LMFDB - Embedded newform 9200.2.a.cz.1.3

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:	$1$
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} - 3x^{6} - 7x^{5} + 24x^{4} + x^{3} - 35x^{2} + 17x - 2 $ x^7 - 3x^6 - 7x^5 + 24x^4 + x^3 - 35x^2 + 17*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 920)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.58319$ 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)

$f(q)$	$=$	$q-1.58319 q^{3} -2.84620 q^{7} -0.493499 q^{9} +O(q^{10})$	$q-1.58319 q^{3} -2.84620 q^{7} -0.493499 q^{9} -1.98637 q^{11} -4.69204 q^{13} -3.16675 q^{17} +6.16110 q^{19} +4.50609 q^{21} +1.00000 q^{23} +5.53088 q^{27} -6.61816 q^{29} +8.29732 q^{31} +3.14481 q^{33} +1.71181 q^{37} +7.42840 q^{39} +6.72440 q^{41} -0.177968 q^{43} +11.4749 q^{47} +1.10086 q^{49} +5.01358 q^{51} -6.18762 q^{53} -9.75422 q^{57} +7.61559 q^{59} +11.8577 q^{61} +1.40460 q^{63} +3.07554 q^{67} -1.58319 q^{69} +1.96195 q^{71} -4.94732 q^{73} +5.65361 q^{77} -8.53182 q^{79} -7.27596 q^{81} +3.49429 q^{83} +10.4778 q^{87} +7.07506 q^{89} +13.3545 q^{91} -13.1363 q^{93} +7.00705 q^{97} +0.980272 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 3 q^{3} - 4 q^{7} + 2 q^{9}+O(q^{10}) $ 7 * q - 3 * q^3 - 4 * q^7 + 2 * q^9	$ 7 q - 3 q^{3} - 4 q^{7} + 2 q^{9} + 7 q^{11} - 7 q^{13} + 7 q^{19} - 6 q^{21} + 7 q^{23} - 11 q^{29} + 10 q^{31} - 19 q^{33} - 19 q^{37} + 24 q^{39} - 16 q^{41} - 6 q^{43} - 6 q^{47} - 17 q^{49} + 7 q^{51} - 15 q^{53} - 8 q^{57} + 11 q^{59} + 5 q^{61} - 13 q^{63} - 9 q^{67} - 3 q^{69} + 14 q^{71} - 10 q^{73} - 6 q^{77} + 32 q^{79} - 5 q^{81} - q^{83} - 10 q^{87} - 24 q^{89} + 7 q^{91} - 26 q^{93} + 7 q^{97} + 61 q^{99}+O(q^{100}) $ 7 * q - 3 * q^3 - 4 * q^7 + 2 * q^9 + 7 * q^11 - 7 * q^13 + 7 * q^19 - 6 * q^21 + 7 * q^23 - 11 * q^29 + 10 * q^31 - 19 * q^33 - 19 * q^37 + 24 * q^39 - 16 * q^41 - 6 * q^43 - 6 * q^47 - 17 * q^49 + 7 * q^51 - 15 * q^53 - 8 * q^57 + 11 * q^59 + 5 * q^61 - 13 * q^63 - 9 * q^67 - 3 * q^69 + 14 * q^71 - 10 * q^73 - 6 * q^77 + 32 * q^79 - 5 * q^81 - q^83 - 10 * q^87 - 24 * q^89 + 7 * q^91 - 26 * q^93 + 7 * q^97 + 61 * 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 $

$2$	0	0
$2$	0	0
$3$	−1.58319	−0.914057	−0.457029	−	0.889452i	$-0.651086\pi$
$3$	−1.58319	−0.914057	−0.457029	+	0.889452i	$0.651086\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.84620	−1.07576	−0.537881	−	0.843020i	$-0.680775\pi$
$7$	−2.84620	−1.07576	−0.537881	+	0.843020i	$0.680775\pi$
$8$	0	0
$9$	−0.493499	−0.164500
$10$	0	0
$11$	−1.98637	−0.598913	−0.299457	−	0.954110i	$-0.596805\pi$
$11$	−1.98637	−0.598913	−0.299457	+	0.954110i	$0.596805\pi$
$12$	0	0
$13$	−4.69204	−1.30134	−0.650668	−	0.759362i	$-0.725511\pi$
$13$	−4.69204	−1.30134	−0.650668	+	0.759362i	$0.725511\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.16675	−0.768050	−0.384025	−	0.923323i	$-0.625462\pi$
$17$	−3.16675	−0.768050	−0.384025	+	0.923323i	$0.625462\pi$
$18$	0	0
$19$	6.16110	1.41345	0.706727	−	0.707486i	$-0.250171\pi$
$19$	6.16110	1.41345	0.706727	+	0.707486i	$0.250171\pi$
$20$	0	0
$21$	4.50609	0.983309
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.53088	1.06442
$28$	0	0
$29$	−6.61816	−1.22896	−0.614481	−	0.788932i	$-0.710635\pi$
$29$	−6.61816	−1.22896	−0.614481	+	0.788932i	$0.710635\pi$
$30$	0	0
$31$	8.29732	1.49024	0.745121	−	0.666929i	$-0.232392\pi$
$31$	8.29732	1.49024	0.745121	+	0.666929i	$0.232392\pi$
$32$	0	0
$33$	3.14481	0.547441
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.71181	0.281420	0.140710	−	0.990051i	$-0.455061\pi$
$37$	1.71181	0.281420	0.140710	+	0.990051i	$0.455061\pi$
$38$	0	0
$39$	7.42840	1.18950
$40$	0	0
$41$	6.72440	1.05018	0.525088	−	0.851048i	$-0.324033\pi$
$41$	6.72440	1.05018	0.525088	+	0.851048i	$0.324033\pi$
$42$	0	0
$43$	−0.177968	−0.0271399	−0.0135700	−	0.999908i	$-0.504320\pi$
$43$	−0.177968	−0.0271399	−0.0135700	+	0.999908i	$0.504320\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	11.4749	1.67378	0.836891	−	0.547369i	$-0.184371\pi$
$47$	11.4749	1.67378	0.836891	+	0.547369i	$0.184371\pi$
$48$	0	0
$49$	1.10086	0.157266
$50$	0	0
$51$	5.01358	0.702042
$52$	0	0
$53$	−6.18762	−0.849934	−0.424967	−	0.905209i	$-0.639714\pi$
$53$	−6.18762	−0.849934	−0.424967	+	0.905209i	$0.639714\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−9.75422	−1.29198
$58$	0	0
$59$	7.61559	0.991465	0.495733	−	0.868475i	$-0.334900\pi$
$59$	7.61559	0.991465	0.495733	+	0.868475i	$0.334900\pi$
$60$	0	0
$61$	11.8577	1.51822	0.759109	−	0.650964i	$-0.225635\pi$
$61$	11.8577	1.51822	0.759109	+	0.650964i	$0.225635\pi$
$62$	0	0
$63$	1.40460	0.176963
$64$	0	0
$65$	0	0
$66$	0	0
$67$	3.07554	0.375737	0.187868	−	0.982194i	$-0.439842\pi$
$67$	3.07554	0.375737	0.187868	+	0.982194i	$0.439842\pi$
$68$	0	0
$69$	−1.58319	−0.190594
$70$	0	0
$71$	1.96195	0.232840	0.116420	−	0.993200i	$-0.462858\pi$
$71$	1.96195	0.232840	0.116420	+	0.993200i	$0.462858\pi$
$72$	0	0
$73$	−4.94732	−0.579041	−0.289520	−	0.957172i	$-0.593496\pi$
$73$	−4.94732	−0.579041	−0.289520	+	0.957172i	$0.593496\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.65361	0.644289
$78$	0	0
$79$	−8.53182	−0.959905	−0.479952	−	0.877295i	$-0.659346\pi$
$79$	−8.53182	−0.959905	−0.479952	+	0.877295i	$0.659346\pi$
$80$	0	0
$81$	−7.27596	−0.808440
$82$	0	0
$83$	3.49429	0.383548	0.191774	−	0.981439i	$-0.438576\pi$
$83$	3.49429	0.383548	0.191774	+	0.981439i	$0.438576\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	10.4778	1.12334
$88$	0	0
$89$	7.07506	0.749954	0.374977	−	0.927034i	$-0.377651\pi$
$89$	7.07506	0.749954	0.374977	+	0.927034i	$0.377651\pi$
$90$	0	0
$91$	13.3545	1.39993
$92$	0	0
$93$	−13.1363	−1.36217
$94$	0	0
$95$	0	0
$96$	0	0
$97$	7.00705	0.711458	0.355729	−	0.934589i	$-0.384233\pi$
$97$	7.00705	0.711458	0.355729	+	0.934589i	$0.384233\pi$
$98$	0	0
$99$	0.980272	0.0985210
$100$	0	0
$101$	7.22362	0.718777	0.359389	−	0.933188i	$-0.382985\pi$
$101$	7.22362	0.718777	0.359389	+	0.933188i	$0.382985\pi$
$102$	0	0
$103$	−12.8178	−1.26297	−0.631487	−	0.775386i	$-0.717555\pi$
$103$	−12.8178	−1.26297	−0.631487	+	0.775386i	$0.717555\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−11.6841	−1.12954	−0.564772	−	0.825247i	$-0.691036\pi$
$107$	−11.6841	−1.12954	−0.564772	+	0.825247i	$0.691036\pi$
$108$	0	0
$109$	−13.6019	−1.30283	−0.651413	−	0.758723i	$-0.725824\pi$
$109$	−13.6019	−1.30283	−0.651413	+	0.758723i	$0.725824\pi$
$110$	0	0
$111$	−2.71013	−0.257234
$112$	0	0
$113$	−1.44526	−0.135959	−0.0679795	−	0.997687i	$-0.521655\pi$
$113$	−1.44526	−0.135959	−0.0679795	+	0.997687i	$0.521655\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.31552	0.214069
$118$	0	0
$119$	9.01322	0.826240
$120$	0	0
$121$	−7.05433	−0.641303
$122$	0	0
$123$	−10.6460	−0.959920
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−9.12328	−0.809561	−0.404780	−	0.914414i	$-0.632652\pi$
$127$	−9.12328	−0.809561	−0.404780	+	0.914414i	$0.632652\pi$
$128$	0	0
$129$	0.281758	0.0248074
$130$	0	0
$131$	−9.33180	−0.815323	−0.407661	−	0.913133i	$-0.633656\pi$
$131$	−9.33180	−0.815323	−0.407661	+	0.913133i	$0.633656\pi$
$132$	0	0
$133$	−17.5357	−1.52054
$134$	0	0
$135$	0	0
$136$	0	0
$137$	21.0013	1.79426	0.897129	−	0.441768i	$-0.145649\pi$
$137$	21.0013	1.79426	0.897129	+	0.441768i	$0.145649\pi$
$138$	0	0
$139$	16.1299	1.36812	0.684061	−	0.729425i	$-0.260212\pi$
$139$	16.1299	1.36812	0.684061	+	0.729425i	$0.260212\pi$
$140$	0	0
$141$	−18.1669	−1.52993
$142$	0	0
$143$	9.32012	0.779388
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−1.74288	−0.143750
$148$	0	0
$149$	−9.57227	−0.784191	−0.392096	−	0.919925i	$-0.628250\pi$
$149$	−9.57227	−0.784191	−0.392096	+	0.919925i	$0.628250\pi$
$150$	0	0
$151$	−3.04120	−0.247489	−0.123745	−	0.992314i	$-0.539490\pi$
$151$	−3.04120	−0.247489	−0.123745	+	0.992314i	$0.539490\pi$
$152$	0	0
$153$	1.56279	0.126344
$154$	0	0
$155$	0	0
$156$	0	0
$157$	12.4786	0.995900	0.497950	−	0.867206i	$-0.334086\pi$
$157$	12.4786	0.995900	0.497950	+	0.867206i	$0.334086\pi$
$158$	0	0
$159$	9.79619	0.776889
$160$	0	0
$161$	−2.84620	−0.224312
$162$	0	0
$163$	−16.3939	−1.28407	−0.642034	−	0.766676i	$-0.721909\pi$
$163$	−16.3939	−1.28407	−0.642034	+	0.766676i	$0.721909\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−8.13265	−0.629323	−0.314662	−	0.949204i	$-0.601891\pi$
$167$	−8.13265	−0.629323	−0.314662	+	0.949204i	$0.601891\pi$
$168$	0	0
$169$	9.01521	0.693477
$170$	0	0
$171$	−3.04050	−0.232513
$172$	0	0
$173$	10.5256	0.800250	0.400125	−	0.916461i	$-0.368967\pi$
$173$	10.5256	0.800250	0.400125	+	0.916461i	$0.368967\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−12.0570	−0.906256
$178$	0	0
$179$	−20.5421	−1.53539	−0.767695	−	0.640816i	$-0.778596\pi$
$179$	−20.5421	−1.53539	−0.767695	+	0.640816i	$0.778596\pi$
$180$	0	0
$181$	−22.0754	−1.64085	−0.820424	−	0.571756i	$-0.806263\pi$
$181$	−22.0754	−1.64085	−0.820424	+	0.571756i	$0.806263\pi$
$182$	0	0
$183$	−18.7730	−1.38774
$184$	0	0
$185$	0	0
$186$	0	0
$187$	6.29034	0.459995
$188$	0	0
$189$	−15.7420	−1.14506
$190$	0	0
$191$	4.50910	0.326267	0.163133	−	0.986604i	$-0.447840\pi$
$191$	4.50910	0.326267	0.163133	+	0.986604i	$0.447840\pi$
$192$	0	0
$193$	−21.1534	−1.52266	−0.761329	−	0.648366i	$-0.775453\pi$
$193$	−21.1534	−1.52266	−0.761329	+	0.648366i	$0.775453\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−10.1475	−0.722980	−0.361490	−	0.932376i	$-0.617732\pi$
$197$	−10.1475	−0.722980	−0.361490	+	0.932376i	$0.617732\pi$
$198$	0	0
$199$	−6.57598	−0.466159	−0.233079	−	0.972458i	$-0.574880\pi$
$199$	−6.57598	−0.466159	−0.233079	+	0.972458i	$0.574880\pi$
$200$	0	0
$201$	−4.86917	−0.343445
$202$	0	0
$203$	18.8366	1.32207
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−0.493499	−0.0343005
$208$	0	0
$209$	−12.2382	−0.846536
$210$	0	0
$211$	26.5357	1.82680	0.913398	−	0.407069i	$-0.133449\pi$
$211$	26.5357	1.82680	0.913398	+	0.407069i	$0.133449\pi$
$212$	0	0
$213$	−3.10614	−0.212829
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−23.6158	−1.60315
$218$	0	0
$219$	7.83257	0.529276
$220$	0	0
$221$	14.8585	0.999492
$222$	0	0
$223$	−20.9231	−1.40111	−0.700557	−	0.713597i	$-0.747065\pi$
$223$	−20.9231	−1.40111	−0.700557	+	0.713597i	$0.747065\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−9.32581	−0.618975	−0.309488	−	0.950903i	$-0.600158\pi$
$227$	−9.32581	−0.618975	−0.309488	+	0.950903i	$0.600158\pi$
$228$	0	0
$229$	29.5189	1.95066	0.975332	−	0.220742i	$-0.0708479\pi$
$229$	29.5189	1.95066	0.975332	+	0.220742i	$0.0708479\pi$
$230$	0	0
$231$	−8.95076	−0.588917
$232$	0	0
$233$	28.9849	1.89886	0.949431	−	0.313974i	$-0.101661\pi$
$233$	28.9849	1.89886	0.949431	+	0.313974i	$0.101661\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	13.5075	0.877408
$238$	0	0
$239$	9.62209	0.622401	0.311201	−	0.950344i	$-0.399269\pi$
$239$	9.62209	0.622401	0.311201	+	0.950344i	$0.399269\pi$
$240$	0	0
$241$	15.5745	1.00324	0.501621	−	0.865088i	$-0.332737\pi$
$241$	15.5745	1.00324	0.501621	+	0.865088i	$0.332737\pi$
$242$	0	0
$243$	−5.07340	−0.325459
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−28.9081	−1.83938
$248$	0	0
$249$	−5.53214	−0.350585
$250$	0	0
$251$	22.8066	1.43954	0.719770	−	0.694213i	$-0.244247\pi$
$251$	22.8066	1.43954	0.719770	+	0.694213i	$0.244247\pi$
$252$	0	0
$253$	−1.98637	−0.124882
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−11.5591	−0.721038	−0.360519	−	0.932752i	$-0.617400\pi$
$257$	−11.5591	−0.721038	−0.360519	+	0.932752i	$0.617400\pi$
$258$	0	0
$259$	−4.87216	−0.302742
$260$	0	0
$261$	3.26606	0.202164
$262$	0	0
$263$	6.57534	0.405453	0.202726	−	0.979235i	$-0.435020\pi$
$263$	6.57534	0.405453	0.202726	+	0.979235i	$0.435020\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−11.2012	−0.685501
$268$	0	0
$269$	−16.6536	−1.01539	−0.507694	−	0.861537i	$-0.669502\pi$
$269$	−16.6536	−1.01539	−0.507694	+	0.861537i	$0.669502\pi$
$270$	0	0
$271$	23.8675	1.44985	0.724925	−	0.688828i	$-0.241874\pi$
$271$	23.8675	1.44985	0.724925	+	0.688828i	$0.241874\pi$
$272$	0	0
$273$	−21.1427	−1.27962
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−5.65369	−0.339697	−0.169849	−	0.985470i	$-0.554328\pi$
$277$	−5.65369	−0.339697	−0.169849	+	0.985470i	$0.554328\pi$
$278$	0	0
$279$	−4.09472	−0.245144
$280$	0	0
$281$	−31.7408	−1.89350	−0.946750	−	0.321971i	$-0.895655\pi$
$281$	−31.7408	−1.89350	−0.946750	+	0.321971i	$0.895655\pi$
$282$	0	0
$283$	−0.998092	−0.0593304	−0.0296652	−	0.999560i	$-0.509444\pi$
$283$	−0.998092	−0.0593304	−0.0296652	+	0.999560i	$0.509444\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−19.1390	−1.12974
$288$	0	0
$289$	−6.97168	−0.410099
$290$	0	0
$291$	−11.0935	−0.650314
$292$	0	0
$293$	−13.8947	−0.811739	−0.405869	−	0.913931i	$-0.633031\pi$
$293$	−13.8947	−0.811739	−0.405869	+	0.913931i	$0.633031\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−10.9864	−0.637495
$298$	0	0
$299$	−4.69204	−0.271347
$300$	0	0
$301$	0.506534	0.0291961
$302$	0	0
$303$	−11.4364	−0.657003
$304$	0	0
$305$	0	0
$306$	0	0
$307$	3.66633	0.209249	0.104624	−	0.994512i	$-0.466636\pi$
$307$	3.66633	0.209249	0.104624	+	0.994512i	$0.466636\pi$
$308$	0	0
$309$	20.2930	1.15443
$310$	0	0
$311$	14.4210	0.817741	0.408870	−	0.912593i	$-0.365923\pi$
$311$	14.4210	0.817741	0.408870	+	0.912593i	$0.365923\pi$
$312$	0	0
$313$	10.5046	0.593754	0.296877	−	0.954916i	$-0.404055\pi$
$313$	10.5046	0.593754	0.296877	+	0.954916i	$0.404055\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−32.0818	−1.80189	−0.900946	−	0.433931i	$-0.857126\pi$
$317$	−32.0818	−1.80189	−0.900946	+	0.433931i	$0.857126\pi$
$318$	0	0
$319$	13.1461	0.736042
$320$	0	0
$321$	18.4982	1.03247
$322$	0	0
$323$	−19.5107	−1.08560
$324$	0	0
$325$	0	0
$326$	0	0
$327$	21.5344	1.19086
$328$	0	0
$329$	−32.6598	−1.80059
$330$	0	0
$331$	−9.71307	−0.533879	−0.266939	−	0.963713i	$-0.586012\pi$
$331$	−9.71307	−0.533879	−0.266939	+	0.963713i	$0.586012\pi$
$332$	0	0
$333$	−0.844778	−0.0462935
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−0.131832	−0.00718136	−0.00359068	−	0.999994i	$-0.501143\pi$
$337$	−0.131832	−0.00718136	−0.00359068	+	0.999994i	$0.501143\pi$
$338$	0	0
$339$	2.28813	0.124274
$340$	0	0
$341$	−16.4815	−0.892526
$342$	0	0
$343$	16.7901	0.906582
$344$	0	0
$345$	0	0
$346$	0	0
$347$	13.1981	0.708512	0.354256	−	0.935148i	$-0.384734\pi$
$347$	13.1981	0.708512	0.354256	+	0.935148i	$0.384734\pi$
$348$	0	0
$349$	−34.0451	−1.82239	−0.911197	−	0.411971i	$-0.864841\pi$
$349$	−34.0451	−1.82239	−0.911197	+	0.411971i	$0.864841\pi$
$350$	0	0
$351$	−25.9511	−1.38517
$352$	0	0
$353$	10.8593	0.577984	0.288992	−	0.957331i	$-0.406680\pi$
$353$	10.8593	0.577984	0.288992	+	0.957331i	$0.406680\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−14.2697	−0.755231
$358$	0	0
$359$	20.3091	1.07188	0.535938	−	0.844257i	$-0.319958\pi$
$359$	20.3091	1.07188	0.535938	+	0.844257i	$0.319958\pi$
$360$	0	0
$361$	18.9592	0.997853
$362$	0	0
$363$	11.1684	0.586188
$364$	0	0
$365$	0	0
$366$	0	0
$367$	18.6731	0.974727	0.487364	−	0.873199i	$-0.337959\pi$
$367$	18.6731	0.974727	0.487364	+	0.873199i	$0.337959\pi$
$368$	0	0
$369$	−3.31848	−0.172753
$370$	0	0
$371$	17.6112	0.914328
$372$	0	0
$373$	9.51979	0.492916	0.246458	−	0.969153i	$-0.420733\pi$
$373$	9.51979	0.492916	0.246458	+	0.969153i	$0.420733\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	31.0527	1.59929
$378$	0	0
$379$	−18.8304	−0.967251	−0.483625	−	0.875275i	$-0.660680\pi$
$379$	−18.8304	−0.967251	−0.483625	+	0.875275i	$0.660680\pi$
$380$	0	0
$381$	14.4439	0.739985
$382$	0	0
$383$	10.9401	0.559015	0.279507	−	0.960144i	$-0.409829\pi$
$383$	10.9401	0.559015	0.279507	+	0.960144i	$0.409829\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0.0878272	0.00446451
$388$	0	0
$389$	22.2830	1.12979	0.564896	−	0.825162i	$-0.308916\pi$
$389$	22.2830	1.12979	0.564896	+	0.825162i	$0.308916\pi$
$390$	0	0
$391$	−3.16675	−0.160150
$392$	0	0
$393$	14.7740	0.745252
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−10.8388	−0.543983	−0.271991	−	0.962300i	$-0.587682\pi$
$397$	−10.8388	−0.543983	−0.271991	+	0.962300i	$0.587682\pi$
$398$	0	0
$399$	27.7625	1.38986
$400$	0	0
$401$	−2.71329	−0.135495	−0.0677476	−	0.997702i	$-0.521581\pi$
$401$	−2.71329	−0.135495	−0.0677476	+	0.997702i	$0.521581\pi$
$402$	0	0
$403$	−38.9313	−1.93931
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−3.40029	−0.168546
$408$	0	0
$409$	21.6271	1.06939	0.534696	−	0.845044i	$-0.320426\pi$
$409$	21.6271	1.06939	0.534696	+	0.845044i	$0.320426\pi$
$410$	0	0
$411$	−33.2490	−1.64005
$412$	0	0
$413$	−21.6755	−1.06658
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−25.5368	−1.25054
$418$	0	0
$419$	−4.00587	−0.195700	−0.0978498	−	0.995201i	$-0.531196\pi$
$419$	−4.00587	−0.195700	−0.0978498	+	0.995201i	$0.531196\pi$
$420$	0	0
$421$	25.4251	1.23914	0.619572	−	0.784940i	$-0.287306\pi$
$421$	25.4251	1.23914	0.619572	+	0.784940i	$0.287306\pi$
$422$	0	0
$423$	−5.66284	−0.275337
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−33.7493	−1.63324
$428$	0	0
$429$	−14.7556	−0.712405
$430$	0	0
$431$	−1.07679	−0.0518673	−0.0259336	−	0.999664i	$-0.508256\pi$
$431$	−1.07679	−0.0518673	−0.0259336	+	0.999664i	$0.508256\pi$
$432$	0	0
$433$	−3.93871	−0.189283	−0.0946413	−	0.995511i	$-0.530170\pi$
$433$	−3.93871	−0.189283	−0.0946413	+	0.995511i	$0.530170\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.16110	0.294726
$438$	0	0
$439$	−13.1539	−0.627802	−0.313901	−	0.949456i	$-0.601636\pi$
$439$	−13.1539	−0.627802	−0.313901	+	0.949456i	$0.601636\pi$
$440$	0	0
$441$	−0.543274	−0.0258702
$442$	0	0
$443$	8.95394	0.425415	0.212707	−	0.977116i	$-0.431772\pi$
$443$	8.95394	0.425415	0.212707	+	0.977116i	$0.431772\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	15.1548	0.716795
$448$	0	0
$449$	8.23267	0.388524	0.194262	−	0.980950i	$-0.437769\pi$
$449$	8.23267	0.388524	0.194262	+	0.980950i	$0.437769\pi$
$450$	0	0
$451$	−13.3571	−0.628964
$452$	0	0
$453$	4.81480	0.226219
$454$	0	0
$455$	0	0
$456$	0	0
$457$	36.3473	1.70026	0.850128	−	0.526576i	$-0.176524\pi$
$457$	36.3473	1.70026	0.850128	+	0.526576i	$0.176524\pi$
$458$	0	0
$459$	−17.5149	−0.817527
$460$	0	0
$461$	−15.2421	−0.709894	−0.354947	−	0.934886i	$-0.615501\pi$
$461$	−15.2421	−0.709894	−0.354947	+	0.934886i	$0.615501\pi$
$462$	0	0
$463$	5.57424	0.259057	0.129528	−	0.991576i	$-0.458654\pi$
$463$	5.57424	0.259057	0.129528	+	0.991576i	$0.458654\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	6.21488	0.287590	0.143795	−	0.989607i	$-0.454069\pi$
$467$	6.21488	0.287590	0.143795	+	0.989607i	$0.454069\pi$
$468$	0	0
$469$	−8.75360	−0.404204
$470$	0	0
$471$	−19.7560	−0.910309
$472$	0	0
$473$	0.353511	0.0162545
$474$	0	0
$475$	0	0
$476$	0	0
$477$	3.05358	0.139814
$478$	0	0
$479$	34.4168	1.57254	0.786272	−	0.617880i	$-0.212008\pi$
$479$	34.4168	1.57254	0.786272	+	0.617880i	$0.212008\pi$
$480$	0	0
$481$	−8.03189	−0.366223
$482$	0	0
$483$	4.50609	0.205034
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−22.7752	−1.03204	−0.516021	−	0.856576i	$-0.672587\pi$
$487$	−22.7752	−1.03204	−0.516021	+	0.856576i	$0.672587\pi$
$488$	0	0
$489$	25.9547	1.17371
$490$	0	0
$491$	−3.16911	−0.143020	−0.0715099	−	0.997440i	$-0.522782\pi$
$491$	−3.16911	−0.143020	−0.0715099	+	0.997440i	$0.522782\pi$
$492$	0	0
$493$	20.9581	0.943905
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−5.58410	−0.250481
$498$	0	0
$499$	14.0809	0.630346	0.315173	−	0.949034i	$-0.397937\pi$
$499$	14.0809	0.630346	0.315173	+	0.949034i	$0.397937\pi$
$500$	0	0
$501$	12.8756	0.575237
$502$	0	0
$503$	−21.4946	−0.958399	−0.479199	−	0.877706i	$-0.659073\pi$
$503$	−21.4946	−0.958399	−0.479199	+	0.877706i	$0.659073\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−14.2728	−0.633878
$508$	0	0
$509$	−34.0952	−1.51124	−0.755622	−	0.655008i	$-0.772665\pi$
$509$	−34.0952	−1.51124	−0.755622	+	0.655008i	$0.772665\pi$
$510$	0	0
$511$	14.0811	0.622910
$512$	0	0
$513$	34.0764	1.50451
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−22.7933	−1.00245
$518$	0	0
$519$	−16.6641	−0.731474
$520$	0	0
$521$	−14.1367	−0.619339	−0.309669	−	0.950844i	$-0.600218\pi$
$521$	−14.1367	−0.619339	−0.309669	+	0.950844i	$0.600218\pi$
$522$	0	0
$523$	11.2020	0.489828	0.244914	−	0.969545i	$-0.421240\pi$
$523$	11.2020	0.489828	0.244914	+	0.969545i	$0.421240\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−26.2756	−1.14458
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−3.75829	−0.163096
$532$	0	0
$533$	−31.5511	−1.36663
$534$	0	0
$535$	0	0
$536$	0	0
$537$	32.5221	1.40343
$538$	0	0
$539$	−2.18672	−0.0941887
$540$	0	0
$541$	−33.6704	−1.44760	−0.723802	−	0.690008i	$-0.757607\pi$
$541$	−33.6704	−1.44760	−0.723802	+	0.690008i	$0.757607\pi$
$542$	0	0
$543$	34.9496	1.49983
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−23.1062	−0.987949	−0.493975	−	0.869476i	$-0.664456\pi$
$547$	−23.1062	−0.987949	−0.493975	+	0.869476i	$0.664456\pi$
$548$	0	0
$549$	−5.85174	−0.249746
$550$	0	0
$551$	−40.7752	−1.73708
$552$	0	0
$553$	24.2833	1.03263
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−3.48937	−0.147849	−0.0739246	−	0.997264i	$-0.523552\pi$
$557$	−3.48937	−0.147849	−0.0739246	+	0.997264i	$0.523552\pi$
$558$	0	0
$559$	0.835034	0.0353182
$560$	0	0
$561$	−9.95883	−0.420462
$562$	0	0
$563$	−38.8593	−1.63772	−0.818862	−	0.573991i	$-0.805394\pi$
$563$	−38.8593	−1.63772	−0.818862	+	0.573991i	$0.805394\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	20.7089	0.869690
$568$	0	0
$569$	−23.2055	−0.972823	−0.486412	−	0.873730i	$-0.661694\pi$
$569$	−23.2055	−0.972823	−0.486412	+	0.873730i	$0.661694\pi$
$570$	0	0
$571$	5.39836	0.225914	0.112957	−	0.993600i	$-0.463968\pi$
$571$	5.39836	0.225914	0.112957	+	0.993600i	$0.463968\pi$
$572$	0	0
$573$	−7.13877	−0.298226
$574$	0	0
$575$	0	0
$576$	0	0
$577$	13.1082	0.545701	0.272850	−	0.962056i	$-0.412034\pi$
$577$	13.1082	0.545701	0.272850	+	0.962056i	$0.412034\pi$
$578$	0	0
$579$	33.4900	1.39180
$580$	0	0
$581$	−9.94546	−0.412607
$582$	0	0
$583$	12.2909	0.509037
$584$	0	0
$585$	0	0
$586$	0	0
$587$	12.9646	0.535108	0.267554	−	0.963543i	$-0.413785\pi$
$587$	12.9646	0.535108	0.267554	+	0.963543i	$0.413785\pi$
$588$	0	0
$589$	51.1206	2.10639
$590$	0	0
$591$	16.0655	0.660845
$592$	0	0
$593$	17.0225	0.699030	0.349515	−	0.936931i	$-0.386346\pi$
$593$	17.0225	0.699030	0.349515	+	0.936931i	$0.386346\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	10.4110	0.426096
$598$	0	0
$599$	5.98111	0.244381	0.122191	−	0.992507i	$-0.461008\pi$
$599$	5.98111	0.244381	0.122191	+	0.992507i	$0.461008\pi$
$600$	0	0
$601$	−21.3680	−0.871617	−0.435809	−	0.900039i	$-0.643538\pi$
$601$	−21.3680	−0.871617	−0.435809	+	0.900039i	$0.643538\pi$
$602$	0	0
$603$	−1.51778	−0.0618086
$604$	0	0
$605$	0	0
$606$	0	0
$607$	17.5464	0.712186	0.356093	−	0.934450i	$-0.384109\pi$
$607$	17.5464	0.712186	0.356093	+	0.934450i	$0.384109\pi$
$608$	0	0
$609$	−29.8220	−1.20845
$610$	0	0
$611$	−53.8405	−2.17815
$612$	0	0
$613$	−45.4601	−1.83612	−0.918059	−	0.396444i	$-0.870244\pi$
$613$	−45.4601	−1.83612	−0.918059	+	0.396444i	$0.870244\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−25.8720	−1.04157	−0.520785	−	0.853688i	$-0.674360\pi$
$617$	−25.8720	−1.04157	−0.520785	+	0.853688i	$0.674360\pi$
$618$	0	0
$619$	3.20381	0.128772	0.0643859	−	0.997925i	$-0.479491\pi$
$619$	3.20381	0.128772	0.0643859	+	0.997925i	$0.479491\pi$
$620$	0	0
$621$	5.53088	0.221947
$622$	0	0
$623$	−20.1370	−0.806773
$624$	0	0
$625$	0	0
$626$	0	0
$627$	19.3755	0.773783
$628$	0	0
$629$	−5.42089	−0.216145
$630$	0	0
$631$	−0.0248650	−0.000989858 0	−0.000494929	−	1.00000i	$-0.500158\pi$
$631$	−0.0248650	−0.000989858 0	−0.000494929		1.00000i	$0.500158\pi$
$632$	0	0
$633$	−42.0112	−1.66980
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−5.16528	−0.204656
$638$	0	0
$639$	−0.968219	−0.0383022
$640$	0	0
$641$	−9.77145	−0.385949	−0.192975	−	0.981204i	$-0.561813\pi$
$641$	−9.77145	−0.385949	−0.192975	+	0.981204i	$0.561813\pi$
$642$	0	0
$643$	−36.8211	−1.45208	−0.726041	−	0.687651i	$-0.758642\pi$
$643$	−36.8211	−1.45208	−0.726041	+	0.687651i	$0.758642\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	46.5570	1.83034	0.915172	−	0.403064i	$-0.132055\pi$
$647$	46.5570	1.83034	0.915172	+	0.403064i	$0.132055\pi$
$648$	0	0
$649$	−15.1274	−0.593802
$650$	0	0
$651$	37.3884	1.46537
$652$	0	0
$653$	−2.76029	−0.108018	−0.0540092	−	0.998540i	$-0.517200\pi$
$653$	−2.76029	−0.108018	−0.0540092	+	0.998540i	$0.517200\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	2.44150	0.0952520
$658$	0	0
$659$	−39.3922	−1.53450	−0.767252	−	0.641346i	$-0.778376\pi$
$659$	−39.3922	−1.53450	−0.767252	+	0.641346i	$0.778376\pi$
$660$	0	0
$661$	−17.7641	−0.690943	−0.345472	−	0.938429i	$-0.612281\pi$
$661$	−17.7641	−0.690943	−0.345472	+	0.938429i	$0.612281\pi$
$662$	0	0
$663$	−23.5239	−0.913593
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−6.61816	−0.256256
$668$	0	0
$669$	33.1253	1.28070
$670$	0	0
$671$	−23.5537	−0.909280
$672$	0	0
$673$	23.7072	0.913847	0.456923	−	0.889506i	$-0.348951\pi$
$673$	23.7072	0.913847	0.456923	+	0.889506i	$0.348951\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−49.3507	−1.89670	−0.948352	−	0.317221i	$-0.897250\pi$
$677$	−49.3507	−1.89670	−0.948352	+	0.317221i	$0.897250\pi$
$678$	0	0
$679$	−19.9435	−0.765361
$680$	0	0
$681$	14.7646	0.565779
$682$	0	0
$683$	15.6461	0.598682	0.299341	−	0.954146i	$-0.403233\pi$
$683$	15.6461	0.598682	0.299341	+	0.954146i	$0.403233\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−46.7341	−1.78302
$688$	0	0
$689$	29.0325	1.10605
$690$	0	0
$691$	−3.79498	−0.144368	−0.0721839	−	0.997391i	$-0.522997\pi$
$691$	−3.79498	−0.144368	−0.0721839	+	0.997391i	$0.522997\pi$
$692$	0	0
$693$	−2.79005	−0.105985
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−21.2945	−0.806587
$698$	0	0
$699$	−45.8887	−1.73567
$700$	0	0
$701$	0.751725	0.0283923	0.0141961	−	0.999899i	$-0.495481\pi$
$701$	0.751725	0.0283923	0.0141961	+	0.999899i	$0.495481\pi$
$702$	0	0
$703$	10.5467	0.397775
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−20.5599	−0.773234
$708$	0	0
$709$	4.52035	0.169765	0.0848827	−	0.996391i	$-0.472948\pi$
$709$	4.52035	0.169765	0.0848827	+	0.996391i	$0.472948\pi$
$710$	0	0
$711$	4.21044	0.157904
$712$	0	0
$713$	8.29732	0.310737
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−15.2336	−0.568910
$718$	0	0
$719$	36.6742	1.36772	0.683859	−	0.729614i	$-0.260300\pi$
$719$	36.6742	1.36772	0.683859	+	0.729614i	$0.260300\pi$
$720$	0	0
$721$	36.4820	1.35866
$722$	0	0
$723$	−24.6574	−0.917020
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−33.6449	−1.24782	−0.623910	−	0.781496i	$-0.714457\pi$
$727$	−33.6449	−1.24782	−0.623910	+	0.781496i	$0.714457\pi$
$728$	0	0
$729$	29.8601	1.10593
$730$	0	0
$731$	0.563582	0.0208448
$732$	0	0
$733$	−47.5745	−1.75720	−0.878602	−	0.477555i	$-0.841523\pi$
$733$	−47.5745	−1.75720	−0.878602	+	0.477555i	$0.841523\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−6.10916	−0.225034
$738$	0	0
$739$	−28.8914	−1.06279	−0.531395	−	0.847124i	$-0.678332\pi$
$739$	−28.8914	−1.06279	−0.531395	+	0.847124i	$0.678332\pi$
$740$	0	0
$741$	45.7672	1.68130
$742$	0	0
$743$	3.61970	0.132794	0.0663969	−	0.997793i	$-0.478850\pi$
$743$	3.61970	0.132794	0.0663969	+	0.997793i	$0.478850\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−1.72443	−0.0630936
$748$	0	0
$749$	33.2553	1.21512
$750$	0	0
$751$	−19.2050	−0.700801	−0.350401	−	0.936600i	$-0.613955\pi$
$751$	−19.2050	−0.700801	−0.350401	+	0.936600i	$0.613955\pi$
$752$	0	0
$753$	−36.1073	−1.31582
$754$	0	0
$755$	0	0
$756$	0	0
$757$	12.6638	0.460275	0.230137	−	0.973158i	$-0.426082\pi$
$757$	12.6638	0.460275	0.230137	+	0.973158i	$0.426082\pi$
$758$	0	0
$759$	3.14481	0.114149
$760$	0	0
$761$	−38.9053	−1.41032	−0.705159	−	0.709050i	$-0.749124\pi$
$761$	−38.9053	−1.41032	−0.705159	+	0.709050i	$0.749124\pi$
$762$	0	0
$763$	38.7138	1.40153
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−35.7326	−1.29023
$768$	0	0
$769$	4.16816	0.150308	0.0751539	−	0.997172i	$-0.476055\pi$
$769$	4.16816	0.150308	0.0751539	+	0.997172i	$0.476055\pi$
$770$	0	0
$771$	18.3003	0.659070
$772$	0	0
$773$	−26.6030	−0.956842	−0.478421	−	0.878131i	$-0.658791\pi$
$773$	−26.6030	−0.956842	−0.478421	+	0.878131i	$0.658791\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	7.71358	0.276723
$778$	0	0
$779$	41.4297	1.48437
$780$	0	0
$781$	−3.89716	−0.139451
$782$	0	0
$783$	−36.6043	−1.30813
$784$	0	0
$785$	0	0
$786$	0	0
$787$	21.7538	0.775440	0.387720	−	0.921777i	$-0.373263\pi$
$787$	21.7538	0.775440	0.387720	+	0.921777i	$0.373263\pi$
$788$	0	0
$789$	−10.4100	−0.370607
$790$	0	0
$791$	4.11351	0.146260
$792$	0	0
$793$	−55.6366	−1.97571
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−10.2490	−0.363039	−0.181519	−	0.983387i	$-0.558102\pi$
$797$	−10.2490	−0.363039	−0.181519	+	0.983387i	$0.558102\pi$
$798$	0	0
$799$	−36.3381	−1.28555
$800$	0	0
$801$	−3.49153	−0.123367
$802$	0	0
$803$	9.82722	0.346795
$804$	0	0
$805$	0	0
$806$	0	0
$807$	26.3659	0.928123
$808$	0	0
$809$	37.0993	1.30434	0.652170	−	0.758072i	$-0.273859\pi$
$809$	37.0993	1.30434	0.652170	+	0.758072i	$0.273859\pi$
$810$	0	0
$811$	−35.8084	−1.25740	−0.628701	−	0.777647i	$-0.716413\pi$
$811$	−35.8084	−1.25740	−0.628701	+	0.777647i	$0.716413\pi$
$812$	0	0
$813$	−37.7869	−1.32525
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−1.09648	−0.0383610
$818$	0	0
$819$	−6.59042	−0.230288
$820$	0	0
$821$	−21.5343	−0.751551	−0.375775	−	0.926711i	$-0.622624\pi$
$821$	−21.5343	−0.751551	−0.375775	+	0.926711i	$0.622624\pi$
$822$	0	0
$823$	−41.1466	−1.43428	−0.717140	−	0.696929i	$-0.754549\pi$
$823$	−41.1466	−1.43428	−0.717140	+	0.696929i	$0.754549\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−9.48319	−0.329763	−0.164881	−	0.986313i	$-0.552724\pi$
$827$	−9.48319	−0.329763	−0.164881	+	0.986313i	$0.552724\pi$
$828$	0	0
$829$	24.0091	0.833872	0.416936	−	0.908936i	$-0.363104\pi$
$829$	24.0091	0.833872	0.416936	+	0.908936i	$0.363104\pi$
$830$	0	0
$831$	8.95088	0.310503
$832$	0	0
$833$	−3.48616	−0.120788
$834$	0	0
$835$	0	0
$836$	0	0
$837$	45.8915	1.58624
$838$	0	0
$839$	−9.89039	−0.341454	−0.170727	−	0.985318i	$-0.554612\pi$
$839$	−9.89039	−0.341454	−0.170727	+	0.985318i	$0.554612\pi$
$840$	0	0
$841$	14.8001	0.510348
$842$	0	0
$843$	50.2519	1.73077
$844$	0	0
$845$	0	0
$846$	0	0
$847$	20.0781	0.689890
$848$	0	0
$849$	1.58017	0.0542314
$850$	0	0
$851$	1.71181	0.0586802
$852$	0	0
$853$	−5.02358	−0.172004	−0.0860021	−	0.996295i	$-0.527409\pi$
$853$	−5.02358	−0.172004	−0.0860021	+	0.996295i	$0.527409\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	26.8011	0.915509	0.457754	−	0.889079i	$-0.348654\pi$
$857$	26.8011	0.915509	0.457754	+	0.889079i	$0.348654\pi$
$858$	0	0
$859$	−27.3018	−0.931525	−0.465763	−	0.884910i	$-0.654220\pi$
$859$	−27.3018	−0.931525	−0.465763	+	0.884910i	$0.654220\pi$
$860$	0	0
$861$	30.3007	1.03265
$862$	0	0
$863$	2.22910	0.0758796	0.0379398	−	0.999280i	$-0.487920\pi$
$863$	2.22910	0.0758796	0.0379398	+	0.999280i	$0.487920\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	11.0375	0.374854
$868$	0	0
$869$	16.9474	0.574900
$870$	0	0
$871$	−14.4305	−0.488960
$872$	0	0
$873$	−3.45797	−0.117035
$874$	0	0
$875$	0	0
$876$	0	0
$877$	46.4517	1.56856	0.784280	−	0.620407i	$-0.213032\pi$
$877$	46.4517	1.56856	0.784280	+	0.620407i	$0.213032\pi$
$878$	0	0
$879$	21.9980	0.741975
$880$	0	0
$881$	−25.6878	−0.865444	−0.432722	−	0.901527i	$-0.642447\pi$
$881$	−25.6878	−0.865444	−0.432722	+	0.901527i	$0.642447\pi$
$882$	0	0
$883$	48.8481	1.64387	0.821934	−	0.569582i	$-0.192895\pi$
$883$	48.8481	1.64387	0.821934	+	0.569582i	$0.192895\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	45.1121	1.51472	0.757358	−	0.653000i	$-0.226490\pi$
$887$	45.1121	1.51472	0.757358	+	0.653000i	$0.226490\pi$
$888$	0	0
$889$	25.9667	0.870895
$890$	0	0
$891$	14.4528	0.484185
$892$	0	0
$893$	70.6979	2.36581
$894$	0	0
$895$	0	0
$896$	0	0
$897$	7.42840	0.248027
$898$	0	0
$899$	−54.9130	−1.83145
$900$	0	0
$901$	19.5947	0.652792
$902$	0	0
$903$	−0.801941	−0.0266869
$904$	0	0
$905$	0	0
$906$	0	0
$907$	15.5516	0.516382	0.258191	−	0.966094i	$-0.416874\pi$
$907$	15.5516	0.516382	0.258191	+	0.966094i	$0.416874\pi$
$908$	0	0
$909$	−3.56485	−0.118239
$910$	0	0
$911$	30.1283	0.998195	0.499097	−	0.866546i	$-0.333665\pi$
$911$	30.1283	0.998195	0.499097	+	0.866546i	$0.333665\pi$
$912$	0	0
$913$	−6.94096	−0.229712
$914$	0	0
$915$	0	0
$916$	0	0
$917$	26.5602	0.877094
$918$	0	0
$919$	−34.2483	−1.12975	−0.564874	−	0.825177i	$-0.691075\pi$
$919$	−34.2483	−1.12975	−0.564874	+	0.825177i	$0.691075\pi$
$920$	0	0
$921$	−5.80452	−0.191265
$922$	0	0
$923$	−9.20553	−0.303004
$924$	0	0
$925$	0	0
$926$	0	0
$927$	6.32556	0.207759
$928$	0	0
$929$	−54.6989	−1.79461	−0.897307	−	0.441408i	$-0.854479\pi$
$929$	−54.6989	−1.79461	−0.897307	+	0.441408i	$0.854479\pi$
$930$	0	0
$931$	6.78252	0.222288
$932$	0	0
$933$	−22.8312	−0.747462
$934$	0	0
$935$	0	0
$936$	0	0
$937$	12.1635	0.397364	0.198682	−	0.980064i	$-0.436334\pi$
$937$	12.1635	0.397364	0.198682	+	0.980064i	$0.436334\pi$
$938$	0	0
$939$	−16.6308	−0.542725
$940$	0	0
$941$	−27.9048	−0.909672	−0.454836	−	0.890575i	$-0.650302\pi$
$941$	−27.9048	−0.909672	−0.454836	+	0.890575i	$0.650302\pi$
$942$	0	0
$943$	6.72440	0.218977
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−10.5104	−0.341541	−0.170770	−	0.985311i	$-0.554626\pi$
$947$	−10.5104	−0.341541	−0.170770	+	0.985311i	$0.554626\pi$
$948$	0	0
$949$	23.2130	0.753527
$950$	0	0
$951$	50.7916	1.64703
$952$	0	0
$953$	−11.7005	−0.379017	−0.189508	−	0.981879i	$-0.560689\pi$
$953$	−11.7005	−0.379017	−0.189508	+	0.981879i	$0.560689\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−20.8129	−0.672784
$958$	0	0
$959$	−59.7738	−1.93020
$960$	0	0
$961$	37.8455	1.22082
$962$	0	0
$963$	5.76609	0.185810
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−11.7133	−0.376673	−0.188337	−	0.982105i	$-0.560310\pi$
$967$	−11.7133	−0.376673	−0.188337	+	0.982105i	$0.560310\pi$
$968$	0	0
$969$	30.8892	0.992304
$970$	0	0
$971$	37.8963	1.21615	0.608075	−	0.793879i	$-0.291942\pi$
$971$	37.8963	1.21615	0.608075	+	0.793879i	$0.291942\pi$
$972$	0	0
$973$	−45.9090	−1.47178
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−2.83500	−0.0906997	−0.0453499	−	0.998971i	$-0.514440\pi$
$977$	−2.83500	−0.0906997	−0.0453499	+	0.998971i	$0.514440\pi$
$978$	0	0
$979$	−14.0537	−0.449158
$980$	0	0
$981$	6.71253	0.214314
$982$	0	0
$983$	4.47696	0.142793	0.0713964	−	0.997448i	$-0.477254\pi$
$983$	4.47696	0.142793	0.0713964	+	0.997448i	$0.477254\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	51.7068	1.64584
$988$	0	0
$989$	−0.177968	−0.00565906
$990$	0	0
$991$	32.2466	1.02435	0.512173	−	0.858882i	$-0.328841\pi$
$991$	32.2466	1.02435	0.512173	+	0.858882i	$0.328841\pi$
$992$	0	0
$993$	15.3777	0.487996
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−27.5708	−0.873177	−0.436589	−	0.899661i	$-0.643813\pi$
$997$	−27.5708	−0.873177	−0.436589	+	0.899661i	$0.643813\pi$
$998$	0	0
$999$	9.46784	0.299549

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9200.2.a.cz.1.3		7
4.3	odd	2		4600.2.a.bi.1.5		7
5.2	odd	4		1840.2.e.g.369.11		14
5.3	odd	4		1840.2.e.g.369.4		14
5.4	even	2		9200.2.a.dc.1.5		7
20.3	even	4		920.2.e.b.369.11	yes	14
20.7	even	4		920.2.e.b.369.4	✓	14
20.19	odd	2		4600.2.a.bh.1.3		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.e.b.369.4	✓	14	20.7	even	4
920.2.e.b.369.11	yes	14	20.3	even	4
1840.2.e.g.369.4		14	5.3	odd	4
1840.2.e.g.369.11		14	5.2	odd	4
4600.2.a.bh.1.3		7	20.19	odd	2
4600.2.a.bi.1.5		7	4.3	odd	2
9200.2.a.cz.1.3		7	1.1	even	1	trivial
9200.2.a.dc.1.5		7	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.58319 q^{3} -2.84620 q^{7} -0.493499 q^{9} +O(q^{10})\)	\(q-1.58319 q^{3} -2.84620 q^{7} -0.493499 q^{9} -1.98637 q^{11} -4.69204 q^{13} -3.16675 q^{17} +6.16110 q^{19} +4.50609 q^{21} +1.00000 q^{23} +5.53088 q^{27} -6.61816 q^{29} +8.29732 q^{31} +3.14481 q^{33} +1.71181 q^{37} +7.42840 q^{39} +6.72440 q^{41} -0.177968 q^{43} +11.4749 q^{47} +1.10086 q^{49} +5.01358 q^{51} -6.18762 q^{53} -9.75422 q^{57} +7.61559 q^{59} +11.8577 q^{61} +1.40460 q^{63} +3.07554 q^{67} -1.58319 q^{69} +1.96195 q^{71} -4.94732 q^{73} +5.65361 q^{77} -8.53182 q^{79} -7.27596 q^{81} +3.49429 q^{83} +10.4778 q^{87} +7.07506 q^{89} +13.3545 q^{91} -13.1363 q^{93} +7.00705 q^{97} +0.980272 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 3 q^{3} - 4 q^{7} + 2 q^{9}+O(q^{10}) \) 7 * q - 3 * q^3 - 4 * q^7 + 2 * q^9	\( 7 q - 3 q^{3} - 4 q^{7} + 2 q^{9} + 7 q^{11} - 7 q^{13} + 7 q^{19} - 6 q^{21} + 7 q^{23} - 11 q^{29} + 10 q^{31} - 19 q^{33} - 19 q^{37} + 24 q^{39} - 16 q^{41} - 6 q^{43} - 6 q^{47} - 17 q^{49} + 7 q^{51} - 15 q^{53} - 8 q^{57} + 11 q^{59} + 5 q^{61} - 13 q^{63} - 9 q^{67} - 3 q^{69} + 14 q^{71} - 10 q^{73} - 6 q^{77} + 32 q^{79} - 5 q^{81} - q^{83} - 10 q^{87} - 24 q^{89} + 7 q^{91} - 26 q^{93} + 7 q^{97} + 61 q^{99}+O(q^{100}) \) 7 * q - 3 * q^3 - 4 * q^7 + 2 * q^9 + 7 * q^11 - 7 * q^13 + 7 * q^19 - 6 * q^21 + 7 * q^23 - 11 * q^29 + 10 * q^31 - 19 * q^33 - 19 * q^37 + 24 * q^39 - 16 * q^41 - 6 * q^43 - 6 * q^47 - 17 * q^49 + 7 * q^51 - 15 * q^53 - 8 * q^57 + 11 * q^59 + 5 * q^61 - 13 * q^63 - 9 * q^67 - 3 * q^69 + 14 * q^71 - 10 * q^73 - 6 * q^77 + 32 * q^79 - 5 * q^81 - q^83 - 10 * q^87 - 24 * q^89 + 7 * q^91 - 26 * q^93 + 7 * q^97 + 61 * q^99

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