88384	0.941920	−	0.335839i	$-0.109020\pi$
$821$	53.9778	1.88384	0.941920	+	0.335839i	$0.109020\pi$
$822$	0	0
$823$	33.4216i	1.16500i	0.812830	+	0.582502i	$0.197926\pi$
$823$	33.4216i	1.16500i	−0.812830	+	0.582502i	$0.802074\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 18.0716i	− 0.628411i	−0.949355	−	0.314205i	$-0.898262\pi$
$827$	− 18.0716i	− 0.628411i	0.949355	−	0.314205i	$-0.101738\pi$
$828$	0	0
$829$	47.0642	1.63461	0.817303	−	0.576208i	$-0.195468\pi$
$829$	47.0642	1.63461	0.817303	+	0.576208i	$0.195468\pi$
$830$	0	0
$831$	33.5800	1.16488
$832$	0	0
$833$	− 0.674992i	− 0.0233871i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 39.0137i	− 1.34851i
$838$	0	0
$839$	10.1102	0.349042	0.174521	−	0.984653i	$-0.444162\pi$
$839$	10.1102	0.349042	0.174521	+	0.984653i	$0.444162\pi$
$840$	0	0
$841$	−18.4442	−0.636007
$842$	0	0
$843$	− 15.7297i	− 0.541759i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 7.37733i	− 0.253488i
$848$	0	0
$849$	−13.5270	−0.464247
$850$	0	0
$851$	26.6135	0.912299
$852$	0	0
$853$	13.1908i	0.451644i	0.974169	+	0.225822i	$0.0725067\pi$
$853$	13.1908i	0.451644i	−0.974169	+	0.225822i	$0.927493\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 27.7419i	− 0.947644i	−0.880621	−	0.473822i	$-0.842874\pi$
$857$	− 27.7419i	− 0.947644i	0.880621	−	0.473822i	$-0.157126\pi$
$858$	0	0
$859$	14.3301	0.488935	0.244468	−	0.969657i	$-0.421387\pi$
$859$	14.3301	0.488935	0.244468	+	0.969657i	$0.421387\pi$
$860$	0	0
$861$	20.0077	0.681862
$862$	0	0
$863$	5.05819i	0.172183i	0.996287	+	0.0860914i	$0.0274377\pi$
$863$	5.05819i	0.172183i	−0.996287	+	0.0860914i	$0.972562\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	31.9222i	1.08414i
$868$	0	0
$869$	3.57903	0.121410
$870$	0	0
$871$	3.75103	0.127099
$872$	0	0
$873$	1.17024i	0.0396068i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	35.0351i	1.18305i	0.806286	+	0.591525i	$0.201474\pi$
$877$	35.0351i	1.18305i	−0.806286	+	0.591525i	$0.798526\pi$
$878$	0	0
$879$	−8.25847	−0.278551
$880$	0	0
$881$	3.36926	0.113513	0.0567566	−	0.998388i	$-0.481924\pi$
$881$	3.36926	0.113513	0.0567566	+	0.998388i	$0.481924\pi$
$882$	0	0
$883$	36.9436i	1.24325i	0.783315	+	0.621625i	$0.213527\pi$
$883$	36.9436i	1.24325i	−0.783315	+	0.621625i	$0.786473\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	4.05232i	0.136064i	0.997683	+	0.0680318i	$0.0216719\pi$
$887$	4.05232i	0.136064i	−0.997683	+	0.0680318i	$0.978328\pi$
$888$	0	0
$889$	7.52940	0.252528
$890$	0	0
$891$	22.5321	0.754853
$892$	0	0
$893$	1.71688i	0.0574532i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	25.7665i	0.860319i
$898$	0	0
$899$	27.3286	0.911461
$900$	0	0
$901$	0.786112	0.0261892
$902$	0	0
$903$	9.04963i	0.301153i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	38.3550i	1.27356i	0.771046	+	0.636779i	$0.219734\pi$
$907$	38.3550i	1.27356i	−0.771046	+	0.636779i	$0.780266\pi$
$908$	0	0
$909$	5.49794	0.182355
$910$	0	0
$911$	3.98721	0.132102	0.0660510	−	0.997816i	$-0.478960\pi$
$911$	3.98721	0.132102	0.0660510	+	0.997816i	$0.478960\pi$
$912$	0	0
$913$	5.34905i	0.177028i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	9.97502i	0.329404i
$918$	0	0
$919$	10.1453	0.334661	0.167331	−	0.985901i	$-0.446485\pi$
$919$	10.1453	0.334661	0.167331	+	0.985901i	$0.446485\pi$
$920$	0	0
$921$	−2.23947	−0.0737932
$922$	0	0
$923$	− 15.6723i	− 0.515860i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	9.32501i	0.306273i
$928$	0	0
$929$	−51.2267	−1.68069	−0.840346	−	0.542050i	$-0.817648\pi$
$929$	−51.2267	−1.68069	−0.840346	+	0.542050i	$0.817648\pi$
$930$	0	0
$931$	−5.59627	−0.183410
$932$	0	0
$933$	− 23.6509i	− 0.774297i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	38.7469i	1.26581i	0.774231	+	0.632903i	$0.218137\pi$
$937$	38.7469i	1.26581i	−0.774231	+	0.632903i	$0.781863\pi$
$938$	0	0
$939$	−8.16075	−0.266316
$940$	0	0
$941$	4.90167	0.159790	0.0798950	−	0.996803i	$-0.474541\pi$
$941$	4.90167	0.159790	0.0798950	+	0.996803i	$0.474541\pi$
$942$	0	0
$943$	71.7529i	2.33660i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 5.00505i	− 0.162642i	−0.996688	−	0.0813212i	$-0.974086\pi$
$947$	− 5.00505i	− 0.162642i	0.996688	−	0.0813212i	$-0.0259139\pi$
$948$	0	0
$949$	1.32770	0.0430988
$950$	0	0
$951$	27.6364	0.896172
$952$	0	0
$953$	60.1489i	1.94841i	0.225657	+	0.974207i	$0.427547\pi$
$953$	60.1489i	1.94841i	−0.225657	+	0.974207i	$0.572453\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	13.3405i	0.431237i
$958$	0	0
$959$	0.764147	0.0246756
$960$	0	0
$961$	39.7529	1.28235
$962$	0	0
$963$	− 1.68273i	− 0.0542253i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 29.8648i	− 0.960388i	−0.877162	−	0.480194i	$-0.840566\pi$
$967$	− 29.8648i	− 0.960388i	0.877162	−	0.480194i	$-0.159434\pi$
$968$	0	0
$969$	−0.226682	−0.00728206
$970$	0	0
$971$	37.9709	1.21854	0.609272	−	0.792961i	$-0.291462\pi$
$971$	37.9709	1.21854	0.609272	+	0.792961i	$0.291462\pi$
$972$	0	0
$973$	− 11.8895i	− 0.381160i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 13.0223i	− 0.416620i	−0.978063	−	0.208310i	$-0.933204\pi$
$977$	− 13.0223i	− 0.416620i	0.978063	−	0.208310i	$-0.0667963\pi$
$978$	0	0
$979$	6.19934	0.198132
$980$	0	0
$981$	−4.66819	−0.149044
$982$	0	0
$983$	− 5.86215i	− 0.186974i	−0.995621	−	0.0934868i	$-0.970199\pi$
$983$	− 5.86215i	− 0.186974i	0.995621	−	0.0934868i	$-0.0298013\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	3.82295i	0.121686i
$988$	0	0
$989$	−32.4543	−1.03199
$990$	0	0
$991$	−1.71925	−0.0546136	−0.0273068	−	0.999627i	$-0.508693\pi$
$991$	−1.71925	−0.0546136	−0.0273068	+	0.999627i	$0.508693\pi$
$992$	0	0
$993$	− 60.3688i	− 1.91574i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	55.4347i	1.75563i	0.478996	+	0.877817i	$0.341001\pi$
$997$	55.4347i	1.75563i	−0.478996	+	0.877817i	$0.658999\pi$
$998$	0	0
$999$	−15.4578	−0.489063

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.2.d.o.3649.6		6
5.2	odd	4		3800.2.a.s.1.3	✓	3
5.3	odd	4		3800.2.a.t.1.1	yes	3
5.4	even	2	inner	3800.2.d.o.3649.1		6
20.3	even	4		7600.2.a.bs.1.3		3
20.7	even	4		7600.2.a.br.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3800.2.a.s.1.3	✓	3	5.2	odd	4
3800.2.a.t.1.1	yes	3	5.3	odd	4
3800.2.d.o.3649.1		6	5.4	even	2	inner
3800.2.d.o.3649.6		6	1.1	even	1	trivial
7600.2.a.br.1.1		3	20.7	even	4
7600.2.a.bs.1.3		3	20.3	even	4

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:	\( N \)	\(=\)	\( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3800.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.87939i q^{3} +1.18479i q^{7} -0.532089 q^{9} +O(q^{10})\)	\(q+1.87939i q^{3} +1.18479i q^{7} -0.532089 q^{9} -2.18479 q^{11} +1.71688i q^{13} -0.120615i q^{17} -1.00000 q^{19} -2.22668 q^{21} -7.98545i q^{23} +4.63816i q^{27} -3.24897 q^{29} -8.41147 q^{31} -4.10607i q^{33} +3.33275i q^{37} -3.22668 q^{39} -8.98545 q^{41} -4.06418i q^{43} -1.71688i q^{47} +5.59627 q^{49} +0.226682 q^{51} +6.51754i q^{53} -1.87939i q^{57} -10.2121 q^{59} +6.53983 q^{61} -0.630415i q^{63} -2.18479i q^{67} +15.0077 q^{69} -9.12836 q^{71} -0.773318i q^{73} -2.58853i q^{77} -1.63816 q^{79} -10.3131 q^{81} -2.44831i q^{83} -6.10607i q^{87} -2.83750 q^{89} -2.03415 q^{91} -15.8084i q^{93} -2.19934i q^{97} +1.16250 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{9}+O(q^{10}) \) 6 * q + 6 * q^9	\( 6 q + 6 q^{9} - 6 q^{11} - 6 q^{19} + 6 q^{29} - 30 q^{31} - 6 q^{39} - 18 q^{41} + 6 q^{49} - 12 q^{51} - 12 q^{59} - 18 q^{61} + 42 q^{69} - 18 q^{71} + 24 q^{79} - 18 q^{81} - 12 q^{89} - 54 q^{91} + 12 q^{99}+O(q^{100}) \) 6 * q + 6 * q^9 - 6 * q^11 - 6 * q^19 + 6 * q^29 - 30 * q^31 - 6 * q^39 - 18 * q^41 + 6 * q^49 - 12 * q^51 - 12 * q^59 - 18 * q^61 + 42 * q^69 - 18 * q^71 + 24 * q^79 - 18 * q^81 - 12 * q^89 - 54 * q^91 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more