LMFDB - Embedded newform 400.3.h.d.399.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,3,Mod(399,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("400.399");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	400.h (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:	$10.8992105744$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.12960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 $ x^8 - 3x^6 + 8x^4 - 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{17}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 80)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			399.8
Root			$-1.40126 - 0.809017i$ of defining polynomial
Character	$\chi$	$=$	400.399
Dual form			400.3.h.d.399.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.60503 q^{3} +1.32317 q^{7} +22.4164 q^{9} +11.2101i q^{11} -17.4164i q^{13} +18.0000i q^{17} -5.29268i q^{19} +7.41641 q^{21} +15.1796 q^{23} +75.1994 q^{27} -8.83282 q^{29} -42.1939i q^{31} +62.8328i q^{33} +33.4164i q^{37} -97.6196i q^{39} -28.2492 q^{41} -25.3788 q^{43} -10.5116 q^{47} -47.2492 q^{49} +100.891i q^{51} +28.2492i q^{53} -29.6656i q^{57} -44.8403i q^{59} -77.4164 q^{61} +29.6607 q^{63} +36.5889 q^{67} +85.0820 q^{69} -97.6196i q^{71} -15.6656i q^{73} +14.8328i q^{77} +112.101i q^{79} +219.748 q^{81} +92.0145 q^{83} -49.5082 q^{87} +59.6656 q^{89} -23.0449i q^{91} -236.498i q^{93} +108.164i q^{97} +251.289i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 72 q^{9} - 48 q^{21} + 144 q^{29} + 96 q^{41} - 56 q^{49} - 512 q^{61} + 144 q^{69} + 792 q^{81} + 48 q^{89}+O(q^{100}) $ 8 * q + 72 * q^9 - 48 * q^21 + 144 * q^29 + 96 * q^41 - 56 * q^49 - 512 * q^61 + 144 * q^69 + 792 * q^81 + 48 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$351$
$\chi(n)$	$1$	$-1$	$-1$

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
$3$	5.60503	1.86834	0.934172	−	0.356822i	$-0.116140\pi$
$3$	5.60503	1.86834	0.934172	+	0.356822i	$0.116140\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.32317	0.189024	0.0945121	−	0.995524i	$-0.469871\pi$
$7$	1.32317	0.189024	0.0945121	+	0.995524i	$0.469871\pi$
$8$	0	0
$9$	22.4164	2.49071
$10$	0	0
$11$	11.2101i	1.01910i	0.860442	+	0.509549i	$0.170188\pi$
$11$	11.2101i	1.01910i	−0.860442	+	0.509549i	$0.829812\pi$
$12$	0	0
$13$	− 17.4164i	− 1.33972i	−0.742486	−	0.669862i	$-0.766353\pi$
$13$	− 17.4164i	− 1.33972i	0.742486	−	0.669862i	$-0.233647\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	18.0000i	1.05882i	0.848365	+	0.529412i	$0.177587\pi$
$17$	18.0000i	1.05882i	−0.848365	+	0.529412i	$0.822413\pi$
$18$	0	0
$19$	− 5.29268i	− 0.278562i	−0.990253	−	0.139281i	$-0.955521\pi$
$19$	− 5.29268i	− 0.278562i	0.990253	−	0.139281i	$-0.0444791\pi$
$20$	0	0
$21$	7.41641	0.353162
$22$	0	0
$23$	15.1796	0.659982	0.329991	−	0.943984i	$-0.392954\pi$
$23$	15.1796	0.659982	0.329991	+	0.943984i	$0.392954\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	75.1994	2.78516
$28$	0	0
$29$	−8.83282	−0.304580	−0.152290	−	0.988336i	$-0.548665\pi$
$29$	−8.83282	−0.304580	−0.152290	+	0.988336i	$0.548665\pi$
$30$	0	0
$31$	− 42.1939i	− 1.36109i	−0.732704	−	0.680547i	$-0.761742\pi$
$31$	− 42.1939i	− 1.36109i	0.732704	−	0.680547i	$-0.238258\pi$
$32$	0	0
$33$	62.8328i	1.90402i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	33.4164i	0.903146i	0.892234	+	0.451573i	$0.149137\pi$
$37$	33.4164i	0.903146i	−0.892234	+	0.451573i	$0.850863\pi$
$38$	0	0
$39$	− 97.6196i	− 2.50307i
$40$	0	0
$41$	−28.2492	−0.689005	−0.344503	−	0.938785i	$-0.611952\pi$
$41$	−28.2492	−0.689005	−0.344503	+	0.938785i	$0.611952\pi$
$42$	0	0
$43$	−25.3788	−0.590205	−0.295103	−	0.955466i	$-0.595354\pi$
$43$	−25.3788	−0.590205	−0.295103	+	0.955466i	$0.595354\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−10.5116	−0.223651	−0.111826	−	0.993728i	$-0.535670\pi$
$47$	−10.5116	−0.223651	−0.111826	+	0.993728i	$0.535670\pi$
$48$	0	0
$49$	−47.2492	−0.964270
$50$	0	0
$51$	100.891i	1.97825i
$52$	0	0
$53$	28.2492i	0.533004i	0.963834	+	0.266502i	$0.0858679\pi$
$53$	28.2492i	0.533004i	−0.963834	+	0.266502i	$0.914132\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 29.6656i	− 0.520450i
$58$	0	0
$59$	− 44.8403i	− 0.760005i	−0.924986	−	0.380002i	$-0.875923\pi$
$59$	− 44.8403i	− 0.760005i	0.924986	−	0.380002i	$-0.124077\pi$
$60$	0	0
$61$	−77.4164	−1.26912	−0.634561	−	0.772873i	$-0.718819\pi$
$61$	−77.4164	−1.26912	−0.634561	+	0.772873i	$0.718819\pi$
$62$	0	0
$63$	29.6607	0.470805
$64$	0	0
$65$	0	0
$66$	0	0
$67$	36.5889	0.546103	0.273051	−	0.961999i	$-0.411967\pi$
$67$	36.5889	0.546103	0.273051	+	0.961999i	$0.411967\pi$
$68$	0	0
$69$	85.0820	1.23307
$70$	0	0
$71$	− 97.6196i	− 1.37492i	−0.726221	−	0.687462i	$-0.758725\pi$
$71$	− 97.6196i	− 1.37492i	0.726221	−	0.687462i	$-0.241275\pi$
$72$	0	0
$73$	− 15.6656i	− 0.214598i	−0.994227	−	0.107299i	$-0.965780\pi$
$73$	− 15.6656i	− 0.214598i	0.994227	−	0.107299i	$-0.0342202\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	14.8328i	0.192634i
$78$	0	0
$79$	112.101i	1.41900i	0.704707	+	0.709498i	$0.251078\pi$
$79$	112.101i	1.41900i	−0.704707	+	0.709498i	$0.748922\pi$
$80$	0	0
$81$	219.748	2.71293
$82$	0	0
$83$	92.0145	1.10861	0.554304	−	0.832314i	$-0.312984\pi$
$83$	92.0145	1.10861	0.554304	+	0.832314i	$0.312984\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−49.5082	−0.569060
$88$	0	0
$89$	59.6656	0.670400	0.335200	−	0.942147i	$-0.391196\pi$
$89$	59.6656	0.670400	0.335200	+	0.942147i	$0.391196\pi$
$90$	0	0
$91$	− 23.0449i	− 0.253240i
$92$	0	0
$93$	− 236.498i	− 2.54299i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	108.164i	1.11509i	0.830146	+	0.557547i	$0.188257\pi$
$97$	108.164i	1.11509i	−0.830146	+	0.557547i	$0.811743\pi$
$98$	0	0
$99$	251.289i	2.53828i
$100$	0	0
$101$	−83.6656	−0.828373	−0.414186	−	0.910192i	$-0.635934\pi$
$101$	−83.6656	−0.828373	−0.414186	+	0.910192i	$0.635934\pi$
$102$	0	0
$103$	−135.219	−1.31281	−0.656404	−	0.754409i	$-0.727923\pi$
$103$	−135.219	−1.31281	−0.656404	+	0.754409i	$0.727923\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−122.374	−1.14368	−0.571840	−	0.820365i	$-0.693770\pi$
$107$	−122.374	−1.14368	−0.571840	+	0.820365i	$0.693770\pi$
$108$	0	0
$109$	−2.58359	−0.0237027	−0.0118513	−	0.999930i	$-0.503772\pi$
$109$	−2.58359	−0.0237027	−0.0118513	+	0.999930i	$0.503772\pi$
$110$	0	0
$111$	187.300i	1.68739i
$112$	0	0
$113$	33.5016i	0.296474i	0.988952	+	0.148237i	$0.0473598\pi$
$113$	33.5016i	0.296474i	−0.988952	+	0.148237i	$0.952640\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 390.413i	− 3.33687i
$118$	0	0
$119$	23.8170i	0.200143i
$120$	0	0
$121$	−4.66563	−0.0385589
$122$	0	0
$123$	−158.338	−1.28730
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−128.677	−1.01321	−0.506603	−	0.862179i	$-0.669099\pi$
$127$	−128.677	−1.01321	−0.506603	+	0.862179i	$0.669099\pi$
$128$	0	0
$129$	−142.249	−1.10271
$130$	0	0
$131$	− 222.327i	− 1.69715i	−0.529072	−	0.848577i	$-0.677460\pi$
$131$	− 222.327i	− 1.69715i	0.529072	−	0.848577i	$-0.322540\pi$
$132$	0	0
$133$	− 7.00311i	− 0.0526549i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	128.833i	0.940386i	0.882564	+	0.470193i	$0.155816\pi$
$137$	128.833i	0.940386i	−0.882564	+	0.470193i	$0.844184\pi$
$138$	0	0
$139$	− 230.743i	− 1.66002i	−0.557745	−	0.830012i	$-0.688333\pi$
$139$	− 230.743i	− 1.66002i	0.557745	−	0.830012i	$-0.311667\pi$
$140$	0	0
$141$	−58.9180	−0.417858
$142$	0	0
$143$	195.239	1.36531
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−264.834	−1.80159
$148$	0	0
$149$	−285.915	−1.91889	−0.959446	−	0.281893i	$-0.909038\pi$
$149$	−285.915	−1.91889	−0.959446	+	0.281893i	$0.909038\pi$
$150$	0	0
$151$	64.6141i	0.427908i	0.976844	+	0.213954i	$0.0686342\pi$
$151$	64.6141i	0.427908i	−0.976844	+	0.213954i	$0.931366\pi$
$152$	0	0
$153$	403.495i	2.63722i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 14.5836i	− 0.0928891i	−0.998921	−	0.0464446i	$-0.985211\pi$
$157$	− 14.5836i	− 0.0928891i	0.998921	−	0.0464446i	$-0.0147891\pi$
$158$	0	0
$159$	158.338i	0.995836i
$160$	0	0
$161$	20.0851	0.124752
$162$	0	0
$163$	−20.8583	−0.127965	−0.0639827	−	0.997951i	$-0.520380\pi$
$163$	−20.8583	−0.127965	−0.0639827	+	0.997951i	$0.520380\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	216.961	1.29917	0.649583	−	0.760290i	$-0.274943\pi$
$167$	216.961	1.29917	0.649583	+	0.760290i	$0.274943\pi$
$168$	0	0
$169$	−134.331	−0.794860
$170$	0	0
$171$	− 118.643i	− 0.693817i
$172$	0	0
$173$	46.5836i	0.269269i	0.990895	+	0.134635i	$0.0429861\pi$
$173$	46.5836i	0.269269i	−0.990895	+	0.134635i	$0.957014\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 251.331i	− 1.41995i
$178$	0	0
$179$	6.54211i	0.0365481i	0.999833	+	0.0182740i	$0.00581713\pi$
$179$	6.54211i	0.0365481i	−0.999833	+	0.0182740i	$0.994183\pi$
$180$	0	0
$181$	−241.331	−1.33332	−0.666661	−	0.745361i	$-0.732277\pi$
$181$	−241.331	−1.33332	−0.666661	+	0.745361i	$0.732277\pi$
$182$	0	0
$183$	−433.922	−2.37116
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−201.781	−1.07904
$188$	0	0
$189$	99.5016	0.526463
$190$	0	0
$191$	59.3214i	0.310583i	0.987869	+	0.155292i	$0.0496317\pi$
$191$	59.3214i	0.310583i	−0.987869	+	0.155292i	$0.950368\pi$
$192$	0	0
$193$	− 234.997i	− 1.21760i	−0.793324	−	0.608800i	$-0.791651\pi$
$193$	− 234.997i	− 1.21760i	0.793324	−	0.608800i	$-0.208349\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	230.912i	1.17214i	0.810260	+	0.586070i	$0.199326\pi$
$197$	230.912i	1.17214i	−0.810260	+	0.586070i	$0.800674\pi$
$198$	0	0
$199$	− 27.7128i	− 0.139260i	−0.997573	−	0.0696302i	$-0.977818\pi$
$199$	− 27.7128i	− 0.139260i	0.997573	−	0.0696302i	$-0.0221819\pi$
$200$	0	0
$201$	205.082	1.02031
$202$	0	0
$203$	−11.6873	−0.0575729
$204$	0	0
$205$	0	0
$206$	0	0
$207$	340.272	1.64382
$208$	0	0
$209$	59.3313	0.283882
$210$	0	0
$211$	93.0991i	0.441228i	0.975361	+	0.220614i	$0.0708061\pi$
$211$	93.0991i	0.441228i	−0.975361	+	0.220614i	$0.929194\pi$
$212$	0	0
$213$	− 547.161i	− 2.56883i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 55.8297i	− 0.257280i
$218$	0	0
$219$	− 87.8064i	− 0.400942i
$220$	0	0
$221$	313.495	1.41853
$222$	0	0
$223$	307.891	1.38068	0.690338	−	0.723487i	$-0.257462\pi$
$223$	307.891	1.38068	0.690338	+	0.723487i	$0.257462\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	255.020	1.12344	0.561719	−	0.827328i	$-0.310140\pi$
$227$	255.020	1.12344	0.561719	+	0.827328i	$0.310140\pi$
$228$	0	0
$229$	411.495	1.79692	0.898461	−	0.439052i	$-0.144686\pi$
$229$	411.495	1.79692	0.898461	+	0.439052i	$0.144686\pi$
$230$	0	0
$231$	83.1384i	0.359907i
$232$	0	0
$233$	332.833i	1.42847i	0.699907	+	0.714234i	$0.253225\pi$
$233$	332.833i	1.42847i	−0.699907	+	0.714234i	$0.746775\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	628.328i	2.65117i
$238$	0	0
$239$	266.248i	1.11401i	0.830510	+	0.557004i	$0.188049\pi$
$239$	266.248i	1.11401i	−0.830510	+	0.557004i	$0.811951\pi$
$240$	0	0
$241$	−78.5836	−0.326073	−0.163036	−	0.986620i	$-0.552129\pi$
$241$	−78.5836	−0.326073	−0.163036	+	0.986620i	$0.552129\pi$
$242$	0	0
$243$	554.898	2.28353
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−92.1794	−0.373196
$248$	0	0
$249$	515.745	2.07126
$250$	0	0
$251$	− 404.482i	− 1.61148i	−0.592268	−	0.805741i	$-0.701767\pi$
$251$	− 404.482i	− 1.61148i	0.592268	−	0.805741i	$-0.298233\pi$
$252$	0	0
$253$	170.164i	0.672585i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 85.8297i	− 0.333968i	−0.985960	−	0.166984i	$-0.946597\pi$
$257$	− 85.8297i	− 0.333968i	0.985960	−	0.166984i	$-0.0534028\pi$
$258$	0	0
$259$	44.2156i	0.170716i
$260$	0	0
$261$	−198.000	−0.758621
$262$	0	0
$263$	440.685	1.67561	0.837804	−	0.545971i	$-0.183839\pi$
$263$	440.685	1.67561	0.837804	+	0.545971i	$0.183839\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	334.428	1.25254
$268$	0	0
$269$	205.416	0.763630	0.381815	−	0.924239i	$-0.375299\pi$
$269$	205.416	0.763630	0.381815	+	0.924239i	$0.375299\pi$
$270$	0	0
$271$	418.044i	1.54260i	0.636474	+	0.771298i	$0.280392\pi$
$271$	418.044i	1.54260i	−0.636474	+	0.771298i	$0.719608\pi$
$272$	0	0
$273$	− 129.167i	− 0.473140i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 511.410i	− 1.84625i	−0.384505	−	0.923123i	$-0.625628\pi$
$277$	− 511.410i	− 1.84625i	0.384505	−	0.923123i	$-0.374372\pi$
$278$	0	0
$279$	− 945.836i	− 3.39009i
$280$	0	0
$281$	115.751	0.411924	0.205962	−	0.978560i	$-0.433968\pi$
$281$	115.751	0.411924	0.205962	+	0.978560i	$0.433968\pi$
$282$	0	0
$283$	−402.001	−1.42050	−0.710249	−	0.703951i	$-0.751417\pi$
$283$	−402.001	−1.42050	−0.710249	+	0.703951i	$0.751417\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−37.3785	−0.130239
$288$	0	0
$289$	−35.0000	−0.121107
$290$	0	0
$291$	606.263i	2.08338i
$292$	0	0
$293$	− 90.4133i	− 0.308578i	−0.988026	−	0.154289i	$-0.950691\pi$
$293$	− 90.4133i	− 0.308578i	0.988026	−	0.154289i	$-0.0493087\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	842.991i	2.83835i
$298$	0	0
$299$	− 264.374i	− 0.884193i
$300$	0	0
$301$	−33.5805	−0.111563
$302$	0	0
$303$	−468.949	−1.54769
$304$	0	0
$305$	0	0
$306$	0	0
$307$	275.419	0.897130	0.448565	−	0.893750i	$-0.351935\pi$
$307$	275.419	0.897130	0.448565	+	0.893750i	$0.351935\pi$
$308$	0	0
$309$	−757.909	−2.45278
$310$	0	0
$311$	− 485.304i	− 1.56046i	−0.625491	−	0.780231i	$-0.715101\pi$
$311$	− 485.304i	− 1.56046i	0.625491	−	0.780231i	$-0.284899\pi$
$312$	0	0
$313$	− 2.00000i	− 0.00638978i	−0.999995	−	0.00319489i	$-0.998983\pi$
$313$	− 2.00000i	− 0.00638978i	0.999995	−	0.00319489i	$-0.00101697\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 394.741i	− 1.24524i	−0.782524	−	0.622621i	$-0.786068\pi$
$317$	− 394.741i	− 1.24524i	0.782524	−	0.622621i	$-0.213932\pi$
$318$	0	0
$319$	− 99.0165i	− 0.310396i
$320$	0	0
$321$	−685.909	−2.13679
$322$	0	0
$323$	95.2682	0.294948
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−14.4811	−0.0442848
$328$	0	0
$329$	−13.9086	−0.0422755
$330$	0	0
$331$	296.130i	0.894652i	0.894371	+	0.447326i	$0.147624\pi$
$331$	296.130i	0.894652i	−0.894371	+	0.447326i	$0.852376\pi$
$332$	0	0
$333$	749.076i	2.24948i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	19.9938i	0.0593288i	0.999560	+	0.0296644i	$0.00944385\pi$
$337$	19.9938i	0.0593288i	−0.999560	+	0.0296644i	$0.990556\pi$
$338$	0	0
$339$	187.777i	0.553915i
$340$	0	0
$341$	472.997	1.38709
$342$	0	0
$343$	−127.354	−0.371294
$344$	0	0
$345$	0	0
$346$	0	0
$347$	546.959	1.57625	0.788126	−	0.615514i	$-0.211052\pi$
$347$	546.959	1.57625	0.788126	+	0.615514i	$0.211052\pi$
$348$	0	0
$349$	−40.8328	−0.116999	−0.0584997	−	0.998287i	$-0.518632\pi$
$349$	−40.8328	−0.116999	−0.0584997	+	0.998287i	$0.518632\pi$
$350$	0	0
$351$	− 1309.70i	− 3.73135i
$352$	0	0
$353$	555.325i	1.57316i	0.617489	+	0.786579i	$0.288150\pi$
$353$	555.325i	1.57316i	−0.617489	+	0.786579i	$0.711850\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	133.495i	0.373937i
$358$	0	0
$359$	326.012i	0.908110i	0.890974	+	0.454055i	$0.150023\pi$
$359$	326.012i	0.908110i	−0.890974	+	0.454055i	$0.849977\pi$
$360$	0	0
$361$	332.988	0.922403
$362$	0	0
$363$	−26.1510	−0.0720414
$364$	0	0
$365$	0	0
$366$	0	0
$367$	13.6352	0.0371531	0.0185766	−	0.999827i	$-0.494087\pi$
$367$	13.6352	0.0371531	0.0185766	+	0.999827i	$0.494087\pi$
$368$	0	0
$369$	−633.246	−1.71611
$370$	0	0
$371$	37.3785i	0.100751i
$372$	0	0
$373$	136.748i	0.366616i	0.983056	+	0.183308i	$0.0586805\pi$
$373$	136.748i	0.366616i	−0.983056	+	0.183308i	$0.941319\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	153.836i	0.408053i
$378$	0	0
$379$	− 186.198i	− 0.491288i	−0.969360	−	0.245644i	$-0.921001\pi$
$379$	− 186.198i	− 0.491288i	0.969360	−	0.245644i	$-0.0789994\pi$
$380$	0	0
$381$	−721.240	−1.89302
$382$	0	0
$383$	475.712	1.24207	0.621034	−	0.783784i	$-0.286713\pi$
$383$	475.712	1.24207	0.621034	+	0.783784i	$0.286713\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−568.902	−1.47003
$388$	0	0
$389$	−11.9211	−0.0306454	−0.0153227	−	0.999883i	$-0.504878\pi$
$389$	−11.9211	−0.0306454	−0.0153227	+	0.999883i	$0.504878\pi$
$390$	0	0
$391$	273.232i	0.698804i
$392$	0	0
$393$	− 1246.15i	− 3.17087i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	250.754i	0.631622i	0.948822	+	0.315811i	$0.102277\pi$
$397$	250.754i	0.631622i	−0.948822	+	0.315811i	$0.897723\pi$
$398$	0	0
$399$	− 39.2526i	− 0.0983776i
$400$	0	0
$401$	550.328	1.37239	0.686195	−	0.727418i	$-0.259280\pi$
$401$	550.328	1.37239	0.686195	+	0.727418i	$0.259280\pi$
$402$	0	0
$403$	−734.867	−1.82349
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−374.600	−0.920394
$408$	0	0
$409$	27.0820	0.0662153	0.0331076	−	0.999452i	$-0.489460\pi$
$409$	27.0820	0.0662153	0.0331076	+	0.999452i	$0.489460\pi$
$410$	0	0
$411$	722.112i	1.75696i
$412$	0	0
$413$	− 59.3313i	− 0.143659i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 1293.33i	− 3.10150i
$418$	0	0
$419$	198.033i	0.472632i	0.971676	+	0.236316i	$0.0759401\pi$
$419$	198.033i	0.472632i	−0.971676	+	0.236316i	$0.924060\pi$
$420$	0	0
$421$	475.580	1.12964	0.564822	−	0.825212i	$-0.308945\pi$
$421$	475.580	1.12964	0.564822	+	0.825212i	$0.308945\pi$
$422$	0	0
$423$	−235.633	−0.557051
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−102.435	−0.239895
$428$	0	0
$429$	1094.32	2.55087
$430$	0	0
$431$	489.052i	1.13469i	0.823479	+	0.567346i	$0.192030\pi$
$431$	489.052i	1.13469i	−0.823479	+	0.567346i	$0.807970\pi$
$432$	0	0
$433$	− 622.498i	− 1.43764i	−0.695196	−	0.718820i	$-0.744682\pi$
$433$	− 622.498i	− 1.43764i	0.695196	−	0.718820i	$-0.255318\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 80.3406i	− 0.183846i
$438$	0	0
$439$	552.712i	1.25902i	0.776990	+	0.629512i	$0.216745\pi$
$439$	552.712i	1.25902i	−0.776990	+	0.629512i	$0.783255\pi$
$440$	0	0
$441$	−1059.16	−2.40172
$442$	0	0
$443$	668.873	1.50987	0.754936	−	0.655799i	$-0.227668\pi$
$443$	668.873	1.50987	0.754936	+	0.655799i	$0.227668\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−1602.56	−3.58515
$448$	0	0
$449$	280.407	0.624515	0.312257	−	0.949998i	$-0.398915\pi$
$449$	280.407	0.624515	0.312257	+	0.949998i	$0.398915\pi$
$450$	0	0
$451$	− 316.676i	− 0.702163i
$452$	0	0
$453$	362.164i	0.799479i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 320.334i	− 0.700950i	−0.936572	−	0.350475i	$-0.886020\pi$
$457$	− 320.334i	− 0.700950i	0.936572	−	0.350475i	$-0.113980\pi$
$458$	0	0
$459$	1353.59i	2.94900i
$460$	0	0
$461$	163.337	0.354311	0.177156	−	0.984183i	$-0.443310\pi$
$461$	163.337	0.354311	0.177156	+	0.984183i	$0.443310\pi$
$462$	0	0
$463$	108.131	0.233545	0.116772	−	0.993159i	$-0.462745\pi$
$463$	108.131	0.233545	0.116772	+	0.993159i	$0.462745\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	207.386	0.444082	0.222041	−	0.975037i	$-0.428728\pi$
$467$	207.386	0.444082	0.222041	+	0.975037i	$0.428728\pi$
$468$	0	0
$469$	48.4133	0.103227
$470$	0	0
$471$	− 81.7415i	− 0.173549i
$472$	0	0
$473$	− 284.498i	− 0.601477i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	633.246i	1.32756i
$478$	0	0
$479$	− 233.537i	− 0.487552i	−0.969832	−	0.243776i	$-0.921614\pi$
$479$	− 233.537i	− 0.487552i	0.969832	−	0.243776i	$-0.0783861\pi$
$480$	0	0
$481$	581.994	1.20997
$482$	0	0
$483$	112.578	0.233081
$484$	0	0
$485$	0	0
$486$	0	0
$487$	308.368	0.633199	0.316600	−	0.948559i	$-0.397459\pi$
$487$	308.368	0.633199	0.316600	+	0.948559i	$0.397459\pi$
$488$	0	0
$489$	−116.912	−0.239083
$490$	0	0
$491$	− 187.777i	− 0.382439i	−0.981547	−	0.191219i	$-0.938756\pi$
$491$	− 187.777i	− 0.382439i	0.981547	−	0.191219i	$-0.0612442\pi$
$492$	0	0
$493$	− 158.991i	− 0.322496i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 129.167i	− 0.259894i
$498$	0	0
$499$	231.038i	0.463003i	0.972835	+	0.231501i	$0.0743638\pi$
$499$	231.038i	0.463003i	−0.972835	+	0.231501i	$0.925636\pi$
$500$	0	0
$501$	1216.07	2.42729
$502$	0	0
$503$	389.780	0.774910	0.387455	−	0.921889i	$-0.373354\pi$
$503$	389.780	0.774910	0.387455	+	0.921889i	$0.373354\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−752.931	−1.48507
$508$	0	0
$509$	−363.167	−0.713492	−0.356746	−	0.934202i	$-0.616114\pi$
$509$	−363.167	−0.713492	−0.356746	+	0.934202i	$0.616114\pi$
$510$	0	0
$511$	− 20.7283i	− 0.0405641i
$512$	0	0
$513$	− 398.006i	− 0.775841i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 117.836i	− 0.227922i
$518$	0	0
$519$	261.103i	0.503088i
$520$	0	0
$521$	−491.666	−0.943696	−0.471848	−	0.881680i	$-0.656413\pi$
$521$	−491.666	−0.943696	−0.471848	+	0.881680i	$0.656413\pi$
$522$	0	0
$523$	−133.914	−0.256049	−0.128024	−	0.991771i	$-0.540864\pi$
$523$	−133.914	−0.256049	−0.128024	+	0.991771i	$0.540864\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	759.491	1.44116
$528$	0	0
$529$	−298.580	−0.564424
$530$	0	0
$531$	− 1005.16i	− 1.89295i
$532$	0	0
$533$	492.000i	0.923077i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	36.6687i	0.0682844i
$538$	0	0
$539$	− 529.667i	− 0.982685i
$540$	0	0
$541$	−303.337	−0.560698	−0.280349	−	0.959898i	$-0.590450\pi$
$541$	−303.337	−0.560698	−0.280349	+	0.959898i	$0.590450\pi$
$542$	0	0
$543$	−1352.67	−2.49110
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−89.6631	−0.163918	−0.0819590	−	0.996636i	$-0.526118\pi$
$547$	−89.6631	−0.163918	−0.0819590	+	0.996636i	$0.526118\pi$
$548$	0	0
$549$	−1735.40	−3.16102
$550$	0	0
$551$	46.7492i	0.0848443i
$552$	0	0
$553$	148.328i	0.268225i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 490.741i	− 0.881044i	−0.897742	−	0.440522i	$-0.854793\pi$
$557$	− 490.741i	− 0.881044i	0.897742	−	0.440522i	$-0.145207\pi$
$558$	0	0
$559$	442.008i	0.790712i
$560$	0	0
$561$	−1130.99	−2.01603
$562$	0	0
$563$	−987.423	−1.75386	−0.876930	−	0.480618i	$-0.840412\pi$
$563$	−987.423	−1.75386	−0.876930	+	0.480618i	$0.840412\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	290.763	0.512810
$568$	0	0
$569$	475.082	0.834942	0.417471	−	0.908690i	$-0.362917\pi$
$569$	475.082	0.834942	0.417471	+	0.908690i	$0.362917\pi$
$570$	0	0
$571$	− 182.485i	− 0.319588i	−0.987150	−	0.159794i	$-0.948917\pi$
$571$	− 182.485i	− 0.319588i	0.987150	−	0.159794i	$-0.0510830\pi$
$572$	0	0
$573$	332.498i	0.580277i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 319.167i	− 0.553149i	−0.960992	−	0.276575i	$-0.910801\pi$
$577$	− 319.167i	− 0.553149i	0.960992	−	0.276575i	$-0.0891993\pi$
$578$	0	0
$579$	− 1317.17i	− 2.27490i
$580$	0	0
$581$	121.751	0.209554
$582$	0	0
$583$	−316.676	−0.543183
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−904.762	−1.54133	−0.770666	−	0.637239i	$-0.780076\pi$
$587$	−904.762	−1.54133	−0.770666	+	0.637239i	$0.780076\pi$
$588$	0	0
$589$	−223.319	−0.379149
$590$	0	0
$591$	1294.27i	2.18996i
$592$	0	0
$593$	747.325i	1.26024i	0.776496	+	0.630122i	$0.216995\pi$
$593$	747.325i	1.26024i	−0.776496	+	0.630122i	$0.783005\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 155.331i	− 0.260186i
$598$	0	0
$599$	874.385i	1.45974i	0.683585	+	0.729871i	$0.260420\pi$
$599$	874.385i	1.45974i	−0.683585	+	0.729871i	$0.739580\pi$
$600$	0	0
$601$	−777.234	−1.29323	−0.646617	−	0.762815i	$-0.723817\pi$
$601$	−777.234	−1.29323	−0.646617	+	0.762815i	$0.723817\pi$
$602$	0	0
$603$	820.192	1.36019
$604$	0	0
$605$	0	0
$606$	0	0
$607$	699.436	1.15228	0.576142	−	0.817350i	$-0.304558\pi$
$607$	699.436	1.15228	0.576142	+	0.817350i	$0.304558\pi$
$608$	0	0
$609$	−65.5078	−0.107566
$610$	0	0
$611$	183.075i	0.299631i
$612$	0	0
$613$	11.0820i	0.0180784i	0.999959	+	0.00903918i	$0.00287730\pi$
$613$	11.0820i	0.0180784i	−0.999959	+	0.00903918i	$0.997123\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1189.16i	1.92733i	0.267119	+	0.963664i	$0.413928\pi$
$617$	1189.16i	1.92733i	−0.267119	+	0.963664i	$0.586072\pi$
$618$	0	0
$619$	− 325.717i	− 0.526198i	−0.964769	−	0.263099i	$-0.915255\pi$
$619$	− 325.717i	− 0.526198i	0.964769	−	0.263099i	$-0.0847446\pi$
$620$	0	0
$621$	1141.50	1.83816
$622$	0	0
$623$	78.9477	0.126722
$624$	0	0
$625$	0	0
$626$	0	0
$627$	332.554	0.530389
$628$	0	0
$629$	−601.495	−0.956272
$630$	0	0
$631$	− 316.823i	− 0.502097i	−0.967975	−	0.251048i	$-0.919225\pi$
$631$	− 316.823i	− 0.502097i	0.967975	−	0.251048i	$-0.0807754\pi$
$632$	0	0
$633$	521.823i	0.824366i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	822.912i	1.29186i
$638$	0	0
$639$	− 2188.28i	− 3.42454i
$640$	0	0
$641$	−149.587	−0.233365	−0.116682	−	0.993169i	$-0.537226\pi$
$641$	−149.587	−0.233365	−0.116682	+	0.993169i	$0.537226\pi$
$642$	0	0
$643$	408.690	0.635599	0.317800	−	0.948158i	$-0.397056\pi$
$643$	408.690	0.635599	0.317800	+	0.948158i	$0.397056\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	104.383	0.161334	0.0806668	−	0.996741i	$-0.474295\pi$
$647$	104.383	0.161334	0.0806668	+	0.996741i	$0.474295\pi$
$648$	0	0
$649$	502.663	0.774519
$650$	0	0
$651$	− 312.927i	− 0.480687i
$652$	0	0
$653$	164.420i	0.251791i	0.992044	+	0.125895i	$0.0401804\pi$
$653$	164.420i	0.251791i	−0.992044	+	0.125895i	$0.959820\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 351.167i	− 0.534501i
$658$	0	0
$659$	− 743.613i	− 1.12840i	−0.825640	−	0.564198i	$-0.809186\pi$
$659$	− 743.613i	− 1.12840i	0.825640	−	0.564198i	$-0.190814\pi$
$660$	0	0
$661$	−758.073	−1.14686	−0.573429	−	0.819255i	$-0.694387\pi$
$661$	−758.073	−1.14686	−0.573429	+	0.819255i	$0.694387\pi$
$662$	0	0
$663$	1757.15	2.65030
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−134.078	−0.201017
$668$	0	0
$669$	1725.74	2.57958
$670$	0	0
$671$	− 867.843i	− 1.29336i
$672$	0	0
$673$	− 364.164i	− 0.541106i	−0.962705	−	0.270553i	$-0.912794\pi$
$673$	− 364.164i	− 0.541106i	0.962705	−	0.270553i	$-0.0872065\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 708.079i	− 1.04591i	−0.852361	−	0.522953i	$-0.824830\pi$
$677$	− 708.079i	− 1.04591i	0.852361	−	0.522953i	$-0.175170\pi$
$678$	0	0
$679$	143.119i	0.210780i
$680$	0	0
$681$	1429.40	2.09897
$682$	0	0
$683$	355.469	0.520452	0.260226	−	0.965548i	$-0.416203\pi$
$683$	355.469	0.520452	0.260226	+	0.965548i	$0.416203\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	2306.45	3.35727
$688$	0	0
$689$	492.000	0.714078
$690$	0	0
$691$	− 116.769i	− 0.168985i	−0.996424	−	0.0844925i	$-0.973073\pi$
$691$	− 116.769i	− 0.168985i	0.996424	−	0.0844925i	$-0.0269269\pi$
$692$	0	0
$693$	332.498i	0.479796i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 508.486i	− 0.729535i
$698$	0	0
$699$	1865.54i	2.66887i
$700$	0	0
$701$	−246.413	−0.351517	−0.175758	−	0.984433i	$-0.556238\pi$
$701$	−246.413	−0.351517	−0.175758	+	0.984433i	$0.556238\pi$
$702$	0	0
$703$	176.862	0.251582
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−110.704	−0.156582
$708$	0	0
$709$	552.820	0.779718	0.389859	−	0.920874i	$-0.372524\pi$
$709$	552.820	0.779718	0.389859	+	0.920874i	$0.372524\pi$
$710$	0	0
$711$	2512.89i	3.53431i
$712$	0	0
$713$	− 640.486i	− 0.898297i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	1492.33i	2.08135i
$718$	0	0
$719$	99.9709i	0.139042i	0.997581	+	0.0695208i	$0.0221470\pi$
$719$	99.9709i	0.139042i	−0.997581	+	0.0695208i	$0.977853\pi$
$720$	0	0
$721$	−178.918	−0.248153
$722$	0	0
$723$	−440.464	−0.609217
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−1081.39	−1.48746	−0.743731	−	0.668479i	$-0.766946\pi$
$727$	−1081.39	−1.48746	−0.743731	+	0.668479i	$0.766946\pi$
$728$	0	0
$729$	1132.50	1.55349
$730$	0	0
$731$	− 456.819i	− 0.624923i
$732$	0	0
$733$	− 380.407i	− 0.518973i	−0.965747	−	0.259486i	$-0.916447\pi$
$733$	− 380.407i	− 0.518973i	0.965747	−	0.259486i	$-0.0835533\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	410.164i	0.556532i
$738$	0	0
$739$	− 1381.60i	− 1.86955i	−0.355240	−	0.934775i	$-0.615601\pi$
$739$	− 1381.60i	− 1.86955i	0.355240	−	0.934775i	$-0.384399\pi$
$740$	0	0
$741$	−516.669	−0.697259
$742$	0	0
$743$	−267.424	−0.359924	−0.179962	−	0.983674i	$-0.557597\pi$
$743$	−267.424	−0.359924	−0.179962	+	0.983674i	$0.557597\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	2062.64	2.76123
$748$	0	0
$749$	−161.921	−0.216183
$750$	0	0
$751$	338.653i	0.450937i	0.974250	+	0.225468i	$0.0723912\pi$
$751$	338.653i	0.450937i	−0.974250	+	0.225468i	$0.927609\pi$
$752$	0	0
$753$	− 2267.14i	− 3.01080i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1247.91i	1.64849i	0.566232	+	0.824246i	$0.308401\pi$
$757$	1247.91i	1.64849i	−0.566232	+	0.824246i	$0.691599\pi$
$758$	0	0
$759$	953.775i	1.25662i
$760$	0	0
$761$	−14.3406	−0.0188444	−0.00942219	−	0.999956i	$-0.502999\pi$
$761$	−14.3406	−0.0188444	−0.00942219	+	0.999956i	$0.502999\pi$
$762$	0	0
$763$	−3.41853	−0.00448038
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−780.956	−1.01820
$768$	0	0
$769$	−1036.99	−1.34849	−0.674246	−	0.738507i	$-0.735531\pi$
$769$	−1036.99	−1.34849	−0.674246	+	0.738507i	$0.735531\pi$
$770$	0	0
$771$	− 481.078i	− 0.623967i
$772$	0	0
$773$	− 1197.91i	− 1.54970i	−0.632148	−	0.774848i	$-0.717826\pi$
$773$	− 1197.91i	− 1.54970i	0.632148	−	0.774848i	$-0.282174\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	247.830i	0.318957i
$778$	0	0
$779$	149.514i	0.191931i
$780$	0	0
$781$	1094.32	1.40118
$782$	0	0
$783$	−664.223	−0.848305
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−99.1814	−0.126025	−0.0630123	−	0.998013i	$-0.520071\pi$
$787$	−99.1814	−0.126025	−0.0630123	+	0.998013i	$0.520071\pi$
$788$	0	0
$789$	2470.05	3.13061
$790$	0	0
$791$	44.3282i	0.0560407i
$792$	0	0
$793$	1348.32i	1.70027i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	139.240i	0.174705i	0.996177	+	0.0873525i	$0.0278407\pi$
$797$	139.240i	0.174705i	−0.996177	+	0.0873525i	$0.972159\pi$
$798$	0	0
$799$	− 189.209i	− 0.236807i
$800$	0	0
$801$	1337.49	1.66977
$802$	0	0
$803$	175.613	0.218696
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1151.37	1.42672
$808$	0	0
$809$	−403.653	−0.498953	−0.249477	−	0.968381i	$-0.580259\pi$
$809$	−403.653	−0.498953	−0.249477	+	0.968381i	$0.580259\pi$
$810$	0	0
$811$	384.561i	0.474181i	0.971488	+	0.237091i	$0.0761938\pi$
$811$	384.561i	0.474181i	−0.971488	+	0.237091i	$0.923806\pi$
$812$	0	0
$813$	2343.15i	2.88210i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	134.322i	0.164409i
$818$	0	0
$819$	− 516.583i	− 0.630748i
$820$	0	0
$821$	−997.732	−1.21526	−0.607632	−	0.794219i	$-0.707881\pi$
$821$	−997.732	−1.21526	−0.607632	+	0.794219i	$0.707881\pi$
$822$	0	0
$823$	955.281	1.16073	0.580365	−	0.814356i	$-0.302910\pi$
$823$	955.281	1.16073	0.580365	+	0.814356i	$0.302910\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	213.451	0.258103	0.129051	−	0.991638i	$-0.458807\pi$
$827$	213.451	0.258103	0.129051	+	0.991638i	$0.458807\pi$
$828$	0	0
$829$	73.4288	0.0885752	0.0442876	−	0.999019i	$-0.485898\pi$
$829$	73.4288	0.0885752	0.0442876	+	0.999019i	$0.485898\pi$
$830$	0	0
$831$	− 2866.47i	− 3.44942i
$832$	0	0
$833$	− 850.486i	− 1.02099i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 3172.96i	− 3.79087i
$838$	0	0
$839$	− 643.642i	− 0.767154i	−0.923509	−	0.383577i	$-0.874692\pi$
$839$	− 643.642i	− 0.767154i	0.923509	−	0.383577i	$-0.125308\pi$
$840$	0	0
$841$	−762.981	−0.907231
$842$	0	0
$843$	648.787	0.769617
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−6.17342	−0.00728857
$848$	0	0
$849$	−2253.23	−2.65398
$850$	0	0
$851$	507.247i	0.596060i
$852$	0	0
$853$	351.410i	0.411970i	0.978555	+	0.205985i	$0.0660398\pi$
$853$	351.410i	0.411970i	−0.978555	+	0.205985i	$0.933960\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1326.16i	1.54744i	0.633526	+	0.773721i	$0.281607\pi$
$857$	1326.16i	1.54744i	−0.633526	+	0.773721i	$0.718393\pi$
$858$	0	0
$859$	343.729i	0.400150i	0.979781	+	0.200075i	$0.0641186\pi$
$859$	343.729i	0.400150i	−0.979781	+	0.200075i	$0.935881\pi$
$860$	0	0
$861$	−209.508	−0.243331
$862$	0	0
$863$	−1136.15	−1.31651	−0.658257	−	0.752793i	$-0.728706\pi$
$863$	−1136.15	−1.31651	−0.658257	+	0.752793i	$0.728706\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−196.176	−0.226270
$868$	0	0
$869$	−1256.66	−1.44609
$870$	0	0
$871$	− 637.247i	− 0.731627i
$872$	0	0
$873$	2424.65i	2.77738i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	769.076i	0.876939i	0.898746	+	0.438470i	$0.144479\pi$
$877$	769.076i	0.876939i	−0.898746	+	0.438470i	$0.855521\pi$
$878$	0	0
$879$	− 506.770i	− 0.576530i
$880$	0	0
$881$	−731.568	−0.830384	−0.415192	−	0.909734i	$-0.636286\pi$
$881$	−731.568	−0.830384	−0.415192	+	0.909734i	$0.636286\pi$
$882$	0	0
$883$	1066.19	1.20746	0.603731	−	0.797188i	$-0.293680\pi$
$883$	1066.19	1.20746	0.603731	+	0.797188i	$0.293680\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−397.242	−0.447848	−0.223924	−	0.974607i	$-0.571887\pi$
$887$	−397.242	−0.447848	−0.223924	+	0.974607i	$0.571887\pi$
$888$	0	0
$889$	−170.262	−0.191520
$890$	0	0
$891$	2463.39i	2.76474i
$892$	0	0
$893$	55.6346i	0.0623008i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 1481.82i	− 1.65198i
$898$	0	0
$899$	372.691i	0.414562i
$900$	0	0
$901$	−508.486	−0.564357
$902$	0	0
$903$	−188.220	−0.208438
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−1021.02	−1.12571	−0.562855	−	0.826556i	$-0.690297\pi$
$907$	−1021.02	−1.12571	−0.562855	+	0.826556i	$0.690297\pi$
$908$	0	0
$909$	−1875.48	−2.06324
$910$	0	0
$911$	1118.66i	1.22794i	0.789328	+	0.613971i	$0.210429\pi$
$911$	1118.66i	1.22794i	−0.789328	+	0.613971i	$0.789571\pi$
$912$	0	0
$913$	1031.49i	1.12978i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 294.177i	− 0.320803i
$918$	0	0
$919$	− 1177.68i	− 1.28148i	−0.767757	−	0.640741i	$-0.778627\pi$
$919$	− 1177.68i	− 1.28148i	0.767757	−	0.640741i	$-0.221373\pi$
$920$	0	0
$921$	1543.73	1.67615
$922$	0	0
$923$	−1700.18	−1.84202
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−3031.13	−3.26983
$928$	0	0
$929$	699.423	0.752877	0.376438	−	0.926442i	$-0.377149\pi$
$929$	699.423	0.752877	0.376438	+	0.926442i	$0.377149\pi$
$930$	0	0
$931$	250.075i	0.268609i
$932$	0	0
$933$	− 2720.15i	− 2.91548i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 332.152i	− 0.354484i	−0.984167	−	0.177242i	$-0.943282\pi$
$937$	− 332.152i	− 0.354484i	0.984167	−	0.177242i	$-0.0567176\pi$
$938$	0	0
$939$	− 11.2101i	− 0.0119383i
$940$	0	0
$941$	−1468.66	−1.56075	−0.780373	−	0.625314i	$-0.784971\pi$
$941$	−1468.66	−1.56075	−0.780373	+	0.625314i	$0.784971\pi$
$942$	0	0
$943$	−428.811	−0.454731
$944$	0	0
$945$	0	0
$946$	0	0
$947$	574.082	0.606211	0.303106	−	0.952957i	$-0.401976\pi$
$947$	574.082	0.606211	0.303106	+	0.952957i	$0.401976\pi$
$948$	0	0
$949$	−272.839	−0.287502
$950$	0	0
$951$	− 2212.54i	− 2.32654i
$952$	0	0
$953$	− 280.663i	− 0.294504i	−0.989099	−	0.147252i	$-0.952957\pi$
$953$	− 280.663i	− 0.294504i	0.989099	−	0.147252i	$-0.0470429\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 554.991i	− 0.579928i
$958$	0	0
$959$	170.468i	0.177756i
$960$	0	0
$961$	−819.328	−0.852579
$962$	0	0
$963$	−2743.18	−2.84858
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−1278.65	−1.32228	−0.661141	−	0.750262i	$-0.729927\pi$
$967$	−1278.65	−1.32228	−0.661141	+	0.750262i	$0.729927\pi$
$968$	0	0
$969$	533.981	0.551064
$970$	0	0
$971$	117.723i	0.121239i	0.998161	+	0.0606195i	$0.0193076\pi$
$971$	117.723i	0.121239i	−0.998161	+	0.0606195i	$0.980692\pi$
$972$	0	0
$973$	− 305.313i	− 0.313785i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	983.155i	1.00630i	0.864199	+	0.503150i	$0.167826\pi$
$977$	983.155i	1.00630i	−0.864199	+	0.503150i	$0.832174\pi$
$978$	0	0
$979$	668.856i	0.683203i
$980$	0	0
$981$	−57.9149	−0.0590365
$982$	0	0
$983$	−607.404	−0.617909	−0.308954	−	0.951077i	$-0.599979\pi$
$983$	−607.404	−0.617909	−0.308954	+	0.951077i	$0.599979\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−77.9584	−0.0789852
$988$	0	0
$989$	−385.240	−0.389525
$990$	0	0
$991$	− 408.413i	− 0.412122i	−0.978539	−	0.206061i	$-0.933935\pi$
$991$	− 408.413i	− 0.412122i	0.978539	−	0.206061i	$-0.0660645\pi$
$992$	0	0
$993$	1659.82i	1.67152i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 1554.91i	− 1.55959i	−0.626035	−	0.779795i	$-0.715323\pi$
$997$	− 1554.91i	− 1.55959i	0.626035	−	0.779795i	$-0.284677\pi$
$998$	0	0
$999$	2512.89i	2.51541i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.3.h.d.399.8		8
3.2	odd	2		3600.3.j.k.1999.5		8
4.3	odd	2	inner	400.3.h.d.399.1		8
5.2	odd	4		400.3.b.g.351.1		4
5.3	odd	4		80.3.b.a.31.4	yes	4
5.4	even	2	inner	400.3.h.d.399.2		8
8.3	odd	2		1600.3.h.p.1599.8		8
8.5	even	2		1600.3.h.p.1599.1		8
12.11	even	2		3600.3.j.k.1999.4		8
15.2	even	4		3600.3.e.bb.3151.3		4
15.8	even	4		720.3.e.c.271.3		4
15.14	odd	2		3600.3.j.k.1999.3		8
20.3	even	4		80.3.b.a.31.1	✓	4
20.7	even	4		400.3.b.g.351.4		4
20.19	odd	2	inner	400.3.h.d.399.7		8
40.3	even	4		320.3.b.a.191.4		4
40.13	odd	4		320.3.b.a.191.1		4
40.19	odd	2		1600.3.h.p.1599.2		8
40.27	even	4		1600.3.b.k.1151.1		4
40.29	even	2		1600.3.h.p.1599.7		8
40.37	odd	4		1600.3.b.k.1151.4		4
60.23	odd	4		720.3.e.c.271.4		4
60.47	odd	4		3600.3.e.bb.3151.2		4
60.59	even	2		3600.3.j.k.1999.6		8
80.3	even	4		1280.3.g.f.1151.8		8
80.13	odd	4		1280.3.g.f.1151.2		8
80.43	even	4		1280.3.g.f.1151.1		8
80.53	odd	4		1280.3.g.f.1151.7		8
120.53	even	4		2880.3.e.b.2431.1		4
120.83	odd	4		2880.3.e.b.2431.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.3.b.a.31.1	✓	4	20.3	even	4
80.3.b.a.31.4	yes	4	5.3	odd	4
320.3.b.a.191.1		4	40.13	odd	4
320.3.b.a.191.4		4	40.3	even	4
400.3.b.g.351.1		4	5.2	odd	4
400.3.b.g.351.4		4	20.7	even	4
400.3.h.d.399.1		8	4.3	odd	2	inner
400.3.h.d.399.2		8	5.4	even	2	inner
400.3.h.d.399.7		8	20.19	odd	2	inner
400.3.h.d.399.8		8	1.1	even	1	trivial
720.3.e.c.271.3		4	15.8	even	4
720.3.e.c.271.4		4	60.23	odd	4
1280.3.g.f.1151.1		8	80.43	even	4
1280.3.g.f.1151.2		8	80.13	odd	4
1280.3.g.f.1151.7		8	80.53	odd	4
1280.3.g.f.1151.8		8	80.3	even	4
1600.3.b.k.1151.1		4	40.27	even	4
1600.3.b.k.1151.4		4	40.37	odd	4
1600.3.h.p.1599.1		8	8.5	even	2
1600.3.h.p.1599.2		8	40.19	odd	2
1600.3.h.p.1599.7		8	40.29	even	2
1600.3.h.p.1599.8		8	8.3	odd	2
2880.3.e.b.2431.1		4	120.53	even	4
2880.3.e.b.2431.2		4	120.83	odd	4
3600.3.e.bb.3151.2		4	60.47	odd	4
3600.3.e.bb.3151.3		4	15.2	even	4
3600.3.j.k.1999.3		8	15.14	odd	2
3600.3.j.k.1999.4		8	12.11	even	2
3600.3.j.k.1999.5		8	3.2	odd	2
3600.3.j.k.1999.6		8	60.59	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	400.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+5.60503 q^{3} +1.32317 q^{7} +22.4164 q^{9} +11.2101i q^{11} -17.4164i q^{13} +18.0000i q^{17} -5.29268i q^{19} +7.41641 q^{21} +15.1796 q^{23} +75.1994 q^{27} -8.83282 q^{29} -42.1939i q^{31} +62.8328i q^{33} +33.4164i q^{37} -97.6196i q^{39} -28.2492 q^{41} -25.3788 q^{43} -10.5116 q^{47} -47.2492 q^{49} +100.891i q^{51} +28.2492i q^{53} -29.6656i q^{57} -44.8403i q^{59} -77.4164 q^{61} +29.6607 q^{63} +36.5889 q^{67} +85.0820 q^{69} -97.6196i q^{71} -15.6656i q^{73} +14.8328i q^{77} +112.101i q^{79} +219.748 q^{81} +92.0145 q^{83} -49.5082 q^{87} +59.6656 q^{89} -23.0449i q^{91} -236.498i q^{93} +108.164i q^{97} +251.289i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 72 q^{9} - 48 q^{21} + 144 q^{29} + 96 q^{41} - 56 q^{49} - 512 q^{61} + 144 q^{69} + 792 q^{81} + 48 q^{89}+O(q^{100}) \) 8 * q + 72 * q^9 - 48 * q^21 + 144 * q^29 + 96 * q^41 - 56 * q^49 - 512 * q^61 + 144 * q^69 + 792 * q^81 + 48 * q^89

Introduction

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