LMFDB - Embedded newform 400.6.a.r.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,6,Mod(1,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("400.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	400.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:	$64.1535279252$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{241}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 60 $ x^2 - x - 60
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 200)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$8.26209$ of defining polynomial
Character	$\chi$	$=$	400.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-19.5242 q^{3} +35.0483 q^{7} +138.193 q^{9} +O(q^{10})$	$q-19.5242 q^{3} +35.0483 q^{7} +138.193 q^{9} +426.008 q^{11} -1103.26 q^{13} -109.387 q^{17} -495.926 q^{19} -684.290 q^{21} +2497.37 q^{23} +2046.26 q^{27} -42.4221 q^{29} +7999.56 q^{31} -8317.45 q^{33} +13763.7 q^{37} +21540.2 q^{39} +11863.6 q^{41} -16816.0 q^{43} -13036.0 q^{47} -15578.6 q^{49} +2135.69 q^{51} -17817.3 q^{53} +9682.55 q^{57} +47346.1 q^{59} -22782.9 q^{61} +4843.45 q^{63} -39418.3 q^{67} -48759.1 q^{69} +1616.32 q^{71} -53293.3 q^{73} +14930.9 q^{77} +6516.98 q^{79} -73532.6 q^{81} +46016.4 q^{83} +828.257 q^{87} +113488. q^{89} -38667.3 q^{91} -156185. q^{93} -107418. q^{97} +58871.4 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{3} + 8 q^{7} + 28 q^{9}+O(q^{10}) $ 2 * q - 8 * q^3 + 8 * q^7 + 28 * q^9	$ 2 q - 8 q^{3} + 8 q^{7} + 28 q^{9} + 200 q^{11} - 592 q^{13} + 278 q^{17} + 840 q^{19} - 996 q^{21} + 1952 q^{23} - 2024 q^{27} - 4680 q^{29} + 5008 q^{31} - 10922 q^{33} + 12500 q^{37} + 27432 q^{39} - 5334 q^{41} - 224 q^{43} - 26072 q^{47} - 31654 q^{49} + 6600 q^{51} - 46812 q^{53} + 25078 q^{57} + 81776 q^{59} - 46932 q^{61} + 7824 q^{63} - 68808 q^{67} - 55044 q^{69} - 7448 q^{71} - 108822 q^{73} + 21044 q^{77} + 108104 q^{79} - 93662 q^{81} - 27224 q^{83} - 52616 q^{87} + 70990 q^{89} - 52496 q^{91} - 190660 q^{93} - 96852 q^{97} + 83776 q^{99}+O(q^{100}) $ 2 * q - 8 * q^3 + 8 * q^7 + 28 * q^9 + 200 * q^11 - 592 * q^13 + 278 * q^17 + 840 * q^19 - 996 * q^21 + 1952 * q^23 - 2024 * q^27 - 4680 * q^29 + 5008 * q^31 - 10922 * q^33 + 12500 * q^37 + 27432 * q^39 - 5334 * q^41 - 224 * q^43 - 26072 * q^47 - 31654 * q^49 + 6600 * q^51 - 46812 * q^53 + 25078 * q^57 + 81776 * q^59 - 46932 * q^61 + 7824 * q^63 - 68808 * q^67 - 55044 * q^69 - 7448 * q^71 - 108822 * q^73 + 21044 * q^77 + 108104 * q^79 - 93662 * q^81 - 27224 * q^83 - 52616 * q^87 + 70990 * q^89 - 52496 * q^91 - 190660 * q^93 - 96852 * q^97 + 83776 * 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
$3$	−19.5242	−1.25248	−0.626238	−	0.779632i	$-0.715406\pi$
$3$	−19.5242	−1.25248	−0.626238	+	0.779632i	$0.715406\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	35.0483	0.270348	0.135174	−	0.990822i	$-0.456841\pi$
$7$	35.0483	0.270348	0.135174	+	0.990822i	$0.456841\pi$
$8$	0	0
$9$	138.193	0.568697
$10$	0	0
$11$	426.008	1.06154	0.530769	−	0.847516i	$-0.321903\pi$
$11$	426.008	1.06154	0.530769	+	0.847516i	$0.321903\pi$
$12$	0	0
$13$	−1103.26	−1.81058	−0.905291	−	0.424791i	$-0.860347\pi$
$13$	−1103.26	−1.81058	−0.905291	+	0.424791i	$0.860347\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−109.387	−0.0918000	−0.0459000	−	0.998946i	$-0.514616\pi$
$17$	−109.387	−0.0918000	−0.0459000	+	0.998946i	$0.514616\pi$
$18$	0	0
$19$	−495.926	−0.315161	−0.157581	−	0.987506i	$-0.550369\pi$
$19$	−495.926	−0.315161	−0.157581	+	0.987506i	$0.550369\pi$
$20$	0	0
$21$	−684.290	−0.338604
$22$	0	0
$23$	2497.37	0.984381	0.492190	−	0.870488i	$-0.336196\pi$
$23$	2497.37	0.984381	0.492190	+	0.870488i	$0.336196\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	2046.26	0.540197
$28$	0	0
$29$	−42.4221	−0.00936694	−0.00468347	−	0.999989i	$-0.501491\pi$
$29$	−42.4221	−0.00936694	−0.00468347	+	0.999989i	$0.501491\pi$
$30$	0	0
$31$	7999.56	1.49507	0.747535	−	0.664222i	$-0.231237\pi$
$31$	7999.56	1.49507	0.747535	+	0.664222i	$0.231237\pi$
$32$	0	0
$33$	−8317.45	−1.32955
$34$	0	0
$35$	0	0
$36$	0	0
$37$	13763.7	1.65284	0.826420	−	0.563054i	$-0.190374\pi$
$37$	13763.7	1.65284	0.826420	+	0.563054i	$0.190374\pi$
$38$	0	0
$39$	21540.2	2.26771
$40$	0	0
$41$	11863.6	1.10219	0.551097	−	0.834441i	$-0.314210\pi$
$41$	11863.6	1.10219	0.551097	+	0.834441i	$0.314210\pi$
$42$	0	0
$43$	−16816.0	−1.38692	−0.693461	−	0.720494i	$-0.743915\pi$
$43$	−16816.0	−1.38692	−0.693461	+	0.720494i	$0.743915\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−13036.0	−0.860795	−0.430397	−	0.902639i	$-0.641627\pi$
$47$	−13036.0	−0.860795	−0.430397	+	0.902639i	$0.641627\pi$
$48$	0	0
$49$	−15578.6	−0.926912
$50$	0	0
$51$	2135.69	0.114977
$52$	0	0
$53$	−17817.3	−0.871269	−0.435634	−	0.900124i	$-0.643476\pi$
$53$	−17817.3	−0.871269	−0.435634	+	0.900124i	$0.643476\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	9682.55	0.394732
$58$	0	0
$59$	47346.1	1.77074	0.885368	−	0.464891i	$-0.153906\pi$
$59$	47346.1	1.77074	0.885368	+	0.464891i	$0.153906\pi$
$60$	0	0
$61$	−22782.9	−0.783944	−0.391972	−	0.919977i	$-0.628207\pi$
$61$	−22782.9	−0.783944	−0.391972	+	0.919977i	$0.628207\pi$
$62$	0	0
$63$	4843.45	0.153746
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−39418.3	−1.07278	−0.536390	−	0.843970i	$-0.680213\pi$
$67$	−39418.3	−1.07278	−0.536390	+	0.843970i	$0.680213\pi$
$68$	0	0
$69$	−48759.1	−1.23291
$70$	0	0
$71$	1616.32	0.0380523	0.0190261	−	0.999819i	$-0.493943\pi$
$71$	1616.32	0.0380523	0.0190261	+	0.999819i	$0.493943\pi$
$72$	0	0
$73$	−53293.3	−1.17048	−0.585242	−	0.810859i	$-0.699000\pi$
$73$	−53293.3	−1.17048	−0.585242	+	0.810859i	$0.699000\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	14930.9	0.286984
$78$	0	0
$79$	6516.98	0.117484	0.0587420	−	0.998273i	$-0.481291\pi$
$79$	6516.98	0.117484	0.0587420	+	0.998273i	$0.481291\pi$
$80$	0	0
$81$	−73532.6	−1.24528
$82$	0	0
$83$	46016.4	0.733191	0.366595	−	0.930380i	$-0.380523\pi$
$83$	46016.4	0.733191	0.366595	+	0.930380i	$0.380523\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	828.257	0.0117319
$88$	0	0
$89$	113488.	1.51872	0.759358	−	0.650673i	$-0.225513\pi$
$89$	113488.	1.51872	0.759358	+	0.650673i	$0.225513\pi$
$90$	0	0
$91$	−38667.3	−0.489487
$92$	0	0
$93$	−156185.	−1.87254
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−107418.	−1.15917	−0.579585	−	0.814912i	$-0.696785\pi$
$97$	−107418.	−1.15917	−0.579585	+	0.814912i	$0.696785\pi$
$98$	0	0
$99$	58871.4	0.603694
$100$	0	0
$101$	−197554.	−1.92700	−0.963502	−	0.267703i	$-0.913736\pi$
$101$	−197554.	−1.92700	−0.963502	+	0.267703i	$0.913736\pi$
$102$	0	0
$103$	−81026.0	−0.752543	−0.376272	−	0.926509i	$-0.622794\pi$
$103$	−81026.0	−0.752543	−0.376272	+	0.926509i	$0.622794\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−137442.	−1.16054	−0.580271	−	0.814423i	$-0.697054\pi$
$107$	−137442.	−1.16054	−0.580271	+	0.814423i	$0.697054\pi$
$108$	0	0
$109$	−68693.1	−0.553792	−0.276896	−	0.960900i	$-0.589306\pi$
$109$	−68693.1	−0.553792	−0.276896	+	0.960900i	$0.589306\pi$
$110$	0	0
$111$	−268725.	−2.07014
$112$	0	0
$113$	139632.	1.02870	0.514352	−	0.857579i	$-0.328033\pi$
$113$	139632.	1.02870	0.514352	+	0.857579i	$0.328033\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−152463.	−1.02967
$118$	0	0
$119$	−3833.83	−0.0248179
$120$	0	0
$121$	20431.5	0.126864
$122$	0	0
$123$	−231628.	−1.38047
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−65230.1	−0.358871	−0.179436	−	0.983770i	$-0.557427\pi$
$127$	−65230.1	−0.358871	−0.179436	+	0.983770i	$0.557427\pi$
$128$	0	0
$129$	328319.	1.73709
$130$	0	0
$131$	−118542.	−0.603524	−0.301762	−	0.953383i	$-0.597575\pi$
$131$	−118542.	−0.603524	−0.301762	+	0.953383i	$0.597575\pi$
$132$	0	0
$133$	−17381.4	−0.0852031
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−84757.8	−0.385814	−0.192907	−	0.981217i	$-0.561792\pi$
$137$	−84757.8	−0.385814	−0.192907	+	0.981217i	$0.561792\pi$
$138$	0	0
$139$	−168334.	−0.738982	−0.369491	−	0.929234i	$-0.620468\pi$
$139$	−168334.	−0.738982	−0.369491	+	0.929234i	$0.620468\pi$
$140$	0	0
$141$	254517.	1.07813
$142$	0	0
$143$	−469996.	−1.92200
$144$	0	0
$145$	0	0
$146$	0	0
$147$	304160.	1.16094
$148$	0	0
$149$	67628.6	0.249554	0.124777	−	0.992185i	$-0.460178\pi$
$149$	67628.6	0.249554	0.124777	+	0.992185i	$0.460178\pi$
$150$	0	0
$151$	65622.3	0.234212	0.117106	−	0.993119i	$-0.462638\pi$
$151$	65622.3	0.234212	0.117106	+	0.993119i	$0.462638\pi$
$152$	0	0
$153$	−15116.5	−0.0522064
$154$	0	0
$155$	0	0
$156$	0	0
$157$	297885.	0.964495	0.482247	−	0.876035i	$-0.339821\pi$
$157$	297885.	0.964495	0.482247	+	0.876035i	$0.339821\pi$
$158$	0	0
$159$	347868.	1.09124
$160$	0	0
$161$	87528.7	0.266125
$162$	0	0
$163$	−473879.	−1.39701	−0.698503	−	0.715607i	$-0.746150\pi$
$163$	−473879.	−1.39701	−0.698503	+	0.715607i	$0.746150\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	267673.	0.742701	0.371351	−	0.928493i	$-0.378895\pi$
$167$	267673.	0.742701	0.371351	+	0.928493i	$0.378895\pi$
$168$	0	0
$169$	845883.	2.27821
$170$	0	0
$171$	−68533.7	−0.179231
$172$	0	0
$173$	−703284.	−1.78655	−0.893276	−	0.449509i	$-0.851599\pi$
$173$	−703284.	−1.78655	−0.893276	+	0.449509i	$0.851599\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−924393.	−2.21780
$178$	0	0
$179$	−635322.	−1.48204	−0.741022	−	0.671481i	$-0.765659\pi$
$179$	−635322.	−1.48204	−0.741022	+	0.671481i	$0.765659\pi$
$180$	0	0
$181$	−547906.	−1.24311	−0.621555	−	0.783370i	$-0.713499\pi$
$181$	−547906.	−1.24311	−0.621555	+	0.783370i	$0.713499\pi$
$182$	0	0
$183$	444818.	0.981872
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−46599.6	−0.0974492
$188$	0	0
$189$	71718.1	0.146041
$190$	0	0
$191$	80607.9	0.159880	0.0799400	−	0.996800i	$-0.474527\pi$
$191$	80607.9	0.159880	0.0799400	+	0.996800i	$0.474527\pi$
$192$	0	0
$193$	26549.3	0.0513049	0.0256525	−	0.999671i	$-0.491834\pi$
$193$	26549.3	0.0513049	0.0256525	+	0.999671i	$0.491834\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	703676.	1.29184	0.645918	−	0.763407i	$-0.276475\pi$
$197$	703676.	1.29184	0.645918	+	0.763407i	$0.276475\pi$
$198$	0	0
$199$	−73599.5	−0.131747	−0.0658737	−	0.997828i	$-0.520983\pi$
$199$	−73599.5	−0.131747	−0.0658737	+	0.997828i	$0.520983\pi$
$200$	0	0
$201$	769610.	1.34363
$202$	0	0
$203$	−1486.83	−0.00253233
$204$	0	0
$205$	0	0
$206$	0	0
$207$	345120.	0.559815
$208$	0	0
$209$	−211268.	−0.334556
$210$	0	0
$211$	1.00808e6	1.55879	0.779395	−	0.626533i	$-0.215526\pi$
$211$	1.00808e6	1.55879	0.779395	+	0.626533i	$0.215526\pi$
$212$	0	0
$213$	−31557.2	−0.0476596
$214$	0	0
$215$	0	0
$216$	0	0
$217$	280371.	0.404189
$218$	0	0
$219$	1.04051e6	1.46600
$220$	0	0
$221$	120682.	0.166211
$222$	0	0
$223$	141067.	0.189961	0.0949805	−	0.995479i	$-0.469721\pi$
$223$	141067.	0.189961	0.0949805	+	0.995479i	$0.469721\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1.14301e6	−1.47226	−0.736129	−	0.676842i	$-0.763348\pi$
$227$	−1.14301e6	−1.47226	−0.736129	+	0.676842i	$0.763348\pi$
$228$	0	0
$229$	−1.29731e6	−1.63476	−0.817380	−	0.576099i	$-0.804574\pi$
$229$	−1.29731e6	−1.63476	−0.817380	+	0.576099i	$0.804574\pi$
$230$	0	0
$231$	−291513.	−0.359441
$232$	0	0
$233$	626232.	0.755693	0.377846	−	0.925868i	$-0.376665\pi$
$233$	626232.	0.755693	0.377846	+	0.925868i	$0.376665\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−127239.	−0.147146
$238$	0	0
$239$	−270281.	−0.306070	−0.153035	−	0.988221i	$-0.548905\pi$
$239$	−270281.	−0.306070	−0.153035	+	0.988221i	$0.548905\pi$
$240$	0	0
$241$	−694339.	−0.770068	−0.385034	−	0.922902i	$-0.625810\pi$
$241$	−694339.	−0.770068	−0.385034	+	0.922902i	$0.625810\pi$
$242$	0	0
$243$	938421.	1.01949
$244$	0	0
$245$	0	0
$246$	0	0
$247$	547134.	0.570626
$248$	0	0
$249$	−898431.	−0.918304
$250$	0	0
$251$	−864085.	−0.865710	−0.432855	−	0.901464i	$-0.642494\pi$
$251$	−864085.	−0.865710	−0.432855	+	0.901464i	$0.642494\pi$
$252$	0	0
$253$	1.06390e6	1.04496
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.00660e6	0.950660	0.475330	−	0.879808i	$-0.342329\pi$
$257$	1.00660e6	0.950660	0.475330	+	0.879808i	$0.342329\pi$
$258$	0	0
$259$	482395.	0.446841
$260$	0	0
$261$	−5862.46	−0.00532695
$262$	0	0
$263$	931373.	0.830299	0.415149	−	0.909753i	$-0.363729\pi$
$263$	931373.	0.830299	0.415149	+	0.909753i	$0.363729\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−2.21577e6	−1.90216
$268$	0	0
$269$	−1.96236e6	−1.65348	−0.826738	−	0.562587i	$-0.809806\pi$
$269$	−1.96236e6	−1.65348	−0.826738	+	0.562587i	$0.809806\pi$
$270$	0	0
$271$	−1.12152e6	−0.927649	−0.463825	−	0.885927i	$-0.653523\pi$
$271$	−1.12152e6	−0.927649	−0.463825	+	0.885927i	$0.653523\pi$
$272$	0	0
$273$	754948.	0.613070
$274$	0	0
$275$	0	0
$276$	0	0
$277$	105917.	0.0829401	0.0414701	−	0.999140i	$-0.486796\pi$
$277$	105917.	0.0829401	0.0414701	+	0.999140i	$0.486796\pi$
$278$	0	0
$279$	1.10549e6	0.850242
$280$	0	0
$281$	−1.05276e6	−0.795362	−0.397681	−	0.917524i	$-0.630185\pi$
$281$	−1.05276e6	−0.795362	−0.397681	+	0.917524i	$0.630185\pi$
$282$	0	0
$283$	950210.	0.705267	0.352633	−	0.935762i	$-0.385286\pi$
$283$	950210.	0.705267	0.352633	+	0.935762i	$0.385286\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	415801.	0.297975
$288$	0	0
$289$	−1.40789e6	−0.991573
$290$	0	0
$291$	2.09725e6	1.45183
$292$	0	0
$293$	−381801.	−0.259817	−0.129909	−	0.991526i	$-0.541468\pi$
$293$	−381801.	−0.259817	−0.129909	+	0.991526i	$0.541468\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	871723.	0.573440
$298$	0	0
$299$	−2.75524e6	−1.78230
$300$	0	0
$301$	−589373.	−0.374951
$302$	0	0
$303$	3.85708e6	2.41353
$304$	0	0
$305$	0	0
$306$	0	0
$307$	1.61048e6	0.975238	0.487619	−	0.873057i	$-0.337866\pi$
$307$	1.61048e6	0.975238	0.487619	+	0.873057i	$0.337866\pi$
$308$	0	0
$309$	1.58197e6	0.942543
$310$	0	0
$311$	2.15652e6	1.26431	0.632153	−	0.774843i	$-0.282171\pi$
$311$	2.15652e6	1.26431	0.632153	+	0.774843i	$0.282171\pi$
$312$	0	0
$313$	−364108.	−0.210073	−0.105036	−	0.994468i	$-0.533496\pi$
$313$	−364108.	−0.210073	−0.105036	+	0.994468i	$0.533496\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1.03846e6	−0.580420	−0.290210	−	0.956963i	$-0.593725\pi$
$317$	−1.03846e6	−0.580420	−0.290210	+	0.956963i	$0.593725\pi$
$318$	0	0
$319$	−18072.2	−0.00994336
$320$	0	0
$321$	2.68345e6	1.45355
$322$	0	0
$323$	54247.8	0.0289318
$324$	0	0
$325$	0	0
$326$	0	0
$327$	1.34118e6	0.693612
$328$	0	0
$329$	−456890.	−0.232714
$330$	0	0
$331$	2.82942e6	1.41948	0.709738	−	0.704466i	$-0.248814\pi$
$331$	2.82942e6	1.41948	0.709738	+	0.704466i	$0.248814\pi$
$332$	0	0
$333$	1.90205e6	0.939966
$334$	0	0
$335$	0	0
$336$	0	0
$337$	804843.	0.386043	0.193022	−	0.981194i	$-0.438171\pi$
$337$	804843.	0.386043	0.193022	+	0.981194i	$0.438171\pi$
$338$	0	0
$339$	−2.72621e6	−1.28843
$340$	0	0
$341$	3.40787e6	1.58708
$342$	0	0
$343$	−1.13506e6	−0.520936
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−759889.	−0.338787	−0.169393	−	0.985549i	$-0.554181\pi$
$347$	−759889.	−0.338787	−0.169393	+	0.985549i	$0.554181\pi$
$348$	0	0
$349$	580673.	0.255193	0.127596	−	0.991826i	$-0.459274\pi$
$349$	580673.	0.255193	0.127596	+	0.991826i	$0.459274\pi$
$350$	0	0
$351$	−2.25755e6	−0.978071
$352$	0	0
$353$	−1.64210e6	−0.701393	−0.350697	−	0.936489i	$-0.614055\pi$
$353$	−1.64210e6	−0.701393	−0.350697	+	0.936489i	$0.614055\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	74852.3	0.0310838
$358$	0	0
$359$	2.88181e6	1.18013	0.590065	−	0.807356i	$-0.299102\pi$
$359$	2.88181e6	1.18013	0.590065	+	0.807356i	$0.299102\pi$
$360$	0	0
$361$	−2.23016e6	−0.900673
$362$	0	0
$363$	−398909.	−0.158894
$364$	0	0
$365$	0	0
$366$	0	0
$367$	3.83418e6	1.48596	0.742980	−	0.669314i	$-0.233412\pi$
$367$	3.83418e6	1.48596	0.742980	+	0.669314i	$0.233412\pi$
$368$	0	0
$369$	1.63947e6	0.626814
$370$	0	0
$371$	−624467.	−0.235545
$372$	0	0
$373$	3.19341e6	1.18845	0.594227	−	0.804298i	$-0.297458\pi$
$373$	3.19341e6	1.18845	0.594227	+	0.804298i	$0.297458\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	46802.5	0.0169596
$378$	0	0
$379$	−3.47338e6	−1.24209	−0.621046	−	0.783774i	$-0.713292\pi$
$379$	−3.47338e6	−1.24209	−0.621046	+	0.783774i	$0.713292\pi$
$380$	0	0
$381$	1.27356e6	0.449478
$382$	0	0
$383$	−1.56341e6	−0.544599	−0.272299	−	0.962213i	$-0.587784\pi$
$383$	−1.56341e6	−0.544599	−0.272299	+	0.962213i	$0.587784\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−2.32386e6	−0.788738
$388$	0	0
$389$	4.98980e6	1.67190	0.835949	−	0.548808i	$-0.184918\pi$
$389$	4.98980e6	1.67190	0.835949	+	0.548808i	$0.184918\pi$
$390$	0	0
$391$	−273179.	−0.0903661
$392$	0	0
$393$	2.31444e6	0.755899
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−1.37245e6	−0.437038	−0.218519	−	0.975833i	$-0.570123\pi$
$397$	−1.37245e6	−0.437038	−0.218519	+	0.975833i	$0.570123\pi$
$398$	0	0
$399$	339357.	0.106715
$400$	0	0
$401$	5.44247e6	1.69019	0.845094	−	0.534617i	$-0.179544\pi$
$401$	5.44247e6	1.69019	0.845094	+	0.534617i	$0.179544\pi$
$402$	0	0
$403$	−8.82557e6	−2.70695
$404$	0	0
$405$	0	0
$406$	0	0
$407$	5.86344e6	1.75455
$408$	0	0
$409$	−6.03830e6	−1.78487	−0.892435	−	0.451176i	$-0.851005\pi$
$409$	−6.03830e6	−1.78487	−0.892435	+	0.451176i	$0.851005\pi$
$410$	0	0
$411$	1.65483e6	0.483223
$412$	0	0
$413$	1.65940e6	0.478714
$414$	0	0
$415$	0	0
$416$	0	0
$417$	3.28657e6	0.925557
$418$	0	0
$419$	−770366.	−0.214369	−0.107184	−	0.994239i	$-0.534184\pi$
$419$	−770366.	−0.214369	−0.107184	+	0.994239i	$0.534184\pi$
$420$	0	0
$421$	2.00249e6	0.550636	0.275318	−	0.961353i	$-0.411217\pi$
$421$	2.00249e6	0.550636	0.275318	+	0.961353i	$0.411217\pi$
$422$	0	0
$423$	−1.80149e6	−0.489532
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−798504.	−0.211937
$428$	0	0
$429$	9.17628e6	2.40726
$430$	0	0
$431$	−5.56690e6	−1.44351	−0.721756	−	0.692148i	$-0.756665\pi$
$431$	−5.56690e6	−1.44351	−0.721756	+	0.692148i	$0.756665\pi$
$432$	0	0
$433$	−2.22941e6	−0.571441	−0.285720	−	0.958313i	$-0.592233\pi$
$433$	−2.22941e6	−0.571441	−0.285720	+	0.958313i	$0.592233\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.23851e6	−0.310239
$438$	0	0
$439$	−543061.	−0.134489	−0.0672446	−	0.997737i	$-0.521421\pi$
$439$	−543061.	−0.134489	−0.0672446	+	0.997737i	$0.521421\pi$
$440$	0	0
$441$	−2.15286e6	−0.527132
$442$	0	0
$443$	4.53358e6	1.09757	0.548785	−	0.835964i	$-0.315091\pi$
$443$	4.53358e6	1.09757	0.548785	+	0.835964i	$0.315091\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−1.32039e6	−0.312561
$448$	0	0
$449$	−3.39468e6	−0.794662	−0.397331	−	0.917675i	$-0.630064\pi$
$449$	−3.39468e6	−0.794662	−0.397331	+	0.917675i	$0.630064\pi$
$450$	0	0
$451$	5.05400e6	1.17002
$452$	0	0
$453$	−1.28122e6	−0.293345
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−8.34100e6	−1.86822	−0.934109	−	0.356988i	$-0.883804\pi$
$457$	−8.34100e6	−1.86822	−0.934109	+	0.356988i	$0.883804\pi$
$458$	0	0
$459$	−223834.	−0.0495900
$460$	0	0
$461$	−3.18453e6	−0.697900	−0.348950	−	0.937141i	$-0.613462\pi$
$461$	−3.18453e6	−0.697900	−0.348950	+	0.937141i	$0.613462\pi$
$462$	0	0
$463$	4.05943e6	0.880061	0.440030	−	0.897983i	$-0.354968\pi$
$463$	4.05943e6	0.880061	0.440030	+	0.897983i	$0.354968\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−7.98706e6	−1.69471	−0.847353	−	0.531029i	$-0.821805\pi$
$467$	−7.98706e6	−1.69471	−0.847353	+	0.531029i	$0.821805\pi$
$468$	0	0
$469$	−1.38155e6	−0.290024
$470$	0	0
$471$	−5.81596e6	−1.20801
$472$	0	0
$473$	−7.16375e6	−1.47227
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−2.46223e6	−0.495488
$478$	0	0
$479$	−5.27553e6	−1.05058	−0.525288	−	0.850924i	$-0.676042\pi$
$479$	−5.27553e6	−1.05058	−0.525288	+	0.850924i	$0.676042\pi$
$480$	0	0
$481$	−1.51849e7	−2.99260
$482$	0	0
$483$	−1.70892e6	−0.333315
$484$	0	0
$485$	0	0
$486$	0	0
$487$	8.76969e6	1.67557	0.837784	−	0.546002i	$-0.183851\pi$
$487$	8.76969e6	1.67557	0.837784	+	0.546002i	$0.183851\pi$
$488$	0	0
$489$	9.25209e6	1.74972
$490$	0	0
$491$	−7.83080e6	−1.46589	−0.732946	−	0.680286i	$-0.761855\pi$
$491$	−7.83080e6	−1.46589	−0.732946	+	0.680286i	$0.761855\pi$
$492$	0	0
$493$	4640.42	0.000859885 0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	56649.2	0.0102873
$498$	0	0
$499$	6.54560e6	1.17679	0.588394	−	0.808574i	$-0.299760\pi$
$499$	6.54560e6	1.17679	0.588394	+	0.808574i	$0.299760\pi$
$500$	0	0
$501$	−5.22610e6	−0.930216
$502$	0	0
$503$	−2.07773e6	−0.366158	−0.183079	−	0.983098i	$-0.558606\pi$
$503$	−2.07773e6	−0.366158	−0.183079	+	0.983098i	$0.558606\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−1.65152e7	−2.85340
$508$	0	0
$509$	−6.25093e6	−1.06942	−0.534712	−	0.845034i	$-0.679580\pi$
$509$	−6.25093e6	−1.06942	−0.534712	+	0.845034i	$0.679580\pi$
$510$	0	0
$511$	−1.86784e6	−0.316437
$512$	0	0
$513$	−1.01480e6	−0.170249
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−5.55344e6	−0.913767
$518$	0	0
$519$	1.37310e7	2.23761
$520$	0	0
$521$	2.19897e6	0.354916	0.177458	−	0.984128i	$-0.443213\pi$
$521$	2.19897e6	0.354916	0.177458	+	0.984128i	$0.443213\pi$
$522$	0	0
$523$	−2.10298e6	−0.336187	−0.168093	−	0.985771i	$-0.553761\pi$
$523$	−2.10298e6	−0.336187	−0.168093	+	0.985771i	$0.553761\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−875046.	−0.137247
$528$	0	0
$529$	−199490.	−0.0309944
$530$	0	0
$531$	6.54291e6	1.00701
$532$	0	0
$533$	−1.30886e7	−1.99561
$534$	0	0
$535$	0	0
$536$	0	0
$537$	1.24041e7	1.85623
$538$	0	0
$539$	−6.63661e6	−0.983953
$540$	0	0
$541$	−1.83729e6	−0.269889	−0.134944	−	0.990853i	$-0.543086\pi$
$541$	−1.83729e6	−0.269889	−0.134944	+	0.990853i	$0.543086\pi$
$542$	0	0
$543$	1.06974e7	1.55697
$544$	0	0
$545$	0	0
$546$	0	0
$547$	4.31872e6	0.617145	0.308572	−	0.951201i	$-0.400149\pi$
$547$	4.31872e6	0.617145	0.308572	+	0.951201i	$0.400149\pi$
$548$	0	0
$549$	−3.14845e6	−0.445827
$550$	0	0
$551$	21038.3	0.00295210
$552$	0	0
$553$	228409.	0.0317615
$554$	0	0
$555$	0	0
$556$	0	0
$557$	4.26837e6	0.582940	0.291470	−	0.956580i	$-0.405856\pi$
$557$	4.26837e6	0.582940	0.291470	+	0.956580i	$0.405856\pi$
$558$	0	0
$559$	1.85524e7	2.51114
$560$	0	0
$561$	909819.	0.122053
$562$	0	0
$563$	−1.26350e7	−1.67998	−0.839988	−	0.542605i	$-0.817438\pi$
$563$	−1.26350e7	−1.67998	−0.839988	+	0.542605i	$0.817438\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−2.57720e6	−0.336659
$568$	0	0
$569$	−8.67409e6	−1.12316	−0.561582	−	0.827421i	$-0.689807\pi$
$569$	−8.67409e6	−1.12316	−0.561582	+	0.827421i	$0.689807\pi$
$570$	0	0
$571$	−3.28781e6	−0.422004	−0.211002	−	0.977486i	$-0.567673\pi$
$571$	−3.28781e6	−0.422004	−0.211002	+	0.977486i	$0.567673\pi$
$572$	0	0
$573$	−1.57380e6	−0.200246
$574$	0	0
$575$	0	0
$576$	0	0
$577$	117668.	0.0147135	0.00735677	−	0.999973i	$-0.497658\pi$
$577$	117668.	0.0147135	0.00735677	+	0.999973i	$0.497658\pi$
$578$	0	0
$579$	−518352.	−0.0642582
$580$	0	0
$581$	1.61280e6	0.198216
$582$	0	0
$583$	−7.59031e6	−0.924885
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1.10151e7	−1.31945	−0.659723	−	0.751509i	$-0.729326\pi$
$587$	−1.10151e7	−1.31945	−0.659723	+	0.751509i	$0.729326\pi$
$588$	0	0
$589$	−3.96719e6	−0.471189
$590$	0	0
$591$	−1.37387e7	−1.61799
$592$	0	0
$593$	5.33879e6	0.623456	0.311728	−	0.950171i	$-0.399092\pi$
$593$	5.33879e6	0.623456	0.311728	+	0.950171i	$0.399092\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.43697e6	0.165011
$598$	0	0
$599$	7.17784e6	0.817385	0.408692	−	0.912672i	$-0.365985\pi$
$599$	7.17784e6	0.817385	0.408692	+	0.912672i	$0.365985\pi$
$600$	0	0
$601$	−809127.	−0.0913756	−0.0456878	−	0.998956i	$-0.514548\pi$
$601$	−809127.	−0.0913756	−0.0456878	+	0.998956i	$0.514548\pi$
$602$	0	0
$603$	−5.44735e6	−0.610087
$604$	0	0
$605$	0	0
$606$	0	0
$607$	591991.	0.0652143	0.0326072	−	0.999468i	$-0.489619\pi$
$607$	591991.	0.0652143	0.0326072	+	0.999468i	$0.489619\pi$
$608$	0	0
$609$	29029.1	0.00317168
$610$	0	0
$611$	1.43821e7	1.55854
$612$	0	0
$613$	4.01173e6	0.431201	0.215601	−	0.976482i	$-0.430829\pi$
$613$	4.01173e6	0.431201	0.215601	+	0.976482i	$0.430829\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−8.24093e6	−0.871491	−0.435746	−	0.900070i	$-0.643515\pi$
$617$	−8.24093e6	−0.871491	−0.435746	+	0.900070i	$0.643515\pi$
$618$	0	0
$619$	−1.96985e6	−0.206636	−0.103318	−	0.994648i	$-0.532946\pi$
$619$	−1.96985e6	−0.206636	−0.103318	+	0.994648i	$0.532946\pi$
$620$	0	0
$621$	5.11027e6	0.531759
$622$	0	0
$623$	3.97758e6	0.410581
$624$	0	0
$625$	0	0
$626$	0	0
$627$	4.12484e6	0.419024
$628$	0	0
$629$	−1.50557e6	−0.151731
$630$	0	0
$631$	1.82360e7	1.82329	0.911646	−	0.410977i	$-0.134813\pi$
$631$	1.82360e7	1.82329	0.911646	+	0.410977i	$0.134813\pi$
$632$	0	0
$633$	−1.96819e7	−1.95235
$634$	0	0
$635$	0	0
$636$	0	0
$637$	1.71872e7	1.67825
$638$	0	0
$639$	223364.	0.0216402
$640$	0	0
$641$	7.65378e6	0.735752	0.367876	−	0.929875i	$-0.380085\pi$
$641$	7.65378e6	0.735752	0.367876	+	0.929875i	$0.380085\pi$
$642$	0	0
$643$	2.47762e6	0.236323	0.118162	−	0.992994i	$-0.462300\pi$
$643$	2.47762e6	0.236323	0.118162	+	0.992994i	$0.462300\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	4.79533e6	0.450358	0.225179	−	0.974317i	$-0.427703\pi$
$647$	4.79533e6	0.450358	0.225179	+	0.974317i	$0.427703\pi$
$648$	0	0
$649$	2.01698e7	1.87970
$650$	0	0
$651$	−5.47402e6	−0.506237
$652$	0	0
$653$	−1.67125e7	−1.53376	−0.766881	−	0.641789i	$-0.778193\pi$
$653$	−1.67125e7	−1.53376	−0.766881	+	0.641789i	$0.778193\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−7.36478e6	−0.665650
$658$	0	0
$659$	−3.28380e6	−0.294553	−0.147277	−	0.989095i	$-0.547051\pi$
$659$	−3.28380e6	−0.294553	−0.147277	+	0.989095i	$0.547051\pi$
$660$	0	0
$661$	−1.98880e7	−1.77047	−0.885234	−	0.465145i	$-0.846002\pi$
$661$	−1.98880e7	−1.77047	−0.885234	+	0.465145i	$0.846002\pi$
$662$	0	0
$663$	−2.35621e6	−0.208176
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−105944.	−0.00922063
$668$	0	0
$669$	−2.75423e6	−0.237922
$670$	0	0
$671$	−9.70571e6	−0.832187
$672$	0	0
$673$	1.39915e7	1.19076	0.595381	−	0.803443i	$-0.297001\pi$
$673$	1.39915e7	1.19076	0.595381	+	0.803443i	$0.297001\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−3.13032e6	−0.262493	−0.131246	−	0.991350i	$-0.541898\pi$
$677$	−3.13032e6	−0.262493	−0.131246	+	0.991350i	$0.541898\pi$
$678$	0	0
$679$	−3.76482e6	−0.313379
$680$	0	0
$681$	2.23162e7	1.84397
$682$	0	0
$683$	8.94034e6	0.733335	0.366667	−	0.930352i	$-0.380499\pi$
$683$	8.94034e6	0.733335	0.366667	+	0.930352i	$0.380499\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	2.53288e7	2.04750
$688$	0	0
$689$	1.96571e7	1.57750
$690$	0	0
$691$	2.38378e7	1.89920	0.949599	−	0.313466i	$-0.101490\pi$
$691$	2.38378e7	1.89920	0.949599	+	0.313466i	$0.101490\pi$
$692$	0	0
$693$	2.06335e6	0.163207
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−1.29772e6	−0.101181
$698$	0	0
$699$	−1.22267e7	−0.946487
$700$	0	0
$701$	1.57226e7	1.20845	0.604225	−	0.796814i	$-0.293483\pi$
$701$	1.57226e7	1.20845	0.604225	+	0.796814i	$0.293483\pi$
$702$	0	0
$703$	−6.82578e6	−0.520912
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−6.92394e6	−0.520961
$708$	0	0
$709$	−3.05866e6	−0.228515	−0.114258	−	0.993451i	$-0.536449\pi$
$709$	−3.05866e6	−0.228515	−0.114258	+	0.993451i	$0.536449\pi$
$710$	0	0
$711$	900603.	0.0668128
$712$	0	0
$713$	1.99778e7	1.47172
$714$	0	0
$715$	0	0
$716$	0	0
$717$	5.27701e6	0.383345
$718$	0	0
$719$	−2.16146e7	−1.55929	−0.779643	−	0.626224i	$-0.784600\pi$
$719$	−2.16146e7	−1.55929	−0.779643	+	0.626224i	$0.784600\pi$
$720$	0	0
$721$	−2.83983e6	−0.203448
$722$	0	0
$723$	1.35564e7	0.964492
$724$	0	0
$725$	0	0
$726$	0	0
$727$	2.73589e7	1.91983	0.959916	−	0.280287i	$-0.0904297\pi$
$727$	2.73589e7	1.91983	0.959916	+	0.280287i	$0.0904297\pi$
$728$	0	0
$729$	−453482.	−0.0316039
$730$	0	0
$731$	1.83945e6	0.127319
$732$	0	0
$733$	1.54693e7	1.06344	0.531718	−	0.846921i	$-0.321547\pi$
$733$	1.54693e7	1.06344	0.531718	+	0.846921i	$0.321547\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.67925e7	−1.13880
$738$	0	0
$739$	−5.56327e6	−0.374731	−0.187365	−	0.982290i	$-0.559995\pi$
$739$	−5.56327e6	−0.374731	−0.187365	+	0.982290i	$0.559995\pi$
$740$	0	0
$741$	−1.06823e7	−0.714695
$742$	0	0
$743$	1.55089e7	1.03064	0.515322	−	0.856997i	$-0.327672\pi$
$743$	1.55089e7	1.03064	0.515322	+	0.856997i	$0.327672\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	6.35916e6	0.416963
$748$	0	0
$749$	−4.81713e6	−0.313750
$750$	0	0
$751$	−2.20221e7	−1.42482	−0.712409	−	0.701765i	$-0.752396\pi$
$751$	−2.20221e7	−1.42482	−0.712409	+	0.701765i	$0.752396\pi$
$752$	0	0
$753$	1.68706e7	1.08428
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−1.75653e6	−0.111408	−0.0557040	−	0.998447i	$-0.517740\pi$
$757$	−1.75653e6	−0.111408	−0.0557040	+	0.998447i	$0.517740\pi$
$758$	0	0
$759$	−2.07717e7	−1.30879
$760$	0	0
$761$	−2.16974e6	−0.135814	−0.0679072	−	0.997692i	$-0.521632\pi$
$761$	−2.16974e6	−0.135814	−0.0679072	+	0.997692i	$0.521632\pi$
$762$	0	0
$763$	−2.40758e6	−0.149716
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−5.22349e7	−3.20606
$768$	0	0
$769$	−1.12131e7	−0.683769	−0.341884	−	0.939742i	$-0.611065\pi$
$769$	−1.12131e7	−0.683769	−0.341884	+	0.939742i	$0.611065\pi$
$770$	0	0
$771$	−1.96531e7	−1.19068
$772$	0	0
$773$	1.43367e7	0.862980	0.431490	−	0.902118i	$-0.357988\pi$
$773$	1.43367e7	0.862980	0.431490	+	0.902118i	$0.357988\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−9.41836e6	−0.559658
$778$	0	0
$779$	−5.88348e6	−0.347369
$780$	0	0
$781$	688563.	0.0403939
$782$	0	0
$783$	−86806.8	−0.00505999
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−2.00549e6	−0.115421	−0.0577103	−	0.998333i	$-0.518380\pi$
$787$	−2.00549e6	−0.115421	−0.0577103	+	0.998333i	$0.518380\pi$
$788$	0	0
$789$	−1.81843e7	−1.03993
$790$	0	0
$791$	4.89389e6	0.278108
$792$	0	0
$793$	2.51354e7	1.41940
$794$	0	0
$795$	0	0
$796$	0	0
$797$	448115.	0.0249887	0.0124944	−	0.999922i	$-0.496023\pi$
$797$	448115.	0.0249887	0.0124944	+	0.999922i	$0.496023\pi$
$798$	0	0
$799$	1.42597e6	0.0790210
$800$	0	0
$801$	1.56834e7	0.863690
$802$	0	0
$803$	−2.27033e7	−1.24251
$804$	0	0
$805$	0	0
$806$	0	0
$807$	3.83134e7	2.07094
$808$	0	0
$809$	1.28425e7	0.689889	0.344944	−	0.938623i	$-0.387898\pi$
$809$	1.28425e7	0.689889	0.344944	+	0.938623i	$0.387898\pi$
$810$	0	0
$811$	−1.78700e7	−0.954054	−0.477027	−	0.878889i	$-0.658286\pi$
$811$	−1.78700e7	−0.954054	−0.477027	+	0.878889i	$0.658286\pi$
$812$	0	0
$813$	2.18968e7	1.16186
$814$	0	0
$815$	0	0
$816$	0	0
$817$	8.33950e6	0.437104
$818$	0	0
$819$	−5.34357e6	−0.278370
$820$	0	0
$821$	2.73336e7	1.41527	0.707634	−	0.706580i	$-0.249763\pi$
$821$	2.73336e7	1.41527	0.707634	+	0.706580i	$0.249763\pi$
$822$	0	0
$823$	−3.05819e6	−0.157385	−0.0786927	−	0.996899i	$-0.525075\pi$
$823$	−3.05819e6	−0.157385	−0.0786927	+	0.996899i	$0.525075\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−1.23961e7	−0.630261	−0.315131	−	0.949048i	$-0.602048\pi$
$827$	−1.23961e7	−0.630261	−0.315131	+	0.949048i	$0.602048\pi$
$828$	0	0
$829$	−399873.	−0.0202086	−0.0101043	−	0.999949i	$-0.503216\pi$
$829$	−399873.	−0.0202086	−0.0101043	+	0.999949i	$0.503216\pi$
$830$	0	0
$831$	−2.06793e6	−0.103881
$832$	0	0
$833$	1.70409e6	0.0850905
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.63692e7	0.807632
$838$	0	0
$839$	2.08411e6	0.102215	0.0511075	−	0.998693i	$-0.483725\pi$
$839$	2.08411e6	0.102215	0.0511075	+	0.998693i	$0.483725\pi$
$840$	0	0
$841$	−2.05093e7	−0.999912
$842$	0	0
$843$	2.05543e7	0.996172
$844$	0	0
$845$	0	0
$846$	0	0
$847$	716092.	0.0342973
$848$	0	0
$849$	−1.85521e7	−0.883330
$850$	0	0
$851$	3.43730e7	1.62702
$852$	0	0
$853$	−2.14298e7	−1.00843	−0.504214	−	0.863579i	$-0.668218\pi$
$853$	−2.14298e7	−1.00843	−0.504214	+	0.863579i	$0.668218\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−3.49123e6	−0.162377	−0.0811887	−	0.996699i	$-0.525872\pi$
$857$	−3.49123e6	−0.162377	−0.0811887	+	0.996699i	$0.525872\pi$
$858$	0	0
$859$	568043.	0.0262662	0.0131331	−	0.999914i	$-0.495819\pi$
$859$	568043.	0.0262662	0.0131331	+	0.999914i	$0.495819\pi$
$860$	0	0
$861$	−8.11816e6	−0.373207
$862$	0	0
$863$	3.28678e7	1.50226	0.751128	−	0.660157i	$-0.229510\pi$
$863$	3.28678e7	1.50226	0.751128	+	0.660157i	$0.229510\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	2.74879e7	1.24192
$868$	0	0
$869$	2.77628e6	0.124714
$870$	0	0
$871$	4.34885e7	1.94236
$872$	0	0
$873$	−1.48444e7	−0.659217
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−2.92945e7	−1.28614	−0.643069	−	0.765808i	$-0.722339\pi$
$877$	−2.92945e7	−1.28614	−0.643069	+	0.765808i	$0.722339\pi$
$878$	0	0
$879$	7.45435e6	0.325415
$880$	0	0
$881$	2.88332e6	0.125156	0.0625782	−	0.998040i	$-0.480068\pi$
$881$	2.88332e6	0.125156	0.0625782	+	0.998040i	$0.480068\pi$
$882$	0	0
$883$	2.79477e7	1.20627	0.603135	−	0.797639i	$-0.293918\pi$
$883$	2.79477e7	1.20627	0.603135	+	0.797639i	$0.293918\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2.89567e7	−1.23578	−0.617888	−	0.786266i	$-0.712011\pi$
$887$	−2.89567e7	−1.23578	−0.617888	+	0.786266i	$0.712011\pi$
$888$	0	0
$889$	−2.28621e6	−0.0970199
$890$	0	0
$891$	−3.13254e7	−1.32191
$892$	0	0
$893$	6.46490e6	0.271289
$894$	0	0
$895$	0	0
$896$	0	0
$897$	5.37938e7	2.23229
$898$	0	0
$899$	−339358.	−0.0140042
$900$	0	0
$901$	1.94898e6	0.0799825
$902$	0	0
$903$	1.15070e7	0.469617
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−1.18508e6	−0.0478333	−0.0239166	−	0.999714i	$-0.507614\pi$
$907$	−1.18508e6	−0.0478333	−0.0239166	+	0.999714i	$0.507614\pi$
$908$	0	0
$909$	−2.73007e7	−1.09588
$910$	0	0
$911$	1.14346e7	0.456485	0.228242	−	0.973604i	$-0.426702\pi$
$911$	1.14346e7	0.456485	0.228242	+	0.973604i	$0.426702\pi$
$912$	0	0
$913$	1.96033e7	0.778310
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−4.15471e6	−0.163161
$918$	0	0
$919$	1.48904e7	0.581589	0.290795	−	0.956785i	$-0.406080\pi$
$919$	1.48904e7	0.581589	0.290795	+	0.956785i	$0.406080\pi$
$920$	0	0
$921$	−3.14434e7	−1.22146
$922$	0	0
$923$	−1.78321e6	−0.0688968
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−1.11973e7	−0.427969
$928$	0	0
$929$	−2.02580e7	−0.770117	−0.385058	−	0.922892i	$-0.625819\pi$
$929$	−2.02580e7	−0.770117	−0.385058	+	0.922892i	$0.625819\pi$
$930$	0	0
$931$	7.72584e6	0.292127
$932$	0	0
$933$	−4.21043e7	−1.58351
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1.96237e7	0.730182	0.365091	−	0.930972i	$-0.381038\pi$
$937$	1.96237e7	0.730182	0.365091	+	0.930972i	$0.381038\pi$
$938$	0	0
$939$	7.10891e6	0.263111
$940$	0	0
$941$	7.62854e6	0.280846	0.140423	−	0.990092i	$-0.455154\pi$
$941$	7.62854e6	0.280846	0.140423	+	0.990092i	$0.455154\pi$
$942$	0	0
$943$	2.96279e7	1.08498
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.93605e7	1.42622	0.713108	−	0.701054i	$-0.247287\pi$
$947$	3.93605e7	1.42622	0.713108	+	0.701054i	$0.247287\pi$
$948$	0	0
$949$	5.87962e7	2.11926
$950$	0	0
$951$	2.02751e7	0.726963
$952$	0	0
$953$	−5.09953e7	−1.81886	−0.909428	−	0.415861i	$-0.863480\pi$
$953$	−5.09953e7	−1.81886	−0.909428	+	0.415861i	$0.863480\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	352844.	0.0124538
$958$	0	0
$959$	−2.97062e6	−0.104304
$960$	0	0
$961$	3.53638e7	1.23524
$962$	0	0
$963$	−1.89936e7	−0.659997
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−5.17740e7	−1.78051	−0.890257	−	0.455458i	$-0.849476\pi$
$967$	−5.17740e7	−1.78051	−0.890257	+	0.455458i	$0.849476\pi$
$968$	0	0
$969$	−1.05914e6	−0.0362364
$970$	0	0
$971$	3.69158e7	1.25651	0.628253	−	0.778009i	$-0.283770\pi$
$971$	3.69158e7	1.25651	0.628253	+	0.778009i	$0.283770\pi$
$972$	0	0
$973$	−5.89981e6	−0.199782
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−3.14468e7	−1.05400	−0.526999	−	0.849866i	$-0.676683\pi$
$977$	−3.14468e7	−1.05400	−0.526999	+	0.849866i	$0.676683\pi$
$978$	0	0
$979$	4.83470e7	1.61218
$980$	0	0
$981$	−9.49293e6	−0.314940
$982$	0	0
$983$	−5.15823e7	−1.70262	−0.851308	−	0.524667i	$-0.824190\pi$
$983$	−5.15823e7	−1.70262	−0.851308	+	0.524667i	$0.824190\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	8.92041e6	0.291469
$988$	0	0
$989$	−4.19958e7	−1.36526
$990$	0	0
$991$	1.30562e7	0.422312	0.211156	−	0.977452i	$-0.432277\pi$
$991$	1.30562e7	0.422312	0.211156	+	0.977452i	$0.432277\pi$
$992$	0	0
$993$	−5.52421e7	−1.77786
$994$	0	0
$995$	0	0
$996$	0	0
$997$	4.97672e7	1.58564	0.792821	−	0.609454i	$-0.208611\pi$
$997$	4.97672e7	1.58564	0.792821	+	0.609454i	$0.208611\pi$
$998$	0	0
$999$	2.81641e7	0.892859

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.6.a.r.1.1		2
4.3	odd	2		200.6.a.f.1.2	yes	2
5.2	odd	4		400.6.c.o.49.4		4
5.3	odd	4		400.6.c.o.49.1		4
5.4	even	2		400.6.a.u.1.2		2
20.3	even	4		200.6.c.f.49.4		4
20.7	even	4		200.6.c.f.49.1		4
20.19	odd	2		200.6.a.e.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
200.6.a.e.1.1	✓	2	20.19	odd	2
200.6.a.f.1.2	yes	2	4.3	odd	2
200.6.c.f.49.1		4	20.7	even	4
200.6.c.f.49.4		4	20.3	even	4
400.6.a.r.1.1		2	1.1	even	1	trivial
400.6.a.u.1.2		2	5.4	even	2
400.6.c.o.49.1		4	5.3	odd	4
400.6.c.o.49.4		4	5.2	odd	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:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	400.a (trivial)

\(f(q)\)	\(=\)	\(q-19.5242 q^{3} +35.0483 q^{7} +138.193 q^{9} +O(q^{10})\)	\(q-19.5242 q^{3} +35.0483 q^{7} +138.193 q^{9} +426.008 q^{11} -1103.26 q^{13} -109.387 q^{17} -495.926 q^{19} -684.290 q^{21} +2497.37 q^{23} +2046.26 q^{27} -42.4221 q^{29} +7999.56 q^{31} -8317.45 q^{33} +13763.7 q^{37} +21540.2 q^{39} +11863.6 q^{41} -16816.0 q^{43} -13036.0 q^{47} -15578.6 q^{49} +2135.69 q^{51} -17817.3 q^{53} +9682.55 q^{57} +47346.1 q^{59} -22782.9 q^{61} +4843.45 q^{63} -39418.3 q^{67} -48759.1 q^{69} +1616.32 q^{71} -53293.3 q^{73} +14930.9 q^{77} +6516.98 q^{79} -73532.6 q^{81} +46016.4 q^{83} +828.257 q^{87} +113488. q^{89} -38667.3 q^{91} -156185. q^{93} -107418. q^{97} +58871.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{3} + 8 q^{7} + 28 q^{9}+O(q^{10}) \) 2 * q - 8 * q^3 + 8 * q^7 + 28 * q^9	\( 2 q - 8 q^{3} + 8 q^{7} + 28 q^{9} + 200 q^{11} - 592 q^{13} + 278 q^{17} + 840 q^{19} - 996 q^{21} + 1952 q^{23} - 2024 q^{27} - 4680 q^{29} + 5008 q^{31} - 10922 q^{33} + 12500 q^{37} + 27432 q^{39} - 5334 q^{41} - 224 q^{43} - 26072 q^{47} - 31654 q^{49} + 6600 q^{51} - 46812 q^{53} + 25078 q^{57} + 81776 q^{59} - 46932 q^{61} + 7824 q^{63} - 68808 q^{67} - 55044 q^{69} - 7448 q^{71} - 108822 q^{73} + 21044 q^{77} + 108104 q^{79} - 93662 q^{81} - 27224 q^{83} - 52616 q^{87} + 70990 q^{89} - 52496 q^{91} - 190660 q^{93} - 96852 q^{97} + 83776 q^{99}+O(q^{100}) \) 2 * q - 8 * q^3 + 8 * q^7 + 28 * q^9 + 200 * q^11 - 592 * q^13 + 278 * q^17 + 840 * q^19 - 996 * q^21 + 1952 * q^23 - 2024 * q^27 - 4680 * q^29 + 5008 * q^31 - 10922 * q^33 + 12500 * q^37 + 27432 * q^39 - 5334 * q^41 - 224 * q^43 - 26072 * q^47 - 31654 * q^49 + 6600 * q^51 - 46812 * q^53 + 25078 * q^57 + 81776 * q^59 - 46932 * q^61 + 7824 * q^63 - 68808 * q^67 - 55044 * q^69 - 7448 * q^71 - 108822 * q^73 + 21044 * q^77 + 108104 * q^79 - 93662 * q^81 - 27224 * q^83 - 52616 * q^87 + 70990 * q^89 - 52496 * q^91 - 190660 * q^93 - 96852 * q^97 + 83776 * q^99

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