LMFDB - Embedded newform 800.4.c.a.449.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,4,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, 4, names="a")

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

chi := DirichletCharacter("800.449");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
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:	$47.2015280046$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 32)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	800.449
Dual form			800.4.c.a.449.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.00000i q^{3} +16.0000i q^{7} -37.0000 q^{9} +O(q^{10})$	$q-8.00000i q^{3} +16.0000i q^{7} -37.0000 q^{9} -40.0000 q^{11} +50.0000i q^{13} -30.0000i q^{17} -40.0000 q^{19} +128.000 q^{21} -48.0000i q^{23} +80.0000i q^{27} +34.0000 q^{29} +320.000 q^{31} +320.000i q^{33} +310.000i q^{37} +400.000 q^{39} +410.000 q^{41} -152.000i q^{43} -416.000i q^{47} +87.0000 q^{49} -240.000 q^{51} +410.000i q^{53} +320.000i q^{57} +200.000 q^{59} +30.0000 q^{61} -592.000i q^{63} +776.000i q^{67} -384.000 q^{69} +400.000 q^{71} +630.000i q^{73} -640.000i q^{77} +1120.00 q^{79} -359.000 q^{81} -552.000i q^{83} -272.000i q^{87} +326.000 q^{89} -800.000 q^{91} -2560.00i q^{93} -110.000i q^{97} +1480.00 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 74 q^{9}+O(q^{10}) $ 2 * q - 74 * q^9	$ 2 q - 74 q^{9} - 80 q^{11} - 80 q^{19} + 256 q^{21} + 68 q^{29} + 640 q^{31} + 800 q^{39} + 820 q^{41} + 174 q^{49} - 480 q^{51} + 400 q^{59} + 60 q^{61} - 768 q^{69} + 800 q^{71} + 2240 q^{79} - 718 q^{81} + 652 q^{89} - 1600 q^{91} + 2960 q^{99}+O(q^{100}) $ 2 * q - 74 * q^9 - 80 * q^11 - 80 * q^19 + 256 * q^21 + 68 * q^29 + 640 * q^31 + 800 * q^39 + 820 * q^41 + 174 * q^49 - 480 * q^51 + 400 * q^59 + 60 * q^61 - 768 * q^69 + 800 * q^71 + 2240 * q^79 - 718 * q^81 + 652 * q^89 - 1600 * q^91 + 2960 * q^99

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
$3$	− 8.00000i	− 1.53960i	−0.638285	−	0.769800i	$-0.720356\pi$
$3$	− 8.00000i	− 1.53960i	0.638285	−	0.769800i	$-0.279644\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	16.0000i	0.863919i	0.901893	+	0.431959i	$0.142178\pi$
$7$	16.0000i	0.863919i	−0.901893	+	0.431959i	$0.857822\pi$
$8$	0	0
$9$	−37.0000	−1.37037
$10$	0	0
$11$	−40.0000	−1.09640	−0.548202	−	0.836346i	$-0.684688\pi$
$11$	−40.0000	−1.09640	−0.548202	+	0.836346i	$0.684688\pi$
$12$	0	0
$13$	50.0000i	1.06673i	0.845885	+	0.533366i	$0.179073\pi$
$13$	50.0000i	1.06673i	−0.845885	+	0.533366i	$0.820927\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 30.0000i	− 0.428004i	−0.976833	−	0.214002i	$-0.931350\pi$
$17$	− 30.0000i	− 0.428004i	0.976833	−	0.214002i	$-0.0686499\pi$
$18$	0	0
$19$	−40.0000	−0.482980	−0.241490	−	0.970403i	$-0.577636\pi$
$19$	−40.0000	−0.482980	−0.241490	+	0.970403i	$0.577636\pi$
$20$	0	0
$21$	128.000	1.33009
$22$	0	0
$23$	− 48.0000i	− 0.435161i	−0.976042	−	0.217580i	$-0.930184\pi$
$23$	− 48.0000i	− 0.435161i	0.976042	−	0.217580i	$-0.0698164\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	80.0000i	0.570222i
$28$	0	0
$29$	34.0000	0.217712	0.108856	−	0.994058i	$-0.465281\pi$
$29$	34.0000	0.217712	0.108856	+	0.994058i	$0.465281\pi$
$30$	0	0
$31$	320.000	1.85399	0.926995	−	0.375073i	$-0.122383\pi$
$31$	320.000	1.85399	0.926995	+	0.375073i	$0.122383\pi$
$32$	0	0
$33$	320.000i	1.68803i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	310.000i	1.37740i	0.725048	+	0.688698i	$0.241818\pi$
$37$	310.000i	1.37740i	−0.725048	+	0.688698i	$0.758182\pi$
$38$	0	0
$39$	400.000	1.64234
$40$	0	0
$41$	410.000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	410.000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	0	0
$43$	− 152.000i	− 0.539065i	−0.962991	−	0.269532i	$-0.913131\pi$
$43$	− 152.000i	− 0.539065i	0.962991	−	0.269532i	$-0.0868691\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 416.000i	− 1.29106i	−0.763735	−	0.645530i	$-0.776636\pi$
$47$	− 416.000i	− 1.29106i	0.763735	−	0.645530i	$-0.223364\pi$
$48$	0	0
$49$	87.0000	0.253644
$50$	0	0
$51$	−240.000	−0.658955
$52$	0	0
$53$	410.000i	1.06260i	0.847184	+	0.531300i	$0.178296\pi$
$53$	410.000i	1.06260i	−0.847184	+	0.531300i	$0.821704\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	320.000i	0.743597i
$58$	0	0
$59$	200.000	0.441318	0.220659	−	0.975351i	$-0.429179\pi$
$59$	200.000	0.441318	0.220659	+	0.975351i	$0.429179\pi$
$60$	0	0
$61$	30.0000	0.0629690	0.0314845	−	0.999504i	$-0.489977\pi$
$61$	30.0000	0.0629690	0.0314845	+	0.999504i	$0.489977\pi$
$62$	0	0
$63$	− 592.000i	− 1.18389i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	776.000i	1.41498i	0.706725	+	0.707489i	$0.250172\pi$
$67$	776.000i	1.41498i	−0.706725	+	0.707489i	$0.749828\pi$
$68$	0	0
$69$	−384.000	−0.669973
$70$	0	0
$71$	400.000	0.668609	0.334305	−	0.942465i	$-0.391499\pi$
$71$	400.000	0.668609	0.334305	+	0.942465i	$0.391499\pi$
$72$	0	0
$73$	630.000i	1.01008i	0.863096	+	0.505041i	$0.168522\pi$
$73$	630.000i	1.01008i	−0.863096	+	0.505041i	$0.831478\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 640.000i	− 0.947205i
$78$	0	0
$79$	1120.00	1.59506	0.797531	−	0.603278i	$-0.206139\pi$
$79$	1120.00	1.59506	0.797531	+	0.603278i	$0.206139\pi$
$80$	0	0
$81$	−359.000	−0.492455
$82$	0	0
$83$	− 552.000i	− 0.729998i	−0.931008	−	0.364999i	$-0.881069\pi$
$83$	− 552.000i	− 0.729998i	0.931008	−	0.364999i	$-0.118931\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 272.000i	− 0.335189i
$88$	0	0
$89$	326.000	0.388269	0.194134	−	0.980975i	$-0.437810\pi$
$89$	326.000	0.388269	0.194134	+	0.980975i	$0.437810\pi$
$90$	0	0
$91$	−800.000	−0.921569
$92$	0	0
$93$	− 2560.00i	− 2.85440i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 110.000i	− 0.115142i	−0.998341	−	0.0575712i	$-0.981664\pi$
$97$	− 110.000i	− 0.115142i	0.998341	−	0.0575712i	$-0.0183356\pi$
$98$	0	0
$99$	1480.00	1.50248
$100$	0	0
$101$	−1098.00	−1.08173	−0.540867	−	0.841108i	$-0.681904\pi$
$101$	−1098.00	−1.08173	−0.540867	+	0.841108i	$0.681904\pi$
$102$	0	0
$103$	48.0000i	0.0459183i	0.999736	+	0.0229591i	$0.00730876\pi$
$103$	48.0000i	0.0459183i	−0.999736	+	0.0229591i	$0.992691\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	664.000i	0.599919i	0.953952	+	0.299959i	$0.0969731\pi$
$107$	664.000i	0.599919i	−0.953952	+	0.299959i	$0.903027\pi$
$108$	0	0
$109$	370.000	0.325134	0.162567	−	0.986698i	$-0.448023\pi$
$109$	370.000	0.325134	0.162567	+	0.986698i	$0.448023\pi$
$110$	0	0
$111$	2480.00	2.12064
$112$	0	0
$113$	− 1490.00i	− 1.24042i	−0.784436	−	0.620210i	$-0.787047\pi$
$113$	− 1490.00i	− 1.24042i	0.784436	−	0.620210i	$-0.212953\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 1850.00i	− 1.46182i
$118$	0	0
$119$	480.000	0.369761
$120$	0	0
$121$	269.000	0.202104
$122$	0	0
$123$	− 3280.00i	− 2.40445i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 1024.00i	− 0.715475i	−0.933822	−	0.357737i	$-0.883548\pi$
$127$	− 1024.00i	− 0.715475i	0.933822	−	0.357737i	$-0.116452\pi$
$128$	0	0
$129$	−1216.00	−0.829944
$130$	0	0
$131$	1160.00	0.773662	0.386831	−	0.922151i	$-0.373570\pi$
$131$	1160.00	0.773662	0.386831	+	0.922151i	$0.373570\pi$
$132$	0	0
$133$	− 640.000i	− 0.417256i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	570.000i	0.355463i	0.984079	+	0.177731i	$0.0568758\pi$
$137$	570.000i	0.355463i	−0.984079	+	0.177731i	$0.943124\pi$
$138$	0	0
$139$	1960.00	1.19601	0.598004	−	0.801493i	$-0.295961\pi$
$139$	1960.00	1.19601	0.598004	+	0.801493i	$0.295961\pi$
$140$	0	0
$141$	−3328.00	−1.98772
$142$	0	0
$143$	− 2000.00i	− 1.16957i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 696.000i	− 0.390511i
$148$	0	0
$149$	2010.00	1.10514	0.552569	−	0.833467i	$-0.313648\pi$
$149$	2010.00	1.10514	0.552569	+	0.833467i	$0.313648\pi$
$150$	0	0
$151$	−720.000	−0.388032	−0.194016	−	0.980998i	$-0.562151\pi$
$151$	−720.000	−0.388032	−0.194016	+	0.980998i	$0.562151\pi$
$152$	0	0
$153$	1110.00i	0.586524i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	1790.00i	0.909921i	0.890512	+	0.454960i	$0.150347\pi$
$157$	1790.00i	0.909921i	−0.890512	+	0.454960i	$0.849653\pi$
$158$	0	0
$159$	3280.00	1.63598
$160$	0	0
$161$	768.000	0.375943
$162$	0	0
$163$	1208.00i	0.580478i	0.956954	+	0.290239i	$0.0937348\pi$
$163$	1208.00i	0.580478i	−0.956954	+	0.290239i	$0.906265\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2896.00i	1.34191i	0.741497	+	0.670956i	$0.234116\pi$
$167$	2896.00i	1.34191i	−0.741497	+	0.670956i	$0.765884\pi$
$168$	0	0
$169$	−303.000	−0.137915
$170$	0	0
$171$	1480.00	0.661862
$172$	0	0
$173$	− 750.000i	− 0.329604i	−0.986327	−	0.164802i	$-0.947302\pi$
$173$	− 750.000i	− 0.329604i	0.986327	−	0.164802i	$-0.0526985\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 1600.00i	− 0.679454i
$178$	0	0
$179$	−2280.00	−0.952040	−0.476020	−	0.879434i	$-0.657921\pi$
$179$	−2280.00	−0.952040	−0.476020	+	0.879434i	$0.657921\pi$
$180$	0	0
$181$	−442.000	−0.181512	−0.0907558	−	0.995873i	$-0.528928\pi$
$181$	−442.000	−0.181512	−0.0907558	+	0.995873i	$0.528928\pi$
$182$	0	0
$183$	− 240.000i	− 0.0969471i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1200.00i	0.469266i
$188$	0	0
$189$	−1280.00	−0.492626
$190$	0	0
$191$	−1920.00	−0.727363	−0.363681	−	0.931523i	$-0.618480\pi$
$191$	−1920.00	−0.727363	−0.363681	+	0.931523i	$0.618480\pi$
$192$	0	0
$193$	5070.00i	1.89091i	0.325746	+	0.945457i	$0.394385\pi$
$193$	5070.00i	1.89091i	−0.325746	+	0.945457i	$0.605615\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1910.00i	0.690771i	0.938461	+	0.345385i	$0.112252\pi$
$197$	1910.00i	0.690771i	−0.938461	+	0.345385i	$0.887748\pi$
$198$	0	0
$199$	−2960.00	−1.05442	−0.527208	−	0.849736i	$-0.676761\pi$
$199$	−2960.00	−1.05442	−0.527208	+	0.849736i	$0.676761\pi$
$200$	0	0
$201$	6208.00	2.17850
$202$	0	0
$203$	544.000i	0.188085i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1776.00i	0.596331i
$208$	0	0
$209$	1600.00	0.529542
$210$	0	0
$211$	40.0000	0.0130508	0.00652539	−	0.999979i	$-0.497923\pi$
$211$	40.0000	0.0130508	0.00652539	+	0.999979i	$0.497923\pi$
$212$	0	0
$213$	− 3200.00i	− 1.02939i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	5120.00i	1.60170i
$218$	0	0
$219$	5040.00	1.55512
$220$	0	0
$221$	1500.00	0.456565
$222$	0	0
$223$	− 4288.00i	− 1.28765i	−0.765173	−	0.643824i	$-0.777347\pi$
$223$	− 4288.00i	− 1.28765i	0.765173	−	0.643824i	$-0.222653\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 6456.00i	− 1.88766i	−0.330425	−	0.943832i	$-0.607192\pi$
$227$	− 6456.00i	− 1.88766i	0.330425	−	0.943832i	$-0.392808\pi$
$228$	0	0
$229$	1066.00	0.307613	0.153806	−	0.988101i	$-0.450847\pi$
$229$	1066.00	0.307613	0.153806	+	0.988101i	$0.450847\pi$
$230$	0	0
$231$	−5120.00	−1.45832
$232$	0	0
$233$	5910.00i	1.66170i	0.556494	+	0.830852i	$0.312146\pi$
$233$	5910.00i	1.66170i	−0.556494	+	0.830852i	$0.687854\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 8960.00i	− 2.45576i
$238$	0	0
$239$	3360.00	0.909374	0.454687	−	0.890651i	$-0.349751\pi$
$239$	3360.00	0.909374	0.454687	+	0.890651i	$0.349751\pi$
$240$	0	0
$241$	3970.00	1.06112	0.530561	−	0.847647i	$-0.321981\pi$
$241$	3970.00	1.06112	0.530561	+	0.847647i	$0.321981\pi$
$242$	0	0
$243$	5032.00i	1.32841i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 2000.00i	− 0.515210i
$248$	0	0
$249$	−4416.00	−1.12391
$250$	0	0
$251$	6840.00	1.72007	0.860034	−	0.510237i	$-0.170442\pi$
$251$	6840.00	1.72007	0.860034	+	0.510237i	$0.170442\pi$
$252$	0	0
$253$	1920.00i	0.477112i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4610.00i	1.11893i	0.828855	+	0.559463i	$0.188993\pi$
$257$	4610.00i	1.11893i	−0.828855	+	0.559463i	$0.811007\pi$
$258$	0	0
$259$	−4960.00	−1.18996
$260$	0	0
$261$	−1258.00	−0.298346
$262$	0	0
$263$	4848.00i	1.13666i	0.822802	+	0.568328i	$0.192409\pi$
$263$	4848.00i	1.13666i	−0.822802	+	0.568328i	$0.807591\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 2608.00i	− 0.597779i
$268$	0	0
$269$	−5550.00	−1.25795	−0.628977	−	0.777424i	$-0.716526\pi$
$269$	−5550.00	−1.25795	−0.628977	+	0.777424i	$0.716526\pi$
$270$	0	0
$271$	−480.000	−0.107594	−0.0537969	−	0.998552i	$-0.517132\pi$
$271$	−480.000	−0.107594	−0.0537969	+	0.998552i	$0.517132\pi$
$272$	0	0
$273$	6400.00i	1.41885i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1030.00i	0.223418i	0.993741	+	0.111709i	$0.0356324\pi$
$277$	1030.00i	0.223418i	−0.993741	+	0.111709i	$0.964368\pi$
$278$	0	0
$279$	−11840.0	−2.54065
$280$	0	0
$281$	−3270.00	−0.694206	−0.347103	−	0.937827i	$-0.612835\pi$
$281$	−3270.00	−0.694206	−0.347103	+	0.937827i	$0.612835\pi$
$282$	0	0
$283$	− 2168.00i	− 0.455386i	−0.973733	−	0.227693i	$-0.926882\pi$
$283$	− 2168.00i	− 0.455386i	0.973733	−	0.227693i	$-0.0731183\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6560.00i	1.34921i
$288$	0	0
$289$	4013.00	0.816813
$290$	0	0
$291$	−880.000	−0.177273
$292$	0	0
$293$	− 2070.00i	− 0.412733i	−0.978475	−	0.206366i	$-0.933836\pi$
$293$	− 2070.00i	− 0.412733i	0.978475	−	0.206366i	$-0.0661639\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 3200.00i	− 0.625195i
$298$	0	0
$299$	2400.00	0.464199
$300$	0	0
$301$	2432.00	0.465708
$302$	0	0
$303$	8784.00i	1.66544i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	1896.00i	0.352477i	0.984347	+	0.176238i	$0.0563930\pi$
$307$	1896.00i	0.352477i	−0.984347	+	0.176238i	$0.943607\pi$
$308$	0	0
$309$	384.000	0.0706958
$310$	0	0
$311$	−1680.00	−0.306315	−0.153158	−	0.988202i	$-0.548944\pi$
$311$	−1680.00	−0.306315	−0.153158	+	0.988202i	$0.548944\pi$
$312$	0	0
$313$	− 970.000i	− 0.175168i	−0.996157	−	0.0875841i	$-0.972085\pi$
$313$	− 970.000i	− 0.175168i	0.996157	−	0.0875841i	$-0.0279146\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	7230.00i	1.28100i	0.767958	+	0.640500i	$0.221273\pi$
$317$	7230.00i	1.28100i	−0.767958	+	0.640500i	$0.778727\pi$
$318$	0	0
$319$	−1360.00	−0.238700
$320$	0	0
$321$	5312.00	0.923635
$322$	0	0
$323$	1200.00i	0.206718i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 2960.00i	− 0.500576i
$328$	0	0
$329$	6656.00	1.11537
$330$	0	0
$331$	−5800.00	−0.963132	−0.481566	−	0.876410i	$-0.659932\pi$
$331$	−5800.00	−0.963132	−0.481566	+	0.876410i	$0.659932\pi$
$332$	0	0
$333$	− 11470.0i	− 1.88754i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 1870.00i	− 0.302271i	−0.988513	−	0.151136i	$-0.951707\pi$
$337$	− 1870.00i	− 0.302271i	0.988513	−	0.151136i	$-0.0482930\pi$
$338$	0	0
$339$	−11920.0	−1.90975
$340$	0	0
$341$	−12800.0	−2.03272
$342$	0	0
$343$	6880.00i	1.08305i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	376.000i	0.0581693i	0.999577	+	0.0290846i	$0.00925923\pi$
$347$	376.000i	0.0581693i	−0.999577	+	0.0290846i	$0.990741\pi$
$348$	0	0
$349$	7586.00	1.16352	0.581761	−	0.813360i	$-0.302364\pi$
$349$	7586.00	1.16352	0.581761	+	0.813360i	$0.302364\pi$
$350$	0	0
$351$	−4000.00	−0.608274
$352$	0	0
$353$	− 2530.00i	− 0.381468i	−0.981642	−	0.190734i	$-0.938913\pi$
$353$	− 2530.00i	− 0.381468i	0.981642	−	0.190734i	$-0.0610868\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 3840.00i	− 0.569284i
$358$	0	0
$359$	−9680.00	−1.42309	−0.711547	−	0.702638i	$-0.752005\pi$
$359$	−9680.00	−1.42309	−0.711547	+	0.702638i	$0.752005\pi$
$360$	0	0
$361$	−5259.00	−0.766730
$362$	0	0
$363$	− 2152.00i	− 0.311159i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	2784.00i	0.395977i	0.980204	+	0.197989i	$0.0634409\pi$
$367$	2784.00i	0.395977i	−0.980204	+	0.197989i	$0.936559\pi$
$368$	0	0
$369$	−15170.0	−2.14016
$370$	0	0
$371$	−6560.00	−0.918001
$372$	0	0
$373$	− 7910.00i	− 1.09803i	−0.835813	−	0.549014i	$-0.815003\pi$
$373$	− 7910.00i	− 1.09803i	0.835813	−	0.549014i	$-0.184997\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1700.00i	0.232240i
$378$	0	0
$379$	−1720.00	−0.233115	−0.116557	−	0.993184i	$-0.537186\pi$
$379$	−1720.00	−0.233115	−0.116557	+	0.993184i	$0.537186\pi$
$380$	0	0
$381$	−8192.00	−1.10155
$382$	0	0
$383$	− 11008.0i	− 1.46862i	−0.678813	−	0.734311i	$-0.737505\pi$
$383$	− 11008.0i	− 1.46862i	0.678813	−	0.734311i	$-0.262495\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	5624.00i	0.738718i
$388$	0	0
$389$	12330.0	1.60708	0.803542	−	0.595248i	$-0.202946\pi$
$389$	12330.0	1.60708	0.803542	+	0.595248i	$0.202946\pi$
$390$	0	0
$391$	−1440.00	−0.186250
$392$	0	0
$393$	− 9280.00i	− 1.19113i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 4370.00i	− 0.552453i	−0.961093	−	0.276227i	$-0.910916\pi$
$397$	− 4370.00i	− 0.552453i	0.961093	−	0.276227i	$-0.0890841\pi$
$398$	0	0
$399$	−5120.00	−0.642408
$400$	0	0
$401$	3298.00	0.410709	0.205354	−	0.978688i	$-0.434165\pi$
$401$	3298.00	0.410709	0.205354	+	0.978688i	$0.434165\pi$
$402$	0	0
$403$	16000.0i	1.97771i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 12400.0i	− 1.51018i
$408$	0	0
$409$	9110.00	1.10137	0.550685	−	0.834713i	$-0.314366\pi$
$409$	9110.00	1.10137	0.550685	+	0.834713i	$0.314366\pi$
$410$	0	0
$411$	4560.00	0.547271
$412$	0	0
$413$	3200.00i	0.381263i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 15680.0i	− 1.84137i
$418$	0	0
$419$	−7880.00	−0.918767	−0.459383	−	0.888238i	$-0.651930\pi$
$419$	−7880.00	−0.918767	−0.459383	+	0.888238i	$0.651930\pi$
$420$	0	0
$421$	−5290.00	−0.612396	−0.306198	−	0.951968i	$-0.599057\pi$
$421$	−5290.00	−0.612396	−0.306198	+	0.951968i	$0.599057\pi$
$422$	0	0
$423$	15392.0i	1.76923i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	480.000i	0.0544001i
$428$	0	0
$429$	−16000.0	−1.80067
$430$	0	0
$431$	13920.0	1.55569	0.777845	−	0.628456i	$-0.216313\pi$
$431$	13920.0	1.55569	0.777845	+	0.628456i	$0.216313\pi$
$432$	0	0
$433$	− 4930.00i	− 0.547161i	−0.961849	−	0.273580i	$-0.911792\pi$
$433$	− 4930.00i	− 0.547161i	0.961849	−	0.273580i	$-0.0882080\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1920.00i	0.210174i
$438$	0	0
$439$	10640.0	1.15676	0.578382	−	0.815766i	$-0.303684\pi$
$439$	10640.0	1.15676	0.578382	+	0.815766i	$0.303684\pi$
$440$	0	0
$441$	−3219.00	−0.347587
$442$	0	0
$443$	9288.00i	0.996131i	0.867139	+	0.498066i	$0.165956\pi$
$443$	9288.00i	0.996131i	−0.867139	+	0.498066i	$0.834044\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 16080.0i	− 1.70147i
$448$	0	0
$449$	−12850.0	−1.35062	−0.675311	−	0.737533i	$-0.735990\pi$
$449$	−12850.0	−1.35062	−0.675311	+	0.737533i	$0.735990\pi$
$450$	0	0
$451$	−16400.0	−1.71230
$452$	0	0
$453$	5760.00i	0.597414i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	10490.0i	1.07375i	0.843663	+	0.536873i	$0.180394\pi$
$457$	10490.0i	1.07375i	−0.843663	+	0.536873i	$0.819606\pi$
$458$	0	0
$459$	2400.00	0.244058
$460$	0	0
$461$	11118.0	1.12325	0.561624	−	0.827393i	$-0.310177\pi$
$461$	11118.0	1.12325	0.561624	+	0.827393i	$0.310177\pi$
$462$	0	0
$463$	− 5792.00i	− 0.581376i	−0.956818	−	0.290688i	$-0.906116\pi$
$463$	− 5792.00i	− 0.581376i	0.956818	−	0.290688i	$-0.0938842\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	2216.00i	0.219581i	0.993955	+	0.109790i	$0.0350180\pi$
$467$	2216.00i	0.219581i	−0.993955	+	0.109790i	$0.964982\pi$
$468$	0	0
$469$	−12416.0	−1.22243
$470$	0	0
$471$	14320.0	1.40091
$472$	0	0
$473$	6080.00i	0.591033i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 15170.0i	− 1.45616i
$478$	0	0
$479$	10560.0	1.00730	0.503652	−	0.863907i	$-0.331989\pi$
$479$	10560.0	1.00730	0.503652	+	0.863907i	$0.331989\pi$
$480$	0	0
$481$	−15500.0	−1.46931
$482$	0	0
$483$	− 6144.00i	− 0.578803i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	13264.0i	1.23419i	0.786890	+	0.617094i	$0.211690\pi$
$487$	13264.0i	1.23419i	−0.786890	+	0.617094i	$0.788310\pi$
$488$	0	0
$489$	9664.00	0.893704
$490$	0	0
$491$	−4840.00	−0.444860	−0.222430	−	0.974949i	$-0.571399\pi$
$491$	−4840.00	−0.444860	−0.222430	+	0.974949i	$0.571399\pi$
$492$	0	0
$493$	− 1020.00i	− 0.0931815i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6400.00i	0.577624i
$498$	0	0
$499$	−19560.0	−1.75476	−0.877381	−	0.479795i	$-0.840711\pi$
$499$	−19560.0	−1.75476	−0.877381	+	0.479795i	$0.840711\pi$
$500$	0	0
$501$	23168.0	2.06601
$502$	0	0
$503$	528.000i	0.0468039i	0.999726	+	0.0234019i	$0.00744975\pi$
$503$	528.000i	0.0468039i	−0.999726	+	0.0234019i	$0.992550\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	2424.00i	0.212335i
$508$	0	0
$509$	19554.0	1.70278	0.851391	−	0.524532i	$-0.175760\pi$
$509$	19554.0	1.70278	0.851391	+	0.524532i	$0.175760\pi$
$510$	0	0
$511$	−10080.0	−0.872628
$512$	0	0
$513$	− 3200.00i	− 0.275406i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	16640.0i	1.41552i
$518$	0	0
$519$	−6000.00	−0.507458
$520$	0	0
$521$	15162.0	1.27497	0.637485	−	0.770463i	$-0.279975\pi$
$521$	15162.0	1.27497	0.637485	+	0.770463i	$0.279975\pi$
$522$	0	0
$523$	− 10968.0i	− 0.917012i	−0.888691	−	0.458506i	$-0.848385\pi$
$523$	− 10968.0i	− 0.917012i	0.888691	−	0.458506i	$-0.151615\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 9600.00i	− 0.793515i
$528$	0	0
$529$	9863.00	0.810635
$530$	0	0
$531$	−7400.00	−0.604770
$532$	0	0
$533$	20500.0i	1.66595i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	18240.0i	1.46576i
$538$	0	0
$539$	−3480.00	−0.278097
$540$	0	0
$541$	−6722.00	−0.534198	−0.267099	−	0.963669i	$-0.586065\pi$
$541$	−6722.00	−0.534198	−0.267099	+	0.963669i	$0.586065\pi$
$542$	0	0
$543$	3536.00i	0.279455i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	20424.0i	1.59647i	0.602348	+	0.798233i	$0.294232\pi$
$547$	20424.0i	1.59647i	−0.602348	+	0.798233i	$0.705768\pi$
$548$	0	0
$549$	−1110.00	−0.0862908
$550$	0	0
$551$	−1360.00	−0.105151
$552$	0	0
$553$	17920.0i	1.37800i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 6610.00i	− 0.502827i	−0.967880	−	0.251414i	$-0.919105\pi$
$557$	− 6610.00i	− 0.502827i	0.967880	−	0.251414i	$-0.0808954\pi$
$558$	0	0
$559$	7600.00	0.575037
$560$	0	0
$561$	9600.00	0.722482
$562$	0	0
$563$	2712.00i	0.203015i	0.994835	+	0.101507i	$0.0323665\pi$
$563$	2712.00i	0.203015i	−0.994835	+	0.101507i	$0.967633\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 5744.00i	− 0.425441i
$568$	0	0
$569$	−3530.00	−0.260080	−0.130040	−	0.991509i	$-0.541511\pi$
$569$	−3530.00	−0.260080	−0.130040	+	0.991509i	$0.541511\pi$
$570$	0	0
$571$	−13640.0	−0.999678	−0.499839	−	0.866118i	$-0.666608\pi$
$571$	−13640.0	−0.999678	−0.499839	+	0.866118i	$0.666608\pi$
$572$	0	0
$573$	15360.0i	1.11985i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 6270.00i	− 0.452380i	−0.974083	−	0.226190i	$-0.927373\pi$
$577$	− 6270.00i	− 0.452380i	0.974083	−	0.226190i	$-0.0726271\pi$
$578$	0	0
$579$	40560.0	2.91125
$580$	0	0
$581$	8832.00	0.630659
$582$	0	0
$583$	− 16400.0i	− 1.16504i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 8616.00i	− 0.605827i	−0.953018	−	0.302913i	$-0.902041\pi$
$587$	− 8616.00i	− 0.605827i	0.953018	−	0.302913i	$-0.0979593\pi$
$588$	0	0
$589$	−12800.0	−0.895441
$590$	0	0
$591$	15280.0	1.06351
$592$	0	0
$593$	− 5490.00i	− 0.380181i	−0.981767	−	0.190090i	$-0.939122\pi$
$593$	− 5490.00i	− 0.380181i	0.981767	−	0.190090i	$-0.0608781\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	23680.0i	1.62338i
$598$	0	0
$599$	15440.0	1.05319	0.526595	−	0.850116i	$-0.323468\pi$
$599$	15440.0	1.05319	0.526595	+	0.850116i	$0.323468\pi$
$600$	0	0
$601$	8890.00	0.603379	0.301689	−	0.953406i	$-0.402449\pi$
$601$	8890.00	0.603379	0.301689	+	0.953406i	$0.402449\pi$
$602$	0	0
$603$	− 28712.0i	− 1.93904i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	23744.0i	1.58771i	0.608108	+	0.793854i	$0.291929\pi$
$607$	23744.0i	1.58771i	−0.608108	+	0.793854i	$0.708071\pi$
$608$	0	0
$609$	4352.00	0.289576
$610$	0	0
$611$	20800.0	1.37721
$612$	0	0
$613$	15210.0i	1.00216i	0.865400	+	0.501082i	$0.167064\pi$
$613$	15210.0i	1.00216i	−0.865400	+	0.501082i	$0.832936\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 12630.0i	− 0.824092i	−0.911163	−	0.412046i	$-0.864814\pi$
$617$	− 12630.0i	− 0.824092i	0.911163	−	0.412046i	$-0.135186\pi$
$618$	0	0
$619$	−11160.0	−0.724650	−0.362325	−	0.932052i	$-0.618017\pi$
$619$	−11160.0	−0.724650	−0.362325	+	0.932052i	$0.618017\pi$
$620$	0	0
$621$	3840.00	0.248138
$622$	0	0
$623$	5216.00i	0.335433i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 12800.0i	− 0.815284i
$628$	0	0
$629$	9300.00	0.589531
$630$	0	0
$631$	13040.0	0.822685	0.411342	−	0.911481i	$-0.365060\pi$
$631$	13040.0	0.822685	0.411342	+	0.911481i	$0.365060\pi$
$632$	0	0
$633$	− 320.000i	− 0.0200930i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	4350.00i	0.270570i
$638$	0	0
$639$	−14800.0	−0.916242
$640$	0	0
$641$	−16910.0	−1.04197	−0.520987	−	0.853565i	$-0.674436\pi$
$641$	−16910.0	−1.04197	−0.520987	+	0.853565i	$0.674436\pi$
$642$	0	0
$643$	− 4488.00i	− 0.275256i	−0.990484	−	0.137628i	$-0.956052\pi$
$643$	− 4488.00i	− 0.275256i	0.990484	−	0.137628i	$-0.0439478\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	2064.00i	0.125416i	0.998032	+	0.0627080i	$0.0199737\pi$
$647$	2064.00i	0.125416i	−0.998032	+	0.0627080i	$0.980026\pi$
$648$	0	0
$649$	−8000.00	−0.483864
$650$	0	0
$651$	40960.0	2.46597
$652$	0	0
$653$	− 4270.00i	− 0.255893i	−0.991781	−	0.127946i	$-0.959161\pi$
$653$	− 4270.00i	− 0.255893i	0.991781	−	0.127946i	$-0.0408386\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 23310.0i	− 1.38419i
$658$	0	0
$659$	19800.0	1.17041	0.585204	−	0.810886i	$-0.301015\pi$
$659$	19800.0	1.17041	0.585204	+	0.810886i	$0.301015\pi$
$660$	0	0
$661$	27110.0	1.59524	0.797622	−	0.603157i	$-0.206091\pi$
$661$	27110.0	1.59524	0.797622	+	0.603157i	$0.206091\pi$
$662$	0	0
$663$	− 12000.0i	− 0.702928i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 1632.00i	− 0.0947396i
$668$	0	0
$669$	−34304.0	−1.98247
$670$	0	0
$671$	−1200.00	−0.0690395
$672$	0	0
$673$	− 32210.0i	− 1.84488i	−0.386140	−	0.922440i	$-0.626192\pi$
$673$	− 32210.0i	− 1.84488i	0.386140	−	0.922440i	$-0.373808\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	27190.0i	1.54357i	0.635884	+	0.771785i	$0.280636\pi$
$677$	27190.0i	1.54357i	−0.635884	+	0.771785i	$0.719364\pi$
$678$	0	0
$679$	1760.00	0.0994736
$680$	0	0
$681$	−51648.0	−2.90625
$682$	0	0
$683$	20328.0i	1.13884i	0.822046	+	0.569421i	$0.192833\pi$
$683$	20328.0i	1.13884i	−0.822046	+	0.569421i	$0.807167\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 8528.00i	− 0.473600i
$688$	0	0
$689$	−20500.0	−1.13351
$690$	0	0
$691$	12520.0	0.689267	0.344633	−	0.938737i	$-0.388003\pi$
$691$	12520.0	0.689267	0.344633	+	0.938737i	$0.388003\pi$
$692$	0	0
$693$	23680.0i	1.29802i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 12300.0i	− 0.668430i
$698$	0	0
$699$	47280.0	2.55836
$700$	0	0
$701$	11550.0	0.622307	0.311154	−	0.950360i	$-0.399285\pi$
$701$	11550.0	0.622307	0.311154	+	0.950360i	$0.399285\pi$
$702$	0	0
$703$	− 12400.0i	− 0.665256i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 17568.0i	− 0.934530i
$708$	0	0
$709$	34154.0	1.80914	0.904570	−	0.426325i	$-0.140192\pi$
$709$	34154.0	1.80914	0.904570	+	0.426325i	$0.140192\pi$
$710$	0	0
$711$	−41440.0	−2.18582
$712$	0	0
$713$	− 15360.0i	− 0.806783i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 26880.0i	− 1.40007i
$718$	0	0
$719$	22880.0	1.18676	0.593380	−	0.804923i	$-0.297793\pi$
$719$	22880.0	1.18676	0.593380	+	0.804923i	$0.297793\pi$
$720$	0	0
$721$	−768.000	−0.0396696
$722$	0	0
$723$	− 31760.0i	− 1.63370i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	10416.0i	0.531373i	0.964060	+	0.265686i	$0.0855986\pi$
$727$	10416.0i	0.531373i	−0.964060	+	0.265686i	$0.914401\pi$
$728$	0	0
$729$	30563.0	1.55276
$730$	0	0
$731$	−4560.00	−0.230722
$732$	0	0
$733$	− 14750.0i	− 0.743252i	−0.928383	−	0.371626i	$-0.878800\pi$
$733$	− 14750.0i	− 0.743252i	0.928383	−	0.371626i	$-0.121200\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 31040.0i	− 1.55139i
$738$	0	0
$739$	2360.00	0.117475	0.0587375	−	0.998273i	$-0.481293\pi$
$739$	2360.00	0.117475	0.0587375	+	0.998273i	$0.481293\pi$
$740$	0	0
$741$	−16000.0	−0.793218
$742$	0	0
$743$	− 32208.0i	− 1.59031i	−0.606409	−	0.795153i	$-0.707391\pi$
$743$	− 32208.0i	− 1.59031i	0.606409	−	0.795153i	$-0.292609\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	20424.0i	1.00037i
$748$	0	0
$749$	−10624.0	−0.518281
$750$	0	0
$751$	−36640.0	−1.78031	−0.890155	−	0.455658i	$-0.849404\pi$
$751$	−36640.0	−1.78031	−0.890155	+	0.455658i	$0.849404\pi$
$752$	0	0
$753$	− 54720.0i	− 2.64822i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 12090.0i	− 0.580474i	−0.956955	−	0.290237i	$-0.906266\pi$
$757$	− 12090.0i	− 0.580474i	0.956955	−	0.290237i	$-0.0937341\pi$
$758$	0	0
$759$	15360.0	0.734562
$760$	0	0
$761$	−3318.00	−0.158052	−0.0790259	−	0.996873i	$-0.525181\pi$
$761$	−3318.00	−0.158052	−0.0790259	+	0.996873i	$0.525181\pi$
$762$	0	0
$763$	5920.00i	0.280889i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	10000.0i	0.470768i
$768$	0	0
$769$	−11506.0	−0.539554	−0.269777	−	0.962923i	$-0.586950\pi$
$769$	−11506.0	−0.539554	−0.269777	+	0.962923i	$0.586950\pi$
$770$	0	0
$771$	36880.0	1.72270
$772$	0	0
$773$	− 22230.0i	− 1.03436i	−0.855878	−	0.517178i	$-0.826982\pi$
$773$	− 22230.0i	− 1.03436i	0.855878	−	0.517178i	$-0.173018\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	39680.0i	1.83206i
$778$	0	0
$779$	−16400.0	−0.754289
$780$	0	0
$781$	−16000.0	−0.733067
$782$	0	0
$783$	2720.00i	0.124144i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 21336.0i	− 0.966387i	−0.875514	−	0.483193i	$-0.839477\pi$
$787$	− 21336.0i	− 0.966387i	0.875514	−	0.483193i	$-0.160523\pi$
$788$	0	0
$789$	38784.0	1.75000
$790$	0	0
$791$	23840.0	1.07162
$792$	0	0
$793$	1500.00i	0.0671709i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 7170.00i	− 0.318663i	−0.987225	−	0.159332i	$-0.949066\pi$
$797$	− 7170.00i	− 0.318663i	0.987225	−	0.159332i	$-0.0509339\pi$
$798$	0	0
$799$	−12480.0	−0.552579
$800$	0	0
$801$	−12062.0	−0.532072
$802$	0	0
$803$	− 25200.0i	− 1.10746i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	44400.0i	1.93675i
$808$	0	0
$809$	23654.0	1.02797	0.513987	−	0.857798i	$-0.328168\pi$
$809$	23654.0	1.02797	0.513987	+	0.857798i	$0.328168\pi$
$810$	0	0
$811$	−30440.0	−1.31799	−0.658997	−	0.752146i	$-0.729019\pi$
$811$	−30440.0	−1.31799	−0.658997	+	0.752146i	$0.729019\pi$
$812$	0	0
$813$	3840.00i	0.165652i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	6080.00i	0.260358i
$818$	0	0
$819$	29600.0	1.26289
$820$	0	0
$821$	−19930.0	−0.847213	−0.423606	−	0.905846i	$-0.639236\pi$
$821$	−19930.0	−0.847213	−0.423606	+	0.905846i	$0.639236\pi$
$822$	0	0
$823$	9872.00i	0.418124i	0.977902	+	0.209062i	$0.0670411\pi$
$823$	9872.00i	0.418124i	−0.977902	+	0.209062i	$0.932959\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 5704.00i	− 0.239840i	−0.992784	−	0.119920i	$-0.961736\pi$
$827$	− 5704.00i	− 0.239840i	0.992784	−	0.119920i	$-0.0382638\pi$
$828$	0	0
$829$	−27230.0	−1.14082	−0.570408	−	0.821361i	$-0.693215\pi$
$829$	−27230.0	−1.14082	−0.570408	+	0.821361i	$0.693215\pi$
$830$	0	0
$831$	8240.00	0.343974
$832$	0	0
$833$	− 2610.00i	− 0.108561i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	25600.0i	1.05719i
$838$	0	0
$839$	18800.0	0.773597	0.386799	−	0.922164i	$-0.373581\pi$
$839$	18800.0	0.773597	0.386799	+	0.922164i	$0.373581\pi$
$840$	0	0
$841$	−23233.0	−0.952602
$842$	0	0
$843$	26160.0i	1.06880i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	4304.00i	0.174601i
$848$	0	0
$849$	−17344.0	−0.701113
$850$	0	0
$851$	14880.0	0.599389
$852$	0	0
$853$	12090.0i	0.485292i	0.970115	+	0.242646i	$0.0780153\pi$
$853$	12090.0i	0.485292i	−0.970115	+	0.242646i	$0.921985\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 470.000i	− 0.0187338i	−0.999956	−	0.00936692i	$-0.997018\pi$
$857$	− 470.000i	− 0.0187338i	0.999956	−	0.00936692i	$-0.00298163\pi$
$858$	0	0
$859$	−24440.0	−0.970759	−0.485380	−	0.874304i	$-0.661319\pi$
$859$	−24440.0	−0.970759	−0.485380	+	0.874304i	$0.661319\pi$
$860$	0	0
$861$	52480.0	2.07725
$862$	0	0
$863$	22592.0i	0.891125i	0.895251	+	0.445562i	$0.146996\pi$
$863$	22592.0i	0.891125i	−0.895251	+	0.445562i	$0.853004\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 32104.0i	− 1.25757i
$868$	0	0
$869$	−44800.0	−1.74883
$870$	0	0
$871$	−38800.0	−1.50940
$872$	0	0
$873$	4070.00i	0.157788i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 17330.0i	− 0.667266i	−0.942703	−	0.333633i	$-0.891725\pi$
$877$	− 17330.0i	− 0.667266i	0.942703	−	0.333633i	$-0.108275\pi$
$878$	0	0
$879$	−16560.0	−0.635444
$880$	0	0
$881$	−31470.0	−1.20346	−0.601732	−	0.798698i	$-0.705522\pi$
$881$	−31470.0	−1.20346	−0.601732	+	0.798698i	$0.705522\pi$
$882$	0	0
$883$	3352.00i	0.127751i	0.997958	+	0.0638753i	$0.0203460\pi$
$883$	3352.00i	0.127751i	−0.997958	+	0.0638753i	$0.979654\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 48144.0i	− 1.82245i	−0.411904	−	0.911227i	$-0.635136\pi$
$887$	− 48144.0i	− 1.82245i	0.411904	−	0.911227i	$-0.364864\pi$
$888$	0	0
$889$	16384.0	0.618112
$890$	0	0
$891$	14360.0	0.539931
$892$	0	0
$893$	16640.0i	0.623557i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 19200.0i	− 0.714682i
$898$	0	0
$899$	10880.0	0.403636
$900$	0	0
$901$	12300.0	0.454797
$902$	0	0
$903$	− 19456.0i	− 0.717005i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	16216.0i	0.593653i	0.954931	+	0.296827i	$0.0959283\pi$
$907$	16216.0i	0.593653i	−0.954931	+	0.296827i	$0.904072\pi$
$908$	0	0
$909$	40626.0	1.48238
$910$	0	0
$911$	49440.0	1.79805	0.899023	−	0.437901i	$-0.144278\pi$
$911$	49440.0	1.79805	0.899023	+	0.437901i	$0.144278\pi$
$912$	0	0
$913$	22080.0i	0.800374i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	18560.0i	0.668381i
$918$	0	0
$919$	16080.0	0.577182	0.288591	−	0.957452i	$-0.406813\pi$
$919$	16080.0	0.577182	0.288591	+	0.957452i	$0.406813\pi$
$920$	0	0
$921$	15168.0	0.542674
$922$	0	0
$923$	20000.0i	0.713226i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 1776.00i	− 0.0629250i
$928$	0	0
$929$	11310.0	0.399428	0.199714	−	0.979854i	$-0.435999\pi$
$929$	11310.0	0.399428	0.199714	+	0.979854i	$0.435999\pi$
$930$	0	0
$931$	−3480.00	−0.122505
$932$	0	0
$933$	13440.0i	0.471603i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	25130.0i	0.876159i	0.898936	+	0.438080i	$0.144341\pi$
$937$	25130.0i	0.876159i	−0.898936	+	0.438080i	$0.855659\pi$
$938$	0	0
$939$	−7760.00	−0.269689
$940$	0	0
$941$	−22322.0	−0.773301	−0.386651	−	0.922226i	$-0.626368\pi$
$941$	−22322.0	−0.773301	−0.386651	+	0.922226i	$0.626368\pi$
$942$	0	0
$943$	− 19680.0i	− 0.679607i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	36456.0i	1.25096i	0.780239	+	0.625481i	$0.215097\pi$
$947$	36456.0i	1.25096i	−0.780239	+	0.625481i	$0.784903\pi$
$948$	0	0
$949$	−31500.0	−1.07749
$950$	0	0
$951$	57840.0	1.97223
$952$	0	0
$953$	− 40650.0i	− 1.38172i	−0.722987	−	0.690862i	$-0.757231\pi$
$953$	− 40650.0i	− 1.38172i	0.722987	−	0.690862i	$-0.242769\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	10880.0i	0.367503i
$958$	0	0
$959$	−9120.00	−0.307091
$960$	0	0
$961$	72609.0	2.43728
$962$	0	0
$963$	− 24568.0i	− 0.822111i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	34704.0i	1.15409i	0.816712	+	0.577045i	$0.195794\pi$
$967$	34704.0i	1.15409i	−0.816712	+	0.577045i	$0.804206\pi$
$968$	0	0
$969$	9600.00	0.318263
$970$	0	0
$971$	−30760.0	−1.01662	−0.508309	−	0.861175i	$-0.669729\pi$
$971$	−30760.0	−1.01662	−0.508309	+	0.861175i	$0.669729\pi$
$972$	0	0
$973$	31360.0i	1.03325i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 38110.0i	− 1.24795i	−0.781444	−	0.623975i	$-0.785517\pi$
$977$	− 38110.0i	− 1.24795i	0.781444	−	0.623975i	$-0.214483\pi$
$978$	0	0
$979$	−13040.0	−0.425700
$980$	0	0
$981$	−13690.0	−0.445554
$982$	0	0
$983$	− 19632.0i	− 0.636992i	−0.947924	−	0.318496i	$-0.896822\pi$
$983$	− 19632.0i	− 0.636992i	0.947924	−	0.318496i	$-0.103178\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 53248.0i	− 1.71723i
$988$	0	0
$989$	−7296.00	−0.234580
$990$	0	0
$991$	−47680.0	−1.52836	−0.764180	−	0.645003i	$-0.776856\pi$
$991$	−47680.0	−1.52836	−0.764180	+	0.645003i	$0.776856\pi$
$992$	0	0
$993$	46400.0i	1.48284i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 39690.0i	− 1.26078i	−0.776280	−	0.630389i	$-0.782896\pi$
$997$	− 39690.0i	− 1.26078i	0.776280	−	0.630389i	$-0.217104\pi$
$998$	0	0
$999$	−24800.0	−0.785423

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.4.c.a.449.1		2
4.3	odd	2		800.4.c.b.449.2		2
5.2	odd	4		800.4.a.a.1.1		1
5.3	odd	4		32.4.a.c.1.1	yes	1
5.4	even	2	inner	800.4.c.a.449.2		2
15.8	even	4		288.4.a.i.1.1		1
20.3	even	4		32.4.a.a.1.1	✓	1
20.7	even	4		800.4.a.k.1.1		1
20.19	odd	2		800.4.c.b.449.1		2
35.13	even	4		1568.4.a.c.1.1		1
40.3	even	4		64.4.a.e.1.1		1
40.13	odd	4		64.4.a.a.1.1		1
40.27	even	4		1600.4.a.e.1.1		1
40.37	odd	4		1600.4.a.bw.1.1		1
60.23	odd	4		288.4.a.h.1.1		1
80.3	even	4		256.4.b.e.129.1		2
80.13	odd	4		256.4.b.c.129.2		2
80.43	even	4		256.4.b.e.129.2		2
80.53	odd	4		256.4.b.c.129.1		2
120.53	even	4		576.4.a.h.1.1		1
120.83	odd	4		576.4.a.g.1.1		1
140.83	odd	4		1568.4.a.o.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.4.a.a.1.1	✓	1	20.3	even	4
32.4.a.c.1.1	yes	1	5.3	odd	4
64.4.a.a.1.1		1	40.13	odd	4
64.4.a.e.1.1		1	40.3	even	4
256.4.b.c.129.1		2	80.53	odd	4
256.4.b.c.129.2		2	80.13	odd	4
256.4.b.e.129.1		2	80.3	even	4
256.4.b.e.129.2		2	80.43	even	4
288.4.a.h.1.1		1	60.23	odd	4
288.4.a.i.1.1		1	15.8	even	4
576.4.a.g.1.1		1	120.83	odd	4
576.4.a.h.1.1		1	120.53	even	4
800.4.a.a.1.1		1	5.2	odd	4
800.4.a.k.1.1		1	20.7	even	4
800.4.c.a.449.1		2	1.1	even	1	trivial
800.4.c.a.449.2		2	5.4	even	2	inner
800.4.c.b.449.1		2	20.19	odd	2
800.4.c.b.449.2		2	4.3	odd	2
1568.4.a.c.1.1		1	35.13	even	4
1568.4.a.o.1.1		1	140.83	odd	4
1600.4.a.e.1.1		1	40.27	even	4
1600.4.a.bw.1.1		1	40.37	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-8.00000i q^{3} +16.0000i q^{7} -37.0000 q^{9} +O(q^{10})\)	\(q-8.00000i q^{3} +16.0000i q^{7} -37.0000 q^{9} -40.0000 q^{11} +50.0000i q^{13} -30.0000i q^{17} -40.0000 q^{19} +128.000 q^{21} -48.0000i q^{23} +80.0000i q^{27} +34.0000 q^{29} +320.000 q^{31} +320.000i q^{33} +310.000i q^{37} +400.000 q^{39} +410.000 q^{41} -152.000i q^{43} -416.000i q^{47} +87.0000 q^{49} -240.000 q^{51} +410.000i q^{53} +320.000i q^{57} +200.000 q^{59} +30.0000 q^{61} -592.000i q^{63} +776.000i q^{67} -384.000 q^{69} +400.000 q^{71} +630.000i q^{73} -640.000i q^{77} +1120.00 q^{79} -359.000 q^{81} -552.000i q^{83} -272.000i q^{87} +326.000 q^{89} -800.000 q^{91} -2560.00i q^{93} -110.000i q^{97} +1480.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 74 q^{9}+O(q^{10}) \) 2 * q - 74 * q^9	\( 2 q - 74 q^{9} - 80 q^{11} - 80 q^{19} + 256 q^{21} + 68 q^{29} + 640 q^{31} + 800 q^{39} + 820 q^{41} + 174 q^{49} - 480 q^{51} + 400 q^{59} + 60 q^{61} - 768 q^{69} + 800 q^{71} + 2240 q^{79} - 718 q^{81} + 652 q^{89} - 1600 q^{91} + 2960 q^{99}+O(q^{100}) \) 2 * q - 74 * q^9 - 80 * q^11 - 80 * q^19 + 256 * q^21 + 68 * q^29 + 640 * q^31 + 800 * q^39 + 820 * q^41 + 174 * q^49 - 480 * q^51 + 400 * q^59 + 60 * q^61 - 768 * q^69 + 800 * q^71 + 2240 * q^79 - 718 * q^81 + 652 * q^89 - 1600 * q^91 + 2960 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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