LMFDB - Embedded newform 10000.2.a.bi.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [10000,2,Mod(1,10000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(10000, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("10000.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 10000 = 2^{4} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	10000.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$79.8504020213$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.14884000000.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 11x^{6} + 36x^{4} - 31x^{2} + 1 $ x^8 - 11x^6 + 36x^4 - 31*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 100)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.183172$ of defining polynomial
Character	$\chi$	$=$	10000.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.183172 q^{3} +1.35874 q^{7} -2.96645 q^{9} +O(q^{10})$	$q-0.183172 q^{3} +1.35874 q^{7} -2.96645 q^{9} -2.79981 q^{11} -2.49487 q^{13} +4.95922 q^{17} +5.28477 q^{19} -0.248884 q^{21} -7.44414 q^{23} +1.09289 q^{27} +0.314862 q^{29} +5.26057 q^{31} +0.512848 q^{33} +0.757065 q^{37} +0.456991 q^{39} -1.20019 q^{41} +10.3054 q^{43} -4.27212 q^{47} -5.15382 q^{49} -0.908390 q^{51} -13.0867 q^{53} -0.968021 q^{57} +5.02074 q^{59} -4.73038 q^{61} -4.03064 q^{63} +13.5268 q^{67} +1.36356 q^{69} -0.185235 q^{71} +9.89749 q^{73} -3.80423 q^{77} +16.3866 q^{79} +8.69916 q^{81} -12.1886 q^{83} -0.0576739 q^{87} -12.6912 q^{89} -3.38989 q^{91} -0.963590 q^{93} -10.2244 q^{97} +8.30550 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{9}+O(q^{10}) $ 8 * q - 2 * q^9	$ 8 q - 2 q^{9} + 10 q^{11} + 12 q^{19} - 22 q^{21} - 32 q^{29} + 2 q^{31} + 2 q^{39} - 42 q^{41} - 14 q^{49} + 14 q^{51} + 24 q^{59} - 34 q^{61} - 36 q^{69} - 4 q^{71} - 4 q^{79} - 28 q^{81} - 58 q^{89} + 18 q^{91} + 20 q^{99}+O(q^{100}) $ 8 * q - 2 * q^9 + 10 * q^11 + 12 * q^19 - 22 * q^21 - 32 * q^29 + 2 * q^31 + 2 * q^39 - 42 * q^41 - 14 * q^49 + 14 * q^51 + 24 * q^59 - 34 * q^61 - 36 * q^69 - 4 * q^71 - 4 * q^79 - 28 * q^81 - 58 * q^89 + 18 * q^91 + 20 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−0.183172	−0.105754	−0.0528772	−	0.998601i	$-0.516839\pi$
$3$	−0.183172	−0.105754	−0.0528772	+	0.998601i	$0.516839\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.35874	0.513556	0.256778	−	0.966470i	$-0.417339\pi$
$7$	1.35874	0.513556	0.256778	+	0.966470i	$0.417339\pi$
$8$	0	0
$9$	−2.96645	−0.988816
$10$	0	0
$11$	−2.79981	−0.844176	−0.422088	−	0.906555i	$-0.638703\pi$
$11$	−2.79981	−0.844176	−0.422088	+	0.906555i	$0.638703\pi$
$12$	0	0
$13$	−2.49487	−0.691953	−0.345976	−	0.938243i	$-0.612452\pi$
$13$	−2.49487	−0.691953	−0.345976	+	0.938243i	$0.612452\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.95922	1.20279	0.601394	−	0.798953i	$-0.294612\pi$
$17$	4.95922	1.20279	0.601394	+	0.798953i	$0.294612\pi$
$18$	0	0
$19$	5.28477	1.21241	0.606204	−	0.795309i	$-0.292691\pi$
$19$	5.28477	1.21241	0.606204	+	0.795309i	$0.292691\pi$
$20$	0	0
$21$	−0.248884	−0.0543109
$22$	0	0
$23$	−7.44414	−1.55221	−0.776106	−	0.630603i	$-0.782808\pi$
$23$	−7.44414	−1.55221	−0.776106	+	0.630603i	$0.782808\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.09289	0.210326
$28$	0	0
$29$	0.314862	0.0584684	0.0292342	−	0.999573i	$-0.490693\pi$
$29$	0.314862	0.0584684	0.0292342	+	0.999573i	$0.490693\pi$
$30$	0	0
$31$	5.26057	0.944827	0.472413	−	0.881377i	$-0.343383\pi$
$31$	5.26057	0.944827	0.472413	+	0.881377i	$0.343383\pi$
$32$	0	0
$33$	0.512848	0.0892753
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0.757065	0.124461	0.0622304	−	0.998062i	$-0.480179\pi$
$37$	0.757065	0.124461	0.0622304	+	0.998062i	$0.480179\pi$
$38$	0	0
$39$	0.456991	0.0731771
$40$	0	0
$41$	−1.20019	−0.187438	−0.0937188	−	0.995599i	$-0.529875\pi$
$41$	−1.20019	−0.187438	−0.0937188	+	0.995599i	$0.529875\pi$
$42$	0	0
$43$	10.3054	1.57155	0.785776	−	0.618511i	$-0.212264\pi$
$43$	10.3054	1.57155	0.785776	+	0.618511i	$0.212264\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.27212	−0.623152	−0.311576	−	0.950221i	$-0.600857\pi$
$47$	−4.27212	−0.623152	−0.311576	+	0.950221i	$0.600857\pi$
$48$	0	0
$49$	−5.15382	−0.736260
$50$	0	0
$51$	−0.908390	−0.127200
$52$	0	0
$53$	−13.0867	−1.79759	−0.898796	−	0.438368i	$-0.855557\pi$
$53$	−13.0867	−1.79759	−0.898796	+	0.438368i	$0.855557\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−0.968021	−0.128218
$58$	0	0
$59$	5.02074	0.653644	0.326822	−	0.945086i	$-0.394022\pi$
$59$	5.02074	0.653644	0.326822	+	0.945086i	$0.394022\pi$
$60$	0	0
$61$	−4.73038	−0.605663	−0.302832	−	0.953044i	$-0.597932\pi$
$61$	−4.73038	−0.605663	−0.302832	+	0.953044i	$0.597932\pi$
$62$	0	0
$63$	−4.03064	−0.507813
$64$	0	0
$65$	0	0
$66$	0	0
$67$	13.5268	1.65256	0.826279	−	0.563261i	$-0.190453\pi$
$67$	13.5268	1.65256	0.826279	+	0.563261i	$0.190453\pi$
$68$	0	0
$69$	1.36356	0.164153
$70$	0	0
$71$	−0.185235	−0.0219834	−0.0109917	−	0.999940i	$-0.503499\pi$
$71$	−0.185235	−0.0219834	−0.0109917	+	0.999940i	$0.503499\pi$
$72$	0	0
$73$	9.89749	1.15841	0.579207	−	0.815181i	$-0.303362\pi$
$73$	9.89749	1.15841	0.579207	+	0.815181i	$0.303362\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−3.80423	−0.433532
$78$	0	0
$79$	16.3866	1.84364	0.921820	−	0.387619i	$-0.126702\pi$
$79$	16.3866	1.84364	0.921820	+	0.387619i	$0.126702\pi$
$80$	0	0
$81$	8.69916	0.966573
$82$	0	0
$83$	−12.1886	−1.33787	−0.668937	−	0.743319i	$-0.733250\pi$
$83$	−12.1886	−1.33787	−0.668937	+	0.743319i	$0.733250\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−0.0576739	−0.00618329
$88$	0	0
$89$	−12.6912	−1.34527	−0.672634	−	0.739975i	$-0.734837\pi$
$89$	−12.6912	−1.34527	−0.672634	+	0.739975i	$0.734837\pi$
$90$	0	0
$91$	−3.38989	−0.355357
$92$	0	0
$93$	−0.963590	−0.0999197
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−10.2244	−1.03813	−0.519065	−	0.854735i	$-0.673720\pi$
$97$	−10.2244	−1.03813	−0.519065	+	0.854735i	$0.673720\pi$
$98$	0	0
$99$	8.30550	0.834734
$100$	0	0
$101$	−13.5302	−1.34630	−0.673152	−	0.739504i	$-0.735060\pi$
$101$	−13.5302	−1.34630	−0.673152	+	0.739504i	$0.735060\pi$
$102$	0	0
$103$	9.78428	0.964074	0.482037	−	0.876151i	$-0.339897\pi$
$103$	9.78428	0.964074	0.482037	+	0.876151i	$0.339897\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−7.96208	−0.769723	−0.384862	−	0.922974i	$-0.625751\pi$
$107$	−7.96208	−0.769723	−0.384862	+	0.922974i	$0.625751\pi$
$108$	0	0
$109$	1.29067	0.123624	0.0618119	−	0.998088i	$-0.480312\pi$
$109$	1.29067	0.123624	0.0618119	+	0.998088i	$0.480312\pi$
$110$	0	0
$111$	−0.138673	−0.0131623
$112$	0	0
$113$	11.5814	1.08949	0.544744	−	0.838602i	$-0.316627\pi$
$113$	11.5814	1.08949	0.544744	+	0.838602i	$0.316627\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	7.40090	0.684214
$118$	0	0
$119$	6.73830	0.617699
$120$	0	0
$121$	−3.16104	−0.287368
$122$	0	0
$123$	0.219841	0.0198224
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−16.9581	−1.50479	−0.752393	−	0.658714i	$-0.771101\pi$
$127$	−16.9581	−1.50479	−0.752393	+	0.658714i	$0.771101\pi$
$128$	0	0
$129$	−1.88765	−0.166199
$130$	0	0
$131$	−8.31719	−0.726676	−0.363338	−	0.931657i	$-0.618363\pi$
$131$	−8.31719	−0.726676	−0.363338	+	0.931657i	$0.618363\pi$
$132$	0	0
$133$	7.18064	0.622640
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−8.83618	−0.754926	−0.377463	−	0.926025i	$-0.623203\pi$
$137$	−8.83618	−0.754926	−0.377463	+	0.926025i	$0.623203\pi$
$138$	0	0
$139$	13.8806	1.17734	0.588670	−	0.808374i	$-0.299652\pi$
$139$	13.8806	1.17734	0.588670	+	0.808374i	$0.299652\pi$
$140$	0	0
$141$	0.782533	0.0659011
$142$	0	0
$143$	6.98517	0.584129
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0.944036	0.0778627
$148$	0	0
$149$	−12.5001	−1.02405	−0.512024	−	0.858971i	$-0.671104\pi$
$149$	−12.5001	−1.02405	−0.512024	+	0.858971i	$0.671104\pi$
$150$	0	0
$151$	−16.5477	−1.34663	−0.673315	−	0.739356i	$-0.735130\pi$
$151$	−16.5477	−1.34663	−0.673315	+	0.739356i	$0.735130\pi$
$152$	0	0
$153$	−14.7113	−1.18934
$154$	0	0
$155$	0	0
$156$	0	0
$157$	6.96182	0.555614	0.277807	−	0.960637i	$-0.410392\pi$
$157$	6.96182	0.555614	0.277807	+	0.960637i	$0.410392\pi$
$158$	0	0
$159$	2.39711	0.190103
$160$	0	0
$161$	−10.1147	−0.797148
$162$	0	0
$163$	7.58408	0.594031	0.297015	−	0.954873i	$-0.404009\pi$
$163$	7.58408	0.594031	0.297015	+	0.954873i	$0.404009\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.52962	0.118366	0.0591829	−	0.998247i	$-0.481150\pi$
$167$	1.52962	0.118366	0.0591829	+	0.998247i	$0.481150\pi$
$168$	0	0
$169$	−6.77562	−0.521202
$170$	0	0
$171$	−15.6770	−1.19885
$172$	0	0
$173$	10.8943	0.828280	0.414140	−	0.910213i	$-0.364082\pi$
$173$	10.8943	0.828280	0.414140	+	0.910213i	$0.364082\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−0.919659	−0.0691258
$178$	0	0
$179$	−3.46655	−0.259102	−0.129551	−	0.991573i	$-0.541354\pi$
$179$	−3.46655	−0.259102	−0.129551	+	0.991573i	$0.541354\pi$
$180$	0	0
$181$	−9.66328	−0.718266	−0.359133	−	0.933286i	$-0.616927\pi$
$181$	−9.66328	−0.718266	−0.359133	+	0.933286i	$0.616927\pi$
$182$	0	0
$183$	0.866474	0.0640516
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−13.8849	−1.01536
$188$	0	0
$189$	1.48495	0.108014
$190$	0	0
$191$	−2.00559	−0.145119	−0.0725597	−	0.997364i	$-0.523117\pi$
$191$	−2.00559	−0.145119	−0.0725597	+	0.997364i	$0.523117\pi$
$192$	0	0
$193$	−13.1910	−0.949505	−0.474753	−	0.880119i	$-0.657462\pi$
$193$	−13.1910	−0.949505	−0.474753	+	0.880119i	$0.657462\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	5.60542	0.399370	0.199685	−	0.979860i	$-0.436008\pi$
$197$	5.60542	0.399370	0.199685	+	0.979860i	$0.436008\pi$
$198$	0	0
$199$	−18.0476	−1.27936	−0.639679	−	0.768642i	$-0.720933\pi$
$199$	−18.0476	−1.27936	−0.639679	+	0.768642i	$0.720933\pi$
$200$	0	0
$201$	−2.47773	−0.174765
$202$	0	0
$203$	0.427816	0.0300268
$204$	0	0
$205$	0	0
$206$	0	0
$207$	22.0827	1.53485
$208$	0	0
$209$	−14.7964	−1.02349
$210$	0	0
$211$	−19.9109	−1.37072	−0.685362	−	0.728203i	$-0.740356\pi$
$211$	−19.9109	−1.37072	−0.685362	+	0.728203i	$0.740356\pi$
$212$	0	0
$213$	0.0339299	0.00232484
$214$	0	0
$215$	0	0
$216$	0	0
$217$	7.14777	0.485222
$218$	0	0
$219$	−1.81294	−0.122507
$220$	0	0
$221$	−12.3726	−0.832272
$222$	0	0
$223$	−22.2531	−1.49018	−0.745090	−	0.666964i	$-0.767593\pi$
$223$	−22.2531	−1.49018	−0.745090	+	0.666964i	$0.767593\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	25.6533	1.70267	0.851334	−	0.524624i	$-0.175794\pi$
$227$	25.6533	1.70267	0.851334	+	0.524624i	$0.175794\pi$
$228$	0	0
$229$	−3.67446	−0.242815	−0.121408	−	0.992603i	$-0.538741\pi$
$229$	−3.67446	−0.242815	−0.121408	+	0.992603i	$0.538741\pi$
$230$	0	0
$231$	0.696828	0.0458479
$232$	0	0
$233$	−7.22388	−0.473252	−0.236626	−	0.971601i	$-0.576042\pi$
$233$	−7.22388	−0.473252	−0.236626	+	0.971601i	$0.576042\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−3.00157	−0.194973
$238$	0	0
$239$	−18.4018	−1.19031	−0.595156	−	0.803610i	$-0.702910\pi$
$239$	−18.4018	−1.19031	−0.595156	+	0.803610i	$0.702910\pi$
$240$	0	0
$241$	−12.6521	−0.814993	−0.407497	−	0.913207i	$-0.633598\pi$
$241$	−12.6521	−0.814993	−0.407497	+	0.913207i	$0.633598\pi$
$242$	0	0
$243$	−4.87210	−0.312546
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−13.1848	−0.838929
$248$	0	0
$249$	2.23261	0.141486
$250$	0	0
$251$	4.16683	0.263008	0.131504	−	0.991316i	$-0.458019\pi$
$251$	4.16683	0.263008	0.131504	+	0.991316i	$0.458019\pi$
$252$	0	0
$253$	20.8422	1.31034
$254$	0	0
$255$	0	0
$256$	0	0
$257$	5.44326	0.339541	0.169771	−	0.985484i	$-0.445697\pi$
$257$	5.44326	0.339541	0.169771	+	0.985484i	$0.445697\pi$
$258$	0	0
$259$	1.02866	0.0639176
$260$	0	0
$261$	−0.934022	−0.0578145
$262$	0	0
$263$	3.65707	0.225504	0.112752	−	0.993623i	$-0.464033\pi$
$263$	3.65707	0.225504	0.112752	+	0.993623i	$0.464033\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2.32468	0.142268
$268$	0	0
$269$	−9.76067	−0.595119	−0.297559	−	0.954703i	$-0.596173\pi$
$269$	−9.76067	−0.595119	−0.297559	+	0.954703i	$0.596173\pi$
$270$	0	0
$271$	14.6378	0.889185	0.444593	−	0.895733i	$-0.353348\pi$
$271$	14.6378	0.889185	0.444593	+	0.895733i	$0.353348\pi$
$272$	0	0
$273$	0.620933	0.0375805
$274$	0	0
$275$	0	0
$276$	0	0
$277$	5.59928	0.336428	0.168214	−	0.985750i	$-0.446200\pi$
$277$	5.59928	0.336428	0.168214	+	0.985750i	$0.446200\pi$
$278$	0	0
$279$	−15.6052	−0.934260
$280$	0	0
$281$	−29.3980	−1.75374	−0.876869	−	0.480730i	$-0.840372\pi$
$281$	−29.3980	−1.75374	−0.876869	+	0.480730i	$0.840372\pi$
$282$	0	0
$283$	−0.389007	−0.0231240	−0.0115620	−	0.999933i	$-0.503680\pi$
$283$	−0.389007	−0.0231240	−0.0115620	+	0.999933i	$0.503680\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.63074	−0.0962598
$288$	0	0
$289$	7.59384	0.446697
$290$	0	0
$291$	1.87282	0.109787
$292$	0	0
$293$	6.66164	0.389177	0.194589	−	0.980885i	$-0.437663\pi$
$293$	6.66164	0.389177	0.194589	+	0.980885i	$0.437663\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−3.05988	−0.177552
$298$	0	0
$299$	18.5722	1.07406
$300$	0	0
$301$	14.0023	0.807081
$302$	0	0
$303$	2.47835	0.142378
$304$	0	0
$305$	0	0
$306$	0	0
$307$	5.74370	0.327810	0.163905	−	0.986476i	$-0.447591\pi$
$307$	5.74370	0.327810	0.163905	+	0.986476i	$0.447591\pi$
$308$	0	0
$309$	−1.79221	−0.101955
$310$	0	0
$311$	5.13164	0.290989	0.145494	−	0.989359i	$-0.453523\pi$
$311$	5.13164	0.290989	0.145494	+	0.989359i	$0.453523\pi$
$312$	0	0
$313$	−16.0600	−0.907766	−0.453883	−	0.891061i	$-0.649962\pi$
$313$	−16.0600	−0.907766	−0.453883	+	0.891061i	$0.649962\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	16.6645	0.935970	0.467985	−	0.883736i	$-0.344980\pi$
$317$	16.6645	0.935970	0.467985	+	0.883736i	$0.344980\pi$
$318$	0	0
$319$	−0.881555	−0.0493576
$320$	0	0
$321$	1.45843	0.0814016
$322$	0	0
$323$	26.2083	1.45827
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−0.236415	−0.0130738
$328$	0	0
$329$	−5.80471	−0.320024
$330$	0	0
$331$	−21.6031	−1.18741	−0.593706	−	0.804682i	$-0.702336\pi$
$331$	−21.6031	−1.18741	−0.593706	+	0.804682i	$0.702336\pi$
$332$	0	0
$333$	−2.24579	−0.123069
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−14.9201	−0.812750	−0.406375	−	0.913706i	$-0.633207\pi$
$337$	−14.9201	−0.812750	−0.406375	+	0.913706i	$0.633207\pi$
$338$	0	0
$339$	−2.12139	−0.115218
$340$	0	0
$341$	−14.7286	−0.797600
$342$	0	0
$343$	−16.5139	−0.891667
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−9.54558	−0.512433	−0.256217	−	0.966619i	$-0.582476\pi$
$347$	−9.54558	−0.512433	−0.256217	+	0.966619i	$0.582476\pi$
$348$	0	0
$349$	14.2777	0.764270	0.382135	−	0.924107i	$-0.375189\pi$
$349$	14.2777	0.764270	0.382135	+	0.924107i	$0.375189\pi$
$350$	0	0
$351$	−2.72661	−0.145536
$352$	0	0
$353$	17.4055	0.926399	0.463199	−	0.886254i	$-0.346701\pi$
$353$	17.4055	0.926399	0.463199	+	0.886254i	$0.346701\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−1.23427	−0.0653244
$358$	0	0
$359$	−14.0117	−0.739509	−0.369754	−	0.929130i	$-0.620558\pi$
$359$	−14.0117	−0.739509	−0.369754	+	0.929130i	$0.620558\pi$
$360$	0	0
$361$	8.92874	0.469934
$362$	0	0
$363$	0.579015	0.0303904
$364$	0	0
$365$	0	0
$366$	0	0
$367$	18.3646	0.958622	0.479311	−	0.877645i	$-0.340887\pi$
$367$	18.3646	0.958622	0.479311	+	0.877645i	$0.340887\pi$
$368$	0	0
$369$	3.56029	0.185341
$370$	0	0
$371$	−17.7814	−0.923165
$372$	0	0
$373$	−21.2501	−1.10029	−0.550145	−	0.835069i	$-0.685427\pi$
$373$	−21.2501	−1.10029	−0.550145	+	0.835069i	$0.685427\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−0.785540	−0.0404574
$378$	0	0
$379$	0.00578502	0.000297156 0	0.000148578	−	1.00000i	$-0.499953\pi$
$379$	0.00578502	0.000297156 0	0.000148578		1.00000i	$0.499953\pi$
$380$	0	0
$381$	3.10625	0.159138
$382$	0	0
$383$	20.8392	1.06483	0.532417	−	0.846482i	$-0.321284\pi$
$383$	20.8392	1.06483	0.532417	+	0.846482i	$0.321284\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−30.5703	−1.55398
$388$	0	0
$389$	3.03375	0.153817	0.0769085	−	0.997038i	$-0.475495\pi$
$389$	3.03375	0.153817	0.0769085	+	0.997038i	$0.475495\pi$
$390$	0	0
$391$	−36.9171	−1.86698
$392$	0	0
$393$	1.52348	0.0768493
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−34.5045	−1.73173	−0.865866	−	0.500276i	$-0.833232\pi$
$397$	−34.5045	−1.73173	−0.865866	+	0.500276i	$0.833232\pi$
$398$	0	0
$399$	−1.31529	−0.0658470
$400$	0	0
$401$	−20.8077	−1.03909	−0.519544	−	0.854443i	$-0.673898\pi$
$401$	−20.8077	−1.03909	−0.519544	+	0.854443i	$0.673898\pi$
$402$	0	0
$403$	−13.1244	−0.653775
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2.11964	−0.105067
$408$	0	0
$409$	24.5103	1.21196	0.605979	−	0.795480i	$-0.292781\pi$
$409$	24.5103	1.21196	0.605979	+	0.795480i	$0.292781\pi$
$410$	0	0
$411$	1.61854	0.0798368
$412$	0	0
$413$	6.82189	0.335683
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−2.54254	−0.124509
$418$	0	0
$419$	−39.3785	−1.92376	−0.961882	−	0.273465i	$-0.911830\pi$
$419$	−39.3785	−1.92376	−0.961882	+	0.273465i	$0.911830\pi$
$420$	0	0
$421$	12.2887	0.598916	0.299458	−	0.954109i	$-0.403194\pi$
$421$	12.2887	0.598916	0.299458	+	0.954109i	$0.403194\pi$
$422$	0	0
$423$	12.6730	0.616183
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−6.42737	−0.311042
$428$	0	0
$429$	−1.27949	−0.0617743
$430$	0	0
$431$	−14.3321	−0.690355	−0.345178	−	0.938537i	$-0.612181\pi$
$431$	−14.3321	−0.690355	−0.345178	+	0.938537i	$0.612181\pi$
$432$	0	0
$433$	5.39959	0.259488	0.129744	−	0.991548i	$-0.458584\pi$
$433$	5.39959	0.259488	0.129744	+	0.991548i	$0.458584\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−39.3406	−1.88191
$438$	0	0
$439$	−0.902291	−0.0430640	−0.0215320	−	0.999768i	$-0.506854\pi$
$439$	−0.902291	−0.0430640	−0.0215320	+	0.999768i	$0.506854\pi$
$440$	0	0
$441$	15.2885	0.728025
$442$	0	0
$443$	−20.1507	−0.957390	−0.478695	−	0.877981i	$-0.658890\pi$
$443$	−20.1507	−0.957390	−0.478695	+	0.877981i	$0.658890\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	2.28967	0.108298
$448$	0	0
$449$	32.9340	1.55425	0.777125	−	0.629346i	$-0.216677\pi$
$449$	32.9340	1.55425	0.777125	+	0.629346i	$0.216677\pi$
$450$	0	0
$451$	3.36030	0.158230
$452$	0	0
$453$	3.03107	0.142412
$454$	0	0
$455$	0	0
$456$	0	0
$457$	17.9433	0.839350	0.419675	−	0.907674i	$-0.362144\pi$
$457$	17.9433	0.839350	0.419675	+	0.907674i	$0.362144\pi$
$458$	0	0
$459$	5.41986	0.252978
$460$	0	0
$461$	−15.8392	−0.737703	−0.368851	−	0.929488i	$-0.620249\pi$
$461$	−15.8392	−0.737703	−0.368851	+	0.929488i	$0.620249\pi$
$462$	0	0
$463$	−24.9003	−1.15721	−0.578607	−	0.815607i	$-0.696403\pi$
$463$	−24.9003	−1.15721	−0.578607	+	0.815607i	$0.696403\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	11.5642	0.535129	0.267565	−	0.963540i	$-0.413781\pi$
$467$	11.5642	0.535129	0.267565	+	0.963540i	$0.413781\pi$
$468$	0	0
$469$	18.3794	0.848682
$470$	0	0
$471$	−1.27521	−0.0587586
$472$	0	0
$473$	−28.8531	−1.32667
$474$	0	0
$475$	0	0
$476$	0	0
$477$	38.8209	1.77749
$478$	0	0
$479$	−16.3751	−0.748195	−0.374098	−	0.927389i	$-0.622048\pi$
$479$	−16.3751	−0.748195	−0.374098	+	0.927389i	$0.622048\pi$
$480$	0	0
$481$	−1.88878	−0.0861209
$482$	0	0
$483$	1.85273	0.0843020
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−7.91481	−0.358654	−0.179327	−	0.983789i	$-0.557392\pi$
$487$	−7.91481	−0.358654	−0.179327	+	0.983789i	$0.557392\pi$
$488$	0	0
$489$	−1.38919	−0.0628214
$490$	0	0
$491$	18.2741	0.824698	0.412349	−	0.911026i	$-0.364708\pi$
$491$	18.2741	0.824698	0.412349	+	0.911026i	$0.364708\pi$
$492$	0	0
$493$	1.56147	0.0703250
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−0.251687	−0.0112897
$498$	0	0
$499$	14.6283	0.654852	0.327426	−	0.944877i	$-0.393819\pi$
$499$	14.6283	0.654852	0.327426	+	0.944877i	$0.393819\pi$
$500$	0	0
$501$	−0.280184	−0.0125177
$502$	0	0
$503$	11.3389	0.505576	0.252788	−	0.967522i	$-0.418653\pi$
$503$	11.3389	0.505576	0.252788	+	0.967522i	$0.418653\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.24110	0.0551194
$508$	0	0
$509$	30.8935	1.36933	0.684666	−	0.728857i	$-0.259948\pi$
$509$	30.8935	1.36933	0.684666	+	0.728857i	$0.259948\pi$
$510$	0	0
$511$	13.4481	0.594911
$512$	0	0
$513$	5.77565	0.255001
$514$	0	0
$515$	0	0
$516$	0	0
$517$	11.9611	0.526050
$518$	0	0
$519$	−1.99553	−0.0875943
$520$	0	0
$521$	−36.1121	−1.58210	−0.791050	−	0.611751i	$-0.790465\pi$
$521$	−36.1121	−1.58210	−0.791050	+	0.611751i	$0.790465\pi$
$522$	0	0
$523$	41.8931	1.83186	0.915929	−	0.401339i	$-0.131455\pi$
$523$	41.8931	1.83186	0.915929	+	0.401339i	$0.131455\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	26.0883	1.13643
$528$	0	0
$529$	32.4153	1.40936
$530$	0	0
$531$	−14.8938	−0.646334
$532$	0	0
$533$	2.99431	0.129698
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0.634974	0.0274012
$538$	0	0
$539$	14.4297	0.621533
$540$	0	0
$541$	−5.91977	−0.254511	−0.127255	−	0.991870i	$-0.540617\pi$
$541$	−5.91977	−0.254511	−0.127255	+	0.991870i	$0.540617\pi$
$542$	0	0
$543$	1.77004	0.0759598
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−11.2774	−0.482187	−0.241094	−	0.970502i	$-0.577506\pi$
$547$	−11.2774	−0.482187	−0.241094	+	0.970502i	$0.577506\pi$
$548$	0	0
$549$	14.0324	0.598889
$550$	0	0
$551$	1.66397	0.0708876
$552$	0	0
$553$	22.2652	0.946813
$554$	0	0
$555$	0	0
$556$	0	0
$557$	43.0087	1.82234	0.911168	−	0.412035i	$-0.135182\pi$
$557$	43.0087	1.82234	0.911168	+	0.412035i	$0.135182\pi$
$558$	0	0
$559$	−25.7105	−1.08744
$560$	0	0
$561$	2.54332	0.107379
$562$	0	0
$563$	10.1466	0.427628	0.213814	−	0.976874i	$-0.431411\pi$
$563$	10.1466	0.427628	0.213814	+	0.976874i	$0.431411\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	11.8199	0.496390
$568$	0	0
$569$	6.62797	0.277859	0.138929	−	0.990302i	$-0.455634\pi$
$569$	6.62797	0.277859	0.138929	+	0.990302i	$0.455634\pi$
$570$	0	0
$571$	−24.8458	−1.03976	−0.519881	−	0.854239i	$-0.674024\pi$
$571$	−24.8458	−1.03976	−0.519881	+	0.854239i	$0.674024\pi$
$572$	0	0
$573$	0.367368	0.0153470
$574$	0	0
$575$	0	0
$576$	0	0
$577$	6.18017	0.257284	0.128642	−	0.991691i	$-0.458938\pi$
$577$	6.18017	0.257284	0.128642	+	0.991691i	$0.458938\pi$
$578$	0	0
$579$	2.41621	0.100414
$580$	0	0
$581$	−16.5612	−0.687074
$582$	0	0
$583$	36.6402	1.51748
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−22.8408	−0.942739	−0.471369	−	0.881936i	$-0.656240\pi$
$587$	−22.8408	−0.942739	−0.471369	+	0.881936i	$0.656240\pi$
$588$	0	0
$589$	27.8009	1.14552
$590$	0	0
$591$	−1.02676	−0.0422351
$592$	0	0
$593$	23.4225	0.961849	0.480924	−	0.876762i	$-0.340301\pi$
$593$	23.4225	0.961849	0.480924	+	0.876762i	$0.340301\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	3.30581	0.135298
$598$	0	0
$599$	−31.8909	−1.30303	−0.651513	−	0.758638i	$-0.725865\pi$
$599$	−31.8909	−1.30303	−0.651513	+	0.758638i	$0.725865\pi$
$600$	0	0
$601$	−9.75803	−0.398038	−0.199019	−	0.979996i	$-0.563776\pi$
$601$	−9.75803	−0.398038	−0.199019	+	0.979996i	$0.563776\pi$
$602$	0	0
$603$	−40.1265	−1.63408
$604$	0	0
$605$	0	0
$606$	0	0
$607$	32.2454	1.30880	0.654400	−	0.756149i	$-0.272921\pi$
$607$	32.2454	1.30880	0.654400	+	0.756149i	$0.272921\pi$
$608$	0	0
$609$	−0.0783640	−0.00317547
$610$	0	0
$611$	10.6584	0.431192
$612$	0	0
$613$	−11.1070	−0.448608	−0.224304	−	0.974519i	$-0.572011\pi$
$613$	−11.1070	−0.448608	−0.224304	+	0.974519i	$0.572011\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−3.74784	−0.150882	−0.0754412	−	0.997150i	$-0.524037\pi$
$617$	−3.74784	−0.150882	−0.0754412	+	0.997150i	$0.524037\pi$
$618$	0	0
$619$	17.6627	0.709925	0.354962	−	0.934881i	$-0.384494\pi$
$619$	17.6627	0.709925	0.354962	+	0.934881i	$0.384494\pi$
$620$	0	0
$621$	−8.13561	−0.326471
$622$	0	0
$623$	−17.2441	−0.690871
$624$	0	0
$625$	0	0
$626$	0	0
$627$	2.71028	0.108238
$628$	0	0
$629$	3.75445	0.149700
$630$	0	0
$631$	−9.29739	−0.370123	−0.185062	−	0.982727i	$-0.559248\pi$
$631$	−9.29739	−0.370123	−0.185062	+	0.982727i	$0.559248\pi$
$632$	0	0
$633$	3.64712	0.144960
$634$	0	0
$635$	0	0
$636$	0	0
$637$	12.8581	0.509457
$638$	0	0
$639$	0.549490	0.0217375
$640$	0	0
$641$	−22.6418	−0.894297	−0.447149	−	0.894460i	$-0.647561\pi$
$641$	−22.6418	−0.894297	−0.447149	+	0.894460i	$0.647561\pi$
$642$	0	0
$643$	−34.9906	−1.37989	−0.689947	−	0.723859i	$-0.742366\pi$
$643$	−34.9906	−1.37989	−0.689947	+	0.723859i	$0.742366\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−40.5960	−1.59599	−0.797997	−	0.602661i	$-0.794107\pi$
$647$	−40.5960	−1.59599	−0.797997	+	0.602661i	$0.794107\pi$
$648$	0	0
$649$	−14.0571	−0.551790
$650$	0	0
$651$	−1.30927	−0.0513144
$652$	0	0
$653$	28.1176	1.10033	0.550164	−	0.835057i	$-0.314565\pi$
$653$	28.1176	1.10033	0.550164	+	0.835057i	$0.314565\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−29.3604	−1.14546
$658$	0	0
$659$	12.9906	0.506041	0.253020	−	0.967461i	$-0.418576\pi$
$659$	12.9906	0.506041	0.253020	+	0.967461i	$0.418576\pi$
$660$	0	0
$661$	25.4514	0.989945	0.494972	−	0.868909i	$-0.335178\pi$
$661$	25.4514	0.989945	0.494972	+	0.868909i	$0.335178\pi$
$662$	0	0
$663$	2.26632	0.0880164
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−2.34388	−0.0907553
$668$	0	0
$669$	4.07615	0.157593
$670$	0	0
$671$	13.2442	0.511286
$672$	0	0
$673$	−27.2429	−1.05014	−0.525068	−	0.851061i	$-0.675960\pi$
$673$	−27.2429	−1.05014	−0.525068	+	0.851061i	$0.675960\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−34.6013	−1.32984	−0.664918	−	0.746916i	$-0.731534\pi$
$677$	−34.6013	−1.32984	−0.664918	+	0.746916i	$0.731534\pi$
$678$	0	0
$679$	−13.8923	−0.533138
$680$	0	0
$681$	−4.69896	−0.180065
$682$	0	0
$683$	−9.59321	−0.367074	−0.183537	−	0.983013i	$-0.558755\pi$
$683$	−9.59321	−0.367074	−0.183537	+	0.983013i	$0.558755\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0.673058	0.0256788
$688$	0	0
$689$	32.6495	1.24385
$690$	0	0
$691$	−22.9093	−0.871510	−0.435755	−	0.900065i	$-0.643519\pi$
$691$	−22.9093	−0.871510	−0.435755	+	0.900065i	$0.643519\pi$
$692$	0	0
$693$	11.2850	0.428683
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−5.95199	−0.225448
$698$	0	0
$699$	1.32321	0.0500485
$700$	0	0
$701$	5.76903	0.217893	0.108947	−	0.994048i	$-0.465252\pi$
$701$	5.76903	0.217893	0.108947	+	0.994048i	$0.465252\pi$
$702$	0	0
$703$	4.00091	0.150897
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−18.3841	−0.691403
$708$	0	0
$709$	−40.9457	−1.53775	−0.768873	−	0.639401i	$-0.779182\pi$
$709$	−40.9457	−1.53775	−0.768873	+	0.639401i	$0.779182\pi$
$710$	0	0
$711$	−48.6101	−1.82302
$712$	0	0
$713$	−39.1605	−1.46657
$714$	0	0
$715$	0	0
$716$	0	0
$717$	3.37069	0.125881
$718$	0	0
$719$	−4.53058	−0.168962	−0.0844810	−	0.996425i	$-0.526923\pi$
$719$	−4.53058	−0.168962	−0.0844810	+	0.996425i	$0.526923\pi$
$720$	0	0
$721$	13.2943	0.495106
$722$	0	0
$723$	2.31751	0.0861891
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−1.05793	−0.0392366	−0.0196183	−	0.999808i	$-0.506245\pi$
$727$	−1.05793	−0.0392366	−0.0196183	+	0.999808i	$0.506245\pi$
$728$	0	0
$729$	−25.2050	−0.933520
$730$	0	0
$731$	51.1065	1.89024
$732$	0	0
$733$	−22.9676	−0.848329	−0.424164	−	0.905585i	$-0.639432\pi$
$733$	−22.9676	−0.848329	−0.424164	+	0.905585i	$0.639432\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−37.8724	−1.39505
$738$	0	0
$739$	−11.7735	−0.433095	−0.216547	−	0.976272i	$-0.569480\pi$
$739$	−11.7735	−0.433095	−0.216547	+	0.976272i	$0.569480\pi$
$740$	0	0
$741$	2.41509	0.0887205
$742$	0	0
$743$	37.0509	1.35926	0.679632	−	0.733553i	$-0.262139\pi$
$743$	37.0509	1.35926	0.679632	+	0.733553i	$0.262139\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	36.1569	1.32291
$748$	0	0
$749$	−10.8184	−0.395296
$750$	0	0
$751$	−30.8637	−1.12623	−0.563115	−	0.826378i	$-0.690397\pi$
$751$	−30.8637	−1.12623	−0.563115	+	0.826378i	$0.690397\pi$
$752$	0	0
$753$	−0.763247	−0.0278143
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−48.9002	−1.77731	−0.888654	−	0.458578i	$-0.848359\pi$
$757$	−48.9002	−1.77731	−0.888654	+	0.458578i	$0.848359\pi$
$758$	0	0
$759$	−3.81771	−0.138574
$760$	0	0
$761$	−39.4445	−1.42986	−0.714931	−	0.699195i	$-0.753542\pi$
$761$	−39.4445	−1.42986	−0.714931	+	0.699195i	$0.753542\pi$
$762$	0	0
$763$	1.75369	0.0634878
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−12.5261	−0.452291
$768$	0	0
$769$	17.2290	0.621293	0.310646	−	0.950526i	$-0.399455\pi$
$769$	17.2290	0.621293	0.310646	+	0.950526i	$0.399455\pi$
$770$	0	0
$771$	−0.997053	−0.0359080
$772$	0	0
$773$	−4.09644	−0.147339	−0.0736693	−	0.997283i	$-0.523471\pi$
$773$	−4.09644	−0.147339	−0.0736693	+	0.997283i	$0.523471\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−0.188421	−0.00675957
$778$	0	0
$779$	−6.34270	−0.227251
$780$	0	0
$781$	0.518624	0.0185578
$782$	0	0
$783$	0.344108	0.0122974
$784$	0	0
$785$	0	0
$786$	0	0
$787$	7.42330	0.264612	0.132306	−	0.991209i	$-0.457762\pi$
$787$	7.42330	0.264612	0.132306	+	0.991209i	$0.457762\pi$
$788$	0	0
$789$	−0.669873	−0.0238481
$790$	0	0
$791$	15.7362	0.559514
$792$	0	0
$793$	11.8017	0.419090
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−7.54949	−0.267417	−0.133708	−	0.991021i	$-0.542689\pi$
$797$	−7.54949	−0.267417	−0.133708	+	0.991021i	$0.542689\pi$
$798$	0	0
$799$	−21.1864	−0.749520
$800$	0	0
$801$	37.6479	1.33022
$802$	0	0
$803$	−27.7111	−0.977904
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.78788	0.0629364
$808$	0	0
$809$	14.5859	0.512812	0.256406	−	0.966569i	$-0.417462\pi$
$809$	14.5859	0.512812	0.256406	+	0.966569i	$0.417462\pi$
$810$	0	0
$811$	8.42274	0.295763	0.147881	−	0.989005i	$-0.452755\pi$
$811$	8.42274	0.295763	0.147881	+	0.989005i	$0.452755\pi$
$812$	0	0
$813$	−2.68124	−0.0940353
$814$	0	0
$815$	0	0
$816$	0	0
$817$	54.4614	1.90536
$818$	0	0
$819$	10.0559	0.351382
$820$	0	0
$821$	−10.4485	−0.364656	−0.182328	−	0.983238i	$-0.558363\pi$
$821$	−10.4485	−0.364656	−0.182328	+	0.983238i	$0.558363\pi$
$822$	0	0
$823$	−31.2866	−1.09058	−0.545292	−	0.838246i	$-0.683581\pi$
$823$	−31.2866	−1.09058	−0.545292	+	0.838246i	$0.683581\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−16.4655	−0.572562	−0.286281	−	0.958146i	$-0.592419\pi$
$827$	−16.4655	−0.572562	−0.286281	+	0.958146i	$0.592419\pi$
$828$	0	0
$829$	15.2643	0.530153	0.265076	−	0.964227i	$-0.414603\pi$
$829$	15.2643	0.530153	0.265076	+	0.964227i	$0.414603\pi$
$830$	0	0
$831$	−1.02563	−0.0355788
$832$	0	0
$833$	−25.5589	−0.885564
$834$	0	0
$835$	0	0
$836$	0	0
$837$	5.74921	0.198722
$838$	0	0
$839$	−43.3558	−1.49681	−0.748404	−	0.663243i	$-0.769180\pi$
$839$	−43.3558	−1.49681	−0.748404	+	0.663243i	$0.769180\pi$
$840$	0	0
$841$	−28.9009	−0.996581
$842$	0	0
$843$	5.38489	0.185466
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−4.29504	−0.147579
$848$	0	0
$849$	0.0712551	0.00244547
$850$	0	0
$851$	−5.63570	−0.193189
$852$	0	0
$853$	−37.9829	−1.30051	−0.650254	−	0.759717i	$-0.725338\pi$
$853$	−37.9829	−1.30051	−0.650254	+	0.759717i	$0.725338\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	40.1917	1.37292	0.686462	−	0.727166i	$-0.259163\pi$
$857$	40.1917	1.37292	0.686462	+	0.727166i	$0.259163\pi$
$858$	0	0
$859$	26.1994	0.893913	0.446957	−	0.894556i	$-0.352508\pi$
$859$	26.1994	0.893913	0.446957	+	0.894556i	$0.352508\pi$
$860$	0	0
$861$	0.298707	0.0101799
$862$	0	0
$863$	−6.91217	−0.235293	−0.117646	−	0.993056i	$-0.537535\pi$
$863$	−6.91217	−0.235293	−0.117646	+	0.993056i	$0.537535\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−1.39098	−0.0472402
$868$	0	0
$869$	−45.8795	−1.55636
$870$	0	0
$871$	−33.7475	−1.14349
$872$	0	0
$873$	30.3301	1.02652
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−19.9630	−0.674103	−0.337052	−	0.941486i	$-0.609430\pi$
$877$	−19.9630	−0.674103	−0.337052	+	0.941486i	$0.609430\pi$
$878$	0	0
$879$	−1.22023	−0.0411572
$880$	0	0
$881$	28.7912	0.969999	0.484999	−	0.874515i	$-0.338820\pi$
$881$	28.7912	0.969999	0.484999	+	0.874515i	$0.338820\pi$
$882$	0	0
$883$	−23.6895	−0.797214	−0.398607	−	0.917122i	$-0.630506\pi$
$883$	−23.6895	−0.797214	−0.398607	+	0.917122i	$0.630506\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	12.0349	0.404094	0.202047	−	0.979376i	$-0.435241\pi$
$887$	12.0349	0.404094	0.202047	+	0.979376i	$0.435241\pi$
$888$	0	0
$889$	−23.0417	−0.772793
$890$	0	0
$891$	−24.3560	−0.815957
$892$	0	0
$893$	−22.5771	−0.755515
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−3.40190	−0.113586
$898$	0	0
$899$	1.65635	0.0552425
$900$	0	0
$901$	−64.8996	−2.16212
$902$	0	0
$903$	−2.56484	−0.0853524
$904$	0	0
$905$	0	0
$906$	0	0
$907$	31.2080	1.03624	0.518122	−	0.855307i	$-0.326631\pi$
$907$	31.2080	1.03624	0.518122	+	0.855307i	$0.326631\pi$
$908$	0	0
$909$	40.1366	1.33125
$910$	0	0
$911$	−43.0992	−1.42794	−0.713971	−	0.700176i	$-0.753105\pi$
$911$	−43.0992	−1.42794	−0.713971	+	0.700176i	$0.753105\pi$
$912$	0	0
$913$	34.1258	1.12940
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−11.3009	−0.373189
$918$	0	0
$919$	−28.7455	−0.948227	−0.474114	−	0.880464i	$-0.657231\pi$
$919$	−28.7455	−0.948227	−0.474114	+	0.880464i	$0.657231\pi$
$920$	0	0
$921$	−1.05209	−0.0346674
$922$	0	0
$923$	0.462137	0.0152114
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−29.0246	−0.953292
$928$	0	0
$929$	4.56464	0.149761	0.0748804	−	0.997193i	$-0.476142\pi$
$929$	4.56464	0.149761	0.0748804	+	0.997193i	$0.476142\pi$
$930$	0	0
$931$	−27.2367	−0.892648
$932$	0	0
$933$	−0.939974	−0.0307734
$934$	0	0
$935$	0	0
$936$	0	0
$937$	50.1341	1.63781	0.818905	−	0.573929i	$-0.194581\pi$
$937$	50.1341	1.63781	0.818905	+	0.573929i	$0.194581\pi$
$938$	0	0
$939$	2.94175	0.0960003
$940$	0	0
$941$	15.4131	0.502454	0.251227	−	0.967928i	$-0.419166\pi$
$941$	15.4131	0.502454	0.251227	+	0.967928i	$0.419166\pi$
$942$	0	0
$943$	8.93436	0.290943
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.85762	−0.0603645	−0.0301823	−	0.999544i	$-0.509609\pi$
$947$	−1.85762	−0.0603645	−0.0301823	+	0.999544i	$0.509609\pi$
$948$	0	0
$949$	−24.6929	−0.801567
$950$	0	0
$951$	−3.05247	−0.0989830
$952$	0	0
$953$	−14.6001	−0.472944	−0.236472	−	0.971638i	$-0.575991\pi$
$953$	−14.6001	−0.472944	−0.236472	+	0.971638i	$0.575991\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0.161476	0.00521979
$958$	0	0
$959$	−12.0061	−0.387697
$960$	0	0
$961$	−3.32636	−0.107302
$962$	0	0
$963$	23.6191	0.761115
$964$	0	0
$965$	0	0
$966$	0	0
$967$	29.6896	0.954754	0.477377	−	0.878699i	$-0.341588\pi$
$967$	29.6896	0.954754	0.477377	+	0.878699i	$0.341588\pi$
$968$	0	0
$969$	−4.80063	−0.154218
$970$	0	0
$971$	47.5397	1.52562	0.762812	−	0.646621i	$-0.223818\pi$
$971$	47.5397	1.52562	0.762812	+	0.646621i	$0.223818\pi$
$972$	0	0
$973$	18.8602	0.604630
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−19.8872	−0.636248	−0.318124	−	0.948049i	$-0.603053\pi$
$977$	−19.8872	−0.636248	−0.318124	+	0.948049i	$0.603053\pi$
$978$	0	0
$979$	35.5331	1.13564
$980$	0	0
$981$	−3.82871	−0.122241
$982$	0	0
$983$	−47.7887	−1.52422	−0.762111	−	0.647446i	$-0.775837\pi$
$983$	−47.7887	−1.52422	−0.762111	+	0.647446i	$0.775837\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1.06326	0.0338440
$988$	0	0
$989$	−76.7146	−2.43938
$990$	0	0
$991$	−12.1198	−0.384997	−0.192499	−	0.981297i	$-0.561659\pi$
$991$	−12.1198	−0.384997	−0.192499	+	0.981297i	$0.561659\pi$
$992$	0	0
$993$	3.95708	0.125574
$994$	0	0
$995$	0	0
$996$	0	0
$997$	25.6615	0.812709	0.406355	−	0.913715i	$-0.366800\pi$
$997$	25.6615	0.812709	0.406355	+	0.913715i	$0.366800\pi$
$998$	0	0
$999$	0.827387	0.0261774

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10000.2.a.bi.1.4		8
4.3	odd	2		2500.2.a.f.1.5		8
5.4	even	2	inner	10000.2.a.bi.1.5		8
20.3	even	4		2500.2.c.b.1249.5		8
20.7	even	4		2500.2.c.b.1249.4		8
20.19	odd	2		2500.2.a.f.1.4		8
25.2	odd	20		400.2.y.b.129.2		8
25.13	odd	20		400.2.y.b.369.2		8
100.11	odd	10		500.2.g.b.101.2		16
100.23	even	20		500.2.i.a.149.2		8
100.27	even	20		100.2.i.a.29.1	✓	8
100.39	odd	10		500.2.g.b.101.3		16
100.59	odd	10		500.2.g.b.401.3		16
100.63	even	20		100.2.i.a.69.1	yes	8
100.87	even	20		500.2.i.a.349.2		8
100.91	odd	10		500.2.g.b.401.2		16
300.227	odd	20		900.2.w.a.829.1		8
300.263	odd	20		900.2.w.a.469.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
100.2.i.a.29.1	✓	8	100.27	even	20
100.2.i.a.69.1	yes	8	100.63	even	20
400.2.y.b.129.2		8	25.2	odd	20
400.2.y.b.369.2		8	25.13	odd	20
500.2.g.b.101.2		16	100.11	odd	10
500.2.g.b.101.3		16	100.39	odd	10
500.2.g.b.401.2		16	100.91	odd	10
500.2.g.b.401.3		16	100.59	odd	10
500.2.i.a.149.2		8	100.23	even	20
500.2.i.a.349.2		8	100.87	even	20
900.2.w.a.469.1		8	300.263	odd	20
900.2.w.a.829.1		8	300.227	odd	20
2500.2.a.f.1.4		8	20.19	odd	2
2500.2.a.f.1.5		8	4.3	odd	2
2500.2.c.b.1249.4		8	20.7	even	4
2500.2.c.b.1249.5		8	20.3	even	4
10000.2.a.bi.1.4		8	1.1	even	1	trivial
10000.2.a.bi.1.5		8	5.4	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:	\( N \)	\(=\)	\( 10000 = 2^{4} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	10000.a (trivial)

\(f(q)\)	\(=\)	\(q-0.183172 q^{3} +1.35874 q^{7} -2.96645 q^{9} +O(q^{10})\)	\(q-0.183172 q^{3} +1.35874 q^{7} -2.96645 q^{9} -2.79981 q^{11} -2.49487 q^{13} +4.95922 q^{17} +5.28477 q^{19} -0.248884 q^{21} -7.44414 q^{23} +1.09289 q^{27} +0.314862 q^{29} +5.26057 q^{31} +0.512848 q^{33} +0.757065 q^{37} +0.456991 q^{39} -1.20019 q^{41} +10.3054 q^{43} -4.27212 q^{47} -5.15382 q^{49} -0.908390 q^{51} -13.0867 q^{53} -0.968021 q^{57} +5.02074 q^{59} -4.73038 q^{61} -4.03064 q^{63} +13.5268 q^{67} +1.36356 q^{69} -0.185235 q^{71} +9.89749 q^{73} -3.80423 q^{77} +16.3866 q^{79} +8.69916 q^{81} -12.1886 q^{83} -0.0576739 q^{87} -12.6912 q^{89} -3.38989 q^{91} -0.963590 q^{93} -10.2244 q^{97} +8.30550 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{9}+O(q^{10}) \) 8 * q - 2 * q^9	\( 8 q - 2 q^{9} + 10 q^{11} + 12 q^{19} - 22 q^{21} - 32 q^{29} + 2 q^{31} + 2 q^{39} - 42 q^{41} - 14 q^{49} + 14 q^{51} + 24 q^{59} - 34 q^{61} - 36 q^{69} - 4 q^{71} - 4 q^{79} - 28 q^{81} - 58 q^{89} + 18 q^{91} + 20 q^{99}+O(q^{100}) \) 8 * q - 2 * q^9 + 10 * q^11 + 12 * q^19 - 22 * q^21 - 32 * q^29 + 2 * q^31 + 2 * q^39 - 42 * q^41 - 14 * q^49 + 14 * q^51 + 24 * q^59 - 34 * q^61 - 36 * q^69 - 4 * q^71 - 4 * q^79 - 28 * q^81 - 58 * q^89 + 18 * q^91 + 20 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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