LMFDB - Embedded newform 800.6.c.i.449.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,6,Mod(449,800)]

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, 0, 1]))

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

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

chi := DirichletCharacter("800.449");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	800.c (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:	$128.307055850$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{10})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 25 $ x^4 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 160)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.1
Root			$1.58114 + 1.58114i$ of defining polynomial
Character	$\chi$	$=$	800.449
Dual form			800.6.c.i.449.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.32456i q^{3} +44.2719i q^{7} +203.000 q^{9} +O(q^{10})$	$q-6.32456i q^{3} +44.2719i q^{7} +203.000 q^{9} -720.999 q^{11} +146.000i q^{13} -702.000i q^{17} -2732.21 q^{19} +280.000 q^{21} +4091.99i q^{23} -2820.75i q^{27} +4010.00 q^{29} +4566.33 q^{31} +4560.00i q^{33} -14778.0i q^{37} +923.385 q^{39} -4350.00 q^{41} -12427.8i q^{43} +6014.65i q^{47} +14847.0 q^{49} -4439.84 q^{51} +18154.0i q^{53} +17280.0i q^{57} +19707.3 q^{59} -42130.0 q^{61} +8987.19i q^{63} +16184.5i q^{67} +25880.0 q^{69} +45448.3 q^{71} -26266.0i q^{73} -31920.0i q^{77} +8677.29 q^{79} +31489.0 q^{81} +98757.9i q^{83} -25361.5i q^{87} -30570.0 q^{89} -6463.70 q^{91} -28880.0i q^{93} +66882.0i q^{97} -146363. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 812 q^{9}+O(q^{10}) $ 4 * q + 812 * q^9	$ 4 q + 812 q^{9} + 1120 q^{21} + 16040 q^{29} - 17400 q^{41} + 59388 q^{49} - 168520 q^{61} + 103520 q^{69} + 125956 q^{81} - 122280 q^{89}+O(q^{100}) $ 4 * q + 812 * q^9 + 1120 * q^21 + 16040 * q^29 - 17400 * q^41 + 59388 * q^49 - 168520 * q^61 + 103520 * q^69 + 125956 * q^81 - 122280 * q^89

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
$3$	− 6.32456i	− 0.405720i	−0.979208	−	0.202860i	$-0.934976\pi$
$3$	− 6.32456i	− 0.405720i	0.979208	−	0.202860i	$-0.0650237\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	44.2719i	0.341494i	0.985315	+	0.170747i	$0.0546181\pi$
$7$	44.2719i	0.341494i	−0.985315	+	0.170747i	$0.945382\pi$
$8$	0	0
$9$	203.000	0.835391
$10$	0	0
$11$	−720.999	−1.79661	−0.898304	−	0.439375i	$-0.855200\pi$
$11$	−720.999	−1.79661	−0.898304	+	0.439375i	$0.855200\pi$
$12$	0	0
$13$	146.000i	0.239604i	0.992798	+	0.119802i	$0.0382260\pi$
$13$	146.000i	0.239604i	−0.992798	+	0.119802i	$0.961774\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 702.000i	− 0.589135i	−0.955631	−	0.294567i	$-0.904824\pi$
$17$	− 702.000i	− 0.589135i	0.955631	−	0.294567i	$-0.0951756\pi$
$18$	0	0
$19$	−2732.21	−1.73632	−0.868160	−	0.496285i	$-0.834697\pi$
$19$	−2732.21	−1.73632	−0.868160	+	0.496285i	$0.834697\pi$
$20$	0	0
$21$	280.000	0.138551
$22$	0	0
$23$	4091.99i	1.61293i	0.591284	+	0.806463i	$0.298621\pi$
$23$	4091.99i	1.61293i	−0.591284	+	0.806463i	$0.701379\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 2820.75i	− 0.744656i
$28$	0	0
$29$	4010.00	0.885420	0.442710	−	0.896665i	$-0.354017\pi$
$29$	4010.00	0.885420	0.442710	+	0.896665i	$0.354017\pi$
$30$	0	0
$31$	4566.33	0.853420	0.426710	−	0.904388i	$-0.359672\pi$
$31$	4566.33	0.853420	0.426710	+	0.904388i	$0.359672\pi$
$32$	0	0
$33$	4560.00i	0.728920i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 14778.0i	− 1.77464i	−0.461150	−	0.887322i	$-0.652563\pi$
$37$	− 14778.0i	− 1.77464i	0.461150	−	0.887322i	$-0.347437\pi$
$38$	0	0
$39$	923.385	0.0972123
$40$	0	0
$41$	−4350.00	−0.404138	−0.202069	−	0.979371i	$-0.564767\pi$
$41$	−4350.00	−0.404138	−0.202069	+	0.979371i	$0.564767\pi$
$42$	0	0
$43$	− 12427.8i	− 1.02499i	−0.858689	−	0.512497i	$-0.828721\pi$
$43$	− 12427.8i	− 1.02499i	0.858689	−	0.512497i	$-0.171279\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6014.65i	0.397160i	0.980085	+	0.198580i	$0.0636330\pi$
$47$	6014.65i	0.397160i	−0.980085	+	0.198580i	$0.936367\pi$
$48$	0	0
$49$	14847.0	0.883382
$50$	0	0
$51$	−4439.84	−0.239024
$52$	0	0
$53$	18154.0i	0.887734i	0.896093	+	0.443867i	$0.146394\pi$
$53$	18154.0i	0.887734i	−0.896093	+	0.443867i	$0.853606\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	17280.0i	0.704460i
$58$	0	0
$59$	19707.3	0.737051	0.368525	−	0.929618i	$-0.379863\pi$
$59$	19707.3	0.737051	0.368525	+	0.929618i	$0.379863\pi$
$60$	0	0
$61$	−42130.0	−1.44966	−0.724831	−	0.688927i	$-0.758082\pi$
$61$	−42130.0	−1.44966	−0.724831	+	0.688927i	$0.758082\pi$
$62$	0	0
$63$	8987.19i	0.285281i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	16184.5i	0.440467i	0.975447	+	0.220233i	$0.0706819\pi$
$67$	16184.5i	0.440467i	−0.975447	+	0.220233i	$0.929318\pi$
$68$	0	0
$69$	25880.0	0.654397
$70$	0	0
$71$	45448.3	1.06997	0.534985	−	0.844862i	$-0.320317\pi$
$71$	45448.3	1.06997	0.534985	+	0.844862i	$0.320317\pi$
$72$	0	0
$73$	− 26266.0i	− 0.576882i	−0.957498	−	0.288441i	$-0.906863\pi$
$73$	− 26266.0i	− 0.576882i	0.957498	−	0.288441i	$-0.0931369\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 31920.0i	− 0.613530i
$78$	0	0
$79$	8677.29	0.156429	0.0782143	−	0.996937i	$-0.475078\pi$
$79$	8677.29	0.156429	0.0782143	+	0.996937i	$0.475078\pi$
$80$	0	0
$81$	31489.0	0.533269
$82$	0	0
$83$	98757.9i	1.57354i	0.617249	+	0.786768i	$0.288247\pi$
$83$	98757.9i	1.57354i	−0.617249	+	0.786768i	$0.711753\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 25361.5i	− 0.359233i
$88$	0	0
$89$	−30570.0	−0.409091	−0.204546	−	0.978857i	$-0.565572\pi$
$89$	−30570.0	−0.409091	−0.204546	+	0.978857i	$0.565572\pi$
$90$	0	0
$91$	−6463.70	−0.0818234
$92$	0	0
$93$	− 28880.0i	− 0.346250i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	66882.0i	0.721739i	0.932616	+	0.360869i	$0.117520\pi$
$97$	66882.0i	0.721739i	−0.932616	+	0.360869i	$0.882480\pi$
$98$	0	0
$99$	−146363.	−1.50087
$100$	0	0
$101$	42798.0	0.417465	0.208732	−	0.977973i	$-0.433066\pi$
$101$	42798.0	0.417465	0.208732	+	0.977973i	$0.433066\pi$
$102$	0	0
$103$	− 10505.1i	− 0.0975678i	−0.998809	−	0.0487839i	$-0.984465\pi$
$103$	− 10505.1i	− 0.0975678i	0.998809	−	0.0487839i	$-0.0155346\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	66856.9i	0.564529i	0.959337	+	0.282265i	$0.0910857\pi$
$107$	66856.9i	0.564529i	−0.959337	+	0.282265i	$0.908914\pi$
$108$	0	0
$109$	111714.	0.900620	0.450310	−	0.892872i	$-0.351314\pi$
$109$	111714.	0.900620	0.450310	+	0.892872i	$0.351314\pi$
$110$	0	0
$111$	−93464.3	−0.720009
$112$	0	0
$113$	− 216834.i	− 1.59746i	−0.601686	−	0.798732i	$-0.705504\pi$
$113$	− 216834.i	− 1.59746i	0.601686	−	0.798732i	$-0.294496\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	29638.0i	0.200163i
$118$	0	0
$119$	31078.9	0.201186
$120$	0	0
$121$	358789.	2.22780
$122$	0	0
$123$	27511.8i	0.163967i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	62277.9i	0.342629i	0.985216	+	0.171315i	$0.0548015\pi$
$127$	62277.9i	0.342629i	−0.985216	+	0.171315i	$0.945199\pi$
$128$	0	0
$129$	−78600.0	−0.415861
$130$	0	0
$131$	176189.	0.897019	0.448510	−	0.893778i	$-0.351955\pi$
$131$	176189.	0.897019	0.448510	+	0.893778i	$0.351955\pi$
$132$	0	0
$133$	− 120960.i	− 0.592943i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	99802.0i	0.454295i	0.973860	+	0.227147i	$0.0729399\pi$
$137$	99802.0i	0.454295i	−0.973860	+	0.227147i	$0.927060\pi$
$138$	0	0
$139$	271475.	1.19177	0.595886	−	0.803069i	$-0.296801\pi$
$139$	271475.	1.19177	0.595886	+	0.803069i	$0.296801\pi$
$140$	0	0
$141$	38040.0	0.161136
$142$	0	0
$143$	− 105266.i	− 0.430475i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 93900.7i	− 0.358406i
$148$	0	0
$149$	413626.	1.52631	0.763154	−	0.646217i	$-0.223650\pi$
$149$	413626.	1.52631	0.763154	+	0.646217i	$0.223650\pi$
$150$	0	0
$151$	172496.	0.615654	0.307827	−	0.951442i	$-0.400398\pi$
$151$	172496.	0.615654	0.307827	+	0.951442i	$0.400398\pi$
$152$	0	0
$153$	− 142506.i	− 0.492158i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	179358.i	0.580726i	0.956917	+	0.290363i	$0.0937761\pi$
$157$	179358.i	0.580726i	−0.956917	+	0.290363i	$0.906224\pi$
$158$	0	0
$159$	114816.	0.360172
$160$	0	0
$161$	−181160.	−0.550805
$162$	0	0
$163$	− 465924.i	− 1.37355i	−0.726868	−	0.686777i	$-0.759025\pi$
$163$	− 465924.i	− 1.37355i	0.726868	−	0.686777i	$-0.240975\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 609567.i	− 1.69134i	−0.533708	−	0.845669i	$-0.679202\pi$
$167$	− 609567.i	− 1.69134i	0.533708	−	0.845669i	$-0.320798\pi$
$168$	0	0
$169$	349977.	0.942590
$170$	0	0
$171$	−554638.	−1.45051
$172$	0	0
$173$	− 591086.i	− 1.50153i	−0.660567	−	0.750767i	$-0.729684\pi$
$173$	− 591086.i	− 1.50153i	0.660567	−	0.750767i	$-0.270316\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 124640.i	− 0.299037i
$178$	0	0
$179$	−211215.	−0.492711	−0.246355	−	0.969180i	$-0.579233\pi$
$179$	−211215.	−0.492711	−0.246355	+	0.969180i	$0.579233\pi$
$180$	0	0
$181$	−97538.0	−0.221298	−0.110649	−	0.993860i	$-0.535293\pi$
$181$	−97538.0	−0.221298	−0.110649	+	0.993860i	$0.535293\pi$
$182$	0	0
$183$	266454.i	0.588158i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	506142.i	1.05844i
$188$	0	0
$189$	124880.	0.254295
$190$	0	0
$191$	307158.	0.609227	0.304613	−	0.952476i	$-0.401473\pi$
$191$	307158.	0.609227	0.304613	+	0.952476i	$0.401473\pi$
$192$	0	0
$193$	− 12434.0i	− 0.0240280i	−0.999928	−	0.0120140i	$-0.996176\pi$
$193$	− 12434.0i	− 0.0240280i	0.999928	−	0.0120140i	$-0.00382427\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 378858.i	− 0.695522i	−0.937583	−	0.347761i	$-0.886942\pi$
$197$	− 378858.i	− 0.695522i	0.937583	−	0.347761i	$-0.113058\pi$
$198$	0	0
$199$	−767194.	−1.37332	−0.686661	−	0.726978i	$-0.740924\pi$
$199$	−767194.	−1.37332	−0.686661	+	0.726978i	$0.740924\pi$
$200$	0	0
$201$	102360.	0.178706
$202$	0	0
$203$	177530.i	0.302366i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	830673.i	1.34742i
$208$	0	0
$209$	1.96992e6	3.11948
$210$	0	0
$211$	−864529.	−1.33682	−0.668411	−	0.743793i	$-0.733025\pi$
$211$	−864529.	−1.33682	−0.668411	+	0.743793i	$0.733025\pi$
$212$	0	0
$213$	− 287440.i	− 0.434108i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	202160.i	0.291438i
$218$	0	0
$219$	−166121.	−0.234053
$220$	0	0
$221$	102492.	0.141159
$222$	0	0
$223$	639710.i	0.861432i	0.902488	+	0.430716i	$0.141739\pi$
$223$	639710.i	0.861432i	−0.902488	+	0.430716i	$0.858261\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 980515.i	− 1.26296i	−0.775392	−	0.631480i	$-0.782448\pi$
$227$	− 980515.i	− 1.26296i	0.775392	−	0.631480i	$-0.217552\pi$
$228$	0	0
$229$	1.01261e6	1.27601	0.638004	−	0.770033i	$-0.279760\pi$
$229$	1.01261e6	1.27601	0.638004	+	0.770033i	$0.279760\pi$
$230$	0	0
$231$	−201880.	−0.248922
$232$	0	0
$233$	706326.i	0.852345i	0.904642	+	0.426172i	$0.140138\pi$
$233$	706326.i	0.852345i	−0.904642	+	0.426172i	$0.859862\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 54880.0i	− 0.0634663i
$238$	0	0
$239$	1.19390e6	1.35199	0.675994	−	0.736907i	$-0.263715\pi$
$239$	1.19390e6	1.35199	0.675994	+	0.736907i	$0.263715\pi$
$240$	0	0
$241$	404410.	0.448517	0.224259	−	0.974530i	$-0.428004\pi$
$241$	404410.	0.448517	0.224259	+	0.974530i	$0.428004\pi$
$242$	0	0
$243$	− 884597.i	− 0.961014i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 398902.i	− 0.416030i
$248$	0	0
$249$	624600.	0.638416
$250$	0	0
$251$	1.43258e6	1.43527	0.717634	−	0.696420i	$-0.245225\pi$
$251$	1.43258e6	1.43527	0.717634	+	0.696420i	$0.245225\pi$
$252$	0	0
$253$	− 2.95032e6i	− 2.89780i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 987982.i	− 0.933074i	−0.884502	−	0.466537i	$-0.845501\pi$
$257$	− 987982.i	− 0.933074i	0.884502	−	0.466537i	$-0.154499\pi$
$258$	0	0
$259$	654250.	0.606030
$260$	0	0
$261$	814030.	0.739672
$262$	0	0
$263$	− 2.06222e6i	− 1.83842i	−0.393767	−	0.919210i	$-0.628828\pi$
$263$	− 2.06222e6i	− 1.83842i	0.393767	−	0.919210i	$-0.371172\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	193342.i	0.165977i
$268$	0	0
$269$	780386.	0.657550	0.328775	−	0.944408i	$-0.393364\pi$
$269$	780386.	0.657550	0.328775	+	0.944408i	$0.393364\pi$
$270$	0	0
$271$	−562291.	−0.465091	−0.232545	−	0.972586i	$-0.574705\pi$
$271$	−562291.	−0.465091	−0.232545	+	0.972586i	$0.574705\pi$
$272$	0	0
$273$	40880.0i	0.0331974i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	386758.i	0.302859i	0.988468	+	0.151429i	$0.0483876\pi$
$277$	386758.i	0.302859i	−0.988468	+	0.151429i	$0.951612\pi$
$278$	0	0
$279$	926965.	0.712940
$280$	0	0
$281$	1.55485e6	1.17469	0.587344	−	0.809337i	$-0.300173\pi$
$281$	1.55485e6	1.17469	0.587344	+	0.809337i	$0.300173\pi$
$282$	0	0
$283$	1.75157e6i	1.30005i	0.759912	+	0.650026i	$0.225242\pi$
$283$	1.75157e6i	1.30005i	−0.759912	+	0.650026i	$0.774758\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 192583.i	− 0.138011i
$288$	0	0
$289$	927053.	0.652920
$290$	0	0
$291$	422999.	0.292824
$292$	0	0
$293$	− 1.55309e6i	− 1.05689i	−0.848968	−	0.528444i	$-0.822776\pi$
$293$	− 1.55309e6i	− 1.05689i	0.848968	−	0.528444i	$-0.177224\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	2.03376e6i	1.33785i
$298$	0	0
$299$	−597430.	−0.386464
$300$	0	0
$301$	550200.	0.350029
$302$	0	0
$303$	− 270678.i	− 0.169374i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	2.60409e6i	1.57692i	0.615085	+	0.788461i	$0.289122\pi$
$307$	2.60409e6i	1.57692i	−0.615085	+	0.788461i	$0.710878\pi$
$308$	0	0
$309$	−66440.0	−0.0395853
$310$	0	0
$311$	−302124.	−0.177127	−0.0885634	−	0.996071i	$-0.528228\pi$
$311$	−302124.	−0.177127	−0.0885634	+	0.996071i	$0.528228\pi$
$312$	0	0
$313$	− 2.09455e6i	− 1.20846i	−0.796812	−	0.604228i	$-0.793482\pi$
$313$	− 2.09455e6i	− 1.20846i	0.796812	−	0.604228i	$-0.206518\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.07624e6i	0.601534i	0.953698	+	0.300767i	$0.0972426\pi$
$317$	1.07624e6i	0.601534i	−0.953698	+	0.300767i	$0.902757\pi$
$318$	0	0
$319$	−2.89121e6	−1.59075
$320$	0	0
$321$	422840.	0.229041
$322$	0	0
$323$	1.91801e6i	1.02293i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 706541.i	− 0.365400i
$328$	0	0
$329$	−266280.	−0.135628
$330$	0	0
$331$	313002.	0.157028	0.0785141	−	0.996913i	$-0.474982\pi$
$331$	313002.	0.157028	0.0785141	+	0.996913i	$0.474982\pi$
$332$	0	0
$333$	− 2.99993e6i	− 1.48252i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	1.55400e6i	0.745378i	0.927956	+	0.372689i	$0.121564\pi$
$337$	1.55400e6i	0.745378i	−0.927956	+	0.372689i	$0.878436\pi$
$338$	0	0
$339$	−1.37138e6	−0.648124
$340$	0	0
$341$	−3.29232e6	−1.53326
$342$	0	0
$343$	1.40138e6i	0.643163i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 2.22208e6i	− 0.990684i	−0.868698	−	0.495342i	$-0.835043\pi$
$347$	− 2.22208e6i	− 0.990684i	0.868698	−	0.495342i	$-0.164957\pi$
$348$	0	0
$349$	774570.	0.340406	0.170203	−	0.985409i	$-0.445558\pi$
$349$	774570.	0.340406	0.170203	+	0.985409i	$0.445558\pi$
$350$	0	0
$351$	411830.	0.178423
$352$	0	0
$353$	2.26217e6i	0.966249i	0.875552	+	0.483125i	$0.160498\pi$
$353$	2.26217e6i	0.966249i	−0.875552	+	0.483125i	$0.839502\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 196560.i	− 0.0816253i
$358$	0	0
$359$	869601.	0.356110	0.178055	−	0.984021i	$-0.443020\pi$
$359$	869601.	0.356110	0.178055	+	0.984021i	$0.443020\pi$
$360$	0	0
$361$	4.98886e6	2.01481
$362$	0	0
$363$	− 2.26918e6i	− 0.903863i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 1.25361e6i	− 0.485844i	−0.970046	−	0.242922i	$-0.921894\pi$
$367$	− 1.25361e6i	− 0.485844i	0.970046	−	0.242922i	$-0.0781059\pi$
$368$	0	0
$369$	−883050.	−0.337613
$370$	0	0
$371$	−803712.	−0.303156
$372$	0	0
$373$	− 175206.i	− 0.0652044i	−0.999468	−	0.0326022i	$-0.989621\pi$
$373$	− 175206.i	− 0.0652044i	0.999468	−	0.0326022i	$-0.0103794\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	585460.i	0.212150i
$378$	0	0
$379$	796945.	0.284990	0.142495	−	0.989795i	$-0.454487\pi$
$379$	796945.	0.284990	0.142495	+	0.989795i	$0.454487\pi$
$380$	0	0
$381$	393880.	0.139012
$382$	0	0
$383$	3.36129e6i	1.17087i	0.810719	+	0.585436i	$0.199077\pi$
$383$	3.36129e6i	1.17087i	−0.810719	+	0.585436i	$0.800923\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 2.52283e6i	− 0.856271i
$388$	0	0
$389$	5.51959e6	1.84941	0.924705	−	0.380685i	$-0.124312\pi$
$389$	5.51959e6	1.84941	0.924705	+	0.380685i	$0.124312\pi$
$390$	0	0
$391$	2.87258e6	0.950232
$392$	0	0
$393$	− 1.11432e6i	− 0.363939i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 1.74738e6i	− 0.556430i	−0.960519	−	0.278215i	$-0.910257\pi$
$397$	− 1.74738e6i	− 0.556430i	0.960519	−	0.278215i	$-0.0897428\pi$
$398$	0	0
$399$	−765018.	−0.240569
$400$	0	0
$401$	541122.	0.168048	0.0840242	−	0.996464i	$-0.473223\pi$
$401$	541122.	0.168048	0.0840242	+	0.996464i	$0.473223\pi$
$402$	0	0
$403$	666684.i	0.204483i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.06549e7i	3.18834i
$408$	0	0
$409$	−3.79699e6	−1.12236	−0.561179	−	0.827694i	$-0.689652\pi$
$409$	−3.79699e6	−1.12236	−0.561179	+	0.827694i	$0.689652\pi$
$410$	0	0
$411$	631203.	0.184317
$412$	0	0
$413$	872480.i	0.251698i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 1.71696e6i	− 0.483526i
$418$	0	0
$419$	3.88108e6	1.07998	0.539992	−	0.841670i	$-0.318427\pi$
$419$	3.88108e6	1.07998	0.539992	+	0.841670i	$0.318427\pi$
$420$	0	0
$421$	−6.66081e6	−1.83156	−0.915781	−	0.401677i	$-0.868427\pi$
$421$	−6.66081e6	−1.83156	−0.915781	+	0.401677i	$0.868427\pi$
$422$	0	0
$423$	1.22097e6i	0.331784i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 1.86517e6i	− 0.495051i
$428$	0	0
$429$	−665760.	−0.174652
$430$	0	0
$431$	−4.65337e6	−1.20663	−0.603315	−	0.797503i	$-0.706154\pi$
$431$	−4.65337e6	−1.20663	−0.603315	+	0.797503i	$0.706154\pi$
$432$	0	0
$433$	6.23873e6i	1.59910i	0.600597	+	0.799552i	$0.294930\pi$
$433$	6.23873e6i	1.59910i	−0.600597	+	0.799552i	$0.705070\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 1.11802e7i	− 2.80056i
$438$	0	0
$439$	4.75743e6	1.17818	0.589089	−	0.808068i	$-0.299487\pi$
$439$	4.75743e6	1.17818	0.589089	+	0.808068i	$0.299487\pi$
$440$	0	0
$441$	3.01394e6	0.737969
$442$	0	0
$443$	− 5.77073e6i	− 1.39708i	−0.715570	−	0.698541i	$-0.753833\pi$
$443$	− 5.77073e6i	− 1.39708i	0.715570	−	0.698541i	$-0.246167\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 2.61600e6i	− 0.619254i
$448$	0	0
$449$	4.22765e6	0.989654	0.494827	−	0.868991i	$-0.335231\pi$
$449$	4.22765e6	0.989654	0.494827	+	0.868991i	$0.335231\pi$
$450$	0	0
$451$	3.13635e6	0.726077
$452$	0	0
$453$	− 1.09096e6i	− 0.249783i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	4.31374e6i	0.966192i	0.875567	+	0.483096i	$0.160488\pi$
$457$	4.31374e6i	0.966192i	−0.875567	+	0.483096i	$0.839512\pi$
$458$	0	0
$459$	−1.98017e6	−0.438703
$460$	0	0
$461$	−5.94820e6	−1.30357	−0.651784	−	0.758405i	$-0.725979\pi$
$461$	−5.94820e6	−1.30357	−0.651784	+	0.758405i	$0.725979\pi$
$462$	0	0
$463$	− 8.39722e6i	− 1.82047i	−0.414094	−	0.910234i	$-0.635902\pi$
$463$	− 8.39722e6i	− 1.82047i	0.414094	−	0.910234i	$-0.364098\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	5.30976e6i	1.12663i	0.826241	+	0.563317i	$0.190475\pi$
$467$	5.30976e6i	1.12663i	−0.826241	+	0.563317i	$0.809525\pi$
$468$	0	0
$469$	−716520.	−0.150417
$470$	0	0
$471$	1.13436e6	0.235613
$472$	0	0
$473$	8.96040e6i	1.84151i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	3.68526e6i	0.741605i
$478$	0	0
$479$	−5.83653e6	−1.16229	−0.581147	−	0.813799i	$-0.697396\pi$
$479$	−5.83653e6	−1.16229	−0.581147	+	0.813799i	$0.697396\pi$
$480$	0	0
$481$	2.15759e6	0.425212
$482$	0	0
$483$	1.14576e6i	0.223473i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 8.75327e6i	− 1.67243i	−0.548402	−	0.836215i	$-0.684764\pi$
$487$	− 8.75327e6i	− 1.67243i	0.548402	−	0.836215i	$-0.315236\pi$
$488$	0	0
$489$	−2.94676e6	−0.557279
$490$	0	0
$491$	−5.16834e6	−0.967492	−0.483746	−	0.875209i	$-0.660724\pi$
$491$	−5.16834e6	−0.967492	−0.483746	+	0.875209i	$0.660724\pi$
$492$	0	0
$493$	− 2.81502e6i	− 0.521632i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.01208e6i	0.365388i
$498$	0	0
$499$	5.36171e6	0.963943	0.481972	−	0.876187i	$-0.339921\pi$
$499$	5.36171e6	0.963943	0.481972	+	0.876187i	$0.339921\pi$
$500$	0	0
$501$	−3.85524e6	−0.686210
$502$	0	0
$503$	5.26635e6i	0.928089i	0.885812	+	0.464045i	$0.153602\pi$
$503$	5.26635e6i	0.928089i	−0.885812	+	0.464045i	$0.846398\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 2.21345e6i	− 0.382428i
$508$	0	0
$509$	−5.46847e6	−0.935559	−0.467780	−	0.883845i	$-0.654946\pi$
$509$	−5.46847e6	−0.935559	−0.467780	+	0.883845i	$0.654946\pi$
$510$	0	0
$511$	1.16285e6	0.197002
$512$	0	0
$513$	7.70688e6i	1.29296i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 4.33656e6i	− 0.713541i
$518$	0	0
$519$	−3.73836e6	−0.609203
$520$	0	0
$521$	−3.05762e6	−0.493503	−0.246751	−	0.969079i	$-0.579363\pi$
$521$	−3.05762e6	−0.493503	−0.246751	+	0.969079i	$0.579363\pi$
$522$	0	0
$523$	9.64117e6i	1.54126i	0.637284	+	0.770629i	$0.280058\pi$
$523$	9.64117e6i	1.54126i	−0.637284	+	0.770629i	$0.719942\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 3.20556e6i	− 0.502780i
$528$	0	0
$529$	−1.03080e7	−1.60153
$530$	0	0
$531$	4.00058e6	0.615726
$532$	0	0
$533$	− 635100.i	− 0.0968332i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	1.33584e6i	0.199903i
$538$	0	0
$539$	−1.07047e7	−1.58709
$540$	0	0
$541$	−4.15820e6	−0.610819	−0.305409	−	0.952221i	$-0.598793\pi$
$541$	−4.15820e6	−0.610819	−0.305409	+	0.952221i	$0.598793\pi$
$542$	0	0
$543$	616884.i	0.0897851i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	5.32628e6i	0.761125i	0.924755	+	0.380562i	$0.124270\pi$
$547$	5.32628e6i	0.761125i	−0.924755	+	0.380562i	$0.875730\pi$
$548$	0	0
$549$	−8.55239e6	−1.21103
$550$	0	0
$551$	−1.09562e7	−1.53737
$552$	0	0
$553$	384160.i	0.0534194i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	999102.i	0.136449i	0.997670	+	0.0682247i	$0.0217335\pi$
$557$	999102.i	0.136449i	−0.997670	+	0.0682247i	$0.978267\pi$
$558$	0	0
$559$	1.81445e6	0.245593
$560$	0	0
$561$	3.20112e6	0.429432
$562$	0	0
$563$	− 5.31886e6i	− 0.707208i	−0.935395	−	0.353604i	$-0.884956\pi$
$563$	− 5.31886e6i	− 0.707208i	0.935395	−	0.353604i	$-0.115044\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.39408e6i	0.182108i
$568$	0	0
$569$	−6.56759e6	−0.850404	−0.425202	−	0.905099i	$-0.639797\pi$
$569$	−6.56759e6	−0.850404	−0.425202	+	0.905099i	$0.639797\pi$
$570$	0	0
$571$	−164173.	−0.0210723	−0.0105361	−	0.999944i	$-0.503354\pi$
$571$	−164173.	−0.0210723	−0.0105361	+	0.999944i	$0.503354\pi$
$572$	0	0
$573$	− 1.94264e6i	− 0.247176i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	9.27762e6i	1.16010i	0.814579	+	0.580052i	$0.196968\pi$
$577$	9.27762e6i	1.16010i	−0.814579	+	0.580052i	$0.803032\pi$
$578$	0	0
$579$	−78639.5	−0.00974865
$580$	0	0
$581$	−4.37220e6	−0.537353
$582$	0	0
$583$	− 1.30890e7i	− 1.59491i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 5.44474e6i	− 0.652202i	−0.945335	−	0.326101i	$-0.894265\pi$
$587$	− 5.44474e6i	− 0.652202i	0.945335	−	0.326101i	$-0.105735\pi$
$588$	0	0
$589$	−1.24762e7	−1.48181
$590$	0	0
$591$	−2.39611e6	−0.282187
$592$	0	0
$593$	1.50978e7i	1.76310i	0.472094	+	0.881548i	$0.343498\pi$
$593$	1.50978e7i	1.76310i	−0.472094	+	0.881548i	$0.656502\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	4.85216e6i	0.557185i
$598$	0	0
$599$	6.98593e6	0.795531	0.397765	−	0.917487i	$-0.369786\pi$
$599$	6.98593e6	0.795531	0.397765	+	0.917487i	$0.369786\pi$
$600$	0	0
$601$	6.16941e6	0.696719	0.348359	−	0.937361i	$-0.386739\pi$
$601$	6.16941e6	0.696719	0.348359	+	0.937361i	$0.386739\pi$
$602$	0	0
$603$	3.28546e6i	0.367962i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	1.00769e7i	1.11009i	0.831822	+	0.555043i	$0.187298\pi$
$607$	1.00769e7i	1.11009i	−0.831822	+	0.555043i	$0.812702\pi$
$608$	0	0
$609$	1.12280e6	0.122676
$610$	0	0
$611$	−878139.	−0.0951613
$612$	0	0
$613$	− 5.26885e6i	− 0.566324i	−0.959072	−	0.283162i	$-0.908617\pi$
$613$	− 5.26885e6i	− 0.566324i	0.959072	−	0.283162i	$-0.0913834\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 1.38112e7i	− 1.46056i	−0.683148	−	0.730280i	$-0.739390\pi$
$617$	− 1.38112e7i	− 1.46056i	0.683148	−	0.730280i	$-0.260610\pi$
$618$	0	0
$619$	3.20903e6	0.336625	0.168313	−	0.985734i	$-0.446168\pi$
$619$	3.20903e6	0.336625	0.168313	+	0.985734i	$0.446168\pi$
$620$	0	0
$621$	1.15425e7	1.20108
$622$	0	0
$623$	− 1.35339e6i	− 0.139702i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 1.24589e7i	− 1.26564i
$628$	0	0
$629$	−1.03742e7	−1.04551
$630$	0	0
$631$	6.07204e6	0.607102	0.303551	−	0.952815i	$-0.401828\pi$
$631$	6.07204e6	0.607102	0.303551	+	0.952815i	$0.401828\pi$
$632$	0	0
$633$	5.46776e6i	0.542376i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	2.16766e6i	0.211662i
$638$	0	0
$639$	9.22600e6	0.893843
$640$	0	0
$641$	−2.02767e6	−0.194918	−0.0974591	−	0.995240i	$-0.531072\pi$
$641$	−2.02767e6	−0.194918	−0.0974591	+	0.995240i	$0.531072\pi$
$642$	0	0
$643$	− 1.19769e7i	− 1.14240i	−0.820811	−	0.571199i	$-0.806478\pi$
$643$	− 1.19769e7i	− 1.14240i	0.820811	−	0.571199i	$-0.193522\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.44768e6i	0.135961i	0.997687	+	0.0679803i	$0.0216555\pi$
$647$	1.44768e6i	0.135961i	−0.997687	+	0.0679803i	$0.978344\pi$
$648$	0	0
$649$	−1.42090e7	−1.32419
$650$	0	0
$651$	1.27857e6	0.118242
$652$	0	0
$653$	− 4.82477e6i	− 0.442785i	−0.975185	−	0.221393i	$-0.928940\pi$
$653$	− 4.82477e6i	− 0.442785i	0.975185	−	0.221393i	$-0.0710603\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 5.33200e6i	− 0.481922i
$658$	0	0
$659$	4.37616e6	0.392536	0.196268	−	0.980550i	$-0.437118\pi$
$659$	4.37616e6	0.392536	0.196268	+	0.980550i	$0.437118\pi$
$660$	0	0
$661$	−7.34953e6	−0.654268	−0.327134	−	0.944978i	$-0.606083\pi$
$661$	−7.34953e6	−0.654268	−0.327134	+	0.944978i	$0.606083\pi$
$662$	0	0
$663$	− 648216.i	− 0.0572712i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.64089e7i	1.42812i
$668$	0	0
$669$	4.04588e6	0.349500
$670$	0	0
$671$	3.03757e7	2.60447
$672$	0	0
$673$	1.75692e7i	1.49525i	0.664119	+	0.747626i	$0.268807\pi$
$673$	1.75692e7i	1.49525i	−0.664119	+	0.747626i	$0.731193\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	5.60338e6i	0.469871i	0.972011	+	0.234935i	$0.0754879\pi$
$677$	5.60338e6i	0.469871i	−0.972011	+	0.234935i	$0.924512\pi$
$678$	0	0
$679$	−2.96099e6	−0.246469
$680$	0	0
$681$	−6.20132e6	−0.512409
$682$	0	0
$683$	5.05809e6i	0.414892i	0.978246	+	0.207446i	$0.0665152\pi$
$683$	5.05809e6i	0.414892i	−0.978246	+	0.207446i	$0.933485\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 6.40431e6i	− 0.517703i
$688$	0	0
$689$	−2.65048e6	−0.212705
$690$	0	0
$691$	−1.47421e7	−1.17453	−0.587267	−	0.809393i	$-0.699796\pi$
$691$	−1.47421e7	−1.17453	−0.587267	+	0.809393i	$0.699796\pi$
$692$	0	0
$693$	− 6.47976e6i	− 0.512538i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	3.05370e6i	0.238092i
$698$	0	0
$699$	4.46720e6	0.345814
$700$	0	0
$701$	−8.83317e6	−0.678925	−0.339462	−	0.940620i	$-0.610245\pi$
$701$	−8.83317e6	−0.678925	−0.339462	+	0.940620i	$0.610245\pi$
$702$	0	0
$703$	4.03766e7i	3.08135i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.89475e6i	0.142562i
$708$	0	0
$709$	1.93101e6	0.144268	0.0721338	−	0.997395i	$-0.477019\pi$
$709$	1.93101e6	0.144268	0.0721338	+	0.997395i	$0.477019\pi$
$710$	0	0
$711$	1.76149e6	0.130679
$712$	0	0
$713$	1.86854e7i	1.37650i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 7.55088e6i	− 0.548529i
$718$	0	0
$719$	−7.22373e6	−0.521122	−0.260561	−	0.965457i	$-0.583907\pi$
$719$	−7.22373e6	−0.521122	−0.260561	+	0.965457i	$0.583907\pi$
$720$	0	0
$721$	465080.	0.0333188
$722$	0	0
$723$	− 2.55771e6i	− 0.181973i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 9.81106e6i	− 0.688462i	−0.938885	−	0.344231i	$-0.888140\pi$
$727$	− 9.81106e6i	− 0.688462i	0.938885	−	0.344231i	$-0.111860\pi$
$728$	0	0
$729$	2.05715e6	0.143366
$730$	0	0
$731$	−8.72428e6	−0.603860
$732$	0	0
$733$	2.40813e7i	1.65547i	0.561122	+	0.827733i	$0.310370\pi$
$733$	2.40813e7i	1.65547i	−0.561122	+	0.827733i	$0.689630\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 1.16690e7i	− 0.791346i
$738$	0	0
$739$	−2.40518e7	−1.62008	−0.810040	−	0.586374i	$-0.800555\pi$
$739$	−2.40518e7	−1.62008	−0.810040	+	0.586374i	$0.800555\pi$
$740$	0	0
$741$	−2.52288e6	−0.168792
$742$	0	0
$743$	− 4.70160e6i	− 0.312445i	−0.987722	−	0.156223i	$-0.950068\pi$
$743$	− 4.70160e6i	− 0.312445i	0.987722	−	0.156223i	$-0.0499317\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	2.00479e7i	1.31452i
$748$	0	0
$749$	−2.95988e6	−0.192783
$750$	0	0
$751$	−2.31282e7	−1.49638	−0.748189	−	0.663485i	$-0.769077\pi$
$751$	−2.31282e7	−1.49638	−0.748189	+	0.663485i	$0.769077\pi$
$752$	0	0
$753$	− 9.06040e6i	− 0.582318i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 1.26635e7i	− 0.803181i	−0.915819	−	0.401591i	$-0.868458\pi$
$757$	− 1.26635e7i	− 0.803181i	0.915819	−	0.401591i	$-0.131542\pi$
$758$	0	0
$759$	−1.86595e7	−1.17570
$760$	0	0
$761$	−4.33524e6	−0.271363	−0.135682	−	0.990752i	$-0.543322\pi$
$761$	−4.33524e6	−0.271363	−0.135682	+	0.990752i	$0.543322\pi$
$762$	0	0
$763$	4.94579e6i	0.307556i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.87727e6i	0.176600i
$768$	0	0
$769$	364270.	0.0222130	0.0111065	−	0.999938i	$-0.496465\pi$
$769$	364270.	0.0222130	0.0111065	+	0.999938i	$0.496465\pi$
$770$	0	0
$771$	−6.24855e6	−0.378567
$772$	0	0
$773$	1.03051e6i	0.0620300i	0.999519	+	0.0310150i	$0.00987396\pi$
$773$	1.03051e6i	0.0620300i	−0.999519	+	0.0310150i	$0.990126\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 4.13784e6i	− 0.245879i
$778$	0	0
$779$	1.18851e7	0.701713
$780$	0	0
$781$	−3.27682e7	−1.92231
$782$	0	0
$783$	− 1.13112e7i	− 0.659333i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 1.95300e7i	− 1.12400i	−0.827138	−	0.561999i	$-0.810032\pi$
$787$	− 1.95300e7i	− 1.12400i	0.827138	−	0.561999i	$-0.189968\pi$
$788$	0	0
$789$	−1.30426e7	−0.745885
$790$	0	0
$791$	9.59965e6	0.545524
$792$	0	0
$793$	− 6.15098e6i	− 0.347345i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	7.86344e6i	0.438497i	0.975669	+	0.219249i	$0.0703605\pi$
$797$	7.86344e6i	0.438497i	−0.975669	+	0.219249i	$0.929639\pi$
$798$	0	0
$799$	4.22229e6	0.233981
$800$	0	0
$801$	−6.20571e6	−0.341751
$802$	0	0
$803$	1.89378e7i	1.03643i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 4.93559e6i	− 0.266781i
$808$	0	0
$809$	1.05014e7	0.564127	0.282064	−	0.959396i	$-0.408981\pi$
$809$	1.05014e7	0.564127	0.282064	+	0.959396i	$0.408981\pi$
$810$	0	0
$811$	1.87803e7	1.00265	0.501325	−	0.865259i	$-0.332846\pi$
$811$	1.87803e7	1.00265	0.501325	+	0.865259i	$0.332846\pi$
$812$	0	0
$813$	3.55624e6i	0.188697i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	3.39552e7i	1.77972i
$818$	0	0
$819$	−1.31213e6	−0.0683545
$820$	0	0
$821$	2.00558e7	1.03844	0.519220	−	0.854641i	$-0.326223\pi$
$821$	2.00558e7	1.03844	0.519220	+	0.854641i	$0.326223\pi$
$822$	0	0
$823$	− 1.02444e7i	− 0.527212i	−0.964630	−	0.263606i	$-0.915088\pi$
$823$	− 1.02444e7i	− 0.527212i	0.964630	−	0.263606i	$-0.0849119\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	827524.i	0.0420743i	0.999779	+	0.0210371i	$0.00669682\pi$
$827$	827524.i	0.0420743i	−0.999779	+	0.0210371i	$0.993303\pi$
$828$	0	0
$829$	3.80328e7	1.92208	0.961041	−	0.276407i	$-0.0891436\pi$
$829$	3.80328e7	1.92208	0.961041	+	0.276407i	$0.0891436\pi$
$830$	0	0
$831$	2.44607e6	0.122876
$832$	0	0
$833$	− 1.04226e7i	− 0.520431i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 1.28805e7i	− 0.635504i
$838$	0	0
$839$	3.57628e7	1.75399	0.876993	−	0.480503i	$-0.159546\pi$
$839$	3.57628e7	1.75399	0.876993	+	0.480503i	$0.159546\pi$
$840$	0	0
$841$	−4.43105e6	−0.216031
$842$	0	0
$843$	− 9.83373e6i	− 0.476595i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1.58843e7i	0.760779i
$848$	0	0
$849$	1.10779e7	0.527457
$850$	0	0
$851$	6.04714e7	2.86237
$852$	0	0
$853$	− 970214.i	− 0.0456557i	−0.999739	−	0.0228278i	$-0.992733\pi$
$853$	− 970214.i	− 0.0456557i	0.999739	−	0.0228278i	$-0.00726696\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.86468e6i	0.0867267i	0.999059	+	0.0433633i	$0.0138073\pi$
$857$	1.86468e6i	0.0867267i	−0.999059	+	0.0433633i	$0.986193\pi$
$858$	0	0
$859$	−1.05785e6	−0.0489147	−0.0244573	−	0.999701i	$-0.507786\pi$
$859$	−1.05785e6	−0.0489147	−0.0244573	+	0.999701i	$0.507786\pi$
$860$	0	0
$861$	−1.21800e6	−0.0559937
$862$	0	0
$863$	− 8.24333e6i	− 0.376769i	−0.982095	−	0.188385i	$-0.939675\pi$
$863$	− 8.24333e6i	− 0.376769i	0.982095	−	0.188385i	$-0.0603252\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 5.86320e6i	− 0.264903i
$868$	0	0
$869$	−6.25632e6	−0.281041
$870$	0	0
$871$	−2.36294e6	−0.105538
$872$	0	0
$873$	1.35770e7i	0.602934i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 2.40362e7i	− 1.05528i	−0.849468	−	0.527640i	$-0.823077\pi$
$877$	− 2.40362e7i	− 1.05528i	0.849468	−	0.527640i	$-0.176923\pi$
$878$	0	0
$879$	−9.82263e6	−0.428801
$880$	0	0
$881$	3.68605e7	1.60000	0.800002	−	0.599997i	$-0.204831\pi$
$881$	3.68605e7	1.60000	0.800002	+	0.599997i	$0.204831\pi$
$882$	0	0
$883$	− 1.67403e7i	− 0.722541i	−0.932461	−	0.361271i	$-0.882343\pi$
$883$	− 1.67403e7i	− 0.722541i	0.932461	−	0.361271i	$-0.117657\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 9.25364e6i	− 0.394915i	−0.980311	−	0.197457i	$-0.936732\pi$
$887$	− 9.25364e6i	− 0.394915i	0.980311	−	0.197457i	$-0.0632684\pi$
$888$	0	0
$889$	−2.75716e6	−0.117006
$890$	0	0
$891$	−2.27035e7	−0.958075
$892$	0	0
$893$	− 1.64333e7i	− 0.689597i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	3.77848e6i	0.156796i
$898$	0	0
$899$	1.83110e7	0.755635
$900$	0	0
$901$	1.27441e7	0.522995
$902$	0	0
$903$	− 3.47977e6i	− 0.142014i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 1.03030e7i	− 0.415859i	−0.978144	−	0.207929i	$-0.933328\pi$
$907$	− 1.03030e7i	− 0.415859i	0.978144	−	0.207929i	$-0.0666724\pi$
$908$	0	0
$909$	8.68799e6	0.348746
$910$	0	0
$911$	−3.82374e7	−1.52649	−0.763243	−	0.646111i	$-0.776394\pi$
$911$	−3.82374e7	−1.52649	−0.763243	+	0.646111i	$0.776394\pi$
$912$	0	0
$913$	− 7.12044e7i	− 2.82703i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	7.80024e6i	0.306327i
$918$	0	0
$919$	−7.72150e6	−0.301587	−0.150794	−	0.988565i	$-0.548183\pi$
$919$	−7.72150e6	−0.301587	−0.150794	+	0.988565i	$0.548183\pi$
$920$	0	0
$921$	1.64697e7	0.639790
$922$	0	0
$923$	6.63545e6i	0.256369i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 2.13253e6i	− 0.0815073i
$928$	0	0
$929$	−3.39039e6	−0.128888	−0.0644438	−	0.997921i	$-0.520527\pi$
$929$	−3.39039e6	−0.128888	−0.0644438	+	0.997921i	$0.520527\pi$
$930$	0	0
$931$	−4.05651e7	−1.53383
$932$	0	0
$933$	1.91080e6i	0.0718640i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 7.67490e6i	− 0.285577i	−0.989753	−	0.142789i	$-0.954393\pi$
$937$	− 7.67490e6i	− 0.285577i	0.989753	−	0.142789i	$-0.0456069\pi$
$938$	0	0
$939$	−1.32471e7	−0.490295
$940$	0	0
$941$	4.55115e7	1.67551	0.837756	−	0.546045i	$-0.183867\pi$
$941$	4.55115e7	1.67551	0.837756	+	0.546045i	$0.183867\pi$
$942$	0	0
$943$	− 1.78001e7i	− 0.651845i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2.28294e7i	0.827218i	0.910455	+	0.413609i	$0.135732\pi$
$947$	2.28294e7i	0.827218i	−0.910455	+	0.413609i	$0.864268\pi$
$948$	0	0
$949$	3.83484e6	0.138223
$950$	0	0
$951$	6.80673e6	0.244055
$952$	0	0
$953$	− 4.64478e7i	− 1.65666i	−0.560241	−	0.828330i	$-0.689291\pi$
$953$	− 4.64478e7i	− 1.65666i	0.560241	−	0.828330i	$-0.310709\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	1.82856e7i	0.645401i
$958$	0	0
$959$	−4.41842e6	−0.155139
$960$	0	0
$961$	−7.77779e6	−0.271674
$962$	0	0
$963$	1.35719e7i	0.471603i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	2.63000e7i	0.904460i	0.891901	+	0.452230i	$0.149371\pi$
$967$	2.63000e7i	0.904460i	−0.891901	+	0.452230i	$0.850629\pi$
$968$	0	0
$969$	1.21306e7	0.415022
$970$	0	0
$971$	−2.19967e7	−0.748704	−0.374352	−	0.927287i	$-0.622135\pi$
$971$	−2.19967e7	−0.748704	−0.374352	+	0.927287i	$0.622135\pi$
$972$	0	0
$973$	1.20187e7i	0.406983i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4.46750e7i	1.49737i	0.662929	+	0.748683i	$0.269313\pi$
$977$	4.46750e7i	1.49737i	−0.662929	+	0.748683i	$0.730687\pi$
$978$	0	0
$979$	2.20409e7	0.734977
$980$	0	0
$981$	2.26779e7	0.752369
$982$	0	0
$983$	2.02740e7i	0.669201i	0.942360	+	0.334600i	$0.108601\pi$
$983$	2.02740e7i	0.669201i	−0.942360	+	0.334600i	$0.891399\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1.68410e6i	0.0550270i
$988$	0	0
$989$	5.08542e7	1.65324
$990$	0	0
$991$	1.49135e7	0.482386	0.241193	−	0.970477i	$-0.422461\pi$
$991$	1.49135e7	0.482386	0.241193	+	0.970477i	$0.422461\pi$
$992$	0	0
$993$	− 1.97960e6i	− 0.0637095i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	3.17032e7i	1.01010i	0.863089	+	0.505052i	$0.168527\pi$
$997$	3.17032e7i	1.01010i	−0.863089	+	0.505052i	$0.831473\pi$
$998$	0	0
$999$	−4.16851e7	−1.32150

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.6.c.i.449.1		4
4.3	odd	2	inner	800.6.c.i.449.4		4
5.2	odd	4		800.6.a.h.1.1		2
5.3	odd	4		160.6.a.d.1.2	yes	2
5.4	even	2	inner	800.6.c.i.449.3		4
20.3	even	4		160.6.a.d.1.1	✓	2
20.7	even	4		800.6.a.h.1.2		2
20.19	odd	2	inner	800.6.c.i.449.2		4
40.3	even	4		320.6.a.s.1.2		2
40.13	odd	4		320.6.a.s.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.6.a.d.1.1	✓	2	20.3	even	4
160.6.a.d.1.2	yes	2	5.3	odd	4
320.6.a.s.1.1		2	40.13	odd	4
320.6.a.s.1.2		2	40.3	even	4
800.6.a.h.1.1		2	5.2	odd	4
800.6.a.h.1.2		2	20.7	even	4
800.6.c.i.449.1		4	1.1	even	1	trivial
800.6.c.i.449.2		4	20.19	odd	2	inner
800.6.c.i.449.3		4	5.4	even	2	inner
800.6.c.i.449.4		4	4.3	odd	2	inner

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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	800.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-6.32456i q^{3} +44.2719i q^{7} +203.000 q^{9} +O(q^{10})\)	\(q-6.32456i q^{3} +44.2719i q^{7} +203.000 q^{9} -720.999 q^{11} +146.000i q^{13} -702.000i q^{17} -2732.21 q^{19} +280.000 q^{21} +4091.99i q^{23} -2820.75i q^{27} +4010.00 q^{29} +4566.33 q^{31} +4560.00i q^{33} -14778.0i q^{37} +923.385 q^{39} -4350.00 q^{41} -12427.8i q^{43} +6014.65i q^{47} +14847.0 q^{49} -4439.84 q^{51} +18154.0i q^{53} +17280.0i q^{57} +19707.3 q^{59} -42130.0 q^{61} +8987.19i q^{63} +16184.5i q^{67} +25880.0 q^{69} +45448.3 q^{71} -26266.0i q^{73} -31920.0i q^{77} +8677.29 q^{79} +31489.0 q^{81} +98757.9i q^{83} -25361.5i q^{87} -30570.0 q^{89} -6463.70 q^{91} -28880.0i q^{93} +66882.0i q^{97} -146363. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 812 q^{9}+O(q^{10}) \) 4 * q + 812 * q^9	\( 4 q + 812 q^{9} + 1120 q^{21} + 16040 q^{29} - 17400 q^{41} + 59388 q^{49} - 168520 q^{61} + 103520 q^{69} + 125956 q^{81} - 122280 q^{89}+O(q^{100}) \) 4 * q + 812 * q^9 + 1120 * q^21 + 16040 * q^29 - 17400 * q^41 + 59388 * q^49 - 168520 * q^61 + 103520 * q^69 + 125956 * q^81 - 122280 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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