LMFDB - Embedded newform 363.6.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [363,6,Mod(1,363)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(363, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("363.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 363 = 3 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	363.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:	$58.2193265921$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 80x^{2} - 30x + 1056 $ x^4 - x^3 - 80x^2 - 30x + 1056
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$8.88086$ of defining polynomial
Character	$\chi$	$=$	363.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-10.8809 q^{2} -9.00000 q^{3} +86.3932 q^{4} +71.5321 q^{5} +97.9278 q^{6} +101.164 q^{7} -591.845 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q-10.8809 q^{2} -9.00000 q^{3} +86.3932 q^{4} +71.5321 q^{5} +97.9278 q^{6} +101.164 q^{7} -591.845 q^{8} +81.0000 q^{9} -778.331 q^{10} -777.539 q^{12} -977.090 q^{13} -1100.76 q^{14} -643.789 q^{15} +3675.20 q^{16} +1292.74 q^{17} -881.350 q^{18} +809.840 q^{19} +6179.89 q^{20} -910.479 q^{21} +3917.04 q^{23} +5326.61 q^{24} +1991.85 q^{25} +10631.6 q^{26} -729.000 q^{27} +8739.91 q^{28} -4632.85 q^{29} +7004.98 q^{30} -4480.07 q^{31} -21050.3 q^{32} -14066.1 q^{34} +7236.50 q^{35} +6997.85 q^{36} +8289.46 q^{37} -8811.76 q^{38} +8793.81 q^{39} -42335.9 q^{40} +14692.5 q^{41} +9906.80 q^{42} +5814.66 q^{43} +5794.10 q^{45} -42620.8 q^{46} +10207.9 q^{47} -33076.8 q^{48} -6572.78 q^{49} -21673.0 q^{50} -11634.6 q^{51} -84413.9 q^{52} +6666.81 q^{53} +7932.15 q^{54} -59873.6 q^{56} -7288.56 q^{57} +50409.4 q^{58} +23296.3 q^{59} -55619.0 q^{60} -32523.2 q^{61} +48747.0 q^{62} +8194.31 q^{63} +111439. q^{64} -69893.3 q^{65} -23585.4 q^{67} +111684. q^{68} -35253.4 q^{69} -78739.4 q^{70} -37017.2 q^{71} -47939.5 q^{72} +37113.5 q^{73} -90196.5 q^{74} -17926.6 q^{75} +69964.7 q^{76} -95684.2 q^{78} -49621.0 q^{79} +262895. q^{80} +6561.00 q^{81} -159867. q^{82} +78347.3 q^{83} -78659.2 q^{84} +92472.3 q^{85} -63268.5 q^{86} +41695.7 q^{87} +5391.49 q^{89} -63044.8 q^{90} -98846.6 q^{91} +338406. q^{92} +40320.6 q^{93} -111071. q^{94} +57929.6 q^{95} +189453. q^{96} +142615. q^{97} +71517.5 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 9 q^{2} - 36 q^{3} + 53 q^{4} + 42 q^{5} + 81 q^{6} + 14 q^{7} - 765 q^{8} + 324 q^{9}+O(q^{10}) $ 4 * q - 9 * q^2 - 36 * q^3 + 53 * q^4 + 42 * q^5 + 81 * q^6 + 14 * q^7 - 765 * q^8 + 324 * q^9	\( 4 q - 9 q^{2} - 36 q^{3} + 53 q^{4} + 42 q^{5} + 81 q^{6} + 14 q^{7} - 765 q^{8} + 324 q^{9} - 589 q^{10} - 477 q^{12} - 886 q^{13} - 1638 q^{14} - 378 q^{15} + 2345 q^{16} + 570 q^{17} - 729 q^{18} - 1338 q^{19} + 5307 q^{20} - 126 q^{21} + 1260 q^{23} + 6885 q^{24} - 1546 q^{25} + 8133 q^{26} - 2916 q^{27} + 7108 q^{28} - 10230 q^{29} + 5301 q^{30} - 8042 q^{31} - 11769 q^{32} - 25931 q^{34} + 24552 q^{35} + 4293 q^{36} + 18936 q^{37} - 16362 q^{38} + 7974 q^{39} - 50055 q^{40} + 3006 q^{41} + 14742 q^{42} + 21504 q^{43} + 3402 q^{45} - 17204 q^{46} + 5916 q^{47} - 21105 q^{48} + 16902 q^{49} - 22614 q^{50} - 5130 q^{51} - 94847 q^{52} + 48414 q^{53} + 6561 q^{54} - 43104 q^{56} + 12042 q^{57} + 59697 q^{58} + 30276 q^{59} - 47763 q^{60} - 106242 q^{61} - 23106 q^{62} + 1134 q^{63} + 185857 q^{64} - 7362 q^{65} - 57538 q^{67} + 114021 q^{68} - 11340 q^{69} - 19148 q^{70} + 45720 q^{71} - 61965 q^{72} + 11426 q^{73} - 30867 q^{74} + 13914 q^{75} + 127752 q^{76} - 73197 q^{78} + 68338 q^{79} + 347487 q^{80} + 26244 q^{81} - 98771 q^{82} - 146748 q^{83} - 63972 q^{84} + 185042 q^{85} + 185952 q^{86} + 92070 q^{87} - 89106 q^{89} - 47709 q^{90} + 187804 q^{91} + 363432 q^{92} + 72378 q^{93} - 285008 q^{94} - 75672 q^{95} + 105921 q^{96} + 386120 q^{97} + 342057 q^{98}+O(q^{100}) \) 4 * q - 9 * q^2 - 36 * q^3 + 53 * q^4 + 42 * q^5 + 81 * q^6 + 14 * q^7 - 765 * q^8 + 324 * q^9 - 589 * q^10 - 477 * q^12 - 886 * q^13 - 1638 * q^14 - 378 * q^15 + 2345 * q^16 + 570 * q^17 - 729 * q^18 - 1338 * q^19 + 5307 * q^20 - 126 * q^21 + 1260 * q^23 + 6885 * q^24 - 1546 * q^25 + 8133 * q^26 - 2916 * q^27 + 7108 * q^28 - 10230 * q^29 + 5301 * q^30 - 8042 * q^31 - 11769 * q^32 - 25931 * q^34 + 24552 * q^35 + 4293 * q^36 + 18936 * q^37 - 16362 * q^38 + 7974 * q^39 - 50055 * q^40 + 3006 * q^41 + 14742 * q^42 + 21504 * q^43 + 3402 * q^45 - 17204 * q^46 + 5916 * q^47 - 21105 * q^48 + 16902 * q^49 - 22614 * q^50 - 5130 * q^51 - 94847 * q^52 + 48414 * q^53 + 6561 * q^54 - 43104 * q^56 + 12042 * q^57 + 59697 * q^58 + 30276 * q^59 - 47763 * q^60 - 106242 * q^61 - 23106 * q^62 + 1134 * q^63 + 185857 * q^64 - 7362 * q^65 - 57538 * q^67 + 114021 * q^68 - 11340 * q^69 - 19148 * q^70 + 45720 * q^71 - 61965 * q^72 + 11426 * q^73 - 30867 * q^74 + 13914 * q^75 + 127752 * q^76 - 73197 * q^78 + 68338 * q^79 + 347487 * q^80 + 26244 * q^81 - 98771 * q^82 - 146748 * q^83 - 63972 * q^84 + 185042 * q^85 + 185952 * q^86 + 92070 * q^87 - 89106 * q^89 - 47709 * q^90 + 187804 * q^91 + 363432 * q^92 + 72378 * q^93 - 285008 * q^94 - 75672 * q^95 + 105921 * q^96 + 386120 * q^97 + 342057 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	−10.8809	−1.92348	−0.961742	−	0.273958i	$-0.911667\pi$
$2$	−10.8809	−1.92348	−0.961742	+	0.273958i	$0.911667\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	86.3932	2.69979
$5$	71.5321	1.27961	0.639803	−	0.768539i	$-0.279016\pi$
$5$	71.5321	1.27961	0.639803	+	0.768539i	$0.279016\pi$
$6$	97.9278	1.11052
$7$	101.164	0.780337	0.390169	−	0.920743i	$-0.372417\pi$
$7$	101.164	0.780337	0.390169	+	0.920743i	$0.372417\pi$
$8$	−591.845	−3.26951
$9$	81.0000	0.333333
$10$	−778.331	−2.46130
$11$	0	0
$11$	0	0
$12$	−777.539	−1.55872
$13$	−977.090	−1.60353	−0.801763	−	0.597642i	$-0.796104\pi$
$13$	−977.090	−1.60353	−0.801763	+	0.597642i	$0.796104\pi$
$14$	−1100.76	−1.50097
$15$	−643.789	−0.738781
$16$	3675.20	3.58907
$17$	1292.74	1.08490	0.542448	−	0.840089i	$-0.317498\pi$
$17$	1292.74	1.08490	0.542448	+	0.840089i	$0.317498\pi$
$18$	−881.350	−0.641161
$19$	809.840	0.514654	0.257327	−	0.966324i	$-0.417158\pi$
$19$	809.840	0.514654	0.257327	+	0.966324i	$0.417158\pi$
$20$	6179.89	3.45466
$21$	−910.479	−0.450528
$22$	0	0
$23$	3917.04	1.54397	0.771985	−	0.635641i	$-0.219264\pi$
$23$	3917.04	1.54397	0.771985	+	0.635641i	$0.219264\pi$
$24$	5326.61	1.88765
$25$	1991.85	0.637391
$26$	10631.6	3.08436
$27$	−729.000	−0.192450
$28$	8739.91	2.10674
$29$	−4632.85	−1.02295	−0.511474	−	0.859299i	$-0.670900\pi$
$29$	−4632.85	−1.02295	−0.511474	+	0.859299i	$0.670900\pi$
$30$	7004.98	1.42103
$31$	−4480.07	−0.837298	−0.418649	−	0.908148i	$-0.637496\pi$
$31$	−4480.07	−0.837298	−0.418649	+	0.908148i	$0.637496\pi$
$32$	−21050.3	−3.63400
$33$	0	0
$34$	−14066.1	−2.08678
$35$	7236.50	0.998524
$36$	6997.85	0.899929
$37$	8289.46	0.995456	0.497728	−	0.867333i	$-0.334168\pi$
$37$	8289.46	0.995456	0.497728	+	0.867333i	$0.334168\pi$
$38$	−8811.76	−0.989928
$39$	8793.81	0.925796
$40$	−42335.9	−4.18369
$41$	14692.5	1.36501	0.682506	−	0.730880i	$-0.260890\pi$
$41$	14692.5	1.36501	0.682506	+	0.730880i	$0.260890\pi$
$42$	9906.80	0.866583
$43$	5814.66	0.479571	0.239786	−	0.970826i	$-0.422923\pi$
$43$	5814.66	0.479571	0.239786	+	0.970826i	$0.422923\pi$
$44$	0	0
$45$	5794.10	0.426535
$46$	−42620.8	−2.96980
$47$	10207.9	0.674048	0.337024	−	0.941496i	$-0.390580\pi$
$47$	10207.9	0.674048	0.337024	+	0.941496i	$0.390580\pi$
$48$	−33076.8	−2.07215
$49$	−6572.78	−0.391074
$50$	−21673.0	−1.22601
$51$	−11634.6	−0.626365
$52$	−84413.9	−4.32918
$53$	6666.81	0.326008	0.163004	−	0.986625i	$-0.447882\pi$
$53$	6666.81	0.326008	0.163004	+	0.986625i	$0.447882\pi$
$54$	7932.15	0.370175
$55$	0	0
$56$	−59873.6	−2.55132
$57$	−7288.56	−0.297136
$58$	50409.4	1.96762
$59$	23296.3	0.871279	0.435639	−	0.900121i	$-0.356522\pi$
$59$	23296.3	0.871279	0.435639	+	0.900121i	$0.356522\pi$
$60$	−55619.0	−1.99455
$61$	−32523.2	−1.11910	−0.559550	−	0.828797i	$-0.689026\pi$
$61$	−32523.2	−1.11910	−0.559550	+	0.828797i	$0.689026\pi$
$62$	48747.0	1.61053
$63$	8194.31	0.260112
$64$	111439.	3.40086
$65$	−69893.3	−2.05188
$66$	0	0
$67$	−23585.4	−0.641884	−0.320942	−	0.947099i	$-0.603999\pi$
$67$	−23585.4	−0.641884	−0.320942	+	0.947099i	$0.603999\pi$
$68$	111684.	2.92899
$69$	−35253.4	−0.891411
$70$	−78739.4	−1.92064
$71$	−37017.2	−0.871480	−0.435740	−	0.900073i	$-0.643513\pi$
$71$	−37017.2	−0.871480	−0.435740	+	0.900073i	$0.643513\pi$
$72$	−47939.5	−1.08984
$73$	37113.5	0.815127	0.407564	−	0.913177i	$-0.366378\pi$
$73$	37113.5	0.815127	0.407564	+	0.913177i	$0.366378\pi$
$74$	−90196.5	−1.91474
$75$	−17926.6	−0.367998
$76$	69964.7	1.38946
$77$	0	0
$78$	−95684.2	−1.78075
$79$	−49621.0	−0.894536	−0.447268	−	0.894400i	$-0.647603\pi$
$79$	−49621.0	−0.894536	−0.447268	+	0.894400i	$0.647603\pi$
$80$	262895.	4.59259
$81$	6561.00	0.111111
$82$	−159867.	−2.62558
$83$	78347.3	1.24833	0.624164	−	0.781293i	$-0.285440\pi$
$83$	78347.3	1.24833	0.624164	+	0.781293i	$0.285440\pi$
$84$	−78659.2	−1.21633
$85$	92472.3	1.38824
$86$	−63268.5	−0.922447
$87$	41695.7	0.590599
$88$	0	0
$89$	5391.49	0.0721496	0.0360748	−	0.999349i	$-0.488515\pi$
$89$	5391.49	0.0721496	0.0360748	+	0.999349i	$0.488515\pi$
$90$	−63044.8	−0.820433
$91$	−98846.6	−1.25129
$92$	338406.	4.16839
$93$	40320.6	0.483414
$94$	−111071.	−1.29652
$95$	57929.6	0.658554
$96$	189453.	2.09809
$97$	142615.	1.53899	0.769495	−	0.638653i	$-0.220508\pi$
$97$	142615.	1.53899	0.769495	+	0.638653i	$0.220508\pi$
$98$	71517.5	0.752224
$99$	0	0
$100$	172082.	1.72082
$101$	63480.2	0.619206	0.309603	−	0.950866i	$-0.399804\pi$
$101$	63480.2	0.619206	0.309603	+	0.950866i	$0.399804\pi$
$102$	126595.	1.20480
$103$	−16807.9	−0.156106	−0.0780530	−	0.996949i	$-0.524870\pi$
$103$	−16807.9	−0.156106	−0.0780530	+	0.996949i	$0.524870\pi$
$104$	578286.	5.24275
$105$	−65128.5	−0.576498
$106$	−72540.6	−0.627071
$107$	15497.9	0.130862	0.0654310	−	0.997857i	$-0.479158\pi$
$107$	15497.9	0.130862	0.0654310	+	0.997857i	$0.479158\pi$
$108$	−62980.6	−0.519574
$109$	−110373.	−0.889811	−0.444905	−	0.895578i	$-0.646763\pi$
$109$	−110373.	−0.889811	−0.444905	+	0.895578i	$0.646763\pi$
$110$	0	0
$111$	−74605.1	−0.574727
$112$	371799.	2.80068
$113$	−113531.	−0.836406	−0.418203	−	0.908354i	$-0.637340\pi$
$113$	−113531.	−0.836406	−0.418203	+	0.908354i	$0.637340\pi$
$114$	79305.8	0.571535
$115$	280194.	1.97567
$116$	−400247.	−2.76174
$117$	−79144.3	−0.534509
$118$	−253484.	−1.67589
$119$	130779.	0.846585
$120$	381023.	2.41545
$121$	0	0
$122$	353881.	2.15257
$123$	−132233.	−0.788090
$124$	−387047.	−2.26053
$125$	−81056.9	−0.463997
$126$	−89161.2	−0.500322
$127$	194368.	1.06934	0.534669	−	0.845061i	$-0.320436\pi$
$127$	194368.	1.06934	0.534669	+	0.845061i	$0.320436\pi$
$128$	−538947.	−2.90751
$129$	−52331.9	−0.276881
$130$	760500.	3.94676
$131$	−66068.1	−0.336367	−0.168184	−	0.985756i	$-0.553790\pi$
$131$	−66068.1	−0.336367	−0.168184	+	0.985756i	$0.553790\pi$
$132$	0	0
$133$	81926.9	0.401603
$134$	256630.	1.23465
$135$	−52146.9	−0.246260
$136$	−765101.	−3.54708
$137$	−153809.	−0.700134	−0.350067	−	0.936725i	$-0.613841\pi$
$137$	−153809.	−0.700134	−0.350067	+	0.936725i	$0.613841\pi$
$138$	383587.	1.71461
$139$	211014.	0.926348	0.463174	−	0.886267i	$-0.346710\pi$
$139$	211014.	0.926348	0.463174	+	0.886267i	$0.346710\pi$
$140$	625184.	2.69580
$141$	−91870.9	−0.389162
$142$	402779.	1.67628
$143$	0	0
$144$	297691.	1.19636
$145$	−331398.	−1.30897
$146$	−403827.	−1.56788
$147$	59155.0	0.225787
$148$	716153.	2.68752
$149$	128322.	0.473516	0.236758	−	0.971569i	$-0.423915\pi$
$149$	128322.	0.473516	0.236758	+	0.971569i	$0.423915\pi$
$150$	195057.	0.707837
$151$	−24669.3	−0.0880468	−0.0440234	−	0.999030i	$-0.514018\pi$
$151$	−24669.3	−0.0880468	−0.0440234	+	0.999030i	$0.514018\pi$
$152$	−479300.	−1.68267
$153$	104712.	0.361632
$154$	0	0
$155$	−320469.	−1.07141
$156$	759725.	2.49945
$157$	−6434.48	−0.0208336	−0.0104168	−	0.999946i	$-0.503316\pi$
$157$	−6434.48	−0.0208336	−0.0104168	+	0.999946i	$0.503316\pi$
$158$	539919.	1.72062
$159$	−60001.3	−0.188221
$160$	−1.50578e6	−4.65008
$161$	396265.	1.20482
$162$	−71389.4	−0.213720
$163$	−617833.	−1.82139	−0.910693	−	0.413083i	$-0.864452\pi$
$163$	−617833.	−1.82139	−0.910693	+	0.413083i	$0.864452\pi$
$164$	1.26933e6	3.68524
$165$	0	0
$166$	−852486.	−2.40114
$167$	129339.	0.358871	0.179436	−	0.983770i	$-0.442573\pi$
$167$	129339.	0.358871	0.179436	+	0.983770i	$0.442573\pi$
$168$	538862.	1.47301
$169$	583411.	1.57130
$170$	−1.00618e6	−2.67025
$171$	65597.0	0.171551
$172$	502347.	1.29474
$173$	522318.	1.32684	0.663422	−	0.748246i	$-0.269104\pi$
$173$	522318.	1.32684	0.663422	+	0.748246i	$0.269104\pi$
$174$	−453685.	−1.13601
$175$	201504.	0.497380
$176$	0	0
$177$	−209667.	−0.503033
$178$	−58664.1	−0.138778
$179$	475949.	1.11027	0.555134	−	0.831761i	$-0.312667\pi$
$179$	475949.	1.11027	0.555134	+	0.831761i	$0.312667\pi$
$180$	500571.	1.15155
$181$	182157.	0.413284	0.206642	−	0.978417i	$-0.433746\pi$
$181$	182157.	0.413284	0.206642	+	0.978417i	$0.433746\pi$
$182$	1.07554e6	2.40684
$183$	292709.	0.646113
$184$	−2.31828e6	−5.04803
$185$	592963.	1.27379
$186$	−438723.	−0.929839
$187$	0	0
$188$	881891.	1.81979
$189$	−73748.8	−0.150176
$190$	−630324.	−1.26672
$191$	−475150.	−0.942426	−0.471213	−	0.882020i	$-0.656184\pi$
$191$	−475150.	−0.942426	−0.471213	+	0.882020i	$0.656184\pi$
$192$	−1.00296e6	−1.96349
$193$	−569921.	−1.10134	−0.550670	−	0.834723i	$-0.685628\pi$
$193$	−569921.	−1.10134	−0.550670	+	0.834723i	$0.685628\pi$
$194$	−1.55177e6	−2.96022
$195$	629040.	1.18465
$196$	−567844.	−1.05582
$197$	871464.	1.59987	0.799934	−	0.600088i	$-0.204868\pi$
$197$	871464.	1.59987	0.799934	+	0.600088i	$0.204868\pi$
$198$	0	0
$199$	−917129.	−1.64171	−0.820857	−	0.571133i	$-0.806504\pi$
$199$	−917129.	−1.64171	−0.820857	+	0.571133i	$0.806504\pi$
$200$	−1.17886e6	−2.08396
$201$	212269.	0.370592
$202$	−690720.	−1.19103
$203$	−468679.	−0.798244
$204$	−1.00515e6	−1.69105
$205$	1.05099e6	1.74668
$206$	182884.	0.300267
$207$	317281.	0.514657
$208$	−3.59100e6	−5.75516
$209$	0	0
$210$	708654.	1.10888
$211$	538516.	0.832707	0.416353	−	0.909203i	$-0.363308\pi$
$211$	538516.	0.832707	0.416353	+	0.909203i	$0.363308\pi$
$212$	575967.	0.880153
$213$	333154.	0.503149
$214$	−168631.	−0.251711
$215$	415935.	0.613662
$216$	431455.	0.629218
$217$	−453223.	−0.653375
$218$	1.20096e6	1.71154
$219$	−334022.	−0.470614
$220$	0	0
$221$	−1.26312e6	−1.73966
$222$	811768.	1.10548
$223$	−182781.	−0.246132	−0.123066	−	0.992398i	$-0.539273\pi$
$223$	−182781.	−0.246132	−0.123066	+	0.992398i	$0.539273\pi$
$224$	−2.12954e6	−2.83574
$225$	161340.	0.212464
$226$	1.23531e6	1.60881
$227$	1.18669e6	1.52852	0.764260	−	0.644908i	$-0.223104\pi$
$227$	1.18669e6	1.52852	0.764260	+	0.644908i	$0.223104\pi$
$228$	−629682.	−0.802203
$229$	1.37491e6	1.73255	0.866273	−	0.499570i	$-0.166509\pi$
$229$	1.37491e6	1.73255	0.866273	+	0.499570i	$0.166509\pi$
$230$	−3.04876e6	−3.80017
$231$	0	0
$232$	2.74193e6	3.34454
$233$	−585649.	−0.706720	−0.353360	−	0.935487i	$-0.614961\pi$
$233$	−585649.	−0.706720	−0.353360	+	0.935487i	$0.614961\pi$
$234$	861158.	1.02812
$235$	730191.	0.862516
$236$	2.01264e6	2.35227
$237$	446589.	0.516460
$238$	−1.42299e6	−1.62839
$239$	705518.	0.798939	0.399469	−	0.916747i	$-0.369194\pi$
$239$	705518.	0.798939	0.399469	+	0.916747i	$0.369194\pi$
$240$	−2.36606e6	−2.65153
$241$	−381652.	−0.423277	−0.211639	−	0.977348i	$-0.567880\pi$
$241$	−381652.	−0.423277	−0.211639	+	0.977348i	$0.567880\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	−2.80978e6	−3.02133
$245$	−470165.	−0.500421
$246$	1.43881e6	1.51588
$247$	−791286.	−0.825261
$248$	2.65150e6	2.73756
$249$	−705126.	−0.720723
$250$	881969.	0.892490
$251$	−600901.	−0.602031	−0.301015	−	0.953619i	$-0.597326\pi$
$251$	−600901.	−0.602031	−0.301015	+	0.953619i	$0.597326\pi$
$252$	707933.	0.702248
$253$	0	0
$254$	−2.11489e6	−2.05686
$255$	−832251.	−0.801500
$256$	2.29814e6	2.19168
$257$	278206.	0.262744	0.131372	−	0.991333i	$-0.458062\pi$
$257$	278206.	0.262744	0.131372	+	0.991333i	$0.458062\pi$
$258$	569417.	0.532575
$259$	838597.	0.776791
$260$	−6.03831e6	−5.53964
$261$	−375261.	−0.340982
$262$	718878.	0.646996
$263$	715382.	0.637747	0.318874	−	0.947797i	$-0.396695\pi$
$263$	715382.	0.637747	0.318874	+	0.947797i	$0.396695\pi$
$264$	0	0
$265$	476891.	0.417162
$266$	−891435.	−0.772477
$267$	−48523.4	−0.0416556
$268$	−2.03762e6	−1.73295
$269$	1.05176e6	0.886212	0.443106	−	0.896469i	$-0.353877\pi$
$269$	1.05176e6	0.886212	0.443106	+	0.896469i	$0.353877\pi$
$270$	567404.	0.473677
$271$	1.43357e6	1.18575	0.592877	−	0.805293i	$-0.297992\pi$
$271$	1.43357e6	1.18575	0.592877	+	0.805293i	$0.297992\pi$
$272$	4.75107e6	3.89376
$273$	889620.	0.722433
$274$	1.67358e6	1.34670
$275$	0	0
$276$	−3.04565e6	−2.40662
$277$	1.35401e6	1.06028	0.530142	−	0.847909i	$-0.322139\pi$
$277$	1.35401e6	1.06028	0.530142	+	0.847909i	$0.322139\pi$
$278$	−2.29601e6	−1.78181
$279$	−362885.	−0.279099
$280$	−4.28289e6	−3.26469
$281$	−706931.	−0.534086	−0.267043	−	0.963685i	$-0.586047\pi$
$281$	−706931.	−0.534086	−0.267043	+	0.963685i	$0.586047\pi$
$282$	999635.	0.748546
$283$	1.11157e6	0.825029	0.412515	−	0.910951i	$-0.364651\pi$
$283$	1.11157e6	0.825029	0.412515	+	0.910951i	$0.364651\pi$
$284$	−3.19803e6	−2.35281
$285$	−521366.	−0.380216
$286$	0	0
$287$	1.48636e6	1.06517
$288$	−1.70508e6	−1.21133
$289$	251314.	0.177000
$290$	3.60589e6	2.51778
$291$	−1.28353e6	−0.888536
$292$	3.20636e6	2.20067
$293$	1.42443e6	0.969331	0.484665	−	0.874700i	$-0.338941\pi$
$293$	1.42443e6	0.969331	0.484665	+	0.874700i	$0.338941\pi$
$294$	−643658.	−0.434297
$295$	1.66643e6	1.11489
$296$	−4.90608e6	−3.25465
$297$	0	0
$298$	−1.39625e6	−0.910800
$299$	−3.82730e6	−2.47580
$300$	−1.54874e6	−0.993516
$301$	588236.	0.374227
$302$	268423.	0.169357
$303$	−571322.	−0.357499
$304$	2.97633e6	1.84713
$305$	−2.32645e6	−1.43201
$306$	−1.13935e6	−0.695593
$307$	−507284.	−0.307188	−0.153594	−	0.988134i	$-0.549085\pi$
$307$	−507284.	−0.307188	−0.153594	+	0.988134i	$0.549085\pi$
$308$	0	0
$309$	151271.	0.0901278
$310$	3.48698e6	2.06084
$311$	808099.	0.473766	0.236883	−	0.971538i	$-0.423874\pi$
$311$	808099.	0.473766	0.236883	+	0.971538i	$0.423874\pi$
$312$	−5.20457e6	−3.02690
$313$	2.37645e6	1.37110	0.685549	−	0.728027i	$-0.259562\pi$
$313$	2.37645e6	1.37110	0.685549	+	0.728027i	$0.259562\pi$
$314$	70012.7	0.0400731
$315$	586156.	0.332841
$316$	−4.28692e6	−2.41506
$317$	3.10494e6	1.73542	0.867711	−	0.497069i	$-0.165590\pi$
$317$	3.10494e6	1.73542	0.867711	+	0.497069i	$0.165590\pi$
$318$	652866.	0.362040
$319$	0	0
$320$	7.97150e6	4.35176
$321$	−139481.	−0.0755532
$322$	−4.31171e6	−2.31745
$323$	1.04691e6	0.558346
$324$	566826.	0.299976
$325$	−1.94621e6	−1.02207
$326$	6.72256e6	3.50341
$327$	993359.	0.513732
$328$	−8.69570e6	−4.46293
$329$	1.03267e6	0.525985
$330$	0	0
$331$	2.67415e6	1.34158	0.670789	−	0.741648i	$-0.265955\pi$
$331$	2.67415e6	1.34158	0.670789	+	0.741648i	$0.265955\pi$
$332$	6.76868e6	3.37022
$333$	671446.	0.331819
$334$	−1.40732e6	−0.690282
$335$	−1.68712e6	−0.821359
$336$	−3.34620e6	−1.61697
$337$	704011.	0.337680	0.168840	−	0.985644i	$-0.445998\pi$
$337$	704011.	0.337680	0.168840	+	0.985644i	$0.445998\pi$
$338$	−6.34802e6	−3.02236
$339$	1.02178e6	0.482899
$340$	7.98898e6	3.74795
$341$	0	0
$342$	−713752.	−0.329976
$343$	−2.36520e6	−1.08551
$344$	−3.44138e6	−1.56796
$345$	−2.52175e6	−1.14066
$346$	−5.68327e6	−2.55216
$347$	−3.31988e6	−1.48013	−0.740064	−	0.672537i	$-0.765205\pi$
$347$	−3.31988e6	−1.48013	−0.740064	+	0.672537i	$0.765205\pi$
$348$	3.60222e6	1.59449
$349$	2.93836e6	1.29134	0.645671	−	0.763616i	$-0.276578\pi$
$349$	2.93836e6	1.29134	0.645671	+	0.763616i	$0.276578\pi$
$350$	−2.19253e6	−0.956701
$351$	712298.	0.308599
$352$	0	0
$353$	−3.46361e6	−1.47942	−0.739711	−	0.672924i	$-0.765038\pi$
$353$	−3.46361e6	−1.47942	−0.739711	+	0.672924i	$0.765038\pi$
$354$	2.28136e6	0.967575
$355$	−2.64792e6	−1.11515
$356$	465788.	0.194789
$357$	−1.17701e6	−0.488776
$358$	−5.17874e6	−2.13558
$359$	−3.56447e6	−1.45969	−0.729843	−	0.683615i	$-0.760407\pi$
$359$	−3.56447e6	−1.45969	−0.729843	+	0.683615i	$0.760407\pi$
$360$	−3.42921e6	−1.39456
$361$	−1.82026e6	−0.735131
$362$	−1.98202e6	−0.794946
$363$	0	0
$364$	−8.53968e6	−3.37822
$365$	2.65481e6	1.04304
$366$	−3.18493e6	−1.24279
$367$	499957.	0.193761	0.0968807	−	0.995296i	$-0.469113\pi$
$367$	499957.	0.193761	0.0968807	+	0.995296i	$0.469113\pi$
$368$	1.43959e7	5.54141
$369$	1.19009e6	0.455004
$370$	−6.45195e6	−2.45011
$371$	674443.	0.254396
$372$	3.48343e6	1.30512
$373$	2.95268e6	1.09886	0.549432	−	0.835538i	$-0.314844\pi$
$373$	2.95268e6	1.09886	0.549432	+	0.835538i	$0.314844\pi$
$374$	0	0
$375$	729512.	0.267889
$376$	−6.04148e6	−2.20381
$377$	4.52671e6	1.64032
$378$	802451.	0.288861
$379$	4.21266e6	1.50646	0.753231	−	0.657756i	$-0.228494\pi$
$379$	4.21266e6	1.50646	0.753231	+	0.657756i	$0.228494\pi$
$380$	5.00472e6	1.77796
$381$	−1.74931e6	−0.617383
$382$	5.17004e6	1.81274
$383$	−4.34970e6	−1.51517	−0.757586	−	0.652735i	$-0.773621\pi$
$383$	−4.34970e6	−1.51517	−0.757586	+	0.652735i	$0.773621\pi$
$384$	4.85052e6	1.67865
$385$	0	0
$386$	6.20124e6	2.11841
$387$	470987.	0.159857
$388$	1.23210e7	4.15495
$389$	3.29686e6	1.10465	0.552327	−	0.833628i	$-0.313740\pi$
$389$	3.29686e6	1.10465	0.552327	+	0.833628i	$0.313740\pi$
$390$	−6.84450e6	−2.27866
$391$	5.06371e6	1.67505
$392$	3.89007e6	1.27862
$393$	594613.	0.194202
$394$	−9.48228e6	−3.07732
$395$	−3.54950e6	−1.14465
$396$	0	0
$397$	4.80212e6	1.52917	0.764587	−	0.644521i	$-0.222943\pi$
$397$	4.80212e6	1.52917	0.764587	+	0.644521i	$0.222943\pi$
$398$	9.97916e6	3.15781
$399$	−737342.	−0.231866
$400$	7.32044e6	2.28764
$401$	1.99240e6	0.618752	0.309376	−	0.950940i	$-0.399880\pi$
$401$	1.99240e6	0.618752	0.309376	+	0.950940i	$0.399880\pi$
$402$	−2.30967e6	−0.712827
$403$	4.37743e6	1.34263
$404$	5.48426e6	1.67172
$405$	469322.	0.142178
$406$	5.09963e6	1.53541
$407$	0	0
$408$	6.88590e6	2.04791
$409$	−2.37439e6	−0.701850	−0.350925	−	0.936404i	$-0.614133\pi$
$409$	−2.37439e6	−0.701850	−0.350925	+	0.936404i	$0.614133\pi$
$410$	−1.14357e7	−3.35971
$411$	1.38428e6	0.404223
$412$	−1.45209e6	−0.421453
$413$	2.35675e6	0.679891
$414$	−3.45229e6	−0.989933
$415$	5.60435e6	1.59737
$416$	2.05681e7	5.82721
$417$	−1.89913e6	−0.534827
$418$	0	0
$419$	−446560.	−0.124264	−0.0621319	−	0.998068i	$-0.519790\pi$
$419$	−446560.	−0.124264	−0.0621319	+	0.998068i	$0.519790\pi$
$420$	−5.62666e6	−1.55642
$421$	4.69041e6	1.28975	0.644875	−	0.764288i	$-0.276910\pi$
$421$	4.69041e6	1.28975	0.644875	+	0.764288i	$0.276910\pi$
$422$	−5.85951e6	−1.60170
$423$	826838.	0.224683
$424$	−3.94572e6	−1.06589
$425$	2.57493e6	0.691503
$426$	−3.62501e6	−0.967799
$427$	−3.29019e6	−0.873275
$428$	1.33891e6	0.353300
$429$	0	0
$430$	−4.52573e6	−1.18037
$431$	−5.04742e6	−1.30881	−0.654404	−	0.756145i	$-0.727081\pi$
$431$	−5.04742e6	−1.30881	−0.654404	+	0.756145i	$0.727081\pi$
$432$	−2.67922e6	−0.690716
$433$	150918.	0.0386832	0.0193416	−	0.999813i	$-0.493843\pi$
$433$	150918.	0.0386832	0.0193416	+	0.999813i	$0.493843\pi$
$434$	4.93146e6	1.25676
$435$	2.98258e6	0.755734
$436$	−9.53550e6	−2.40230
$437$	3.17218e6	0.794610
$438$	3.63445e6	0.905218
$439$	2.53024e6	0.626614	0.313307	−	0.949652i	$-0.398563\pi$
$439$	2.53024e6	0.626614	0.313307	+	0.949652i	$0.398563\pi$
$440$	0	0
$441$	−532395.	−0.130358
$442$	1.37438e7	3.34621
$443$	2.00429e6	0.485233	0.242616	−	0.970122i	$-0.421994\pi$
$443$	2.00429e6	0.485233	0.242616	+	0.970122i	$0.421994\pi$
$444$	−6.44538e6	−1.55164
$445$	385665.	0.0923230
$446$	1.98881e6	0.473432
$447$	−1.15490e6	−0.273385
$448$	1.12737e7	2.65382
$449$	4.88928e6	1.14453	0.572267	−	0.820067i	$-0.306064\pi$
$449$	4.88928e6	1.14453	0.572267	+	0.820067i	$0.306064\pi$
$450$	−1.75551e6	−0.408670
$451$	0	0
$452$	−9.80828e6	−2.25812
$453$	222023.	0.0508339
$454$	−1.29122e7	−2.94008
$455$	−7.07071e6	−1.60116
$456$	4.31370e6	0.971488
$457$	−6.10529e6	−1.36746	−0.683732	−	0.729733i	$-0.739644\pi$
$457$	−6.10529e6	−1.36746	−0.683732	+	0.729733i	$0.739644\pi$
$458$	−1.49602e7	−3.33252
$459$	−942406.	−0.208788
$460$	2.42069e7	5.33390
$461$	−3.29721e6	−0.722593	−0.361296	−	0.932451i	$-0.617666\pi$
$461$	−3.29721e6	−0.722593	−0.361296	+	0.932451i	$0.617666\pi$
$462$	0	0
$463$	3.14407e6	0.681616	0.340808	−	0.940133i	$-0.389299\pi$
$463$	3.14407e6	0.681616	0.340808	+	0.940133i	$0.389299\pi$
$464$	−1.70267e7	−3.67143
$465$	2.88422e6	0.618580
$466$	6.37236e6	1.35936
$467$	−40824.0	−0.00866210	−0.00433105	−	0.999991i	$-0.501379\pi$
$467$	−40824.0	−0.00866210	−0.00433105	+	0.999991i	$0.501379\pi$
$468$	−6.83753e6	−1.44306
$469$	−2.38600e6	−0.500886
$470$	−7.94511e6	−1.65903
$471$	57910.3	0.0120283
$472$	−1.37878e7	−2.84866
$473$	0	0
$474$	−4.85927e6	−0.993403
$475$	1.61308e6	0.328036
$476$	1.12984e7	2.28560
$477$	540012.	0.108669
$478$	−7.67665e6	−1.53675
$479$	−475207.	−0.0946333	−0.0473166	−	0.998880i	$-0.515067\pi$
$479$	−475207.	−0.0946333	−0.0473166	+	0.998880i	$0.515067\pi$
$480$	1.35520e7	2.68473
$481$	−8.09954e6	−1.59624
$482$	4.15270e6	0.814166
$483$	−3.56639e6	−0.695601
$484$	0	0
$485$	1.02016e7	1.96930
$486$	642504.	0.123392
$487$	3.30753e6	0.631949	0.315974	−	0.948768i	$-0.397669\pi$
$487$	3.30753e6	0.631949	0.315974	+	0.948768i	$0.397669\pi$
$488$	1.92487e7	3.65891
$489$	5.56050e6	1.05158
$490$	5.11580e6	0.962551
$491$	−6.14340e6	−1.15002	−0.575010	−	0.818146i	$-0.695002\pi$
$491$	−6.14340e6	−1.15002	−0.575010	+	0.818146i	$0.695002\pi$
$492$	−1.14240e7	−2.12768
$493$	−5.98906e6	−1.10979
$494$	8.60988e6	1.58738
$495$	0	0
$496$	−1.64652e7	−3.00512
$497$	−3.74482e6	−0.680048
$498$	7.67238e6	1.38630
$499$	−1.07231e7	−1.92783	−0.963913	−	0.266219i	$-0.914226\pi$
$499$	−1.07231e7	−1.92783	−0.963913	+	0.266219i	$0.914226\pi$
$500$	−7.00277e6	−1.25269
$501$	−1.16405e6	−0.207194
$502$	6.53832e6	1.15800
$503$	−9.57862e6	−1.68804	−0.844021	−	0.536311i	$-0.819818\pi$
$503$	−9.57862e6	−1.68804	−0.844021	+	0.536311i	$0.819818\pi$
$504$	−4.84976e6	−0.850441
$505$	4.54088e6	0.792339
$506$	0	0
$507$	−5.25070e6	−0.907188
$508$	1.67921e7	2.88699
$509$	5.39913e6	0.923696	0.461848	−	0.886959i	$-0.347187\pi$
$509$	5.39913e6	0.923696	0.461848	+	0.886959i	$0.347187\pi$
$510$	9.05561e6	1.54167
$511$	3.75457e6	0.636074
$512$	−7.75948e6	−1.30815
$513$	−590373.	−0.0990452
$514$	−3.02712e6	−0.505384
$515$	−1.20230e6	−0.199754
$516$	−4.52112e6	−0.747519
$517$	0	0
$518$	−9.12466e6	−1.49414
$519$	−4.70086e6	−0.766053
$520$	4.13660e7	6.70865
$521$	−2.45658e6	−0.396494	−0.198247	−	0.980152i	$-0.563525\pi$
$521$	−2.45658e6	−0.396494	−0.198247	+	0.980152i	$0.563525\pi$
$522$	4.08316e6	0.655874
$523$	−6.83211e6	−1.09220	−0.546098	−	0.837721i	$-0.683887\pi$
$523$	−6.83211e6	−1.09220	−0.546098	+	0.837721i	$0.683887\pi$
$524$	−5.70783e6	−0.908120
$525$	−1.81353e6	−0.287162
$526$	−7.78397e6	−1.22670
$527$	−5.79155e6	−0.908382
$528$	0	0
$529$	8.90689e6	1.38384
$530$	−5.18899e6	−0.802404
$531$	1.88700e6	0.290426
$532$	7.07793e6	1.08424
$533$	−1.43559e7	−2.18883
$534$	527977.	0.0801238
$535$	1.10860e6	0.167452
$536$	1.39589e7	2.09865
$537$	−4.28354e6	−0.641014
$538$	−1.14441e7	−1.70461
$539$	0	0
$540$	−4.50514e6	−0.664850
$541$	7.47535e6	1.09809	0.549045	−	0.835792i	$-0.314991\pi$
$541$	7.47535e6	1.09809	0.549045	+	0.835792i	$0.314991\pi$
$542$	−1.55984e7	−2.28078
$543$	−1.63941e6	−0.238610
$544$	−2.72126e7	−3.94251
$545$	−7.89523e6	−1.13861
$546$	−9.67983e6	−1.38959
$547$	4.77332e6	0.682106	0.341053	−	0.940044i	$-0.389216\pi$
$547$	4.77332e6	0.682106	0.341053	+	0.940044i	$0.389216\pi$
$548$	−1.32881e7	−1.89021
$549$	−2.63438e6	−0.373033
$550$	0	0
$551$	−3.75187e6	−0.526464
$552$	2.08645e7	2.91448
$553$	−5.01987e6	−0.698039
$554$	−1.47328e7	−2.03944
$555$	−5.33666e6	−0.735423
$556$	1.82302e7	2.50094
$557$	1.30189e6	0.177802	0.0889012	−	0.996040i	$-0.471664\pi$
$557$	1.30189e6	0.177802	0.0889012	+	0.996040i	$0.471664\pi$
$558$	3.94851e6	0.536843
$559$	−5.68144e6	−0.769005
$560$	2.65956e7	3.58377
$561$	0	0
$562$	7.69202e6	1.02730
$563$	5.52661e6	0.734831	0.367415	−	0.930057i	$-0.380243\pi$
$563$	5.52661e6	0.734831	0.367415	+	0.930057i	$0.380243\pi$
$564$	−7.93702e6	−1.05065
$565$	−8.12109e6	−1.07027
$566$	−1.20948e7	−1.58693
$567$	663739.	0.0867041
$568$	2.19084e7	2.84931
$569$	5.86760e6	0.759766	0.379883	−	0.925035i	$-0.375964\pi$
$569$	5.86760e6	0.759766	0.379883	+	0.925035i	$0.375964\pi$
$570$	5.67291e6	0.731340
$571$	3.98064e6	0.510931	0.255466	−	0.966818i	$-0.417771\pi$
$571$	3.98064e6	0.510931	0.255466	+	0.966818i	$0.417771\pi$
$572$	0	0
$573$	4.27635e6	0.544110
$574$	−1.61729e7	−2.04884
$575$	7.80215e6	0.984112
$576$	9.02660e6	1.13362
$577$	7.03892e6	0.880170	0.440085	−	0.897956i	$-0.354948\pi$
$577$	7.03892e6	0.880170	0.440085	+	0.897956i	$0.354948\pi$
$578$	−2.73451e6	−0.340456
$579$	5.12929e6	0.635859
$580$	−2.86305e7	−3.53394
$581$	7.92595e6	0.974117
$582$	1.39660e7	1.70908
$583$	0	0
$584$	−2.19655e7	−2.66507
$585$	−5.66136e6	−0.683960
$586$	−1.54990e7	−1.86449
$587$	−1.50660e7	−1.80469	−0.902347	−	0.431011i	$-0.858157\pi$
$587$	−1.50660e7	−1.80469	−0.902347	+	0.431011i	$0.858157\pi$
$588$	5.11059e6	0.609576
$589$	−3.62814e6	−0.430919
$590$	−1.81322e7	−2.14448
$591$	−7.84318e6	−0.923684
$592$	3.04654e7	3.57276
$593$	−5.95733e6	−0.695689	−0.347844	−	0.937552i	$-0.613086\pi$
$593$	−5.95733e6	−0.695689	−0.347844	+	0.937552i	$0.613086\pi$
$594$	0	0
$595$	9.35490e6	1.08329
$596$	1.10861e7	1.27839
$597$	8.25416e6	0.947845
$598$	4.16444e7	4.76215
$599$	−749998.	−0.0854069	−0.0427034	−	0.999088i	$-0.513597\pi$
$599$	−749998.	−0.0854069	−0.0427034	+	0.999088i	$0.513597\pi$
$600$	1.06098e7	1.20317
$601$	3.96452e6	0.447718	0.223859	−	0.974622i	$-0.428134\pi$
$601$	3.96452e6	0.447718	0.223859	+	0.974622i	$0.428134\pi$
$602$	−6.40052e6	−0.719820
$603$	−1.91042e6	−0.213961
$604$	−2.13126e6	−0.237708
$605$	0	0
$606$	6.21648e6	0.687643
$607$	−3.66853e6	−0.404129	−0.202064	−	0.979372i	$-0.564765\pi$
$607$	−3.66853e6	−0.404129	−0.202064	+	0.979372i	$0.564765\pi$
$608$	−1.70474e7	−1.87025
$609$	4.21811e6	0.460866
$610$	2.53138e7	2.75444
$611$	−9.97401e6	−1.08085
$612$	9.04639e6	0.976330
$613$	−1.87382e6	−0.201408	−0.100704	−	0.994916i	$-0.532109\pi$
$613$	−1.87382e6	−0.201408	−0.100704	+	0.994916i	$0.532109\pi$
$614$	5.51968e6	0.590872
$615$	−9.45889e6	−1.00844
$616$	0	0
$617$	6.16970e6	0.652456	0.326228	−	0.945291i	$-0.394222\pi$
$617$	6.16970e6	0.652456	0.326228	+	0.945291i	$0.394222\pi$
$618$	−1.64596e6	−0.173359
$619$	−1.14718e7	−1.20339	−0.601694	−	0.798727i	$-0.705507\pi$
$619$	−1.14718e7	−1.20339	−0.601694	+	0.798727i	$0.705507\pi$
$620$	−2.76863e7	−2.89258
$621$	−2.85552e6	−0.297137
$622$	−8.79281e6	−0.911280
$623$	545426.	0.0563010
$624$	3.23190e7	3.32274
$625$	−1.20227e7	−1.23112
$626$	−2.58579e7	−2.63728
$627$	0	0
$628$	−555896.	−0.0562463
$629$	1.07161e7	1.07997
$630$	−6.37789e6	−0.640215
$631$	−9.52206e6	−0.952045	−0.476022	−	0.879433i	$-0.657922\pi$
$631$	−9.52206e6	−0.952045	−0.476022	+	0.879433i	$0.657922\pi$
$632$	2.93679e7	2.92470
$633$	−4.84664e6	−0.480764
$634$	−3.37844e7	−3.33806
$635$	1.39036e7	1.36833
$636$	−5.18370e6	−0.508156
$637$	6.42220e6	0.627097
$638$	0	0
$639$	−2.99839e6	−0.290493
$640$	−3.85520e7	−3.72046
$641$	9.76811e6	0.939000	0.469500	−	0.882932i	$-0.344434\pi$
$641$	9.76811e6	0.939000	0.469500	+	0.882932i	$0.344434\pi$
$642$	1.51768e6	0.145325
$643$	1.35976e7	1.29698	0.648492	−	0.761222i	$-0.275400\pi$
$643$	1.35976e7	1.29698	0.648492	+	0.761222i	$0.275400\pi$
$644$	3.42346e7	3.25275
$645$	−3.74341e6	−0.354298
$646$	−1.13913e7	−1.07397
$647$	6.75765e6	0.634651	0.317325	−	0.948317i	$-0.397215\pi$
$647$	6.75765e6	0.634651	0.317325	+	0.948317i	$0.397215\pi$
$648$	−3.88310e6	−0.363279
$649$	0	0
$650$	2.11765e7	1.96594
$651$	4.07900e6	0.377226
$652$	−5.33766e7	−4.91736
$653$	1.55844e7	1.43023	0.715116	−	0.699006i	$-0.246374\pi$
$653$	1.55844e7	1.43023	0.715116	+	0.699006i	$0.246374\pi$
$654$	−1.08086e7	−0.988156
$655$	−4.72599e6	−0.430417
$656$	5.39980e7	4.89912
$657$	3.00620e6	0.271709
$658$	−1.12364e7	−1.01172
$659$	1.35146e7	1.21224	0.606121	−	0.795373i	$-0.292725\pi$
$659$	1.35146e7	1.21224	0.606121	+	0.795373i	$0.292725\pi$
$660$	0	0
$661$	−1.95264e7	−1.73827	−0.869136	−	0.494574i	$-0.835324\pi$
$661$	−1.95264e7	−1.73827	−0.869136	+	0.494574i	$0.835324\pi$
$662$	−2.90971e7	−2.58050
$663$	1.13681e7	1.00439
$664$	−4.63695e7	−4.08143
$665$	5.86041e6	0.513894
$666$	−7.30591e6	−0.638247
$667$	−1.81471e7	−1.57940
$668$	1.11740e7	0.968876
$669$	1.64503e6	0.142105
$670$	1.83573e7	1.57987
$671$	0	0
$672$	1.91659e7	1.63722
$673$	−1.26791e7	−1.07907	−0.539536	−	0.841963i	$-0.681400\pi$
$673$	−1.26791e7	−1.07907	−0.539536	+	0.841963i	$0.681400\pi$
$674$	−7.66025e6	−0.649521
$675$	−1.45206e6	−0.122666
$676$	5.04028e7	4.24217
$677$	2.09878e6	0.175993	0.0879965	−	0.996121i	$-0.471954\pi$
$677$	2.09878e6	0.175993	0.0879965	+	0.996121i	$0.471954\pi$
$678$	−1.11178e7	−0.928849
$679$	1.44275e7	1.20093
$680$	−5.47293e7	−4.53887
$681$	−1.06802e7	−0.882492
$682$	0	0
$683$	2.91186e6	0.238847	0.119423	−	0.992843i	$-0.461895\pi$
$683$	2.91186e6	0.238847	0.119423	+	0.992843i	$0.461895\pi$
$684$	5.66714e6	0.463152
$685$	−1.10023e7	−0.895896
$686$	2.57354e7	2.08795
$687$	−1.23742e7	−1.00029
$688$	2.13701e7	1.72121
$689$	−6.51407e6	−0.522762
$690$	2.74388e7	2.19403
$691$	1.09825e7	0.874992	0.437496	−	0.899220i	$-0.355865\pi$
$691$	1.09825e7	0.874992	0.437496	+	0.899220i	$0.355865\pi$
$692$	4.51247e7	3.58220
$693$	0	0
$694$	3.61232e7	2.84700
$695$	1.50943e7	1.18536
$696$	−2.46774e7	−1.93097
$697$	1.89936e7	1.48090
$698$	−3.19719e7	−2.48388
$699$	5.27084e6	0.408025
$700$	1.74086e7	1.34282
$701$	−49155.8	−0.00377816	−0.00188908	−	0.999998i	$-0.500601\pi$
$701$	−49155.8	−0.00377816	−0.00188908	+	0.999998i	$0.500601\pi$
$702$	−7.75042e6	−0.593585
$703$	6.71313e6	0.512315
$704$	0	0
$705$	−6.57172e6	−0.497974
$706$	3.76871e7	2.84564
$707$	6.42194e6	0.483189
$708$	−1.81138e7	−1.35808
$709$	−1.25234e7	−0.935638	−0.467819	−	0.883824i	$-0.654960\pi$
$709$	−1.25234e7	−0.935638	−0.467819	+	0.883824i	$0.654960\pi$
$710$	2.88116e7	2.14497
$711$	−4.01930e6	−0.298179
$712$	−3.19093e6	−0.235894
$713$	−1.75486e7	−1.29276
$714$	1.28069e7	0.940152
$715$	0	0
$716$	4.11188e7	2.99749
$717$	−6.34967e6	−0.461268
$718$	3.87846e7	2.80768
$719$	1.61469e7	1.16484	0.582422	−	0.812887i	$-0.302105\pi$
$719$	1.61469e7	1.16484	0.582422	+	0.812887i	$0.302105\pi$
$720$	2.12945e7	1.53086
$721$	−1.70036e6	−0.121815
$722$	1.98060e7	1.41401
$723$	3.43487e6	0.244379
$724$	1.57371e7	1.11578
$725$	−9.22792e6	−0.652017
$726$	0	0
$727$	1.64742e7	1.15603	0.578015	−	0.816026i	$-0.303828\pi$
$727$	1.64742e7	1.15603	0.578015	+	0.816026i	$0.303828\pi$
$728$	5.85019e7	4.09111
$729$	531441.	0.0370370
$730$	−2.88866e7	−2.00627
$731$	7.51683e6	0.520285
$732$	2.52881e7	1.74437
$733$	1.87814e7	1.29112	0.645561	−	0.763709i	$-0.276624\pi$
$733$	1.87814e7	1.29112	0.645561	+	0.763709i	$0.276624\pi$
$734$	−5.43996e6	−0.372697
$735$	4.23149e6	0.288918
$736$	−8.24551e7	−5.61078
$737$	0	0
$738$	−1.29493e7	−0.875193
$739$	−1.12716e7	−0.759235	−0.379618	−	0.925144i	$-0.623944\pi$
$739$	−1.12716e7	−0.759235	−0.379618	+	0.925144i	$0.623944\pi$
$740$	5.12279e7	3.43896
$741$	7.12158e6	0.476465
$742$	−7.33852e6	−0.489327
$743$	−8.81711e6	−0.585942	−0.292971	−	0.956121i	$-0.594644\pi$
$743$	−8.81711e6	−0.585942	−0.292971	+	0.956121i	$0.594644\pi$
$744$	−2.38635e7	−1.58053
$745$	9.17913e6	0.605914
$746$	−3.21277e7	−2.11365
$747$	6.34613e6	0.416109
$748$	0	0
$749$	1.56783e6	0.102116
$750$	−7.93772e6	−0.515279
$751$	−2.08903e7	−1.35159	−0.675795	−	0.737090i	$-0.736200\pi$
$751$	−2.08903e7	−1.35159	−0.675795	+	0.737090i	$0.736200\pi$
$752$	3.75160e7	2.41920
$753$	5.40811e6	0.347583
$754$	−4.92545e7	−3.15513
$755$	−1.76464e6	−0.112665
$756$	−6.37139e6	−0.405443
$757$	−2.42308e7	−1.53684	−0.768418	−	0.639948i	$-0.778956\pi$
$757$	−2.42308e7	−1.53684	−0.768418	+	0.639948i	$0.778956\pi$
$758$	−4.58373e7	−2.89765
$759$	0	0
$760$	−3.42853e7	−2.15315
$761$	−8.72110e6	−0.545895	−0.272948	−	0.962029i	$-0.587999\pi$
$761$	−8.72110e6	−0.545895	−0.272948	+	0.962029i	$0.587999\pi$
$762$	1.90340e7	1.18753
$763$	−1.11658e7	−0.694352
$764$	−4.10497e7	−2.54435
$765$	7.49026e6	0.462746
$766$	4.73285e7	2.91441
$767$	−2.27626e7	−1.39712
$768$	−2.06833e7	−1.26537
$769$	2.61417e7	1.59411	0.797053	−	0.603909i	$-0.206391\pi$
$769$	2.61417e7	1.59411	0.797053	+	0.603909i	$0.206391\pi$
$770$	0	0
$771$	−2.50385e6	−0.151696
$772$	−4.92373e7	−2.97339
$773$	−4.51924e6	−0.272030	−0.136015	−	0.990707i	$-0.543429\pi$
$773$	−4.51924e6	−0.272030	−0.136015	+	0.990707i	$0.543429\pi$
$774$	−5.12475e6	−0.307482
$775$	−8.92360e6	−0.533686
$776$	−8.44060e7	−5.03175
$777$	−7.54738e6	−0.448480
$778$	−3.58727e7	−2.12478
$779$	1.18986e7	0.702509
$780$	5.43448e7	3.19831
$781$	0	0
$782$	−5.50975e7	−3.22192
$783$	3.37735e6	0.196866
$784$	−2.41563e7	−1.40359
$785$	−460272.	−0.0266588
$786$	−6.46990e6	−0.373544
$787$	−4.82287e6	−0.277568	−0.138784	−	0.990323i	$-0.544319\pi$
$787$	−4.82287e6	−0.277568	−0.138784	+	0.990323i	$0.544319\pi$
$788$	7.52886e7	4.31930
$789$	−6.43844e6	−0.368204
$790$	3.86216e7	2.20172
$791$	−1.14853e7	−0.652679
$792$	0	0
$793$	3.17781e7	1.79451
$794$	−5.22512e7	−2.94134
$795$	−4.29202e6	−0.240848
$796$	−7.92337e7	−4.43228
$797$	2.13395e7	1.18998	0.594988	−	0.803735i	$-0.297157\pi$
$797$	2.13395e7	1.18998	0.594988	+	0.803735i	$0.297157\pi$
$798$	8.02292e6	0.445990
$799$	1.31961e7	0.731272
$800$	−4.19291e7	−2.31627
$801$	436711.	0.0240499
$802$	−2.16791e7	−1.19016
$803$	0	0
$804$	1.83386e7	1.00052
$805$	2.83457e7	1.54169
$806$	−4.76302e7	−2.58253
$807$	−9.46587e6	−0.511654
$808$	−3.75705e7	−2.02450
$809$	2.86686e7	1.54005	0.770026	−	0.638013i	$-0.220243\pi$
$809$	2.86686e7	1.54005	0.770026	+	0.638013i	$0.220243\pi$
$810$	−5.10663e6	−0.273478
$811$	−7.47496e6	−0.399077	−0.199539	−	0.979890i	$-0.563944\pi$
$811$	−7.47496e6	−0.399077	−0.199539	+	0.979890i	$0.563944\pi$
$812$	−4.04907e7	−2.15509
$813$	−1.29021e7	−0.684595
$814$	0	0
$815$	−4.41949e7	−2.33066
$816$	−4.27597e7	−2.24807
$817$	4.70894e6	0.246813
$818$	2.58354e7	1.35000
$819$	−8.00658e6	−0.417097
$820$	9.07982e7	4.71566
$821$	−3.31673e7	−1.71732	−0.858661	−	0.512544i	$-0.828703\pi$
$821$	−3.31673e7	−1.71732	−0.858661	+	0.512544i	$0.828703\pi$
$822$	−1.50622e7	−0.777516
$823$	2.57844e7	1.32696	0.663478	−	0.748196i	$-0.269080\pi$
$823$	2.57844e7	1.32696	0.663478	+	0.748196i	$0.269080\pi$
$824$	9.94765e6	0.510390
$825$	0	0
$826$	−2.56435e7	−1.30776
$827$	−2.10625e7	−1.07089	−0.535446	−	0.844569i	$-0.679856\pi$
$827$	−2.10625e7	−1.07089	−0.535446	+	0.844569i	$0.679856\pi$
$828$	2.74109e7	1.38946
$829$	−3.04962e7	−1.54120	−0.770600	−	0.637319i	$-0.780043\pi$
$829$	−3.04962e7	−1.54120	−0.770600	+	0.637319i	$0.780043\pi$
$830$	−6.09802e7	−3.07251
$831$	−1.21861e7	−0.612155
$832$	−1.08886e8	−5.45337
$833$	−8.49688e6	−0.424275
$834$	2.06641e7	1.02873
$835$	9.25190e6	0.459213
$836$	0	0
$837$	3.26597e6	0.161138
$838$	4.85896e6	0.239019
$839$	1.47153e7	0.721715	0.360857	−	0.932621i	$-0.382484\pi$
$839$	1.47153e7	0.721715	0.360857	+	0.932621i	$0.382484\pi$
$840$	3.85460e7	1.88487
$841$	952151.	0.0464211
$842$	−5.10357e7	−2.48081
$843$	6.36238e6	0.308355
$844$	4.65241e7	2.24813
$845$	4.17327e7	2.01064
$846$	−8.99671e6	−0.432173
$847$	0	0
$848$	2.45019e7	1.17006
$849$	−1.00041e7	−0.476331
$850$	−2.80175e7	−1.33009
$851$	3.24702e7	1.53695
$852$	2.87823e7	1.35840
$853$	2.23479e7	1.05163	0.525817	−	0.850598i	$-0.323760\pi$
$853$	2.23479e7	1.05163	0.525817	+	0.850598i	$0.323760\pi$
$854$	3.58001e7	1.67973
$855$	4.69230e6	0.219518
$856$	−9.17236e6	−0.427855
$857$	−1.87213e7	−0.870730	−0.435365	−	0.900254i	$-0.643381\pi$
$857$	−1.87213e7	−0.870730	−0.435365	+	0.900254i	$0.643381\pi$
$858$	0	0
$859$	−2.83191e7	−1.30947	−0.654737	−	0.755857i	$-0.727221\pi$
$859$	−2.83191e7	−1.30947	−0.654737	+	0.755857i	$0.727221\pi$
$860$	3.59340e7	1.65676
$861$	−1.33772e7	−0.614976
$862$	5.49203e7	2.51747
$863$	3.60603e7	1.64817	0.824086	−	0.566464i	$-0.191689\pi$
$863$	3.60603e7	1.64817	0.824086	+	0.566464i	$0.191689\pi$
$864$	1.53457e7	0.699363
$865$	3.73625e7	1.69784
$866$	−1.64212e6	−0.0744065
$867$	−2.26183e6	−0.102191
$868$	−3.91554e7	−1.76397
$869$	0	0
$870$	−3.24530e7	−1.45364
$871$	2.30451e7	1.02928
$872$	6.53239e7	2.90925
$873$	1.15518e7	0.512997
$874$	−3.45160e7	−1.52842
$875$	−8.20007e6	−0.362074
$876$	−2.88572e7	−1.27056
$877$	−2.13724e7	−0.938328	−0.469164	−	0.883111i	$-0.655445\pi$
$877$	−2.13724e7	−0.938328	−0.469164	+	0.883111i	$0.655445\pi$
$878$	−2.75312e7	−1.20528
$879$	−1.28199e7	−0.559643
$880$	0	0
$881$	−2.08494e7	−0.905009	−0.452504	−	0.891762i	$-0.649469\pi$
$881$	−2.08494e7	−0.905009	−0.452504	+	0.891762i	$0.649469\pi$
$882$	5.79292e6	0.250741
$883$	−1.71241e7	−0.739103	−0.369552	−	0.929210i	$-0.620489\pi$
$883$	−1.71241e7	−0.739103	−0.369552	+	0.929210i	$0.620489\pi$
$884$	−1.09125e8	−4.69671
$885$	−1.49979e7	−0.643684
$886$	−2.18084e7	−0.933337
$887$	720665.	0.0307556	0.0153778	−	0.999882i	$-0.495105\pi$
$887$	720665.	0.0307556	0.0153778	+	0.999882i	$0.495105\pi$
$888$	4.41547e7	1.87908
$889$	1.96631e7	0.834445
$890$	−4.19637e6	−0.177582
$891$	0	0
$892$	−1.57910e7	−0.664505
$893$	8.26675e6	0.346901
$894$	1.25663e7	0.525851
$895$	3.40457e7	1.42071
$896$	−5.45222e7	−2.26884
$897$	3.44457e7	1.42940
$898$	−5.31995e7	−2.20149
$899$	2.07555e7	0.856512
$900$	1.39386e7	0.573607
$901$	8.61844e6	0.353685
$902$	0	0
$903$	−5.29412e6	−0.216060
$904$	6.71926e7	2.73464
$905$	1.30301e7	0.528841
$906$	−2.41581e6	−0.0977781
$907$	3.36500e7	1.35821	0.679106	−	0.734041i	$-0.262368\pi$
$907$	3.36500e7	1.35821	0.679106	+	0.734041i	$0.262368\pi$
$908$	1.02522e8	4.12668
$909$	5.14190e6	0.206402
$910$	7.69354e7	3.07980
$911$	1.65103e7	0.659110	0.329555	−	0.944136i	$-0.393101\pi$
$911$	1.65103e7	0.659110	0.329555	+	0.944136i	$0.393101\pi$
$912$	−2.67869e7	−1.06644
$913$	0	0
$914$	6.64309e7	2.63030
$915$	2.09381e7	0.826769
$916$	1.18783e8	4.67751
$917$	−6.68373e6	−0.262480
$918$	1.02542e7	0.401601
$919$	−1.97825e7	−0.772669	−0.386334	−	0.922359i	$-0.626259\pi$
$919$	−1.97825e7	−0.772669	−0.386334	+	0.922359i	$0.626259\pi$
$920$	−1.65832e8	−6.45949
$921$	4.56555e6	0.177355
$922$	3.58764e7	1.38990
$923$	3.61691e7	1.39744
$924$	0	0
$925$	1.65113e7	0.634494
$926$	−3.42102e7	−1.31108
$927$	−1.36144e6	−0.0520353
$928$	9.75231e7	3.71739
$929$	−2.01941e7	−0.767689	−0.383844	−	0.923398i	$-0.625400\pi$
$929$	−2.01941e7	−0.767689	−0.383844	+	0.923398i	$0.625400\pi$
$930$	−3.13828e7	−1.18983
$931$	−5.32290e6	−0.201268
$932$	−5.05961e7	−1.90799
$933$	−7.27289e6	−0.273529
$934$	444200.	0.0166614
$935$	0	0
$936$	4.68411e7	1.74758
$937$	−4.42407e7	−1.64616	−0.823082	−	0.567922i	$-0.807747\pi$
$937$	−4.42407e7	−1.64616	−0.823082	+	0.567922i	$0.807747\pi$
$938$	2.59618e7	0.963446
$939$	−2.13881e7	−0.791603
$940$	6.30836e7	2.32861
$941$	4.64018e7	1.70829	0.854143	−	0.520038i	$-0.174082\pi$
$941$	4.64018e7	1.70829	0.854143	+	0.520038i	$0.174082\pi$
$942$	−630115.	−0.0231362
$943$	5.75512e7	2.10754
$944$	8.56187e7	3.12708
$945$	−5.27541e6	−0.192166
$946$	0	0
$947$	−4.43015e7	−1.60525	−0.802626	−	0.596483i	$-0.796564\pi$
$947$	−4.43015e7	−1.60525	−0.802626	+	0.596483i	$0.796564\pi$
$948$	3.85822e7	1.39433
$949$	−3.62633e7	−1.30708
$950$	−1.75517e7	−0.630971
$951$	−2.79445e7	−1.00195
$952$	−7.74009e7	−2.76792
$953$	4.41215e6	0.157368	0.0786842	−	0.996900i	$-0.474928\pi$
$953$	4.41215e6	0.157368	0.0786842	+	0.996900i	$0.474928\pi$
$954$	−5.87579e6	−0.209024
$955$	−3.39885e7	−1.20593
$956$	6.09520e7	2.15697
$957$	0	0
$958$	5.17066e6	0.182026
$959$	−1.55600e7	−0.546341
$960$	−7.17435e7	−2.51249
$961$	−8.55816e6	−0.298932
$962$	8.81300e7	3.07034
$963$	1.25533e6	0.0436207
$964$	−3.29721e7	−1.14276
$965$	−4.07677e7	−1.40928
$966$	3.88054e7	1.33798
$967$	−2.63479e7	−0.906107	−0.453054	−	0.891483i	$-0.649665\pi$
$967$	−2.63479e7	−0.906107	−0.453054	+	0.891483i	$0.649665\pi$
$968$	0	0
$969$	−9.42220e6	−0.322361
$970$	−1.11002e8	−3.78792
$971$	−3.64284e7	−1.23991	−0.619957	−	0.784636i	$-0.712850\pi$
$971$	−3.64284e7	−1.23991	−0.619957	+	0.784636i	$0.712850\pi$
$972$	−5.10143e6	−0.173191
$973$	2.13471e7	0.722864
$974$	−3.59888e7	−1.21554
$975$	1.75159e7	0.590094
$976$	−1.19529e8	−4.01652
$977$	1.84287e7	0.617672	0.308836	−	0.951115i	$-0.400061\pi$
$977$	1.84287e7	0.617672	0.308836	+	0.951115i	$0.400061\pi$
$978$	−6.05030e7	−2.02269
$979$	0	0
$980$	−4.06191e7	−1.35103
$981$	−8.94023e6	−0.296604
$982$	6.68455e7	2.21204
$983$	5.22282e6	0.172394	0.0861968	−	0.996278i	$-0.472529\pi$
$983$	5.22282e6	0.172394	0.0861968	+	0.996278i	$0.472529\pi$
$984$	7.82613e7	2.57667
$985$	6.23377e7	2.04720
$986$	6.51662e7	2.13467
$987$	−9.29406e6	−0.303677
$988$	−6.83618e7	−2.22803
$989$	2.27763e7	0.740443
$990$	0	0
$991$	−3.07737e6	−0.0995395	−0.0497697	−	0.998761i	$-0.515849\pi$
$991$	−3.07737e6	−0.0995395	−0.0497697	+	0.998761i	$0.515849\pi$
$992$	9.43069e7	3.04274
$993$	−2.40674e7	−0.774561
$994$	4.07468e7	1.30806
$995$	−6.56042e7	−2.10075
$996$	−6.09181e7	−1.94580
$997$	4.89985e7	1.56115	0.780575	−	0.625062i	$-0.214926\pi$
$997$	4.89985e7	1.56115	0.780575	+	0.625062i	$0.214926\pi$
$998$	1.16676e8	3.70814
$999$	−6.04302e6	−0.191576

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	363.6.a.m.1.1	✓	4
3.2	odd	2		1089.6.a.u.1.4		4
11.10	odd	2		363.6.a.n.1.4	yes	4
33.32	even	2		1089.6.a.t.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
363.6.a.m.1.1	✓	4	1.1	even	1	trivial
363.6.a.n.1.4	yes	4	11.10	odd	2
1089.6.a.t.1.1		4	33.32	even	2
1089.6.a.u.1.4		4	3.2	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 363 = 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	363.a (trivial)

\(f(q)\)	\(=\)	\(q-10.8809 q^{2} -9.00000 q^{3} +86.3932 q^{4} +71.5321 q^{5} +97.9278 q^{6} +101.164 q^{7} -591.845 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q-10.8809 q^{2} -9.00000 q^{3} +86.3932 q^{4} +71.5321 q^{5} +97.9278 q^{6} +101.164 q^{7} -591.845 q^{8} +81.0000 q^{9} -778.331 q^{10} -777.539 q^{12} -977.090 q^{13} -1100.76 q^{14} -643.789 q^{15} +3675.20 q^{16} +1292.74 q^{17} -881.350 q^{18} +809.840 q^{19} +6179.89 q^{20} -910.479 q^{21} +3917.04 q^{23} +5326.61 q^{24} +1991.85 q^{25} +10631.6 q^{26} -729.000 q^{27} +8739.91 q^{28} -4632.85 q^{29} +7004.98 q^{30} -4480.07 q^{31} -21050.3 q^{32} -14066.1 q^{34} +7236.50 q^{35} +6997.85 q^{36} +8289.46 q^{37} -8811.76 q^{38} +8793.81 q^{39} -42335.9 q^{40} +14692.5 q^{41} +9906.80 q^{42} +5814.66 q^{43} +5794.10 q^{45} -42620.8 q^{46} +10207.9 q^{47} -33076.8 q^{48} -6572.78 q^{49} -21673.0 q^{50} -11634.6 q^{51} -84413.9 q^{52} +6666.81 q^{53} +7932.15 q^{54} -59873.6 q^{56} -7288.56 q^{57} +50409.4 q^{58} +23296.3 q^{59} -55619.0 q^{60} -32523.2 q^{61} +48747.0 q^{62} +8194.31 q^{63} +111439. q^{64} -69893.3 q^{65} -23585.4 q^{67} +111684. q^{68} -35253.4 q^{69} -78739.4 q^{70} -37017.2 q^{71} -47939.5 q^{72} +37113.5 q^{73} -90196.5 q^{74} -17926.6 q^{75} +69964.7 q^{76} -95684.2 q^{78} -49621.0 q^{79} +262895. q^{80} +6561.00 q^{81} -159867. q^{82} +78347.3 q^{83} -78659.2 q^{84} +92472.3 q^{85} -63268.5 q^{86} +41695.7 q^{87} +5391.49 q^{89} -63044.8 q^{90} -98846.6 q^{91} +338406. q^{92} +40320.6 q^{93} -111071. q^{94} +57929.6 q^{95} +189453. q^{96} +142615. q^{97} +71517.5 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 9 q^{2} - 36 q^{3} + 53 q^{4} + 42 q^{5} + 81 q^{6} + 14 q^{7} - 765 q^{8} + 324 q^{9}+O(q^{10}) \) 4 * q - 9 * q^2 - 36 * q^3 + 53 * q^4 + 42 * q^5 + 81 * q^6 + 14 * q^7 - 765 * q^8 + 324 * q^9	\( 4 q - 9 q^{2} - 36 q^{3} + 53 q^{4} + 42 q^{5} + 81 q^{6} + 14 q^{7} - 765 q^{8} + 324 q^{9} - 589 q^{10} - 477 q^{12} - 886 q^{13} - 1638 q^{14} - 378 q^{15} + 2345 q^{16} + 570 q^{17} - 729 q^{18} - 1338 q^{19} + 5307 q^{20} - 126 q^{21} + 1260 q^{23} + 6885 q^{24} - 1546 q^{25} + 8133 q^{26} - 2916 q^{27} + 7108 q^{28} - 10230 q^{29} + 5301 q^{30} - 8042 q^{31} - 11769 q^{32} - 25931 q^{34} + 24552 q^{35} + 4293 q^{36} + 18936 q^{37} - 16362 q^{38} + 7974 q^{39} - 50055 q^{40} + 3006 q^{41} + 14742 q^{42} + 21504 q^{43} + 3402 q^{45} - 17204 q^{46} + 5916 q^{47} - 21105 q^{48} + 16902 q^{49} - 22614 q^{50} - 5130 q^{51} - 94847 q^{52} + 48414 q^{53} + 6561 q^{54} - 43104 q^{56} + 12042 q^{57} + 59697 q^{58} + 30276 q^{59} - 47763 q^{60} - 106242 q^{61} - 23106 q^{62} + 1134 q^{63} + 185857 q^{64} - 7362 q^{65} - 57538 q^{67} + 114021 q^{68} - 11340 q^{69} - 19148 q^{70} + 45720 q^{71} - 61965 q^{72} + 11426 q^{73} - 30867 q^{74} + 13914 q^{75} + 127752 q^{76} - 73197 q^{78} + 68338 q^{79} + 347487 q^{80} + 26244 q^{81} - 98771 q^{82} - 146748 q^{83} - 63972 q^{84} + 185042 q^{85} + 185952 q^{86} + 92070 q^{87} - 89106 q^{89} - 47709 q^{90} + 187804 q^{91} + 363432 q^{92} + 72378 q^{93} - 285008 q^{94} - 75672 q^{95} + 105921 q^{96} + 386120 q^{97} + 342057 q^{98}+O(q^{100}) \) 4 * q - 9 * q^2 - 36 * q^3 + 53 * q^4 + 42 * q^5 + 81 * q^6 + 14 * q^7 - 765 * q^8 + 324 * q^9 - 589 * q^10 - 477 * q^12 - 886 * q^13 - 1638 * q^14 - 378 * q^15 + 2345 * q^16 + 570 * q^17 - 729 * q^18 - 1338 * q^19 + 5307 * q^20 - 126 * q^21 + 1260 * q^23 + 6885 * q^24 - 1546 * q^25 + 8133 * q^26 - 2916 * q^27 + 7108 * q^28 - 10230 * q^29 + 5301 * q^30 - 8042 * q^31 - 11769 * q^32 - 25931 * q^34 + 24552 * q^35 + 4293 * q^36 + 18936 * q^37 - 16362 * q^38 + 7974 * q^39 - 50055 * q^40 + 3006 * q^41 + 14742 * q^42 + 21504 * q^43 + 3402 * q^45 - 17204 * q^46 + 5916 * q^47 - 21105 * q^48 + 16902 * q^49 - 22614 * q^50 - 5130 * q^51 - 94847 * q^52 + 48414 * q^53 + 6561 * q^54 - 43104 * q^56 + 12042 * q^57 + 59697 * q^58 + 30276 * q^59 - 47763 * q^60 - 106242 * q^61 - 23106 * q^62 + 1134 * q^63 + 185857 * q^64 - 7362 * q^65 - 57538 * q^67 + 114021 * q^68 - 11340 * q^69 - 19148 * q^70 + 45720 * q^71 - 61965 * q^72 + 11426 * q^73 - 30867 * q^74 + 13914 * q^75 + 127752 * q^76 - 73197 * q^78 + 68338 * q^79 + 347487 * q^80 + 26244 * q^81 - 98771 * q^82 - 146748 * q^83 - 63972 * q^84 + 185042 * q^85 + 185952 * q^86 + 92070 * q^87 - 89106 * q^89 - 47709 * q^90 + 187804 * q^91 + 363432 * q^92 + 72378 * q^93 - 285008 * q^94 - 75672 * q^95 + 105921 * q^96 + 386120 * q^97 + 342057 * q^98

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