LMFDB - Embedded newform 800.6.f.d.49.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,6,Mod(49,800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("800.49");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	800.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$128.307055850$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	no (minimal twist has level 200)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.18
Character	$\chi$	$=$	800.49
Dual form			800.6.f.d.49.17

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.58812 q^{3} -27.4015i q^{7} -236.302 q^{9} +O(q^{10})$	$q-2.58812 q^{3} -27.4015i q^{7} -236.302 q^{9} +313.336i q^{11} +627.354 q^{13} +1598.17i q^{17} -219.049i q^{19} +70.9184i q^{21} +1007.11i q^{23} +1240.49 q^{27} -6140.83i q^{29} +2244.61 q^{31} -810.951i q^{33} +3591.40 q^{37} -1623.67 q^{39} -608.697 q^{41} -11410.6 q^{43} -16909.4i q^{47} +16056.2 q^{49} -4136.26i q^{51} +12783.9 q^{53} +566.923i q^{57} +29917.1i q^{59} +17985.1i q^{61} +6475.03i q^{63} -48881.2 q^{67} -2606.52i q^{69} -37296.0 q^{71} -26092.7i q^{73} +8585.89 q^{77} -77636.5 q^{79} +54210.8 q^{81} -58419.8 q^{83} +15893.2i q^{87} +81513.8 q^{89} -17190.5i q^{91} -5809.30 q^{93} +28396.6i q^{97} -74041.9i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 3240 q^{9}+O(q^{10}) $ 40 * q + 3240 * q^9	$ 40 q + 3240 q^{9} - 14320 q^{31} + 44904 q^{39} - 11608 q^{41} - 125304 q^{49} + 15448 q^{71} - 15560 q^{79} + 193968 q^{81} + 6320 q^{89}+O(q^{100}) $ 40 * q + 3240 * q^9 - 14320 * q^31 + 44904 * q^39 - 11608 * q^41 - 125304 * q^49 + 15448 * q^71 - 15560 * q^79 + 193968 * q^81 + 6320 * 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
$3$	−2.58812	−0.166028	−0.0830139	−	0.996548i	$-0.526455\pi$
$3$	−2.58812	−0.166028	−0.0830139	+	0.996548i	$0.526455\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 27.4015i	− 0.211363i	−0.994400	−	0.105682i	$-0.966298\pi$
$7$	− 27.4015i	− 0.211363i	0.994400	−	0.105682i	$-0.0337025\pi$
$8$	0	0
$9$	−236.302	−0.972435
$10$	0	0
$11$	313.336i	0.780780i	0.920650	+	0.390390i	$0.127660\pi$
$11$	313.336i	0.780780i	−0.920650	+	0.390390i	$0.872340\pi$
$12$	0	0
$13$	627.354	1.02957	0.514783	−	0.857320i	$-0.327872\pi$
$13$	627.354	1.02957	0.514783	+	0.857320i	$0.327872\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1598.17i	1.34122i	0.741809	+	0.670611i	$0.233968\pi$
$17$	1598.17i	1.34122i	−0.741809	+	0.670611i	$0.766032\pi$
$18$	0	0
$19$	− 219.049i	− 0.139205i	−0.997575	−	0.0696027i	$-0.977827\pi$
$19$	− 219.049i	− 0.139205i	0.997575	−	0.0696027i	$-0.0221732\pi$
$20$	0	0
$21$	70.9184i	0.0350922i
$22$	0	0
$23$	1007.11i	0.396970i	0.980104	+	0.198485i	$0.0636021\pi$
$23$	1007.11i	0.396970i	−0.980104	+	0.198485i	$0.936398\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1240.49	0.327479
$28$	0	0
$29$	− 6140.83i	− 1.35591i	−0.735102	−	0.677957i	$-0.762866\pi$
$29$	− 6140.83i	− 1.35591i	0.735102	−	0.677957i	$-0.237134\pi$
$30$	0	0
$31$	2244.61	0.419504	0.209752	−	0.977755i	$-0.432734\pi$
$31$	2244.61	0.419504	0.209752	+	0.977755i	$0.432734\pi$
$32$	0	0
$33$	− 810.951i	− 0.129631i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3591.40	0.431280	0.215640	−	0.976473i	$-0.430816\pi$
$37$	3591.40	0.431280	0.215640	+	0.976473i	$0.430816\pi$
$38$	0	0
$39$	−1623.67	−0.170937
$40$	0	0
$41$	−608.697	−0.0565511	−0.0282756	−	0.999600i	$-0.509002\pi$
$41$	−608.697	−0.0565511	−0.0282756	+	0.999600i	$0.509002\pi$
$42$	0	0
$43$	−11410.6	−0.941105	−0.470553	−	0.882372i	$-0.655945\pi$
$43$	−11410.6	−0.941105	−0.470553	+	0.882372i	$0.655945\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 16909.4i	− 1.11656i	−0.829652	−	0.558280i	$-0.811461\pi$
$47$	− 16909.4i	− 1.11656i	0.829652	−	0.558280i	$-0.188539\pi$
$48$	0	0
$49$	16056.2	0.955326
$50$	0	0
$51$	− 4136.26i	− 0.222680i
$52$	0	0
$53$	12783.9	0.625134	0.312567	−	0.949896i	$-0.398811\pi$
$53$	12783.9	0.625134	0.312567	+	0.949896i	$0.398811\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	566.923i	0.0231120i
$58$	0	0
$59$	29917.1i	1.11889i	0.828866	+	0.559447i	$0.188986\pi$
$59$	29917.1i	1.11889i	−0.828866	+	0.559447i	$0.811014\pi$
$60$	0	0
$61$	17985.1i	0.618852i	0.950923	+	0.309426i	$0.100137\pi$
$61$	17985.1i	0.618852i	−0.950923	+	0.309426i	$0.899863\pi$
$62$	0	0
$63$	6475.03i	0.205537i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−48881.2	−1.33032	−0.665158	−	0.746703i	$-0.731636\pi$
$67$	−48881.2	−1.33032	−0.665158	+	0.746703i	$0.731636\pi$
$68$	0	0
$69$	− 2606.52i	− 0.0659081i
$70$	0	0
$71$	−37296.0	−0.878044	−0.439022	−	0.898476i	$-0.644675\pi$
$71$	−37296.0	−0.878044	−0.439022	+	0.898476i	$0.644675\pi$
$72$	0	0
$73$	− 26092.7i	− 0.573075i	−0.958069	−	0.286538i	$-0.907496\pi$
$73$	− 26092.7i	− 0.573075i	0.958069	−	0.286538i	$-0.0925043\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	8585.89	0.165028
$78$	0	0
$79$	−77636.5	−1.39958	−0.699791	−	0.714348i	$-0.746723\pi$
$79$	−77636.5	−1.39958	−0.699791	+	0.714348i	$0.746723\pi$
$80$	0	0
$81$	54210.8	0.918064
$82$	0	0
$83$	−58419.8	−0.930818	−0.465409	−	0.885096i	$-0.654093\pi$
$83$	−58419.8	−0.930818	−0.465409	+	0.885096i	$0.654093\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	15893.2i	0.225120i
$88$	0	0
$89$	81513.8	1.09083	0.545414	−	0.838167i	$-0.316373\pi$
$89$	81513.8	1.09083	0.545414	+	0.838167i	$0.316373\pi$
$90$	0	0
$91$	− 17190.5i	− 0.217613i
$92$	0	0
$93$	−5809.30	−0.0696493
$94$	0	0
$95$	0	0
$96$	0	0
$97$	28396.6i	0.306434i	0.988193	+	0.153217i	$0.0489634\pi$
$97$	28396.6i	0.306434i	−0.988193	+	0.153217i	$0.951037\pi$
$98$	0	0
$99$	− 74041.9i	− 0.759258i
$100$	0	0
$101$	136025.i	1.32683i	0.748253	+	0.663413i	$0.230893\pi$
$101$	136025.i	1.32683i	−0.748253	+	0.663413i	$0.769107\pi$
$102$	0	0
$103$	166600.i	1.54733i	0.633595	+	0.773665i	$0.281579\pi$
$103$	166600.i	1.54733i	−0.633595	+	0.773665i	$0.718421\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	67774.4	0.572277	0.286138	−	0.958188i	$-0.407628\pi$
$107$	67774.4	0.572277	0.286138	+	0.958188i	$0.407628\pi$
$108$	0	0
$109$	− 55567.6i	− 0.447977i	−0.974592	−	0.223988i	$-0.928092\pi$
$109$	− 55567.6i	− 0.447977i	0.974592	−	0.223988i	$-0.0719077\pi$
$110$	0	0
$111$	−9294.96	−0.0716044
$112$	0	0
$113$	− 170497.i	− 1.25609i	−0.778177	−	0.628045i	$-0.783855\pi$
$113$	− 170497.i	− 1.25609i	0.778177	−	0.628045i	$-0.216145\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−148245.	−1.00119
$118$	0	0
$119$	43792.3	0.283485
$120$	0	0
$121$	62871.4	0.390382
$122$	0	0
$123$	1575.38	0.00938906
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 313764.i	− 1.72621i	−0.505022	−	0.863106i	$-0.668516\pi$
$127$	− 313764.i	− 1.72621i	0.505022	−	0.863106i	$-0.331484\pi$
$128$	0	0
$129$	29532.0	0.156250
$130$	0	0
$131$	353658.i	1.80055i	0.435322	+	0.900275i	$0.356634\pi$
$131$	353658.i	1.80055i	−0.435322	+	0.900275i	$0.643366\pi$
$132$	0	0
$133$	−6002.26	−0.0294229
$134$	0	0
$135$	0	0
$136$	0	0
$137$	72750.1i	0.331156i	0.986197	+	0.165578i	$0.0529489\pi$
$137$	72750.1i	0.331156i	−0.986197	+	0.165578i	$0.947051\pi$
$138$	0	0
$139$	− 259480.i	− 1.13911i	−0.821952	−	0.569557i	$-0.807115\pi$
$139$	− 259480.i	− 1.13911i	0.821952	−	0.569557i	$-0.192885\pi$
$140$	0	0
$141$	43763.4i	0.185380i
$142$	0	0
$143$	196573.i	0.803865i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−41555.2	−0.158611
$148$	0	0
$149$	206318.i	0.761328i	0.924713	+	0.380664i	$0.124305\pi$
$149$	206318.i	0.761328i	−0.924713	+	0.380664i	$0.875695\pi$
$150$	0	0
$151$	−435318.	−1.55369	−0.776845	−	0.629692i	$-0.783181\pi$
$151$	−435318.	−1.55369	−0.776845	+	0.629692i	$0.783181\pi$
$152$	0	0
$153$	− 377650.i	− 1.30425i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−341575.	−1.10595	−0.552977	−	0.833196i	$-0.686508\pi$
$157$	−341575.	−1.10595	−0.552977	+	0.833196i	$0.686508\pi$
$158$	0	0
$159$	−33086.2	−0.103790
$160$	0	0
$161$	27596.4	0.0839049
$162$	0	0
$163$	−458266.	−1.35098	−0.675489	−	0.737370i	$-0.736068\pi$
$163$	−458266.	−1.35098	−0.675489	+	0.737370i	$0.736068\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	443800.i	1.23139i	0.787984	+	0.615696i	$0.211125\pi$
$167$	443800.i	1.23139i	−0.787984	+	0.615696i	$0.788875\pi$
$168$	0	0
$169$	22280.4	0.0600075
$170$	0	0
$171$	51761.5i	0.135368i
$172$	0	0
$173$	−731330.	−1.85780	−0.928898	−	0.370336i	$-0.879243\pi$
$173$	−731330.	−1.85780	−0.928898	+	0.370336i	$0.879243\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 77428.9i	− 0.185768i
$178$	0	0
$179$	− 318951.i	− 0.744032i	−0.928226	−	0.372016i	$-0.878667\pi$
$179$	− 318951.i	− 0.744032i	0.928226	−	0.372016i	$-0.121333\pi$
$180$	0	0
$181$	586939.i	1.33167i	0.746099	+	0.665835i	$0.231924\pi$
$181$	586939.i	1.33167i	−0.746099	+	0.665835i	$0.768076\pi$
$182$	0	0
$183$	− 46547.4i	− 0.102747i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−500765.	−1.04720
$188$	0	0
$189$	− 33991.3i	− 0.0692171i
$190$	0	0
$191$	97320.1	0.193027	0.0965137	−	0.995332i	$-0.469231\pi$
$191$	97320.1	0.193027	0.0965137	+	0.995332i	$0.469231\pi$
$192$	0	0
$193$	− 52528.5i	− 0.101508i	−0.998711	−	0.0507541i	$-0.983838\pi$
$193$	− 52528.5i	− 0.101508i	0.998711	−	0.0507541i	$-0.0161625\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−446496.	−0.819695	−0.409847	−	0.912154i	$-0.634418\pi$
$197$	−446496.	−0.819695	−0.409847	+	0.912154i	$0.634418\pi$
$198$	0	0
$199$	−518787.	−0.928659	−0.464329	−	0.885663i	$-0.653705\pi$
$199$	−518787.	−0.928659	−0.464329	+	0.885663i	$0.653705\pi$
$200$	0	0
$201$	126510.	0.220870
$202$	0	0
$203$	−168268.	−0.286591
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 237982.i	− 0.386028i
$208$	0	0
$209$	68635.8	0.108689
$210$	0	0
$211$	1.12592e6i	1.74101i	0.492163	+	0.870503i	$0.336206\pi$
$211$	1.12592e6i	1.74101i	−0.492163	+	0.870503i	$0.663794\pi$
$212$	0	0
$213$	96526.5	0.145780
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 61505.6i	− 0.0886677i
$218$	0	0
$219$	67530.9i	0.0951464i
$220$	0	0
$221$	1.00262e6i	1.38088i
$222$	0	0
$223$	− 138775.i	− 0.186874i	−0.995625	−	0.0934368i	$-0.970215\pi$
$223$	− 138775.i	− 0.186874i	0.995625	−	0.0934368i	$-0.0297853\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−966829.	−1.24533	−0.622666	−	0.782488i	$-0.713950\pi$
$227$	−966829.	−1.24533	−0.622666	+	0.782488i	$0.713950\pi$
$228$	0	0
$229$	− 840679.i	− 1.05935i	−0.848199	−	0.529677i	$-0.822313\pi$
$229$	− 840679.i	− 1.05935i	0.848199	−	0.529677i	$-0.177687\pi$
$230$	0	0
$231$	−22221.3	−0.0273993
$232$	0	0
$233$	977446.i	1.17951i	0.807581	+	0.589757i	$0.200776\pi$
$233$	977446.i	1.17951i	−0.807581	+	0.589757i	$0.799224\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	200932.	0.232370
$238$	0	0
$239$	1.17543e6	1.33108	0.665538	−	0.746364i	$-0.268202\pi$
$239$	1.17543e6	1.33108	0.665538	+	0.746364i	$0.268202\pi$
$240$	0	0
$241$	−615094.	−0.682180	−0.341090	−	0.940031i	$-0.610796\pi$
$241$	−615094.	−0.682180	−0.341090	+	0.940031i	$0.610796\pi$
$242$	0	0
$243$	−441743.	−0.479903
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 137421.i	− 0.143321i
$248$	0	0
$249$	151197.	0.154542
$250$	0	0
$251$	1.26252e6i	1.26490i	0.774602	+	0.632448i	$0.217950\pi$
$251$	1.26252e6i	1.26490i	−0.774602	+	0.632448i	$0.782050\pi$
$252$	0	0
$253$	−315564.	−0.309946
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 346343.i	− 0.327095i	−0.986535	−	0.163547i	$-0.947706\pi$
$257$	− 346343.i	− 0.327095i	0.986535	−	0.163547i	$-0.0522937\pi$
$258$	0	0
$259$	− 98409.8i	− 0.0911567i
$260$	0	0
$261$	1.45109e6i	1.31854i
$262$	0	0
$263$	− 1.93221e6i	− 1.72252i	−0.508162	−	0.861262i	$-0.669675\pi$
$263$	− 1.93221e6i	− 1.72252i	0.508162	−	0.861262i	$-0.330325\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−210967.	−0.181108
$268$	0	0
$269$	− 2.22703e6i	− 1.87649i	−0.345976	−	0.938243i	$-0.612452\pi$
$269$	− 2.22703e6i	− 1.87649i	0.345976	−	0.938243i	$-0.387548\pi$
$270$	0	0
$271$	512488.	0.423897	0.211948	−	0.977281i	$-0.432019\pi$
$271$	512488.	0.423897	0.211948	+	0.977281i	$0.432019\pi$
$272$	0	0
$273$	44491.0i	0.0361298i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−825684.	−0.646568	−0.323284	−	0.946302i	$-0.604787\pi$
$277$	−825684.	−0.646568	−0.323284	+	0.946302i	$0.604787\pi$
$278$	0	0
$279$	−530404.	−0.407940
$280$	0	0
$281$	−2.18865e6	−1.65352	−0.826762	−	0.562552i	$-0.809820\pi$
$281$	−2.18865e6	−1.65352	−0.826762	+	0.562552i	$0.809820\pi$
$282$	0	0
$283$	−1.64409e6	−1.22028	−0.610141	−	0.792293i	$-0.708887\pi$
$283$	−1.64409e6	−1.22028	−0.610141	+	0.792293i	$0.708887\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	16679.2i	0.0119528i
$288$	0	0
$289$	−1.13429e6	−0.798879
$290$	0	0
$291$	− 73493.9i	− 0.0508767i
$292$	0	0
$293$	−1.79254e6	−1.21983	−0.609917	−	0.792465i	$-0.708797\pi$
$293$	−1.79254e6	−1.21983	−0.609917	+	0.792465i	$0.708797\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	388690.i	0.255689i
$298$	0	0
$299$	631815.i	0.408707i
$300$	0	0
$301$	312669.i	0.198915i
$302$	0	0
$303$	− 352048.i	− 0.220290i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−1.30565e6	−0.790646	−0.395323	−	0.918542i	$-0.629367\pi$
$307$	−1.30565e6	−0.790646	−0.395323	+	0.918542i	$0.629367\pi$
$308$	0	0
$309$	− 431182.i	− 0.256900i
$310$	0	0
$311$	996126.	0.584001	0.292000	−	0.956418i	$-0.405679\pi$
$311$	996126.	0.584001	0.292000	+	0.956418i	$0.405679\pi$
$312$	0	0
$313$	887076.i	0.511800i	0.966703	+	0.255900i	$0.0823717\pi$
$313$	887076.i	0.511800i	−0.966703	+	0.255900i	$0.917628\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1.23286e6	−0.689075	−0.344538	−	0.938773i	$-0.611964\pi$
$317$	−1.23286e6	−0.689075	−0.344538	+	0.938773i	$0.611964\pi$
$318$	0	0
$319$	1.92414e6	1.05867
$320$	0	0
$321$	−175408.	−0.0950139
$322$	0	0
$323$	350077.	0.186706
$324$	0	0
$325$	0	0
$326$	0	0
$327$	143815.i	0.0743766i
$328$	0	0
$329$	−463342.	−0.236000
$330$	0	0
$331$	3.32521e6i	1.66820i	0.551612	+	0.834101i	$0.314013\pi$
$331$	3.32521e6i	1.66820i	−0.551612	+	0.834101i	$0.685987\pi$
$332$	0	0
$333$	−848653.	−0.419391
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 902443.i	− 0.432858i	−0.976298	−	0.216429i	$-0.930559\pi$
$337$	− 902443.i	− 0.432858i	0.976298	−	0.216429i	$-0.0694409\pi$
$338$	0	0
$339$	441267.i	0.208546i
$340$	0	0
$341$	703316.i	0.327540i
$342$	0	0
$343$	− 900501.i	− 0.413284i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.99415e6	−0.889066	−0.444533	−	0.895763i	$-0.646630\pi$
$347$	−1.99415e6	−0.889066	−0.444533	+	0.895763i	$0.646630\pi$
$348$	0	0
$349$	3.06586e6i	1.34737i	0.739017	+	0.673687i	$0.235290\pi$
$349$	3.06586e6i	1.34737i	−0.739017	+	0.673687i	$0.764710\pi$
$350$	0	0
$351$	778226.	0.337162
$352$	0	0
$353$	2.66290e6i	1.13741i	0.822541	+	0.568705i	$0.192556\pi$
$353$	2.66290e6i	1.13741i	−0.822541	+	0.568705i	$0.807444\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−113340.	−0.0470665
$358$	0	0
$359$	39323.6	0.0161034	0.00805169	−	0.999968i	$-0.497437\pi$
$359$	39323.6	0.0161034	0.00805169	+	0.999968i	$0.497437\pi$
$360$	0	0
$361$	2.42812e6	0.980622
$362$	0	0
$363$	−162719.	−0.0648143
$364$	0	0
$365$	0	0
$366$	0	0
$367$	851976.i	0.330189i	0.986278	+	0.165094i	$0.0527929\pi$
$367$	851976.i	0.330189i	−0.986278	+	0.165094i	$0.947207\pi$
$368$	0	0
$369$	143836.	0.0549923
$370$	0	0
$371$	− 350298.i	− 0.132130i
$372$	0	0
$373$	−1.78163e6	−0.663047	−0.331524	−	0.943447i	$-0.607563\pi$
$373$	−1.78163e6	−0.663047	−0.331524	+	0.943447i	$0.607563\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 3.85248e6i	− 1.39600i
$378$	0	0
$379$	− 4.29296e6i	− 1.53518i	−0.640941	−	0.767590i	$-0.721456\pi$
$379$	− 4.29296e6i	− 1.53518i	0.640941	−	0.767590i	$-0.278544\pi$
$380$	0	0
$381$	812059.i	0.286599i
$382$	0	0
$383$	− 1.06598e6i	− 0.371322i	−0.982614	−	0.185661i	$-0.940557\pi$
$383$	− 1.06598e6i	− 0.371322i	0.982614	−	0.185661i	$-0.0594426\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2.69635e6	0.915164
$388$	0	0
$389$	− 251531.i	− 0.0842786i	−0.999112	−	0.0421393i	$-0.986583\pi$
$389$	− 251531.i	− 0.0842786i	0.999112	−	0.0421393i	$-0.0134173\pi$
$390$	0	0
$391$	−1.60954e6	−0.532425
$392$	0	0
$393$	− 915308.i	− 0.298941i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	560036.	0.178336	0.0891682	−	0.996017i	$-0.471579\pi$
$397$	560036.	0.178336	0.0891682	+	0.996017i	$0.471579\pi$
$398$	0	0
$399$	15534.6	0.00488503
$400$	0	0
$401$	319882.	0.0993410	0.0496705	−	0.998766i	$-0.484183\pi$
$401$	319882.	0.0993410	0.0496705	+	0.998766i	$0.484183\pi$
$402$	0	0
$403$	1.40816e6	0.431907
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.12531e6i	0.336735i
$408$	0	0
$409$	221487.	0.0654695	0.0327348	−	0.999464i	$-0.489578\pi$
$409$	221487.	0.0654695	0.0327348	+	0.999464i	$0.489578\pi$
$410$	0	0
$411$	− 188286.i	− 0.0549811i
$412$	0	0
$413$	819773.	0.236493
$414$	0	0
$415$	0	0
$416$	0	0
$417$	671565.i	0.189125i
$418$	0	0
$419$	− 1.61245e6i	− 0.448695i	−0.974509	−	0.224347i	$-0.927975\pi$
$419$	− 1.61245e6i	− 0.448695i	0.974509	−	0.224347i	$-0.0720250\pi$
$420$	0	0
$421$	− 3.39591e6i	− 0.933795i	−0.884312	−	0.466897i	$-0.845372\pi$
$421$	− 3.39591e6i	− 0.933795i	0.884312	−	0.466897i	$-0.154628\pi$
$422$	0	0
$423$	3.99571e6i	1.08578i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	492818.	0.130803
$428$	0	0
$429$	− 508754.i	− 0.133464i
$430$	0	0
$431$	−2.08841e6	−0.541530	−0.270765	−	0.962645i	$-0.587277\pi$
$431$	−2.08841e6	−0.541530	−0.270765	+	0.962645i	$0.587277\pi$
$432$	0	0
$433$	− 4.35432e6i	− 1.11609i	−0.829810	−	0.558047i	$-0.811551\pi$
$433$	− 4.35432e6i	− 1.11609i	0.829810	−	0.558047i	$-0.188449\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	220606.	0.0552604
$438$	0	0
$439$	3.21572e6	0.796375	0.398187	−	0.917304i	$-0.369639\pi$
$439$	3.21572e6	0.796375	0.398187	+	0.917304i	$0.369639\pi$
$440$	0	0
$441$	−3.79410e6	−0.928992
$442$	0	0
$443$	3.36028e6	0.813516	0.406758	−	0.913536i	$-0.366659\pi$
$443$	3.36028e6	0.813516	0.406758	+	0.913536i	$0.366659\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 533976.i	− 0.126402i
$448$	0	0
$449$	−963097.	−0.225452	−0.112726	−	0.993626i	$-0.535958\pi$
$449$	−963097.	−0.225452	−0.112726	+	0.993626i	$0.535958\pi$
$450$	0	0
$451$	− 190727.i	− 0.0441540i
$452$	0	0
$453$	1.12665e6	0.257956
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6.88516e6i	1.54214i	0.636752	+	0.771069i	$0.280278\pi$
$457$	6.88516e6i	1.54214i	−0.636752	+	0.771069i	$0.719722\pi$
$458$	0	0
$459$	1.98251e6i	0.439222i
$460$	0	0
$461$	686048.i	0.150349i	0.997170	+	0.0751747i	$0.0239515\pi$
$461$	686048.i	0.150349i	−0.997170	+	0.0751747i	$0.976049\pi$
$462$	0	0
$463$	− 104740.i	− 0.0227070i	−0.999936	−	0.0113535i	$-0.996386\pi$
$463$	− 104740.i	− 0.0227070i	0.999936	−	0.0113535i	$-0.00361401\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	5.79633e6	1.22987	0.614937	−	0.788576i	$-0.289181\pi$
$467$	5.79633e6	1.22987	0.614937	+	0.788576i	$0.289181\pi$
$468$	0	0
$469$	1.33942e6i	0.281180i
$470$	0	0
$471$	884037.	0.183619
$472$	0	0
$473$	− 3.57536e6i	− 0.734797i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−3.02085e6	−0.607902
$478$	0	0
$479$	−9.29136e6	−1.85029	−0.925146	−	0.379611i	$-0.876058\pi$
$479$	−9.29136e6	−1.85029	−0.925146	+	0.379611i	$0.876058\pi$
$480$	0	0
$481$	2.25308e6	0.444031
$482$	0	0
$483$	−71422.7	−0.0139306
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 6.53708e6i	− 1.24900i	−0.781026	−	0.624498i	$-0.785304\pi$
$487$	− 6.53708e6i	− 1.24900i	0.781026	−	0.624498i	$-0.214696\pi$
$488$	0	0
$489$	1.18605e6	0.224300
$490$	0	0
$491$	− 6.62569e6i	− 1.24030i	−0.784483	−	0.620151i	$-0.787071\pi$
$491$	− 6.62569e6i	− 1.24030i	0.784483	−	0.620151i	$-0.212929\pi$
$492$	0	0
$493$	9.81410e6	1.81858
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.02197e6i	0.185586i
$498$	0	0
$499$	5.25473e6i	0.944710i	0.881408	+	0.472355i	$0.156596\pi$
$499$	5.25473e6i	0.944710i	−0.881408	+	0.472355i	$0.843404\pi$
$500$	0	0
$501$	− 1.14861e6i	− 0.204445i
$502$	0	0
$503$	8.78835e6i	1.54877i	0.632713	+	0.774386i	$0.281941\pi$
$503$	8.78835e6i	1.54877i	−0.632713	+	0.774386i	$0.718059\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−57664.2	−0.00996292
$508$	0	0
$509$	5.24405e6i	0.897164i	0.893742	+	0.448582i	$0.148071\pi$
$509$	5.24405e6i	0.897164i	−0.893742	+	0.448582i	$0.851929\pi$
$510$	0	0
$511$	−714979.	−0.121127
$512$	0	0
$513$	− 271727.i	− 0.0455869i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	5.29831e6	0.871788
$518$	0	0
$519$	1.89277e6	0.308446
$520$	0	0
$521$	1.02485e7	1.65412	0.827059	−	0.562115i	$-0.190012\pi$
$521$	1.02485e7	1.65412	0.827059	+	0.562115i	$0.190012\pi$
$522$	0	0
$523$	−5.94878e6	−0.950985	−0.475493	−	0.879720i	$-0.657730\pi$
$523$	−5.94878e6	−0.950985	−0.475493	+	0.879720i	$0.657730\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.58726e6i	0.562648i
$528$	0	0
$529$	5.42207e6	0.842415
$530$	0	0
$531$	− 7.06945e6i	− 1.08805i
$532$	0	0
$533$	−381868.	−0.0582231
$534$	0	0
$535$	0	0
$536$	0	0
$537$	825483.i	0.123530i
$538$	0	0
$539$	5.03097e6i	0.745899i
$540$	0	0
$541$	− 1.29778e7i	− 1.90637i	−0.302385	−	0.953186i	$-0.597783\pi$
$541$	− 1.29778e7i	− 1.90637i	0.302385	−	0.953186i	$-0.402217\pi$
$542$	0	0
$543$	− 1.51907e6i	− 0.221094i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−6.10894e6	−0.872966	−0.436483	−	0.899712i	$-0.643776\pi$
$547$	−6.10894e6	−0.872966	−0.436483	+	0.899712i	$0.643776\pi$
$548$	0	0
$549$	− 4.24990e6i	− 0.601794i
$550$	0	0
$551$	−1.34514e6	−0.188751
$552$	0	0
$553$	2.12736e6i	0.295820i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.01541e7	1.38676	0.693382	−	0.720570i	$-0.256120\pi$
$557$	1.01541e7	1.38676	0.693382	+	0.720570i	$0.256120\pi$
$558$	0	0
$559$	−7.15850e6	−0.968931
$560$	0	0
$561$	1.29604e6	0.173864
$562$	0	0
$563$	−229520.	−0.0305175	−0.0152588	−	0.999884i	$-0.504857\pi$
$563$	−229520.	−0.0305175	−0.0152588	+	0.999884i	$0.504857\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 1.48546e6i	− 0.194045i
$568$	0	0
$569$	5.11894e6	0.662826	0.331413	−	0.943486i	$-0.392475\pi$
$569$	5.11894e6	0.662826	0.331413	+	0.943486i	$0.392475\pi$
$570$	0	0
$571$	− 1.09781e7i	− 1.40909i	−0.709660	−	0.704544i	$-0.751152\pi$
$571$	− 1.09781e7i	− 1.40909i	0.709660	−	0.704544i	$-0.248848\pi$
$572$	0	0
$573$	−251876.	−0.0320479
$574$	0	0
$575$	0	0
$576$	0	0
$577$	3.09452e6i	0.386949i	0.981105	+	0.193474i	$0.0619756\pi$
$577$	3.09452e6i	0.386949i	−0.981105	+	0.193474i	$0.938024\pi$
$578$	0	0
$579$	135950.i	0.0168532i
$580$	0	0
$581$	1.60079e6i	0.196741i
$582$	0	0
$583$	4.00565e6i	0.488093i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.24448e7	1.49071	0.745357	−	0.666666i	$-0.232279\pi$
$587$	1.24448e7	1.49071	0.745357	+	0.666666i	$0.232279\pi$
$588$	0	0
$589$	− 491677.i	− 0.0583972i
$590$	0	0
$591$	1.15558e6	0.136092
$592$	0	0
$593$	− 7.81508e6i	− 0.912634i	−0.889817	−	0.456317i	$-0.849168\pi$
$593$	− 7.81508e6i	− 0.912634i	0.889817	−	0.456317i	$-0.150832\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.34268e6	0.154183
$598$	0	0
$599$	−8.68576e6	−0.989101	−0.494550	−	0.869149i	$-0.664667\pi$
$599$	−8.68576e6	−0.989101	−0.494550	+	0.869149i	$0.664667\pi$
$600$	0	0
$601$	−3.99770e6	−0.451465	−0.225733	−	0.974189i	$-0.572478\pi$
$601$	−3.99770e6	−0.451465	−0.225733	+	0.974189i	$0.572478\pi$
$602$	0	0
$603$	1.15507e7	1.29365
$604$	0	0
$605$	0	0
$606$	0	0
$607$	2.96166e6i	0.326259i	0.986605	+	0.163130i	$0.0521589\pi$
$607$	2.96166e6i	0.326259i	−0.986605	+	0.163130i	$0.947841\pi$
$608$	0	0
$609$	435498.	0.0475820
$610$	0	0
$611$	− 1.06082e7i	− 1.14957i
$612$	0	0
$613$	1.42575e7	1.53247	0.766235	−	0.642561i	$-0.222128\pi$
$613$	1.42575e7	1.53247	0.766235	+	0.642561i	$0.222128\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 4.11383e6i	− 0.435044i	−0.976055	−	0.217522i	$-0.930203\pi$
$617$	− 4.11383e6i	− 0.435044i	0.976055	−	0.217522i	$-0.0697974\pi$
$618$	0	0
$619$	6.74663e6i	0.707718i	0.935299	+	0.353859i	$0.115131\pi$
$619$	6.74663e6i	0.707718i	−0.935299	+	0.353859i	$0.884869\pi$
$620$	0	0
$621$	1.24931e6i	0.129999i
$622$	0	0
$623$	− 2.23360e6i	− 0.230561i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−177638.	−0.0180454
$628$	0	0
$629$	5.73967e6i	0.578442i
$630$	0	0
$631$	−5.01676e6	−0.501591	−0.250795	−	0.968040i	$-0.580692\pi$
$631$	−5.01676e6	−0.501591	−0.250795	+	0.968040i	$0.580692\pi$
$632$	0	0
$633$	− 2.91401e6i	− 0.289055i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	1.00729e7	0.983571
$638$	0	0
$639$	8.81311e6	0.853841
$640$	0	0
$641$	−1.79114e6	−0.172180	−0.0860902	−	0.996287i	$-0.527437\pi$
$641$	−1.79114e6	−0.172180	−0.0860902	+	0.996287i	$0.527437\pi$
$642$	0	0
$643$	4.49516e6	0.428763	0.214382	−	0.976750i	$-0.431226\pi$
$643$	4.49516e6	0.428763	0.214382	+	0.976750i	$0.431226\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	2.47054e6i	0.232023i	0.993248	+	0.116012i	$0.0370109\pi$
$647$	2.47054e6i	0.232023i	−0.993248	+	0.116012i	$0.962989\pi$
$648$	0	0
$649$	−9.37410e6	−0.873610
$650$	0	0
$651$	159184.i	0.0147213i
$652$	0	0
$653$	1.26068e7	1.15697	0.578484	−	0.815693i	$-0.303644\pi$
$653$	1.26068e7	1.15697	0.578484	+	0.815693i	$0.303644\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	6.16574e6i	0.557278i
$658$	0	0
$659$	− 7.51240e6i	− 0.673853i	−0.941531	−	0.336927i	$-0.890613\pi$
$659$	− 7.51240e6i	− 0.673853i	0.941531	−	0.336927i	$-0.109387\pi$
$660$	0	0
$661$	− 1.66707e7i	− 1.48405i	−0.670371	−	0.742026i	$-0.733865\pi$
$661$	− 1.66707e7i	− 1.48405i	0.670371	−	0.742026i	$-0.266135\pi$
$662$	0	0
$663$	− 2.59490e6i	− 0.229264i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	6.18450e6	0.538257
$668$	0	0
$669$	359165.i	0.0310262i
$670$	0	0
$671$	−5.63537e6	−0.483188
$672$	0	0
$673$	1.75455e7i	1.49323i	0.665254	+	0.746617i	$0.268323\pi$
$673$	1.75455e7i	1.49323i	−0.665254	+	0.746617i	$0.731677\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	681435.	0.0571416	0.0285708	−	0.999592i	$-0.490904\pi$
$677$	681435.	0.0571416	0.0285708	+	0.999592i	$0.490904\pi$
$678$	0	0
$679$	778111.	0.0647690
$680$	0	0
$681$	2.50227e6	0.206760
$682$	0	0
$683$	1.08945e7	0.893629	0.446815	−	0.894627i	$-0.352558\pi$
$683$	1.08945e7	0.893629	0.446815	+	0.894627i	$0.352558\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	2.17578e6i	0.175882i
$688$	0	0
$689$	8.02003e6	0.643617
$690$	0	0
$691$	1.31179e7i	1.04513i	0.852599	+	0.522565i	$0.175025\pi$
$691$	1.31179e7i	1.04513i	−0.852599	+	0.522565i	$0.824975\pi$
$692$	0	0
$693$	−2.02886e6	−0.160479
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 972801.i	− 0.0758477i
$698$	0	0
$699$	− 2.52975e6i	− 0.195832i
$700$	0	0
$701$	− 1.34915e7i	− 1.03697i	−0.855087	−	0.518485i	$-0.826496\pi$
$701$	− 1.34915e7i	− 1.03697i	0.855087	−	0.518485i	$-0.173504\pi$
$702$	0	0
$703$	− 786690.i	− 0.0600365i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.72728e6	0.280443
$708$	0	0
$709$	1.23849e7i	0.925290i	0.886544	+	0.462645i	$0.153100\pi$
$709$	1.23849e7i	0.925290i	−0.886544	+	0.462645i	$0.846900\pi$
$710$	0	0
$711$	1.83456e7	1.36100
$712$	0	0
$713$	2.26057e6i	0.166530i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−3.04216e6	−0.220996
$718$	0	0
$719$	1.59608e7	1.15142	0.575710	−	0.817654i	$-0.304726\pi$
$719$	1.59608e7	1.15142	0.575710	+	0.817654i	$0.304726\pi$
$720$	0	0
$721$	4.56511e6	0.327049
$722$	0	0
$723$	1.59194e6	0.113261
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 1.58056e7i	− 1.10911i	−0.832148	−	0.554554i	$-0.812889\pi$
$727$	− 1.58056e7i	− 1.10911i	0.832148	−	0.554554i	$-0.187111\pi$
$728$	0	0
$729$	−1.20299e7	−0.838387
$730$	0	0
$731$	− 1.82361e7i	− 1.26223i
$732$	0	0
$733$	−2.47037e7	−1.69825	−0.849126	−	0.528190i	$-0.822871\pi$
$733$	−2.47037e7	−1.69825	−0.849126	+	0.528190i	$0.822871\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 1.53162e7i	− 1.03868i
$738$	0	0
$739$	8.37792e6i	0.564319i	0.959367	+	0.282160i	$0.0910508\pi$
$739$	8.37792e6i	0.564319i	−0.959367	+	0.282160i	$0.908949\pi$
$740$	0	0
$741$	355662.i	0.0237953i
$742$	0	0
$743$	1.37134e6i	0.0911323i	0.998961	+	0.0455662i	$0.0145092\pi$
$743$	1.37134e6i	0.0911323i	−0.998961	+	0.0455662i	$0.985491\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	1.38047e7	0.905160
$748$	0	0
$749$	− 1.85712e6i	− 0.120958i
$750$	0	0
$751$	2.32593e7	1.50486	0.752431	−	0.658671i	$-0.228881\pi$
$751$	2.32593e7	1.50486	0.752431	+	0.658671i	$0.228881\pi$
$752$	0	0
$753$	− 3.26756e6i	− 0.210008i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−9.50255e6	−0.602699	−0.301350	−	0.953514i	$-0.597437\pi$
$757$	−9.50255e6	−0.602699	−0.301350	+	0.953514i	$0.597437\pi$
$758$	0	0
$759$	816718.	0.0514597
$760$	0	0
$761$	−1.85691e6	−0.116233	−0.0581164	−	0.998310i	$-0.518509\pi$
$761$	−1.85691e6	−0.116233	−0.0581164	+	0.998310i	$0.518509\pi$
$762$	0	0
$763$	−1.52264e6	−0.0946858
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.87686e7i	1.15198i
$768$	0	0
$769$	−3.11565e7	−1.89991	−0.949955	−	0.312386i	$-0.898872\pi$
$769$	−3.11565e7	−1.89991	−0.949955	+	0.312386i	$0.898872\pi$
$770$	0	0
$771$	896377.i	0.0543069i
$772$	0	0
$773$	−3.47037e6	−0.208894	−0.104447	−	0.994530i	$-0.533307\pi$
$773$	−3.47037e6	−0.208894	−0.104447	+	0.994530i	$0.533307\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	254696.i	0.0151346i
$778$	0	0
$779$	133334.i	0.00787223i
$780$	0	0
$781$	− 1.16862e7i	− 0.685560i
$782$	0	0
$783$	− 7.61764e6i	− 0.444034i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	8.99343e6	0.517593	0.258797	−	0.965932i	$-0.416674\pi$
$787$	8.99343e6	0.517593	0.258797	+	0.965932i	$0.416674\pi$
$788$	0	0
$789$	5.00079e6i	0.285987i
$790$	0	0
$791$	−4.67188e6	−0.265491
$792$	0	0
$793$	1.12830e7i	0.637150i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−6.61412e6	−0.368830	−0.184415	−	0.982848i	$-0.559039\pi$
$797$	−6.61412e6	−0.368830	−0.184415	+	0.982848i	$0.559039\pi$
$798$	0	0
$799$	2.70240e7	1.49756
$800$	0	0
$801$	−1.92618e7	−1.06076
$802$	0	0
$803$	8.17578e6	0.447446
$804$	0	0
$805$	0	0
$806$	0	0
$807$	5.76382e6i	0.311549i
$808$	0	0
$809$	1.45192e7	0.779957	0.389978	−	0.920824i	$-0.372482\pi$
$809$	1.45192e7	0.779957	0.389978	+	0.920824i	$0.372482\pi$
$810$	0	0
$811$	1.23447e7i	0.659068i	0.944144	+	0.329534i	$0.106892\pi$
$811$	1.23447e7i	0.659068i	−0.944144	+	0.329534i	$0.893108\pi$
$812$	0	0
$813$	−1.32638e6	−0.0703787
$814$	0	0
$815$	0	0
$816$	0	0
$817$	2.49948e6i	0.131007i
$818$	0	0
$819$	4.06214e6i	0.211614i
$820$	0	0
$821$	1.98036e7i	1.02538i	0.858573	+	0.512692i	$0.171352\pi$
$821$	1.98036e7i	1.02538i	−0.858573	+	0.512692i	$0.828648\pi$
$822$	0	0
$823$	− 1.28299e7i	− 0.660275i	−0.943933	−	0.330137i	$-0.892905\pi$
$823$	− 1.28299e7i	− 0.660275i	0.943933	−	0.330137i	$-0.107095\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−3.83991e7	−1.95235	−0.976174	−	0.216989i	$-0.930376\pi$
$827$	−3.83991e7	−1.95235	−0.976174	+	0.216989i	$0.930376\pi$
$828$	0	0
$829$	1.17104e7i	0.591813i	0.955217	+	0.295907i	$0.0956217\pi$
$829$	1.17104e7i	0.591813i	−0.955217	+	0.295907i	$0.904378\pi$
$830$	0	0
$831$	2.13697e6	0.107348
$832$	0	0
$833$	2.56605e7i	1.28130i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	2.78441e6	0.137379
$838$	0	0
$839$	−2.39114e7	−1.17273	−0.586367	−	0.810046i	$-0.699442\pi$
$839$	−2.39114e7	−1.17273	−0.586367	+	0.810046i	$0.699442\pi$
$840$	0	0
$841$	−1.71987e7	−0.838503
$842$	0	0
$843$	5.66448e6	0.274531
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 1.72277e6i	− 0.0825125i
$848$	0	0
$849$	4.25511e6	0.202601
$850$	0	0
$851$	3.61694e6i	0.171205i
$852$	0	0
$853$	−1.27294e6	−0.0599010	−0.0299505	−	0.999551i	$-0.509535\pi$
$853$	−1.27294e6	−0.0599010	−0.0299505	+	0.999551i	$0.509535\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 1.88053e7i	− 0.874636i	−0.899307	−	0.437318i	$-0.855928\pi$
$857$	− 1.88053e7i	− 0.874636i	0.899307	−	0.437318i	$-0.144072\pi$
$858$	0	0
$859$	− 1.00554e7i	− 0.464960i	−0.972601	−	0.232480i	$-0.925316\pi$
$859$	− 1.00554e7i	− 0.464960i	0.972601	−	0.232480i	$-0.0746840\pi$
$860$	0	0
$861$	− 43167.8i	− 0.00198450i
$862$	0	0
$863$	− 2.96146e7i	− 1.35357i	−0.736183	−	0.676783i	$-0.763374\pi$
$863$	− 2.96146e7i	− 1.35357i	0.736183	−	0.676783i	$-0.236626\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	2.93569e6	0.132636
$868$	0	0
$869$	− 2.43263e7i	− 1.09277i
$870$	0	0
$871$	−3.06658e7	−1.36965
$872$	0	0
$873$	− 6.71017e6i	− 0.297988i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1.42067e7	0.623727	0.311863	−	0.950127i	$-0.399047\pi$
$877$	1.42067e7	0.623727	0.311863	+	0.950127i	$0.399047\pi$
$878$	0	0
$879$	4.63932e6	0.202526
$880$	0	0
$881$	1.88875e7	0.819851	0.409925	−	0.912119i	$-0.365555\pi$
$881$	1.88875e7	0.819851	0.409925	+	0.912119i	$0.365555\pi$
$882$	0	0
$883$	−9.53131e6	−0.411387	−0.205694	−	0.978616i	$-0.565945\pi$
$883$	−9.53131e6	−0.411387	−0.205694	+	0.978616i	$0.565945\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	3.28115e7i	1.40029i	0.714002	+	0.700143i	$0.246881\pi$
$887$	3.28115e7i	1.40029i	−0.714002	+	0.700143i	$0.753119\pi$
$888$	0	0
$889$	−8.59762e6	−0.364858
$890$	0	0
$891$	1.69862e7i	0.716806i
$892$	0	0
$893$	−3.70397e6	−0.155431
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 1.63521e6i	− 0.0678568i
$898$	0	0
$899$	− 1.37837e7i	− 0.568811i
$900$	0	0
$901$	2.04308e7i	0.838444i
$902$	0	0
$903$	− 809223.i	− 0.0330255i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−1.53129e7	−0.618073	−0.309037	−	0.951050i	$-0.600007\pi$
$907$	−1.53129e7	−0.618073	−0.309037	+	0.951050i	$0.600007\pi$
$908$	0	0
$909$	− 3.21429e7i	− 1.29025i
$910$	0	0
$911$	5.70920e6	0.227918	0.113959	−	0.993485i	$-0.463647\pi$
$911$	5.70920e6	0.227918	0.113959	+	0.993485i	$0.463647\pi$
$912$	0	0
$913$	− 1.83050e7i	− 0.726764i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	9.69077e6	0.380570
$918$	0	0
$919$	−9.51702e6	−0.371717	−0.185858	−	0.982577i	$-0.559507\pi$
$919$	−9.51702e6	−0.371717	−0.185858	+	0.982577i	$0.559507\pi$
$920$	0	0
$921$	3.37919e6	0.131269
$922$	0	0
$923$	−2.33978e7	−0.904005
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 3.93680e7i	− 1.50468i
$928$	0	0
$929$	2.79927e7	1.06416	0.532078	−	0.846695i	$-0.321411\pi$
$929$	2.79927e7	1.06416	0.532078	+	0.846695i	$0.321411\pi$
$930$	0	0
$931$	− 3.51708e6i	− 0.132987i
$932$	0	0
$933$	−2.57809e6	−0.0969604
$934$	0	0
$935$	0	0
$936$	0	0
$937$	6.87551e6i	0.255832i	0.991785	+	0.127916i	$0.0408289\pi$
$937$	6.87551e6i	0.255832i	−0.991785	+	0.127916i	$0.959171\pi$
$938$	0	0
$939$	− 2.29586e6i	− 0.0849730i
$940$	0	0
$941$	2.68027e7i	0.986743i	0.869819	+	0.493371i	$0.164236\pi$
$941$	2.68027e7i	0.986743i	−0.869819	+	0.493371i	$0.835764\pi$
$942$	0	0
$943$	− 613025.i	− 0.0224491i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−8.91973e6	−0.323204	−0.161602	−	0.986856i	$-0.551666\pi$
$947$	−8.91973e6	−0.323204	−0.161602	+	0.986856i	$0.551666\pi$
$948$	0	0
$949$	− 1.63694e7i	− 0.590019i
$950$	0	0
$951$	3.19080e6	0.114406
$952$	0	0
$953$	3.25154e7i	1.15973i	0.814713	+	0.579865i	$0.196895\pi$
$953$	3.25154e7i	1.15973i	−0.814713	+	0.579865i	$0.803105\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−4.97991e6	−0.175769
$958$	0	0
$959$	1.99347e6	0.0699942
$960$	0	0
$961$	−2.35909e7	−0.824017
$962$	0	0
$963$	−1.60152e7	−0.556502
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 1.43390e7i	− 0.493121i	−0.969127	−	0.246561i	$-0.920700\pi$
$967$	− 1.43390e7i	− 0.493121i	0.969127	−	0.246561i	$-0.0793005\pi$
$968$	0	0
$969$	−906041.	−0.0309983
$970$	0	0
$971$	4.37060e7i	1.48762i	0.668390	+	0.743811i	$0.266984\pi$
$971$	4.37060e7i	1.48762i	−0.668390	+	0.743811i	$0.733016\pi$
$972$	0	0
$973$	−7.11015e6	−0.240767
$974$	0	0
$975$	0	0
$976$	0	0
$977$	7.03004e6i	0.235625i	0.993036	+	0.117813i	$0.0375882\pi$
$977$	7.03004e6i	0.235625i	−0.993036	+	0.117813i	$0.962412\pi$
$978$	0	0
$979$	2.55412e7i	0.851696i
$980$	0	0
$981$	1.31307e7i	0.435628i
$982$	0	0
$983$	3.83715e7i	1.26656i	0.773924	+	0.633279i	$0.218291\pi$
$983$	3.83715e7i	1.26656i	−0.773924	+	0.633279i	$0.781709\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1.19918e6	0.0391826
$988$	0	0
$989$	− 1.14918e7i	− 0.373591i
$990$	0	0
$991$	5.24440e7	1.69634	0.848168	−	0.529727i	$-0.177706\pi$
$991$	5.24440e7	1.69634	0.848168	+	0.529727i	$0.177706\pi$
$992$	0	0
$993$	− 8.60602e6i	− 0.276968i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−4.88081e7	−1.55508	−0.777542	−	0.628832i	$-0.783533\pi$
$997$	−4.88081e7	−1.55508	−0.777542	+	0.628832i	$0.783533\pi$
$998$	0	0
$999$	4.45509e6	0.141235

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.6.f.d.49.18		40
4.3	odd	2		200.6.f.d.149.2		40
5.2	odd	4		800.6.d.d.401.12		20
5.3	odd	4		800.6.d.b.401.9		20
5.4	even	2	inner	800.6.f.d.49.23		40
8.3	odd	2		200.6.f.d.149.40		40
8.5	even	2	inner	800.6.f.d.49.24		40
20.3	even	4		200.6.d.d.101.12	yes	20
20.7	even	4		200.6.d.c.101.9	✓	20
20.19	odd	2		200.6.f.d.149.39		40
40.3	even	4		200.6.d.d.101.11	yes	20
40.13	odd	4		800.6.d.b.401.12		20
40.19	odd	2		200.6.f.d.149.1		40
40.27	even	4		200.6.d.c.101.10	yes	20
40.29	even	2	inner	800.6.f.d.49.17		40
40.37	odd	4		800.6.d.d.401.9		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
200.6.d.c.101.9	✓	20	20.7	even	4
200.6.d.c.101.10	yes	20	40.27	even	4
200.6.d.d.101.11	yes	20	40.3	even	4
200.6.d.d.101.12	yes	20	20.3	even	4
200.6.f.d.149.1		40	40.19	odd	2
200.6.f.d.149.2		40	4.3	odd	2
200.6.f.d.149.39		40	20.19	odd	2
200.6.f.d.149.40		40	8.3	odd	2
800.6.d.b.401.9		20	5.3	odd	4
800.6.d.b.401.12		20	40.13	odd	4
800.6.d.d.401.9		20	40.37	odd	4
800.6.d.d.401.12		20	5.2	odd	4
800.6.f.d.49.17		40	40.29	even	2	inner
800.6.f.d.49.18		40	1.1	even	1	trivial
800.6.f.d.49.23		40	5.4	even	2	inner
800.6.f.d.49.24		40	8.5	even	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.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.58812 q^{3} -27.4015i q^{7} -236.302 q^{9} +O(q^{10})\)	\(q-2.58812 q^{3} -27.4015i q^{7} -236.302 q^{9} +313.336i q^{11} +627.354 q^{13} +1598.17i q^{17} -219.049i q^{19} +70.9184i q^{21} +1007.11i q^{23} +1240.49 q^{27} -6140.83i q^{29} +2244.61 q^{31} -810.951i q^{33} +3591.40 q^{37} -1623.67 q^{39} -608.697 q^{41} -11410.6 q^{43} -16909.4i q^{47} +16056.2 q^{49} -4136.26i q^{51} +12783.9 q^{53} +566.923i q^{57} +29917.1i q^{59} +17985.1i q^{61} +6475.03i q^{63} -48881.2 q^{67} -2606.52i q^{69} -37296.0 q^{71} -26092.7i q^{73} +8585.89 q^{77} -77636.5 q^{79} +54210.8 q^{81} -58419.8 q^{83} +15893.2i q^{87} +81513.8 q^{89} -17190.5i q^{91} -5809.30 q^{93} +28396.6i q^{97} -74041.9i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 3240 q^{9}+O(q^{10}) \) 40 * q + 3240 * q^9	\( 40 q + 3240 q^{9} - 14320 q^{31} + 44904 q^{39} - 11608 q^{41} - 125304 q^{49} + 15448 q^{71} - 15560 q^{79} + 193968 q^{81} + 6320 q^{89}+O(q^{100}) \) 40 * q + 3240 * q^9 - 14320 * q^31 + 44904 * q^39 - 11608 * q^41 - 125304 * q^49 + 15448 * q^71 - 15560 * q^79 + 193968 * q^81 + 6320 * q^89

Introduction

L-functions

Modular forms

Varieties

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Database

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