LMFDB - Embedded newform 196.6.a.k.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [196,6,Mod(1,196)] 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("196.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(196, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 196 = 2^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	196.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$31.4352286833$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{1177})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 591x^{2} + 592x + 85262 $ x^4 - 2x^3 - 591x^2 + 592*x + 85262
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 7^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-18.0679$ of defining polynomial
Character	$\chi$	$=$	196.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+29.2088 q^{3} -98.5052 q^{5} +610.152 q^{9} +129.152 q^{11} -126.140 q^{13} -2877.22 q^{15} +1614.04 q^{17} +1399.47 q^{19} +2773.22 q^{23} +6578.28 q^{25} +10724.1 q^{27} +3499.94 q^{29} +710.456 q^{31} +3772.37 q^{33} -3051.10 q^{37} -3684.39 q^{39} -15263.7 q^{41} +7880.58 q^{43} -60103.2 q^{45} +16787.7 q^{47} +47144.1 q^{51} -3230.13 q^{53} -12722.2 q^{55} +40876.7 q^{57} -12163.3 q^{59} -51007.4 q^{61} +12425.4 q^{65} +27232.3 q^{67} +81002.2 q^{69} -13313.9 q^{71} -49712.9 q^{73} +192143. q^{75} +21105.9 q^{79} +164970. q^{81} +79860.2 q^{83} -158991. q^{85} +102229. q^{87} -36846.5 q^{89} +20751.5 q^{93} -137855. q^{95} -27968.5 q^{97} +78802.4 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 1480 q^{9} - 444 q^{11} - 3824 q^{15} + 3408 q^{23} + 11904 q^{25} + 20724 q^{29} + 13732 q^{37} + 25608 q^{39} - 28996 q^{43} + 181852 q^{51} + 528 q^{53} + 89540 q^{57} + 110220 q^{65} + 195384 q^{67}+ \cdots + 66412 q^{99}+O(q^{100}) $ 4 * q + 1480 * q^9 - 444 * q^11 - 3824 * q^15 + 3408 * q^23 + 11904 * q^25 + 20724 * q^29 + 13732 * q^37 + 25608 * q^39 - 28996 * q^43 + 181852 * q^51 + 528 * q^53 + 89540 * q^57 + 110220 * q^65 + 195384 * q^67 + 21672 * q^71 - 117304 * q^79 + 182456 * q^81 - 91300 * q^85 + 250152 * q^93 - 255552 * q^95 + 66412 * q^99

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
$3$	29.2088	1.87374	0.936872	−	0.349674i	$-0.113707\pi$
$3$	29.2088	1.87374	0.936872	+	0.349674i	$0.113707\pi$
$4$	0	0
$5$	−98.5052	−1.76212	−0.881058	−	0.473009i	$-0.843168\pi$
$5$	−98.5052	−1.76212	−0.881058	+	0.473009i	$0.843168\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	610.152	2.51091
$10$	0	0
$11$	129.152	0.321825	0.160912	−	0.986969i	$-0.448556\pi$
$11$	129.152	0.321825	0.160912	+	0.986969i	$0.448556\pi$
$12$	0	0
$13$	−126.140	−0.207011	−0.103506	−	0.994629i	$-0.533006\pi$
$13$	−126.140	−0.207011	−0.103506	+	0.994629i	$0.533006\pi$
$14$	0	0
$15$	−2877.22	−3.30175
$16$	0	0
$17$	1614.04	1.35454	0.677269	−	0.735735i	$-0.263163\pi$
$17$	1614.04	1.35454	0.677269	+	0.735735i	$0.263163\pi$
$18$	0	0
$19$	1399.47	0.889362	0.444681	−	0.895689i	$-0.353317\pi$
$19$	1399.47	0.889362	0.444681	+	0.895689i	$0.353317\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2773.22	1.09311	0.546555	−	0.837423i	$-0.315939\pi$
$23$	2773.22	1.09311	0.546555	+	0.837423i	$0.315939\pi$
$24$	0	0
$25$	6578.28	2.10505
$26$	0	0
$27$	10724.1	2.83106
$28$	0	0
$29$	3499.94	0.772796	0.386398	−	0.922332i	$-0.373719\pi$
$29$	3499.94	0.772796	0.386398	+	0.922332i	$0.373719\pi$
$30$	0	0
$31$	710.456	0.132780	0.0663901	−	0.997794i	$-0.478852\pi$
$31$	710.456	0.132780	0.0663901	+	0.997794i	$0.478852\pi$
$32$	0	0
$33$	3772.37	0.603017
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−3051.10	−0.366398	−0.183199	−	0.983076i	$-0.558645\pi$
$37$	−3051.10	−0.366398	−0.183199	+	0.983076i	$0.558645\pi$
$38$	0	0
$39$	−3684.39	−0.387885
$40$	0	0
$41$	−15263.7	−1.41808	−0.709041	−	0.705168i	$-0.750872\pi$
$41$	−15263.7	−1.41808	−0.709041	+	0.705168i	$0.750872\pi$
$42$	0	0
$43$	7880.58	0.649960	0.324980	−	0.945721i	$-0.394642\pi$
$43$	7880.58	0.649960	0.324980	+	0.945721i	$0.394642\pi$
$44$	0	0
$45$	−60103.2	−4.42452
$46$	0	0
$47$	16787.7	1.10852	0.554262	−	0.832342i	$-0.313000\pi$
$47$	16787.7	1.10852	0.554262	+	0.832342i	$0.313000\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	47144.1	2.53806
$52$	0	0
$53$	−3230.13	−0.157954	−0.0789769	−	0.996876i	$-0.525165\pi$
$53$	−3230.13	−0.157954	−0.0789769	+	0.996876i	$0.525165\pi$
$54$	0	0
$55$	−12722.2	−0.567092
$56$	0	0
$57$	40876.7	1.66644
$58$	0	0
$59$	−12163.3	−0.454907	−0.227453	−	0.973789i	$-0.573040\pi$
$59$	−12163.3	−0.454907	−0.227453	+	0.973789i	$0.573040\pi$
$60$	0	0
$61$	−51007.4	−1.75513	−0.877563	−	0.479461i	$-0.840832\pi$
$61$	−51007.4	−1.75513	−0.877563	+	0.479461i	$0.840832\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	12425.4	0.364777
$66$	0	0
$67$	27232.3	0.741135	0.370568	−	0.928805i	$-0.379163\pi$
$67$	27232.3	0.741135	0.370568	+	0.928805i	$0.379163\pi$
$68$	0	0
$69$	81002.2	2.04821
$70$	0	0
$71$	−13313.9	−0.313443	−0.156721	−	0.987643i	$-0.550092\pi$
$71$	−13313.9	−0.313443	−0.156721	+	0.987643i	$0.550092\pi$
$72$	0	0
$73$	−49712.9	−1.09185	−0.545924	−	0.837835i	$-0.683821\pi$
$73$	−49712.9	−1.09185	−0.545924	+	0.837835i	$0.683821\pi$
$74$	0	0
$75$	192143.	3.94432
$76$	0	0
$77$	0	0
$78$	0	0
$79$	21105.9	0.380484	0.190242	−	0.981737i	$-0.439073\pi$
$79$	21105.9	0.380484	0.190242	+	0.981737i	$0.439073\pi$
$80$	0	0
$81$	164970.	2.79377
$82$	0	0
$83$	79860.2	1.27243	0.636216	−	0.771511i	$-0.280499\pi$
$83$	79860.2	1.27243	0.636216	+	0.771511i	$0.280499\pi$
$84$	0	0
$85$	−158991.	−2.38685
$86$	0	0
$87$	102229.	1.44802
$88$	0	0
$89$	−36846.5	−0.493085	−0.246542	−	0.969132i	$-0.579294\pi$
$89$	−36846.5	−0.493085	−0.246542	+	0.969132i	$0.579294\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	20751.5	0.248796
$94$	0	0
$95$	−137855.	−1.56716
$96$	0	0
$97$	−27968.5	−0.301814	−0.150907	−	0.988548i	$-0.548219\pi$
$97$	−27968.5	−0.301814	−0.150907	+	0.988548i	$0.548219\pi$
$98$	0	0
$99$	78802.4	0.808074
$100$	0	0
$101$	93531.5	0.912336	0.456168	−	0.889894i	$-0.349222\pi$
$101$	93531.5	0.912336	0.456168	+	0.889894i	$0.349222\pi$
$102$	0	0
$103$	−179414.	−1.66634	−0.833168	−	0.553020i	$-0.813475\pi$
$103$	−179414.	−1.66634	−0.833168	+	0.553020i	$0.813475\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	188903.	1.59507	0.797534	−	0.603274i	$-0.206137\pi$
$107$	188903.	1.59507	0.797534	+	0.603274i	$0.206137\pi$
$108$	0	0
$109$	104680.	0.843909	0.421954	−	0.906617i	$-0.361344\pi$
$109$	104680.	0.843909	0.421954	+	0.906617i	$0.361344\pi$
$110$	0	0
$111$	−89119.0	−0.686535
$112$	0	0
$113$	25226.6	0.185850	0.0929252	−	0.995673i	$-0.470378\pi$
$113$	25226.6	0.185850	0.0929252	+	0.995673i	$0.470378\pi$
$114$	0	0
$115$	−273176.	−1.92619
$116$	0	0
$117$	−76964.4	−0.519787
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−144371.	−0.896429
$122$	0	0
$123$	−445835.	−2.65712
$124$	0	0
$125$	−340166.	−1.94722
$126$	0	0
$127$	5481.01	0.0301544	0.0150772	−	0.999886i	$-0.495201\pi$
$127$	5481.01	0.0301544	0.0150772	+	0.999886i	$0.495201\pi$
$128$	0	0
$129$	230182.	1.21786
$130$	0	0
$131$	195305.	0.994339	0.497169	−	0.867653i	$-0.334373\pi$
$131$	195305.	0.994339	0.497169	+	0.867653i	$0.334373\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−1.05638e6	−4.98866
$136$	0	0
$137$	−173837.	−0.791299	−0.395650	−	0.918402i	$-0.629481\pi$
$137$	−173837.	−0.791299	−0.395650	+	0.918402i	$0.629481\pi$
$138$	0	0
$139$	−283445.	−1.24432	−0.622160	−	0.782890i	$-0.713745\pi$
$139$	−283445.	−1.24432	−0.622160	+	0.782890i	$0.713745\pi$
$140$	0	0
$141$	490347.	2.07709
$142$	0	0
$143$	−16291.2	−0.0666213
$144$	0	0
$145$	−344762.	−1.36176
$146$	0	0
$147$	0	0
$148$	0	0
$149$	227056.	0.837852	0.418926	−	0.908020i	$-0.362407\pi$
$149$	227056.	0.837852	0.418926	+	0.908020i	$0.362407\pi$
$150$	0	0
$151$	332707.	1.18746	0.593731	−	0.804663i	$-0.297654\pi$
$151$	332707.	1.18746	0.593731	+	0.804663i	$0.297654\pi$
$152$	0	0
$153$	984809.	3.40113
$154$	0	0
$155$	−69983.6	−0.233974
$156$	0	0
$157$	443481.	1.43590	0.717952	−	0.696093i	$-0.245080\pi$
$157$	443481.	1.43590	0.717952	+	0.696093i	$0.245080\pi$
$158$	0	0
$159$	−94348.1	−0.295965
$160$	0	0
$161$	0	0
$162$	0	0
$163$	201725.	0.594690	0.297345	−	0.954770i	$-0.403899\pi$
$163$	201725.	0.594690	0.297345	+	0.954770i	$0.403899\pi$
$164$	0	0
$165$	−371598.	−1.06259
$166$	0	0
$167$	−567965.	−1.57591	−0.787953	−	0.615735i	$-0.788859\pi$
$167$	−567965.	−1.57591	−0.787953	+	0.615735i	$0.788859\pi$
$168$	0	0
$169$	−355382.	−0.957146
$170$	0	0
$171$	853888.	2.23311
$172$	0	0
$173$	−301586.	−0.766118	−0.383059	−	0.923724i	$-0.625129\pi$
$173$	−301586.	−0.766118	−0.383059	+	0.923724i	$0.625129\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−355276.	−0.852378
$178$	0	0
$179$	466742.	1.08879	0.544395	−	0.838829i	$-0.316759\pi$
$179$	466742.	1.08879	0.544395	+	0.838829i	$0.316759\pi$
$180$	0	0
$181$	−530851.	−1.20442	−0.602208	−	0.798340i	$-0.705712\pi$
$181$	−530851.	−1.20442	−0.602208	+	0.798340i	$0.705712\pi$
$182$	0	0
$183$	−1.48986e6	−3.28866
$184$	0	0
$185$	300550.	0.645635
$186$	0	0
$187$	208456.	0.435924
$188$	0	0
$189$	0	0
$190$	0	0
$191$	523209.	1.03775	0.518874	−	0.854851i	$-0.326351\pi$
$191$	523209.	1.03775	0.518874	+	0.854851i	$0.326351\pi$
$192$	0	0
$193$	−956688.	−1.84875	−0.924373	−	0.381490i	$-0.875411\pi$
$193$	−956688.	−1.84875	−0.924373	+	0.381490i	$0.875411\pi$
$194$	0	0
$195$	362931.	0.683499
$196$	0	0
$197$	−84916.4	−0.155893	−0.0779464	−	0.996958i	$-0.524836\pi$
$197$	−84916.4	−0.155893	−0.0779464	+	0.996958i	$0.524836\pi$
$198$	0	0
$199$	−565514.	−1.01230	−0.506152	−	0.862444i	$-0.668933\pi$
$199$	−565514.	−1.01230	−0.506152	+	0.862444i	$0.668933\pi$
$200$	0	0
$201$	795422.	1.38870
$202$	0	0
$203$	0	0
$204$	0	0
$205$	1.50356e6	2.49882
$206$	0	0
$207$	1.69208e6	2.74471
$208$	0	0
$209$	180744.	0.286219
$210$	0	0
$211$	−303148.	−0.468758	−0.234379	−	0.972145i	$-0.575306\pi$
$211$	−303148.	−0.468758	−0.234379	+	0.972145i	$0.575306\pi$
$212$	0	0
$213$	−388881.	−0.587311
$214$	0	0
$215$	−776278.	−1.14531
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−1.45205e6	−2.04584
$220$	0	0
$221$	−203594.	−0.280404
$222$	0	0
$223$	796568.	1.07266	0.536328	−	0.844009i	$-0.319811\pi$
$223$	796568.	1.07266	0.536328	+	0.844009i	$0.319811\pi$
$224$	0	0
$225$	4.01375e6	5.28560
$226$	0	0
$227$	−673345.	−0.867308	−0.433654	−	0.901080i	$-0.642776\pi$
$227$	−673345.	−0.867308	−0.433654	+	0.901080i	$0.642776\pi$
$228$	0	0
$229$	−590794.	−0.744471	−0.372235	−	0.928138i	$-0.621409\pi$
$229$	−590794.	−0.744471	−0.372235	+	0.928138i	$0.621409\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−479099.	−0.578143	−0.289072	−	0.957307i	$-0.593347\pi$
$233$	−479099.	−0.578143	−0.289072	+	0.957307i	$0.593347\pi$
$234$	0	0
$235$	−1.65367e6	−1.95335
$236$	0	0
$237$	616478.	0.712930
$238$	0	0
$239$	−349286.	−0.395536	−0.197768	−	0.980249i	$-0.563369\pi$
$239$	−349286.	−0.395536	−0.197768	+	0.980249i	$0.563369\pi$
$240$	0	0
$241$	−337648.	−0.374474	−0.187237	−	0.982315i	$-0.559953\pi$
$241$	−337648.	−0.374474	−0.187237	+	0.982315i	$0.559953\pi$
$242$	0	0
$243$	2.21261e6	2.40375
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−176528.	−0.184108
$248$	0	0
$249$	2.33262e6	2.38421
$250$	0	0
$251$	−1.67737e6	−1.68053	−0.840264	−	0.542178i	$-0.817600\pi$
$251$	−1.67737e6	−1.68053	−0.840264	+	0.542178i	$0.817600\pi$
$252$	0	0
$253$	358167.	0.351790
$254$	0	0
$255$	−4.64394e6	−4.47235
$256$	0	0
$257$	755490.	0.713503	0.356752	−	0.934199i	$-0.383884\pi$
$257$	755490.	0.713503	0.356752	+	0.934199i	$0.383884\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	2.13549e6	1.94043
$262$	0	0
$263$	74806.9	0.0666887	0.0333443	−	0.999444i	$-0.489384\pi$
$263$	74806.9	0.0666887	0.0333443	+	0.999444i	$0.489384\pi$
$264$	0	0
$265$	318185.	0.278333
$266$	0	0
$267$	−1.07624e6	−0.923914
$268$	0	0
$269$	−607218.	−0.511639	−0.255820	−	0.966725i	$-0.582345\pi$
$269$	−607218.	−0.511639	−0.255820	+	0.966725i	$0.582345\pi$
$270$	0	0
$271$	1.48029e6	1.22440	0.612198	−	0.790704i	$-0.290285\pi$
$271$	1.48029e6	1.22440	0.612198	+	0.790704i	$0.290285\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	849598.	0.677457
$276$	0	0
$277$	−1.32490e6	−1.03749	−0.518746	−	0.854928i	$-0.673601\pi$
$277$	−1.32490e6	−1.03749	−0.518746	+	0.854928i	$0.673601\pi$
$278$	0	0
$279$	433486.	0.333399
$280$	0	0
$281$	−649433.	−0.490646	−0.245323	−	0.969441i	$-0.578894\pi$
$281$	−649433.	−0.490646	−0.245323	+	0.969441i	$0.578894\pi$
$282$	0	0
$283$	2.21531e6	1.64426	0.822128	−	0.569303i	$-0.192787\pi$
$283$	2.21531e6	1.64426	0.822128	+	0.569303i	$0.192787\pi$
$284$	0	0
$285$	−4.02657e6	−2.93645
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.18526e6	0.834776
$290$	0	0
$291$	−816924.	−0.565522
$292$	0	0
$293$	−513832.	−0.349665	−0.174833	−	0.984598i	$-0.555938\pi$
$293$	−513832.	−0.349665	−0.174833	+	0.984598i	$0.555938\pi$
$294$	0	0
$295$	1.19815e6	0.801598
$296$	0	0
$297$	1.38503e6	0.911107
$298$	0	0
$299$	−349813.	−0.226286
$300$	0	0
$301$	0	0
$302$	0	0
$303$	2.73194e6	1.70948
$304$	0	0
$305$	5.02449e6	3.09274
$306$	0	0
$307$	835478.	0.505928	0.252964	−	0.967476i	$-0.418594\pi$
$307$	835478.	0.505928	0.252964	+	0.967476i	$0.418594\pi$
$308$	0	0
$309$	−5.24045e6	−3.12229
$310$	0	0
$311$	2.11820e6	1.24184	0.620920	−	0.783874i	$-0.286759\pi$
$311$	2.11820e6	1.24184	0.620920	+	0.783874i	$0.286759\pi$
$312$	0	0
$313$	1.69582e6	0.978406	0.489203	−	0.872170i	$-0.337288\pi$
$313$	1.69582e6	0.978406	0.489203	+	0.872170i	$0.337288\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.10549e6	0.617886	0.308943	−	0.951081i	$-0.400025\pi$
$317$	1.10549e6	0.617886	0.308943	+	0.951081i	$0.400025\pi$
$318$	0	0
$319$	452024.	0.248705
$320$	0	0
$321$	5.51762e6	2.98875
$322$	0	0
$323$	2.25879e6	1.20468
$324$	0	0
$325$	−829782.	−0.435769
$326$	0	0
$327$	3.05756e6	1.58127
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−2.87642e6	−1.44305	−0.721526	−	0.692387i	$-0.756559\pi$
$331$	−2.87642e6	−1.44305	−0.721526	+	0.692387i	$0.756559\pi$
$332$	0	0
$333$	−1.86164e6	−0.919993
$334$	0	0
$335$	−2.68253e6	−1.30597
$336$	0	0
$337$	−1.71881e6	−0.824430	−0.412215	−	0.911087i	$-0.635245\pi$
$337$	−1.71881e6	−0.824430	−0.412215	+	0.911087i	$0.635245\pi$
$338$	0	0
$339$	736839.	0.348236
$340$	0	0
$341$	91756.8	0.0427319
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−7.97914e6	−3.60918
$346$	0	0
$347$	235874.	0.105162	0.0525808	−	0.998617i	$-0.483255\pi$
$347$	235874.	0.105162	0.0525808	+	0.998617i	$0.483255\pi$
$348$	0	0
$349$	−611300.	−0.268652	−0.134326	−	0.990937i	$-0.542887\pi$
$349$	−611300.	−0.268652	−0.134326	+	0.990937i	$0.542887\pi$
$350$	0	0
$351$	−1.35273e6	−0.586062
$352$	0	0
$353$	3.26751e6	1.39566	0.697831	−	0.716263i	$-0.254149\pi$
$353$	3.26751e6	1.39566	0.697831	+	0.716263i	$0.254149\pi$
$354$	0	0
$355$	1.31148e6	0.552322
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3.44281e6	−1.40986	−0.704932	−	0.709275i	$-0.749022\pi$
$359$	−3.44281e6	−1.40986	−0.704932	+	0.709275i	$0.749022\pi$
$360$	0	0
$361$	−517591.	−0.209035
$362$	0	0
$363$	−4.21689e6	−1.67968
$364$	0	0
$365$	4.89698e6	1.92396
$366$	0	0
$367$	−141610.	−0.0548819	−0.0274410	−	0.999623i	$-0.508736\pi$
$367$	−141610.	−0.0548819	−0.0274410	+	0.999623i	$0.508736\pi$
$368$	0	0
$369$	−9.31320e6	−3.56068
$370$	0	0
$371$	0	0
$372$	0	0
$373$	2.90684e6	1.08181	0.540903	−	0.841085i	$-0.318083\pi$
$373$	2.90684e6	1.08181	0.540903	+	0.841085i	$0.318083\pi$
$374$	0	0
$375$	−9.93583e6	−3.64860
$376$	0	0
$377$	−441481.	−0.159977
$378$	0	0
$379$	1.38009e6	0.493526	0.246763	−	0.969076i	$-0.420633\pi$
$379$	1.38009e6	0.493526	0.246763	+	0.969076i	$0.420633\pi$
$380$	0	0
$381$	160094.	0.0565017
$382$	0	0
$383$	1.95357e6	0.680506	0.340253	−	0.940334i	$-0.389487\pi$
$383$	1.95357e6	0.680506	0.340253	+	0.940334i	$0.389487\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4.80835e6	1.63199
$388$	0	0
$389$	4.18943e6	1.40372	0.701860	−	0.712315i	$-0.252353\pi$
$389$	4.18943e6	1.40372	0.701860	+	0.712315i	$0.252353\pi$
$390$	0	0
$391$	4.47608e6	1.48066
$392$	0	0
$393$	5.70461e6	1.86314
$394$	0	0
$395$	−2.07904e6	−0.670457
$396$	0	0
$397$	−5.87109e6	−1.86957	−0.934786	−	0.355211i	$-0.884409\pi$
$397$	−5.87109e6	−1.86957	−0.934786	+	0.355211i	$0.884409\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−2.70005e6	−0.838516	−0.419258	−	0.907867i	$-0.637710\pi$
$401$	−2.70005e6	−0.838516	−0.419258	+	0.907867i	$0.637710\pi$
$402$	0	0
$403$	−89616.7	−0.0274869
$404$	0	0
$405$	−1.62504e7	−4.92295
$406$	0	0
$407$	−394056.	−0.117916
$408$	0	0
$409$	−5.19426e6	−1.53538	−0.767690	−	0.640822i	$-0.778594\pi$
$409$	−5.19426e6	−1.53538	−0.767690	+	0.640822i	$0.778594\pi$
$410$	0	0
$411$	−5.07756e6	−1.48269
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−7.86664e6	−2.24217
$416$	0	0
$417$	−8.27909e6	−2.33154
$418$	0	0
$419$	−1.48129e6	−0.412198	−0.206099	−	0.978531i	$-0.566077\pi$
$419$	−1.48129e6	−0.412198	−0.206099	+	0.978531i	$0.566077\pi$
$420$	0	0
$421$	−4.52469e6	−1.24418	−0.622090	−	0.782946i	$-0.713716\pi$
$421$	−4.52469e6	−1.24418	−0.622090	+	0.782946i	$0.713716\pi$
$422$	0	0
$423$	1.02430e7	2.78341
$424$	0	0
$425$	1.06176e7	2.85137
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−475846.	−0.124831
$430$	0	0
$431$	−4.23835e6	−1.09902	−0.549508	−	0.835488i	$-0.685185\pi$
$431$	−4.23835e6	−1.09902	−0.549508	+	0.835488i	$0.685185\pi$
$432$	0	0
$433$	3.31287e6	0.849152	0.424576	−	0.905392i	$-0.360423\pi$
$433$	3.31287e6	0.849152	0.424576	+	0.905392i	$0.360423\pi$
$434$	0	0
$435$	−1.00701e7	−2.55158
$436$	0	0
$437$	3.88102e6	0.972171
$438$	0	0
$439$	−6.60667e6	−1.63614	−0.818072	−	0.575116i	$-0.804957\pi$
$439$	−6.60667e6	−1.63614	−0.818072	+	0.575116i	$0.804957\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1.57041e6	0.380192	0.190096	−	0.981766i	$-0.439120\pi$
$443$	1.57041e6	0.380192	0.190096	+	0.981766i	$0.439120\pi$
$444$	0	0
$445$	3.62958e6	0.868872
$446$	0	0
$447$	6.63203e6	1.56992
$448$	0	0
$449$	2.10265e6	0.492211	0.246106	−	0.969243i	$-0.420849\pi$
$449$	2.10265e6	0.492211	0.246106	+	0.969243i	$0.420849\pi$
$450$	0	0
$451$	−1.97134e6	−0.456374
$452$	0	0
$453$	9.71797e6	2.22500
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−6.31909e6	−1.41535	−0.707675	−	0.706538i	$-0.750256\pi$
$457$	−6.31909e6	−1.41535	−0.707675	+	0.706538i	$0.750256\pi$
$458$	0	0
$459$	1.73090e7	3.83479
$460$	0	0
$461$	5.38278e6	1.17965	0.589826	−	0.807530i	$-0.299196\pi$
$461$	5.38278e6	1.17965	0.589826	+	0.807530i	$0.299196\pi$
$462$	0	0
$463$	2.50782e6	0.543682	0.271841	−	0.962342i	$-0.412368\pi$
$463$	2.50782e6	0.543682	0.271841	+	0.962342i	$0.412368\pi$
$464$	0	0
$465$	−2.04414e6	−0.438407
$466$	0	0
$467$	1.59292e6	0.337988	0.168994	−	0.985617i	$-0.445948\pi$
$467$	1.59292e6	0.337988	0.168994	+	0.985617i	$0.445948\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	1.29535e7	2.69052
$472$	0	0
$473$	1.01779e6	0.209173
$474$	0	0
$475$	9.20609e6	1.87215
$476$	0	0
$477$	−1.97087e6	−0.396608
$478$	0	0
$479$	9.00646e6	1.79356	0.896779	−	0.442479i	$-0.145901\pi$
$479$	9.00646e6	1.79356	0.896779	+	0.442479i	$0.145901\pi$
$480$	0	0
$481$	384866.	0.0758484
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.75504e6	0.531831
$486$	0	0
$487$	4.59261e6	0.877479	0.438740	−	0.898614i	$-0.355425\pi$
$487$	4.59261e6	0.877479	0.438740	+	0.898614i	$0.355425\pi$
$488$	0	0
$489$	5.89214e6	1.11430
$490$	0	0
$491$	8.27422e6	1.54890	0.774450	−	0.632635i	$-0.218027\pi$
$491$	8.27422e6	1.54890	0.774450	+	0.632635i	$0.218027\pi$
$492$	0	0
$493$	5.64903e6	1.04678
$494$	0	0
$495$	−7.76245e6	−1.42392
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−8.01038e6	−1.44013	−0.720065	−	0.693906i	$-0.755888\pi$
$499$	−8.01038e6	−1.44013	−0.720065	+	0.693906i	$0.755888\pi$
$500$	0	0
$501$	−1.65896e7	−2.95284
$502$	0	0
$503$	−8.70461e6	−1.53401	−0.767007	−	0.641639i	$-0.778255\pi$
$503$	−8.70461e6	−1.53401	−0.767007	+	0.641639i	$0.778255\pi$
$504$	0	0
$505$	−9.21335e6	−1.60764
$506$	0	0
$507$	−1.03803e7	−1.79345
$508$	0	0
$509$	−244907.	−0.0418992	−0.0209496	−	0.999781i	$-0.506669\pi$
$509$	−244907.	−0.0418992	−0.0209496	+	0.999781i	$0.506669\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	1.50080e7	2.51784
$514$	0	0
$515$	1.76732e7	2.93628
$516$	0	0
$517$	2.16816e6	0.356751
$518$	0	0
$519$	−8.80895e6	−1.43551
$520$	0	0
$521$	2.21278e6	0.357144	0.178572	−	0.983927i	$-0.442852\pi$
$521$	2.21278e6	0.357144	0.178572	+	0.983927i	$0.442852\pi$
$522$	0	0
$523$	5.09897e6	0.815133	0.407566	−	0.913176i	$-0.366378\pi$
$523$	5.09897e6	0.815133	0.407566	+	0.913176i	$0.366378\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.14670e6	0.179856
$528$	0	0
$529$	1.25439e6	0.194891
$530$	0	0
$531$	−7.42148e6	−1.14223
$532$	0	0
$533$	1.92536e6	0.293558
$534$	0	0
$535$	−1.86079e7	−2.81069
$536$	0	0
$537$	1.36330e7	2.04011
$538$	0	0
$539$	0	0
$540$	0	0
$541$	5.26596e6	0.773543	0.386771	−	0.922176i	$-0.373590\pi$
$541$	5.26596e6	0.773543	0.386771	+	0.922176i	$0.373590\pi$
$542$	0	0
$543$	−1.55055e7	−2.25676
$544$	0	0
$545$	−1.03115e7	−1.48706
$546$	0	0
$547$	8.54636e6	1.22127	0.610636	−	0.791911i	$-0.290914\pi$
$547$	8.54636e6	1.22127	0.610636	+	0.791911i	$0.290914\pi$
$548$	0	0
$549$	−3.11223e7	−4.40697
$550$	0	0
$551$	4.89804e6	0.687296
$552$	0	0
$553$	0	0
$554$	0	0
$555$	8.77869e6	1.20975
$556$	0	0
$557$	1.59694e6	0.218097	0.109049	−	0.994036i	$-0.465220\pi$
$557$	1.59694e6	0.218097	0.109049	+	0.994036i	$0.465220\pi$
$558$	0	0
$559$	−994054.	−0.134549
$560$	0	0
$561$	6.08875e6	0.816810
$562$	0	0
$563$	−1.79904e6	−0.239205	−0.119603	−	0.992822i	$-0.538162\pi$
$563$	−1.79904e6	−0.239205	−0.119603	+	0.992822i	$0.538162\pi$
$564$	0	0
$565$	−2.48496e6	−0.327490
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−2.05359e6	−0.265909	−0.132954	−	0.991122i	$-0.542446\pi$
$569$	−2.05359e6	−0.265909	−0.132954	+	0.991122i	$0.542446\pi$
$570$	0	0
$571$	−3.22858e6	−0.414402	−0.207201	−	0.978298i	$-0.566435\pi$
$571$	−3.22858e6	−0.414402	−0.207201	+	0.978298i	$0.566435\pi$
$572$	0	0
$573$	1.52823e7	1.94447
$574$	0	0
$575$	1.82430e7	2.30105
$576$	0	0
$577$	8.26800e6	1.03386	0.516929	−	0.856028i	$-0.327075\pi$
$577$	8.26800e6	1.03386	0.516929	+	0.856028i	$0.327075\pi$
$578$	0	0
$579$	−2.79437e7	−3.46408
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−417178.	−0.0508335
$584$	0	0
$585$	7.58140e6	0.915924
$586$	0	0
$587$	−1.12852e7	−1.35180	−0.675901	−	0.736993i	$-0.736245\pi$
$587$	−1.12852e7	−1.35180	−0.675901	+	0.736993i	$0.736245\pi$
$588$	0	0
$589$	994260.	0.118090
$590$	0	0
$591$	−2.48030e6	−0.292103
$592$	0	0
$593$	9.01399e6	1.05264	0.526320	−	0.850286i	$-0.323571\pi$
$593$	9.01399e6	1.05264	0.526320	+	0.850286i	$0.323571\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−1.65180e7	−1.89680
$598$	0	0
$599$	9.92639e6	1.13038	0.565190	−	0.824961i	$-0.308803\pi$
$599$	9.92639e6	1.13038	0.565190	+	0.824961i	$0.308803\pi$
$600$	0	0
$601$	−7.45600e6	−0.842015	−0.421008	−	0.907057i	$-0.638323\pi$
$601$	−7.45600e6	−0.842015	−0.421008	+	0.907057i	$0.638323\pi$
$602$	0	0
$603$	1.66159e7	1.86093
$604$	0	0
$605$	1.42213e7	1.57961
$606$	0	0
$607$	−247024.	−0.0272124	−0.0136062	−	0.999907i	$-0.504331\pi$
$607$	−247024.	−0.0272124	−0.0136062	+	0.999907i	$0.504331\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−2.11759e6	−0.229477
$612$	0	0
$613$	−9.21880e6	−0.990885	−0.495443	−	0.868641i	$-0.664994\pi$
$613$	−9.21880e6	−0.990885	−0.495443	+	0.868641i	$0.664994\pi$
$614$	0	0
$615$	4.39171e7	4.68215
$616$	0	0
$617$	−1.17574e7	−1.24336	−0.621682	−	0.783270i	$-0.713550\pi$
$617$	−1.17574e7	−1.24336	−0.621682	+	0.783270i	$0.713550\pi$
$618$	0	0
$619$	−2.63128e6	−0.276020	−0.138010	−	0.990431i	$-0.544071\pi$
$619$	−2.63128e6	−0.276020	−0.138010	+	0.990431i	$0.544071\pi$
$620$	0	0
$621$	2.97401e7	3.09467
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.29510e7	1.32618
$626$	0	0
$627$	5.27931e6	0.536301
$628$	0	0
$629$	−4.92460e6	−0.496300
$630$	0	0
$631$	−2.00147e6	−0.200113	−0.100057	−	0.994982i	$-0.531902\pi$
$631$	−2.00147e6	−0.200113	−0.100057	+	0.994982i	$0.531902\pi$
$632$	0	0
$633$	−8.85458e6	−0.878332
$634$	0	0
$635$	−539908.	−0.0531356
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−8.12348e6	−0.787027
$640$	0	0
$641$	1.54877e7	1.48882	0.744410	−	0.667723i	$-0.232731\pi$
$641$	1.54877e7	1.48882	0.744410	+	0.667723i	$0.232731\pi$
$642$	0	0
$643$	4.97821e6	0.474838	0.237419	−	0.971407i	$-0.423699\pi$
$643$	4.97821e6	0.474838	0.237419	+	0.971407i	$0.423699\pi$
$644$	0	0
$645$	−2.26741e7	−2.14601
$646$	0	0
$647$	−1.28137e7	−1.20341	−0.601704	−	0.798719i	$-0.705511\pi$
$647$	−1.28137e7	−1.20341	−0.601704	+	0.798719i	$0.705511\pi$
$648$	0	0
$649$	−1.57092e6	−0.146400
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−3.81501e6	−0.350116	−0.175058	−	0.984558i	$-0.556011\pi$
$653$	−3.81501e6	−0.350116	−0.175058	+	0.984558i	$0.556011\pi$
$654$	0	0
$655$	−1.92385e7	−1.75214
$656$	0	0
$657$	−3.03324e7	−2.74154
$658$	0	0
$659$	−1.25889e7	−1.12921	−0.564604	−	0.825362i	$-0.690971\pi$
$659$	−1.25889e7	−1.12921	−0.564604	+	0.825362i	$0.690971\pi$
$660$	0	0
$661$	1.75130e7	1.55904	0.779518	−	0.626380i	$-0.215464\pi$
$661$	1.75130e7	1.55904	0.779518	+	0.626380i	$0.215464\pi$
$662$	0	0
$663$	−5.94674e6	−0.525406
$664$	0	0
$665$	0	0
$666$	0	0
$667$	9.70608e6	0.844752
$668$	0	0
$669$	2.32668e7	2.00988
$670$	0	0
$671$	−6.58771e6	−0.564843
$672$	0	0
$673$	−2.20008e7	−1.87241	−0.936205	−	0.351453i	$-0.885688\pi$
$673$	−2.20008e7	−1.87241	−0.936205	+	0.351453i	$0.885688\pi$
$674$	0	0
$675$	7.05459e7	5.95953
$676$	0	0
$677$	−1.02906e7	−0.862920	−0.431460	−	0.902132i	$-0.642001\pi$
$677$	−1.02906e7	−0.862920	−0.431460	+	0.902132i	$0.642001\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−1.96676e7	−1.62511
$682$	0	0
$683$	−2.15320e7	−1.76617	−0.883085	−	0.469213i	$-0.844538\pi$
$683$	−2.15320e7	−1.76617	−0.883085	+	0.469213i	$0.844538\pi$
$684$	0	0
$685$	1.71239e7	1.39436
$686$	0	0
$687$	−1.72564e7	−1.39495
$688$	0	0
$689$	407447.	0.0326982
$690$	0	0
$691$	8.86868e6	0.706584	0.353292	−	0.935513i	$-0.385062\pi$
$691$	8.86868e6	0.706584	0.353292	+	0.935513i	$0.385062\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2.79209e7	2.19264
$696$	0	0
$697$	−2.46362e7	−1.92085
$698$	0	0
$699$	−1.39939e7	−1.08329
$700$	0	0
$701$	1.79181e7	1.37720	0.688598	−	0.725143i	$-0.258227\pi$
$701$	1.79181e7	1.37720	0.688598	+	0.725143i	$0.258227\pi$
$702$	0	0
$703$	−4.26992e6	−0.325860
$704$	0	0
$705$	−4.83017e7	−3.66007
$706$	0	0
$707$	0	0
$708$	0	0
$709$	2.48783e6	0.185868	0.0929341	−	0.995672i	$-0.470375\pi$
$709$	2.48783e6	0.185868	0.0929341	+	0.995672i	$0.470375\pi$
$710$	0	0
$711$	1.28778e7	0.955363
$712$	0	0
$713$	1.97025e6	0.145143
$714$	0	0
$715$	1.60477e6	0.117394
$716$	0	0
$717$	−1.02022e7	−0.741133
$718$	0	0
$719$	−1.29696e7	−0.935632	−0.467816	−	0.883826i	$-0.654959\pi$
$719$	−1.29696e7	−0.935632	−0.467816	+	0.883826i	$0.654959\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−9.86228e6	−0.701668
$724$	0	0
$725$	2.30236e7	1.62677
$726$	0	0
$727$	3.25820e6	0.228634	0.114317	−	0.993444i	$-0.463532\pi$
$727$	3.25820e6	0.228634	0.114317	+	0.993444i	$0.463532\pi$
$728$	0	0
$729$	2.45400e7	1.71024
$730$	0	0
$731$	1.27196e7	0.880397
$732$	0	0
$733$	−6.50445e6	−0.447148	−0.223574	−	0.974687i	$-0.571772\pi$
$733$	−6.50445e6	−0.447148	−0.223574	+	0.974687i	$0.571772\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.51711e6	0.238516
$738$	0	0
$739$	−1.58555e7	−1.06800	−0.533998	−	0.845486i	$-0.679311\pi$
$739$	−1.58555e7	−1.06800	−0.533998	+	0.845486i	$0.679311\pi$
$740$	0	0
$741$	−5.15618e6	−0.344971
$742$	0	0
$743$	2.11717e6	0.140696	0.0703482	−	0.997522i	$-0.477589\pi$
$743$	2.11717e6	0.140696	0.0703482	+	0.997522i	$0.477589\pi$
$744$	0	0
$745$	−2.23662e7	−1.47639
$746$	0	0
$747$	4.87268e7	3.19497
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−4.55362e6	−0.294616	−0.147308	−	0.989091i	$-0.547061\pi$
$751$	−4.55362e6	−0.294616	−0.147308	+	0.989091i	$0.547061\pi$
$752$	0	0
$753$	−4.89940e7	−3.14888
$754$	0	0
$755$	−3.27734e7	−2.09245
$756$	0	0
$757$	6.04759e6	0.383568	0.191784	−	0.981437i	$-0.438573\pi$
$757$	6.04759e6	0.383568	0.191784	+	0.981437i	$0.438573\pi$
$758$	0	0
$759$	1.04616e7	0.659164
$760$	0	0
$761$	−1.35226e7	−0.846445	−0.423222	−	0.906026i	$-0.639101\pi$
$761$	−1.35226e7	−0.846445	−0.423222	+	0.906026i	$0.639101\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−9.70088e7	−5.99318
$766$	0	0
$767$	1.53428e6	0.0941707
$768$	0	0
$769$	−6.83797e6	−0.416976	−0.208488	−	0.978025i	$-0.566854\pi$
$769$	−6.83797e6	−0.416976	−0.208488	+	0.978025i	$0.566854\pi$
$770$	0	0
$771$	2.20669e7	1.33692
$772$	0	0
$773$	−5.26405e6	−0.316863	−0.158431	−	0.987370i	$-0.550644\pi$
$773$	−5.26405e6	−0.316863	−0.158431	+	0.987370i	$0.550644\pi$
$774$	0	0
$775$	4.67358e6	0.279509
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.13611e7	−1.26119
$780$	0	0
$781$	−1.71951e6	−0.100874
$782$	0	0
$783$	3.75335e7	2.18784
$784$	0	0
$785$	−4.36852e7	−2.53023
$786$	0	0
$787$	1.87800e7	1.08083	0.540416	−	0.841398i	$-0.318267\pi$
$787$	1.87800e7	1.08083	0.540416	+	0.841398i	$0.318267\pi$
$788$	0	0
$789$	2.18502e6	0.124957
$790$	0	0
$791$	0	0
$792$	0	0
$793$	6.43406e6	0.363331
$794$	0	0
$795$	9.29378e6	0.521524
$796$	0	0
$797$	−2.71298e7	−1.51286	−0.756432	−	0.654072i	$-0.773059\pi$
$797$	−2.71298e7	−1.51286	−0.756432	+	0.654072i	$0.773059\pi$
$798$	0	0
$799$	2.70959e7	1.50154
$800$	0	0
$801$	−2.24820e7	−1.23809
$802$	0	0
$803$	−6.42052e6	−0.351384
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−1.77361e7	−0.958680
$808$	0	0
$809$	4.53380e6	0.243552	0.121776	−	0.992558i	$-0.461141\pi$
$809$	4.53380e6	0.243552	0.121776	+	0.992558i	$0.461141\pi$
$810$	0	0
$811$	−1.91621e7	−1.02303	−0.511517	−	0.859273i	$-0.670916\pi$
$811$	−1.91621e7	−1.02303	−0.511517	+	0.859273i	$0.670916\pi$
$812$	0	0
$813$	4.32373e7	2.29421
$814$	0	0
$815$	−1.98710e7	−1.04791
$816$	0	0
$817$	1.10286e7	0.578050
$818$	0	0
$819$	0	0
$820$	0	0
$821$	3.22138e7	1.66795	0.833977	−	0.551800i	$-0.186059\pi$
$821$	3.22138e7	1.66795	0.833977	+	0.551800i	$0.186059\pi$
$822$	0	0
$823$	−2.90394e6	−0.149448	−0.0747238	−	0.997204i	$-0.523808\pi$
$823$	−2.90394e6	−0.149448	−0.0747238	+	0.997204i	$0.523808\pi$
$824$	0	0
$825$	2.48157e7	1.26938
$826$	0	0
$827$	3.30382e7	1.67978	0.839890	−	0.542756i	$-0.182619\pi$
$827$	3.30382e7	1.67978	0.839890	+	0.542756i	$0.182619\pi$
$828$	0	0
$829$	3.55877e6	0.179851	0.0899257	−	0.995948i	$-0.471337\pi$
$829$	3.55877e6	0.179851	0.0899257	+	0.995948i	$0.471337\pi$
$830$	0	0
$831$	−3.86988e7	−1.94399
$832$	0	0
$833$	0	0
$834$	0	0
$835$	5.59475e7	2.77693
$836$	0	0
$837$	7.61897e6	0.375909
$838$	0	0
$839$	7.33186e6	0.359591	0.179796	−	0.983704i	$-0.442456\pi$
$839$	7.33186e6	0.359591	0.179796	+	0.983704i	$0.442456\pi$
$840$	0	0
$841$	−8.26160e6	−0.402786
$842$	0	0
$843$	−1.89691e7	−0.919345
$844$	0	0
$845$	3.50070e7	1.68660
$846$	0	0
$847$	0	0
$848$	0	0
$849$	6.47066e7	3.08091
$850$	0	0
$851$	−8.46137e6	−0.400513
$852$	0	0
$853$	3.34015e6	0.157178	0.0785892	−	0.996907i	$-0.474958\pi$
$853$	3.34015e6	0.157178	0.0785892	+	0.996907i	$0.474958\pi$
$854$	0	0
$855$	−8.41124e7	−3.93500
$856$	0	0
$857$	2.96675e7	1.37984	0.689921	−	0.723885i	$-0.257645\pi$
$857$	2.96675e7	1.37984	0.689921	+	0.723885i	$0.257645\pi$
$858$	0	0
$859$	−3.04453e7	−1.40779	−0.703895	−	0.710305i	$-0.748557\pi$
$859$	−3.04453e7	−1.40779	−0.703895	+	0.710305i	$0.748557\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−2.20240e7	−1.00663	−0.503314	−	0.864103i	$-0.667886\pi$
$863$	−2.20240e7	−1.00663	−0.503314	+	0.864103i	$0.667886\pi$
$864$	0	0
$865$	2.97078e7	1.34999
$866$	0	0
$867$	3.46200e7	1.56416
$868$	0	0
$869$	2.72587e6	0.122449
$870$	0	0
$871$	−3.43508e6	−0.153423
$872$	0	0
$873$	−1.70650e7	−0.757829
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−1.60857e7	−0.706220	−0.353110	−	0.935582i	$-0.614876\pi$
$877$	−1.60857e7	−0.706220	−0.353110	+	0.935582i	$0.614876\pi$
$878$	0	0
$879$	−1.50084e7	−0.655183
$880$	0	0
$881$	−2.03474e7	−0.883221	−0.441610	−	0.897207i	$-0.645593\pi$
$881$	−2.03474e7	−0.883221	−0.441610	+	0.897207i	$0.645593\pi$
$882$	0	0
$883$	−3.79836e7	−1.63943	−0.819717	−	0.572769i	$-0.805869\pi$
$883$	−3.79836e7	−1.63943	−0.819717	+	0.572769i	$0.805869\pi$
$884$	0	0
$885$	3.49965e7	1.50199
$886$	0	0
$887$	8.34995e6	0.356349	0.178174	−	0.983999i	$-0.442981\pi$
$887$	8.34995e6	0.356349	0.178174	+	0.983999i	$0.442981\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	2.13062e7	0.899106
$892$	0	0
$893$	2.34938e7	0.985880
$894$	0	0
$895$	−4.59765e7	−1.91857
$896$	0	0
$897$	−1.02176e7	−0.424002
$898$	0	0
$899$	2.48655e6	0.102612
$900$	0	0
$901$	−5.21355e6	−0.213955
$902$	0	0
$903$	0	0
$904$	0	0
$905$	5.22916e7	2.12232
$906$	0	0
$907$	2.69605e7	1.08820	0.544101	−	0.839020i	$-0.316871\pi$
$907$	2.69605e7	1.08820	0.544101	+	0.839020i	$0.316871\pi$
$908$	0	0
$909$	5.70685e7	2.29080
$910$	0	0
$911$	−3.84152e7	−1.53358	−0.766792	−	0.641896i	$-0.778148\pi$
$911$	−3.84152e7	−1.53358	−0.766792	+	0.641896i	$0.778148\pi$
$912$	0	0
$913$	1.03141e7	0.409501
$914$	0	0
$915$	1.46759e8	5.79499
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−2.52762e7	−0.987241	−0.493620	−	0.869678i	$-0.664327\pi$
$919$	−2.52762e7	−0.987241	−0.493620	+	0.869678i	$0.664327\pi$
$920$	0	0
$921$	2.44033e7	0.947980
$922$	0	0
$923$	1.67941e6	0.0648861
$924$	0	0
$925$	−2.00710e7	−0.771286
$926$	0	0
$927$	−1.09470e8	−4.18403
$928$	0	0
$929$	9.16787e6	0.348521	0.174261	−	0.984700i	$-0.444247\pi$
$929$	9.16787e6	0.348521	0.174261	+	0.984700i	$0.444247\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	6.18700e7	2.32689
$934$	0	0
$935$	−2.05340e7	−0.768149
$936$	0	0
$937$	7.16427e6	0.266577	0.133289	−	0.991077i	$-0.457446\pi$
$937$	7.16427e6	0.266577	0.133289	+	0.991077i	$0.457446\pi$
$938$	0	0
$939$	4.95328e7	1.83328
$940$	0	0
$941$	−8.49562e6	−0.312767	−0.156384	−	0.987696i	$-0.549984\pi$
$941$	−8.49562e6	−0.312767	−0.156384	+	0.987696i	$0.549984\pi$
$942$	0	0
$943$	−4.23296e7	−1.55012
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.98201e7	−0.718177	−0.359088	−	0.933304i	$-0.616912\pi$
$947$	−1.98201e7	−0.718177	−0.359088	+	0.933304i	$0.616912\pi$
$948$	0	0
$949$	6.27077e6	0.226025
$950$	0	0
$951$	3.22901e7	1.15776
$952$	0	0
$953$	4.97501e7	1.77444	0.887221	−	0.461344i	$-0.152633\pi$
$953$	4.97501e7	1.77444	0.887221	+	0.461344i	$0.152633\pi$
$954$	0	0
$955$	−5.15388e7	−1.82863
$956$	0	0
$957$	1.32031e7	0.466009
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−2.81244e7	−0.982369
$962$	0	0
$963$	1.15260e8	4.00508
$964$	0	0
$965$	9.42388e7	3.25770
$966$	0	0
$967$	−1.50099e7	−0.516192	−0.258096	−	0.966119i	$-0.583095\pi$
$967$	−1.50099e7	−0.516192	−0.258096	+	0.966119i	$0.583095\pi$
$968$	0	0
$969$	6.59766e7	2.25725
$970$	0	0
$971$	−2.58655e7	−0.880386	−0.440193	−	0.897903i	$-0.645090\pi$
$971$	−2.58655e7	−0.880386	−0.440193	+	0.897903i	$0.645090\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−2.42369e7	−0.816518
$976$	0	0
$977$	−5.08788e7	−1.70530	−0.852649	−	0.522485i	$-0.825005\pi$
$977$	−5.08788e7	−1.70530	−0.852649	+	0.522485i	$0.825005\pi$
$978$	0	0
$979$	−4.75880e6	−0.158687
$980$	0	0
$981$	6.38704e7	2.11898
$982$	0	0
$983$	5.03176e7	1.66087	0.830437	−	0.557113i	$-0.188091\pi$
$983$	5.03176e7	1.66087	0.830437	+	0.557113i	$0.188091\pi$
$984$	0	0
$985$	8.36471e6	0.274701
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2.18545e7	0.710479
$990$	0	0
$991$	−1.41832e7	−0.458765	−0.229382	−	0.973336i	$-0.573671\pi$
$991$	−1.41832e7	−0.458765	−0.229382	+	0.973336i	$0.573671\pi$
$992$	0	0
$993$	−8.40167e7	−2.70391
$994$	0	0
$995$	5.57061e7	1.78380
$996$	0	0
$997$	−1.63176e7	−0.519897	−0.259949	−	0.965622i	$-0.583706\pi$
$997$	−1.63176e7	−0.519897	−0.259949	+	0.965622i	$0.583706\pi$
$998$	0	0
$999$	−3.27202e7	−1.03730

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	196.6.a.k.1.4	yes	4
4.3	odd	2		784.6.a.bi.1.1		4
7.2	even	3		196.6.e.l.165.1		8
7.3	odd	6		196.6.e.l.177.4		8
7.4	even	3		196.6.e.l.177.1		8
7.5	odd	6		196.6.e.l.165.4		8
7.6	odd	2	inner	196.6.a.k.1.1	✓	4
28.27	even	2		784.6.a.bi.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
196.6.a.k.1.1	✓	4	7.6	odd	2	inner
196.6.a.k.1.4	yes	4	1.1	even	1	trivial
196.6.e.l.165.1		8	7.2	even	3
196.6.e.l.165.4		8	7.5	odd	6
196.6.e.l.177.1		8	7.4	even	3
196.6.e.l.177.4		8	7.3	odd	6
784.6.a.bi.1.1		4	4.3	odd	2
784.6.a.bi.1.4		4	28.27	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:	\( N \)	\(=\)	\( 196 = 2^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	196.a (trivial)

\(f(q)\)	\(=\)	\(q+29.2088 q^{3} -98.5052 q^{5} +610.152 q^{9} +129.152 q^{11} -126.140 q^{13} -2877.22 q^{15} +1614.04 q^{17} +1399.47 q^{19} +2773.22 q^{23} +6578.28 q^{25} +10724.1 q^{27} +3499.94 q^{29} +710.456 q^{31} +3772.37 q^{33} -3051.10 q^{37} -3684.39 q^{39} -15263.7 q^{41} +7880.58 q^{43} -60103.2 q^{45} +16787.7 q^{47} +47144.1 q^{51} -3230.13 q^{53} -12722.2 q^{55} +40876.7 q^{57} -12163.3 q^{59} -51007.4 q^{61} +12425.4 q^{65} +27232.3 q^{67} +81002.2 q^{69} -13313.9 q^{71} -49712.9 q^{73} +192143. q^{75} +21105.9 q^{79} +164970. q^{81} +79860.2 q^{83} -158991. q^{85} +102229. q^{87} -36846.5 q^{89} +20751.5 q^{93} -137855. q^{95} -27968.5 q^{97} +78802.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 1480 q^{9} - 444 q^{11} - 3824 q^{15} + 3408 q^{23} + 11904 q^{25} + 20724 q^{29} + 13732 q^{37} + 25608 q^{39} - 28996 q^{43} + 181852 q^{51} + 528 q^{53} + 89540 q^{57} + 110220 q^{65} + 195384 q^{67}+ \cdots + 66412 q^{99}+O(q^{100}) \) 4 * q + 1480 * q^9 - 444 * q^11 - 3824 * q^15 + 3408 * q^23 + 11904 * q^25 + 20724 * q^29 + 13732 * q^37 + 25608 * q^39 - 28996 * q^43 + 181852 * q^51 + 528 * q^53 + 89540 * q^57 + 110220 * q^65 + 195384 * q^67 + 21672 * q^71 - 117304 * q^79 + 182456 * q^81 - 91300 * q^85 + 250152 * q^93 - 255552 * q^95 + 66412 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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