LMFDB - Embedded newform 9200.2.a.da.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9200,2,Mod(1,9200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9200, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("9200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9200 = 2^{4} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9200.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:	$73.4623698596$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 12x^{5} + 9x^{4} + 43x^{3} - 14x^{2} - 49x - 9 $ x^7 - x^6 - 12x^5 + 9x^4 + 43x^3 - 14x^2 - 49*x - 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 575)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.202227$ of defining polynomial
Character	$\chi$	$=$	9200.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.69619 q^{3} +2.81698 q^{7} +4.26945 q^{9} +1.84786 q^{11} -6.49089 q^{13} +7.06930 q^{17} +0.252319 q^{19} +7.59513 q^{21} +1.00000 q^{23} +3.42269 q^{27} +4.12351 q^{29} -3.54811 q^{31} +4.98220 q^{33} -7.91248 q^{37} -17.5007 q^{39} +6.53833 q^{41} -4.93835 q^{43} +0.851917 q^{47} +0.935388 q^{49} +19.0602 q^{51} +12.5831 q^{53} +0.680301 q^{57} +10.3616 q^{59} +1.01207 q^{61} +12.0270 q^{63} +3.37930 q^{67} +2.69619 q^{69} +0.851917 q^{71} +9.75748 q^{73} +5.20540 q^{77} +16.5320 q^{79} -3.58013 q^{81} +0.696772 q^{83} +11.1178 q^{87} +13.1835 q^{89} -18.2847 q^{91} -9.56638 q^{93} +5.48684 q^{97} +7.88937 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 3 q^{7} + 15 q^{9} + q^{11} - 3 q^{13} + 10 q^{17} - 15 q^{19} + 2 q^{21} + 7 q^{23} + 3 q^{29} - 14 q^{31} + 6 q^{33} - 10 q^{37} + 8 q^{39} + 19 q^{41} - 5 q^{43} + 14 q^{47} + 40 q^{49} - 2 q^{51}+ \cdots + 53 q^{99}+O(q^{100}) $ 7 * q - 3 * q^7 + 15 * q^9 + q^11 - 3 * q^13 + 10 * q^17 - 15 * q^19 + 2 * q^21 + 7 * q^23 + 3 * q^29 - 14 * q^31 + 6 * q^33 - 10 * q^37 + 8 * q^39 + 19 * q^41 - 5 * q^43 + 14 * q^47 + 40 * q^49 - 2 * q^51 + 4 * q^53 - 4 * q^57 + 16 * q^59 + 40 * q^61 - 53 * q^63 + 4 * q^67 + 14 * q^71 - 3 * q^73 - 17 * q^77 + q^79 + 47 * q^81 - 17 * q^83 + 56 * q^87 + 16 * q^89 - 25 * q^91 + 14 * q^93 - 24 * q^97 + 53 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	2.69619	1.55665	0.778324	−	0.627863i	$-0.216070\pi$
$3$	2.69619	1.55665	0.778324	+	0.627863i	$0.216070\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.81698	1.06472	0.532360	−	0.846518i	$-0.321305\pi$
$7$	2.81698	1.06472	0.532360	+	0.846518i	$0.321305\pi$
$8$	0	0
$9$	4.26945	1.42315
$10$	0	0
$11$	1.84786	0.557152	0.278576	−	0.960414i	$-0.410138\pi$
$11$	1.84786	0.557152	0.278576	+	0.960414i	$0.410138\pi$
$12$	0	0
$13$	−6.49089	−1.80025	−0.900124	−	0.435633i	$-0.856525\pi$
$13$	−6.49089	−1.80025	−0.900124	+	0.435633i	$0.856525\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.06930	1.71456	0.857279	−	0.514853i	$-0.172153\pi$
$17$	7.06930	1.71456	0.857279	+	0.514853i	$0.172153\pi$
$18$	0	0
$19$	0.252319	0.0578860	0.0289430	−	0.999581i	$-0.490786\pi$
$19$	0.252319	0.0578860	0.0289430	+	0.999581i	$0.490786\pi$
$20$	0	0
$21$	7.59513	1.65739
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	3.42269	0.658696
$28$	0	0
$29$	4.12351	0.765717	0.382858	−	0.923807i	$-0.374940\pi$
$29$	4.12351	0.765717	0.382858	+	0.923807i	$0.374940\pi$
$30$	0	0
$31$	−3.54811	−0.637259	−0.318630	−	0.947879i	$-0.603223\pi$
$31$	−3.54811	−0.637259	−0.318630	+	0.947879i	$0.603223\pi$
$32$	0	0
$33$	4.98220	0.867289
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−7.91248	−1.30080	−0.650402	−	0.759591i	$-0.725399\pi$
$37$	−7.91248	−1.30080	−0.650402	+	0.759591i	$0.725399\pi$
$38$	0	0
$39$	−17.5007	−2.80235
$40$	0	0
$41$	6.53833	1.02111	0.510557	−	0.859844i	$-0.329439\pi$
$41$	6.53833	1.02111	0.510557	+	0.859844i	$0.329439\pi$
$42$	0	0
$43$	−4.93835	−0.753092	−0.376546	−	0.926398i	$-0.622888\pi$
$43$	−4.93835	−0.753092	−0.376546	+	0.926398i	$0.622888\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.851917	0.124265	0.0621324	−	0.998068i	$-0.480210\pi$
$47$	0.851917	0.124265	0.0621324	+	0.998068i	$0.480210\pi$
$48$	0	0
$49$	0.935388	0.133627
$50$	0	0
$51$	19.0602	2.66896
$52$	0	0
$53$	12.5831	1.72842	0.864208	−	0.503135i	$-0.167820\pi$
$53$	12.5831	1.72842	0.864208	+	0.503135i	$0.167820\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0.680301	0.0901080
$58$	0	0
$59$	10.3616	1.34897	0.674484	−	0.738290i	$-0.264366\pi$
$59$	10.3616	1.34897	0.674484	+	0.738290i	$0.264366\pi$
$60$	0	0
$61$	1.01207	0.129582	0.0647911	−	0.997899i	$-0.479362\pi$
$61$	1.01207	0.129582	0.0647911	+	0.997899i	$0.479362\pi$
$62$	0	0
$63$	12.0270	1.51526
$64$	0	0
$65$	0	0
$66$	0	0
$67$	3.37930	0.412848	0.206424	−	0.978463i	$-0.433817\pi$
$67$	3.37930	0.412848	0.206424	+	0.978463i	$0.433817\pi$
$68$	0	0
$69$	2.69619	0.324583
$70$	0	0
$71$	0.851917	0.101104	0.0505520	−	0.998721i	$-0.483902\pi$
$71$	0.851917	0.101104	0.0505520	+	0.998721i	$0.483902\pi$
$72$	0	0
$73$	9.75748	1.14203	0.571013	−	0.820941i	$-0.306551\pi$
$73$	9.75748	1.14203	0.571013	+	0.820941i	$0.306551\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.20540	0.593211
$78$	0	0
$79$	16.5320	1.86000	0.929999	−	0.367561i	$-0.119807\pi$
$79$	16.5320	1.86000	0.929999	+	0.367561i	$0.119807\pi$
$80$	0	0
$81$	−3.58013	−0.397793
$82$	0	0
$83$	0.696772	0.0764806	0.0382403	−	0.999269i	$-0.487825\pi$
$83$	0.696772	0.0764806	0.0382403	+	0.999269i	$0.487825\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	11.1178	1.19195
$88$	0	0
$89$	13.1835	1.39745	0.698724	−	0.715392i	$-0.253752\pi$
$89$	13.1835	1.39745	0.698724	+	0.715392i	$0.253752\pi$
$90$	0	0
$91$	−18.2847	−1.91676
$92$	0	0
$93$	−9.56638	−0.991988
$94$	0	0
$95$	0	0
$96$	0	0
$97$	5.48684	0.557104	0.278552	−	0.960421i	$-0.410146\pi$
$97$	5.48684	0.557104	0.278552	+	0.960421i	$0.410146\pi$
$98$	0	0
$99$	7.88937	0.792911
$100$	0	0
$101$	−9.56925	−0.952176	−0.476088	−	0.879398i	$-0.657946\pi$
$101$	−9.56925	−0.952176	−0.476088	+	0.879398i	$0.657946\pi$
$102$	0	0
$103$	−9.10123	−0.896770	−0.448385	−	0.893840i	$-0.648001\pi$
$103$	−9.10123	−0.896770	−0.448385	+	0.893840i	$0.648001\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	11.7093	1.13198	0.565991	−	0.824411i	$-0.308494\pi$
$107$	11.7093	1.13198	0.565991	+	0.824411i	$0.308494\pi$
$108$	0	0
$109$	1.46844	0.140651	0.0703257	−	0.997524i	$-0.477596\pi$
$109$	1.46844	0.140651	0.0703257	+	0.997524i	$0.477596\pi$
$110$	0	0
$111$	−21.3336	−2.02489
$112$	0	0
$113$	−1.62939	−0.153280	−0.0766401	−	0.997059i	$-0.524419\pi$
$113$	−1.62939	−0.153280	−0.0766401	+	0.997059i	$0.524419\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−27.7125	−2.56203
$118$	0	0
$119$	19.9141	1.82552
$120$	0	0
$121$	−7.58540	−0.689581
$122$	0	0
$123$	17.6286	1.58952
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−8.59258	−0.762468	−0.381234	−	0.924479i	$-0.624501\pi$
$127$	−8.59258	−0.762468	−0.381234	+	0.924479i	$0.624501\pi$
$128$	0	0
$129$	−13.3147	−1.17230
$130$	0	0
$131$	−2.93777	−0.256674	−0.128337	−	0.991731i	$-0.540964\pi$
$131$	−2.93777	−0.256674	−0.128337	+	0.991731i	$0.540964\pi$
$132$	0	0
$133$	0.710778	0.0616323
$134$	0	0
$135$	0	0
$136$	0	0
$137$	21.9366	1.87417	0.937085	−	0.349101i	$-0.113513\pi$
$137$	21.9366	1.87417	0.937085	+	0.349101i	$0.113513\pi$
$138$	0	0
$139$	−15.3324	−1.30048	−0.650239	−	0.759730i	$-0.725331\pi$
$139$	−15.3324	−1.30048	−0.650239	+	0.759730i	$0.725331\pi$
$140$	0	0
$141$	2.29693	0.193437
$142$	0	0
$143$	−11.9943	−1.00301
$144$	0	0
$145$	0	0
$146$	0	0
$147$	2.52199	0.208010
$148$	0	0
$149$	−15.0218	−1.23063	−0.615316	−	0.788281i	$-0.710972\pi$
$149$	−15.0218	−1.23063	−0.615316	+	0.788281i	$0.710972\pi$
$150$	0	0
$151$	−17.5294	−1.42652	−0.713262	−	0.700897i	$-0.752783\pi$
$151$	−17.5294	−1.42652	−0.713262	+	0.700897i	$0.752783\pi$
$152$	0	0
$153$	30.1820	2.44007
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−7.10315	−0.566893	−0.283446	−	0.958988i	$-0.591478\pi$
$157$	−7.10315	−0.566893	−0.283446	+	0.958988i	$0.591478\pi$
$158$	0	0
$159$	33.9263	2.69053
$160$	0	0
$161$	2.81698	0.222009
$162$	0	0
$163$	−14.5905	−1.14282	−0.571408	−	0.820666i	$-0.693603\pi$
$163$	−14.5905	−1.14282	−0.571408	+	0.820666i	$0.693603\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−9.45066	−0.731314	−0.365657	−	0.930750i	$-0.619156\pi$
$167$	−9.45066	−0.731314	−0.365657	+	0.930750i	$0.619156\pi$
$168$	0	0
$169$	29.1316	2.24090
$170$	0	0
$171$	1.07726	0.0823805
$172$	0	0
$173$	23.3055	1.77188	0.885940	−	0.463799i	$-0.153514\pi$
$173$	23.3055	1.77188	0.885940	+	0.463799i	$0.153514\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	27.9369	2.09987
$178$	0	0
$179$	−23.5562	−1.76068	−0.880338	−	0.474347i	$-0.842684\pi$
$179$	−23.5562	−1.76068	−0.880338	+	0.474347i	$0.842684\pi$
$180$	0	0
$181$	0.505059	0.0375407	0.0187704	−	0.999824i	$-0.494025\pi$
$181$	0.505059	0.0375407	0.0187704	+	0.999824i	$0.494025\pi$
$182$	0	0
$183$	2.72874	0.201714
$184$	0	0
$185$	0	0
$186$	0	0
$187$	13.0631	0.955269
$188$	0	0
$189$	9.64165	0.701327
$190$	0	0
$191$	−2.51313	−0.181844	−0.0909219	−	0.995858i	$-0.528981\pi$
$191$	−2.51313	−0.181844	−0.0909219	+	0.995858i	$0.528981\pi$
$192$	0	0
$193$	4.17037	0.300190	0.150095	−	0.988672i	$-0.452042\pi$
$193$	4.17037	0.300190	0.150095	+	0.988672i	$0.452042\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.60102	0.470303	0.235151	−	0.971959i	$-0.424441\pi$
$197$	6.60102	0.470303	0.235151	+	0.971959i	$0.424441\pi$
$198$	0	0
$199$	9.91175	0.702626	0.351313	−	0.936258i	$-0.385735\pi$
$199$	9.91175	0.702626	0.351313	+	0.936258i	$0.385735\pi$
$200$	0	0
$201$	9.11125	0.642658
$202$	0	0
$203$	11.6159	0.815273
$204$	0	0
$205$	0	0
$206$	0	0
$207$	4.26945	0.296747
$208$	0	0
$209$	0.466251	0.0322513
$210$	0	0
$211$	−8.33546	−0.573837	−0.286918	−	0.957955i	$-0.592631\pi$
$211$	−8.33546	−0.573837	−0.286918	+	0.957955i	$0.592631\pi$
$212$	0	0
$213$	2.29693	0.157383
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−9.99496	−0.678502
$218$	0	0
$219$	26.3080	1.77773
$220$	0	0
$221$	−45.8860	−3.08663
$222$	0	0
$223$	13.4909	0.903420	0.451710	−	0.892165i	$-0.350814\pi$
$223$	13.4909	0.903420	0.451710	+	0.892165i	$0.350814\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−5.77415	−0.383244	−0.191622	−	0.981469i	$-0.561375\pi$
$227$	−5.77415	−0.383244	−0.191622	+	0.981469i	$0.561375\pi$
$228$	0	0
$229$	22.3181	1.47482	0.737412	−	0.675444i	$-0.236048\pi$
$229$	22.3181	1.47482	0.737412	+	0.675444i	$0.236048\pi$
$230$	0	0
$231$	14.0348	0.923420
$232$	0	0
$233$	−21.8227	−1.42965	−0.714825	−	0.699303i	$-0.753494\pi$
$233$	−21.8227	−1.42965	−0.714825	+	0.699303i	$0.753494\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	44.5735	2.89536
$238$	0	0
$239$	26.7115	1.72782	0.863912	−	0.503644i	$-0.168007\pi$
$239$	26.7115	1.72782	0.863912	+	0.503644i	$0.168007\pi$
$240$	0	0
$241$	15.1565	0.976318	0.488159	−	0.872755i	$-0.337669\pi$
$241$	15.1565	0.976318	0.488159	+	0.872755i	$0.337669\pi$
$242$	0	0
$243$	−19.9208	−1.27792
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.63778	−0.104209
$248$	0	0
$249$	1.87863	0.119053
$250$	0	0
$251$	13.2026	0.833340	0.416670	−	0.909058i	$-0.363197\pi$
$251$	13.2026	0.833340	0.416670	+	0.909058i	$0.363197\pi$
$252$	0	0
$253$	1.84786	0.116174
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−13.9450	−0.869866	−0.434933	−	0.900463i	$-0.643228\pi$
$257$	−13.9450	−0.869866	−0.434933	+	0.900463i	$0.643228\pi$
$258$	0	0
$259$	−22.2893	−1.38499
$260$	0	0
$261$	17.6051	1.08973
$262$	0	0
$263$	21.7883	1.34352	0.671762	−	0.740767i	$-0.265538\pi$
$263$	21.7883	1.34352	0.671762	+	0.740767i	$0.265538\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	35.5452	2.17533
$268$	0	0
$269$	10.2596	0.625540	0.312770	−	0.949829i	$-0.398743\pi$
$269$	10.2596	0.625540	0.312770	+	0.949829i	$0.398743\pi$
$270$	0	0
$271$	−4.11888	−0.250204	−0.125102	−	0.992144i	$-0.539926\pi$
$271$	−4.11888	−0.250204	−0.125102	+	0.992144i	$0.539926\pi$
$272$	0	0
$273$	−49.2991	−2.98372
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−9.62736	−0.578452	−0.289226	−	0.957261i	$-0.593398\pi$
$277$	−9.62736	−0.578452	−0.289226	+	0.957261i	$0.593398\pi$
$278$	0	0
$279$	−15.1485	−0.906916
$280$	0	0
$281$	16.9191	1.00931	0.504654	−	0.863322i	$-0.331620\pi$
$281$	16.9191	1.00931	0.504654	+	0.863322i	$0.331620\pi$
$282$	0	0
$283$	−7.51375	−0.446646	−0.223323	−	0.974744i	$-0.571691\pi$
$283$	−7.51375	−0.446646	−0.223323	+	0.974744i	$0.571691\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	18.4183	1.08720
$288$	0	0
$289$	32.9750	1.93971
$290$	0	0
$291$	14.7936	0.867214
$292$	0	0
$293$	−15.7923	−0.922596	−0.461298	−	0.887245i	$-0.652616\pi$
$293$	−15.7923	−0.922596	−0.461298	+	0.887245i	$0.652616\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	6.32466	0.366994
$298$	0	0
$299$	−6.49089	−0.375378
$300$	0	0
$301$	−13.9112	−0.801831
$302$	0	0
$303$	−25.8005	−1.48220
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−25.0861	−1.43174	−0.715870	−	0.698234i	$-0.753970\pi$
$307$	−25.0861	−1.43174	−0.715870	+	0.698234i	$0.753970\pi$
$308$	0	0
$309$	−24.5387	−1.39596
$310$	0	0
$311$	−18.2599	−1.03542	−0.517711	−	0.855555i	$-0.673216\pi$
$311$	−18.2599	−1.03542	−0.517711	+	0.855555i	$0.673216\pi$
$312$	0	0
$313$	−11.3712	−0.642738	−0.321369	−	0.946954i	$-0.604143\pi$
$313$	−11.3712	−0.642738	−0.321369	+	0.946954i	$0.604143\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−15.7315	−0.883570	−0.441785	−	0.897121i	$-0.645655\pi$
$317$	−15.7315	−0.883570	−0.441785	+	0.897121i	$0.645655\pi$
$318$	0	0
$319$	7.61969	0.426621
$320$	0	0
$321$	31.5706	1.76210
$322$	0	0
$323$	1.78372	0.0992488
$324$	0	0
$325$	0	0
$326$	0	0
$327$	3.95921	0.218945
$328$	0	0
$329$	2.39984	0.132307
$330$	0	0
$331$	−12.5991	−0.692507	−0.346253	−	0.938141i	$-0.612546\pi$
$331$	−12.5991	−0.692507	−0.346253	+	0.938141i	$0.612546\pi$
$332$	0	0
$333$	−33.7819	−1.85124
$334$	0	0
$335$	0	0
$336$	0	0
$337$	16.5473	0.901388	0.450694	−	0.892679i	$-0.351177\pi$
$337$	16.5473	0.901388	0.450694	+	0.892679i	$0.351177\pi$
$338$	0	0
$339$	−4.39315	−0.238603
$340$	0	0
$341$	−6.55643	−0.355050
$342$	0	0
$343$	−17.0839	−0.922444
$344$	0	0
$345$	0	0
$346$	0	0
$347$	8.43461	0.452794	0.226397	−	0.974035i	$-0.427305\pi$
$347$	8.43461	0.452794	0.226397	+	0.974035i	$0.427305\pi$
$348$	0	0
$349$	−13.2201	−0.707658	−0.353829	−	0.935310i	$-0.615121\pi$
$349$	−13.2201	−0.707658	−0.353829	+	0.935310i	$0.615121\pi$
$350$	0	0
$351$	−22.2163	−1.18582
$352$	0	0
$353$	1.28663	0.0684803	0.0342401	−	0.999414i	$-0.489099\pi$
$353$	1.28663	0.0684803	0.0342401	+	0.999414i	$0.489099\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	53.6922	2.84169
$358$	0	0
$359$	−14.7493	−0.778438	−0.389219	−	0.921145i	$-0.627255\pi$
$359$	−14.7493	−0.778438	−0.389219	+	0.921145i	$0.627255\pi$
$360$	0	0
$361$	−18.9363	−0.996649
$362$	0	0
$363$	−20.4517	−1.07344
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−25.9403	−1.35407	−0.677036	−	0.735950i	$-0.736736\pi$
$367$	−25.9403	−1.35407	−0.677036	+	0.735950i	$0.736736\pi$
$368$	0	0
$369$	27.9151	1.45320
$370$	0	0
$371$	35.4462	1.84028
$372$	0	0
$373$	−13.1177	−0.679209	−0.339604	−	0.940568i	$-0.610293\pi$
$373$	−13.1177	−0.679209	−0.339604	+	0.940568i	$0.610293\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−26.7653	−1.37848
$378$	0	0
$379$	−9.63674	−0.495006	−0.247503	−	0.968887i	$-0.579610\pi$
$379$	−9.63674	−0.495006	−0.247503	+	0.968887i	$0.579610\pi$
$380$	0	0
$381$	−23.1672	−1.18689
$382$	0	0
$383$	−16.1595	−0.825711	−0.412856	−	0.910797i	$-0.635469\pi$
$383$	−16.1595	−0.825711	−0.412856	+	0.910797i	$0.635469\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−21.0841	−1.07176
$388$	0	0
$389$	27.6271	1.40075	0.700374	−	0.713776i	$-0.253017\pi$
$389$	27.6271	1.40075	0.700374	+	0.713776i	$0.253017\pi$
$390$	0	0
$391$	7.06930	0.357510
$392$	0	0
$393$	−7.92080	−0.399551
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−10.7707	−0.540565	−0.270282	−	0.962781i	$-0.587117\pi$
$397$	−10.7707	−0.540565	−0.270282	+	0.962781i	$0.587117\pi$
$398$	0	0
$399$	1.91639	0.0959397
$400$	0	0
$401$	−31.5635	−1.57621	−0.788104	−	0.615542i	$-0.788937\pi$
$401$	−31.5635	−1.57621	−0.788104	+	0.615542i	$0.788937\pi$
$402$	0	0
$403$	23.0304	1.14723
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−14.6212	−0.724745
$408$	0	0
$409$	15.6426	0.773479	0.386740	−	0.922189i	$-0.373601\pi$
$409$	15.6426	0.773479	0.386740	+	0.922189i	$0.373601\pi$
$410$	0	0
$411$	59.1453	2.91742
$412$	0	0
$413$	29.1885	1.43627
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−41.3391	−2.02439
$418$	0	0
$419$	11.1799	0.546175	0.273088	−	0.961989i	$-0.411955\pi$
$419$	11.1799	0.546175	0.273088	+	0.961989i	$0.411955\pi$
$420$	0	0
$421$	0.661200	0.0322249	0.0161125	−	0.999870i	$-0.494871\pi$
$421$	0.661200	0.0322249	0.0161125	+	0.999870i	$0.494871\pi$
$422$	0	0
$423$	3.63722	0.176848
$424$	0	0
$425$	0	0
$426$	0	0
$427$	2.85098	0.137969
$428$	0	0
$429$	−32.3389	−1.56134
$430$	0	0
$431$	15.2276	0.733488	0.366744	−	0.930322i	$-0.380473\pi$
$431$	15.2276	0.733488	0.366744	+	0.930322i	$0.380473\pi$
$432$	0	0
$433$	22.2662	1.07005	0.535023	−	0.844838i	$-0.320303\pi$
$433$	22.2662	1.07005	0.535023	+	0.844838i	$0.320303\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.252319	0.0120701
$438$	0	0
$439$	−21.1491	−1.00939	−0.504695	−	0.863298i	$-0.668395\pi$
$439$	−21.1491	−1.00939	−0.504695	+	0.863298i	$0.668395\pi$
$440$	0	0
$441$	3.99359	0.190171
$442$	0	0
$443$	−9.94721	−0.472606	−0.236303	−	0.971679i	$-0.575936\pi$
$443$	−9.94721	−0.472606	−0.236303	+	0.971679i	$0.575936\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−40.5016	−1.91566
$448$	0	0
$449$	−8.32870	−0.393056	−0.196528	−	0.980498i	$-0.562967\pi$
$449$	−8.32870	−0.393056	−0.196528	+	0.980498i	$0.562967\pi$
$450$	0	0
$451$	12.0819	0.568916
$452$	0	0
$453$	−47.2627	−2.22060
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−33.2599	−1.55583	−0.777916	−	0.628369i	$-0.783723\pi$
$457$	−33.2599	−1.55583	−0.777916	+	0.628369i	$0.783723\pi$
$458$	0	0
$459$	24.1960	1.12937
$460$	0	0
$461$	14.8481	0.691544	0.345772	−	0.938318i	$-0.387617\pi$
$461$	14.8481	0.691544	0.345772	+	0.938318i	$0.387617\pi$
$462$	0	0
$463$	15.6422	0.726956	0.363478	−	0.931603i	$-0.381589\pi$
$463$	15.6422	0.726956	0.363478	+	0.931603i	$0.381589\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	5.26987	0.243861	0.121930	−	0.992539i	$-0.461092\pi$
$467$	5.26987	0.243861	0.121930	+	0.992539i	$0.461092\pi$
$468$	0	0
$469$	9.51944	0.439567
$470$	0	0
$471$	−19.1514	−0.882452
$472$	0	0
$473$	−9.12541	−0.419587
$474$	0	0
$475$	0	0
$476$	0	0
$477$	53.7228	2.45980
$478$	0	0
$479$	−0.404422	−0.0184785	−0.00923927	−	0.999957i	$-0.502941\pi$
$479$	−0.404422	−0.0184785	−0.00923927	+	0.999957i	$0.502941\pi$
$480$	0	0
$481$	51.3590	2.34177
$482$	0	0
$483$	7.59513	0.345590
$484$	0	0
$485$	0	0
$486$	0	0
$487$	17.0409	0.772198	0.386099	−	0.922457i	$-0.373822\pi$
$487$	17.0409	0.772198	0.386099	+	0.922457i	$0.373822\pi$
$488$	0	0
$489$	−39.3388	−1.77896
$490$	0	0
$491$	9.42821	0.425489	0.212745	−	0.977108i	$-0.431760\pi$
$491$	9.42821	0.425489	0.212745	+	0.977108i	$0.431760\pi$
$492$	0	0
$493$	29.1503	1.31287
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.39984	0.107647
$498$	0	0
$499$	25.7959	1.15478	0.577391	−	0.816468i	$-0.304071\pi$
$499$	25.7959	1.15478	0.577391	+	0.816468i	$0.304071\pi$
$500$	0	0
$501$	−25.4808	−1.13840
$502$	0	0
$503$	−21.6111	−0.963589	−0.481795	−	0.876284i	$-0.660015\pi$
$503$	−21.6111	−0.963589	−0.481795	+	0.876284i	$0.660015\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	78.5445	3.48828
$508$	0	0
$509$	−6.13182	−0.271788	−0.135894	−	0.990723i	$-0.543391\pi$
$509$	−6.13182	−0.271788	−0.135894	+	0.990723i	$0.543391\pi$
$510$	0	0
$511$	27.4866	1.21594
$512$	0	0
$513$	0.863609	0.0381293
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1.57423	0.0692345
$518$	0	0
$519$	62.8360	2.75819
$520$	0	0
$521$	22.4021	0.981454	0.490727	−	0.871313i	$-0.336731\pi$
$521$	22.4021	0.981454	0.490727	+	0.871313i	$0.336731\pi$
$522$	0	0
$523$	−8.25468	−0.360952	−0.180476	−	0.983579i	$-0.557764\pi$
$523$	−8.25468	−0.360952	−0.180476	+	0.983579i	$0.557764\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−25.0827	−1.09262
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	44.2384	1.91978
$532$	0	0
$533$	−42.4395	−1.83826
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−63.5121	−2.74075
$538$	0	0
$539$	1.72847	0.0744505
$540$	0	0
$541$	24.9079	1.07087	0.535437	−	0.844575i	$-0.320147\pi$
$541$	24.9079	1.07087	0.535437	+	0.844575i	$0.320147\pi$
$542$	0	0
$543$	1.36174	0.0584376
$544$	0	0
$545$	0	0
$546$	0	0
$547$	19.8049	0.846795	0.423398	−	0.905944i	$-0.360837\pi$
$547$	19.8049	0.846795	0.423398	+	0.905944i	$0.360837\pi$
$548$	0	0
$549$	4.32098	0.184415
$550$	0	0
$551$	1.04044	0.0443243
$552$	0	0
$553$	46.5704	1.98038
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−36.6380	−1.55240	−0.776201	−	0.630486i	$-0.782856\pi$
$557$	−36.6380	−1.55240	−0.776201	+	0.630486i	$0.782856\pi$
$558$	0	0
$559$	32.0543	1.35575
$560$	0	0
$561$	35.2207	1.48702
$562$	0	0
$563$	−19.0783	−0.804054	−0.402027	−	0.915628i	$-0.631694\pi$
$563$	−19.0783	−0.804054	−0.402027	+	0.915628i	$0.631694\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−10.0852	−0.423537
$568$	0	0
$569$	4.50086	0.188686	0.0943429	−	0.995540i	$-0.469925\pi$
$569$	4.50086	0.188686	0.0943429	+	0.995540i	$0.469925\pi$
$570$	0	0
$571$	−1.25913	−0.0526930	−0.0263465	−	0.999653i	$-0.508387\pi$
$571$	−1.25913	−0.0526930	−0.0263465	+	0.999653i	$0.508387\pi$
$572$	0	0
$573$	−6.77589	−0.283067
$574$	0	0
$575$	0	0
$576$	0	0
$577$	1.33814	0.0557077	0.0278539	−	0.999612i	$-0.491133\pi$
$577$	1.33814	0.0557077	0.0278539	+	0.999612i	$0.491133\pi$
$578$	0	0
$579$	11.2441	0.467289
$580$	0	0
$581$	1.96279	0.0814304
$582$	0	0
$583$	23.2518	0.962990
$584$	0	0
$585$	0	0
$586$	0	0
$587$	22.4997	0.928664	0.464332	−	0.885661i	$-0.346295\pi$
$587$	22.4997	0.928664	0.464332	+	0.885661i	$0.346295\pi$
$588$	0	0
$589$	−0.895256	−0.0368884
$590$	0	0
$591$	17.7976	0.732096
$592$	0	0
$593$	−30.8988	−1.26886	−0.634432	−	0.772979i	$-0.718766\pi$
$593$	−30.8988	−1.26886	−0.634432	+	0.772979i	$0.718766\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	26.7240	1.09374
$598$	0	0
$599$	7.34413	0.300073	0.150036	−	0.988680i	$-0.452061\pi$
$599$	7.34413	0.300073	0.150036	+	0.988680i	$0.452061\pi$
$600$	0	0
$601$	−5.10253	−0.208136	−0.104068	−	0.994570i	$-0.533186\pi$
$601$	−5.10253	−0.208136	−0.104068	+	0.994570i	$0.533186\pi$
$602$	0	0
$603$	14.4278	0.587544
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−21.4965	−0.872518	−0.436259	−	0.899821i	$-0.643697\pi$
$607$	−21.4965	−0.872518	−0.436259	+	0.899821i	$0.643697\pi$
$608$	0	0
$609$	31.3186	1.26909
$610$	0	0
$611$	−5.52970	−0.223708
$612$	0	0
$613$	−44.4369	−1.79479	−0.897396	−	0.441227i	$-0.854543\pi$
$613$	−44.4369	−1.79479	−0.897396	+	0.441227i	$0.854543\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	25.0662	1.00913	0.504564	−	0.863375i	$-0.331653\pi$
$617$	25.0662	1.00913	0.504564	+	0.863375i	$0.331653\pi$
$618$	0	0
$619$	11.6002	0.466252	0.233126	−	0.972447i	$-0.425105\pi$
$619$	11.6002	0.466252	0.233126	+	0.972447i	$0.425105\pi$
$620$	0	0
$621$	3.42269	0.137348
$622$	0	0
$623$	37.1377	1.48789
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1.25710	0.0502039
$628$	0	0
$629$	−55.9357	−2.23030
$630$	0	0
$631$	9.17437	0.365226	0.182613	−	0.983185i	$-0.441544\pi$
$631$	9.17437	0.365226	0.182613	+	0.983185i	$0.441544\pi$
$632$	0	0
$633$	−22.4740	−0.893261
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−6.07150	−0.240562
$638$	0	0
$639$	3.63722	0.143886
$640$	0	0
$641$	3.98400	0.157359	0.0786793	−	0.996900i	$-0.474930\pi$
$641$	3.98400	0.157359	0.0786793	+	0.996900i	$0.474930\pi$
$642$	0	0
$643$	−13.3005	−0.524520	−0.262260	−	0.964997i	$-0.584468\pi$
$643$	−13.3005	−0.524520	−0.262260	+	0.964997i	$0.584468\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−3.02871	−0.119071	−0.0595354	−	0.998226i	$-0.518962\pi$
$647$	−3.02871	−0.119071	−0.0595354	+	0.998226i	$0.518962\pi$
$648$	0	0
$649$	19.1469	0.751580
$650$	0	0
$651$	−26.9483	−1.05619
$652$	0	0
$653$	24.8569	0.972727	0.486363	−	0.873757i	$-0.338323\pi$
$653$	24.8569	0.972727	0.486363	+	0.873757i	$0.338323\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	41.6591	1.62528
$658$	0	0
$659$	−30.2901	−1.17994	−0.589968	−	0.807427i	$-0.700860\pi$
$659$	−30.2901	−1.17994	−0.589968	+	0.807427i	$0.700860\pi$
$660$	0	0
$661$	−41.7574	−1.62418	−0.812088	−	0.583535i	$-0.801669\pi$
$661$	−41.7574	−1.62418	−0.812088	+	0.583535i	$0.801669\pi$
$662$	0	0
$663$	−123.718	−4.80479
$664$	0	0
$665$	0	0
$666$	0	0
$667$	4.12351	0.159663
$668$	0	0
$669$	36.3742	1.40631
$670$	0	0
$671$	1.87017	0.0721971
$672$	0	0
$673$	−18.1446	−0.699424	−0.349712	−	0.936857i	$-0.613721\pi$
$673$	−18.1446	−0.699424	−0.349712	+	0.936857i	$0.613721\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	16.4647	0.632790	0.316395	−	0.948627i	$-0.397527\pi$
$677$	16.4647	0.632790	0.316395	+	0.948627i	$0.397527\pi$
$678$	0	0
$679$	15.4563	0.593159
$680$	0	0
$681$	−15.5682	−0.596575
$682$	0	0
$683$	−46.7050	−1.78712	−0.893558	−	0.448948i	$-0.851799\pi$
$683$	−46.7050	−1.78712	−0.893558	+	0.448948i	$0.851799\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	60.1740	2.29578
$688$	0	0
$689$	−81.6752	−3.11158
$690$	0	0
$691$	11.5366	0.438872	0.219436	−	0.975627i	$-0.429578\pi$
$691$	11.5366	0.438872	0.219436	+	0.975627i	$0.429578\pi$
$692$	0	0
$693$	22.2242	0.844228
$694$	0	0
$695$	0	0
$696$	0	0
$697$	46.2214	1.75076
$698$	0	0
$699$	−58.8381	−2.22546
$700$	0	0
$701$	31.8694	1.20369	0.601845	−	0.798613i	$-0.294432\pi$
$701$	31.8694	1.20369	0.601845	+	0.798613i	$0.294432\pi$
$702$	0	0
$703$	−1.99647	−0.0752982
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−26.9564	−1.01380
$708$	0	0
$709$	5.06884	0.190364	0.0951822	−	0.995460i	$-0.469657\pi$
$709$	5.06884	0.190364	0.0951822	+	0.995460i	$0.469657\pi$
$710$	0	0
$711$	70.5827	2.64706
$712$	0	0
$713$	−3.54811	−0.132878
$714$	0	0
$715$	0	0
$716$	0	0
$717$	72.0193	2.68961
$718$	0	0
$719$	−6.53346	−0.243657	−0.121828	−	0.992551i	$-0.538876\pi$
$719$	−6.53346	−0.243657	−0.121828	+	0.992551i	$0.538876\pi$
$720$	0	0
$721$	−25.6380	−0.954809
$722$	0	0
$723$	40.8649	1.51978
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−46.2193	−1.71418	−0.857089	−	0.515169i	$-0.827729\pi$
$727$	−46.2193	−1.71418	−0.857089	+	0.515169i	$0.827729\pi$
$728$	0	0
$729$	−42.9699	−1.59148
$730$	0	0
$731$	−34.9107	−1.29122
$732$	0	0
$733$	20.2561	0.748177	0.374089	−	0.927393i	$-0.377956\pi$
$733$	20.2561	0.748177	0.374089	+	0.927393i	$0.377956\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6.24450	0.230019
$738$	0	0
$739$	−0.454209	−0.0167083	−0.00835417	−	0.999965i	$-0.502659\pi$
$739$	−0.454209	−0.0167083	−0.00835417	+	0.999965i	$0.502659\pi$
$740$	0	0
$741$	−4.41576	−0.162217
$742$	0	0
$743$	−17.9525	−0.658614	−0.329307	−	0.944223i	$-0.606815\pi$
$743$	−17.9525	−0.658614	−0.329307	+	0.944223i	$0.606815\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	2.97483	0.108843
$748$	0	0
$749$	32.9849	1.20524
$750$	0	0
$751$	−14.8866	−0.543219	−0.271610	−	0.962408i	$-0.587556\pi$
$751$	−14.8866	−0.543219	−0.271610	+	0.962408i	$0.587556\pi$
$752$	0	0
$753$	35.5967	1.29722
$754$	0	0
$755$	0	0
$756$	0	0
$757$	31.7028	1.15226	0.576129	−	0.817358i	$-0.304562\pi$
$757$	31.7028	1.15226	0.576129	+	0.817358i	$0.304562\pi$
$758$	0	0
$759$	4.98220	0.180842
$760$	0	0
$761$	−6.80194	−0.246570	−0.123285	−	0.992371i	$-0.539343\pi$
$761$	−6.80194	−0.246570	−0.123285	+	0.992371i	$0.539343\pi$
$762$	0	0
$763$	4.13658	0.149754
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−67.2561	−2.42848
$768$	0	0
$769$	19.2067	0.692612	0.346306	−	0.938122i	$-0.387436\pi$
$769$	19.2067	0.692612	0.346306	+	0.938122i	$0.387436\pi$
$770$	0	0
$771$	−37.5985	−1.35408
$772$	0	0
$773$	−27.6037	−0.992835	−0.496418	−	0.868084i	$-0.665351\pi$
$773$	−27.6037	−0.992835	−0.496418	+	0.868084i	$0.665351\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−60.0962	−2.15594
$778$	0	0
$779$	1.64974	0.0591082
$780$	0	0
$781$	1.57423	0.0563303
$782$	0	0
$783$	14.1135	0.504375
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−35.6271	−1.26997	−0.634984	−	0.772525i	$-0.718993\pi$
$787$	−35.6271	−1.26997	−0.634984	+	0.772525i	$0.718993\pi$
$788$	0	0
$789$	58.7455	2.09139
$790$	0	0
$791$	−4.58997	−0.163200
$792$	0	0
$793$	−6.56923	−0.233280
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−9.85579	−0.349110	−0.174555	−	0.984647i	$-0.555849\pi$
$797$	−9.85579	−0.349110	−0.174555	+	0.984647i	$0.555849\pi$
$798$	0	0
$799$	6.02246	0.213059
$800$	0	0
$801$	56.2863	1.98878
$802$	0	0
$803$	18.0305	0.636282
$804$	0	0
$805$	0	0
$806$	0	0
$807$	27.6619	0.973745
$808$	0	0
$809$	35.4171	1.24520	0.622599	−	0.782541i	$-0.286077\pi$
$809$	35.4171	1.24520	0.622599	+	0.782541i	$0.286077\pi$
$810$	0	0
$811$	23.6023	0.828790	0.414395	−	0.910097i	$-0.363993\pi$
$811$	23.6023	0.828790	0.414395	+	0.910097i	$0.363993\pi$
$812$	0	0
$813$	−11.1053	−0.389480
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−1.24604	−0.0435934
$818$	0	0
$819$	−78.0657	−2.72784
$820$	0	0
$821$	−21.1904	−0.739548	−0.369774	−	0.929122i	$-0.620565\pi$
$821$	−21.1904	−0.739548	−0.369774	+	0.929122i	$0.620565\pi$
$822$	0	0
$823$	28.8463	1.00552	0.502759	−	0.864427i	$-0.332318\pi$
$823$	28.8463	1.00552	0.502759	+	0.864427i	$0.332318\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	7.85313	0.273080	0.136540	−	0.990635i	$-0.456402\pi$
$827$	7.85313	0.273080	0.136540	+	0.990635i	$0.456402\pi$
$828$	0	0
$829$	−7.58148	−0.263316	−0.131658	−	0.991295i	$-0.542030\pi$
$829$	−7.58148	−0.263316	−0.131658	+	0.991295i	$0.542030\pi$
$830$	0	0
$831$	−25.9572	−0.900446
$832$	0	0
$833$	6.61254	0.229111
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−12.1441	−0.419760
$838$	0	0
$839$	−33.2424	−1.14765	−0.573827	−	0.818977i	$-0.694542\pi$
$839$	−33.2424	−1.14765	−0.573827	+	0.818977i	$0.694542\pi$
$840$	0	0
$841$	−11.9967	−0.413678
$842$	0	0
$843$	45.6171	1.57114
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−21.3679	−0.734211
$848$	0	0
$849$	−20.2585	−0.695271
$850$	0	0
$851$	−7.91248	−0.271236
$852$	0	0
$853$	−21.1798	−0.725183	−0.362591	−	0.931948i	$-0.618108\pi$
$853$	−21.1798	−0.725183	−0.362591	+	0.931948i	$0.618108\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	13.2340	0.452066	0.226033	−	0.974120i	$-0.427424\pi$
$857$	13.2340	0.452066	0.226033	+	0.974120i	$0.427424\pi$
$858$	0	0
$859$	31.6863	1.08112	0.540561	−	0.841305i	$-0.318212\pi$
$859$	31.6863	1.08112	0.540561	+	0.841305i	$0.318212\pi$
$860$	0	0
$861$	49.6594	1.69239
$862$	0	0
$863$	−1.12760	−0.0383840	−0.0191920	−	0.999816i	$-0.506109\pi$
$863$	−1.12760	−0.0383840	−0.0191920	+	0.999816i	$0.506109\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	88.9070	3.01944
$868$	0	0
$869$	30.5490	1.03630
$870$	0	0
$871$	−21.9347	−0.743228
$872$	0	0
$873$	23.4258	0.792843
$874$	0	0
$875$	0	0
$876$	0	0
$877$	30.9499	1.04510	0.522552	−	0.852608i	$-0.324980\pi$
$877$	30.9499	1.04510	0.522552	+	0.852608i	$0.324980\pi$
$878$	0	0
$879$	−42.5791	−1.43616
$880$	0	0
$881$	45.9333	1.54753	0.773765	−	0.633472i	$-0.218371\pi$
$881$	45.9333	1.54753	0.773765	+	0.633472i	$0.218371\pi$
$882$	0	0
$883$	−14.0404	−0.472496	−0.236248	−	0.971693i	$-0.575918\pi$
$883$	−14.0404	−0.472496	−0.236248	+	0.971693i	$0.575918\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	33.4833	1.12426	0.562129	−	0.827049i	$-0.309982\pi$
$887$	33.4833	1.12426	0.562129	+	0.827049i	$0.309982\pi$
$888$	0	0
$889$	−24.2051	−0.811814
$890$	0	0
$891$	−6.61560	−0.221631
$892$	0	0
$893$	0.214955	0.00719319
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−17.5007	−0.584331
$898$	0	0
$899$	−14.6307	−0.487960
$900$	0	0
$901$	88.9534	2.96347
$902$	0	0
$903$	−37.5074	−1.24817
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−22.0732	−0.732929	−0.366464	−	0.930432i	$-0.619432\pi$
$907$	−22.0732	−0.732929	−0.366464	+	0.930432i	$0.619432\pi$
$908$	0	0
$909$	−40.8555	−1.35509
$910$	0	0
$911$	29.9624	0.992699	0.496349	−	0.868123i	$-0.334673\pi$
$911$	29.9624	0.992699	0.496349	+	0.868123i	$0.334673\pi$
$912$	0	0
$913$	1.28754	0.0426113
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−8.27565	−0.273286
$918$	0	0
$919$	−27.8980	−0.920269	−0.460134	−	0.887849i	$-0.652199\pi$
$919$	−27.8980	−0.920269	−0.460134	+	0.887849i	$0.652199\pi$
$920$	0	0
$921$	−67.6370	−2.22871
$922$	0	0
$923$	−5.52970	−0.182012
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−38.8573	−1.27624
$928$	0	0
$929$	−25.6063	−0.840115	−0.420057	−	0.907498i	$-0.637990\pi$
$929$	−25.6063	−0.840115	−0.420057	+	0.907498i	$0.637990\pi$
$930$	0	0
$931$	0.236016	0.00773512
$932$	0	0
$933$	−49.2322	−1.61179
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−23.1946	−0.757736	−0.378868	−	0.925451i	$-0.623687\pi$
$937$	−23.1946	−0.757736	−0.378868	+	0.925451i	$0.623687\pi$
$938$	0	0
$939$	−30.6589	−1.00052
$940$	0	0
$941$	8.04530	0.262269	0.131135	−	0.991365i	$-0.458138\pi$
$941$	8.04530	0.262269	0.131135	+	0.991365i	$0.458138\pi$
$942$	0	0
$943$	6.53833	0.212917
$944$	0	0
$945$	0	0
$946$	0	0
$947$	42.2271	1.37220	0.686098	−	0.727509i	$-0.259322\pi$
$947$	42.2271	1.37220	0.686098	+	0.727509i	$0.259322\pi$
$948$	0	0
$949$	−63.3347	−2.05593
$950$	0	0
$951$	−42.4152	−1.37541
$952$	0	0
$953$	13.3175	0.431396	0.215698	−	0.976460i	$-0.430797\pi$
$953$	13.3175	0.431396	0.215698	+	0.976460i	$0.430797\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	20.5442	0.664098
$958$	0	0
$959$	61.7950	1.99546
$960$	0	0
$961$	−18.4109	−0.593901
$962$	0	0
$963$	49.9924	1.61098
$964$	0	0
$965$	0	0
$966$	0	0
$967$	42.9969	1.38269	0.691344	−	0.722526i	$-0.257019\pi$
$967$	42.9969	1.38269	0.691344	+	0.722526i	$0.257019\pi$
$968$	0	0
$969$	4.80925	0.154495
$970$	0	0
$971$	9.01250	0.289225	0.144612	−	0.989488i	$-0.453806\pi$
$971$	9.01250	0.289225	0.144612	+	0.989488i	$0.453806\pi$
$972$	0	0
$973$	−43.1911	−1.38464
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−41.1074	−1.31514	−0.657571	−	0.753392i	$-0.728416\pi$
$977$	−41.1074	−1.31514	−0.657571	+	0.753392i	$0.728416\pi$
$978$	0	0
$979$	24.3613	0.778591
$980$	0	0
$981$	6.26945	0.200168
$982$	0	0
$983$	−59.8861	−1.91007	−0.955035	−	0.296493i	$-0.904183\pi$
$983$	−59.8861	−1.91007	−0.955035	+	0.296493i	$0.904183\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	6.47042	0.205956
$988$	0	0
$989$	−4.93835	−0.157030
$990$	0	0
$991$	−20.8275	−0.661607	−0.330804	−	0.943700i	$-0.607320\pi$
$991$	−20.8275	−0.661607	−0.330804	+	0.943700i	$0.607320\pi$
$992$	0	0
$993$	−33.9695	−1.07799
$994$	0	0
$995$	0	0
$996$	0	0
$997$	12.7634	0.404222	0.202111	−	0.979363i	$-0.435220\pi$
$997$	12.7634	0.404222	0.202111	+	0.979363i	$0.435220\pi$
$998$	0	0
$999$	−27.0819	−0.856834

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9200.2.a.da.1.6		7
4.3	odd	2		575.2.a.k.1.4	✓	7
5.4	even	2		9200.2.a.db.1.2		7
12.11	even	2		5175.2.a.cg.1.4		7
20.3	even	4		575.2.b.f.24.7		14
20.7	even	4		575.2.b.f.24.8		14
20.19	odd	2		575.2.a.l.1.4	yes	7
60.59	even	2		5175.2.a.cb.1.4		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
575.2.a.k.1.4	✓	7	4.3	odd	2
575.2.a.l.1.4	yes	7	20.19	odd	2
575.2.b.f.24.7		14	20.3	even	4
575.2.b.f.24.8		14	20.7	even	4
5175.2.a.cb.1.4		7	60.59	even	2
5175.2.a.cg.1.4		7	12.11	even	2
9200.2.a.da.1.6		7	1.1	even	1	trivial
9200.2.a.db.1.2		7	5.4	even	2

Overview	Random
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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:	\( N \)	\(=\)	\( 9200 = 2^{4} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9200.a (trivial)

\(f(q)\)	\(=\)	\(q+2.69619 q^{3} +2.81698 q^{7} +4.26945 q^{9} +1.84786 q^{11} -6.49089 q^{13} +7.06930 q^{17} +0.252319 q^{19} +7.59513 q^{21} +1.00000 q^{23} +3.42269 q^{27} +4.12351 q^{29} -3.54811 q^{31} +4.98220 q^{33} -7.91248 q^{37} -17.5007 q^{39} +6.53833 q^{41} -4.93835 q^{43} +0.851917 q^{47} +0.935388 q^{49} +19.0602 q^{51} +12.5831 q^{53} +0.680301 q^{57} +10.3616 q^{59} +1.01207 q^{61} +12.0270 q^{63} +3.37930 q^{67} +2.69619 q^{69} +0.851917 q^{71} +9.75748 q^{73} +5.20540 q^{77} +16.5320 q^{79} -3.58013 q^{81} +0.696772 q^{83} +11.1178 q^{87} +13.1835 q^{89} -18.2847 q^{91} -9.56638 q^{93} +5.48684 q^{97} +7.88937 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 3 q^{7} + 15 q^{9} + q^{11} - 3 q^{13} + 10 q^{17} - 15 q^{19} + 2 q^{21} + 7 q^{23} + 3 q^{29} - 14 q^{31} + 6 q^{33} - 10 q^{37} + 8 q^{39} + 19 q^{41} - 5 q^{43} + 14 q^{47} + 40 q^{49} - 2 q^{51}+ \cdots + 53 q^{99}+O(q^{100}) \) 7 * q - 3 * q^7 + 15 * q^9 + q^11 - 3 * q^13 + 10 * q^17 - 15 * q^19 + 2 * q^21 + 7 * q^23 + 3 * q^29 - 14 * q^31 + 6 * q^33 - 10 * q^37 + 8 * q^39 + 19 * q^41 - 5 * q^43 + 14 * q^47 + 40 * q^49 - 2 * q^51 + 4 * q^53 - 4 * q^57 + 16 * q^59 + 40 * q^61 - 53 * q^63 + 4 * q^67 + 14 * q^71 - 3 * q^73 - 17 * q^77 + q^79 + 47 * q^81 - 17 * q^83 + 56 * q^87 + 16 * q^89 - 25 * q^91 + 14 * q^93 - 24 * q^97 + 53 * q^99

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