LMFDB - Embedded newform 2300.3.d.c.1149.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2300,3,Mod(1149,2300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2300, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2300.1149");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2300 = 2^{2} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2300.d (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:	$62.6704608029$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1149.2
Character	$\chi$	$=$	2300.1149
Dual form			2300.3.d.c.1149.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.43352i q^{3} +12.3431 q^{7} -20.5231 q^{9} +O(q^{10})$	$q+5.43352i q^{3} +12.3431 q^{7} -20.5231 q^{9} +0.283034i q^{11} +18.2116i q^{13} +24.7396 q^{17} -5.66988i q^{19} +67.0662i q^{21} +(22.2237 + 5.92519i) q^{23} -62.6111i q^{27} +36.9609 q^{29} -13.8040 q^{31} -1.53787 q^{33} +48.0466 q^{37} -98.9528 q^{39} -30.8988 q^{41} +10.2166 q^{43} +16.8186i q^{47} +103.351 q^{49} +134.423i q^{51} -20.2654 q^{53} +30.8074 q^{57} +12.1433 q^{59} -109.987i q^{61} -253.318 q^{63} -46.9397 q^{67} +(-32.1946 + 120.753i) q^{69} +110.708 q^{71} +102.561i q^{73} +3.49351i q^{77} +88.9924i q^{79} +155.491 q^{81} +58.8907 q^{83} +200.828i q^{87} +150.764i q^{89} +224.786i q^{91} -75.0045i q^{93} +37.2144 q^{97} -5.80875i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 128 q^{9}+O(q^{10}) $ 32 * q - 128 * q^9	$ 32 q - 128 q^{9} + 12 q^{29} + 56 q^{31} - 148 q^{39} - 180 q^{41} + 380 q^{49} + 348 q^{59} - 100 q^{69} + 232 q^{71} + 112 q^{81}+O(q^{100}) $ 32 * q - 128 * q^9 + 12 * q^29 + 56 * q^31 - 148 * q^39 - 180 * q^41 + 380 * q^49 + 348 * q^59 - 100 * q^69 + 232 * q^71 + 112 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2300\mathbb{Z}\right)^\times$.

$n$	$277$	$1151$	$1201$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$			5.43352i			1.81117i	0.424162	+	0.905586i	$0.360569\pi$
$3$			5.43352i			1.81117i	−0.424162	+	0.905586i	$0.639431\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	12.3431			1.76329			0.881647	−	0.471910i	$-0.156435\pi$
$7$	12.3431			1.76329			0.881647	+	0.471910i	$0.156435\pi$
$8$		0			0
$9$	−20.5231			−2.28035
$10$		0			0
$11$			0.283034i			0.0257304i	0.999917	+	0.0128652i	$0.00409523\pi$
$11$			0.283034i			0.0257304i	−0.999917	+	0.0128652i	$0.995905\pi$
$12$		0			0
$13$			18.2116i			1.40089i	0.713707	+	0.700444i	$0.247015\pi$
$13$			18.2116i			1.40089i	−0.713707	+	0.700444i	$0.752985\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	24.7396			1.45527			0.727636	−	0.685963i	$-0.240619\pi$
$17$	24.7396			1.45527			0.727636	+	0.685963i	$0.240619\pi$
$18$		0			0
$19$		−	5.66988i		−	0.298415i	−0.988806	−	0.149207i	$-0.952328\pi$
$19$		−	5.66988i		−	0.298415i	0.988806	−	0.149207i	$-0.0476722\pi$
$20$		0			0
$21$			67.0662i			3.19363i
$22$		0			0
$23$	22.2237	+	5.92519i	0.966247	+	0.257617i
$23$	22.2237	+	5.92519i	0.966247	+	0.257617i
$24$		0			0
$25$		0			0
$26$		0			0
$27$		−	62.6111i		−	2.31893i
$28$		0			0
$29$	36.9609			1.27451			0.637257	−	0.770651i	$-0.280069\pi$
$29$	36.9609			1.27451			0.637257	+	0.770651i	$0.280069\pi$
$30$		0			0
$31$	−13.8040			−0.445292			−0.222646	−	0.974899i	$-0.571469\pi$
$31$	−13.8040			−0.445292			−0.222646	+	0.974899i	$0.571469\pi$
$32$		0			0
$33$	−1.53787			−0.0466022
$34$		0			0
$35$		0			0
$36$		0			0
$37$	48.0466			1.29856			0.649278	−	0.760551i	$-0.275071\pi$
$37$	48.0466			1.29856			0.649278	+	0.760551i	$0.275071\pi$
$38$		0			0
$39$	−98.9528			−2.53725
$40$		0			0
$41$	−30.8988			−0.753629			−0.376815	−	0.926289i	$-0.622981\pi$
$41$	−30.8988			−0.753629			−0.376815	+	0.926289i	$0.622981\pi$
$42$		0			0
$43$	10.2166			0.237596			0.118798	−	0.992918i	$-0.462096\pi$
$43$	10.2166			0.237596			0.118798	+	0.992918i	$0.462096\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			16.8186i			0.357842i	0.983863	+	0.178921i	$0.0572606\pi$
$47$			16.8186i			0.357842i	−0.983863	+	0.178921i	$0.942739\pi$
$48$		0			0
$49$	103.351			2.10920
$50$		0			0
$51$			134.423i			2.63575i
$52$		0			0
$53$	−20.2654			−0.382367			−0.191183	−	0.981554i	$-0.561232\pi$
$53$	−20.2654			−0.382367			−0.191183	+	0.981554i	$0.561232\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	30.8074			0.540481
$58$		0			0
$59$	12.1433			0.205818			0.102909	−	0.994691i	$-0.467185\pi$
$59$	12.1433			0.205818			0.102909	+	0.994691i	$0.467185\pi$
$60$		0			0
$61$		−	109.987i		−	1.80306i	−0.432717	−	0.901530i	$-0.642445\pi$
$61$		−	109.987i		−	1.80306i	0.432717	−	0.901530i	$-0.357555\pi$
$62$		0			0
$63$	−253.318			−4.02092
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−46.9397			−0.700592			−0.350296	−	0.936639i	$-0.613919\pi$
$67$	−46.9397			−0.700592			−0.350296	+	0.936639i	$0.613919\pi$
$68$		0			0
$69$	−32.1946	+	120.753i	−0.466589	+	1.75004i
$70$		0			0
$71$	110.708			1.55927			0.779634	−	0.626236i	$-0.215405\pi$
$71$	110.708			1.55927			0.779634	+	0.626236i	$0.215405\pi$
$72$		0			0
$73$			102.561i			1.40494i	0.711714	+	0.702469i	$0.247919\pi$
$73$			102.561i			1.40494i	−0.711714	+	0.702469i	$0.752081\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			3.49351i			0.0453702i
$78$		0			0
$79$			88.9924i			1.12649i	0.826291	+	0.563243i	$0.190447\pi$
$79$			88.9924i			1.12649i	−0.826291	+	0.563243i	$0.809553\pi$
$80$		0			0
$81$	155.491			1.91964
$82$		0			0
$83$	58.8907			0.709526			0.354763	−	0.934956i	$-0.384562\pi$
$83$	58.8907			0.709526			0.354763	+	0.934956i	$0.384562\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$			200.828i			2.30837i
$88$		0			0
$89$			150.764i			1.69398i	0.531612	+	0.846988i	$0.321586\pi$
$89$			150.764i			1.69398i	−0.531612	+	0.846988i	$0.678414\pi$
$90$		0			0
$91$			224.786i			2.47018i
$92$		0			0
$93$		−	75.0045i		−	0.806500i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	37.2144			0.383654			0.191827	−	0.981429i	$-0.438559\pi$
$97$	37.2144			0.383654			0.191827	+	0.981429i	$0.438559\pi$
$98$		0			0
$99$		−	5.80875i		−	0.0586742i
$100$		0			0
$101$	−102.910			−1.01891			−0.509456	−	0.860497i	$-0.670153\pi$
$101$	−102.910			−1.01891			−0.509456	+	0.860497i	$0.670153\pi$
$102$		0			0
$103$	−141.910			−1.37777			−0.688884	−	0.724872i	$-0.741899\pi$
$103$	−141.910			−1.37777			−0.688884	+	0.724872i	$0.741899\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	80.5269			0.752588			0.376294	−	0.926500i	$-0.377198\pi$
$107$	80.5269			0.752588			0.376294	+	0.926500i	$0.377198\pi$
$108$		0			0
$109$		−	189.748i		−	1.74081i	−0.492338	−	0.870404i	$-0.663858\pi$
$109$		−	189.748i		−	1.74081i	0.492338	−	0.870404i	$-0.336142\pi$
$110$		0			0
$111$			261.062i			2.35191i
$112$		0			0
$113$	−52.8872			−0.468029			−0.234014	−	0.972233i	$-0.575186\pi$
$113$	−52.8872			−0.468029			−0.234014	+	0.972233i	$0.575186\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		−	373.758i		−	3.19451i
$118$		0			0
$119$	305.363			2.56607
$120$		0			0
$121$	120.920			0.999338
$122$		0			0
$123$		−	167.889i		−	1.36495i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	144.493i		−	1.13774i	−0.822427	−	0.568870i	$-0.807381\pi$
$127$		−	144.493i		−	1.13774i	0.822427	−	0.568870i	$-0.192619\pi$
$128$		0			0
$129$			55.5123i			0.430328i
$130$		0			0
$131$	−98.2044			−0.749652			−0.374826	−	0.927095i	$-0.622298\pi$
$131$	−98.2044			−0.749652			−0.374826	+	0.927095i	$0.622298\pi$
$132$		0			0
$133$		−	69.9837i		−	0.526193i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−118.258			−0.863199			−0.431599	−	0.902065i	$-0.642051\pi$
$137$	−118.258			−0.863199			−0.431599	+	0.902065i	$0.642051\pi$
$138$		0			0
$139$	−19.6663			−0.141484			−0.0707419	−	0.997495i	$-0.522537\pi$
$139$	−19.6663			−0.141484			−0.0707419	+	0.997495i	$0.522537\pi$
$140$		0			0
$141$	−91.3839			−0.648113
$142$		0			0
$143$	−5.15449			−0.0360454
$144$		0			0
$145$		0			0
$146$		0			0
$147$			561.559i			3.82013i
$148$		0			0
$149$		−	179.682i		−	1.20592i	−0.797770	−	0.602961i	$-0.793987\pi$
$149$		−	179.682i		−	1.20592i	0.797770	−	0.602961i	$-0.206013\pi$
$150$		0			0
$151$	77.2560			0.511629			0.255814	−	0.966726i	$-0.417656\pi$
$151$	77.2560			0.511629			0.255814	+	0.966726i	$0.417656\pi$
$152$		0			0
$153$	−507.735			−3.31853
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−106.730			−0.679811			−0.339905	−	0.940460i	$-0.610395\pi$
$157$	−106.730			−0.679811			−0.339905	+	0.940460i	$0.610395\pi$
$158$		0			0
$159$		−	110.113i		−	0.692532i
$160$		0			0
$161$	274.308	+	73.1350i	1.70378	+	0.454254i
$162$		0			0
$163$			158.131i			0.970128i	0.874479	+	0.485064i	$0.161204\pi$
$163$			158.131i			0.970128i	−0.874479	+	0.485064i	$0.838796\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	12.9250i		−	0.0773954i	−0.999251	−	0.0386977i	$-0.987679\pi$
$167$		−	12.9250i		−	0.0773954i	0.999251	−	0.0386977i	$-0.0123209\pi$
$168$		0			0
$169$	−162.661			−0.962489
$170$		0			0
$171$			116.364i			0.680490i
$172$		0			0
$173$		−	188.860i		−	1.09168i	−0.837890	−	0.545839i	$-0.816211\pi$
$173$		−	188.860i		−	1.09168i	0.837890	−	0.545839i	$-0.183789\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$			65.9806i			0.372772i
$178$		0			0
$179$	−278.276			−1.55461			−0.777306	−	0.629123i	$-0.783414\pi$
$179$	−278.276			−1.55461			−0.777306	+	0.629123i	$0.783414\pi$
$180$		0			0
$181$		−	9.26126i		−	0.0511672i	−0.999673	−	0.0255836i	$-0.991856\pi$
$181$		−	9.26126i		−	0.0511672i	0.999673	−	0.0255836i	$-0.00814440\pi$
$182$		0			0
$183$	597.614			3.26565
$184$		0			0
$185$		0			0
$186$		0			0
$187$			7.00216i			0.0374447i
$188$		0			0
$189$		−	772.813i		−	4.08896i
$190$		0			0
$191$			266.850i			1.39712i	0.715550	+	0.698561i	$0.246176\pi$
$191$			266.850i			1.39712i	−0.715550	+	0.698561i	$0.753824\pi$
$192$		0			0
$193$			186.351i			0.965548i	0.875745	+	0.482774i	$0.160371\pi$
$193$			186.351i			0.965548i	−0.875745	+	0.482774i	$0.839629\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			8.45729i			0.0429304i	0.999770	+	0.0214652i	$0.00683311\pi$
$197$			8.45729i			0.0429304i	−0.999770	+	0.0214652i	$0.993167\pi$
$198$		0			0
$199$		−	319.726i		−	1.60666i	−0.595532	−	0.803331i	$-0.703059\pi$
$199$		−	319.726i		−	1.60666i	0.595532	−	0.803331i	$-0.296941\pi$
$200$		0			0
$201$		−	255.048i		−	1.26889i
$202$		0			0
$203$	456.210			2.24734
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−456.099	−	121.603i	−2.20338	−	0.587456i
$208$		0			0
$209$	1.60477			0.00767833
$210$		0			0
$211$	−134.246			−0.636235			−0.318117	−	0.948051i	$-0.603051\pi$
$211$	−134.246			−0.636235			−0.318117	+	0.948051i	$0.603051\pi$
$212$		0			0
$213$			601.534i			2.82410i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−170.384			−0.785180
$218$		0			0
$219$	−557.265			−2.54459
$220$		0			0
$221$			450.547i			2.03867i
$222$		0			0
$223$		−	304.584i		−	1.36585i	−0.730490	−	0.682924i	$-0.760708\pi$
$223$		−	304.584i		−	1.36585i	0.730490	−	0.682924i	$-0.239292\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−382.831			−1.68648			−0.843241	−	0.537536i	$-0.819355\pi$
$227$	−382.831			−1.68648			−0.843241	+	0.537536i	$0.819355\pi$
$228$		0			0
$229$			76.1267i			0.332431i	0.986089	+	0.166215i	$0.0531547\pi$
$229$			76.1267i			0.332431i	−0.986089	+	0.166215i	$0.946845\pi$
$230$		0			0
$231$	−18.9820			−0.0821733
$232$		0			0
$233$			250.390i			1.07464i	0.843380	+	0.537318i	$0.180563\pi$
$233$			250.390i			1.07464i	−0.843380	+	0.537318i	$0.819437\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−483.542			−2.04026
$238$		0			0
$239$	−365.524			−1.52939			−0.764695	−	0.644393i	$-0.777110\pi$
$239$	−365.524			−1.52939			−0.764695	+	0.644393i	$0.777110\pi$
$240$		0			0
$241$		−	245.881i		−	1.02026i	−0.860099	−	0.510128i	$-0.829598\pi$
$241$		−	245.881i		−	1.02026i	0.860099	−	0.510128i	$-0.170402\pi$
$242$		0			0
$243$			281.361i			1.15786i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	103.257			0.418046
$248$		0			0
$249$			319.984i			1.28507i
$250$		0			0
$251$		−	131.566i		−	0.524168i	−0.965045	−	0.262084i	$-0.915590\pi$
$251$		−	131.566i		−	0.524168i	0.965045	−	0.262084i	$-0.0844097\pi$
$252$		0			0
$253$	−1.67703	+	6.29006i	−0.00662858	+	0.0248619i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			355.534i			1.38340i	0.722185	+	0.691700i	$0.243138\pi$
$257$			355.534i			1.38340i	−0.722185	+	0.691700i	$0.756862\pi$
$258$		0			0
$259$	593.041			2.28974
$260$		0			0
$261$	−758.553			−2.90633
$262$		0			0
$263$	−254.503			−0.967692			−0.483846	−	0.875153i	$-0.660761\pi$
$263$	−254.503			−0.967692			−0.483846	+	0.875153i	$0.660761\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−819.178			−3.06808
$268$		0			0
$269$	−153.131			−0.569262			−0.284631	−	0.958637i	$-0.591871\pi$
$269$	−153.131			−0.569262			−0.284631	+	0.958637i	$0.591871\pi$
$270$		0			0
$271$	175.523			0.647686			0.323843	−	0.946111i	$-0.395025\pi$
$271$	175.523			0.647686			0.323843	+	0.946111i	$0.395025\pi$
$272$		0			0
$273$	−1221.38			−4.47392
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	453.306i		−	1.63649i	−0.574873	−	0.818243i	$-0.694948\pi$
$277$		−	453.306i		−	1.63649i	0.574873	−	0.818243i	$-0.305052\pi$
$278$		0			0
$279$	283.302			1.01542
$280$		0			0
$281$		−	400.509i		−	1.42530i	−0.701520	−	0.712650i	$-0.747495\pi$
$281$		−	400.509i		−	1.42530i	0.701520	−	0.712650i	$-0.252505\pi$
$282$		0			0
$283$	416.425			1.47147			0.735734	−	0.677271i	$-0.236837\pi$
$283$	416.425			1.47147			0.735734	+	0.677271i	$0.236837\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−381.386			−1.32887
$288$		0			0
$289$	323.049			1.11782
$290$		0			0
$291$			202.205i			0.694864i
$292$		0			0
$293$	−53.2145			−0.181620			−0.0908098	−	0.995868i	$-0.528946\pi$
$293$	−53.2145			−0.181620			−0.0908098	+	0.995868i	$0.528946\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	17.7211			0.0596670
$298$		0			0
$299$	−107.907	+	404.728i	−0.360893	+	1.35360i
$300$		0			0
$301$	126.104			0.418952
$302$		0			0
$303$		−	559.164i		−	1.84543i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	170.742i		−	0.556162i	−0.960558	−	0.278081i	$-0.910302\pi$
$307$		−	170.742i		−	0.556162i	0.960558	−	0.278081i	$-0.0896984\pi$
$308$		0			0
$309$		−	771.071i		−	2.49538i
$310$		0			0
$311$	−592.486			−1.90510			−0.952550	−	0.304383i	$-0.901550\pi$
$311$	−592.486			−1.90510			−0.952550	+	0.304383i	$0.901550\pi$
$312$		0			0
$313$	183.270			0.585527			0.292764	−	0.956185i	$-0.405425\pi$
$313$	183.270			0.585527			0.292764	+	0.956185i	$0.405425\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			278.672i			0.879092i	0.898220	+	0.439546i	$0.144861\pi$
$317$			278.672i			0.879092i	−0.898220	+	0.439546i	$0.855139\pi$
$318$		0			0
$319$			10.4612i			0.0327937i
$320$		0			0
$321$			437.544i			1.36307i
$322$		0			0
$323$		−	140.271i		−	0.434275i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	1031.00			3.15290
$328$		0			0
$329$			207.592i			0.630980i
$330$		0			0
$331$	252.479			0.762775			0.381388	−	0.924415i	$-0.375446\pi$
$331$	252.479			0.762775			0.381388	+	0.924415i	$0.375446\pi$
$332$		0			0
$333$	−986.066			−2.96116
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−568.229			−1.68614			−0.843070	−	0.537804i	$-0.819254\pi$
$337$	−568.229			−1.68614			−0.843070	+	0.537804i	$0.819254\pi$
$338$		0			0
$339$		−	287.364i		−	0.847681i
$340$		0			0
$341$		−	3.90702i		−	0.0114575i
$342$		0			0
$343$	670.857			1.95585
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	310.626i		−	0.895177i	−0.894240	−	0.447589i	$-0.852283\pi$
$347$		−	310.626i		−	0.895177i	0.894240	−	0.447589i	$-0.147717\pi$
$348$		0			0
$349$	−533.091			−1.52748			−0.763741	−	0.645523i	$-0.776640\pi$
$349$	−533.091			−1.52748			−0.763741	+	0.645523i	$0.776640\pi$
$350$		0			0
$351$	1140.25			3.24856
$352$		0			0
$353$		−	405.742i		−	1.14941i	−0.818361	−	0.574705i	$-0.805117\pi$
$353$		−	405.742i		−	1.14941i	0.818361	−	0.574705i	$-0.194883\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$			1659.19i			4.64760i
$358$		0			0
$359$			512.842i			1.42853i	0.699876	+	0.714265i	$0.253239\pi$
$359$			512.842i			1.42853i	−0.699876	+	0.714265i	$0.746761\pi$
$360$		0			0
$361$	328.852			0.910949
$362$		0			0
$363$			657.021i			1.80997i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	226.315			0.616662			0.308331	−	0.951279i	$-0.400230\pi$
$367$	226.315			0.616662			0.308331	+	0.951279i	$0.400230\pi$
$368$		0			0
$369$	634.140			1.71854
$370$		0			0
$371$	−250.137			−0.674224
$372$		0			0
$373$	60.8146			0.163042			0.0815209	−	0.996672i	$-0.474022\pi$
$373$	60.8146			0.163042			0.0815209	+	0.996672i	$0.474022\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			673.115i			1.78545i
$378$		0			0
$379$			265.878i			0.701525i	0.936464	+	0.350762i	$0.114078\pi$
$379$			265.878i			0.701525i	−0.936464	+	0.350762i	$0.885922\pi$
$380$		0			0
$381$	785.106			2.06065
$382$		0			0
$383$	128.454			0.335390			0.167695	−	0.985839i	$-0.446368\pi$
$383$	128.454			0.335390			0.167695	+	0.985839i	$0.446368\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−209.677			−0.541802
$388$		0			0
$389$			172.037i			0.442255i	0.975245	+	0.221128i	$0.0709737\pi$
$389$			172.037i			0.442255i	−0.975245	+	0.221128i	$0.929026\pi$
$390$		0			0
$391$	549.806	+	146.587i	1.40615	+	0.374903i
$392$		0			0
$393$		−	533.596i		−	1.35775i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	672.498i		−	1.69395i	−0.531633	−	0.846975i	$-0.678421\pi$
$397$		−	672.498i		−	1.69395i	0.531633	−	0.846975i	$-0.321579\pi$
$398$		0			0
$399$	380.258			0.953026
$400$		0			0
$401$			9.53573i			0.0237799i	0.999929	+	0.0118899i	$0.00378477\pi$
$401$			9.53573i			0.0237799i	−0.999929	+	0.0118899i	$0.996215\pi$
$402$		0			0
$403$		−	251.393i		−	0.623804i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			13.5988i			0.0334123i
$408$		0			0
$409$	36.7819			0.0899313			0.0449656	−	0.998989i	$-0.485682\pi$
$409$	36.7819			0.0899313			0.0449656	+	0.998989i	$0.485682\pi$
$410$		0			0
$411$		−	642.558i		−	1.56340i
$412$		0			0
$413$	149.885			0.362917
$414$		0			0
$415$		0			0
$416$		0			0
$417$		−	106.857i		−	0.256252i
$418$		0			0
$419$		−	544.930i		−	1.30055i	−0.759700	−	0.650274i	$-0.774654\pi$
$419$		−	544.930i		−	1.30055i	0.759700	−	0.650274i	$-0.225346\pi$
$420$		0			0
$421$			36.3125i			0.0862529i	0.999070	+	0.0431264i	$0.0137318\pi$
$421$			36.3125i			0.0862529i	−0.999070	+	0.0431264i	$0.986268\pi$
$422$		0			0
$423$		−	345.169i		−	0.816003i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		−	1357.57i		−	3.17932i
$428$		0			0
$429$		−	28.0070i		−	0.0652845i
$430$		0			0
$431$			629.315i			1.46013i	0.683379	+	0.730063i	$0.260509\pi$
$431$			629.315i			1.46013i	−0.683379	+	0.730063i	$0.739491\pi$
$432$		0			0
$433$	−190.197			−0.439254			−0.219627	−	0.975584i	$-0.570484\pi$
$433$	−190.197			−0.439254			−0.219627	+	0.975584i	$0.570484\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	33.5951	−	126.006i	0.0768768	−	0.288342i
$438$		0			0
$439$	258.812			0.589549			0.294775	−	0.955567i	$-0.404755\pi$
$439$	258.812			0.589549			0.294775	+	0.955567i	$0.404755\pi$
$440$		0			0
$441$	−2121.09			−4.80972
$442$		0			0
$443$		−	298.428i		−	0.673653i	−0.941567	−	0.336826i	$-0.890646\pi$
$443$		−	298.428i		−	0.673653i	0.941567	−	0.336826i	$-0.109354\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	976.308			2.18413
$448$		0			0
$449$	−137.787			−0.306875			−0.153437	−	0.988158i	$-0.549034\pi$
$449$	−137.787			−0.306875			−0.153437	+	0.988158i	$0.549034\pi$
$450$		0			0
$451$		−	8.74542i		−	0.0193912i
$452$		0			0
$453$			419.772i			0.926648i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−88.3845			−0.193401			−0.0967007	−	0.995314i	$-0.530829\pi$
$457$	−88.3845			−0.193401			−0.0967007	+	0.995314i	$0.530829\pi$
$458$		0			0
$459$		−	1548.98i		−	3.37468i
$460$		0			0
$461$	417.531			0.905707			0.452854	−	0.891585i	$-0.350406\pi$
$461$	417.531			0.905707			0.452854	+	0.891585i	$0.350406\pi$
$462$		0			0
$463$		−	720.784i		−	1.55677i	−0.627789	−	0.778384i	$-0.716040\pi$
$463$		−	720.784i		−	1.55677i	0.627789	−	0.778384i	$-0.283960\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	821.497			1.75909			0.879547	−	0.475812i	$-0.157846\pi$
$467$	821.497			1.75909			0.879547	+	0.475812i	$0.157846\pi$
$468$		0			0
$469$	−579.379			−1.23535
$470$		0			0
$471$		−	579.921i		−	1.23125i
$472$		0			0
$473$			2.89166i			0.00611344i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	415.910			0.871929
$478$		0			0
$479$			431.638i			0.901124i	0.892745	+	0.450562i	$0.148776\pi$
$479$			431.638i			0.901124i	−0.892745	+	0.450562i	$0.851224\pi$
$480$		0			0
$481$			875.003i			1.81913i
$482$		0			0
$483$	−397.380	+	1490.46i	−0.822733	+	3.08584i
$484$		0			0
$485$		0			0
$486$		0			0
$487$			493.280i			1.01289i	0.862271	+	0.506447i	$0.169041\pi$
$487$			493.280i			1.01289i	−0.862271	+	0.506447i	$0.830959\pi$
$488$		0			0
$489$	−859.207			−1.75707
$490$		0			0
$491$	−106.888			−0.217694			−0.108847	−	0.994059i	$-0.534716\pi$
$491$	−106.888			−0.217694			−0.108847	+	0.994059i	$0.534716\pi$
$492$		0			0
$493$	914.399			1.85476
$494$		0			0
$495$		0			0
$496$		0			0
$497$	1366.47			2.74945
$498$		0			0
$499$	393.734			0.789045			0.394523	−	0.918886i	$-0.370910\pi$
$499$	393.734			0.789045			0.394523	+	0.918886i	$0.370910\pi$
$500$		0			0
$501$	70.2284			0.140177
$502$		0			0
$503$	519.084			1.03198			0.515988	−	0.856596i	$-0.327425\pi$
$503$	519.084			1.03198			0.515988	+	0.856596i	$0.327425\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		−	883.820i		−	1.74323i
$508$		0			0
$509$	−527.334			−1.03602			−0.518010	−	0.855375i	$-0.673327\pi$
$509$	−527.334			−1.03602			−0.518010	+	0.855375i	$0.673327\pi$
$510$		0			0
$511$			1265.91i			2.47732i
$512$		0			0
$513$	−354.998			−0.692003
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−4.76023			−0.00920740
$518$		0			0
$519$	1026.18			1.97722
$520$		0			0
$521$		−	527.852i		−	1.01315i	−0.862196	−	0.506575i	$-0.830911\pi$
$521$		−	527.852i		−	1.01315i	0.862196	−	0.506575i	$-0.169089\pi$
$522$		0			0
$523$	0.882342			0.00168708			0.000843540	−	1.00000i	$-0.499731\pi$
$523$	0.882342			0.00168708			0.000843540		1.00000i	$0.499731\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−341.507			−0.648021
$528$		0			0
$529$	458.784	+	263.359i	0.867267	+	0.497843i
$530$		0			0
$531$	−249.218			−0.469336
$532$		0			0
$533$		−	562.715i		−	1.05575i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		−	1512.02i		−	2.81567i
$538$		0			0
$539$			29.2519i			0.0542706i
$540$		0			0
$541$	−455.910			−0.842716			−0.421358	−	0.906894i	$-0.638446\pi$
$541$	−455.910			−0.842716			−0.421358	+	0.906894i	$0.638446\pi$
$542$		0			0
$543$	50.3212			0.0926726
$544$		0			0
$545$		0			0
$546$		0			0
$547$			856.872i			1.56649i	0.621710	+	0.783247i	$0.286438\pi$
$547$			856.872i			1.56649i	−0.621710	+	0.783247i	$0.713562\pi$
$548$		0			0
$549$			2257.27i			4.11160i
$550$		0			0
$551$		−	209.564i		−	0.380334i
$552$		0			0
$553$			1098.44i			1.98633i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−618.036			−1.10958			−0.554790	−	0.831991i	$-0.687201\pi$
$557$	−618.036			−1.10958			−0.554790	+	0.831991i	$0.687201\pi$
$558$		0			0
$559$			186.061i			0.332846i
$560$		0			0
$561$	−38.0464			−0.0678189
$562$		0			0
$563$	170.840			0.303446			0.151723	−	0.988423i	$-0.451518\pi$
$563$	170.840			0.303446			0.151723	+	0.988423i	$0.451518\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	1919.23			3.38488
$568$		0			0
$569$			652.044i			1.14595i	0.819574	+	0.572973i	$0.194210\pi$
$569$			652.044i			1.14595i	−0.819574	+	0.572973i	$0.805790\pi$
$570$		0			0
$571$			64.7144i			0.113335i	0.998393	+	0.0566676i	$0.0180475\pi$
$571$			64.7144i			0.113335i	−0.998393	+	0.0566676i	$0.981952\pi$
$572$		0			0
$573$	−1449.94			−2.53043
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	30.8641i		−	0.0534907i	−0.999642	−	0.0267453i	$-0.991486\pi$
$577$		−	30.8641i		−	0.0534907i	0.999642	−	0.0267453i	$-0.00851432\pi$
$578$		0			0
$579$	−1012.54			−1.74878
$580$		0			0
$581$	726.891			1.25110
$582$		0			0
$583$		−	5.73581i		−	0.00983844i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			494.110i			0.841755i	0.907118	+	0.420877i	$0.138278\pi$
$587$			494.110i			0.841755i	−0.907118	+	0.420877i	$0.861722\pi$
$588$		0			0
$589$			78.2673i			0.132882i
$590$		0			0
$591$	−45.9528			−0.0777544
$592$		0			0
$593$		−	446.646i		−	0.753198i	−0.926376	−	0.376599i	$-0.877093\pi$
$593$		−	446.646i		−	0.753198i	0.926376	−	0.376599i	$-0.122907\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	1737.24			2.90994
$598$		0			0
$599$	744.158			1.24233			0.621167	−	0.783678i	$-0.286659\pi$
$599$	744.158			1.24233			0.621167	+	0.783678i	$0.286659\pi$
$600$		0			0
$601$	451.657			0.751509			0.375754	−	0.926719i	$-0.377384\pi$
$601$	451.657			0.751509			0.375754	+	0.926719i	$0.377384\pi$
$602$		0			0
$603$	963.349			1.59759
$604$		0			0
$605$		0			0
$606$		0			0
$607$			438.471i			0.722358i	0.932496	+	0.361179i	$0.117626\pi$
$607$			438.471i			0.722358i	−0.932496	+	0.361179i	$0.882374\pi$
$608$		0			0
$609$			2478.83i			4.07032i
$610$		0			0
$611$	−306.292			−0.501296
$612$		0			0
$613$	1051.54			1.71541			0.857704	−	0.514144i	$-0.171890\pi$
$613$	1051.54			1.71541			0.857704	+	0.514144i	$0.171890\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−567.429			−0.919658			−0.459829	−	0.888007i	$-0.652089\pi$
$617$	−567.429			−0.919658			−0.459829	+	0.888007i	$0.652089\pi$
$618$		0			0
$619$		−	185.256i		−	0.299282i	−0.988740	−	0.149641i	$-0.952188\pi$
$619$		−	185.256i		−	0.299282i	0.988740	−	0.149641i	$-0.0478118\pi$
$620$		0			0
$621$	370.983	−	1391.45i	0.597396	−	2.24066i
$622$		0			0
$623$			1860.89i			2.98698i
$624$		0			0
$625$		0			0
$626$		0			0
$627$			8.71955i			0.0139068i
$628$		0			0
$629$	1188.65			1.88975
$630$		0			0
$631$			1053.90i			1.67020i	0.550095	+	0.835102i	$0.314591\pi$
$631$			1053.90i			1.67020i	−0.550095	+	0.835102i	$0.685409\pi$
$632$		0			0
$633$		−	729.426i		−	1.15233i
$634$		0			0
$635$		0			0
$636$		0			0
$637$			1882.18i			2.95476i
$638$		0			0
$639$	−2272.07			−3.55567
$640$		0			0
$641$			120.122i			0.187397i	0.995601	+	0.0936987i	$0.0298690\pi$
$641$			120.122i			0.187397i	−0.995601	+	0.0936987i	$0.970131\pi$
$642$		0			0
$643$	190.393			0.296102			0.148051	−	0.988980i	$-0.452700\pi$
$643$	190.393			0.296102			0.148051	+	0.988980i	$0.452700\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	992.764i		−	1.53441i	−0.641401	−	0.767206i	$-0.721647\pi$
$647$		−	992.764i		−	1.53441i	0.641401	−	0.767206i	$-0.278353\pi$
$648$		0			0
$649$			3.43696i			0.00529577i
$650$		0			0
$651$		−	925.785i		−	1.42210i
$652$		0			0
$653$			130.683i			0.200127i	0.994981	+	0.100063i	$0.0319046\pi$
$653$			130.683i			0.200127i	−0.994981	+	0.100063i	$0.968095\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		−	2104.86i		−	3.20375i
$658$		0			0
$659$			307.482i			0.466589i	0.972406	+	0.233295i	$0.0749507\pi$
$659$			307.482i			0.466589i	−0.972406	+	0.233295i	$0.925049\pi$
$660$		0			0
$661$		−	714.537i		−	1.08099i	−0.841346	−	0.540497i	$-0.818236\pi$
$661$		−	714.537i		−	1.08099i	0.841346	−	0.540497i	$-0.181764\pi$
$662$		0			0
$663$	−2448.06			−3.69239
$664$		0			0
$665$		0			0
$666$		0			0
$667$	821.407	+	219.000i	1.23150	+	0.328336i
$668$		0			0
$669$	1654.96			2.47379
$670$		0			0
$671$	31.1300			0.0463934
$672$		0			0
$673$			921.119i			1.36868i	0.729165	+	0.684338i	$0.239909\pi$
$673$			921.119i			1.36868i	−0.729165	+	0.684338i	$0.760091\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	98.4249			0.145384			0.0726919	−	0.997354i	$-0.476841\pi$
$677$	98.4249			0.145384			0.0726919	+	0.997354i	$0.476841\pi$
$678$		0			0
$679$	459.340			0.676494
$680$		0			0
$681$		−	2080.12i		−	3.05451i
$682$		0			0
$683$			13.3472i			0.0195420i	0.999952	+	0.00977100i	$0.00311026\pi$
$683$			13.3472i			0.0195420i	−0.999952	+	0.00977100i	$0.996890\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−413.636			−0.602090
$688$		0			0
$689$		−	369.065i		−	0.535653i
$690$		0			0
$691$	−791.448			−1.14537			−0.572683	−	0.819777i	$-0.694098\pi$
$691$	−791.448			−1.14537			−0.572683	+	0.819777i	$0.694098\pi$
$692$		0			0
$693$		−	71.6977i		−	0.103460i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−764.425			−1.09674
$698$		0			0
$699$	−1360.50			−1.94635
$700$		0			0
$701$			891.507i			1.27177i	0.771786	+	0.635883i	$0.219364\pi$
$701$			891.507i			1.27177i	−0.771786	+	0.635883i	$0.780636\pi$
$702$		0			0
$703$		−	272.418i		−	0.387508i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1270.23			−1.79664
$708$		0			0
$709$			123.075i			0.173590i	0.996226	+	0.0867949i	$0.0276625\pi$
$709$			123.075i			0.173590i	−0.996226	+	0.0867949i	$0.972338\pi$
$710$		0			0
$711$		−	1826.40i		−	2.56878i
$712$		0			0
$713$	−306.777	−	81.7916i	−0.430262	−	0.114715i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		−	1986.08i		−	2.76999i
$718$		0			0
$719$	754.620			1.04954			0.524771	−	0.851244i	$-0.324151\pi$
$719$	754.620			1.04954			0.524771	+	0.851244i	$0.324151\pi$
$720$		0			0
$721$	−1751.60			−2.42941
$722$		0			0
$723$	1336.00			1.84786
$724$		0			0
$725$		0			0
$726$		0			0
$727$	509.926			0.701411			0.350706	−	0.936486i	$-0.385942\pi$
$727$	509.926			0.701411			0.350706	+	0.936486i	$0.385942\pi$
$728$		0			0
$729$	−129.365			−0.177455
$730$		0			0
$731$	252.756			0.345767
$732$		0			0
$733$	1327.62			1.81122			0.905608	−	0.424116i	$-0.139415\pi$
$733$	1327.62			1.81122			0.905608	+	0.424116i	$0.139415\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	13.2855i		−	0.0180265i
$738$		0			0
$739$	−738.809			−0.999741			−0.499870	−	0.866100i	$-0.666619\pi$
$739$	−738.809			−0.999741			−0.499870	+	0.866100i	$0.666619\pi$
$740$		0			0
$741$			561.051i			0.757154i
$742$		0			0
$743$	−506.757			−0.682041			−0.341021	−	0.940056i	$-0.610773\pi$
$743$	−506.757			−0.682041			−0.341021	+	0.940056i	$0.610773\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−1208.62			−1.61797
$748$		0			0
$749$	993.948			1.32703
$750$		0			0
$751$			432.369i			0.575724i	0.957672	+	0.287862i	$0.0929445\pi$
$751$			432.369i			0.575724i	−0.957672	+	0.287862i	$0.907056\pi$
$752$		0			0
$753$	714.867			0.949358
$754$		0			0
$755$		0			0
$756$		0			0
$757$	554.856			0.732967			0.366483	−	0.930425i	$-0.380562\pi$
$757$	554.856			0.732967			0.366483	+	0.930425i	$0.380562\pi$
$758$		0			0
$759$	−34.1772	−	9.11219i	−0.0450292	−	0.0120055i
$760$		0			0
$761$	−993.358			−1.30533			−0.652666	−	0.757645i	$-0.726350\pi$
$761$	−993.358			−1.30533			−0.652666	+	0.757645i	$0.726350\pi$
$762$		0			0
$763$		−	2342.07i		−	3.06956i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			221.147i			0.288328i
$768$		0			0
$769$		−	356.084i		−	0.463048i	−0.972829	−	0.231524i	$-0.925629\pi$
$769$		−	356.084i		−	0.463048i	0.972829	−	0.231524i	$-0.0743711\pi$
$770$		0			0
$771$	−1931.80			−2.50557
$772$		0			0
$773$	−940.527			−1.21672			−0.608362	−	0.793660i	$-0.708173\pi$
$773$	−940.527			−1.21672			−0.608362	+	0.793660i	$0.708173\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$			3222.30i			4.14711i
$778$		0			0
$779$			175.193i			0.224894i
$780$		0			0
$781$			31.3341i			0.0401205i
$782$		0			0
$783$		−	2314.16i		−	2.95551i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	325.692			0.413839			0.206920	−	0.978358i	$-0.433656\pi$
$787$	325.692			0.413839			0.206920	+	0.978358i	$0.433656\pi$
$788$		0			0
$789$		−	1382.85i		−	1.75266i
$790$		0			0
$791$	−652.790			−0.825272
$792$		0			0
$793$	2003.03			2.52589
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−503.556			−0.631814			−0.315907	−	0.948790i	$-0.602309\pi$
$797$	−503.556			−0.631814			−0.315907	+	0.948790i	$0.602309\pi$
$798$		0			0
$799$			416.085i			0.520757i
$800$		0			0
$801$		−	3094.15i		−	3.86285i
$802$		0			0
$803$	−29.0281			−0.0361496
$804$		0			0
$805$		0			0
$806$		0			0
$807$		−	832.043i		−	1.03103i
$808$		0			0
$809$	1210.56			1.49636			0.748181	−	0.663495i	$-0.230928\pi$
$809$	1210.56			1.49636			0.748181	+	0.663495i	$0.230928\pi$
$810$		0			0
$811$	857.571			1.05742			0.528712	−	0.848801i	$-0.322675\pi$
$811$	857.571			1.05742			0.528712	+	0.848801i	$0.322675\pi$
$812$		0			0
$813$			953.707i			1.17307i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	57.9271i		−	0.0709022i
$818$		0			0
$819$		−	4613.32i		−	5.63286i
$820$		0			0
$821$	1551.02			1.88918			0.944591	−	0.328249i	$-0.106459\pi$
$821$	1551.02			1.88918			0.944591	+	0.328249i	$0.106459\pi$
$822$		0			0
$823$		−	51.2070i		−	0.0622199i	−0.999516	−	0.0311100i	$-0.990096\pi$
$823$		−	51.2070i		−	0.0622199i	0.999516	−	0.0311100i	$-0.00990420\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−464.159			−0.561256			−0.280628	−	0.959817i	$-0.590543\pi$
$827$	−464.159			−0.561256			−0.280628	+	0.959817i	$0.590543\pi$
$828$		0			0
$829$	−1416.57			−1.70877			−0.854386	−	0.519639i	$-0.826066\pi$
$829$	−1416.57			−1.70877			−0.854386	+	0.519639i	$0.826066\pi$
$830$		0			0
$831$	2463.05			2.96396
$832$		0			0
$833$	2556.86			3.06947
$834$		0			0
$835$		0			0
$836$		0			0
$837$			864.287i			1.03260i
$838$		0			0
$839$			174.721i			0.208249i	0.994564	+	0.104125i	$0.0332041\pi$
$839$			174.721i			0.208249i	−0.994564	+	0.104125i	$0.966796\pi$
$840$		0			0
$841$	525.108			0.624386
$842$		0			0
$843$	2176.17			2.58146
$844$		0			0
$845$		0			0
$846$		0			0
$847$	1492.52			1.76213
$848$		0			0
$849$			2262.65i			2.66508i
$850$		0			0
$851$	1067.77	+	284.685i	1.25473	+	0.334530i
$852$		0			0
$853$		−	1224.84i		−	1.43592i	−0.696083	−	0.717961i	$-0.745076\pi$
$853$		−	1224.84i		−	1.43592i	0.696083	−	0.717961i	$-0.254924\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	83.8220i		−	0.0978086i	−0.998803	−	0.0489043i	$-0.984427\pi$
$857$		−	83.8220i		−	0.0978086i	0.998803	−	0.0489043i	$-0.0155729\pi$
$858$		0			0
$859$	−5.35668			−0.00623595			−0.00311797	−	0.999995i	$-0.500992\pi$
$859$	−5.35668			−0.00623595			−0.00311797	+	0.999995i	$0.500992\pi$
$860$		0			0
$861$		−	2072.27i		−	2.40681i
$862$		0			0
$863$		−	730.205i		−	0.846124i	−0.906101	−	0.423062i	$-0.860955\pi$
$863$		−	730.205i		−	0.846124i	0.906101	−	0.423062i	$-0.139045\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$			1755.29i			2.02456i
$868$		0			0
$869$	−25.1879			−0.0289849
$870$		0			0
$871$		−	854.844i		−	0.981451i
$872$		0			0
$873$	−763.756			−0.874864
$874$		0			0
$875$		0			0
$876$		0			0
$877$			197.939i			0.225701i	0.993612	+	0.112850i	$0.0359981\pi$
$877$			197.939i			0.225701i	−0.993612	+	0.112850i	$0.964002\pi$
$878$		0			0
$879$		−	289.142i		−	0.328944i
$880$		0			0
$881$		−	485.658i		−	0.551257i	−0.961264	−	0.275629i	$-0.911114\pi$
$881$		−	485.658i		−	0.551257i	0.961264	−	0.275629i	$-0.0888860\pi$
$882$		0			0
$883$			294.830i			0.333895i	0.985966	+	0.166948i	$0.0533911\pi$
$883$			294.830i			0.333895i	−0.985966	+	0.166948i	$0.946609\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	509.245i		−	0.574121i	−0.957912	−	0.287060i	$-0.907322\pi$
$887$		−	509.245i		−	0.574121i	0.957912	−	0.287060i	$-0.0926780\pi$
$888$		0			0
$889$		−	1783.49i		−	2.00617i
$890$		0			0
$891$			44.0092i			0.0493930i
$892$		0			0
$893$	95.3592			0.106785
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−2199.10	−	586.314i	−2.45161	−	0.653639i
$898$		0			0
$899$	−510.210			−0.567530
$900$		0			0
$901$	−501.359			−0.556447
$902$		0			0
$903$			685.191i			0.758794i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	852.611			0.940034			0.470017	−	0.882657i	$-0.344248\pi$
$907$	852.611			0.940034			0.470017	+	0.882657i	$0.344248\pi$
$908$		0			0
$909$	2112.04			2.32347
$910$		0			0
$911$			1166.56i			1.28053i	0.768154	+	0.640265i	$0.221175\pi$
$911$			1166.56i			1.28053i	−0.768154	+	0.640265i	$0.778825\pi$
$912$		0			0
$913$			16.6681i			0.0182564i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1212.14			−1.32186
$918$		0			0
$919$		−	1437.90i		−	1.56463i	−0.622880	−	0.782317i	$-0.714038\pi$
$919$		−	1437.90i		−	1.56463i	0.622880	−	0.782317i	$-0.285962\pi$
$920$		0			0
$921$	927.728			1.00731
$922$		0			0
$923$			2016.16i			2.18436i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	2912.44			3.14179
$928$		0			0
$929$	128.712			0.138549			0.0692746	−	0.997598i	$-0.477932\pi$
$929$	128.712			0.138549			0.0692746	+	0.997598i	$0.477932\pi$
$930$		0			0
$931$		−	585.988i		−	0.629418i
$932$		0			0
$933$		−	3219.28i		−	3.45046i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−376.018			−0.401300			−0.200650	−	0.979663i	$-0.564305\pi$
$937$	−376.018			−0.401300			−0.200650	+	0.979663i	$0.564305\pi$
$938$		0			0
$939$			995.801i			1.06049i
$940$		0			0
$941$		−	1296.54i		−	1.37784i	−0.724839	−	0.688919i	$-0.758086\pi$
$941$		−	1296.54i		−	1.37784i	0.724839	−	0.688919i	$-0.241914\pi$
$942$		0			0
$943$	−686.685	−	183.081i	−0.728192	−	0.194148i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	602.245i		−	0.635950i	−0.948099	−	0.317975i	$-0.896997\pi$
$947$		−	602.245i		−	0.635950i	0.948099	−	0.317975i	$-0.103003\pi$
$948$		0			0
$949$	−1867.79			−1.96816
$950$		0			0
$951$	−1514.17			−1.59219
$952$		0			0
$953$	−96.0530			−0.100790			−0.0503951	−	0.998729i	$-0.516048\pi$
$953$	−96.0530			−0.100790			−0.0503951	+	0.998729i	$0.516048\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−56.8411			−0.0593951
$958$		0			0
$959$	−1459.67			−1.52207
$960$		0			0
$961$	−770.448			−0.801715
$962$		0			0
$963$	−1652.66			−1.71616
$964$		0			0
$965$		0			0
$966$		0			0
$967$			1635.70i			1.69152i	0.533564	+	0.845760i	$0.320852\pi$
$967$			1635.70i			1.69152i	−0.533564	+	0.845760i	$0.679148\pi$
$968$		0			0
$969$	762.164			0.786547
$970$		0			0
$971$		−	1549.29i		−	1.59556i	−0.602946	−	0.797782i	$-0.706006\pi$
$971$		−	1549.29i		−	1.59556i	0.602946	−	0.797782i	$-0.293994\pi$
$972$		0			0
$973$	−242.742			−0.249478
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−775.876			−0.794141			−0.397071	−	0.917788i	$-0.629973\pi$
$977$	−775.876			−0.794141			−0.397071	+	0.917788i	$0.629973\pi$
$978$		0			0
$979$	−42.6713			−0.0435866
$980$		0			0
$981$			3894.22i			3.96965i
$982$		0			0
$983$	1601.98			1.62969			0.814844	−	0.579680i	$-0.196823\pi$
$983$	1601.98			1.62969			0.814844	+	0.579680i	$0.196823\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−1127.96			−1.14281
$988$		0			0
$989$	227.051	+	60.5355i	0.229577	+	0.0612088i
$990$		0			0
$991$	−271.297			−0.273761			−0.136880	−	0.990588i	$-0.543708\pi$
$991$	−271.297			−0.273761			−0.136880	+	0.990588i	$0.543708\pi$
$992$		0			0
$993$			1371.85i			1.38152i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	338.386i		−	0.339404i	−0.985495	−	0.169702i	$-0.945719\pi$
$997$		−	338.386i		−	0.339404i	0.985495	−	0.169702i	$-0.0542806\pi$
$998$		0			0
$999$		−	3008.25i		−	3.01126i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2300.3.d.c.1149.2		32
5.2	odd	4		2300.3.f.d.1701.16	yes	16
5.3	odd	4		2300.3.f.c.1701.1	✓	16
5.4	even	2	inner	2300.3.d.c.1149.31		32
23.22	odd	2	inner	2300.3.d.c.1149.32		32
115.22	even	4		2300.3.f.d.1701.15	yes	16
115.68	even	4		2300.3.f.c.1701.2	yes	16
115.114	odd	2	inner	2300.3.d.c.1149.1		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2300.3.d.c.1149.1		32	115.114	odd	2	inner
2300.3.d.c.1149.2		32	1.1	even	1	trivial
2300.3.d.c.1149.31		32	5.4	even	2	inner
2300.3.d.c.1149.32		32	23.22	odd	2	inner
2300.3.f.c.1701.1	✓	16	5.3	odd	4
2300.3.f.c.1701.2	yes	16	115.68	even	4
2300.3.f.d.1701.15	yes	16	115.22	even	4
2300.3.f.d.1701.16	yes	16	5.2	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:	\( N \)	\(=\)	\( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2300.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+5.43352i q^{3} +12.3431 q^{7} -20.5231 q^{9} +O(q^{10})\)	\(q+5.43352i q^{3} +12.3431 q^{7} -20.5231 q^{9} +0.283034i q^{11} +18.2116i q^{13} +24.7396 q^{17} -5.66988i q^{19} +67.0662i q^{21} +(22.2237 + 5.92519i) q^{23} -62.6111i q^{27} +36.9609 q^{29} -13.8040 q^{31} -1.53787 q^{33} +48.0466 q^{37} -98.9528 q^{39} -30.8988 q^{41} +10.2166 q^{43} +16.8186i q^{47} +103.351 q^{49} +134.423i q^{51} -20.2654 q^{53} +30.8074 q^{57} +12.1433 q^{59} -109.987i q^{61} -253.318 q^{63} -46.9397 q^{67} +(-32.1946 + 120.753i) q^{69} +110.708 q^{71} +102.561i q^{73} +3.49351i q^{77} +88.9924i q^{79} +155.491 q^{81} +58.8907 q^{83} +200.828i q^{87} +150.764i q^{89} +224.786i q^{91} -75.0045i q^{93} +37.2144 q^{97} -5.80875i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 128 q^{9}+O(q^{10}) \) 32 * q - 128 * q^9	\( 32 q - 128 q^{9} + 12 q^{29} + 56 q^{31} - 148 q^{39} - 180 q^{41} + 380 q^{49} + 348 q^{59} - 100 q^{69} + 232 q^{71} + 112 q^{81}+O(q^{100}) \) 32 * q - 128 * q^9 + 12 * q^29 + 56 * q^31 - 148 * q^39 - 180 * q^41 + 380 * q^49 + 348 * q^59 - 100 * q^69 + 232 * q^71 + 112 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more