LMFDB - Embedded newform 800.4.f.c.49.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,4,Mod(49,800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("800.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	800.f (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$47.2015280046$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4x^{11} + 7x^{10} - 12x^{9} + 21x^{8} - 68x^{6} + 336x^{4} - 768x^{3} + 1792x^{2} - 4096x + 4096 $ x^12 - 4x^11 + 7x^10 - 12x^9 + 21x^8 - 68x^6 + 336x^4 - 768x^3 + 1792x^2 - 4096*x + 4096
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{32} $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.10
Root			$-1.86176 - 0.730647i$ of defining polynomial
Character	$\chi$	$=$	800.49
Dual form			800.4.f.c.49.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+6.25785 q^{3} +34.6280i q^{7} +12.1606 q^{9} +O(q^{10})$	$q+6.25785 q^{3} +34.6280i q^{7} +12.1606 q^{9} +7.91595i q^{11} -11.0346 q^{13} +57.6152i q^{17} -141.133i q^{19} +216.696i q^{21} +129.328i q^{23} -92.8625 q^{27} +89.1664i q^{29} +78.3307 q^{31} +49.5368i q^{33} -249.332 q^{37} -69.0530 q^{39} +91.8705 q^{41} -194.184 q^{43} +72.6149i q^{47} -856.096 q^{49} +360.547i q^{51} -456.782 q^{53} -883.187i q^{57} +341.098i q^{59} -217.067i q^{61} +421.098i q^{63} +529.237 q^{67} +809.314i q^{69} -381.540 q^{71} +876.902i q^{73} -274.113 q^{77} -203.950 q^{79} -909.456 q^{81} +996.654 q^{83} +557.989i q^{87} -172.456 q^{89} -382.107i q^{91} +490.182 q^{93} +1058.49i q^{97} +96.2629i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{3} + 108 q^{9}+O(q^{10}) $ 12 * q + 12 * q^3 + 108 * q^9	$ 12 q + 12 q^{3} + 108 q^{9} + 432 q^{27} + 264 q^{31} - 136 q^{37} + 600 q^{39} + 40 q^{41} - 1204 q^{43} - 1308 q^{49} - 1056 q^{53} - 2412 q^{67} + 1592 q^{71} - 824 q^{77} + 2016 q^{79} + 2508 q^{81} + 3556 q^{83} + 424 q^{89} - 2784 q^{93}+O(q^{100}) $ 12 * q + 12 * q^3 + 108 * q^9 + 432 * q^27 + 264 * q^31 - 136 * q^37 + 600 * q^39 + 40 * q^41 - 1204 * q^43 - 1308 * q^49 - 1056 * q^53 - 2412 * q^67 + 1592 * q^71 - 824 * q^77 + 2016 * q^79 + 2508 * q^81 + 3556 * q^83 + 424 * q^89 - 2784 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/800\mathbb{Z}\right)^\times$.

$n$	$101$	$351$	$577$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	6.25785	1.20432	0.602161	−	0.798374i	$-0.294306\pi$
$3$	6.25785	1.20432	0.602161	+	0.798374i	$0.294306\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	34.6280i	1.86973i	0.354998	+	0.934867i	$0.384482\pi$
$7$	34.6280i	1.86973i	−0.354998	+	0.934867i	$0.615518\pi$
$8$	0	0
$9$	12.1606	0.450394
$10$	0	0
$11$	7.91595i	0.216977i	0.994098	+	0.108489i	$0.0346011\pi$
$11$	7.91595i	0.216977i	−0.994098	+	0.108489i	$0.965399\pi$
$12$	0	0
$13$	−11.0346	−0.235420	−0.117710	−	0.993048i	$-0.537555\pi$
$13$	−11.0346	−0.235420	−0.117710	+	0.993048i	$0.537555\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	57.6152i	0.821985i	0.911639	+	0.410992i	$0.134818\pi$
$17$	57.6152i	0.821985i	−0.911639	+	0.410992i	$0.865182\pi$
$18$	0	0
$19$	− 141.133i	− 1.70411i	−0.523453	−	0.852055i	$-0.675356\pi$
$19$	− 141.133i	− 1.70411i	0.523453	−	0.852055i	$-0.324644\pi$
$20$	0	0
$21$	216.696i	2.25176i
$22$	0	0
$23$	129.328i	1.17247i	0.810143	+	0.586233i	$0.199390\pi$
$23$	129.328i	1.17247i	−0.810143	+	0.586233i	$0.800610\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−92.8625	−0.661903
$28$	0	0
$29$	89.1664i	0.570958i	0.958385	+	0.285479i	$0.0921527\pi$
$29$	89.1664i	0.570958i	−0.958385	+	0.285479i	$0.907847\pi$
$30$	0	0
$31$	78.3307	0.453826	0.226913	−	0.973915i	$-0.427137\pi$
$31$	78.3307	0.453826	0.226913	+	0.973915i	$0.427137\pi$
$32$	0	0
$33$	49.5368i	0.261310i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−249.332	−1.10783	−0.553917	−	0.832572i	$-0.686868\pi$
$37$	−249.332	−1.10783	−0.553917	+	0.832572i	$0.686868\pi$
$38$	0	0
$39$	−69.0530	−0.283521
$40$	0	0
$41$	91.8705	0.349946	0.174973	−	0.984573i	$-0.444016\pi$
$41$	91.8705	0.349946	0.174973	+	0.984573i	$0.444016\pi$
$42$	0	0
$43$	−194.184	−0.688668	−0.344334	−	0.938847i	$-0.611895\pi$
$43$	−194.184	−0.688668	−0.344334	+	0.938847i	$0.611895\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	72.6149i	0.225361i	0.993631	+	0.112680i	$0.0359437\pi$
$47$	72.6149i	0.225361i	−0.993631	+	0.112680i	$0.964056\pi$
$48$	0	0
$49$	−856.096	−2.49591
$50$	0	0
$51$	360.547i	0.989935i
$52$	0	0
$53$	−456.782	−1.18385	−0.591923	−	0.805995i	$-0.701631\pi$
$53$	−456.782	−1.18385	−0.591923	+	0.805995i	$0.701631\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 883.187i	− 2.05230i
$58$	0	0
$59$	341.098i	0.752664i	0.926485	+	0.376332i	$0.122815\pi$
$59$	341.098i	0.752664i	−0.926485	+	0.376332i	$0.877185\pi$
$60$	0	0
$61$	− 217.067i	− 0.455616i	−0.973706	−	0.227808i	$-0.926844\pi$
$61$	− 217.067i	− 0.455616i	0.973706	−	0.227808i	$-0.0731559\pi$
$62$	0	0
$63$	421.098i	0.842117i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	529.237	0.965024	0.482512	−	0.875889i	$-0.339724\pi$
$67$	529.237	0.965024	0.482512	+	0.875889i	$0.339724\pi$
$68$	0	0
$69$	809.314i	1.41203i
$70$	0	0
$71$	−381.540	−0.637754	−0.318877	−	0.947796i	$-0.603306\pi$
$71$	−381.540	−0.637754	−0.318877	+	0.947796i	$0.603306\pi$
$72$	0	0
$73$	876.902i	1.40594i	0.711220	+	0.702970i	$0.248143\pi$
$73$	876.902i	1.40594i	−0.711220	+	0.702970i	$0.751857\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−274.113	−0.405690
$78$	0	0
$79$	−203.950	−0.290458	−0.145229	−	0.989398i	$-0.546392\pi$
$79$	−203.950	−0.290458	−0.145229	+	0.989398i	$0.546392\pi$
$80$	0	0
$81$	−909.456	−1.24754
$82$	0	0
$83$	996.654	1.31804	0.659018	−	0.752127i	$-0.270972\pi$
$83$	996.654	1.31804	0.659018	+	0.752127i	$0.270972\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	557.989i	0.687618i
$88$	0	0
$89$	−172.456	−0.205396	−0.102698	−	0.994713i	$-0.532748\pi$
$89$	−172.456	−0.205396	−0.102698	+	0.994713i	$0.532748\pi$
$90$	0	0
$91$	− 382.107i	− 0.440172i
$92$	0	0
$93$	490.182	0.546553
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1058.49i	1.10797i	0.832527	+	0.553984i	$0.186893\pi$
$97$	1058.49i	1.10797i	−0.832527	+	0.553984i	$0.813107\pi$
$98$	0	0
$99$	96.2629i	0.0977251i
$100$	0	0
$101$	− 689.992i	− 0.679770i	−0.940467	−	0.339885i	$-0.889612\pi$
$101$	− 689.992i	− 0.679770i	0.940467	−	0.339885i	$-0.110388\pi$
$102$	0	0
$103$	1289.96i	1.23402i	0.786956	+	0.617009i	$0.211656\pi$
$103$	1289.96i	1.23402i	−0.786956	+	0.617009i	$0.788344\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1843.74	1.66580	0.832902	−	0.553420i	$-0.186678\pi$
$107$	1843.74	1.66580	0.832902	+	0.553420i	$0.186678\pi$
$108$	0	0
$109$	− 340.598i	− 0.299297i	−0.988739	−	0.149648i	$-0.952186\pi$
$109$	− 340.598i	− 0.299297i	0.988739	−	0.149648i	$-0.0478142\pi$
$110$	0	0
$111$	−1560.28	−1.33419
$112$	0	0
$113$	− 157.831i	− 0.131393i	−0.997840	−	0.0656967i	$-0.979073\pi$
$113$	− 157.831i	− 0.131393i	0.997840	−	0.0656967i	$-0.0209270\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−134.188	−0.106031
$118$	0	0
$119$	−1995.10	−1.53689
$120$	0	0
$121$	1268.34	0.952921
$122$	0	0
$123$	574.912	0.421447
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 494.704i	− 0.345652i	−0.984952	−	0.172826i	$-0.944710\pi$
$127$	− 494.704i	− 0.345652i	0.984952	−	0.172826i	$-0.0552899\pi$
$128$	0	0
$129$	−1215.17	−0.829379
$130$	0	0
$131$	488.783i	0.325993i	0.986627	+	0.162997i	$0.0521160\pi$
$131$	488.783i	0.325993i	−0.986627	+	0.162997i	$0.947884\pi$
$132$	0	0
$133$	4887.14	3.18623
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 1532.13i	− 0.955464i	−0.878506	−	0.477732i	$-0.841459\pi$
$137$	− 1532.13i	− 0.955464i	0.878506	−	0.477732i	$-0.158541\pi$
$138$	0	0
$139$	755.095i	0.460765i	0.973100	+	0.230382i	$0.0739977\pi$
$139$	755.095i	0.460765i	−0.973100	+	0.230382i	$0.926002\pi$
$140$	0	0
$141$	454.413i	0.271407i
$142$	0	0
$143$	− 87.3495i	− 0.0510807i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−5357.32	−3.00588
$148$	0	0
$149$	1446.67i	0.795410i	0.917513	+	0.397705i	$0.130193\pi$
$149$	1446.67i	0.795410i	−0.917513	+	0.397705i	$0.869807\pi$
$150$	0	0
$151$	1230.20	0.662998	0.331499	−	0.943456i	$-0.392446\pi$
$151$	1230.20	0.662998	0.331499	+	0.943456i	$0.392446\pi$
$152$	0	0
$153$	700.637i	0.370217i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	1773.74	0.901657	0.450828	−	0.892611i	$-0.351129\pi$
$157$	1773.74	0.901657	0.450828	+	0.892611i	$0.351129\pi$
$158$	0	0
$159$	−2858.47	−1.42573
$160$	0	0
$161$	−4478.36	−2.19220
$162$	0	0
$163$	1711.28	0.822319	0.411159	−	0.911563i	$-0.365124\pi$
$163$	1711.28	0.822319	0.411159	+	0.911563i	$0.365124\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 3913.26i	− 1.81328i	−0.421909	−	0.906638i	$-0.638640\pi$
$167$	− 3913.26i	− 1.81328i	0.421909	−	0.906638i	$-0.361360\pi$
$168$	0	0
$169$	−2075.24	−0.944578
$170$	0	0
$171$	− 1716.26i	− 0.767520i
$172$	0	0
$173$	−3985.34	−1.75144	−0.875722	−	0.482815i	$-0.839614\pi$
$173$	−3985.34	−1.75144	−0.875722	+	0.482815i	$0.839614\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2134.54i	0.906450i
$178$	0	0
$179$	884.734i	0.369431i	0.982792	+	0.184715i	$0.0591363\pi$
$179$	884.734i	0.369431i	−0.982792	+	0.184715i	$0.940864\pi$
$180$	0	0
$181$	2810.77i	1.15427i	0.816649	+	0.577134i	$0.195829\pi$
$181$	2810.77i	1.15427i	−0.816649	+	0.577134i	$0.804171\pi$
$182$	0	0
$183$	− 1358.37i	− 0.548709i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−456.079	−0.178352
$188$	0	0
$189$	− 3215.64i	− 1.23758i
$190$	0	0
$191$	749.321	0.283869	0.141935	−	0.989876i	$-0.454668\pi$
$191$	749.321	0.283869	0.141935	+	0.989876i	$0.454668\pi$
$192$	0	0
$193$	3969.83i	1.48060i	0.672279	+	0.740298i	$0.265315\pi$
$193$	3969.83i	1.48060i	−0.672279	+	0.740298i	$0.734685\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2243.20	0.811275	0.405638	−	0.914034i	$-0.367049\pi$
$197$	2243.20	0.811275	0.405638	+	0.914034i	$0.367049\pi$
$198$	0	0
$199$	−672.030	−0.239392	−0.119696	−	0.992811i	$-0.538192\pi$
$199$	−672.030	−0.239392	−0.119696	+	0.992811i	$0.538192\pi$
$200$	0	0
$201$	3311.89	1.16220
$202$	0	0
$203$	−3087.65	−1.06754
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1572.71i	0.528071i
$208$	0	0
$209$	1117.20	0.369753
$210$	0	0
$211$	1935.07i	0.631356i	0.948866	+	0.315678i	$0.102232\pi$
$211$	1935.07i	0.631356i	−0.948866	+	0.315678i	$0.897768\pi$
$212$	0	0
$213$	−2387.62	−0.768061
$214$	0	0
$215$	0	0
$216$	0	0
$217$	2712.43i	0.848535i
$218$	0	0
$219$	5487.52i	1.69321i
$220$	0	0
$221$	− 635.762i	− 0.193511i
$222$	0	0
$223$	2492.56i	0.748494i	0.927329	+	0.374247i	$0.122099\pi$
$223$	2492.56i	0.748494i	−0.927329	+	0.374247i	$0.877901\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3155.27	0.922567	0.461283	−	0.887253i	$-0.347389\pi$
$227$	3155.27	0.922567	0.461283	+	0.887253i	$0.347389\pi$
$228$	0	0
$229$	2299.43i	0.663539i	0.943360	+	0.331770i	$0.107646\pi$
$229$	2299.43i	0.663539i	−0.943360	+	0.331770i	$0.892354\pi$
$230$	0	0
$231$	−1715.36	−0.488581
$232$	0	0
$233$	− 741.991i	− 0.208624i	−0.994545	−	0.104312i	$-0.966736\pi$
$233$	− 741.991i	− 0.208624i	0.994545	−	0.104312i	$-0.0332641\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−1276.29	−0.349806
$238$	0	0
$239$	5786.01	1.56597	0.782984	−	0.622042i	$-0.213697\pi$
$239$	5786.01	1.56597	0.782984	+	0.622042i	$0.213697\pi$
$240$	0	0
$241$	265.054	0.0708449	0.0354224	−	0.999372i	$-0.488722\pi$
$241$	265.054	0.0708449	0.0354224	+	0.999372i	$0.488722\pi$
$242$	0	0
$243$	−3183.95	−0.840537
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1557.35i	0.401181i
$248$	0	0
$249$	6236.91	1.58734
$250$	0	0
$251$	− 1762.02i	− 0.443098i	−0.975149	−	0.221549i	$-0.928889\pi$
$251$	− 1762.02i	− 0.443098i	0.975149	−	0.221549i	$-0.0711113\pi$
$252$	0	0
$253$	−1023.75	−0.254398
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 1507.84i	− 0.365980i	−0.983115	−	0.182990i	$-0.941422\pi$
$257$	− 1507.84i	− 0.365980i	0.983115	−	0.182990i	$-0.0585775\pi$
$258$	0	0
$259$	− 8633.85i	− 2.07136i
$260$	0	0
$261$	1084.32i	0.257156i
$262$	0	0
$263$	2772.54i	0.650046i	0.945706	+	0.325023i	$0.105372\pi$
$263$	2772.54i	0.650046i	−0.945706	+	0.325023i	$0.894628\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−1079.20	−0.247364
$268$	0	0
$269$	− 5166.61i	− 1.17106i	−0.810652	−	0.585528i	$-0.800887\pi$
$269$	− 5166.61i	− 1.17106i	0.810652	−	0.585528i	$-0.199113\pi$
$270$	0	0
$271$	1458.79	0.326994	0.163497	−	0.986544i	$-0.447723\pi$
$271$	1458.79	0.326994	0.163497	+	0.986544i	$0.447723\pi$
$272$	0	0
$273$	− 2391.16i	− 0.530109i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−1994.60	−0.432650	−0.216325	−	0.976321i	$-0.569407\pi$
$277$	−1994.60	−0.432650	−0.216325	+	0.976321i	$0.569407\pi$
$278$	0	0
$279$	952.551	0.204400
$280$	0	0
$281$	−311.583	−0.0661477	−0.0330739	−	0.999453i	$-0.510530\pi$
$281$	−311.583	−0.0661477	−0.0330739	+	0.999453i	$0.510530\pi$
$282$	0	0
$283$	−6072.33	−1.27549	−0.637743	−	0.770249i	$-0.720132\pi$
$283$	−6072.33	−1.27549	−0.637743	+	0.770249i	$0.720132\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3181.29i	0.654305i
$288$	0	0
$289$	1593.49	0.324341
$290$	0	0
$291$	6623.84i	1.33435i
$292$	0	0
$293$	2321.06	0.462791	0.231395	−	0.972860i	$-0.425671\pi$
$293$	2321.06	0.462791	0.231395	+	0.972860i	$0.425671\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 735.095i	− 0.143618i
$298$	0	0
$299$	− 1427.08i	− 0.276021i
$300$	0	0
$301$	− 6724.19i	− 1.28763i
$302$	0	0
$303$	− 4317.86i	− 0.818662i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	2499.43	0.464658	0.232329	−	0.972637i	$-0.425365\pi$
$307$	2499.43	0.464658	0.232329	+	0.972637i	$0.425365\pi$
$308$	0	0
$309$	8072.39i	1.48616i
$310$	0	0
$311$	3052.83	0.556624	0.278312	−	0.960491i	$-0.410225\pi$
$311$	3052.83	0.556624	0.278312	+	0.960491i	$0.410225\pi$
$312$	0	0
$313$	− 6179.23i	− 1.11588i	−0.829881	−	0.557941i	$-0.811592\pi$
$313$	− 6179.23i	− 1.11588i	0.829881	−	0.557941i	$-0.188408\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3116.20	−0.552124	−0.276062	−	0.961140i	$-0.589029\pi$
$317$	−3116.20	−0.552124	−0.276062	+	0.961140i	$0.589029\pi$
$318$	0	0
$319$	−705.836	−0.123885
$320$	0	0
$321$	11537.8	2.00617
$322$	0	0
$323$	8131.39	1.40075
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 2131.41i	− 0.360450i
$328$	0	0
$329$	−2514.51	−0.421365
$330$	0	0
$331$	4157.26i	0.690343i	0.938540	+	0.345171i	$0.112179\pi$
$331$	4157.26i	0.690343i	−0.938540	+	0.345171i	$0.887821\pi$
$332$	0	0
$333$	−3032.03	−0.498962
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 8123.42i	− 1.31309i	−0.754287	−	0.656544i	$-0.772017\pi$
$337$	− 8123.42i	− 1.31309i	0.754287	−	0.656544i	$-0.227983\pi$
$338$	0	0
$339$	− 987.679i	− 0.158240i
$340$	0	0
$341$	620.062i	0.0984699i
$342$	0	0
$343$	− 17767.5i	− 2.79695i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4620.55	0.714824	0.357412	−	0.933947i	$-0.383659\pi$
$347$	4620.55	0.714824	0.357412	+	0.933947i	$0.383659\pi$
$348$	0	0
$349$	− 5560.98i	− 0.852929i	−0.904504	−	0.426465i	$-0.859759\pi$
$349$	− 5560.98i	− 0.852929i	0.904504	−	0.426465i	$-0.140241\pi$
$350$	0	0
$351$	1024.70	0.155825
$352$	0	0
$353$	− 12031.8i	− 1.81413i	−0.420989	−	0.907066i	$-0.638317\pi$
$353$	− 12031.8i	− 1.81413i	0.420989	−	0.907066i	$-0.361683\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−12485.0	−1.85092
$358$	0	0
$359$	1533.24	0.225408	0.112704	−	0.993629i	$-0.464049\pi$
$359$	1533.24	0.225408	0.112704	+	0.993629i	$0.464049\pi$
$360$	0	0
$361$	−13059.5	−1.90399
$362$	0	0
$363$	7937.06	1.14762
$364$	0	0
$365$	0	0
$366$	0	0
$367$	5480.04i	0.779444i	0.920933	+	0.389722i	$0.127429\pi$
$367$	5480.04i	0.779444i	−0.920933	+	0.389722i	$0.872571\pi$
$368$	0	0
$369$	1117.20	0.157613
$370$	0	0
$371$	− 15817.4i	− 2.21348i
$372$	0	0
$373$	6225.70	0.864221	0.432111	−	0.901821i	$-0.357769\pi$
$373$	6225.70	0.864221	0.432111	+	0.901821i	$0.357769\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 983.917i	− 0.134415i
$378$	0	0
$379$	− 11172.0i	− 1.51416i	−0.653325	−	0.757078i	$-0.726626\pi$
$379$	− 11172.0i	− 1.51416i	0.653325	−	0.757078i	$-0.273374\pi$
$380$	0	0
$381$	− 3095.78i	− 0.416277i
$382$	0	0
$383$	7621.03i	1.01675i	0.861135	+	0.508376i	$0.169754\pi$
$383$	7621.03i	1.01675i	−0.861135	+	0.508376i	$0.830246\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−2361.40	−0.310172
$388$	0	0
$389$	− 5546.31i	− 0.722903i	−0.932391	−	0.361451i	$-0.882281\pi$
$389$	− 5546.31i	− 0.722903i	0.932391	−	0.361451i	$-0.117719\pi$
$390$	0	0
$391$	−7451.25	−0.963749
$392$	0	0
$393$	3058.73i	0.392601i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	11025.2	1.39380	0.696900	−	0.717169i	$-0.254562\pi$
$397$	11025.2	1.39380	0.696900	+	0.717169i	$0.254562\pi$
$398$	0	0
$399$	30583.0	3.83725
$400$	0	0
$401$	10522.3	1.31037	0.655186	−	0.755467i	$-0.272590\pi$
$401$	10522.3	1.31037	0.655186	+	0.755467i	$0.272590\pi$
$402$	0	0
$403$	−864.350	−0.106840
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 1973.70i	− 0.240375i
$408$	0	0
$409$	−2320.17	−0.280502	−0.140251	−	0.990116i	$-0.544791\pi$
$409$	−2320.17	−0.280502	−0.140251	+	0.990116i	$0.544791\pi$
$410$	0	0
$411$	− 9587.82i	− 1.15069i
$412$	0	0
$413$	−11811.5	−1.40728
$414$	0	0
$415$	0	0
$416$	0	0
$417$	4725.27i	0.554910i
$418$	0	0
$419$	9212.17i	1.07409i	0.843554	+	0.537045i	$0.180459\pi$
$419$	9212.17i	1.07409i	−0.843554	+	0.537045i	$0.819541\pi$
$420$	0	0
$421$	6967.70i	0.806615i	0.915064	+	0.403308i	$0.132140\pi$
$421$	6967.70i	0.806615i	−0.915064	+	0.403308i	$0.867860\pi$
$422$	0	0
$423$	883.042i	0.101501i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	7516.59	0.851882
$428$	0	0
$429$	− 546.620i	− 0.0615176i
$430$	0	0
$431$	11247.1	1.25697	0.628486	−	0.777821i	$-0.283675\pi$
$431$	11247.1	1.25697	0.628486	+	0.777821i	$0.283675\pi$
$432$	0	0
$433$	− 2589.27i	− 0.287372i	−0.989623	−	0.143686i	$-0.954104\pi$
$433$	− 2589.27i	− 0.287372i	0.989623	−	0.143686i	$-0.0458956\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	18252.4	1.99801
$438$	0	0
$439$	4220.01	0.458793	0.229396	−	0.973333i	$-0.426325\pi$
$439$	4220.01	0.458793	0.229396	+	0.973333i	$0.426325\pi$
$440$	0	0
$441$	−10410.7	−1.12414
$442$	0	0
$443$	9764.04	1.04719	0.523593	−	0.851969i	$-0.324591\pi$
$443$	9764.04	1.04719	0.523593	+	0.851969i	$0.324591\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	9053.06i	0.957931i
$448$	0	0
$449$	17159.3	1.80355	0.901777	−	0.432202i	$-0.142263\pi$
$449$	17159.3	1.80355	0.901777	+	0.432202i	$0.142263\pi$
$450$	0	0
$451$	727.242i	0.0759302i
$452$	0	0
$453$	7698.43	0.798463
$454$	0	0
$455$	0	0
$456$	0	0
$457$	13027.3i	1.33346i	0.745299	+	0.666730i	$0.232307\pi$
$457$	13027.3i	1.33346i	−0.745299	+	0.666730i	$0.767693\pi$
$458$	0	0
$459$	− 5350.29i	− 0.544075i
$460$	0	0
$461$	− 313.396i	− 0.0316623i	−0.999875	−	0.0158312i	$-0.994961\pi$
$461$	− 313.396i	− 0.0316623i	0.999875	−	0.0158312i	$-0.00503943\pi$
$462$	0	0
$463$	− 12166.5i	− 1.22123i	−0.791930	−	0.610613i	$-0.790923\pi$
$463$	− 12166.5i	− 1.22123i	0.791930	−	0.610613i	$-0.209077\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1844.42	−0.182761	−0.0913806	−	0.995816i	$-0.529128\pi$
$467$	−1844.42	−0.182761	−0.0913806	+	0.995816i	$0.529128\pi$
$468$	0	0
$469$	18326.4i	1.80434i
$470$	0	0
$471$	11099.8	1.08589
$472$	0	0
$473$	− 1537.15i	− 0.149425i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−5554.75	−0.533196
$478$	0	0
$479$	−16355.9	−1.56017	−0.780084	−	0.625675i	$-0.784824\pi$
$479$	−16355.9	−1.56017	−0.780084	+	0.625675i	$0.784824\pi$
$480$	0	0
$481$	2751.28	0.260806
$482$	0	0
$483$	−28024.9	−2.64012
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 11824.6i	− 1.10025i	−0.835082	−	0.550126i	$-0.814580\pi$
$487$	− 11824.6i	− 1.10025i	0.835082	−	0.550126i	$-0.185420\pi$
$488$	0	0
$489$	10708.9	0.990337
$490$	0	0
$491$	7931.40i	0.729000i	0.931203	+	0.364500i	$0.118760\pi$
$491$	7931.40i	0.729000i	−0.931203	+	0.364500i	$0.881240\pi$
$492$	0	0
$493$	−5137.34	−0.469319
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 13212.0i	− 1.19243i
$498$	0	0
$499$	2734.69i	0.245333i	0.992448	+	0.122667i	$0.0391446\pi$
$499$	2734.69i	0.245333i	−0.992448	+	0.122667i	$0.960855\pi$
$500$	0	0
$501$	− 24488.6i	− 2.18377i
$502$	0	0
$503$	7297.64i	0.646890i	0.946247	+	0.323445i	$0.104841\pi$
$503$	7297.64i	0.646890i	−0.946247	+	0.323445i	$0.895159\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−12986.5	−1.13758
$508$	0	0
$509$	− 10972.4i	− 0.955486i	−0.878500	−	0.477743i	$-0.841455\pi$
$509$	− 10972.4i	− 0.955486i	0.878500	−	0.477743i	$-0.158545\pi$
$510$	0	0
$511$	−30365.3	−2.62873
$512$	0	0
$513$	13105.9i	1.12796i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−574.815	−0.0488982
$518$	0	0
$519$	−24939.7	−2.10931
$520$	0	0
$521$	7693.43	0.646939	0.323470	−	0.946239i	$-0.395151\pi$
$521$	7693.43	0.646939	0.323470	+	0.946239i	$0.395151\pi$
$522$	0	0
$523$	−14535.3	−1.21527	−0.607634	−	0.794217i	$-0.707881\pi$
$523$	−14535.3	−1.21527	−0.607634	+	0.794217i	$0.707881\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4513.04i	0.373038i
$528$	0	0
$529$	−4558.69	−0.374676
$530$	0	0
$531$	4147.97i	0.338995i
$532$	0	0
$533$	−1013.76	−0.0823840
$534$	0	0
$535$	0	0
$536$	0	0
$537$	5536.53i	0.444914i
$538$	0	0
$539$	− 6776.81i	− 0.541555i
$540$	0	0
$541$	19131.8i	1.52040i	0.649686	+	0.760202i	$0.274900\pi$
$541$	19131.8i	1.52040i	−0.649686	+	0.760202i	$0.725100\pi$
$542$	0	0
$543$	17589.3i	1.39011i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	7142.78	0.558324	0.279162	−	0.960244i	$-0.409943\pi$
$547$	7142.78	0.558324	0.279162	+	0.960244i	$0.409943\pi$
$548$	0	0
$549$	− 2639.67i	− 0.205207i
$550$	0	0
$551$	12584.3	0.972974
$552$	0	0
$553$	− 7062.39i	− 0.543080i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−11347.8	−0.863236	−0.431618	−	0.902057i	$-0.642057\pi$
$557$	−11347.8	−0.863236	−0.431618	+	0.902057i	$0.642057\pi$
$558$	0	0
$559$	2142.74	0.162126
$560$	0	0
$561$	−2854.07	−0.214793
$562$	0	0
$563$	8802.19	0.658913	0.329456	−	0.944171i	$-0.393135\pi$
$563$	8802.19	0.658913	0.329456	+	0.944171i	$0.393135\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 31492.6i	− 2.33257i
$568$	0	0
$569$	−16714.7	−1.23149	−0.615744	−	0.787946i	$-0.711145\pi$
$569$	−16714.7	−1.23149	−0.615744	+	0.787946i	$0.711145\pi$
$570$	0	0
$571$	17386.8i	1.27428i	0.770746	+	0.637142i	$0.219884\pi$
$571$	17386.8i	1.27428i	−0.770746	+	0.637142i	$0.780116\pi$
$572$	0	0
$573$	4689.14	0.341870
$574$	0	0
$575$	0	0
$576$	0	0
$577$	2260.52i	0.163097i	0.996669	+	0.0815483i	$0.0259865\pi$
$577$	2260.52i	0.163097i	−0.996669	+	0.0815483i	$0.974014\pi$
$578$	0	0
$579$	24842.6i	1.78311i
$580$	0	0
$581$	34512.1i	2.46438i
$582$	0	0
$583$	− 3615.86i	− 0.256867i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−25591.0	−1.79941	−0.899705	−	0.436498i	$-0.856219\pi$
$587$	−25591.0	−1.79941	−0.899705	+	0.436498i	$0.856219\pi$
$588$	0	0
$589$	− 11055.0i	− 0.773369i
$590$	0	0
$591$	14037.6	0.977037
$592$	0	0
$593$	10053.7i	0.696218i	0.937454	+	0.348109i	$0.113176\pi$
$593$	10053.7i	0.696218i	−0.937454	+	0.348109i	$0.886824\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−4205.46	−0.288305
$598$	0	0
$599$	−7086.68	−0.483395	−0.241698	−	0.970352i	$-0.577704\pi$
$599$	−7086.68	−0.483395	−0.241698	+	0.970352i	$0.577704\pi$
$600$	0	0
$601$	6673.11	0.452915	0.226458	−	0.974021i	$-0.427286\pi$
$601$	6673.11	0.452915	0.226458	+	0.974021i	$0.427286\pi$
$602$	0	0
$603$	6435.86	0.434641
$604$	0	0
$605$	0	0
$606$	0	0
$607$	15205.6i	1.01677i	0.861131	+	0.508383i	$0.169757\pi$
$607$	15205.6i	1.01677i	−0.861131	+	0.508383i	$0.830243\pi$
$608$	0	0
$609$	−19322.0	−1.28566
$610$	0	0
$611$	− 801.278i	− 0.0530544i
$612$	0	0
$613$	−8147.64	−0.536836	−0.268418	−	0.963303i	$-0.586501\pi$
$613$	−8147.64	−0.536836	−0.268418	+	0.963303i	$0.586501\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	449.110i	0.0293039i	0.999893	+	0.0146519i	$0.00466402\pi$
$617$	449.110i	0.0293039i	−0.999893	+	0.0146519i	$0.995336\pi$
$618$	0	0
$619$	27800.4i	1.80516i	0.430524	+	0.902579i	$0.358329\pi$
$619$	27800.4i	1.80516i	−0.430524	+	0.902579i	$0.641671\pi$
$620$	0	0
$621$	− 12009.7i	− 0.776059i
$622$	0	0
$623$	− 5971.79i	− 0.384037i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	6991.26	0.445302
$628$	0	0
$629$	− 14365.3i	− 0.910623i
$630$	0	0
$631$	−14793.5	−0.933315	−0.466657	−	0.884438i	$-0.654542\pi$
$631$	−14793.5	−0.933315	−0.466657	+	0.884438i	$0.654542\pi$
$632$	0	0
$633$	12109.4i	0.760356i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	9446.70	0.587585
$638$	0	0
$639$	−4639.77	−0.287240
$640$	0	0
$641$	−13853.8	−0.853655	−0.426828	−	0.904333i	$-0.640369\pi$
$641$	−13853.8	−0.853655	−0.426828	+	0.904333i	$0.640369\pi$
$642$	0	0
$643$	4978.60	0.305345	0.152672	−	0.988277i	$-0.451212\pi$
$643$	4978.60	0.305345	0.152672	+	0.988277i	$0.451212\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1903.39i	0.115657i	0.998327	+	0.0578284i	$0.0184176\pi$
$647$	1903.39i	0.115657i	−0.998327	+	0.0578284i	$0.981582\pi$
$648$	0	0
$649$	−2700.11	−0.163311
$650$	0	0
$651$	16974.0i	1.02191i
$652$	0	0
$653$	10877.0	0.651839	0.325920	−	0.945397i	$-0.394326\pi$
$653$	10877.0	0.651839	0.325920	+	0.945397i	$0.394326\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	10663.7i	0.633226i
$658$	0	0
$659$	− 30611.3i	− 1.80948i	−0.425965	−	0.904739i	$-0.640065\pi$
$659$	− 30611.3i	− 1.80948i	0.425965	−	0.904739i	$-0.359935\pi$
$660$	0	0
$661$	− 16098.8i	− 0.947310i	−0.880710	−	0.473655i	$-0.842934\pi$
$661$	− 16098.8i	− 0.947310i	0.880710	−	0.473655i	$-0.157066\pi$
$662$	0	0
$663$	− 3978.50i	− 0.233050i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−11531.7	−0.669429
$668$	0	0
$669$	15598.0i	0.901428i
$670$	0	0
$671$	1718.29	0.0988583
$672$	0	0
$673$	23837.8i	1.36535i	0.730724	+	0.682673i	$0.239183\pi$
$673$	23837.8i	1.36535i	−0.730724	+	0.682673i	$0.760817\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−25235.3	−1.43260	−0.716302	−	0.697791i	$-0.754167\pi$
$677$	−25235.3	−1.43260	−0.716302	+	0.697791i	$0.754167\pi$
$678$	0	0
$679$	−36653.2	−2.07161
$680$	0	0
$681$	19745.2	1.11107
$682$	0	0
$683$	19975.3	1.11909	0.559543	−	0.828802i	$-0.310977\pi$
$683$	19975.3	1.11909	0.559543	+	0.828802i	$0.310977\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	14389.5i	0.799115i
$688$	0	0
$689$	5040.42	0.278700
$690$	0	0
$691$	− 13772.3i	− 0.758209i	−0.925354	−	0.379104i	$-0.876232\pi$
$691$	− 13772.3i	− 0.758209i	0.925354	−	0.379104i	$-0.123768\pi$
$692$	0	0
$693$	−3333.39	−0.182720
$694$	0	0
$695$	0	0
$696$	0	0
$697$	5293.14i	0.287650i
$698$	0	0
$699$	− 4643.26i	− 0.251251i
$700$	0	0
$701$	− 6347.90i	− 0.342021i	−0.985269	−	0.171011i	$-0.945297\pi$
$701$	− 6347.90i	− 0.342021i	0.985269	−	0.171011i	$-0.0547032\pi$
$702$	0	0
$703$	35188.9i	1.88787i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	23893.0	1.27099
$708$	0	0
$709$	− 17910.5i	− 0.948719i	−0.880331	−	0.474360i	$-0.842680\pi$
$709$	− 17910.5i	− 0.948719i	0.880331	−	0.474360i	$-0.157320\pi$
$710$	0	0
$711$	−2480.16	−0.130821
$712$	0	0
$713$	10130.3i	0.532096i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	36208.0	1.88593
$718$	0	0
$719$	36601.6	1.89849	0.949243	−	0.314545i	$-0.101852\pi$
$719$	36601.6	1.89849	0.949243	+	0.314545i	$0.101852\pi$
$720$	0	0
$721$	−44668.8	−2.30728
$722$	0	0
$723$	1658.67	0.0853201
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 2644.18i	− 0.134893i	−0.997723	−	0.0674466i	$-0.978515\pi$
$727$	− 2644.18i	− 0.134893i	0.997723	−	0.0674466i	$-0.0214852\pi$
$728$	0	0
$729$	4630.66	0.235262
$730$	0	0
$731$	− 11187.9i	− 0.566075i
$732$	0	0
$733$	33452.1	1.68565	0.842825	−	0.538188i	$-0.180891\pi$
$733$	33452.1	1.68565	0.842825	+	0.538188i	$0.180891\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4189.42i	0.209388i
$738$	0	0
$739$	25834.9i	1.28600i	0.765867	+	0.642999i	$0.222310\pi$
$739$	25834.9i	1.28600i	−0.765867	+	0.642999i	$0.777690\pi$
$740$	0	0
$741$	9745.64i	0.483151i
$742$	0	0
$743$	7625.09i	0.376497i	0.982121	+	0.188249i	$0.0602811\pi$
$743$	7625.09i	0.376497i	−0.982121	+	0.188249i	$0.939719\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	12119.9	0.593635
$748$	0	0
$749$	63845.0i	3.11461i
$750$	0	0
$751$	−25362.7	−1.23235	−0.616177	−	0.787607i	$-0.711320\pi$
$751$	−25362.7	−1.23235	−0.616177	+	0.787607i	$0.711320\pi$
$752$	0	0
$753$	− 11026.4i	− 0.533633i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	38948.0	1.87000	0.935000	−	0.354648i	$-0.115399\pi$
$757$	38948.0	1.87000	0.935000	+	0.354648i	$0.115399\pi$
$758$	0	0
$759$	−6406.48	−0.306378
$760$	0	0
$761$	−13722.2	−0.653654	−0.326827	−	0.945084i	$-0.605979\pi$
$761$	−13722.2	−0.653654	−0.326827	+	0.945084i	$0.605979\pi$
$762$	0	0
$763$	11794.2	0.559606
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 3763.89i	− 0.177192i
$768$	0	0
$769$	−18689.5	−0.876414	−0.438207	−	0.898874i	$-0.644386\pi$
$769$	−18689.5	−0.876414	−0.438207	+	0.898874i	$0.644386\pi$
$770$	0	0
$771$	− 9435.86i	− 0.440758i
$772$	0	0
$773$	6386.49	0.297162	0.148581	−	0.988900i	$-0.452529\pi$
$773$	6386.49	0.297162	0.148581	+	0.988900i	$0.452529\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 54029.3i	− 2.49458i
$778$	0	0
$779$	− 12965.9i	− 0.596345i
$780$	0	0
$781$	− 3020.25i	− 0.138378i
$782$	0	0
$783$	− 8280.21i	− 0.377919i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−19410.3	−0.879167	−0.439583	−	0.898202i	$-0.644874\pi$
$787$	−19410.3	−0.879167	−0.439583	+	0.898202i	$0.644874\pi$
$788$	0	0
$789$	17350.1i	0.782865i
$790$	0	0
$791$	5465.35	0.245671
$792$	0	0
$793$	2395.25i	0.107261i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−30740.8	−1.36624	−0.683121	−	0.730305i	$-0.739378\pi$
$797$	−30740.8	−1.36624	−0.683121	+	0.730305i	$0.739378\pi$
$798$	0	0
$799$	−4183.72	−0.185243
$800$	0	0
$801$	−2097.17	−0.0925092
$802$	0	0
$803$	−6941.51	−0.305057
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 32331.9i	− 1.41033i
$808$	0	0
$809$	−31805.5	−1.38223	−0.691114	−	0.722746i	$-0.742880\pi$
$809$	−31805.5	−1.38223	−0.691114	+	0.722746i	$0.742880\pi$
$810$	0	0
$811$	− 31014.7i	− 1.34288i	−0.741060	−	0.671439i	$-0.765677\pi$
$811$	− 31014.7i	− 1.34288i	0.741060	−	0.671439i	$-0.234323\pi$
$812$	0	0
$813$	9128.90	0.393806
$814$	0	0
$815$	0	0
$816$	0	0
$817$	27405.7i	1.17357i
$818$	0	0
$819$	− 4646.66i	− 0.198251i
$820$	0	0
$821$	− 12021.4i	− 0.511021i	−0.966806	−	0.255511i	$-0.917756\pi$
$821$	− 12021.4i	− 0.511021i	0.966806	−	0.255511i	$-0.0822436\pi$
$822$	0	0
$823$	15489.0i	0.656032i	0.944672	+	0.328016i	$0.106380\pi$
$823$	15489.0i	0.656032i	−0.944672	+	0.328016i	$0.893620\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	17097.2	0.718898	0.359449	−	0.933165i	$-0.382965\pi$
$827$	17097.2	0.718898	0.359449	+	0.933165i	$0.382965\pi$
$828$	0	0
$829$	9885.97i	0.414178i	0.978322	+	0.207089i	$0.0663990\pi$
$829$	9885.97i	0.414178i	−0.978322	+	0.207089i	$0.933601\pi$
$830$	0	0
$831$	−12481.9	−0.521050
$832$	0	0
$833$	− 49324.2i	− 2.05160i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−7273.99	−0.300389
$838$	0	0
$839$	−39204.3	−1.61321	−0.806606	−	0.591090i	$-0.798698\pi$
$839$	−39204.3	−1.61321	−0.806606	+	0.591090i	$0.798698\pi$
$840$	0	0
$841$	16438.4	0.674007
$842$	0	0
$843$	−1949.84	−0.0796632
$844$	0	0
$845$	0	0
$846$	0	0
$847$	43920.0i	1.78171i
$848$	0	0
$849$	−37999.7	−1.53610
$850$	0	0
$851$	− 32245.5i	− 1.29890i
$852$	0	0
$853$	−45054.1	−1.80847	−0.904234	−	0.427038i	$-0.859557\pi$
$853$	−45054.1	−1.80847	−0.904234	+	0.427038i	$0.859557\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 48464.2i	− 1.93174i	−0.259021	−	0.965872i	$-0.583400\pi$
$857$	− 48464.2i	− 1.93174i	0.259021	−	0.965872i	$-0.416600\pi$
$858$	0	0
$859$	1000.76i	0.0397503i	0.999802	+	0.0198751i	$0.00632687\pi$
$859$	1000.76i	0.0397503i	−0.999802	+	0.0198751i	$0.993673\pi$
$860$	0	0
$861$	19908.0i	0.787995i
$862$	0	0
$863$	22485.5i	0.886922i	0.896294	+	0.443461i	$0.146250\pi$
$863$	22485.5i	0.886922i	−0.896294	+	0.443461i	$0.853750\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	9971.79	0.390611
$868$	0	0
$869$	− 1614.46i	− 0.0630228i
$870$	0	0
$871$	−5839.94	−0.227186
$872$	0	0
$873$	12871.9i	0.499022i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−43726.2	−1.68361	−0.841806	−	0.539780i	$-0.818507\pi$
$877$	−43726.2	−1.68361	−0.841806	+	0.539780i	$0.818507\pi$
$878$	0	0
$879$	14524.8	0.557349
$880$	0	0
$881$	20290.0	0.775923	0.387962	−	0.921675i	$-0.373179\pi$
$881$	20290.0	0.775923	0.387962	+	0.921675i	$0.373179\pi$
$882$	0	0
$883$	14887.3	0.567382	0.283691	−	0.958916i	$-0.408441\pi$
$883$	14887.3	0.567382	0.283691	+	0.958916i	$0.408441\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 34359.3i	− 1.30065i	−0.759658	−	0.650323i	$-0.774633\pi$
$887$	− 34359.3i	− 1.30065i	0.759658	−	0.650323i	$-0.225367\pi$
$888$	0	0
$889$	17130.6	0.646278
$890$	0	0
$891$	− 7199.21i	− 0.270687i
$892$	0	0
$893$	10248.3	0.384040
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 8930.47i	− 0.332419i
$898$	0	0
$899$	6984.47i	0.259116i
$900$	0	0
$901$	− 26317.6i	− 0.973103i
$902$	0	0
$903$	− 42078.9i	− 1.55072i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	53363.9	1.95360	0.976802	−	0.214144i	$-0.0686963\pi$
$907$	53363.9	1.95360	0.976802	+	0.214144i	$0.0686963\pi$
$908$	0	0
$909$	− 8390.73i	− 0.306164i
$910$	0	0
$911$	25213.9	0.916984	0.458492	−	0.888699i	$-0.348390\pi$
$911$	25213.9	0.916984	0.458492	+	0.888699i	$0.348390\pi$
$912$	0	0
$913$	7889.46i	0.285984i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−16925.5	−0.609521
$918$	0	0
$919$	3747.02	0.134497	0.0672485	−	0.997736i	$-0.478578\pi$
$919$	3747.02	0.134497	0.0672485	+	0.997736i	$0.478578\pi$
$920$	0	0
$921$	15641.1	0.559599
$922$	0	0
$923$	4210.15	0.150140
$924$	0	0
$925$	0	0
$926$	0	0
$927$	15686.8i	0.555794i
$928$	0	0
$929$	27726.7	0.979209	0.489604	−	0.871945i	$-0.337141\pi$
$929$	27726.7	0.979209	0.489604	+	0.871945i	$0.337141\pi$
$930$	0	0
$931$	120823.i	4.25330i
$932$	0	0
$933$	19104.1	0.670355
$934$	0	0
$935$	0	0
$936$	0	0
$937$	29333.2i	1.02270i	0.859371	+	0.511352i	$0.170855\pi$
$937$	29333.2i	1.02270i	−0.859371	+	0.511352i	$0.829145\pi$
$938$	0	0
$939$	− 38668.7i	− 1.34388i
$940$	0	0
$941$	24218.4i	0.838998i	0.907756	+	0.419499i	$0.137794\pi$
$941$	24218.4i	0.838998i	−0.907756	+	0.419499i	$0.862206\pi$
$942$	0	0
$943$	11881.4i	0.410299i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−50824.6	−1.74401	−0.872005	−	0.489497i	$-0.837180\pi$
$947$	−50824.6	−1.74401	−0.872005	+	0.489497i	$0.837180\pi$
$948$	0	0
$949$	− 9676.28i	− 0.330986i
$950$	0	0
$951$	−19500.7	−0.664935
$952$	0	0
$953$	10155.1i	0.345180i	0.984994	+	0.172590i	$0.0552136\pi$
$953$	10155.1i	0.345180i	−0.984994	+	0.172590i	$0.944786\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−4417.01	−0.149197
$958$	0	0
$959$	53054.5	1.78646
$960$	0	0
$961$	−23655.3	−0.794042
$962$	0	0
$963$	22421.0	0.750268
$964$	0	0
$965$	0	0
$966$	0	0
$967$	28225.7i	0.938652i	0.883025	+	0.469326i	$0.155503\pi$
$967$	28225.7i	0.938652i	−0.883025	+	0.469326i	$0.844497\pi$
$968$	0	0
$969$	50885.0	1.68696
$970$	0	0
$971$	47631.4i	1.57422i	0.616814	+	0.787109i	$0.288423\pi$
$971$	47631.4i	1.57422i	−0.616814	+	0.787109i	$0.711577\pi$
$972$	0	0
$973$	−26147.4	−0.861508
$974$	0	0
$975$	0	0
$976$	0	0
$977$	13542.4i	0.443460i	0.975108	+	0.221730i	$0.0711703\pi$
$977$	13542.4i	0.443460i	−0.975108	+	0.221730i	$0.928830\pi$
$978$	0	0
$979$	− 1365.15i	− 0.0445663i
$980$	0	0
$981$	− 4141.88i	− 0.134801i
$982$	0	0
$983$	− 35108.5i	− 1.13915i	−0.821938	−	0.569576i	$-0.807107\pi$
$983$	− 35108.5i	− 1.13915i	0.821938	−	0.569576i	$-0.192893\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−15735.4	−0.507460
$988$	0	0
$989$	− 25113.4i	− 0.807440i
$990$	0	0
$991$	42816.1	1.37245	0.686225	−	0.727390i	$-0.259267\pi$
$991$	42816.1	1.37245	0.686225	+	0.727390i	$0.259267\pi$
$992$	0	0
$993$	26015.5i	0.831396i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	13029.0	0.413873	0.206936	−	0.978354i	$-0.433651\pi$
$997$	13029.0	0.413873	0.206936	+	0.978354i	$0.433651\pi$
$998$	0	0
$999$	23153.6	0.733280

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.4.f.c.49.10		12
4.3	odd	2		200.4.f.b.149.9		12
5.2	odd	4		800.4.d.d.401.3		12
5.3	odd	4		160.4.d.a.81.10		12
5.4	even	2		800.4.f.b.49.3		12
8.3	odd	2		200.4.f.c.149.3		12
8.5	even	2		800.4.f.b.49.4		12
15.8	even	4		1440.4.k.c.721.12		12
20.3	even	4		40.4.d.a.21.1	✓	12
20.7	even	4		200.4.d.b.101.12		12
20.19	odd	2		200.4.f.c.149.4		12
40.3	even	4		40.4.d.a.21.2	yes	12
40.13	odd	4		160.4.d.a.81.3		12
40.19	odd	2		200.4.f.b.149.10		12
40.27	even	4		200.4.d.b.101.11		12
40.29	even	2	inner	800.4.f.c.49.9		12
40.37	odd	4		800.4.d.d.401.10		12
60.23	odd	4		360.4.k.c.181.12		12
80.3	even	4		1280.4.a.bc.1.5		6
80.13	odd	4		1280.4.a.ba.1.2		6
80.43	even	4		1280.4.a.bb.1.2		6
80.53	odd	4		1280.4.a.bd.1.5		6
120.53	even	4		1440.4.k.c.721.6		12
120.83	odd	4		360.4.k.c.181.11		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.4.d.a.21.1	✓	12	20.3	even	4
40.4.d.a.21.2	yes	12	40.3	even	4
160.4.d.a.81.3		12	40.13	odd	4
160.4.d.a.81.10		12	5.3	odd	4
200.4.d.b.101.11		12	40.27	even	4
200.4.d.b.101.12		12	20.7	even	4
200.4.f.b.149.9		12	4.3	odd	2
200.4.f.b.149.10		12	40.19	odd	2
200.4.f.c.149.3		12	8.3	odd	2
200.4.f.c.149.4		12	20.19	odd	2
360.4.k.c.181.11		12	120.83	odd	4
360.4.k.c.181.12		12	60.23	odd	4
800.4.d.d.401.3		12	5.2	odd	4
800.4.d.d.401.10		12	40.37	odd	4
800.4.f.b.49.3		12	5.4	even	2
800.4.f.b.49.4		12	8.5	even	2
800.4.f.c.49.9		12	40.29	even	2	inner
800.4.f.c.49.10		12	1.1	even	1	trivial
1280.4.a.ba.1.2		6	80.13	odd	4
1280.4.a.bb.1.2		6	80.43	even	4
1280.4.a.bc.1.5		6	80.3	even	4
1280.4.a.bd.1.5		6	80.53	odd	4
1440.4.k.c.721.6		12	120.53	even	4
1440.4.k.c.721.12		12	15.8	even	4

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:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	800.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+6.25785 q^{3} +34.6280i q^{7} +12.1606 q^{9} +O(q^{10})\)	\(q+6.25785 q^{3} +34.6280i q^{7} +12.1606 q^{9} +7.91595i q^{11} -11.0346 q^{13} +57.6152i q^{17} -141.133i q^{19} +216.696i q^{21} +129.328i q^{23} -92.8625 q^{27} +89.1664i q^{29} +78.3307 q^{31} +49.5368i q^{33} -249.332 q^{37} -69.0530 q^{39} +91.8705 q^{41} -194.184 q^{43} +72.6149i q^{47} -856.096 q^{49} +360.547i q^{51} -456.782 q^{53} -883.187i q^{57} +341.098i q^{59} -217.067i q^{61} +421.098i q^{63} +529.237 q^{67} +809.314i q^{69} -381.540 q^{71} +876.902i q^{73} -274.113 q^{77} -203.950 q^{79} -909.456 q^{81} +996.654 q^{83} +557.989i q^{87} -172.456 q^{89} -382.107i q^{91} +490.182 q^{93} +1058.49i q^{97} +96.2629i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{3} + 108 q^{9}+O(q^{10}) \) 12 * q + 12 * q^3 + 108 * q^9	\( 12 q + 12 q^{3} + 108 q^{9} + 432 q^{27} + 264 q^{31} - 136 q^{37} + 600 q^{39} + 40 q^{41} - 1204 q^{43} - 1308 q^{49} - 1056 q^{53} - 2412 q^{67} + 1592 q^{71} - 824 q^{77} + 2016 q^{79} + 2508 q^{81} + 3556 q^{83} + 424 q^{89} - 2784 q^{93}+O(q^{100}) \) 12 * q + 12 * q^3 + 108 * q^9 + 432 * q^27 + 264 * q^31 - 136 * q^37 + 600 * q^39 + 40 * q^41 - 1204 * q^43 - 1308 * q^49 - 1056 * q^53 - 2412 * q^67 + 1592 * q^71 - 824 * q^77 + 2016 * q^79 + 2508 * q^81 + 3556 * q^83 + 424 * q^89 - 2784 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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