LMFDB - Embedded newform 400.6.a.q.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{129}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 32 $ x^2 - x - 32
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 40)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$6.17891$ 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)

$f(q)$	$=$	$q-28.7156 q^{3} -42.1469 q^{7} +581.588 q^{9} +O(q^{10})$	$q-28.7156 q^{3} -42.1469 q^{7} +581.588 q^{9} -416.294 q^{11} -966.588 q^{13} +1834.11 q^{17} -317.763 q^{19} +1210.27 q^{21} +1568.02 q^{23} -9722.76 q^{27} +7757.28 q^{29} -102.644 q^{31} +11954.1 q^{33} -1936.58 q^{37} +27756.2 q^{39} +7994.36 q^{41} +16542.6 q^{43} +18649.3 q^{47} -15030.6 q^{49} -52667.7 q^{51} +14972.4 q^{53} +9124.76 q^{57} -19843.3 q^{59} -18024.1 q^{61} -24512.1 q^{63} -55040.6 q^{67} -45026.8 q^{69} -11201.3 q^{71} +4013.95 q^{73} +17545.5 q^{77} -24018.8 q^{79} +137869. q^{81} +70512.8 q^{83} -222755. q^{87} -60765.7 q^{89} +40738.7 q^{91} +2947.49 q^{93} +31112.2 q^{97} -242111. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 12 q^{3} + 52 q^{7} + 618 q^{9}+O(q^{10}) $ 2 * q - 12 * q^3 + 52 * q^7 + 618 * q^9	$ 2 q - 12 q^{3} + 52 q^{7} + 618 q^{9} - 560 q^{11} - 1388 q^{13} - 148 q^{17} + 1000 q^{19} + 2784 q^{21} - 2452 q^{23} - 13176 q^{27} + 1340 q^{29} + 2248 q^{31} + 9552 q^{33} + 5940 q^{37} + 20712 q^{39} + 23076 q^{41} + 17684 q^{43} - 2908 q^{47} - 22974 q^{49} - 85800 q^{51} + 5412 q^{53} + 31152 q^{57} - 62584 q^{59} + 14108 q^{61} - 21084 q^{63} - 85412 q^{67} - 112224 q^{69} - 47208 q^{71} + 67452 q^{73} + 4016 q^{77} + 65904 q^{79} + 71298 q^{81} + 108724 q^{83} - 330024 q^{87} - 55020 q^{89} + 1064 q^{91} + 42240 q^{93} - 147668 q^{97} - 247344 q^{99}+O(q^{100}) $ 2 * q - 12 * q^3 + 52 * q^7 + 618 * q^9 - 560 * q^11 - 1388 * q^13 - 148 * q^17 + 1000 * q^19 + 2784 * q^21 - 2452 * q^23 - 13176 * q^27 + 1340 * q^29 + 2248 * q^31 + 9552 * q^33 + 5940 * q^37 + 20712 * q^39 + 23076 * q^41 + 17684 * q^43 - 2908 * q^47 - 22974 * q^49 - 85800 * q^51 + 5412 * q^53 + 31152 * q^57 - 62584 * q^59 + 14108 * q^61 - 21084 * q^63 - 85412 * q^67 - 112224 * q^69 - 47208 * q^71 + 67452 * q^73 + 4016 * q^77 + 65904 * q^79 + 71298 * q^81 + 108724 * q^83 - 330024 * q^87 - 55020 * q^89 + 1064 * q^91 + 42240 * q^93 - 147668 * q^97 - 247344 * 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
$3$	−28.7156	−1.84211	−0.921054	−	0.389434i	$-0.872671\pi$
$3$	−28.7156	−1.84211	−0.921054	+	0.389434i	$0.872671\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−42.1469	−0.325103	−0.162551	−	0.986700i	$-0.551972\pi$
$7$	−42.1469	−0.325103	−0.162551	+	0.986700i	$0.551972\pi$
$8$	0	0
$9$	581.588	2.39336
$10$	0	0
$11$	−416.294	−1.03733	−0.518667	−	0.854977i	$-0.673571\pi$
$11$	−416.294	−1.03733	−0.518667	+	0.854977i	$0.673571\pi$
$12$	0	0
$13$	−966.588	−1.58629	−0.793145	−	0.609032i	$-0.791558\pi$
$13$	−966.588	−1.58629	−0.793145	+	0.609032i	$0.791558\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1834.11	1.53923	0.769616	−	0.638508i	$-0.220448\pi$
$17$	1834.11	1.53923	0.769616	+	0.638508i	$0.220448\pi$
$18$	0	0
$19$	−317.763	−0.201938	−0.100969	−	0.994890i	$-0.532194\pi$
$19$	−317.763	−0.201938	−0.100969	+	0.994890i	$0.532194\pi$
$20$	0	0
$21$	1210.27	0.598874
$22$	0	0
$23$	1568.02	0.618063	0.309032	−	0.951052i	$-0.399995\pi$
$23$	1568.02	0.618063	0.309032	+	0.951052i	$0.399995\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−9722.76	−2.56673
$28$	0	0
$29$	7757.28	1.71283	0.856415	−	0.516288i	$-0.172687\pi$
$29$	7757.28	1.71283	0.856415	+	0.516288i	$0.172687\pi$
$30$	0	0
$31$	−102.644	−0.0191836	−0.00959180	−	0.999954i	$-0.503053\pi$
$31$	−102.644	−0.0191836	−0.00959180	+	0.999954i	$0.503053\pi$
$32$	0	0
$33$	11954.1	1.91088
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−1936.58	−0.232558	−0.116279	−	0.993217i	$-0.537097\pi$
$37$	−1936.58	−0.232558	−0.116279	+	0.993217i	$0.537097\pi$
$38$	0	0
$39$	27756.2	2.92212
$40$	0	0
$41$	7994.36	0.742718	0.371359	−	0.928489i	$-0.378892\pi$
$41$	7994.36	0.742718	0.371359	+	0.928489i	$0.378892\pi$
$42$	0	0
$43$	16542.6	1.36437	0.682186	−	0.731179i	$-0.261030\pi$
$43$	16542.6	1.36437	0.682186	+	0.731179i	$0.261030\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	18649.3	1.23146	0.615728	−	0.787959i	$-0.288862\pi$
$47$	18649.3	1.23146	0.615728	+	0.787959i	$0.288862\pi$
$48$	0	0
$49$	−15030.6	−0.894308
$50$	0	0
$51$	−52667.7	−2.83543
$52$	0	0
$53$	14972.4	0.732155	0.366077	−	0.930584i	$-0.380701\pi$
$53$	14972.4	0.732155	0.366077	+	0.930584i	$0.380701\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	9124.76	0.371993
$58$	0	0
$59$	−19843.3	−0.742137	−0.371069	−	0.928605i	$-0.621009\pi$
$59$	−19843.3	−0.742137	−0.371069	+	0.928605i	$0.621009\pi$
$60$	0	0
$61$	−18024.1	−0.620195	−0.310097	−	0.950705i	$-0.600362\pi$
$61$	−18024.1	−0.620195	−0.310097	+	0.950705i	$0.600362\pi$
$62$	0	0
$63$	−24512.1	−0.778089
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−55040.6	−1.49795	−0.748973	−	0.662601i	$-0.769453\pi$
$67$	−55040.6	−1.49795	−0.748973	+	0.662601i	$0.769453\pi$
$68$	0	0
$69$	−45026.8	−1.13854
$70$	0	0
$71$	−11201.3	−0.263707	−0.131853	−	0.991269i	$-0.542093\pi$
$71$	−11201.3	−0.263707	−0.131853	+	0.991269i	$0.542093\pi$
$72$	0	0
$73$	4013.95	0.0881587	0.0440793	−	0.999028i	$-0.485965\pi$
$73$	4013.95	0.0881587	0.0440793	+	0.999028i	$0.485965\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	17545.5	0.337240
$78$	0	0
$79$	−24018.8	−0.432996	−0.216498	−	0.976283i	$-0.569463\pi$
$79$	−24018.8	−0.432996	−0.216498	+	0.976283i	$0.569463\pi$
$80$	0	0
$81$	137869.	2.33483
$82$	0	0
$83$	70512.8	1.12350	0.561750	−	0.827307i	$-0.310128\pi$
$83$	70512.8	1.12350	0.561750	+	0.827307i	$0.310128\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−222755.	−3.15522
$88$	0	0
$89$	−60765.7	−0.813174	−0.406587	−	0.913612i	$-0.633281\pi$
$89$	−60765.7	−0.813174	−0.406587	+	0.913612i	$0.633281\pi$
$90$	0	0
$91$	40738.7	0.515707
$92$	0	0
$93$	2947.49	0.0353383
$94$	0	0
$95$	0	0
$96$	0	0
$97$	31112.2	0.335739	0.167869	−	0.985809i	$-0.446311\pi$
$97$	31112.2	0.335739	0.167869	+	0.985809i	$0.446311\pi$
$98$	0	0
$99$	−242111.	−2.48272
$100$	0	0
$101$	−49491.4	−0.482755	−0.241377	−	0.970431i	$-0.577599\pi$
$101$	−49491.4	−0.482755	−0.241377	+	0.970431i	$0.577599\pi$
$102$	0	0
$103$	−95117.9	−0.883424	−0.441712	−	0.897157i	$-0.645629\pi$
$103$	−95117.9	−0.883424	−0.441712	+	0.897157i	$0.645629\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	88429.8	0.746688	0.373344	−	0.927693i	$-0.378211\pi$
$107$	88429.8	0.746688	0.373344	+	0.927693i	$0.378211\pi$
$108$	0	0
$109$	−10598.9	−0.0854464	−0.0427232	−	0.999087i	$-0.513603\pi$
$109$	−10598.9	−0.0854464	−0.0427232	+	0.999087i	$0.513603\pi$
$110$	0	0
$111$	55610.0	0.428396
$112$	0	0
$113$	−235124.	−1.73221	−0.866107	−	0.499859i	$-0.833385\pi$
$113$	−235124.	−1.73221	−0.866107	+	0.499859i	$0.833385\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−562155.	−3.79657
$118$	0	0
$119$	−77302.2	−0.500408
$120$	0	0
$121$	12249.5	0.0760599
$122$	0	0
$123$	−229563.	−1.36817
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−102533.	−0.564098	−0.282049	−	0.959400i	$-0.591014\pi$
$127$	−102533.	−0.564098	−0.282049	+	0.959400i	$0.591014\pi$
$128$	0	0
$129$	−475031.	−2.51332
$130$	0	0
$131$	−50603.0	−0.257631	−0.128815	−	0.991669i	$-0.541117\pi$
$131$	−50603.0	−0.257631	−0.128815	+	0.991669i	$0.541117\pi$
$132$	0	0
$133$	13392.7	0.0656507
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−47811.7	−0.217637	−0.108818	−	0.994062i	$-0.534707\pi$
$137$	−47811.7	−0.217637	−0.108818	+	0.994062i	$0.534707\pi$
$138$	0	0
$139$	220190.	0.966629	0.483314	−	0.875447i	$-0.339433\pi$
$139$	220190.	0.966629	0.483314	+	0.875447i	$0.339433\pi$
$140$	0	0
$141$	−535527.	−2.26847
$142$	0	0
$143$	402384.	1.64551
$144$	0	0
$145$	0	0
$146$	0	0
$147$	431614.	1.64741
$148$	0	0
$149$	154533.	0.570236	0.285118	−	0.958492i	$-0.407967\pi$
$149$	154533.	0.570236	0.285118	+	0.958492i	$0.407967\pi$
$150$	0	0
$151$	455395.	1.62534	0.812672	−	0.582721i	$-0.198012\pi$
$151$	455395.	1.62534	0.812672	+	0.582721i	$0.198012\pi$
$152$	0	0
$153$	1.06670e6	3.68394
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−361690.	−1.17108	−0.585541	−	0.810643i	$-0.699118\pi$
$157$	−361690.	−1.17108	−0.585541	+	0.810643i	$0.699118\pi$
$158$	0	0
$159$	−429943.	−1.34871
$160$	0	0
$161$	−66087.3	−0.200934
$162$	0	0
$163$	490843.	1.44702	0.723508	−	0.690316i	$-0.242528\pi$
$163$	490843.	1.44702	0.723508	+	0.690316i	$0.242528\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−196273.	−0.544591	−0.272295	−	0.962214i	$-0.587783\pi$
$167$	−196273.	−0.544591	−0.272295	+	0.962214i	$0.587783\pi$
$168$	0	0
$169$	562999.	1.51632
$170$	0	0
$171$	−184807.	−0.483312
$172$	0	0
$173$	−183395.	−0.465877	−0.232938	−	0.972491i	$-0.574834\pi$
$173$	−183395.	−0.465877	−0.232938	+	0.972491i	$0.574834\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	569814.	1.36710
$178$	0	0
$179$	38660.0	0.0901839	0.0450920	−	0.998983i	$-0.485642\pi$
$179$	38660.0	0.0901839	0.0450920	+	0.998983i	$0.485642\pi$
$180$	0	0
$181$	−561287.	−1.27347	−0.636735	−	0.771083i	$-0.719715\pi$
$181$	−561287.	−1.27347	−0.636735	+	0.771083i	$0.719715\pi$
$182$	0	0
$183$	517572.	1.14247
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−763530.	−1.59670
$188$	0	0
$189$	409784.	0.834450
$190$	0	0
$191$	−265393.	−0.526387	−0.263194	−	0.964743i	$-0.584776\pi$
$191$	−265393.	−0.526387	−0.263194	+	0.964743i	$0.584776\pi$
$192$	0	0
$193$	−863148.	−1.66798	−0.833992	−	0.551776i	$-0.813950\pi$
$193$	−863148.	−1.66798	−0.833992	+	0.551776i	$0.813950\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−281871.	−0.517469	−0.258734	−	0.965949i	$-0.583305\pi$
$197$	−281871.	−0.517469	−0.258734	+	0.965949i	$0.583305\pi$
$198$	0	0
$199$	192798.	0.345120	0.172560	−	0.984999i	$-0.444796\pi$
$199$	192798.	0.345120	0.172560	+	0.984999i	$0.444796\pi$
$200$	0	0
$201$	1.58053e6	2.75938
$202$	0	0
$203$	−326945.	−0.556846
$204$	0	0
$205$	0	0
$206$	0	0
$207$	911943.	1.47925
$208$	0	0
$209$	132283.	0.209477
$210$	0	0
$211$	−44631.4	−0.0690136	−0.0345068	−	0.999404i	$-0.510986\pi$
$211$	−44631.4	−0.0690136	−0.0345068	+	0.999404i	$0.510986\pi$
$212$	0	0
$213$	321651.	0.485776
$214$	0	0
$215$	0	0
$216$	0	0
$217$	4326.13	0.00623664
$218$	0	0
$219$	−115263.	−0.162398
$220$	0	0
$221$	−1.77283e6	−2.44167
$222$	0	0
$223$	−904469.	−1.21796	−0.608978	−	0.793187i	$-0.708420\pi$
$223$	−904469.	−1.21796	−0.608978	+	0.793187i	$0.708420\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1.02684e6	−1.32263	−0.661314	−	0.750109i	$-0.730001\pi$
$227$	−1.02684e6	−1.32263	−0.661314	+	0.750109i	$0.730001\pi$
$228$	0	0
$229$	−101521.	−0.127928	−0.0639641	−	0.997952i	$-0.520374\pi$
$229$	−101521.	−0.127928	−0.0639641	+	0.997952i	$0.520374\pi$
$230$	0	0
$231$	−503830.	−0.621232
$232$	0	0
$233$	313713.	0.378567	0.189283	−	0.981923i	$-0.439384\pi$
$233$	313713.	0.378567	0.189283	+	0.981923i	$0.439384\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	689715.	0.797625
$238$	0	0
$239$	1.53093e6	1.73365	0.866825	−	0.498612i	$-0.166157\pi$
$239$	1.53093e6	1.73365	0.866825	+	0.498612i	$0.166157\pi$
$240$	0	0
$241$	506999.	0.562296	0.281148	−	0.959664i	$-0.409285\pi$
$241$	506999.	0.562296	0.281148	+	0.959664i	$0.409285\pi$
$242$	0	0
$243$	−1.59638e6	−1.73428
$244$	0	0
$245$	0	0
$246$	0	0
$247$	307146.	0.320333
$248$	0	0
$249$	−2.02482e6	−2.06961
$250$	0	0
$251$	511659.	0.512621	0.256311	−	0.966594i	$-0.417493\pi$
$251$	511659.	0.512621	0.256311	+	0.966594i	$0.417493\pi$
$252$	0	0
$253$	−652758.	−0.641137
$254$	0	0
$255$	0	0
$256$	0	0
$257$	256082.	0.241850	0.120925	−	0.992662i	$-0.461414\pi$
$257$	256082.	0.241850	0.120925	+	0.992662i	$0.461414\pi$
$258$	0	0
$259$	81620.7	0.0756051
$260$	0	0
$261$	4.51154e6	4.09943
$262$	0	0
$263$	1.32451e6	1.18077	0.590386	−	0.807121i	$-0.298976\pi$
$263$	1.32451e6	1.18077	0.590386	+	0.807121i	$0.298976\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	1.74493e6	1.49795
$268$	0	0
$269$	−1.42989e6	−1.20482	−0.602410	−	0.798187i	$-0.705793\pi$
$269$	−1.42989e6	−1.20482	−0.602410	+	0.798187i	$0.705793\pi$
$270$	0	0
$271$	706426.	0.584310	0.292155	−	0.956371i	$-0.405628\pi$
$271$	706426.	0.584310	0.292155	+	0.956371i	$0.405628\pi$
$272$	0	0
$273$	−1.16984e6	−0.949989
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−314677.	−0.246414	−0.123207	−	0.992381i	$-0.539318\pi$
$277$	−314677.	−0.246414	−0.123207	+	0.992381i	$0.539318\pi$
$278$	0	0
$279$	−59696.6	−0.0459134
$280$	0	0
$281$	−437793.	−0.330752	−0.165376	−	0.986231i	$-0.552884\pi$
$281$	−437793.	−0.330752	−0.165376	+	0.986231i	$0.552884\pi$
$282$	0	0
$283$	−2.08248e6	−1.54566	−0.772831	−	0.634612i	$-0.781160\pi$
$283$	−2.08248e6	−1.54566	−0.772831	+	0.634612i	$0.781160\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−336938.	−0.241460
$288$	0	0
$289$	1.94411e6	1.36923
$290$	0	0
$291$	−893407.	−0.618468
$292$	0	0
$293$	−90716.3	−0.0617329	−0.0308664	−	0.999524i	$-0.509827\pi$
$293$	−90716.3	−0.0617329	−0.0308664	+	0.999524i	$0.509827\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	4.04752e6	2.66255
$298$	0	0
$299$	−1.51563e6	−0.980428
$300$	0	0
$301$	−697219.	−0.443561
$302$	0	0
$303$	1.42118e6	0.889286
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−571699.	−0.346196	−0.173098	−	0.984905i	$-0.555378\pi$
$307$	−571699.	−0.346196	−0.173098	+	0.984905i	$0.555378\pi$
$308$	0	0
$309$	2.73137e6	1.62736
$310$	0	0
$311$	−2.59515e6	−1.52147	−0.760733	−	0.649065i	$-0.775160\pi$
$311$	−2.59515e6	−1.52147	−0.760733	+	0.649065i	$0.775160\pi$
$312$	0	0
$313$	510659.	0.294625	0.147313	−	0.989090i	$-0.452938\pi$
$313$	510659.	0.294625	0.147313	+	0.989090i	$0.452938\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.37030e6	1.88374	0.941868	−	0.335984i	$-0.109069\pi$
$317$	3.37030e6	1.88374	0.941868	+	0.335984i	$0.109069\pi$
$318$	0	0
$319$	−3.22931e6	−1.77678
$320$	0	0
$321$	−2.53932e6	−1.37548
$322$	0	0
$323$	−582813.	−0.310830
$324$	0	0
$325$	0	0
$326$	0	0
$327$	304354.	0.157402
$328$	0	0
$329$	−786012.	−0.400349
$330$	0	0
$331$	−3.80172e6	−1.90726	−0.953632	−	0.300976i	$-0.902688\pi$
$331$	−3.80172e6	−1.90726	−0.953632	+	0.300976i	$0.902688\pi$
$332$	0	0
$333$	−1.12629e6	−0.556595
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−2.06627e6	−0.991088	−0.495544	−	0.868583i	$-0.665031\pi$
$337$	−2.06627e6	−0.991088	−0.495544	+	0.868583i	$0.665031\pi$
$338$	0	0
$339$	6.75174e6	3.19093
$340$	0	0
$341$	42730.1	0.0198998
$342$	0	0
$343$	1.34186e6	0.615845
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.35066e6	1.04801	0.524005	−	0.851715i	$-0.324437\pi$
$347$	2.35066e6	1.04801	0.524005	+	0.851715i	$0.324437\pi$
$348$	0	0
$349$	1.67220e6	0.734896	0.367448	−	0.930044i	$-0.380232\pi$
$349$	1.67220e6	0.734896	0.367448	+	0.930044i	$0.380232\pi$
$350$	0	0
$351$	9.39790e6	4.07158
$352$	0	0
$353$	−1.71355e6	−0.731914	−0.365957	−	0.930632i	$-0.619258\pi$
$353$	−1.71355e6	−0.731914	−0.365957	+	0.930632i	$0.619258\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	2.21978e6	0.921806
$358$	0	0
$359$	3.85773e6	1.57978	0.789888	−	0.613251i	$-0.210139\pi$
$359$	3.85773e6	1.57978	0.789888	+	0.613251i	$0.210139\pi$
$360$	0	0
$361$	−2.37513e6	−0.959221
$362$	0	0
$363$	−351753.	−0.140111
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−3.18109e6	−1.23285	−0.616427	−	0.787412i	$-0.711420\pi$
$367$	−3.18109e6	−1.23285	−0.616427	+	0.787412i	$0.711420\pi$
$368$	0	0
$369$	4.64942e6	1.77760
$370$	0	0
$371$	−631042.	−0.238026
$372$	0	0
$373$	4.79295e6	1.78374	0.891868	−	0.452295i	$-0.149395\pi$
$373$	4.79295e6	1.78374	0.891868	+	0.452295i	$0.149395\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−7.49809e6	−2.71705
$378$	0	0
$379$	−7018.36	−0.00250979	−0.00125490	−	0.999999i	$-0.500399\pi$
$379$	−7018.36	−0.00250979	−0.00125490	+	0.999999i	$0.500399\pi$
$380$	0	0
$381$	2.94430e6	1.03913
$382$	0	0
$383$	−694105.	−0.241784	−0.120892	−	0.992666i	$-0.538576\pi$
$383$	−694105.	−0.241784	−0.120892	+	0.992666i	$0.538576\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	9.62097e6	3.26544
$388$	0	0
$389$	−53514.1	−0.0179306	−0.00896529	−	0.999960i	$-0.502854\pi$
$389$	−53514.1	−0.0179306	−0.00896529	+	0.999960i	$0.502854\pi$
$390$	0	0
$391$	2.87593e6	0.951342
$392$	0	0
$393$	1.45310e6	0.474584
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−907937.	−0.289121	−0.144560	−	0.989496i	$-0.546177\pi$
$397$	−907937.	−0.289121	−0.144560	+	0.989496i	$0.546177\pi$
$398$	0	0
$399$	−384580.	−0.120936
$400$	0	0
$401$	−514404.	−0.159751	−0.0798755	−	0.996805i	$-0.525452\pi$
$401$	−514404.	−0.159751	−0.0798755	+	0.996805i	$0.525452\pi$
$402$	0	0
$403$	99214.6	0.0304308
$404$	0	0
$405$	0	0
$406$	0	0
$407$	806185.	0.241240
$408$	0	0
$409$	5.61814e6	1.66067	0.830337	−	0.557262i	$-0.188148\pi$
$409$	5.61814e6	1.66067	0.830337	+	0.557262i	$0.188148\pi$
$410$	0	0
$411$	1.37294e6	0.400911
$412$	0	0
$413$	836334.	0.241271
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−6.32288e6	−1.78064
$418$	0	0
$419$	−708382.	−0.197121	−0.0985605	−	0.995131i	$-0.531424\pi$
$419$	−708382.	−0.197121	−0.0985605	+	0.995131i	$0.531424\pi$
$420$	0	0
$421$	−3.91741e6	−1.07719	−0.538597	−	0.842563i	$-0.681046\pi$
$421$	−3.91741e6	−1.07719	−0.538597	+	0.842563i	$0.681046\pi$
$422$	0	0
$423$	1.08462e7	2.94732
$424$	0	0
$425$	0	0
$426$	0	0
$427$	759658.	0.201627
$428$	0	0
$429$	−1.15547e7	−3.03121
$430$	0	0
$431$	−4.42095e6	−1.14636	−0.573181	−	0.819429i	$-0.694291\pi$
$431$	−4.42095e6	−1.14636	−0.573181	+	0.819429i	$0.694291\pi$
$432$	0	0
$433$	4.01914e6	1.03018	0.515090	−	0.857136i	$-0.327758\pi$
$433$	4.01914e6	1.03018	0.515090	+	0.857136i	$0.327758\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−498259.	−0.124811
$438$	0	0
$439$	5.41795e6	1.34176	0.670879	−	0.741567i	$-0.265917\pi$
$439$	5.41795e6	1.34176	0.670879	+	0.741567i	$0.265917\pi$
$440$	0	0
$441$	−8.74163e6	−2.14041
$442$	0	0
$443$	−7.21518e6	−1.74678	−0.873389	−	0.487023i	$-0.838083\pi$
$443$	−7.21518e6	−1.74678	−0.873389	+	0.487023i	$0.838083\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−4.43751e6	−1.05044
$448$	0	0
$449$	−203792.	−0.0477057	−0.0238529	−	0.999715i	$-0.507593\pi$
$449$	−203792.	−0.0477057	−0.0238529	+	0.999715i	$0.507593\pi$
$450$	0	0
$451$	−3.32800e6	−0.770446
$452$	0	0
$453$	−1.30769e7	−2.99406
$454$	0	0
$455$	0	0
$456$	0	0
$457$	3.45642e6	0.774169	0.387085	−	0.922044i	$-0.373482\pi$
$457$	3.45642e6	0.774169	0.387085	+	0.922044i	$0.373482\pi$
$458$	0	0
$459$	−1.78326e7	−3.95079
$460$	0	0
$461$	6.40596e6	1.40389	0.701944	−	0.712233i	$-0.252316\pi$
$461$	6.40596e6	1.40389	0.701944	+	0.712233i	$0.252316\pi$
$462$	0	0
$463$	−7.59550e6	−1.64666	−0.823330	−	0.567563i	$-0.807887\pi$
$463$	−7.59550e6	−1.64666	−0.823330	+	0.567563i	$0.807887\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−4.58166e6	−0.972145	−0.486072	−	0.873919i	$-0.661571\pi$
$467$	−4.58166e6	−0.972145	−0.486072	+	0.873919i	$0.661571\pi$
$468$	0	0
$469$	2.31979e6	0.486986
$470$	0	0
$471$	1.03862e7	2.15726
$472$	0	0
$473$	−6.88658e6	−1.41531
$474$	0	0
$475$	0	0
$476$	0	0
$477$	8.70779e6	1.75231
$478$	0	0
$479$	−5.01195e6	−0.998085	−0.499043	−	0.866577i	$-0.666315\pi$
$479$	−5.01195e6	−0.998085	−0.499043	+	0.866577i	$0.666315\pi$
$480$	0	0
$481$	1.87187e6	0.368904
$482$	0	0
$483$	1.89774e6	0.370142
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−2.10670e6	−0.402514	−0.201257	−	0.979539i	$-0.564503\pi$
$487$	−2.10670e6	−0.402514	−0.201257	+	0.979539i	$0.564503\pi$
$488$	0	0
$489$	−1.40949e7	−2.66556
$490$	0	0
$491$	−5.43322e6	−1.01708	−0.508538	−	0.861040i	$-0.669814\pi$
$491$	−5.43322e6	−1.01708	−0.508538	+	0.861040i	$0.669814\pi$
$492$	0	0
$493$	1.42277e7	2.63644
$494$	0	0
$495$	0	0
$496$	0	0
$497$	472099.	0.0857318
$498$	0	0
$499$	−9.65183e6	−1.73523	−0.867617	−	0.497233i	$-0.834350\pi$
$499$	−9.65183e6	−1.73523	−0.867617	+	0.497233i	$0.834350\pi$
$500$	0	0
$501$	5.63611e6	1.00319
$502$	0	0
$503$	1.33954e6	0.236067	0.118034	−	0.993010i	$-0.462341\pi$
$503$	1.33954e6	0.236067	0.118034	+	0.993010i	$0.462341\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−1.61669e7	−2.79322
$508$	0	0
$509$	4.60771e6	0.788299	0.394150	−	0.919046i	$-0.371039\pi$
$509$	4.60771e6	0.788299	0.394150	+	0.919046i	$0.371039\pi$
$510$	0	0
$511$	−169176.	−0.0286606
$512$	0	0
$513$	3.08953e6	0.518321
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−7.76360e6	−1.27743
$518$	0	0
$519$	5.26629e6	0.858196
$520$	0	0
$521$	−5.80941e6	−0.937644	−0.468822	−	0.883293i	$-0.655321\pi$
$521$	−5.80941e6	−0.937644	−0.468822	+	0.883293i	$0.655321\pi$
$522$	0	0
$523$	−3.83877e6	−0.613674	−0.306837	−	0.951762i	$-0.599271\pi$
$523$	−3.83877e6	−0.613674	−0.306837	+	0.951762i	$0.599271\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−188261.	−0.0295280
$528$	0	0
$529$	−3.97765e6	−0.617998
$530$	0	0
$531$	−1.15406e7	−1.77621
$532$	0	0
$533$	−7.72725e6	−1.17817
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−1.11015e6	−0.166129
$538$	0	0
$539$	6.25716e6	0.927696
$540$	0	0
$541$	6.28830e6	0.923720	0.461860	−	0.886953i	$-0.347182\pi$
$541$	6.28830e6	0.923720	0.461860	+	0.886953i	$0.347182\pi$
$542$	0	0
$543$	1.61177e7	2.34587
$544$	0	0
$545$	0	0
$546$	0	0
$547$	4.54365e6	0.649287	0.324644	−	0.945836i	$-0.394756\pi$
$547$	4.54365e6	0.649287	0.324644	+	0.945836i	$0.394756\pi$
$548$	0	0
$549$	−1.04826e7	−1.48435
$550$	0	0
$551$	−2.46497e6	−0.345886
$552$	0	0
$553$	1.01232e6	0.140768
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1.20198e7	−1.64157	−0.820783	−	0.571239i	$-0.806463\pi$
$557$	−1.20198e7	−1.64157	−0.820783	+	0.571239i	$0.806463\pi$
$558$	0	0
$559$	−1.59899e7	−2.16429
$560$	0	0
$561$	2.19252e7	2.94129
$562$	0	0
$563$	3.65741e6	0.486298	0.243149	−	0.969989i	$-0.421820\pi$
$563$	3.65741e6	0.486298	0.243149	+	0.969989i	$0.421820\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−5.81077e6	−0.759059
$568$	0	0
$569$	−2.80214e6	−0.362835	−0.181418	−	0.983406i	$-0.558069\pi$
$569$	−2.80214e6	−0.362835	−0.181418	+	0.983406i	$0.558069\pi$
$570$	0	0
$571$	6.78472e6	0.870846	0.435423	−	0.900226i	$-0.356599\pi$
$571$	6.78472e6	0.870846	0.435423	+	0.900226i	$0.356599\pi$
$572$	0	0
$573$	7.62091e6	0.969662
$574$	0	0
$575$	0	0
$576$	0	0
$577$	1.25416e6	0.156825	0.0784123	−	0.996921i	$-0.475015\pi$
$577$	1.25416e6	0.156825	0.0784123	+	0.996921i	$0.475015\pi$
$578$	0	0
$579$	2.47858e7	3.07261
$580$	0	0
$581$	−2.97190e6	−0.365253
$582$	0	0
$583$	−6.23293e6	−0.759488
$584$	0	0
$585$	0	0
$586$	0	0
$587$	726476.	0.0870213	0.0435107	−	0.999053i	$-0.486146\pi$
$587$	726476.	0.0870213	0.0435107	+	0.999053i	$0.486146\pi$
$588$	0	0
$589$	32616.5	0.00387391
$590$	0	0
$591$	8.09409e6	0.953234
$592$	0	0
$593$	−933494.	−0.109012	−0.0545060	−	0.998513i	$-0.517358\pi$
$593$	−933494.	−0.109012	−0.0545060	+	0.998513i	$0.517358\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−5.53632e6	−0.635749
$598$	0	0
$599$	1.27354e7	1.45025	0.725127	−	0.688615i	$-0.241781\pi$
$599$	1.27354e7	1.45025	0.725127	+	0.688615i	$0.241781\pi$
$600$	0	0
$601$	−6.87190e6	−0.776052	−0.388026	−	0.921648i	$-0.626843\pi$
$601$	−6.87190e6	−0.776052	−0.388026	+	0.921648i	$0.626843\pi$
$602$	0	0
$603$	−3.20109e7	−3.58513
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−3.87130e6	−0.426467	−0.213233	−	0.977001i	$-0.568399\pi$
$607$	−3.87130e6	−0.426467	−0.213233	+	0.977001i	$0.568399\pi$
$608$	0	0
$609$	9.38844e6	1.02577
$610$	0	0
$611$	−1.80262e7	−1.95345
$612$	0	0
$613$	2.38824e6	0.256701	0.128350	−	0.991729i	$-0.459032\pi$
$613$	2.38824e6	0.256701	0.128350	+	0.991729i	$0.459032\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	3.07299e6	0.324974	0.162487	−	0.986711i	$-0.448048\pi$
$617$	3.07299e6	0.324974	0.162487	+	0.986711i	$0.448048\pi$
$618$	0	0
$619$	−8.52257e6	−0.894013	−0.447007	−	0.894531i	$-0.647510\pi$
$619$	−8.52257e6	−0.894013	−0.447007	+	0.894531i	$0.647510\pi$
$620$	0	0
$621$	−1.52455e7	−1.58640
$622$	0	0
$623$	2.56109e6	0.264365
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−3.79858e6	−0.385880
$628$	0	0
$629$	−3.55190e6	−0.357960
$630$	0	0
$631$	−8.54170e6	−0.854026	−0.427013	−	0.904246i	$-0.640434\pi$
$631$	−8.54170e6	−0.854026	−0.427013	+	0.904246i	$0.640434\pi$
$632$	0	0
$633$	1.28162e6	0.127131
$634$	0	0
$635$	0	0
$636$	0	0
$637$	1.45284e7	1.41863
$638$	0	0
$639$	−6.51452e6	−0.631146
$640$	0	0
$641$	3.81006e6	0.366257	0.183129	−	0.983089i	$-0.441377\pi$
$641$	3.81006e6	0.366257	0.183129	+	0.983089i	$0.441377\pi$
$642$	0	0
$643$	1.40516e7	1.34029	0.670144	−	0.742231i	$-0.266232\pi$
$643$	1.40516e7	1.34029	0.670144	+	0.742231i	$0.266232\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	962366.	0.0903815	0.0451908	−	0.998978i	$-0.485610\pi$
$647$	962366.	0.0903815	0.0451908	+	0.998978i	$0.485610\pi$
$648$	0	0
$649$	8.26065e6	0.769844
$650$	0	0
$651$	−124228.	−0.0114886
$652$	0	0
$653$	413989.	0.0379932	0.0189966	−	0.999820i	$-0.493953\pi$
$653$	413989.	0.0379932	0.0189966	+	0.999820i	$0.493953\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	2.33446e6	0.210996
$658$	0	0
$659$	1.92401e7	1.72581	0.862905	−	0.505367i	$-0.168643\pi$
$659$	1.92401e7	1.72581	0.862905	+	0.505367i	$0.168643\pi$
$660$	0	0
$661$	−2.04652e7	−1.82185	−0.910924	−	0.412574i	$-0.864630\pi$
$661$	−2.04652e7	−1.82185	−0.910924	+	0.412574i	$0.864630\pi$
$662$	0	0
$663$	5.09080e7	4.49782
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.21636e7	1.05864
$668$	0	0
$669$	2.59724e7	2.24361
$670$	0	0
$671$	7.50330e6	0.643348
$672$	0	0
$673$	9.12086e6	0.776244	0.388122	−	0.921608i	$-0.373124\pi$
$673$	9.12086e6	0.776244	0.388122	+	0.921608i	$0.373124\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.29457e7	1.08556	0.542778	−	0.839876i	$-0.317372\pi$
$677$	1.29457e7	1.08556	0.542778	+	0.839876i	$0.317372\pi$
$678$	0	0
$679$	−1.31128e6	−0.109150
$680$	0	0
$681$	2.94863e7	2.43642
$682$	0	0
$683$	−8.56637e6	−0.702660	−0.351330	−	0.936252i	$-0.614270\pi$
$683$	−8.56637e6	−0.702660	−0.351330	+	0.936252i	$0.614270\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	2.91523e6	0.235658
$688$	0	0
$689$	−1.44722e7	−1.16141
$690$	0	0
$691$	1.01163e7	0.805984	0.402992	−	0.915204i	$-0.367970\pi$
$691$	1.01163e7	0.805984	0.402992	+	0.915204i	$0.367970\pi$
$692$	0	0
$693$	1.02042e7	0.807138
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1.46626e7	1.14322
$698$	0	0
$699$	−9.00846e6	−0.697361
$700$	0	0
$701$	−1.95732e7	−1.50441	−0.752207	−	0.658927i	$-0.771011\pi$
$701$	−1.95732e7	−1.50441	−0.752207	+	0.658927i	$0.771011\pi$
$702$	0	0
$703$	615372.	0.0469623
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.08591e6	0.156945
$708$	0	0
$709$	2.36252e7	1.76506	0.882531	−	0.470254i	$-0.155838\pi$
$709$	2.36252e7	1.76506	0.882531	+	0.470254i	$0.155838\pi$
$710$	0	0
$711$	−1.39690e7	−1.03632
$712$	0	0
$713$	−160948.	−0.0118567
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−4.39617e7	−3.19357
$718$	0	0
$719$	−2.44994e7	−1.76740	−0.883698	−	0.468058i	$-0.844954\pi$
$719$	−2.44994e7	−1.76740	−0.883698	+	0.468058i	$0.844954\pi$
$720$	0	0
$721$	4.00892e6	0.287204
$722$	0	0
$723$	−1.45588e7	−1.03581
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−1.75199e7	−1.22941	−0.614703	−	0.788759i	$-0.710724\pi$
$727$	−1.75199e7	−1.22941	−0.614703	+	0.788759i	$0.710724\pi$
$728$	0	0
$729$	1.23387e7	0.859904
$730$	0	0
$731$	3.03410e7	2.10008
$732$	0	0
$733$	−2.29025e7	−1.57443	−0.787215	−	0.616679i	$-0.788478\pi$
$733$	−2.29025e7	−1.57443	−0.787215	+	0.616679i	$0.788478\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2.29131e7	1.55387
$738$	0	0
$739$	−1.31983e7	−0.889010	−0.444505	−	0.895776i	$-0.646620\pi$
$739$	−1.31983e7	−0.889010	−0.444505	+	0.895776i	$0.646620\pi$
$740$	0	0
$741$	−8.81988e6	−0.590089
$742$	0	0
$743$	−1.89399e6	−0.125865	−0.0629326	−	0.998018i	$-0.520045\pi$
$743$	−1.89399e6	−0.125865	−0.0629326	+	0.998018i	$0.520045\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	4.10094e7	2.68894
$748$	0	0
$749$	−3.72704e6	−0.242750
$750$	0	0
$751$	1.71988e7	1.11275	0.556376	−	0.830930i	$-0.312191\pi$
$751$	1.71988e7	1.11275	0.556376	+	0.830930i	$0.312191\pi$
$752$	0	0
$753$	−1.46926e7	−0.944304
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−2.02697e7	−1.28561	−0.642804	−	0.766031i	$-0.722229\pi$
$757$	−2.02697e7	−1.28561	−0.642804	+	0.766031i	$0.722229\pi$
$758$	0	0
$759$	1.87444e7	1.18104
$760$	0	0
$761$	−6.54894e6	−0.409929	−0.204965	−	0.978769i	$-0.565708\pi$
$761$	−6.54894e6	−0.409929	−0.204965	+	0.978769i	$0.565708\pi$
$762$	0	0
$763$	446710.	0.0277789
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.91803e7	1.17725
$768$	0	0
$769$	−2.48043e7	−1.51256	−0.756278	−	0.654251i	$-0.772984\pi$
$769$	−2.48043e7	−1.51256	−0.756278	+	0.654251i	$0.772984\pi$
$770$	0	0
$771$	−7.35357e6	−0.445515
$772$	0	0
$773$	−2.27616e7	−1.37010	−0.685052	−	0.728495i	$-0.740220\pi$
$773$	−2.27616e7	−1.37010	−0.685052	+	0.728495i	$0.740220\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−2.34379e6	−0.139273
$778$	0	0
$779$	−2.54031e6	−0.149983
$780$	0	0
$781$	4.66302e6	0.273552
$782$	0	0
$783$	−7.54221e7	−4.39637
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1.56494e7	0.900658	0.450329	−	0.892863i	$-0.351307\pi$
$787$	1.56494e7	0.900658	0.450329	+	0.892863i	$0.351307\pi$
$788$	0	0
$789$	−3.80341e7	−2.17511
$790$	0	0
$791$	9.90976e6	0.563147
$792$	0	0
$793$	1.74218e7	0.983809
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2.26205e7	1.26141	0.630704	−	0.776023i	$-0.282766\pi$
$797$	2.26205e7	1.26141	0.630704	+	0.776023i	$0.282766\pi$
$798$	0	0
$799$	3.42050e7	1.89549
$800$	0	0
$801$	−3.53406e7	−1.94622
$802$	0	0
$803$	−1.67098e6	−0.0914499
$804$	0	0
$805$	0	0
$806$	0	0
$807$	4.10602e7	2.21941
$808$	0	0
$809$	−2.25367e7	−1.21065	−0.605326	−	0.795977i	$-0.706957\pi$
$809$	−2.25367e7	−1.21065	−0.605326	+	0.795977i	$0.706957\pi$
$810$	0	0
$811$	410196.	0.0218997	0.0109499	−	0.999940i	$-0.496514\pi$
$811$	410196.	0.0218997	0.0109499	+	0.999940i	$0.496514\pi$
$812$	0	0
$813$	−2.02855e7	−1.07636
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−5.25662e6	−0.275519
$818$	0	0
$819$	2.36931e7	1.23428
$820$	0	0
$821$	1.62749e7	0.842678	0.421339	−	0.906903i	$-0.361560\pi$
$821$	1.62749e7	0.842678	0.421339	+	0.906903i	$0.361560\pi$
$822$	0	0
$823$	−628881.	−0.0323645	−0.0161823	−	0.999869i	$-0.505151\pi$
$823$	−628881.	−0.0323645	−0.0161823	+	0.999869i	$0.505151\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−1.94403e7	−0.988417	−0.494209	−	0.869343i	$-0.664542\pi$
$827$	−1.94403e7	−0.988417	−0.494209	+	0.869343i	$0.664542\pi$
$828$	0	0
$829$	3.39088e7	1.71366	0.856832	−	0.515595i	$-0.172429\pi$
$829$	3.39088e7	1.71366	0.856832	+	0.515595i	$0.172429\pi$
$830$	0	0
$831$	9.03614e6	0.453921
$832$	0	0
$833$	−2.75679e7	−1.37655
$834$	0	0
$835$	0	0
$836$	0	0
$837$	997985.	0.0492391
$838$	0	0
$839$	−6.84583e6	−0.335754	−0.167877	−	0.985808i	$-0.553691\pi$
$839$	−6.84583e6	−0.335754	−0.167877	+	0.985808i	$0.553691\pi$
$840$	0	0
$841$	3.96642e7	1.93379
$842$	0	0
$843$	1.25715e7	0.609282
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−516280.	−0.0247273
$848$	0	0
$849$	5.97997e7	2.84728
$850$	0	0
$851$	−3.03660e6	−0.143735
$852$	0	0
$853$	2.67852e7	1.26044	0.630221	−	0.776416i	$-0.282964\pi$
$853$	2.67852e7	1.26044	0.630221	+	0.776416i	$0.282964\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−9.32339e6	−0.433632	−0.216816	−	0.976212i	$-0.569567\pi$
$857$	−9.32339e6	−0.433632	−0.216816	+	0.976212i	$0.569567\pi$
$858$	0	0
$859$	−3.73055e7	−1.72500	−0.862502	−	0.506054i	$-0.831104\pi$
$859$	−3.73055e7	−1.72500	−0.862502	+	0.506054i	$0.831104\pi$
$860$	0	0
$861$	9.67537e6	0.444795
$862$	0	0
$863$	−5.18636e6	−0.237048	−0.118524	−	0.992951i	$-0.537816\pi$
$863$	−5.18636e6	−0.237048	−0.118524	+	0.992951i	$0.537816\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−5.58265e7	−2.52228
$868$	0	0
$869$	9.99888e6	0.449161
$870$	0	0
$871$	5.32016e7	2.37618
$872$	0	0
$873$	1.80945e7	0.803546
$874$	0	0
$875$	0	0
$876$	0	0
$877$	5.63396e6	0.247352	0.123676	−	0.992323i	$-0.460532\pi$
$877$	5.63396e6	0.247352	0.123676	+	0.992323i	$0.460532\pi$
$878$	0	0
$879$	2.60498e6	0.113719
$880$	0	0
$881$	−5.76880e6	−0.250407	−0.125203	−	0.992131i	$-0.539958\pi$
$881$	−5.76880e6	−0.250407	−0.125203	+	0.992131i	$0.539958\pi$
$882$	0	0
$883$	2.60630e7	1.12492	0.562462	−	0.826823i	$-0.309854\pi$
$883$	2.60630e7	1.12492	0.562462	+	0.826823i	$0.309854\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	3.24889e7	1.38652	0.693259	−	0.720688i	$-0.256174\pi$
$887$	3.24889e7	1.38652	0.693259	+	0.720688i	$0.256174\pi$
$888$	0	0
$889$	4.32145e6	0.183390
$890$	0	0
$891$	−5.73942e7	−2.42200
$892$	0	0
$893$	−5.92607e6	−0.248678
$894$	0	0
$895$	0	0
$896$	0	0
$897$	4.35223e7	1.80605
$898$	0	0
$899$	−796240.	−0.0328583
$900$	0	0
$901$	2.74612e7	1.12696
$902$	0	0
$903$	2.00211e7	0.817087
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−3.11745e7	−1.25829	−0.629145	−	0.777288i	$-0.716595\pi$
$907$	−3.11745e7	−1.25829	−0.629145	+	0.777288i	$0.716595\pi$
$908$	0	0
$909$	−2.87836e7	−1.15541
$910$	0	0
$911$	−3.58254e7	−1.43019	−0.715097	−	0.699025i	$-0.753618\pi$
$911$	−3.58254e7	−1.43019	−0.715097	+	0.699025i	$0.753618\pi$
$912$	0	0
$913$	−2.93540e7	−1.16544
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.13276e6	0.0837565
$918$	0	0
$919$	3.14710e7	1.22920	0.614598	−	0.788840i	$-0.289318\pi$
$919$	3.14710e7	1.22920	0.614598	+	0.788840i	$0.289318\pi$
$920$	0	0
$921$	1.64167e7	0.637730
$922$	0	0
$923$	1.08270e7	0.418316
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−5.53194e7	−2.11436
$928$	0	0
$929$	532600.	0.0202471	0.0101235	−	0.999949i	$-0.496778\pi$
$929$	532600.	0.0202471	0.0101235	+	0.999949i	$0.496778\pi$
$930$	0	0
$931$	4.77618e6	0.180595
$932$	0	0
$933$	7.45215e7	2.80270
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−1.97320e7	−0.734214	−0.367107	−	0.930179i	$-0.619652\pi$
$937$	−1.97320e7	−0.734214	−0.367107	+	0.930179i	$0.619652\pi$
$938$	0	0
$939$	−1.46639e7	−0.542732
$940$	0	0
$941$	−2.12060e7	−0.780702	−0.390351	−	0.920666i	$-0.627646\pi$
$941$	−2.12060e7	−0.780702	−0.390351	+	0.920666i	$0.627646\pi$
$942$	0	0
$943$	1.25353e7	0.459047
$944$	0	0
$945$	0	0
$946$	0	0
$947$	4.60224e7	1.66761	0.833804	−	0.552060i	$-0.186158\pi$
$947$	4.60224e7	1.66761	0.833804	+	0.552060i	$0.186158\pi$
$948$	0	0
$949$	−3.87984e6	−0.139845
$950$	0	0
$951$	−9.67802e7	−3.47004
$952$	0	0
$953$	−4.76525e7	−1.69963	−0.849813	−	0.527085i	$-0.823285\pi$
$953$	−4.76525e7	−1.69963	−0.849813	+	0.527085i	$0.823285\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	9.27316e7	3.27301
$958$	0	0
$959$	2.01511e6	0.0707543
$960$	0	0
$961$	−2.86186e7	−0.999632
$962$	0	0
$963$	5.14297e7	1.78710
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−2.29686e7	−0.789893	−0.394946	−	0.918704i	$-0.629237\pi$
$967$	−2.29686e7	−0.789893	−0.394946	+	0.918704i	$0.629237\pi$
$968$	0	0
$969$	1.67358e7	0.572583
$970$	0	0
$971$	−2.22231e7	−0.756408	−0.378204	−	0.925722i	$-0.623458\pi$
$971$	−2.22231e7	−0.756408	−0.378204	+	0.925722i	$0.623458\pi$
$972$	0	0
$973$	−9.28031e6	−0.314254
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3.68581e7	1.23537	0.617684	−	0.786426i	$-0.288071\pi$
$977$	3.68581e7	1.23537	0.617684	+	0.786426i	$0.288071\pi$
$978$	0	0
$979$	2.52964e7	0.843532
$980$	0	0
$981$	−6.16418e6	−0.204504
$982$	0	0
$983$	−4.53136e7	−1.49570	−0.747850	−	0.663868i	$-0.768914\pi$
$983$	−4.53136e7	−1.49570	−0.747850	+	0.663868i	$0.768914\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	2.25708e7	0.737487
$988$	0	0
$989$	2.59392e7	0.843268
$990$	0	0
$991$	3.68582e7	1.19220	0.596102	−	0.802909i	$-0.296716\pi$
$991$	3.68582e7	1.19220	0.596102	+	0.802909i	$0.296716\pi$
$992$	0	0
$993$	1.09169e8	3.51339
$994$	0	0
$995$	0	0
$996$	0	0
$997$	4.19245e7	1.33577	0.667883	−	0.744266i	$-0.267201\pi$
$997$	4.19245e7	1.33577	0.667883	+	0.744266i	$0.267201\pi$
$998$	0	0
$999$	1.88289e7	0.596912

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.6.a.q.1.1		2
4.3	odd	2		200.6.a.g.1.2		2
5.2	odd	4		400.6.c.l.49.4		4
5.3	odd	4		400.6.c.l.49.1		4
5.4	even	2		80.6.a.i.1.2		2
15.14	odd	2		720.6.a.z.1.2		2
20.3	even	4		200.6.c.e.49.4		4
20.7	even	4		200.6.c.e.49.1		4
20.19	odd	2		40.6.a.d.1.1	✓	2
40.19	odd	2		320.6.a.w.1.2		2
40.29	even	2		320.6.a.q.1.1		2
60.59	even	2		360.6.a.l.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.6.a.d.1.1	✓	2	20.19	odd	2
80.6.a.i.1.2		2	5.4	even	2
200.6.a.g.1.2		2	4.3	odd	2
200.6.c.e.49.1		4	20.7	even	4
200.6.c.e.49.4		4	20.3	even	4
320.6.a.q.1.1		2	40.29	even	2
320.6.a.w.1.2		2	40.19	odd	2
360.6.a.l.1.1		2	60.59	even	2
400.6.a.q.1.1		2	1.1	even	1	trivial
400.6.c.l.49.1		4	5.3	odd	4
400.6.c.l.49.4		4	5.2	odd	4
720.6.a.z.1.2		2	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	400.a (trivial)

\(f(q)\)	\(=\)	\(q-28.7156 q^{3} -42.1469 q^{7} +581.588 q^{9} +O(q^{10})\)	\(q-28.7156 q^{3} -42.1469 q^{7} +581.588 q^{9} -416.294 q^{11} -966.588 q^{13} +1834.11 q^{17} -317.763 q^{19} +1210.27 q^{21} +1568.02 q^{23} -9722.76 q^{27} +7757.28 q^{29} -102.644 q^{31} +11954.1 q^{33} -1936.58 q^{37} +27756.2 q^{39} +7994.36 q^{41} +16542.6 q^{43} +18649.3 q^{47} -15030.6 q^{49} -52667.7 q^{51} +14972.4 q^{53} +9124.76 q^{57} -19843.3 q^{59} -18024.1 q^{61} -24512.1 q^{63} -55040.6 q^{67} -45026.8 q^{69} -11201.3 q^{71} +4013.95 q^{73} +17545.5 q^{77} -24018.8 q^{79} +137869. q^{81} +70512.8 q^{83} -222755. q^{87} -60765.7 q^{89} +40738.7 q^{91} +2947.49 q^{93} +31112.2 q^{97} -242111. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 12 q^{3} + 52 q^{7} + 618 q^{9}+O(q^{10}) \) 2 * q - 12 * q^3 + 52 * q^7 + 618 * q^9	\( 2 q - 12 q^{3} + 52 q^{7} + 618 q^{9} - 560 q^{11} - 1388 q^{13} - 148 q^{17} + 1000 q^{19} + 2784 q^{21} - 2452 q^{23} - 13176 q^{27} + 1340 q^{29} + 2248 q^{31} + 9552 q^{33} + 5940 q^{37} + 20712 q^{39} + 23076 q^{41} + 17684 q^{43} - 2908 q^{47} - 22974 q^{49} - 85800 q^{51} + 5412 q^{53} + 31152 q^{57} - 62584 q^{59} + 14108 q^{61} - 21084 q^{63} - 85412 q^{67} - 112224 q^{69} - 47208 q^{71} + 67452 q^{73} + 4016 q^{77} + 65904 q^{79} + 71298 q^{81} + 108724 q^{83} - 330024 q^{87} - 55020 q^{89} + 1064 q^{91} + 42240 q^{93} - 147668 q^{97} - 247344 q^{99}+O(q^{100}) \) 2 * q - 12 * q^3 + 52 * q^7 + 618 * q^9 - 560 * q^11 - 1388 * q^13 - 148 * q^17 + 1000 * q^19 + 2784 * q^21 - 2452 * q^23 - 13176 * q^27 + 1340 * q^29 + 2248 * q^31 + 9552 * q^33 + 5940 * q^37 + 20712 * q^39 + 23076 * q^41 + 17684 * q^43 - 2908 * q^47 - 22974 * q^49 - 85800 * q^51 + 5412 * q^53 + 31152 * q^57 - 62584 * q^59 + 14108 * q^61 - 21084 * q^63 - 85412 * q^67 - 112224 * q^69 - 47208 * q^71 + 67452 * q^73 + 4016 * q^77 + 65904 * q^79 + 71298 * q^81 + 108724 * q^83 - 330024 * q^87 - 55020 * q^89 + 1064 * q^91 + 42240 * q^93 - 147668 * q^97 - 247344 * q^99

Introduction

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