LMFDB - Embedded newform 9800.2.a.cf.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9800,2,Mod(1,9800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9800, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("9800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$9800 = 2^{3} \cdot 5^{2} \cdot 7^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	9800.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$78.2533939809$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.1944.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{3} - 9x - 6$ x^3 - 9*x - 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 280)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.28995$ of defining polynomial
Character	$\chi$	$=$	9800.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.28995 q^{3} +7.82374 q^{9} +5.82374 q^{11} +2.75615 q^{13} +2.00000 q^{17} +0.756152 q^{19} +0.533794 q^{23} -15.8698 q^{27} -0.823739 q^{29} +2.57989 q^{31} -19.1598 q^{33} -4.75615 q^{37} -9.06759 q^{39} -6.06759 q^{41} -0.710055 q^{43} -12.8913 q^{47} -6.57989 q^{51} +8.40363 q^{53} -2.48770 q^{57} -8.00000 q^{59} -9.40363 q^{61} -11.8698 q^{67} -1.75615 q^{69} +3.51230 q^{73} -9.51230 q^{79} +28.7397 q^{81} -6.71005 q^{83} +2.71005 q^{87} -1.75615 q^{89} -8.48770 q^{93} -2.00000 q^{97} +45.5634 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$3 q + 9 q^{9} + 3 q^{11} + 3 q^{13} + 6 q^{17} - 3 q^{19} - 3 q^{23} - 18 q^{27} + 12 q^{29} - 12 q^{31} - 18 q^{33} - 9 q^{37} - 18 q^{39} - 9 q^{41} - 12 q^{43} - 15 q^{47} - 9 q^{53} - 18 q^{57} - 24 q^{59}+ \cdots + 63 q^{99}+O(q^{100})$ 3 * q + 9 * q^9 + 3 * q^11 + 3 * q^13 + 6 * q^17 - 3 * q^19 - 3 * q^23 - 18 * q^27 + 12 * q^29 - 12 * q^31 - 18 * q^33 - 9 * q^37 - 18 * q^39 - 9 * q^41 - 12 * q^43 - 15 * q^47 - 9 * q^53 - 18 * q^57 - 24 * q^59 + 6 * q^61 - 6 * q^67 - 18 * q^79 + 27 * q^81 - 30 * q^83 + 18 * q^87 - 36 * q^93 - 6 * q^97 + 63 * 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
$3$	−3.28995	−1.89945	−0.949725	−	0.313084i	$-0.898638\pi$
$3$	−3.28995	−1.89945	−0.949725	+	0.313084i	$0.898638\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	7.82374	2.60791
$10$	0	0
$11$	5.82374	1.75592	0.877962	−	0.478731i	$-0.158903\pi$
$11$	5.82374	1.75592	0.877962	+	0.478731i	$0.158903\pi$
$12$	0	0
$13$	2.75615	0.764419	0.382209	−	0.924076i	$-0.375163\pi$
$13$	2.75615	0.764419	0.382209	+	0.924076i	$0.375163\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	0.756152	0.173473	0.0867365	−	0.996231i	$-0.472356\pi$
$19$	0.756152	0.173473	0.0867365	+	0.996231i	$0.472356\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0.533794	0.111304	0.0556518	−	0.998450i	$-0.482276\pi$
$23$	0.533794	0.111304	0.0556518	+	0.998450i	$0.482276\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−15.8698	−3.05415
$28$	0	0
$29$	−0.823739	−0.152964	−0.0764822	−	0.997071i	$-0.524369\pi$
$29$	−0.823739	−0.152964	−0.0764822	+	0.997071i	$0.524369\pi$
$30$	0	0
$31$	2.57989	0.463362	0.231681	−	0.972792i	$-0.425577\pi$
$31$	2.57989	0.463362	0.231681	+	0.972792i	$0.425577\pi$
$32$	0	0
$33$	−19.1598	−3.33529
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4.75615	−0.781906	−0.390953	−	0.920411i	$-0.627855\pi$
$37$	−4.75615	−0.781906	−0.390953	+	0.920411i	$0.627855\pi$
$38$	0	0
$39$	−9.06759	−1.45198
$40$	0	0
$41$	−6.06759	−0.947598	−0.473799	−	0.880633i	$-0.657118\pi$
$41$	−6.06759	−0.947598	−0.473799	+	0.880633i	$0.657118\pi$
$42$	0	0
$43$	−0.710055	−0.108282	−0.0541412	−	0.998533i	$-0.517242\pi$
$43$	−0.710055	−0.108282	−0.0541412	+	0.998533i	$0.517242\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−12.8913	−1.88039	−0.940197	−	0.340632i	$-0.889359\pi$
$47$	−12.8913	−1.88039	−0.940197	+	0.340632i	$0.889359\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−6.57989	−0.921369
$52$	0	0
$53$	8.40363	1.15433	0.577164	−	0.816629i	$-0.304159\pi$
$53$	8.40363	1.15433	0.577164	+	0.816629i	$0.304159\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−2.48770	−0.329504
$58$	0	0
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	−9.40363	−1.20401	−0.602006	−	0.798492i	$-0.705632\pi$
$61$	−9.40363	−1.20401	−0.602006	+	0.798492i	$0.705632\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−11.8698	−1.45013	−0.725066	−	0.688680i	$-0.758191\pi$
$67$	−11.8698	−1.45013	−0.725066	+	0.688680i	$0.758191\pi$
$68$	0	0
$69$	−1.75615	−0.211416
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	3.51230	0.411084	0.205542	−	0.978648i	$-0.434104\pi$
$73$	3.51230	0.411084	0.205542	+	0.978648i	$0.434104\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−9.51230	−1.07022	−0.535109	−	0.844783i	$-0.679729\pi$
$79$	−9.51230	−1.07022	−0.535109	+	0.844783i	$0.679729\pi$
$80$	0	0
$81$	28.7397	3.19330
$82$	0	0
$83$	−6.71005	−0.736524	−0.368262	−	0.929722i	$-0.620047\pi$
$83$	−6.71005	−0.736524	−0.368262	+	0.929722i	$0.620047\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	2.71005	0.290548
$88$	0	0
$89$	−1.75615	−0.186152	−0.0930758	−	0.995659i	$-0.529670\pi$
$89$	−1.75615	−0.186152	−0.0930758	+	0.995659i	$0.529670\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−8.48770	−0.880133
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	45.5634	4.57929
$100$	0	0
$101$	−2.33604	−0.232445	−0.116222	−	0.993223i	$-0.537079\pi$
$101$	−2.33604	−0.232445	−0.116222	+	0.993223i	$0.537079\pi$
$102$	0	0
$103$	−2.22236	−0.218975	−0.109488	−	0.993988i	$-0.534921\pi$
$103$	−2.22236	−0.218975	−0.109488	+	0.993988i	$0.534921\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−15.8698	−1.53419	−0.767097	−	0.641531i	$-0.778300\pi$
$107$	−15.8698	−1.53419	−0.767097	+	0.641531i	$0.778300\pi$
$108$	0	0
$109$	−3.26845	−0.313061	−0.156531	−	0.987673i	$-0.550031\pi$
$109$	−3.26845	−0.313061	−0.156531	+	0.987673i	$0.550031\pi$
$110$	0	0
$111$	15.6475	1.48519
$112$	0	0
$113$	13.1598	1.23797	0.618984	−	0.785404i	$-0.287545\pi$
$113$	13.1598	1.23797	0.618984	+	0.785404i	$0.287545\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	21.5634	1.99354
$118$	0	0
$119$	0	0
$120$	0	0
$121$	22.9159	2.08327
$122$	0	0
$123$	19.9620	1.79992
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−11.9159	−1.05737	−0.528684	−	0.848819i	$-0.677314\pi$
$127$	−11.9159	−1.05737	−0.528684	+	0.848819i	$0.677314\pi$
$128$	0	0
$129$	2.33604	0.205677
$130$	0	0
$131$	−21.5634	−1.88400	−0.942002	−	0.335608i	$-0.891058\pi$
$131$	−21.5634	−1.88400	−0.942002	+	0.335608i	$0.891058\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−4.00000	−0.341743	−0.170872	−	0.985293i	$-0.554658\pi$
$137$	−4.00000	−0.341743	−0.170872	+	0.985293i	$0.554658\pi$
$138$	0	0
$139$	−6.57989	−0.558099	−0.279049	−	0.960277i	$-0.590019\pi$
$139$	−6.57989	−0.558099	−0.279049	+	0.960277i	$0.590019\pi$
$140$	0	0
$141$	42.4118	3.57171
$142$	0	0
$143$	16.0511	1.34226
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0.108674	0.00890294	0.00445147	−	0.999990i	$-0.498583\pi$
$149$	0.108674	0.00890294	0.00445147	+	0.999990i	$0.498583\pi$
$150$	0	0
$151$	−13.0676	−1.06343	−0.531713	−	0.846925i	$-0.678451\pi$
$151$	−13.0676	−1.06343	−0.531713	+	0.846925i	$0.678451\pi$
$152$	0	0
$153$	15.6475	1.26502
$154$	0	0
$155$	0	0
$156$	0	0
$157$	8.89133	0.709605	0.354803	−	0.934941i	$-0.384548\pi$
$157$	8.89133	0.709605	0.354803	+	0.934941i	$0.384548\pi$
$158$	0	0
$159$	−27.6475	−2.19259
$160$	0	0
$161$	0	0
$162$	0	0
$163$	11.5123	0.901713	0.450857	−	0.892596i	$-0.351119\pi$
$163$	11.5123	0.901713	0.450857	+	0.892596i	$0.351119\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.46621	0.113458	0.0567292	−	0.998390i	$-0.481933\pi$
$167$	1.46621	0.113458	0.0567292	+	0.998390i	$0.481933\pi$
$168$	0	0
$169$	−5.40363	−0.415664
$170$	0	0
$171$	5.91593	0.452403
$172$	0	0
$173$	−11.2438	−0.854854	−0.427427	−	0.904050i	$-0.640580\pi$
$173$	−11.2438	−0.854854	−0.427427	+	0.904050i	$0.640580\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	26.3196	1.97830
$178$	0	0
$179$	7.33604	0.548321	0.274161	−	0.961684i	$-0.411600\pi$
$179$	7.33604	0.548321	0.274161	+	0.961684i	$0.411600\pi$
$180$	0	0
$181$	18.4712	1.37295	0.686477	−	0.727151i	$-0.259156\pi$
$181$	18.4712	1.37295	0.686477	+	0.727151i	$0.259156\pi$
$182$	0	0
$183$	30.9374	2.28696
$184$	0	0
$185$	0	0
$186$	0	0
$187$	11.6475	0.851748
$188$	0	0
$189$	0	0
$190$	0	0
$191$	14.5799	1.05496	0.527482	−	0.849566i	$-0.323136\pi$
$191$	14.5799	1.05496	0.527482	+	0.849566i	$0.323136\pi$
$192$	0	0
$193$	18.8073	1.35378	0.676888	−	0.736086i	$-0.263328\pi$
$193$	18.8073	1.35378	0.676888	+	0.736086i	$0.263328\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2.75615	−0.196368	−0.0981838	−	0.995168i	$-0.531303\pi$
$197$	−2.75615	−0.196368	−0.0981838	+	0.995168i	$0.531303\pi$
$198$	0	0
$199$	−15.0246	−1.06507	−0.532533	−	0.846409i	$-0.678760\pi$
$199$	−15.0246	−1.06507	−0.532533	+	0.846409i	$0.678760\pi$
$200$	0	0
$201$	39.0511	2.75445
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	4.17626	0.290270
$208$	0	0
$209$	4.40363	0.304605
$210$	0	0
$211$	−21.9159	−1.50875	−0.754377	−	0.656441i	$-0.772061\pi$
$211$	−21.9159	−1.50875	−0.754377	+	0.656441i	$0.772061\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−11.5553	−0.780834
$220$	0	0
$221$	5.51230	0.370798
$222$	0	0
$223$	18.8073	1.25943	0.629714	−	0.776827i	$-0.283172\pi$
$223$	18.8073	1.25943	0.629714	+	0.776827i	$0.283172\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	16.6721	1.10657	0.553283	−	0.832994i	$-0.313375\pi$
$227$	16.6721	1.10657	0.553283	+	0.832994i	$0.313375\pi$
$228$	0	0
$229$	−19.5123	−1.28941	−0.644705	−	0.764432i	$-0.723020\pi$
$229$	−19.5123	−1.28941	−0.644705	+	0.764432i	$0.723020\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	31.2950	2.03283
$238$	0	0
$239$	−16.0922	−1.04092	−0.520459	−	0.853887i	$-0.674239\pi$
$239$	−16.0922	−1.04092	−0.520459	+	0.853887i	$0.674239\pi$
$240$	0	0
$241$	−0.891326	−0.0574153	−0.0287077	−	0.999588i	$-0.509139\pi$
$241$	−0.891326	−0.0574153	−0.0287077	+	0.999588i	$0.509139\pi$
$242$	0	0
$243$	−46.9424	−3.01136
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.08407	0.132606
$248$	0	0
$249$	22.0757	1.39899
$250$	0	0
$251$	−17.4712	−1.10277	−0.551387	−	0.834250i	$-0.685901\pi$
$251$	−17.4712	−1.10277	−0.551387	+	0.834250i	$0.685901\pi$
$252$	0	0
$253$	3.10867	0.195441
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−7.02461	−0.438183	−0.219091	−	0.975704i	$-0.570309\pi$
$257$	−7.02461	−0.438183	−0.219091	+	0.975704i	$0.570309\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−6.44472	−0.398918
$262$	0	0
$263$	−17.7776	−1.09622	−0.548108	−	0.836407i	$-0.684652\pi$
$263$	−17.7776	−1.09622	−0.548108	+	0.836407i	$0.684652\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	5.77764	0.353586
$268$	0	0
$269$	9.40363	0.573349	0.286675	−	0.958028i	$-0.407450\pi$
$269$	9.40363	0.573349	0.286675	+	0.958028i	$0.407450\pi$
$270$	0	0
$271$	−7.02461	−0.426714	−0.213357	−	0.976974i	$-0.568440\pi$
$271$	−7.02461	−0.426714	−0.213357	+	0.976974i	$0.568440\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−27.1598	−1.63187	−0.815937	−	0.578141i	$-0.803778\pi$
$277$	−27.1598	−1.63187	−0.815937	+	0.578141i	$0.803778\pi$
$278$	0	0
$279$	20.1844	1.20841
$280$	0	0
$281$	−0.620977	−0.0370444	−0.0185222	−	0.999828i	$-0.505896\pi$
$281$	−0.620977	−0.0370444	−0.0185222	+	0.999828i	$0.505896\pi$
$282$	0	0
$283$	3.15978	0.187829	0.0939147	−	0.995580i	$-0.470062\pi$
$283$	3.15978	0.187829	0.0939147	+	0.995580i	$0.470062\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	6.57989	0.385720
$292$	0	0
$293$	11.2438	0.656873	0.328436	−	0.944526i	$-0.393478\pi$
$293$	11.2438	0.656873	0.328436	+	0.944526i	$0.393478\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−92.4218	−5.36286
$298$	0	0
$299$	1.47122	0.0850826
$300$	0	0
$301$	0	0
$302$	0	0
$303$	7.68545	0.441518
$304$	0	0
$305$	0	0
$306$	0	0
$307$	23.5173	1.34220	0.671102	−	0.741365i	$-0.265821\pi$
$307$	23.5173	1.34220	0.671102	+	0.741365i	$0.265821\pi$
$308$	0	0
$309$	7.31144	0.415933
$310$	0	0
$311$	29.1598	1.65350	0.826750	−	0.562570i	$-0.190187\pi$
$311$	29.1598	1.65350	0.826750	+	0.562570i	$0.190187\pi$
$312$	0	0
$313$	−24.1844	−1.36698	−0.683491	−	0.729959i	$-0.739539\pi$
$313$	−24.1844	−1.36698	−0.683491	+	0.729959i	$0.739539\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−12.1352	−0.681579	−0.340790	−	0.940140i	$-0.610694\pi$
$317$	−12.1352	−0.681579	−0.340790	+	0.940140i	$0.610694\pi$
$318$	0	0
$319$	−4.79724	−0.268594
$320$	0	0
$321$	52.2109	2.91413
$322$	0	0
$323$	1.51230	0.0841468
$324$	0	0
$325$	0	0
$326$	0	0
$327$	10.7530	0.594644
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−1.82374	−0.100242	−0.0501209	−	0.998743i	$-0.515961\pi$
$331$	−1.82374	−0.100242	−0.0501209	+	0.998743i	$0.515961\pi$
$332$	0	0
$333$	−37.2109	−2.03914
$334$	0	0
$335$	0	0
$336$	0	0
$337$	17.7827	0.968683	0.484341	−	0.874879i	$-0.339059\pi$
$337$	17.7827	0.968683	0.484341	+	0.874879i	$0.339059\pi$
$338$	0	0
$339$	−43.2950	−2.35146
$340$	0	0
$341$	15.0246	0.813628
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.28995	0.0692479	0.0346239	−	0.999400i	$-0.488977\pi$
$347$	1.28995	0.0692479	0.0346239	+	0.999400i	$0.488977\pi$
$348$	0	0
$349$	−1.31144	−0.0701995	−0.0350998	−	0.999384i	$-0.511175\pi$
$349$	−1.31144	−0.0701995	−0.0350998	+	0.999384i	$0.511175\pi$
$350$	0	0
$351$	−43.7397	−2.33465
$352$	0	0
$353$	−11.0246	−0.586781	−0.293390	−	0.955993i	$-0.594784\pi$
$353$	−11.0246	−0.586781	−0.293390	+	0.955993i	$0.594784\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	10.2274	0.539780	0.269890	−	0.962891i	$-0.413013\pi$
$359$	10.2274	0.539780	0.269890	+	0.962891i	$0.413013\pi$
$360$	0	0
$361$	−18.4282	−0.969907
$362$	0	0
$363$	−75.3922	−3.95706
$364$	0	0
$365$	0	0
$366$	0	0
$367$	17.3411	0.905196	0.452598	−	0.891715i	$-0.350497\pi$
$367$	17.3411	0.905196	0.452598	+	0.891715i	$0.350497\pi$
$368$	0	0
$369$	−47.4712	−2.47125
$370$	0	0
$371$	0	0
$372$	0	0
$373$	17.7827	0.920751	0.460375	−	0.887724i	$-0.347715\pi$
$373$	17.7827	0.920751	0.460375	+	0.887724i	$0.347715\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.27035	−0.116929
$378$	0	0
$379$	−33.2109	−1.70593	−0.852964	−	0.521969i	$-0.825198\pi$
$379$	−33.2109	−1.70593	−0.852964	+	0.521969i	$0.825198\pi$
$380$	0	0
$381$	39.2028	2.00842
$382$	0	0
$383$	14.7612	0.754260	0.377130	−	0.926160i	$-0.376911\pi$
$383$	14.7612	0.754260	0.377130	+	0.926160i	$0.376911\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−5.55528	−0.282391
$388$	0	0
$389$	−22.8073	−1.15637	−0.578187	−	0.815904i	$-0.696240\pi$
$389$	−22.8073	−1.15637	−0.578187	+	0.815904i	$0.696240\pi$
$390$	0	0
$391$	1.06759	0.0539902
$392$	0	0
$393$	70.9424	3.57857
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−14.2703	−0.716208	−0.358104	−	0.933682i	$-0.616577\pi$
$397$	−14.2703	−0.716208	−0.358104	+	0.933682i	$0.616577\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−11.4877	−0.573668	−0.286834	−	0.957980i	$-0.592603\pi$
$401$	−11.4877	−0.573668	−0.286834	+	0.957980i	$0.592603\pi$
$402$	0	0
$403$	7.11057	0.354203
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−27.6986	−1.37297
$408$	0	0
$409$	−1.79913	−0.0889614	−0.0444807	−	0.999010i	$-0.514163\pi$
$409$	−1.79913	−0.0889614	−0.0444807	+	0.999010i	$0.514163\pi$
$410$	0	0
$411$	13.1598	0.649124
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	21.6475	1.06008
$418$	0	0
$419$	−27.4282	−1.33996	−0.669978	−	0.742381i	$-0.733697\pi$
$419$	−27.4282	−1.33996	−0.669978	+	0.742381i	$0.733697\pi$
$420$	0	0
$421$	2.91593	0.142114	0.0710569	−	0.997472i	$-0.477363\pi$
$421$	2.91593	0.142114	0.0710569	+	0.997472i	$0.477363\pi$
$422$	0	0
$423$	−100.858	−4.90390
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−52.8073	−2.54956
$430$	0	0
$431$	14.5799	0.702289	0.351144	−	0.936321i	$-0.385793\pi$
$431$	14.5799	0.702289	0.351144	+	0.936321i	$0.385793\pi$
$432$	0	0
$433$	21.1598	1.01687	0.508437	−	0.861099i	$-0.330223\pi$
$433$	21.1598	1.01687	0.508437	+	0.861099i	$0.330223\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.403629	0.0193082
$438$	0	0
$439$	30.9424	1.47680	0.738401	−	0.674362i	$-0.235581\pi$
$439$	30.9424	1.47680	0.738401	+	0.674362i	$0.235581\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−37.5173	−1.78250	−0.891251	−	0.453511i	$-0.850171\pi$
$443$	−37.5173	−1.78250	−0.891251	+	0.453511i	$0.850171\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−0.357532	−0.0169107
$448$	0	0
$449$	31.0922	1.46733	0.733666	−	0.679511i	$-0.237808\pi$
$449$	31.0922	1.46733	0.733666	+	0.679511i	$0.237808\pi$
$450$	0	0
$451$	−35.3360	−1.66391
$452$	0	0
$453$	42.9916	2.01992
$454$	0	0
$455$	0	0
$456$	0	0
$457$	4.35252	0.203602	0.101801	−	0.994805i	$-0.467539\pi$
$457$	4.35252	0.203602	0.101801	+	0.994805i	$0.467539\pi$
$458$	0	0
$459$	−31.7397	−1.48148
$460$	0	0
$461$	−37.2950	−1.73700	−0.868500	−	0.495690i	$-0.834915\pi$
$461$	−37.2950	−1.73700	−0.868500	+	0.495690i	$0.834915\pi$
$462$	0	0
$463$	9.81873	0.456315	0.228158	−	0.973624i	$-0.426730\pi$
$463$	9.81873	0.456315	0.228158	+	0.973624i	$0.426730\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−22.8022	−1.05516	−0.527581	−	0.849505i	$-0.676901\pi$
$467$	−22.8022	−1.05516	−0.527581	+	0.849505i	$0.676901\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−29.2520	−1.34786
$472$	0	0
$473$	−4.13517	−0.190136
$474$	0	0
$475$	0	0
$476$	0	0
$477$	65.7478	3.01038
$478$	0	0
$479$	20.4447	0.934143	0.467071	−	0.884220i	$-0.345309\pi$
$479$	20.4447	0.934143	0.467071	+	0.884220i	$0.345309\pi$
$480$	0	0
$481$	−13.1087	−0.597704
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	42.8073	1.93978	0.969891	−	0.243539i	$-0.0783085\pi$
$487$	42.8073	1.93978	0.969891	+	0.243539i	$0.0783085\pi$
$488$	0	0
$489$	−37.8748	−1.71276
$490$	0	0
$491$	−18.8502	−0.850699	−0.425350	−	0.905029i	$-0.639849\pi$
$491$	−18.8502	−0.850699	−0.425350	+	0.905029i	$0.639849\pi$
$492$	0	0
$493$	−1.64748	−0.0741986
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	27.7397	1.24180	0.620899	−	0.783890i	$-0.286768\pi$
$499$	27.7397	1.24180	0.620899	+	0.783890i	$0.286768\pi$
$500$	0	0
$501$	−4.82374	−0.215509
$502$	0	0
$503$	14.7101	0.655889	0.327944	−	0.944697i	$-0.393644\pi$
$503$	14.7101	0.655889	0.327944	+	0.944697i	$0.393644\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	17.7776	0.789533
$508$	0	0
$509$	13.9835	0.619809	0.309904	−	0.950768i	$-0.399703\pi$
$509$	13.9835	0.619809	0.309904	+	0.950768i	$0.399703\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−12.0000	−0.529813
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−75.0757	−3.30183
$518$	0	0
$519$	36.9916	1.62375
$520$	0	0
$521$	−20.7232	−0.907899	−0.453950	−	0.891027i	$-0.649985\pi$
$521$	−20.7232	−0.907899	−0.453950	+	0.891027i	$0.649985\pi$
$522$	0	0
$523$	−22.3525	−0.977408	−0.488704	−	0.872450i	$-0.662530\pi$
$523$	−22.3525	−0.977408	−0.488704	+	0.872450i	$0.662530\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.15978	0.224764
$528$	0	0
$529$	−22.7151	−0.987611
$530$	0	0
$531$	−62.5899	−2.71617
$532$	0	0
$533$	−16.7232	−0.724362
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−24.1352	−1.04151
$538$	0	0
$539$	0	0
$540$	0	0
$541$	32.6556	1.40397	0.701987	−	0.712190i	$-0.252296\pi$
$541$	32.6556	1.40397	0.701987	+	0.712190i	$0.252296\pi$
$542$	0	0
$543$	−60.7693	−2.60786
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−31.1648	−1.33251	−0.666255	−	0.745724i	$-0.732104\pi$
$547$	−31.1648	−1.33251	−0.666255	+	0.745724i	$0.732104\pi$
$548$	0	0
$549$	−73.5715	−3.13996
$550$	0	0
$551$	−0.622871	−0.0265352
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	44.7232	1.89498	0.947491	−	0.319782i	$-0.103610\pi$
$557$	44.7232	1.89498	0.947491	+	0.319782i	$0.103610\pi$
$558$	0	0
$559$	−1.95702	−0.0827731
$560$	0	0
$561$	−38.3196	−1.61785
$562$	0	0
$563$	−32.4927	−1.36940	−0.684702	−	0.728823i	$-0.740068\pi$
$563$	−32.4927	−1.36940	−0.684702	+	0.728823i	$0.740068\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	34.6740	1.45361	0.726804	−	0.686845i	$-0.241005\pi$
$569$	34.6740	1.45361	0.726804	+	0.686845i	$0.241005\pi$
$570$	0	0
$571$	3.11057	0.130173	0.0650866	−	0.997880i	$-0.479268\pi$
$571$	3.11057	0.130173	0.0650866	+	0.997880i	$0.479268\pi$
$572$	0	0
$573$	−47.9670	−2.00385
$574$	0	0
$575$	0	0
$576$	0	0
$577$	0.672083	0.0279792	0.0139896	−	0.999902i	$-0.495547\pi$
$577$	0.672083	0.0279792	0.0139896	+	0.999902i	$0.495547\pi$
$578$	0	0
$579$	−61.8748	−2.57143
$580$	0	0
$581$	0	0
$582$	0	0
$583$	48.9405	2.02691
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−4.67208	−0.192838	−0.0964188	−	0.995341i	$-0.530739\pi$
$587$	−4.67208	−0.192838	−0.0964188	+	0.995341i	$0.530739\pi$
$588$	0	0
$589$	1.95079	0.0803808
$590$	0	0
$591$	9.06759	0.372991
$592$	0	0
$593$	20.1844	0.828873	0.414437	−	0.910078i	$-0.363979\pi$
$593$	20.1844	0.828873	0.414437	+	0.910078i	$0.363979\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	49.4301	2.02304
$598$	0	0
$599$	21.5123	0.878969	0.439484	−	0.898250i	$-0.355161\pi$
$599$	21.5123	0.878969	0.439484	+	0.898250i	$0.355161\pi$
$600$	0	0
$601$	−5.29495	−0.215986	−0.107993	−	0.994152i	$-0.534442\pi$
$601$	−5.29495	−0.215986	−0.107993	+	0.994152i	$0.534442\pi$
$602$	0	0
$603$	−92.8665	−3.78182
$604$	0	0
$605$	0	0
$606$	0	0
$607$	36.7182	1.49034	0.745172	−	0.666872i	$-0.232367\pi$
$607$	36.7182	1.49034	0.745172	+	0.666872i	$0.232367\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−35.5304	−1.43741
$612$	0	0
$613$	−40.7232	−1.64479	−0.822397	−	0.568914i	$-0.807364\pi$
$613$	−40.7232	−1.64479	−0.822397	+	0.568914i	$0.807364\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−16.8073	−0.676635	−0.338317	−	0.941032i	$-0.609858\pi$
$617$	−16.8073	−0.676635	−0.338317	+	0.941032i	$0.609858\pi$
$618$	0	0
$619$	35.7908	1.43855	0.719276	−	0.694724i	$-0.244474\pi$
$619$	35.7908	1.43855	0.719276	+	0.694724i	$0.244474\pi$
$620$	0	0
$621$	−8.47122	−0.339938
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−14.4877	−0.578583
$628$	0	0
$629$	−9.51230	−0.379280
$630$	0	0
$631$	−32.4447	−1.29160	−0.645802	−	0.763505i	$-0.723477\pi$
$631$	−32.4447	−1.29160	−0.645802	+	0.763505i	$0.723477\pi$
$632$	0	0
$633$	72.1022	2.86581
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	16.9078	0.667818	0.333909	−	0.942605i	$-0.391632\pi$
$641$	16.9078	0.667818	0.333909	+	0.942605i	$0.391632\pi$
$642$	0	0
$643$	0.135174	0.00533075	0.00266538	−	0.999996i	$-0.499152\pi$
$643$	0.135174	0.00533075	0.00266538	+	0.999996i	$0.499152\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−4.08908	−0.160758	−0.0803791	−	0.996764i	$-0.525613\pi$
$647$	−4.08908	−0.160758	−0.0803791	+	0.996764i	$0.525613\pi$
$648$	0	0
$649$	−46.5899	−1.82881
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−31.9159	−1.24897	−0.624483	−	0.781038i	$-0.714690\pi$
$653$	−31.9159	−1.24897	−0.624483	+	0.781038i	$0.714690\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	27.4793	1.07207
$658$	0	0
$659$	4.70505	0.183283	0.0916413	−	0.995792i	$-0.470789\pi$
$659$	4.70505	0.183283	0.0916413	+	0.995792i	$0.470789\pi$
$660$	0	0
$661$	−4.51420	−0.175582	−0.0877910	−	0.996139i	$-0.527981\pi$
$661$	−4.51420	−0.175582	−0.0877910	+	0.996139i	$0.527981\pi$
$662$	0	0
$663$	−18.1352	−0.704312
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−0.439706	−0.0170255
$668$	0	0
$669$	−61.8748	−2.39222
$670$	0	0
$671$	−54.7643	−2.11415
$672$	0	0
$673$	−46.9424	−1.80950	−0.904749	−	0.425945i	$-0.859942\pi$
$673$	−46.9424	−1.80950	−0.904749	+	0.425945i	$0.859942\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−10.4036	−0.399844	−0.199922	−	0.979812i	$-0.564069\pi$
$677$	−10.4036	−0.399844	−0.199922	+	0.979812i	$0.564069\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−54.8502	−2.10187
$682$	0	0
$683$	−5.06258	−0.193714	−0.0968571	−	0.995298i	$-0.530879\pi$
$683$	−5.06258	−0.193714	−0.0968571	+	0.995298i	$0.530879\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	64.1944	2.44917
$688$	0	0
$689$	23.1617	0.882390
$690$	0	0
$691$	4.44472	0.169085	0.0845425	−	0.996420i	$-0.473057\pi$
$691$	4.44472	0.169085	0.0845425	+	0.996420i	$0.473057\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−12.1352	−0.459653
$698$	0	0
$699$	59.2190	2.23987
$700$	0	0
$701$	21.7562	0.821719	0.410859	−	0.911699i	$-0.365229\pi$
$701$	21.7562	0.821719	0.410859	+	0.911699i	$0.365229\pi$
$702$	0	0
$703$	−3.59637	−0.135640
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−41.0081	−1.54009	−0.770046	−	0.637988i	$-0.779767\pi$
$709$	−41.0081	−1.54009	−0.770046	+	0.637988i	$0.779767\pi$
$710$	0	0
$711$	−74.4218	−2.79103
$712$	0	0
$713$	1.37713	0.0515739
$714$	0	0
$715$	0	0
$716$	0	0
$717$	52.9424	1.97717
$718$	0	0
$719$	−23.7397	−0.885340	−0.442670	−	0.896685i	$-0.645969\pi$
$719$	−23.7397	−0.885340	−0.442670	+	0.896685i	$0.645969\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	2.93241	0.109058
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−28.8534	−1.07011	−0.535056	−	0.844817i	$-0.679709\pi$
$727$	−28.8534	−1.07011	−0.535056	+	0.844817i	$0.679709\pi$
$728$	0	0
$729$	68.2190	2.52663
$730$	0	0
$731$	−1.42011	−0.0525247
$732$	0	0
$733$	46.8584	1.73075	0.865377	−	0.501122i	$-0.167079\pi$
$733$	46.8584	1.73075	0.865377	+	0.501122i	$0.167079\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−69.1268	−2.54632
$738$	0	0
$739$	16.1433	0.593841	0.296920	−	0.954902i	$-0.404040\pi$
$739$	16.1433	0.593841	0.296920	+	0.954902i	$0.404040\pi$
$740$	0	0
$741$	−6.85647	−0.251879
$742$	0	0
$743$	16.2243	0.595210	0.297605	−	0.954689i	$-0.403812\pi$
$743$	16.2243	0.595210	0.297605	+	0.954689i	$0.403812\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−52.4977	−1.92079
$748$	0	0
$749$	0	0
$750$	0	0
$751$	21.5123	0.784995	0.392498	−	0.919753i	$-0.371611\pi$
$751$	21.5123	0.784995	0.392498	+	0.919753i	$0.371611\pi$
$752$	0	0
$753$	57.4793	2.09466
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−0.840220	−0.0305383	−0.0152692	−	0.999883i	$-0.504861\pi$
$757$	−0.840220	−0.0305383	−0.0152692	+	0.999883i	$0.504861\pi$
$758$	0	0
$759$	−10.2274	−0.371230
$760$	0	0
$761$	28.8913	1.04731	0.523655	−	0.851930i	$-0.324568\pi$
$761$	28.8913	1.04731	0.523655	+	0.851930i	$0.324568\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−22.0492	−0.796151
$768$	0	0
$769$	27.3790	0.987313	0.493656	−	0.869657i	$-0.335660\pi$
$769$	27.3790	0.987313	0.493656	+	0.869657i	$0.335660\pi$
$770$	0	0
$771$	23.1106	0.832307
$772$	0	0
$773$	−38.2355	−1.37524	−0.687618	−	0.726073i	$-0.741343\pi$
$773$	−38.2355	−1.37524	−0.687618	+	0.726073i	$0.741343\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−4.58802	−0.164383
$780$	0	0
$781$	0	0
$782$	0	0
$783$	13.0726	0.467176
$784$	0	0
$785$	0	0
$786$	0	0
$787$	7.33293	0.261391	0.130695	−	0.991423i	$-0.458279\pi$
$787$	7.33293	0.261391	0.130695	+	0.991423i	$0.458279\pi$
$788$	0	0
$789$	58.4875	2.08221
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−25.9178	−0.920369
$794$	0	0
$795$	0	0
$796$	0	0
$797$	14.4877	0.513181	0.256590	−	0.966520i	$-0.417401\pi$
$797$	14.4877	0.513181	0.256590	+	0.966520i	$0.417401\pi$
$798$	0	0
$799$	−25.7827	−0.912125
$800$	0	0
$801$	−13.7397	−0.485467
$802$	0	0
$803$	20.4547	0.721832
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−30.9374	−1.08905
$808$	0	0
$809$	−11.3525	−0.399133	−0.199567	−	0.979884i	$-0.563953\pi$
$809$	−11.3525	−0.399133	−0.199567	+	0.979884i	$0.563953\pi$
$810$	0	0
$811$	28.7662	1.01012	0.505058	−	0.863085i	$-0.331471\pi$
$811$	28.7662	1.01012	0.505058	+	0.863085i	$0.331471\pi$
$812$	0	0
$813$	23.1106	0.810523
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−0.536909	−0.0187841
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−16.0492	−0.560121	−0.280061	−	0.959982i	$-0.590355\pi$
$821$	−16.0492	−0.560121	−0.280061	+	0.959982i	$0.590355\pi$
$822$	0	0
$823$	−14.0050	−0.488184	−0.244092	−	0.969752i	$-0.578490\pi$
$823$	−14.0050	−0.488184	−0.244092	+	0.969752i	$0.578490\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−15.2899	−0.531683	−0.265842	−	0.964017i	$-0.585650\pi$
$827$	−15.2899	−0.531683	−0.265842	+	0.964017i	$0.585650\pi$
$828$	0	0
$829$	21.2950	0.739604	0.369802	−	0.929111i	$-0.379425\pi$
$829$	21.2950	0.739604	0.369802	+	0.929111i	$0.379425\pi$
$830$	0	0
$831$	89.3542	3.09966
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−40.9424	−1.41518
$838$	0	0
$839$	−42.2274	−1.45785	−0.728925	−	0.684593i	$-0.759980\pi$
$839$	−42.2274	−1.45785	−0.728925	+	0.684593i	$0.759980\pi$
$840$	0	0
$841$	−28.3215	−0.976602
$842$	0	0
$843$	2.04298	0.0703640
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−10.3955	−0.356773
$850$	0	0
$851$	−2.53880	−0.0870290
$852$	0	0
$853$	−13.2931	−0.455146	−0.227573	−	0.973761i	$-0.573079\pi$
$853$	−13.2931	−0.455146	−0.227573	+	0.973761i	$0.573079\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	38.4547	1.31359	0.656794	−	0.754070i	$-0.271912\pi$
$857$	38.4547	1.31359	0.656794	+	0.754070i	$0.271912\pi$
$858$	0	0
$859$	−8.00000	−0.272956	−0.136478	−	0.990643i	$-0.543578\pi$
$859$	−8.00000	−0.272956	−0.136478	+	0.990643i	$0.543578\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	47.0807	1.60265	0.801323	−	0.598232i	$-0.204130\pi$
$863$	47.0807	1.60265	0.801323	+	0.598232i	$0.204130\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	42.7693	1.45252
$868$	0	0
$869$	−55.3972	−1.87922
$870$	0	0
$871$	−32.7151	−1.10851
$872$	0	0
$873$	−15.6475	−0.529587
$874$	0	0
$875$	0	0
$876$	0	0
$877$	4.57177	0.154378	0.0771888	−	0.997016i	$-0.475406\pi$
$877$	4.57177	0.154378	0.0771888	+	0.997016i	$0.475406\pi$
$878$	0	0
$879$	−36.9916	−1.24770
$880$	0	0
$881$	14.2849	0.481272	0.240636	−	0.970615i	$-0.422644\pi$
$881$	14.2849	0.481272	0.240636	+	0.970615i	$0.422644\pi$
$882$	0	0
$883$	15.9670	0.537334	0.268667	−	0.963233i	$-0.413417\pi$
$883$	15.9670	0.537334	0.268667	+	0.963233i	$0.413417\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−7.16479	−0.240570	−0.120285	−	0.992739i	$-0.538381\pi$
$887$	−7.16479	−0.240570	−0.120285	+	0.992739i	$0.538381\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	167.372	5.60718
$892$	0	0
$893$	−9.74780	−0.326198
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−4.84022	−0.161610
$898$	0	0
$899$	−2.12516	−0.0708779
$900$	0	0
$901$	16.8073	0.559931
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−8.26534	−0.274446	−0.137223	−	0.990540i	$-0.543818\pi$
$907$	−8.26534	−0.274446	−0.137223	+	0.990540i	$0.543818\pi$
$908$	0	0
$909$	−18.2766	−0.606196
$910$	0	0
$911$	26.2274	0.868951	0.434476	−	0.900684i	$-0.356934\pi$
$911$	26.2274	0.868951	0.434476	+	0.900684i	$0.356934\pi$
$912$	0	0
$913$	−39.0776	−1.29328
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−18.7580	−0.618771	−0.309385	−	0.950937i	$-0.600123\pi$
$919$	−18.7580	−0.618771	−0.309385	+	0.950937i	$0.600123\pi$
$920$	0	0
$921$	−77.3707	−2.54945
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−17.3871	−0.571069
$928$	0	0
$929$	12.9078	0.423491	0.211746	−	0.977325i	$-0.432085\pi$
$929$	12.9078	0.423491	0.211746	+	0.977325i	$0.432085\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−95.9341	−3.14074
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1.78265	0.0582367	0.0291183	−	0.999576i	$-0.490730\pi$
$937$	1.78265	0.0582367	0.0291183	+	0.999576i	$0.490730\pi$
$938$	0	0
$939$	79.5653	2.59652
$940$	0	0
$941$	45.1268	1.47109	0.735546	−	0.677475i	$-0.236926\pi$
$941$	45.1268	1.47109	0.735546	+	0.677475i	$0.236926\pi$
$942$	0	0
$943$	−3.23884	−0.105471
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−22.0872	−0.717737	−0.358869	−	0.933388i	$-0.616837\pi$
$947$	−22.0872	−0.717737	−0.358869	+	0.933388i	$0.616837\pi$
$948$	0	0
$949$	9.68044	0.314240
$950$	0	0
$951$	39.9241	1.29463
$952$	0	0
$953$	45.3442	1.46884	0.734421	−	0.678694i	$-0.237454\pi$
$953$	45.3442	1.46884	0.734421	+	0.678694i	$0.237454\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	15.7827	0.510181
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−24.3442	−0.785296
$962$	0	0
$963$	−124.161	−4.00105
$964$	0	0
$965$	0	0
$966$	0	0
$967$	17.0296	0.547636	0.273818	−	0.961782i	$-0.411713\pi$
$967$	17.0296	0.547636	0.273818	+	0.961782i	$0.411713\pi$
$968$	0	0
$969$	−4.97539	−0.159833
$970$	0	0
$971$	29.2109	0.937422	0.468711	−	0.883352i	$-0.344719\pi$
$971$	29.2109	0.937422	0.468711	+	0.883352i	$0.344719\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−0.840220	−0.0268810	−0.0134405	−	0.999910i	$-0.504278\pi$
$977$	−0.840220	−0.0268810	−0.0134405	+	0.999910i	$0.504278\pi$
$978$	0	0
$979$	−10.2274	−0.326868
$980$	0	0
$981$	−25.5715	−0.816436
$982$	0	0
$983$	8.31645	0.265253	0.132627	−	0.991166i	$-0.457659\pi$
$983$	8.31645	0.265253	0.132627	+	0.991166i	$0.457659\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−0.379023	−0.0120522
$990$	0	0
$991$	−20.2603	−0.643591	−0.321795	−	0.946809i	$-0.604286\pi$
$991$	−20.2603	−0.643591	−0.321795	+	0.946809i	$0.604286\pi$
$992$	0	0
$993$	6.00000	0.190404
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−53.0776	−1.68098	−0.840492	−	0.541823i	$-0.817734\pi$
$997$	−53.0776	−1.68098	−0.840492	+	0.541823i	$0.817734\pi$
$998$	0	0
$999$	75.4793	2.38806

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9800.2.a.cf.1.1		3
5.4	even	2		1960.2.a.v.1.3		3
7.3	odd	6		1400.2.q.j.401.1		6
7.5	odd	6		1400.2.q.j.1201.1		6
7.6	odd	2		9800.2.a.ce.1.3		3
20.19	odd	2		3920.2.a.cb.1.1		3
35.3	even	12		1400.2.bh.i.849.6		12
35.4	even	6		1960.2.q.w.961.1		6
35.9	even	6		1960.2.q.w.361.1		6
35.12	even	12		1400.2.bh.i.249.6		12
35.17	even	12		1400.2.bh.i.849.1		12
35.19	odd	6		280.2.q.e.81.3	✓	6
35.24	odd	6		280.2.q.e.121.3	yes	6
35.33	even	12		1400.2.bh.i.249.1		12
35.34	odd	2		1960.2.a.w.1.1		3
105.59	even	6		2520.2.bi.q.1801.3		6
105.89	even	6		2520.2.bi.q.361.3		6
140.19	even	6		560.2.q.l.81.1		6
140.59	even	6		560.2.q.l.401.1		6
140.139	even	2		3920.2.a.cc.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.q.e.81.3	✓	6	35.19	odd	6
280.2.q.e.121.3	yes	6	35.24	odd	6
560.2.q.l.81.1		6	140.19	even	6
560.2.q.l.401.1		6	140.59	even	6
1400.2.q.j.401.1		6	7.3	odd	6
1400.2.q.j.1201.1		6	7.5	odd	6
1400.2.bh.i.249.1		12	35.33	even	12
1400.2.bh.i.249.6		12	35.12	even	12
1400.2.bh.i.849.1		12	35.17	even	12
1400.2.bh.i.849.6		12	35.3	even	12
1960.2.a.v.1.3		3	5.4	even	2
1960.2.a.w.1.1		3	35.34	odd	2
1960.2.q.w.361.1		6	35.9	even	6
1960.2.q.w.961.1		6	35.4	even	6
2520.2.bi.q.361.3		6	105.89	even	6
2520.2.bi.q.1801.3		6	105.59	even	6
3920.2.a.cb.1.1		3	20.19	odd	2
3920.2.a.cc.1.3		3	140.139	even	2
9800.2.a.ce.1.3		3	7.6	odd	2
9800.2.a.cf.1.1		3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	9800.2.a.cf.1.1

Level	$9800$
Weight	$2$
Character	9800.1
Self dual	yes
Analytic conductor	$78.253$
Analytic rank	$1$
Dimension	$3$
CM	no
Inner twists	$1$