LMFDB - Embedded newform 147.8.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [147,8,Mod(1,147)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(147, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("147.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 147 = 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	147.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:	$45.9205987462$
Analytic rank:	$1$
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} - 2x^{3} - 299x^{2} - 1110x + 1890 $ x^4 - 2x^3 - 299x^2 - 1110*x + 1890
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2}\cdot 7 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.26815$ of defining polynomial
Character	$\chi$	$=$	147.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-21.6612 q^{2} -27.0000 q^{3} +341.209 q^{4} -516.937 q^{5} +584.853 q^{6} -4618.37 q^{8} +729.000 q^{9} +O(q^{10})$	\(q-21.6612 q^{2} -27.0000 q^{3} +341.209 q^{4} -516.937 q^{5} +584.853 q^{6} -4618.37 q^{8} +729.000 q^{9} +11197.5 q^{10} +662.581 q^{11} -9212.65 q^{12} -7851.30 q^{13} +13957.3 q^{15} +56364.9 q^{16} -11632.6 q^{17} -15791.0 q^{18} -14985.6 q^{19} -176384. q^{20} -14352.3 q^{22} -26185.0 q^{23} +124696. q^{24} +189099. q^{25} +170069. q^{26} -19683.0 q^{27} +108612. q^{29} -302332. q^{30} +195632. q^{31} -629782. q^{32} -17889.7 q^{33} +251976. q^{34} +248741. q^{36} +326569. q^{37} +324606. q^{38} +211985. q^{39} +2.38741e6 q^{40} +79193.0 q^{41} +638173. q^{43} +226079. q^{44} -376847. q^{45} +567199. q^{46} +776840. q^{47} -1.52185e6 q^{48} -4.09611e6 q^{50} +314079. q^{51} -2.67894e6 q^{52} -928216. q^{53} +426358. q^{54} -342513. q^{55} +404610. q^{57} -2.35267e6 q^{58} +2.29493e6 q^{59} +4.76236e6 q^{60} -2.43649e6 q^{61} -4.23764e6 q^{62} +6.42715e6 q^{64} +4.05863e6 q^{65} +387513. q^{66} -1.75090e6 q^{67} -3.96914e6 q^{68} +706995. q^{69} -1.22315e6 q^{71} -3.36679e6 q^{72} -464159. q^{73} -7.07389e6 q^{74} -5.10566e6 q^{75} -5.11321e6 q^{76} -4.59186e6 q^{78} +1.73244e6 q^{79} -2.91371e7 q^{80} +531441. q^{81} -1.71542e6 q^{82} -3.98160e6 q^{83} +6.01330e6 q^{85} -1.38236e7 q^{86} -2.93252e6 q^{87} -3.06005e6 q^{88} +7.83254e6 q^{89} +8.16297e6 q^{90} -8.93456e6 q^{92} -5.28208e6 q^{93} -1.68273e7 q^{94} +7.74659e6 q^{95} +1.70041e7 q^{96} +4.49108e6 q^{97} +483022. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 15 q^{2} - 108 q^{3} + 437 q^{4} - 504 q^{5} + 405 q^{6} - 2145 q^{8} + 2916 q^{9}+O(q^{10}) $ 4 * q - 15 * q^2 - 108 * q^3 + 437 * q^4 - 504 * q^5 + 405 * q^6 - 2145 * q^8 + 2916 * q^9	\( 4 q - 15 q^{2} - 108 q^{3} + 437 q^{4} - 504 q^{5} + 405 q^{6} - 2145 q^{8} + 2916 q^{9} + 5724 q^{10} - 1920 q^{11} - 11799 q^{12} - 18144 q^{13} + 13608 q^{15} + 52529 q^{16} + 19584 q^{17} - 10935 q^{18} - 31320 q^{19} - 169812 q^{20} + 111940 q^{22} + 101160 q^{23} + 57915 q^{24} + 74788 q^{25} + 107820 q^{26} - 78732 q^{27} + 194832 q^{29} - 154548 q^{30} - 78840 q^{31} - 732585 q^{32} + 51840 q^{33} + 77760 q^{34} + 318573 q^{36} + 128640 q^{37} - 716292 q^{38} + 489888 q^{39} + 2649780 q^{40} - 365040 q^{41} + 449520 q^{43} + 113220 q^{44} - 367416 q^{45} + 1033664 q^{46} + 1575792 q^{47} - 1418283 q^{48} - 5391933 q^{50} - 528768 q^{51} - 5747652 q^{52} - 1448160 q^{53} + 295245 q^{54} - 3083400 q^{55} + 845640 q^{57} - 2570950 q^{58} - 3280320 q^{59} + 4584924 q^{60} - 606960 q^{61} - 12064536 q^{62} + 2609137 q^{64} + 1318464 q^{65} - 3022380 q^{66} + 3492880 q^{67} + 1476432 q^{68} - 2731320 q^{69} + 984 q^{71} - 1563705 q^{72} - 10981440 q^{73} - 14177022 q^{74} - 2019276 q^{75} - 18964260 q^{76} - 2911140 q^{78} + 4654544 q^{79} - 37026324 q^{80} + 2125764 q^{81} + 10402560 q^{82} - 8126496 q^{83} + 16792056 q^{85} - 6392244 q^{86} - 5260464 q^{87} - 10447140 q^{88} + 11272320 q^{89} + 4172796 q^{90} + 3289440 q^{92} + 2128680 q^{93} - 37697400 q^{94} + 11132208 q^{95} + 19779795 q^{96} - 6572448 q^{97} - 1399680 q^{99}+O(q^{100}) \) 4 * q - 15 * q^2 - 108 * q^3 + 437 * q^4 - 504 * q^5 + 405 * q^6 - 2145 * q^8 + 2916 * q^9 + 5724 * q^10 - 1920 * q^11 - 11799 * q^12 - 18144 * q^13 + 13608 * q^15 + 52529 * q^16 + 19584 * q^17 - 10935 * q^18 - 31320 * q^19 - 169812 * q^20 + 111940 * q^22 + 101160 * q^23 + 57915 * q^24 + 74788 * q^25 + 107820 * q^26 - 78732 * q^27 + 194832 * q^29 - 154548 * q^30 - 78840 * q^31 - 732585 * q^32 + 51840 * q^33 + 77760 * q^34 + 318573 * q^36 + 128640 * q^37 - 716292 * q^38 + 489888 * q^39 + 2649780 * q^40 - 365040 * q^41 + 449520 * q^43 + 113220 * q^44 - 367416 * q^45 + 1033664 * q^46 + 1575792 * q^47 - 1418283 * q^48 - 5391933 * q^50 - 528768 * q^51 - 5747652 * q^52 - 1448160 * q^53 + 295245 * q^54 - 3083400 * q^55 + 845640 * q^57 - 2570950 * q^58 - 3280320 * q^59 + 4584924 * q^60 - 606960 * q^61 - 12064536 * q^62 + 2609137 * q^64 + 1318464 * q^65 - 3022380 * q^66 + 3492880 * q^67 + 1476432 * q^68 - 2731320 * q^69 + 984 * q^71 - 1563705 * q^72 - 10981440 * q^73 - 14177022 * q^74 - 2019276 * q^75 - 18964260 * q^76 - 2911140 * q^78 + 4654544 * q^79 - 37026324 * q^80 + 2125764 * q^81 + 10402560 * q^82 - 8126496 * q^83 + 16792056 * q^85 - 6392244 * q^86 - 5260464 * q^87 - 10447140 * q^88 + 11272320 * q^89 + 4172796 * q^90 + 3289440 * q^92 + 2128680 * q^93 - 37697400 * q^94 + 11132208 * q^95 + 19779795 * q^96 - 6572448 * q^97 - 1399680 * q^99

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 $

$2$	−21.6612	−1.91460	−0.957300	−	0.289095i	$-0.906646\pi$
$2$	−21.6612	−1.91460	−0.957300	+	0.289095i	$0.906646\pi$
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	341.209	2.66570
$5$	−516.937	−1.84945	−0.924725	−	0.380637i	$-0.875705\pi$
$5$	−516.937	−1.84945	−0.924725	+	0.380637i	$0.875705\pi$
$6$	584.853	1.10540
$7$	0	0
$7$	0	0
$8$	−4618.37	−3.18914
$9$	729.000	0.333333
$10$	11197.5	3.54096
$11$	662.581	0.150095	0.0750473	−	0.997180i	$-0.476089\pi$
$11$	662.581	0.150095	0.0750473	+	0.997180i	$0.476089\pi$
$12$	−9212.65	−1.53904
$13$	−7851.30	−0.991151	−0.495576	−	0.868565i	$-0.665043\pi$
$13$	−7851.30	−0.991151	−0.495576	+	0.868565i	$0.665043\pi$
$14$	0	0
$15$	13957.3	1.06778
$16$	56364.9	3.44024
$17$	−11632.6	−0.574255	−0.287127	−	0.957892i	$-0.592700\pi$
$17$	−11632.6	−0.574255	−0.287127	+	0.957892i	$0.592700\pi$
$18$	−15791.0	−0.638200
$19$	−14985.6	−0.501228	−0.250614	−	0.968087i	$-0.580633\pi$
$19$	−14985.6	−0.501228	−0.250614	+	0.968087i	$0.580633\pi$
$20$	−176384.	−4.93007
$21$	0	0
$22$	−14352.3	−0.287371
$23$	−26185.0	−0.448750	−0.224375	−	0.974503i	$-0.572034\pi$
$23$	−26185.0	−0.448750	−0.224375	+	0.974503i	$0.572034\pi$
$24$	124696.	1.84125
$25$	189099.	2.42046
$26$	170069.	1.89766
$27$	−19683.0	−0.192450
$28$	0	0
$29$	108612.	0.826959	0.413480	−	0.910513i	$-0.364313\pi$
$29$	108612.	0.826959	0.413480	+	0.910513i	$0.364313\pi$
$30$	−302332.	−2.04437
$31$	195632.	1.17944	0.589719	−	0.807609i	$-0.299239\pi$
$31$	195632.	1.17944	0.589719	+	0.807609i	$0.299239\pi$
$32$	−629782.	−3.39755
$33$	−17889.7	−0.0866571
$34$	251976.	1.09947
$35$	0	0
$36$	248741.	0.888566
$37$	326569.	1.05991	0.529955	−	0.848026i	$-0.322209\pi$
$37$	326569.	1.05991	0.529955	+	0.848026i	$0.322209\pi$
$38$	324606.	0.959652
$39$	211985.	0.572242
$40$	2.38741e6	5.89816
$41$	79193.0	0.179450	0.0897249	−	0.995967i	$-0.471401\pi$
$41$	79193.0	0.179450	0.0897249	+	0.995967i	$0.471401\pi$
$42$	0	0
$43$	638173.	1.22405	0.612025	−	0.790838i	$-0.290355\pi$
$43$	638173.	1.22405	0.612025	+	0.790838i	$0.290355\pi$
$44$	226079.	0.400106
$45$	−376847.	−0.616483
$46$	567199.	0.859178
$47$	776840.	1.09141	0.545707	−	0.837976i	$-0.316261\pi$
$47$	776840.	1.09141	0.545707	+	0.837976i	$0.316261\pi$
$48$	−1.52185e6	−1.98622
$49$	0	0
$50$	−4.09611e6	−4.63422
$51$	314079.	0.331546
$52$	−2.67894e6	−2.64211
$53$	−928216.	−0.856414	−0.428207	−	0.903681i	$-0.640855\pi$
$53$	−928216.	−0.856414	−0.428207	+	0.903681i	$0.640855\pi$
$54$	426358.	0.368465
$55$	−342513.	−0.277592
$56$	0	0
$57$	404610.	0.289384
$58$	−2.35267e6	−1.58330
$59$	2.29493e6	1.45475	0.727374	−	0.686242i	$-0.240741\pi$
$59$	2.29493e6	1.45475	0.727374	+	0.686242i	$0.240741\pi$
$60$	4.76236e6	2.84638
$61$	−2.43649e6	−1.37439	−0.687194	−	0.726474i	$-0.741158\pi$
$61$	−2.43649e6	−1.37439	−0.687194	+	0.726474i	$0.741158\pi$
$62$	−4.23764e6	−2.25815
$63$	0	0
$64$	6.42715e6	3.06470
$65$	4.05863e6	1.83308
$66$	387513.	0.165914
$67$	−1.75090e6	−0.711213	−0.355606	−	0.934636i	$-0.615726\pi$
$67$	−1.75090e6	−0.711213	−0.355606	+	0.934636i	$0.615726\pi$
$68$	−3.96914e6	−1.53079
$69$	706995.	0.259086
$70$	0	0
$71$	−1.22315e6	−0.405581	−0.202790	−	0.979222i	$-0.565001\pi$
$71$	−1.22315e6	−0.405581	−0.202790	+	0.979222i	$0.565001\pi$
$72$	−3.36679e6	−1.06305
$73$	−464159.	−0.139649	−0.0698244	−	0.997559i	$-0.522244\pi$
$73$	−464159.	−0.139649	−0.0698244	+	0.997559i	$0.522244\pi$
$74$	−7.07389e6	−2.02930
$75$	−5.10566e6	−1.39746
$76$	−5.11321e6	−1.33612
$77$	0	0
$78$	−4.59186e6	−1.09561
$79$	1.73244e6	0.395332	0.197666	−	0.980269i	$-0.436664\pi$
$79$	1.73244e6	0.395332	0.197666	+	0.980269i	$0.436664\pi$
$80$	−2.91371e7	−6.36255
$81$	531441.	0.111111
$82$	−1.71542e6	−0.343575
$83$	−3.98160e6	−0.764337	−0.382168	−	0.924093i	$-0.624823\pi$
$83$	−3.98160e6	−0.764337	−0.382168	+	0.924093i	$0.624823\pi$
$84$	0	0
$85$	6.01330e6	1.06205
$86$	−1.38236e7	−2.34357
$87$	−2.93252e6	−0.477445
$88$	−3.06005e6	−0.478673
$89$	7.83254e6	1.17771	0.588854	−	0.808240i	$-0.299579\pi$
$89$	7.83254e6	1.17771	0.588854	+	0.808240i	$0.299579\pi$
$90$	8.16297e6	1.18032
$91$	0	0
$92$	−8.93456e6	−1.19623
$93$	−5.28208e6	−0.680949
$94$	−1.68273e7	−2.08962
$95$	7.74659e6	0.926996
$96$	1.70041e7	1.96157
$97$	4.49108e6	0.499632	0.249816	−	0.968293i	$-0.419630\pi$
$97$	4.49108e6	0.499632	0.249816	+	0.968293i	$0.419630\pi$
$98$	0	0
$99$	483022.	0.0500315
$100$	6.45222e7	6.45222
$101$	1.24245e7	1.19993	0.599964	−	0.800027i	$-0.295182\pi$
$101$	1.24245e7	1.19993	0.599964	+	0.800027i	$0.295182\pi$
$102$	−6.80335e6	−0.634778
$103$	−2.14293e7	−1.93231	−0.966157	−	0.257955i	$-0.916952\pi$
$103$	−2.14293e7	−1.93231	−0.966157	+	0.257955i	$0.916952\pi$
$104$	3.62603e7	3.16092
$105$	0	0
$106$	2.01063e7	1.63969
$107$	−2.22446e7	−1.75542	−0.877710	−	0.479193i	$-0.840930\pi$
$107$	−2.22446e7	−1.75542	−0.877710	+	0.479193i	$0.840930\pi$
$108$	−6.71602e6	−0.513014
$109$	9.90130e6	0.732318	0.366159	−	0.930552i	$-0.380673\pi$
$109$	9.90130e6	0.732318	0.366159	+	0.930552i	$0.380673\pi$
$110$	7.41925e6	0.531478
$111$	−8.81736e6	−0.611939
$112$	0	0
$113$	1.91928e7	1.25130	0.625652	−	0.780102i	$-0.284833\pi$
$113$	1.91928e7	1.25130	0.625652	+	0.780102i	$0.284833\pi$
$114$	−8.76436e6	−0.554055
$115$	1.35360e7	0.829941
$116$	3.70593e7	2.20442
$117$	−5.72360e6	−0.330384
$118$	−4.97110e7	−2.78526
$119$	0	0
$120$	−6.44600e7	−3.40530
$121$	−1.90482e7	−0.977472
$122$	5.27773e7	2.63141
$123$	−2.13821e6	−0.103605
$124$	6.67516e7	3.14402
$125$	−5.73664e7	−2.62707
$126$	0	0
$127$	−1.43563e7	−0.621912	−0.310956	−	0.950424i	$-0.600649\pi$
$127$	−1.43563e7	−0.621912	−0.310956	+	0.950424i	$0.600649\pi$
$128$	−5.86078e7	−2.47014
$129$	−1.72307e7	−0.706706
$130$	−8.79149e7	−3.50963
$131$	−1.82094e7	−0.707696	−0.353848	−	0.935303i	$-0.615127\pi$
$131$	−1.82094e7	−0.707696	−0.353848	+	0.935303i	$0.615127\pi$
$132$	−6.10413e6	−0.231002
$133$	0	0
$134$	3.79267e7	1.36169
$135$	1.01749e7	0.355927
$136$	5.37236e7	1.83138
$137$	2.45269e7	0.814932	0.407466	−	0.913220i	$-0.366413\pi$
$137$	2.45269e7	0.814932	0.407466	+	0.913220i	$0.366413\pi$
$138$	−1.53144e7	−0.496047
$139$	−1.18402e7	−0.373946	−0.186973	−	0.982365i	$-0.559868\pi$
$139$	−1.18402e7	−0.373946	−0.186973	+	0.982365i	$0.559868\pi$
$140$	0	0
$141$	−2.09747e7	−0.630128
$142$	2.64950e7	0.776525
$143$	−5.20213e6	−0.148766
$144$	4.10900e7	1.14675
$145$	−5.61454e7	−1.52942
$146$	1.00543e7	0.267372
$147$	0	0
$148$	1.11428e8	2.82540
$149$	5.63211e6	0.139482	0.0697412	−	0.997565i	$-0.477783\pi$
$149$	5.63211e6	0.139482	0.0697412	+	0.997565i	$0.477783\pi$
$150$	1.10595e8	2.67557
$151$	−6.07958e7	−1.43699	−0.718495	−	0.695532i	$-0.755169\pi$
$151$	−6.07958e7	−1.43699	−0.718495	+	0.695532i	$0.755169\pi$
$152$	6.92089e7	1.59849
$153$	−8.48014e6	−0.191418
$154$	0	0
$155$	−1.01130e8	−2.18131
$156$	7.23313e7	1.52542
$157$	5.63347e7	1.16179	0.580895	−	0.813979i	$-0.302703\pi$
$157$	5.63347e7	1.16179	0.580895	+	0.813979i	$0.302703\pi$
$158$	−3.75267e7	−0.756904
$159$	2.50618e7	0.494451
$160$	3.25557e8	6.28359
$161$	0	0
$162$	−1.15117e7	−0.212733
$163$	5.26719e7	0.952627	0.476313	−	0.879276i	$-0.341973\pi$
$163$	5.26719e7	0.952627	0.476313	+	0.879276i	$0.341973\pi$
$164$	2.70214e7	0.478359
$165$	9.24784e6	0.160268
$166$	8.62464e7	1.46340
$167$	−2.07427e7	−0.344634	−0.172317	−	0.985042i	$-0.555125\pi$
$167$	−2.07427e7	−0.344634	−0.172317	+	0.985042i	$0.555125\pi$
$168$	0	0
$169$	−1.10556e6	−0.0176189
$170$	−1.30256e8	−2.03341
$171$	−1.09245e7	−0.167076
$172$	2.17751e8	3.26295
$173$	−1.30053e8	−1.90967	−0.954834	−	0.297140i	$-0.903967\pi$
$173$	−1.30053e8	−1.90967	−0.954834	+	0.297140i	$0.903967\pi$
$174$	6.35220e7	0.914117
$175$	0	0
$176$	3.73463e7	0.516361
$177$	−6.19631e7	−0.839899
$178$	−1.69662e8	−2.25484
$179$	−1.53454e7	−0.199983	−0.0999914	−	0.994988i	$-0.531882\pi$
$179$	−1.53454e7	−0.199983	−0.0999914	+	0.994988i	$0.531882\pi$
$180$	−1.28584e8	−1.64336
$181$	−7.02000e7	−0.879959	−0.439979	−	0.898008i	$-0.645014\pi$
$181$	−7.02000e7	−0.879959	−0.439979	+	0.898008i	$0.645014\pi$
$182$	0	0
$183$	6.57851e7	0.793504
$184$	1.20932e8	1.43113
$185$	−1.68815e8	−1.96025
$186$	1.14416e8	1.30375
$187$	−7.70753e6	−0.0861925
$188$	2.65065e8	2.90938
$189$	0	0
$190$	−1.67801e8	−1.77483
$191$	1.69111e8	1.75613	0.878063	−	0.478545i	$-0.158836\pi$
$191$	1.69111e8	1.75613	0.878063	+	0.478545i	$0.158836\pi$
$192$	−1.73533e8	−1.76941
$193$	9.96399e7	0.997660	0.498830	−	0.866700i	$-0.333763\pi$
$193$	9.96399e7	0.997660	0.498830	+	0.866700i	$0.333763\pi$
$194$	−9.72824e7	−0.956595
$195$	−1.09583e8	−1.05833
$196$	0	0
$197$	1.73109e8	1.61319	0.806597	−	0.591101i	$-0.201307\pi$
$197$	1.73109e8	1.61319	0.806597	+	0.591101i	$0.201307\pi$
$198$	−1.04629e7	−0.0957904
$199$	9.43518e7	0.848720	0.424360	−	0.905494i	$-0.360499\pi$
$199$	9.43518e7	0.848720	0.424360	+	0.905494i	$0.360499\pi$
$200$	−8.73328e8	−7.71921
$201$	4.72743e7	0.410619
$202$	−2.69131e8	−2.29738
$203$	0	0
$204$	1.07167e8	0.883801
$205$	−4.09378e7	−0.331884
$206$	4.64185e8	3.69961
$207$	−1.90889e7	−0.149583
$208$	−4.42538e8	−3.40980
$209$	−9.92916e6	−0.0752316
$210$	0	0
$211$	−5.13471e6	−0.0376294	−0.0188147	−	0.999823i	$-0.505989\pi$
$211$	−5.13471e6	−0.0376294	−0.0188147	+	0.999823i	$0.505989\pi$
$212$	−3.16716e8	−2.28294
$213$	3.30252e7	0.234162
$214$	4.81845e8	3.36093
$215$	−3.29895e8	−2.26382
$216$	9.09035e7	0.613751
$217$	0	0
$218$	−2.14474e8	−1.40210
$219$	1.25323e7	0.0806262
$220$	−1.16868e8	−0.739977
$221$	9.13308e7	0.569173
$222$	1.90995e8	1.17162
$223$	1.73853e8	1.04982	0.524909	−	0.851158i	$-0.324099\pi$
$223$	1.73853e8	1.04982	0.524909	+	0.851158i	$0.324099\pi$
$224$	0	0
$225$	1.37853e8	0.806821
$226$	−4.15739e8	−2.39575
$227$	−1.70253e8	−0.966063	−0.483032	−	0.875603i	$-0.660464\pi$
$227$	−1.70253e8	−0.966063	−0.483032	+	0.875603i	$0.660464\pi$
$228$	1.38057e8	0.771411
$229$	−2.47698e8	−1.36301	−0.681504	−	0.731814i	$-0.738674\pi$
$229$	−2.47698e8	−1.36301	−0.681504	+	0.731814i	$0.738674\pi$
$230$	−2.93206e8	−1.58901
$231$	0	0
$232$	−5.01610e8	−2.63729
$233$	−1.14411e8	−0.592548	−0.296274	−	0.955103i	$-0.595744\pi$
$233$	−1.14411e8	−0.592548	−0.296274	+	0.955103i	$0.595744\pi$
$234$	1.23980e8	0.632553
$235$	−4.01577e8	−2.01851
$236$	7.83051e8	3.87791
$237$	−4.67758e7	−0.228245
$238$	0	0
$239$	−8.83010e7	−0.418382	−0.209191	−	0.977875i	$-0.567083\pi$
$239$	−8.83010e7	−0.418382	−0.209191	+	0.977875i	$0.567083\pi$
$240$	7.86702e8	3.67342
$241$	1.55646e8	0.716271	0.358135	−	0.933670i	$-0.383413\pi$
$241$	1.55646e8	0.716271	0.358135	+	0.933670i	$0.383413\pi$
$242$	4.12607e8	1.87147
$243$	−1.43489e7	−0.0641500
$244$	−8.31351e8	−3.66370
$245$	0	0
$246$	4.63163e7	0.198363
$247$	1.17656e8	0.496793
$248$	−9.03504e8	−3.76140
$249$	1.07503e8	0.441290
$250$	1.24263e9	5.02980
$251$	1.07520e8	0.429171	0.214585	−	0.976705i	$-0.431160\pi$
$251$	1.07520e8	0.429171	0.214585	+	0.976705i	$0.431160\pi$
$252$	0	0
$253$	−1.73497e7	−0.0673550
$254$	3.10975e8	1.19071
$255$	−1.62359e8	−0.613178
$256$	4.46844e8	1.66462
$257$	4.03178e8	1.48160	0.740800	−	0.671726i	$-0.234447\pi$
$257$	4.03178e8	1.48160	0.740800	+	0.671726i	$0.234447\pi$
$258$	3.73238e8	1.35306
$259$	0	0
$260$	1.38484e9	4.88645
$261$	7.91780e7	0.275653
$262$	3.94439e8	1.35496
$263$	−2.73338e8	−0.926520	−0.463260	−	0.886223i	$-0.653320\pi$
$263$	−2.73338e8	−0.926520	−0.463260	+	0.886223i	$0.653320\pi$
$264$	8.26213e7	0.276362
$265$	4.79829e8	1.58389
$266$	0	0
$267$	−2.11478e8	−0.679950
$268$	−5.97423e8	−1.89588
$269$	−1.03641e8	−0.324638	−0.162319	−	0.986738i	$-0.551897\pi$
$269$	−1.03641e8	−0.324638	−0.162319	+	0.986738i	$0.551897\pi$
$270$	−2.20400e8	−0.681458
$271$	1.98124e7	0.0604706	0.0302353	−	0.999543i	$-0.490374\pi$
$271$	1.98124e7	0.0604706	0.0302353	+	0.999543i	$0.490374\pi$
$272$	−6.55669e8	−1.97557
$273$	0	0
$274$	−5.31284e8	−1.56027
$275$	1.25293e8	0.363298
$276$	2.41233e8	0.690645
$277$	−3.56304e8	−1.00726	−0.503630	−	0.863919i	$-0.668003\pi$
$277$	−3.56304e8	−1.00726	−0.503630	+	0.863919i	$0.668003\pi$
$278$	2.56474e8	0.715957
$279$	1.42616e8	0.393146
$280$	0	0
$281$	−1.59268e7	−0.0428210	−0.0214105	−	0.999771i	$-0.506816\pi$
$281$	−1.59268e7	−0.0428210	−0.0214105	+	0.999771i	$0.506816\pi$
$282$	4.54338e8	1.20644
$283$	−4.82361e8	−1.26508	−0.632542	−	0.774526i	$-0.717989\pi$
$283$	−4.82361e8	−1.26508	−0.632542	+	0.774526i	$0.717989\pi$
$284$	−4.17352e8	−1.08115
$285$	−2.09158e8	−0.535202
$286$	1.12685e8	0.284828
$287$	0	0
$288$	−4.59111e8	−1.13252
$289$	−2.75022e8	−0.670232
$290$	1.21618e9	2.92823
$291$	−1.21259e8	−0.288462
$292$	−1.58375e8	−0.372261
$293$	4.43360e8	1.02972	0.514860	−	0.857274i	$-0.327844\pi$
$293$	4.43360e8	1.02972	0.514860	+	0.857274i	$0.327844\pi$
$294$	0	0
$295$	−1.18633e9	−2.69048
$296$	−1.50822e9	−3.38020
$297$	−1.30416e7	−0.0288857
$298$	−1.21999e8	−0.267053
$299$	2.05586e8	0.444780
$300$	−1.74210e9	−3.72519
$301$	0	0
$302$	1.31691e9	2.75126
$303$	−3.35462e8	−0.692778
$304$	−8.44660e8	−1.72435
$305$	1.25951e9	2.54186
$306$	1.83690e8	0.366489
$307$	−7.75755e7	−0.153017	−0.0765086	−	0.997069i	$-0.524377\pi$
$307$	−7.75755e7	−0.153017	−0.0765086	+	0.997069i	$0.524377\pi$
$308$	0	0
$309$	5.78591e8	1.11562
$310$	2.19059e9	4.17634
$311$	4.42636e8	0.834421	0.417211	−	0.908810i	$-0.363008\pi$
$311$	4.42636e8	0.834421	0.417211	+	0.908810i	$0.363008\pi$
$312$	−9.79027e8	−1.82496
$313$	−5.63026e8	−1.03782	−0.518912	−	0.854828i	$-0.673663\pi$
$313$	−5.63026e8	−1.03782	−0.518912	+	0.854828i	$0.673663\pi$
$314$	−1.22028e9	−2.22436
$315$	0	0
$316$	5.91123e8	1.05384
$317$	−7.63549e8	−1.34626	−0.673130	−	0.739524i	$-0.735051\pi$
$317$	−7.63549e8	−1.34626	−0.673130	+	0.739524i	$0.735051\pi$
$318$	−5.42871e8	−0.946676
$319$	7.19642e7	0.124122
$320$	−3.32243e9	−5.66801
$321$	6.00604e8	1.01349
$322$	0	0
$323$	1.74321e8	0.287833
$324$	1.81333e8	0.296189
$325$	−1.48467e9	−2.39905
$326$	−1.14094e9	−1.82390
$327$	−2.67335e8	−0.422804
$328$	−3.65743e8	−0.572292
$329$	0	0
$330$	−2.00320e8	−0.306849
$331$	4.28687e8	0.649745	0.324872	−	0.945758i	$-0.394679\pi$
$331$	4.28687e8	0.649745	0.324872	+	0.945758i	$0.394679\pi$
$332$	−1.35856e9	−2.03749
$333$	2.38069e8	0.353303
$334$	4.49313e8	0.659836
$335$	9.05105e8	1.31535
$336$	0	0
$337$	7.56118e8	1.07618	0.538090	−	0.842887i	$-0.319146\pi$
$337$	7.56118e8	1.07618	0.538090	+	0.842887i	$0.319146\pi$
$338$	2.39477e7	0.0337331
$339$	−5.18205e8	−0.722441
$340$	2.05179e9	2.83112
$341$	1.29622e8	0.177027
$342$	2.36638e8	0.319884
$343$	0	0
$344$	−2.94732e9	−3.90367
$345$	−3.65472e8	−0.479167
$346$	2.81710e9	3.65625
$347$	1.07256e9	1.37807	0.689033	−	0.724730i	$-0.258035\pi$
$347$	1.07256e9	1.37807	0.689033	+	0.724730i	$0.258035\pi$
$348$	−1.00060e9	−1.27272
$349$	8.07851e8	1.01728	0.508642	−	0.860978i	$-0.330148\pi$
$349$	8.07851e8	1.01728	0.508642	+	0.860978i	$0.330148\pi$
$350$	0	0
$351$	1.54537e8	0.190747
$352$	−4.17282e8	−0.509953
$353$	−3.52640e8	−0.426697	−0.213349	−	0.976976i	$-0.568437\pi$
$353$	−3.52640e8	−0.426697	−0.213349	+	0.976976i	$0.568437\pi$
$354$	1.34220e9	1.60807
$355$	6.32294e8	0.750101
$356$	2.67253e9	3.13941
$357$	0	0
$358$	3.32400e8	0.382887
$359$	4.73467e8	0.540082	0.270041	−	0.962849i	$-0.412963\pi$
$359$	4.73467e8	0.540082	0.270041	+	0.962849i	$0.412963\pi$
$360$	1.74042e9	1.96605
$361$	−6.69305e8	−0.748770
$362$	1.52062e9	1.68477
$363$	5.14300e8	0.564344
$364$	0	0
$365$	2.39941e8	0.258273
$366$	−1.42499e9	−1.51924
$367$	−9.74512e8	−1.02910	−0.514548	−	0.857462i	$-0.672040\pi$
$367$	−9.74512e8	−1.02910	−0.514548	+	0.857462i	$0.672040\pi$
$368$	−1.47591e9	−1.54381
$369$	5.77317e7	0.0598166
$370$	3.65675e9	3.75309
$371$	0	0
$372$	−1.80229e9	−1.81520
$373$	5.80133e8	0.578824	0.289412	−	0.957205i	$-0.406540\pi$
$373$	5.80133e8	0.578824	0.289412	+	0.957205i	$0.406540\pi$
$374$	1.66955e8	0.165024
$375$	1.54889e9	1.51674
$376$	−3.58774e9	−3.48067
$377$	−8.52744e8	−0.819642
$378$	0	0
$379$	−5.10991e8	−0.482143	−0.241072	−	0.970507i	$-0.577499\pi$
$379$	−5.10991e8	−0.482143	−0.241072	+	0.970507i	$0.577499\pi$
$380$	2.64321e9	2.47109
$381$	3.87619e8	0.359061
$382$	−3.66316e9	−3.36228
$383$	2.36881e6	0.00215444	0.00107722	−	0.999999i	$-0.499657\pi$
$383$	2.36881e6	0.00215444	0.00107722	+	0.999999i	$0.499657\pi$
$384$	1.58241e9	1.42613
$385$	0	0
$386$	−2.15832e9	−1.91012
$387$	4.65228e8	0.408017
$388$	1.53240e9	1.33187
$389$	−1.34362e9	−1.15732	−0.578660	−	0.815569i	$-0.696424\pi$
$389$	−1.34362e9	−1.15732	−0.578660	+	0.815569i	$0.696424\pi$
$390$	2.37370e9	2.02628
$391$	3.04599e8	0.257697
$392$	0	0
$393$	4.91655e8	0.408589
$394$	−3.74975e9	−3.08862
$395$	−8.95560e8	−0.731147
$396$	1.64811e8	0.133369
$397$	8.36087e8	0.670633	0.335317	−	0.942106i	$-0.391157\pi$
$397$	8.36087e8	0.670633	0.335317	+	0.942106i	$0.391157\pi$
$398$	−2.04378e9	−1.62496
$399$	0	0
$400$	1.06585e10	8.32698
$401$	−2.41709e8	−0.187192	−0.0935962	−	0.995610i	$-0.529836\pi$
$401$	−2.41709e8	−0.187192	−0.0935962	+	0.995610i	$0.529836\pi$
$402$	−1.02402e9	−0.786171
$403$	−1.53597e9	−1.16900
$404$	4.23936e9	3.19864
$405$	−2.74721e8	−0.205494
$406$	0	0
$407$	2.16378e8	0.159087
$408$	−1.45054e9	−1.05735
$409$	−2.67400e8	−0.193255	−0.0966273	−	0.995321i	$-0.530806\pi$
$409$	−2.67400e8	−0.193255	−0.0966273	+	0.995321i	$0.530806\pi$
$410$	8.86763e8	0.635424
$411$	−6.62227e8	−0.470501
$412$	−7.31187e9	−5.15096
$413$	0	0
$414$	4.13488e8	0.286393
$415$	2.05824e9	1.41360
$416$	4.94461e9	3.36748
$417$	3.19687e8	0.215898
$418$	2.15078e8	0.144039
$419$	1.59619e9	1.06007	0.530036	−	0.847975i	$-0.322178\pi$
$419$	1.59619e9	1.06007	0.530036	+	0.847975i	$0.322178\pi$
$420$	0	0
$421$	−1.89643e9	−1.23865	−0.619327	−	0.785133i	$-0.712595\pi$
$421$	−1.89643e9	−1.23865	−0.619327	+	0.785133i	$0.712595\pi$
$422$	1.11224e8	0.0720454
$423$	5.66316e8	0.363804
$424$	4.28685e9	2.73123
$425$	−2.19970e9	−1.38996
$426$	−7.15366e8	−0.448327
$427$	0	0
$428$	−7.59005e9	−4.67942
$429$	1.40457e8	0.0858903
$430$	7.14594e9	4.33431
$431$	−4.19921e8	−0.252637	−0.126319	−	0.991990i	$-0.540316\pi$
$431$	−4.19921e8	−0.252637	−0.126319	+	0.991990i	$0.540316\pi$
$432$	−1.10943e9	−0.662075
$433$	−2.39286e9	−1.41648	−0.708238	−	0.705974i	$-0.750509\pi$
$433$	−2.39286e9	−1.41648	−0.708238	+	0.705974i	$0.750509\pi$
$434$	0	0
$435$	1.51593e9	0.883010
$436$	3.37841e9	1.95214
$437$	3.92397e8	0.224926
$438$	−2.71465e8	−0.154367
$439$	5.68006e8	0.320426	0.160213	−	0.987082i	$-0.448782\pi$
$439$	5.68006e8	0.320426	0.160213	+	0.987082i	$0.448782\pi$
$440$	1.58185e9	0.885282
$441$	0	0
$442$	−1.97834e9	−1.08974
$443$	−2.26339e9	−1.23693	−0.618467	−	0.785811i	$-0.712246\pi$
$443$	−2.26339e9	−1.23693	−0.618467	+	0.785811i	$0.712246\pi$
$444$	−3.00856e9	−1.63124
$445$	−4.04893e9	−2.17811
$446$	−3.76586e9	−2.00998
$447$	−1.52067e8	−0.0805302
$448$	0	0
$449$	7.86592e8	0.410098	0.205049	−	0.978752i	$-0.434265\pi$
$449$	7.86592e8	0.410098	0.205049	+	0.978752i	$0.434265\pi$
$450$	−2.98607e9	−1.54474
$451$	5.24718e7	0.0269344
$452$	6.54875e9	3.33560
$453$	1.64149e9	0.829647
$454$	3.68790e9	1.84963
$455$	0	0
$456$	−1.86864e9	−0.922888
$457$	1.14935e9	0.563310	0.281655	−	0.959516i	$-0.409117\pi$
$457$	1.14935e9	0.563310	0.281655	+	0.959516i	$0.409117\pi$
$458$	5.36545e9	2.60962
$459$	2.28964e8	0.110515
$460$	4.61860e9	2.21237
$461$	7.67125e8	0.364681	0.182340	−	0.983235i	$-0.441633\pi$
$461$	7.67125e8	0.364681	0.182340	+	0.983235i	$0.441633\pi$
$462$	0	0
$463$	2.43312e9	1.13928	0.569640	−	0.821894i	$-0.307082\pi$
$463$	2.43312e9	1.13928	0.569640	+	0.821894i	$0.307082\pi$
$464$	6.12190e9	2.84494
$465$	2.73050e9	1.25938
$466$	2.47829e9	1.13449
$467$	−2.38932e9	−1.08559	−0.542794	−	0.839866i	$-0.682634\pi$
$467$	−2.38932e9	−1.08559	−0.542794	+	0.839866i	$0.682634\pi$
$468$	−1.95294e9	−0.880703
$469$	0	0
$470$	8.69866e9	3.86465
$471$	−1.52104e9	−0.670759
$472$	−1.05988e10	−4.63940
$473$	4.22842e8	0.183723
$474$	1.01322e9	0.436999
$475$	−2.83375e9	−1.21320
$476$	0	0
$477$	−6.76670e8	−0.285471
$478$	1.91271e9	0.801034
$479$	−2.23197e9	−0.927927	−0.463964	−	0.885854i	$-0.653573\pi$
$479$	−2.23197e9	−0.927927	−0.463964	+	0.885854i	$0.653573\pi$
$480$	−8.79005e9	−3.62783
$481$	−2.56399e9	−1.05053
$482$	−3.37148e9	−1.37137
$483$	0	0
$484$	−6.49941e9	−2.60564
$485$	−2.32161e9	−0.924043
$486$	3.10815e8	0.122822
$487$	2.07827e8	0.0815361	0.0407681	−	0.999169i	$-0.487020\pi$
$487$	2.07827e8	0.0815361	0.0407681	+	0.999169i	$0.487020\pi$
$488$	1.12526e10	4.38312
$489$	−1.42214e9	−0.549999
$490$	0	0
$491$	1.49013e9	0.568119	0.284060	−	0.958807i	$-0.408319\pi$
$491$	1.49013e9	0.568119	0.284060	+	0.958807i	$0.408319\pi$
$492$	−7.29577e8	−0.276181
$493$	−1.26343e9	−0.474885
$494$	−2.54858e9	−0.951161
$495$	−2.49692e8	−0.0925307
$496$	1.10268e10	4.05755
$497$	0	0
$498$	−2.32865e9	−0.844894
$499$	1.77157e9	0.638274	0.319137	−	0.947709i	$-0.396607\pi$
$499$	1.77157e9	0.638274	0.319137	+	0.947709i	$0.396607\pi$
$500$	−1.95739e10	−7.00298
$501$	5.60053e8	0.198974
$502$	−2.32901e9	−0.821691
$503$	2.60286e8	0.0911933	0.0455966	−	0.998960i	$-0.485481\pi$
$503$	2.60286e8	0.0911933	0.0455966	+	0.998960i	$0.485481\pi$
$504$	0	0
$505$	−6.42270e9	−2.21921
$506$	3.75816e8	0.128958
$507$	2.98501e7	0.0101723
$508$	−4.89849e9	−1.65783
$509$	−1.21248e9	−0.407534	−0.203767	−	0.979019i	$-0.565318\pi$
$509$	−1.21248e9	−0.407534	−0.203767	+	0.979019i	$0.565318\pi$
$510$	3.51690e9	1.17399
$511$	0	0
$512$	−2.17738e9	−0.716951
$513$	2.94961e8	0.0964614
$514$	−8.73333e9	−2.83667
$515$	1.10776e10	3.57372
$516$	−5.87927e9	−1.88386
$517$	5.14720e8	0.163815
$518$	0	0
$519$	3.51142e9	1.10255
$520$	−1.87443e10	−5.84597
$521$	2.57432e9	0.797499	0.398749	−	0.917060i	$-0.369444\pi$
$521$	2.57432e9	0.797499	0.398749	+	0.917060i	$0.369444\pi$
$522$	−1.71509e9	−0.527766
$523$	2.72268e9	0.832226	0.416113	−	0.909313i	$-0.363392\pi$
$523$	2.72268e9	0.832226	0.416113	+	0.909313i	$0.363392\pi$
$524$	−6.21323e9	−1.88650
$525$	0	0
$526$	5.92084e9	1.77392
$527$	−2.27571e9	−0.677298
$528$	−1.00835e9	−0.298121
$529$	−2.71917e9	−0.798623
$530$	−1.03937e10	−3.03253
$531$	1.67300e9	0.484916
$532$	0	0
$533$	−6.21768e8	−0.177862
$534$	4.58089e9	1.30183
$535$	1.14990e10	3.24656
$536$	8.08631e9	2.26816
$537$	4.14326e8	0.115460
$538$	2.24500e9	0.621552
$539$	0	0
$540$	3.47176e9	0.948793
$541$	−1.54196e9	−0.418681	−0.209340	−	0.977843i	$-0.567132\pi$
$541$	−1.54196e9	−0.418681	−0.209340	+	0.977843i	$0.567132\pi$
$542$	−4.29161e8	−0.115777
$543$	1.89540e9	0.508044
$544$	7.32598e9	1.95106
$545$	−5.11835e9	−1.35438
$546$	0	0
$547$	−5.49200e9	−1.43475	−0.717373	−	0.696689i	$-0.754656\pi$
$547$	−5.49200e9	−1.43475	−0.717373	+	0.696689i	$0.754656\pi$
$548$	8.36881e9	2.17236
$549$	−1.77620e9	−0.458130
$550$	−2.71401e9	−0.695571
$551$	−1.62761e9	−0.414495
$552$	−3.26517e9	−0.826263
$553$	0	0
$554$	7.71799e9	1.92850
$555$	4.55802e9	1.13175
$556$	−4.04000e9	−0.996827
$557$	−6.14019e9	−1.50553	−0.752764	−	0.658291i	$-0.771280\pi$
$557$	−6.14019e9	−1.50553	−0.752764	+	0.658291i	$0.771280\pi$
$558$	−3.08924e9	−0.752718
$559$	−5.01049e9	−1.21322
$560$	0	0
$561$	2.08103e8	0.0497632
$562$	3.44994e8	0.0819851
$563$	3.24766e9	0.766993	0.383497	−	0.923542i	$-0.374720\pi$
$563$	3.24766e9	0.766993	0.383497	+	0.923542i	$0.374720\pi$
$564$	−7.15675e9	−1.67973
$565$	−9.92145e9	−2.31422
$566$	1.04485e10	2.42213
$567$	0	0
$568$	5.64899e9	1.29346
$569$	2.01161e9	0.457774	0.228887	−	0.973453i	$-0.426491\pi$
$569$	2.01161e9	0.457774	0.228887	+	0.973453i	$0.426491\pi$
$570$	4.53062e9	1.02470
$571$	−3.66896e9	−0.824738	−0.412369	−	0.911017i	$-0.635299\pi$
$571$	−3.66896e9	−0.824738	−0.412369	+	0.911017i	$0.635299\pi$
$572$	−1.77501e9	−0.396566
$573$	−4.56600e9	−1.01390
$574$	0	0
$575$	−4.95155e9	−1.08618
$576$	4.68539e9	1.02157
$577$	−3.43622e9	−0.744673	−0.372337	−	0.928098i	$-0.621443\pi$
$577$	−3.43622e9	−0.744673	−0.372337	+	0.928098i	$0.621443\pi$
$578$	5.95732e9	1.28323
$579$	−2.69028e9	−0.575999
$580$	−1.91573e10	−4.07697
$581$	0	0
$582$	2.62662e9	0.552290
$583$	−6.15019e8	−0.128543
$584$	2.14366e9	0.445360
$585$	2.95874e9	0.611028
$586$	−9.60372e9	−1.97150
$587$	−8.47232e9	−1.72890	−0.864449	−	0.502721i	$-0.832332\pi$
$587$	−8.47232e9	−1.72890	−0.864449	+	0.502721i	$0.832332\pi$
$588$	0	0
$589$	−2.93166e9	−0.591168
$590$	2.56975e10	5.15120
$591$	−4.67393e9	−0.931379
$592$	1.84070e10	3.64634
$593$	9.58929e9	1.88840	0.944202	−	0.329367i	$-0.106835\pi$
$593$	9.58929e9	1.88840	0.944202	+	0.329367i	$0.106835\pi$
$594$	2.82497e8	0.0553046
$595$	0	0
$596$	1.92173e9	0.371818
$597$	−2.54750e9	−0.490009
$598$	−4.45325e9	−0.851576
$599$	−3.16634e9	−0.601955	−0.300978	−	0.953631i	$-0.597313\pi$
$599$	−3.16634e9	−0.601955	−0.300978	+	0.953631i	$0.597313\pi$
$600$	2.35799e10	4.45669
$601$	5.32513e9	1.00062	0.500310	−	0.865846i	$-0.333219\pi$
$601$	5.32513e9	1.00062	0.500310	+	0.865846i	$0.333219\pi$
$602$	0	0
$603$	−1.27641e9	−0.237071
$604$	−2.07441e10	−3.83058
$605$	9.84669e9	1.80778
$606$	7.26653e9	1.32639
$607$	5.32039e9	0.965569	0.482785	−	0.875739i	$-0.339625\pi$
$607$	5.32039e9	0.965569	0.482785	+	0.875739i	$0.339625\pi$
$608$	9.43764e9	1.70295
$609$	0	0
$610$	−2.72825e10	−4.86665
$611$	−6.09921e9	−1.08176
$612$	−2.89350e9	−0.510263
$613$	−6.15195e9	−1.07870	−0.539351	−	0.842081i	$-0.681330\pi$
$613$	−6.15195e9	−1.07870	−0.539351	+	0.842081i	$0.681330\pi$
$614$	1.68038e9	0.292967
$615$	1.10532e9	0.191613
$616$	0	0
$617$	−4.29619e9	−0.736352	−0.368176	−	0.929756i	$-0.620018\pi$
$617$	−4.29619e9	−0.736352	−0.368176	+	0.929756i	$0.620018\pi$
$618$	−1.25330e10	−2.13597
$619$	3.90151e9	0.661173	0.330587	−	0.943776i	$-0.392753\pi$
$619$	3.90151e9	0.661173	0.330587	+	0.943776i	$0.392753\pi$
$620$	−3.45064e10	−5.81471
$621$	5.15399e8	0.0863621
$622$	−9.58804e9	−1.59758
$623$	0	0
$624$	1.19485e10	1.96865
$625$	1.48815e10	2.43818
$626$	1.21958e10	1.98702
$627$	2.68087e8	0.0434350
$628$	1.92219e10	3.09698
$629$	−3.79884e9	−0.608658
$630$	0	0
$631$	5.79183e9	0.917726	0.458863	−	0.888507i	$-0.348257\pi$
$631$	5.79183e9	0.917726	0.458863	+	0.888507i	$0.348257\pi$
$632$	−8.00104e9	−1.26077
$633$	1.38637e8	0.0217254
$634$	1.65394e10	2.57755
$635$	7.42129e9	1.15019
$636$	8.55133e9	1.31806
$637$	0	0
$638$	−1.55883e9	−0.237644
$639$	−8.91680e8	−0.135194
$640$	3.02966e10	4.56839
$641$	3.31891e9	0.497729	0.248864	−	0.968538i	$-0.419943\pi$
$641$	3.31891e9	0.497729	0.248864	+	0.968538i	$0.419943\pi$
$642$	−1.30098e10	−1.94043
$643$	1.23513e10	1.83220	0.916100	−	0.400949i	$-0.131319\pi$
$643$	1.23513e10	1.83220	0.916100	+	0.400949i	$0.131319\pi$
$644$	0	0
$645$	8.90718e9	1.30702
$646$	−3.77600e9	−0.551085
$647$	−8.67679e9	−1.25949	−0.629744	−	0.776802i	$-0.716840\pi$
$647$	−8.67679e9	−1.25949	−0.629744	+	0.776802i	$0.716840\pi$
$648$	−2.45439e9	−0.354349
$649$	1.52058e9	0.218350
$650$	3.21598e10	4.59321
$651$	0	0
$652$	1.79722e10	2.53941
$653$	−5.08215e9	−0.714251	−0.357126	−	0.934056i	$-0.616243\pi$
$653$	−5.08215e9	−0.714251	−0.357126	+	0.934056i	$0.616243\pi$
$654$	5.79081e9	0.809500
$655$	9.41313e9	1.30885
$656$	4.46371e9	0.617351
$657$	−3.38372e8	−0.0465496
$658$	0	0
$659$	7.83208e9	1.06605	0.533026	−	0.846099i	$-0.321055\pi$
$659$	7.83208e9	1.06605	0.533026	+	0.846099i	$0.321055\pi$
$660$	3.15545e9	0.427226
$661$	6.36676e9	0.857459	0.428729	−	0.903433i	$-0.358961\pi$
$661$	6.36676e9	0.857459	0.428729	+	0.903433i	$0.358961\pi$
$662$	−9.28590e9	−1.24400
$663$	−2.46593e9	−0.328612
$664$	1.83885e10	2.43758
$665$	0	0
$666$	−5.15686e9	−0.676434
$667$	−2.84400e9	−0.371098
$668$	−7.07760e9	−0.918689
$669$	−4.69402e9	−0.606113
$670$	−1.96057e10	−2.51837
$671$	−1.61437e9	−0.206288
$672$	0	0
$673$	1.11438e10	1.40923	0.704615	−	0.709590i	$-0.251120\pi$
$673$	1.11438e10	1.40923	0.704615	+	0.709590i	$0.251120\pi$
$674$	−1.63785e10	−2.06046
$675$	−3.72203e9	−0.465818
$676$	−3.77226e8	−0.0469665
$677$	7.73190e9	0.957693	0.478846	−	0.877899i	$-0.341055\pi$
$677$	7.73190e9	0.957693	0.478846	+	0.877899i	$0.341055\pi$
$678$	1.12250e10	1.38319
$679$	0	0
$680$	−2.77717e10	−3.38705
$681$	4.59684e9	0.557757
$682$	−2.80778e9	−0.338936
$683$	3.26590e9	0.392220	0.196110	−	0.980582i	$-0.437169\pi$
$683$	3.26590e9	0.392220	0.196110	+	0.980582i	$0.437169\pi$
$684$	−3.72753e9	−0.445374
$685$	−1.26789e10	−1.50717
$686$	0	0
$687$	6.68785e9	0.786933
$688$	3.59706e10	4.21103
$689$	7.28771e9	0.848836
$690$	7.91657e9	0.917413
$691$	1.44199e10	1.66261	0.831303	−	0.555819i	$-0.187595\pi$
$691$	1.44199e10	1.66261	0.831303	+	0.555819i	$0.187595\pi$
$692$	−4.43752e10	−5.09060
$693$	0	0
$694$	−2.32331e10	−2.63845
$695$	6.12066e9	0.691594
$696$	1.35435e10	1.52264
$697$	−9.21218e8	−0.103050
$698$	−1.74990e10	−1.94769
$699$	3.08911e9	0.342108
$700$	0	0
$701$	1.37465e9	0.150723	0.0753615	−	0.997156i	$-0.475989\pi$
$701$	1.37465e9	0.150723	0.0753615	+	0.997156i	$0.475989\pi$
$702$	−3.34747e9	−0.365205
$703$	−4.89382e9	−0.531257
$704$	4.25851e9	0.459995
$705$	1.08426e10	1.16539
$706$	7.63862e9	0.816955
$707$	0	0
$708$	−2.11424e10	−2.23892
$709$	−1.41543e10	−1.49151	−0.745753	−	0.666222i	$-0.767910\pi$
$709$	−1.41543e10	−1.49151	−0.745753	+	0.666222i	$0.767910\pi$
$710$	−1.36963e10	−1.43614
$711$	1.26295e9	0.131777
$712$	−3.61736e10	−3.75588
$713$	−5.12263e9	−0.529273
$714$	0	0
$715$	2.68917e9	0.275136
$716$	−5.23599e9	−0.533094
$717$	2.38413e9	0.241553
$718$	−1.02559e10	−1.03404
$719$	6.64223e9	0.666442	0.333221	−	0.942849i	$-0.391864\pi$
$719$	6.64223e9	0.666442	0.333221	+	0.942849i	$0.391864\pi$
$720$	−2.12409e10	−2.12085
$721$	0	0
$722$	1.44980e10	1.43360
$723$	−4.20243e9	−0.413539
$724$	−2.39529e10	−2.34570
$725$	2.05383e10	2.00162
$726$	−1.11404e10	−1.08049
$727$	7.79081e9	0.751990	0.375995	−	0.926622i	$-0.377301\pi$
$727$	7.79081e9	0.751990	0.375995	+	0.926622i	$0.377301\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	−5.19742e9	−0.494490
$731$	−7.42360e9	−0.702916
$732$	2.24465e10	2.11524
$733$	−1.92152e10	−1.80210	−0.901052	−	0.433711i	$-0.857204\pi$
$733$	−1.92152e10	−1.80210	−0.901052	+	0.433711i	$0.857204\pi$
$734$	2.11091e10	1.97031
$735$	0	0
$736$	1.64908e10	1.52465
$737$	−1.16011e9	−0.106749
$738$	−1.25054e9	−0.114525
$739$	−1.13260e10	−1.03233	−0.516166	−	0.856489i	$-0.672641\pi$
$739$	−1.13260e10	−1.03233	−0.516166	+	0.856489i	$0.672641\pi$
$740$	−5.76014e10	−5.22543
$741$	−3.17672e9	−0.286824
$742$	0	0
$743$	−9.10743e9	−0.814582	−0.407291	−	0.913298i	$-0.633527\pi$
$743$	−9.10743e9	−0.814582	−0.407291	+	0.913298i	$0.633527\pi$
$744$	2.43946e10	2.17164
$745$	−2.91145e9	−0.257966
$746$	−1.25664e10	−1.10822
$747$	−2.90259e9	−0.254779
$748$	−2.62988e9	−0.229763
$749$	0	0
$750$	−3.35509e10	−2.90396
$751$	1.65376e9	0.142473	0.0712365	−	0.997459i	$-0.477305\pi$
$751$	1.65376e9	0.142473	0.0712365	+	0.997459i	$0.477305\pi$
$752$	4.37865e10	3.75472
$753$	−2.90303e9	−0.247782
$754$	1.84715e10	1.56929
$755$	3.14276e10	2.65764
$756$	0	0
$757$	1.48318e10	1.24268	0.621338	−	0.783543i	$-0.286590\pi$
$757$	1.48318e10	1.24268	0.621338	+	0.783543i	$0.286590\pi$
$758$	1.10687e10	0.923111
$759$	4.68441e8	0.0388874
$760$	−3.57767e10	−2.95632
$761$	−2.80637e9	−0.230833	−0.115417	−	0.993317i	$-0.536820\pi$
$761$	−2.80637e9	−0.230833	−0.115417	+	0.993317i	$0.536820\pi$
$762$	−8.39631e9	−0.687458
$763$	0	0
$764$	5.77023e10	4.68130
$765$	4.38370e9	0.354018
$766$	−5.13115e7	−0.00412490
$767$	−1.80182e10	−1.44187
$768$	−1.20648e10	−0.961070
$769$	1.56499e10	1.24099	0.620496	−	0.784210i	$-0.286932\pi$
$769$	1.56499e10	1.24099	0.620496	+	0.784210i	$0.286932\pi$
$770$	0	0
$771$	−1.08858e10	−0.855402
$772$	3.39980e10	2.65946
$773$	−4.08905e9	−0.318416	−0.159208	−	0.987245i	$-0.550894\pi$
$773$	−4.08905e9	−0.318416	−0.159208	+	0.987245i	$0.550894\pi$
$774$	−1.00774e10	−0.781189
$775$	3.69938e10	2.85479
$776$	−2.07415e10	−1.59340
$777$	0	0
$778$	2.91045e10	2.21580
$779$	−1.18675e9	−0.0899454
$780$	−3.73907e10	−2.82119
$781$	−8.10440e8	−0.0608754
$782$	−6.59798e9	−0.493387
$783$	−2.13781e9	−0.159148
$784$	0	0
$785$	−2.91215e10	−2.14867
$786$	−1.06499e10	−0.782284
$787$	2.52055e10	1.84325	0.921623	−	0.388087i	$-0.126864\pi$
$787$	2.52055e10	1.84325	0.921623	+	0.388087i	$0.126864\pi$
$788$	5.90662e10	4.30029
$789$	7.38012e9	0.534926
$790$	1.93989e10	1.39986
$791$	0	0
$792$	−2.23078e9	−0.159558
$793$	1.91296e10	1.36223
$794$	−1.81107e10	−1.28399
$795$	−1.29554e10	−0.914462
$796$	3.21937e10	2.26243
$797$	6.07194e9	0.424838	0.212419	−	0.977179i	$-0.431866\pi$
$797$	6.07194e9	0.424838	0.212419	+	0.977179i	$0.431866\pi$
$798$	0	0
$799$	−9.03665e9	−0.626749
$800$	−1.19091e11	−8.22363
$801$	5.70992e9	0.392569
$802$	5.23573e9	0.358399
$803$	−3.07543e8	−0.0209605
$804$	1.61304e10	1.09459
$805$	0	0
$806$	3.32710e10	2.23817
$807$	2.79831e9	0.187430
$808$	−5.73811e10	−3.82674
$809$	2.26249e10	1.50234	0.751168	−	0.660111i	$-0.229491\pi$
$809$	2.26249e10	1.50234	0.751168	+	0.660111i	$0.229491\pi$
$810$	5.95081e9	0.393440
$811$	−1.70292e10	−1.12104	−0.560520	−	0.828141i	$-0.689399\pi$
$811$	−1.70292e10	−1.12104	−0.560520	+	0.828141i	$0.689399\pi$
$812$	0	0
$813$	−5.34935e8	−0.0349127
$814$	−4.68702e9	−0.304587
$815$	−2.72281e10	−1.76184
$816$	1.77031e10	1.14060
$817$	−9.56339e9	−0.613528
$818$	5.79222e9	0.370006
$819$	0	0
$820$	−1.39683e10	−0.884701
$821$	−2.99513e10	−1.88893	−0.944463	−	0.328617i	$-0.893418\pi$
$821$	−2.99513e10	−1.88893	−0.944463	+	0.328617i	$0.893418\pi$
$822$	1.43447e10	0.900822
$823$	−1.10149e10	−0.688782	−0.344391	−	0.938826i	$-0.611915\pi$
$823$	−1.10149e10	−0.688782	−0.344391	+	0.938826i	$0.611915\pi$
$824$	9.89685e10	6.16243
$825$	−3.38292e9	−0.209750
$826$	0	0
$827$	−1.67625e10	−1.03055	−0.515276	−	0.857024i	$-0.672311\pi$
$827$	−1.67625e10	−1.03055	−0.515276	+	0.857024i	$0.672311\pi$
$828$	−6.51329e9	−0.398744
$829$	−3.00024e10	−1.82901	−0.914504	−	0.404577i	$-0.867419\pi$
$829$	−3.00024e10	−1.82901	−0.914504	+	0.404577i	$0.867419\pi$
$830$	−4.45840e10	−2.70648
$831$	9.62022e9	0.581542
$832$	−5.04615e10	−3.03758
$833$	0	0
$834$	−6.92481e9	−0.413358
$835$	1.07227e10	0.637383
$836$	−3.38792e9	−0.200545
$837$	−3.85063e9	−0.226983
$838$	−3.45754e10	−2.02961
$839$	2.30538e10	1.34764	0.673821	−	0.738894i	$-0.264652\pi$
$839$	2.30538e10	1.34764	0.673821	+	0.738894i	$0.264652\pi$
$840$	0	0
$841$	−5.45335e9	−0.316139
$842$	4.10791e10	2.37153
$843$	4.30024e8	0.0247227
$844$	−1.75201e9	−0.100309
$845$	5.71503e8	0.0325852
$846$	−1.22671e10	−0.696540
$847$	0	0
$848$	−5.23188e10	−2.94627
$849$	1.30237e10	0.730397
$850$	4.76483e10	2.66122
$851$	−8.55120e9	−0.475635
$852$	1.12685e10	0.624205
$853$	5.09026e9	0.280814	0.140407	−	0.990094i	$-0.455159\pi$
$853$	5.09026e9	0.280814	0.140407	+	0.990094i	$0.455159\pi$
$854$	0	0
$855$	5.64726e9	0.308999
$856$	1.02734e11	5.59829
$857$	1.39679e10	0.758052	0.379026	−	0.925386i	$-0.376259\pi$
$857$	1.39679e10	0.758052	0.379026	+	0.925386i	$0.376259\pi$
$858$	−3.04248e9	−0.164446
$859$	2.23443e10	1.20279	0.601396	−	0.798951i	$-0.294611\pi$
$859$	2.23443e10	1.20279	0.601396	+	0.798951i	$0.294611\pi$
$860$	−1.12563e11	−6.03465
$861$	0	0
$862$	9.09602e9	0.483700
$863$	−6.66234e9	−0.352849	−0.176425	−	0.984314i	$-0.556453\pi$
$863$	−6.66234e9	−0.352849	−0.176425	+	0.984314i	$0.556453\pi$
$864$	1.23960e10	0.653858
$865$	6.72290e10	3.53183
$866$	5.18322e10	2.71198
$867$	7.42559e9	0.386958
$868$	0	0
$869$	1.14788e9	0.0593372
$870$	−3.28369e10	−1.69061
$871$	1.37468e10	0.704920
$872$	−4.57279e10	−2.33547
$873$	3.27400e9	0.166544
$874$	−8.49980e9	−0.430644
$875$	0	0
$876$	4.27614e9	0.214925
$877$	−2.98982e10	−1.49674	−0.748370	−	0.663282i	$-0.769163\pi$
$877$	−2.98982e10	−1.49674	−0.748370	+	0.663282i	$0.769163\pi$
$878$	−1.23037e10	−0.613487
$879$	−1.19707e10	−0.594510
$880$	−1.93057e10	−0.954984
$881$	−1.24713e10	−0.614466	−0.307233	−	0.951634i	$-0.599403\pi$
$881$	−1.24713e10	−0.614466	−0.307233	+	0.951634i	$0.599403\pi$
$882$	0	0
$883$	1.37548e10	0.672345	0.336173	−	0.941800i	$-0.390867\pi$
$883$	1.37548e10	0.672345	0.336173	+	0.941800i	$0.390867\pi$
$884$	3.11629e10	1.51724
$885$	3.20310e10	1.55335
$886$	4.90278e10	2.36823
$887$	−3.13536e10	−1.50853	−0.754265	−	0.656570i	$-0.772007\pi$
$887$	−3.13536e10	−1.50853	−0.754265	+	0.656570i	$0.772007\pi$
$888$	4.07219e10	1.95156
$889$	0	0
$890$	8.77048e10	4.17021
$891$	3.52123e8	0.0166772
$892$	5.93201e10	2.79850
$893$	−1.16414e10	−0.547047
$894$	3.29396e9	0.154183
$895$	7.93261e9	0.369858
$896$	0	0
$897$	−5.55083e9	−0.256794
$898$	−1.70386e10	−0.785174
$899$	2.12480e10	0.975347
$900$	4.70367e10	2.15074
$901$	1.07975e10	0.491800
$902$	−1.13660e9	−0.0515687
$903$	0	0
$904$	−8.86394e10	−3.99059
$905$	3.62890e10	1.62744
$906$	−3.55566e10	−1.58844
$907$	−1.12062e9	−0.0498694	−0.0249347	−	0.999689i	$-0.507938\pi$
$907$	−1.12062e9	−0.0498694	−0.0249347	+	0.999689i	$0.507938\pi$
$908$	−5.80920e10	−2.57523
$909$	9.05748e9	0.399976
$910$	0	0
$911$	−3.97792e10	−1.74318	−0.871589	−	0.490238i	$-0.836910\pi$
$911$	−3.97792e10	−1.74318	−0.871589	+	0.490238i	$0.836910\pi$
$912$	2.28058e10	0.995552
$913$	−2.63814e9	−0.114723
$914$	−2.48964e10	−1.07851
$915$	−3.40068e10	−1.46754
$916$	−8.45169e10	−3.63337
$917$	0	0
$918$	−4.95964e9	−0.211593
$919$	−1.44290e10	−0.613244	−0.306622	−	0.951831i	$-0.599199\pi$
$919$	−1.44290e10	−0.613244	−0.306622	+	0.951831i	$0.599199\pi$
$920$	−6.25142e10	−2.64680
$921$	2.09454e9	0.0883446
$922$	−1.66169e10	−0.698218
$923$	9.60336e9	0.401992
$924$	0	0
$925$	6.17537e10	2.56547
$926$	−5.27045e10	−2.18127
$927$	−1.56220e10	−0.644105
$928$	−6.84018e10	−2.80963
$929$	−2.66062e10	−1.08875	−0.544374	−	0.838842i	$-0.683233\pi$
$929$	−2.66062e10	−1.08875	−0.544374	+	0.838842i	$0.683233\pi$
$930$	−5.91460e10	−2.41121
$931$	0	0
$932$	−3.90382e10	−1.57955
$933$	−1.19512e10	−0.481753
$934$	5.17556e10	2.07847
$935$	3.98430e9	0.159409
$936$	2.64337e10	1.05364
$937$	−4.46208e10	−1.77194	−0.885970	−	0.463742i	$-0.846506\pi$
$937$	−4.46208e10	−1.77194	−0.885970	+	0.463742i	$0.846506\pi$
$938$	0	0
$939$	1.52017e10	0.599188
$940$	−1.37022e11	−5.38074
$941$	−4.28359e9	−0.167588	−0.0837942	−	0.996483i	$-0.526704\pi$
$941$	−4.28359e9	−0.167588	−0.0837942	+	0.996483i	$0.526704\pi$
$942$	3.29475e10	1.28424
$943$	−2.07367e9	−0.0805282
$944$	1.29354e11	5.00468
$945$	0	0
$946$	−9.15928e9	−0.351757
$947$	1.78755e9	0.0683965	0.0341982	−	0.999415i	$-0.489112\pi$
$947$	1.78755e9	0.0683965	0.0341982	+	0.999415i	$0.489112\pi$
$948$	−1.59603e10	−0.608433
$949$	3.64425e9	0.138413
$950$	6.13825e10	2.32280
$951$	2.06158e10	0.777264
$952$	0	0
$953$	−1.13652e10	−0.425356	−0.212678	−	0.977122i	$-0.568219\pi$
$953$	−1.13652e10	−0.425356	−0.212678	+	0.977122i	$0.568219\pi$
$954$	1.46575e10	0.546564
$955$	−8.74198e10	−3.24787
$956$	−3.01291e10	−1.11528
$957$	−1.94303e9	−0.0716619
$958$	4.83472e10	1.77661
$959$	0	0
$960$	8.97056e10	3.27243
$961$	1.07595e10	0.391074
$962$	5.55392e10	2.01135
$963$	−1.62163e10	−0.585140
$964$	5.31077e10	1.90936
$965$	−5.15075e10	−1.84512
$966$	0	0
$967$	4.31418e10	1.53428	0.767141	−	0.641478i	$-0.221678\pi$
$967$	4.31418e10	1.53428	0.767141	+	0.641478i	$0.221678\pi$
$968$	8.79715e10	3.11730
$969$	−4.70666e9	−0.166180
$970$	5.02888e10	1.76917
$971$	1.88022e10	0.659086	0.329543	−	0.944141i	$-0.393105\pi$
$971$	1.88022e10	0.659086	0.329543	+	0.944141i	$0.393105\pi$
$972$	−4.89598e9	−0.171005
$973$	0	0
$974$	−4.50179e9	−0.156109
$975$	4.00861e10	1.38509
$976$	−1.37332e11	−4.72823
$977$	−2.32046e9	−0.0796056	−0.0398028	−	0.999208i	$-0.512673\pi$
$977$	−2.32046e9	−0.0796056	−0.0398028	+	0.999208i	$0.512673\pi$
$978$	3.08054e10	1.05303
$979$	5.18969e9	0.176767
$980$	0	0
$981$	7.21805e9	0.244106
$982$	−3.22781e10	−1.08772
$983$	2.43379e10	0.817233	0.408616	−	0.912706i	$-0.366011\pi$
$983$	2.43379e10	0.817233	0.408616	+	0.912706i	$0.366011\pi$
$984$	9.87506e9	0.330413
$985$	−8.94862e10	−2.98352
$986$	2.73676e10	0.909215
$987$	0	0
$988$	4.01454e10	1.32430
$989$	−1.67106e10	−0.549293
$990$	5.40863e9	0.177159
$991$	−2.72855e10	−0.890582	−0.445291	−	0.895386i	$-0.646900\pi$
$991$	−2.72855e10	−0.890582	−0.445291	+	0.895386i	$0.646900\pi$
$992$	−1.23206e11	−4.00719
$993$	−1.15746e10	−0.375130
$994$	0	0
$995$	−4.87739e10	−1.56966
$996$	3.66811e10	1.17635
$997$	1.81964e9	0.0581503	0.0290751	−	0.999577i	$-0.490744\pi$
$997$	1.81964e9	0.0581503	0.0290751	+	0.999577i	$0.490744\pi$
$998$	−3.83744e10	−1.22204
$999$	−6.42785e9	−0.203980

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.8.a.f.1.1	✓	4
3.2	odd	2		441.8.a.v.1.4		4
7.2	even	3		147.8.e.m.67.4		8
7.3	odd	6		147.8.e.l.79.4		8
7.4	even	3		147.8.e.m.79.4		8
7.5	odd	6		147.8.e.l.67.4		8
7.6	odd	2		147.8.a.g.1.1	yes	4
21.20	even	2		441.8.a.u.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.8.a.f.1.1	✓	4	1.1	even	1	trivial
147.8.a.g.1.1	yes	4	7.6	odd	2
147.8.e.l.67.4		8	7.5	odd	6
147.8.e.l.79.4		8	7.3	odd	6
147.8.e.m.67.4		8	7.2	even	3
147.8.e.m.79.4		8	7.4	even	3
441.8.a.u.1.4		4	21.20	even	2
441.8.a.v.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:	\( N \)	\(=\)	\( 147 = 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	147.a (trivial)

\(f(q)\)	\(=\)	\(q-21.6612 q^{2} -27.0000 q^{3} +341.209 q^{4} -516.937 q^{5} +584.853 q^{6} -4618.37 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q-21.6612 q^{2} -27.0000 q^{3} +341.209 q^{4} -516.937 q^{5} +584.853 q^{6} -4618.37 q^{8} +729.000 q^{9} +11197.5 q^{10} +662.581 q^{11} -9212.65 q^{12} -7851.30 q^{13} +13957.3 q^{15} +56364.9 q^{16} -11632.6 q^{17} -15791.0 q^{18} -14985.6 q^{19} -176384. q^{20} -14352.3 q^{22} -26185.0 q^{23} +124696. q^{24} +189099. q^{25} +170069. q^{26} -19683.0 q^{27} +108612. q^{29} -302332. q^{30} +195632. q^{31} -629782. q^{32} -17889.7 q^{33} +251976. q^{34} +248741. q^{36} +326569. q^{37} +324606. q^{38} +211985. q^{39} +2.38741e6 q^{40} +79193.0 q^{41} +638173. q^{43} +226079. q^{44} -376847. q^{45} +567199. q^{46} +776840. q^{47} -1.52185e6 q^{48} -4.09611e6 q^{50} +314079. q^{51} -2.67894e6 q^{52} -928216. q^{53} +426358. q^{54} -342513. q^{55} +404610. q^{57} -2.35267e6 q^{58} +2.29493e6 q^{59} +4.76236e6 q^{60} -2.43649e6 q^{61} -4.23764e6 q^{62} +6.42715e6 q^{64} +4.05863e6 q^{65} +387513. q^{66} -1.75090e6 q^{67} -3.96914e6 q^{68} +706995. q^{69} -1.22315e6 q^{71} -3.36679e6 q^{72} -464159. q^{73} -7.07389e6 q^{74} -5.10566e6 q^{75} -5.11321e6 q^{76} -4.59186e6 q^{78} +1.73244e6 q^{79} -2.91371e7 q^{80} +531441. q^{81} -1.71542e6 q^{82} -3.98160e6 q^{83} +6.01330e6 q^{85} -1.38236e7 q^{86} -2.93252e6 q^{87} -3.06005e6 q^{88} +7.83254e6 q^{89} +8.16297e6 q^{90} -8.93456e6 q^{92} -5.28208e6 q^{93} -1.68273e7 q^{94} +7.74659e6 q^{95} +1.70041e7 q^{96} +4.49108e6 q^{97} +483022. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 15 q^{2} - 108 q^{3} + 437 q^{4} - 504 q^{5} + 405 q^{6} - 2145 q^{8} + 2916 q^{9}+O(q^{10}) \) 4 * q - 15 * q^2 - 108 * q^3 + 437 * q^4 - 504 * q^5 + 405 * q^6 - 2145 * q^8 + 2916 * q^9	\( 4 q - 15 q^{2} - 108 q^{3} + 437 q^{4} - 504 q^{5} + 405 q^{6} - 2145 q^{8} + 2916 q^{9} + 5724 q^{10} - 1920 q^{11} - 11799 q^{12} - 18144 q^{13} + 13608 q^{15} + 52529 q^{16} + 19584 q^{17} - 10935 q^{18} - 31320 q^{19} - 169812 q^{20} + 111940 q^{22} + 101160 q^{23} + 57915 q^{24} + 74788 q^{25} + 107820 q^{26} - 78732 q^{27} + 194832 q^{29} - 154548 q^{30} - 78840 q^{31} - 732585 q^{32} + 51840 q^{33} + 77760 q^{34} + 318573 q^{36} + 128640 q^{37} - 716292 q^{38} + 489888 q^{39} + 2649780 q^{40} - 365040 q^{41} + 449520 q^{43} + 113220 q^{44} - 367416 q^{45} + 1033664 q^{46} + 1575792 q^{47} - 1418283 q^{48} - 5391933 q^{50} - 528768 q^{51} - 5747652 q^{52} - 1448160 q^{53} + 295245 q^{54} - 3083400 q^{55} + 845640 q^{57} - 2570950 q^{58} - 3280320 q^{59} + 4584924 q^{60} - 606960 q^{61} - 12064536 q^{62} + 2609137 q^{64} + 1318464 q^{65} - 3022380 q^{66} + 3492880 q^{67} + 1476432 q^{68} - 2731320 q^{69} + 984 q^{71} - 1563705 q^{72} - 10981440 q^{73} - 14177022 q^{74} - 2019276 q^{75} - 18964260 q^{76} - 2911140 q^{78} + 4654544 q^{79} - 37026324 q^{80} + 2125764 q^{81} + 10402560 q^{82} - 8126496 q^{83} + 16792056 q^{85} - 6392244 q^{86} - 5260464 q^{87} - 10447140 q^{88} + 11272320 q^{89} + 4172796 q^{90} + 3289440 q^{92} + 2128680 q^{93} - 37697400 q^{94} + 11132208 q^{95} + 19779795 q^{96} - 6572448 q^{97} - 1399680 q^{99}+O(q^{100}) \) 4 * q - 15 * q^2 - 108 * q^3 + 437 * q^4 - 504 * q^5 + 405 * q^6 - 2145 * q^8 + 2916 * q^9 + 5724 * q^10 - 1920 * q^11 - 11799 * q^12 - 18144 * q^13 + 13608 * q^15 + 52529 * q^16 + 19584 * q^17 - 10935 * q^18 - 31320 * q^19 - 169812 * q^20 + 111940 * q^22 + 101160 * q^23 + 57915 * q^24 + 74788 * q^25 + 107820 * q^26 - 78732 * q^27 + 194832 * q^29 - 154548 * q^30 - 78840 * q^31 - 732585 * q^32 + 51840 * q^33 + 77760 * q^34 + 318573 * q^36 + 128640 * q^37 - 716292 * q^38 + 489888 * q^39 + 2649780 * q^40 - 365040 * q^41 + 449520 * q^43 + 113220 * q^44 - 367416 * q^45 + 1033664 * q^46 + 1575792 * q^47 - 1418283 * q^48 - 5391933 * q^50 - 528768 * q^51 - 5747652 * q^52 - 1448160 * q^53 + 295245 * q^54 - 3083400 * q^55 + 845640 * q^57 - 2570950 * q^58 - 3280320 * q^59 + 4584924 * q^60 - 606960 * q^61 - 12064536 * q^62 + 2609137 * q^64 + 1318464 * q^65 - 3022380 * q^66 + 3492880 * q^67 + 1476432 * q^68 - 2731320 * q^69 + 984 * q^71 - 1563705 * q^72 - 10981440 * q^73 - 14177022 * q^74 - 2019276 * q^75 - 18964260 * q^76 - 2911140 * q^78 + 4654544 * q^79 - 37026324 * q^80 + 2125764 * q^81 + 10402560 * q^82 - 8126496 * q^83 + 16792056 * q^85 - 6392244 * q^86 - 5260464 * q^87 - 10447140 * q^88 + 11272320 * q^89 + 4172796 * q^90 + 3289440 * q^92 + 2128680 * q^93 - 37697400 * q^94 + 11132208 * q^95 + 19779795 * q^96 - 6572448 * q^97 - 1399680 * q^99

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