LMFDB - Embedded newform 2300.3.f.c.1701.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2300,3,Mod(1701,2300)]

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, 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("2300.1701");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2300 = 2^{2} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2300.f (of order $2$, degree $1$, 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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 487 x^{14} + 91703 x^{12} + 8599549 x^{10} + 437649516 x^{8} + 12136718132 x^{6} + \cdots + 1845424439296 $ x^16 + 487x^14 + 91703x^12 + 8599549x^10 + 437649516x^8 + 12136718132x^6 + 169813416064x^4 + 973986747392*x^2 + 1845424439296
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1701.1
Root			$-12.3431i$ of defining polynomial
Character	$\chi$	$=$	2300.1701
Dual form			2300.3.f.c.1701.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.43352 q^{3} -12.3431i q^{7} +20.5231 q^{9} +O(q^{10})$	$q-5.43352 q^{3} -12.3431i q^{7} +20.5231 q^{9} +0.283034i q^{11} -18.2116 q^{13} -24.7396i q^{17} +5.66988i q^{19} +67.0662i q^{21} +(-5.92519 + 22.2237i) q^{23} -62.6111 q^{27} -36.9609 q^{29} -13.8040 q^{31} -1.53787i q^{33} -48.0466i q^{37} +98.9528 q^{39} -30.8988 q^{41} +10.2166i q^{43} +16.8186 q^{47} -103.351 q^{49} +134.423i q^{51} -20.2654i q^{53} -30.8074i q^{57} -12.1433 q^{59} -109.987i q^{61} -253.318i q^{63} +46.9397i q^{67} +(32.1946 - 120.753i) q^{69} +110.708 q^{71} -102.561 q^{73} +3.49351 q^{77} -88.9924i q^{79} +155.491 q^{81} +58.8907i q^{83} +200.828 q^{87} -150.764i q^{89} +224.786i q^{91} +75.0045 q^{93} -37.2144i q^{97} +5.80875i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 64 q^{9}+O(q^{10}) $ 16 * q + 64 * q^9	$ 16 q + 64 q^{9} - 6 q^{13} + 8 q^{23} - 66 q^{27} - 6 q^{29} + 28 q^{31} + 74 q^{39} - 90 q^{41} - 40 q^{47} - 190 q^{49} - 174 q^{59} + 50 q^{69} + 116 q^{71} - 110 q^{73} - 198 q^{77} + 56 q^{81} + 362 q^{87} + 50 q^{93}+O(q^{100}) $ 16 * q + 64 * q^9 - 6 * q^13 + 8 * q^23 - 66 * q^27 - 6 * q^29 + 28 * q^31 + 74 * q^39 - 90 * q^41 - 40 * q^47 - 190 * q^49 - 174 * q^59 + 50 * q^69 + 116 * q^71 - 110 * q^73 - 198 * q^77 + 56 * q^81 + 362 * q^87 + 50 * q^93

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
$3$	−5.43352			−1.81117			−0.905586	−	0.424162i	$-0.860569\pi$
$3$	−5.43352			−1.81117			−0.905586	+	0.424162i	$0.860569\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$		−	12.3431i		−	1.76329i	−0.471910	−	0.881647i	$-0.656435\pi$
$7$		−	12.3431i		−	1.76329i	0.471910	−	0.881647i	$-0.343565\pi$
$8$		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
$13$	−18.2116			−1.40089			−0.700444	−	0.713707i	$-0.747015\pi$
$13$	−18.2116			−1.40089			−0.700444	+	0.713707i	$0.747015\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	24.7396i		−	1.45527i	−0.685963	−	0.727636i	$-0.740619\pi$
$17$		−	24.7396i		−	1.45527i	0.685963	−	0.727636i	$-0.259381\pi$
$18$		0			0
$19$			5.66988i			0.298415i	0.988806	+	0.149207i	$0.0476722\pi$
$19$			5.66988i			0.298415i	−0.988806	+	0.149207i	$0.952328\pi$
$20$		0			0
$21$			67.0662i			3.19363i
$22$		0			0
$23$	−5.92519	+	22.2237i	−0.257617	+	0.966247i
$23$	−5.92519	+	22.2237i	−0.257617	+	0.966247i
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−62.6111			−2.31893
$28$		0			0
$29$	−36.9609			−1.27451			−0.637257	−	0.770651i	$-0.719931\pi$
$29$	−36.9609			−1.27451			−0.637257	+	0.770651i	$0.719931\pi$
$30$		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
$33$		−	1.53787i		−	0.0466022i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	48.0466i		−	1.29856i	−0.760551	−	0.649278i	$-0.775071\pi$
$37$		−	48.0466i		−	1.29856i	0.760551	−	0.649278i	$-0.224929\pi$
$38$		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
$43$			10.2166i			0.237596i	0.992918	+	0.118798i	$0.0379041\pi$
$43$			10.2166i			0.237596i	−0.992918	+	0.118798i	$0.962096\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	16.8186			0.357842			0.178921	−	0.983863i	$-0.442739\pi$
$47$	16.8186			0.357842			0.178921	+	0.983863i	$0.442739\pi$
$48$		0			0
$49$	−103.351			−2.10920
$50$		0			0
$51$			134.423i			2.63575i
$52$		0			0
$53$		−	20.2654i		−	0.382367i	−0.981554	−	0.191183i	$-0.938768\pi$
$53$		−	20.2654i		−	0.382367i	0.981554	−	0.191183i	$-0.0612325\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		−	30.8074i		−	0.540481i
$58$		0			0
$59$	−12.1433			−0.205818			−0.102909	−	0.994691i	$-0.532815\pi$
$59$	−12.1433			−0.205818			−0.102909	+	0.994691i	$0.532815\pi$
$60$		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
$63$		−	253.318i		−	4.02092i
$64$		0			0
$65$		0			0
$66$		0			0
$67$			46.9397i			0.700592i	0.936639	+	0.350296i	$0.113919\pi$
$67$			46.9397i			0.700592i	−0.936639	+	0.350296i	$0.886081\pi$
$68$		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
$73$	−102.561			−1.40494			−0.702469	−	0.711714i	$-0.747919\pi$
$73$	−102.561			−1.40494			−0.702469	+	0.711714i	$0.747919\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	3.49351			0.0453702
$78$		0			0
$79$		−	88.9924i		−	1.12649i	−0.826291	−	0.563243i	$-0.809553\pi$
$79$		−	88.9924i		−	1.12649i	0.826291	−	0.563243i	$-0.190447\pi$
$80$		0			0
$81$	155.491			1.91964
$82$		0			0
$83$			58.8907i			0.709526i	0.934956	+	0.354763i	$0.115438\pi$
$83$			58.8907i			0.709526i	−0.934956	+	0.354763i	$0.884562\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	200.828			2.30837
$88$		0			0
$89$		−	150.764i		−	1.69398i	−0.531612	−	0.846988i	$-0.678414\pi$
$89$		−	150.764i		−	1.69398i	0.531612	−	0.846988i	$-0.321586\pi$
$90$		0			0
$91$			224.786i			2.47018i
$92$		0			0
$93$	75.0045			0.806500
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	37.2144i		−	0.383654i	−0.981429	−	0.191827i	$-0.938559\pi$
$97$		−	37.2144i		−	0.383654i	0.981429	−	0.191827i	$-0.0614412\pi$
$98$		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
$103$		−	141.910i		−	1.37777i	−0.724872	−	0.688884i	$-0.758101\pi$
$103$		−	141.910i		−	1.37777i	0.724872	−	0.688884i	$-0.241899\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	80.5269i		−	0.752588i	−0.926500	−	0.376294i	$-0.877198\pi$
$107$		−	80.5269i		−	0.752588i	0.926500	−	0.376294i	$-0.122802\pi$
$108$		0			0
$109$			189.748i			1.74081i	0.492338	+	0.870404i	$0.336142\pi$
$109$			189.748i			1.74081i	−0.492338	+	0.870404i	$0.663858\pi$
$110$		0			0
$111$			261.062i			2.35191i
$112$		0			0
$113$		−	52.8872i		−	0.468029i	−0.972233	−	0.234014i	$-0.924814\pi$
$113$		−	52.8872i		−	0.468029i	0.972233	−	0.234014i	$-0.0751863\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−373.758			−3.19451
$118$		0			0
$119$	−305.363			−2.56607
$120$		0			0
$121$	120.920			0.999338
$122$		0			0
$123$	167.889			1.36495
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−144.493			−1.13774			−0.568870	−	0.822427i	$-0.692619\pi$
$127$	−144.493			−1.13774			−0.568870	+	0.822427i	$0.692619\pi$
$128$		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
$133$	69.9837			0.526193
$134$		0			0
$135$		0			0
$136$		0			0
$137$			118.258i			0.863199i	0.902065	+	0.431599i	$0.142051\pi$
$137$			118.258i			0.863199i	−0.902065	+	0.431599i	$0.857949\pi$
$138$		0			0
$139$	19.6663			0.141484			0.0707419	−	0.997495i	$-0.477463\pi$
$139$	19.6663			0.141484			0.0707419	+	0.997495i	$0.477463\pi$
$140$		0			0
$141$	−91.3839			−0.648113
$142$		0			0
$143$		−	5.15449i		−	0.0360454i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	561.559			3.82013
$148$		0			0
$149$			179.682i			1.20592i	0.797770	+	0.602961i	$0.206013\pi$
$149$			179.682i			1.20592i	−0.797770	+	0.602961i	$0.793987\pi$
$150$		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
$153$		−	507.735i		−	3.31853i
$154$		0			0
$155$		0			0
$156$		0			0
$157$			106.730i			0.679811i	0.940460	+	0.339905i	$0.110395\pi$
$157$			106.730i			0.679811i	−0.940460	+	0.339905i	$0.889605\pi$
$158$		0			0
$159$			110.113i			0.692532i
$160$		0			0
$161$	274.308	+	73.1350i	1.70378	+	0.454254i
$162$		0			0
$163$	−158.131			−0.970128			−0.485064	−	0.874479i	$-0.661204\pi$
$163$	−158.131			−0.970128			−0.485064	+	0.874479i	$0.661204\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−12.9250			−0.0773954			−0.0386977	−	0.999251i	$-0.512321\pi$
$167$	−12.9250			−0.0773954			−0.0386977	+	0.999251i	$0.512321\pi$
$168$		0			0
$169$	162.661			0.962489
$170$		0			0
$171$			116.364i			0.680490i
$172$		0			0
$173$	188.860			1.09168			0.545839	−	0.837890i	$-0.316211\pi$
$173$	188.860			1.09168			0.545839	+	0.837890i	$0.316211\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	65.9806			0.372772
$178$		0			0
$179$	278.276			1.55461			0.777306	−	0.629123i	$-0.216586\pi$
$179$	278.276			1.55461			0.777306	+	0.629123i	$0.216586\pi$
$180$		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
$183$			597.614i			3.26565i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	7.00216			0.0374447
$188$		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
$193$	−186.351			−0.965548			−0.482774	−	0.875745i	$-0.660371\pi$
$193$	−186.351			−0.965548			−0.482774	+	0.875745i	$0.660371\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	8.45729			0.0429304			0.0214652	−	0.999770i	$-0.493167\pi$
$197$	8.45729			0.0429304			0.0214652	+	0.999770i	$0.493167\pi$
$198$		0			0
$199$			319.726i			1.60666i	0.595532	+	0.803331i	$0.296941\pi$
$199$			319.726i			1.60666i	−0.595532	+	0.803331i	$0.703059\pi$
$200$		0			0
$201$		−	255.048i		−	1.26889i
$202$		0			0
$203$			456.210i			2.24734i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−121.603	+	456.099i	−0.587456	+	2.20338i
$208$		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
$213$	−601.534			−2.82410
$214$		0			0
$215$		0			0
$216$		0			0
$217$			170.384i			0.785180i
$218$		0			0
$219$	557.265			2.54459
$220$		0			0
$221$			450.547i			2.03867i
$222$		0			0
$223$	304.584			1.36585			0.682924	−	0.730490i	$-0.260708\pi$
$223$	304.584			1.36585			0.682924	+	0.730490i	$0.260708\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			382.831i			1.68648i	0.537536	+	0.843241i	$0.319355\pi$
$227$			382.831i			1.68648i	−0.537536	+	0.843241i	$0.680645\pi$
$228$		0			0
$229$		−	76.1267i		−	0.332431i	−0.986089	−	0.166215i	$-0.946845\pi$
$229$		−	76.1267i		−	0.332431i	0.986089	−	0.166215i	$-0.0531547\pi$
$230$		0			0
$231$	−18.9820			−0.0821733
$232$		0			0
$233$	−250.390			−1.07464			−0.537318	−	0.843380i	$-0.680563\pi$
$233$	−250.390			−1.07464			−0.537318	+	0.843380i	$0.680563\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$			483.542i			2.04026i
$238$		0			0
$239$	365.524			1.52939			0.764695	−	0.644393i	$-0.222890\pi$
$239$	365.524			1.52939			0.764695	+	0.644393i	$0.222890\pi$
$240$		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
$243$	−281.361			−1.15786
$244$		0			0
$245$		0			0
$246$		0			0
$247$		−	103.257i		−	0.418046i
$248$		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
$253$	−6.29006	−	1.67703i	−0.0248619	−	0.00662858i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	355.534			1.38340			0.691700	−	0.722185i	$-0.256862\pi$
$257$	355.534			1.38340			0.691700	+	0.722185i	$0.256862\pi$
$258$		0			0
$259$	−593.041			−2.28974
$260$		0			0
$261$	−758.553			−2.90633
$262$		0			0
$263$		−	254.503i		−	0.967692i	−0.875153	−	0.483846i	$-0.839239\pi$
$263$		−	254.503i		−	0.967692i	0.875153	−	0.483846i	$-0.160761\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$			819.178i			3.06808i
$268$		0			0
$269$	153.131			0.569262			0.284631	−	0.958637i	$-0.408129\pi$
$269$	153.131			0.569262			0.284631	+	0.958637i	$0.408129\pi$
$270$		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
$273$		−	1221.38i		−	4.47392i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−453.306			−1.63649			−0.818243	−	0.574873i	$-0.805052\pi$
$277$	−453.306			−1.63649			−0.818243	+	0.574873i	$0.805052\pi$
$278$		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
$283$			416.425i			1.47147i	0.677271	+	0.735734i	$0.263163\pi$
$283$			416.425i			1.47147i	−0.677271	+	0.735734i	$0.736837\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			381.386i			1.32887i
$288$		0			0
$289$	−323.049			−1.11782
$290$		0			0
$291$			202.205i			0.694864i
$292$		0			0
$293$		−	53.2145i		−	0.181620i	−0.995868	−	0.0908098i	$-0.971054\pi$
$293$		−	53.2145i		−	0.181620i	0.995868	−	0.0908098i	$-0.0289455\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		−	17.7211i		−	0.0596670i
$298$		0			0
$299$	107.907	−	404.728i	0.360893	−	1.35360i
$300$		0			0
$301$	126.104			0.418952
$302$		0			0
$303$	559.164			1.84543
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−170.742			−0.556162			−0.278081	−	0.960558i	$-0.589698\pi$
$307$	−170.742			−0.556162			−0.278081	+	0.960558i	$0.589698\pi$
$308$		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
$313$			183.270i			0.585527i	0.956185	+	0.292764i	$0.0945749\pi$
$313$			183.270i			0.585527i	−0.956185	+	0.292764i	$0.905425\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	278.672			0.879092			0.439546	−	0.898220i	$-0.355139\pi$
$317$	278.672			0.879092			0.439546	+	0.898220i	$0.355139\pi$
$318$		0			0
$319$		−	10.4612i		−	0.0327937i
$320$		0			0
$321$			437.544i			1.36307i
$322$		0			0
$323$	140.271			0.434275
$324$		0			0
$325$		0			0
$326$		0			0
$327$		−	1031.00i		−	3.15290i
$328$		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
$333$		−	986.066i		−	2.96116i
$334$		0			0
$335$		0			0
$336$		0			0
$337$			568.229i			1.68614i	0.537804	+	0.843070i	$0.319254\pi$
$337$			568.229i			1.68614i	−0.537804	+	0.843070i	$0.680746\pi$
$338$		0			0
$339$			287.364i			0.847681i
$340$		0			0
$341$		−	3.90702i		−	0.0114575i
$342$		0			0
$343$			670.857i			1.95585i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−310.626			−0.895177			−0.447589	−	0.894240i	$-0.647717\pi$
$347$	−310.626			−0.895177			−0.447589	+	0.894240i	$0.647717\pi$
$348$		0			0
$349$	533.091			1.52748			0.763741	−	0.645523i	$-0.223360\pi$
$349$	533.091			1.52748			0.763741	+	0.645523i	$0.223360\pi$
$350$		0			0
$351$	1140.25			3.24856
$352$		0			0
$353$	405.742			1.14941			0.574705	−	0.818361i	$-0.305117\pi$
$353$	405.742			1.14941			0.574705	+	0.818361i	$0.305117\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	1659.19			4.64760
$358$		0			0
$359$		−	512.842i		−	1.42853i	−0.699876	−	0.714265i	$-0.746761\pi$
$359$		−	512.842i		−	1.42853i	0.699876	−	0.714265i	$-0.253239\pi$
$360$		0			0
$361$	328.852			0.910949
$362$		0			0
$363$	−657.021			−1.80997
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	226.315i		−	0.616662i	−0.951279	−	0.308331i	$-0.900230\pi$
$367$		−	226.315i		−	0.616662i	0.951279	−	0.308331i	$-0.0997705\pi$
$368$		0			0
$369$	−634.140			−1.71854
$370$		0			0
$371$	−250.137			−0.674224
$372$		0			0
$373$			60.8146i			0.163042i	0.996672	+	0.0815209i	$0.0259777\pi$
$373$			60.8146i			0.163042i	−0.996672	+	0.0815209i	$0.974022\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	673.115			1.78545
$378$		0			0
$379$		−	265.878i		−	0.701525i	−0.936464	−	0.350762i	$-0.885922\pi$
$379$		−	265.878i		−	0.701525i	0.936464	−	0.350762i	$-0.114078\pi$
$380$		0			0
$381$	785.106			2.06065
$382$		0			0
$383$			128.454i			0.335390i	0.985839	+	0.167695i	$0.0536324\pi$
$383$			128.454i			0.335390i	−0.985839	+	0.167695i	$0.946368\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$			209.677i			0.541802i
$388$		0			0
$389$		−	172.037i		−	0.442255i	−0.975245	−	0.221128i	$-0.929026\pi$
$389$		−	172.037i		−	0.442255i	0.975245	−	0.221128i	$-0.0709737\pi$
$390$		0			0
$391$	549.806	+	146.587i	1.40615	+	0.374903i
$392$		0			0
$393$	533.596			1.35775
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−672.498			−1.69395			−0.846975	−	0.531633i	$-0.821579\pi$
$397$	−672.498			−1.69395			−0.846975	+	0.531633i	$0.821579\pi$
$398$		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
$403$	251.393			0.623804
$404$		0			0
$405$		0			0
$406$		0			0
$407$	13.5988			0.0334123
$408$		0			0
$409$	−36.7819			−0.0899313			−0.0449656	−	0.998989i	$-0.514318\pi$
$409$	−36.7819			−0.0899313			−0.0449656	+	0.998989i	$0.514318\pi$
$410$		0			0
$411$		−	642.558i		−	1.56340i
$412$		0			0
$413$			149.885i			0.362917i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−106.857			−0.256252
$418$		0			0
$419$			544.930i			1.30055i	0.759700	+	0.650274i	$0.225346\pi$
$419$			544.930i			1.30055i	−0.759700	+	0.650274i	$0.774654\pi$
$420$		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
$423$	345.169			0.816003
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−1357.57			−3.17932
$428$		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
$433$		−	190.197i		−	0.439254i	−0.975584	−	0.219627i	$-0.929516\pi$
$433$		−	190.197i		−	0.439254i	0.975584	−	0.219627i	$-0.0704840\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−126.006	−	33.5951i	−0.288342	−	0.0768768i
$438$		0			0
$439$	−258.812			−0.589549			−0.294775	−	0.955567i	$-0.595245\pi$
$439$	−258.812			−0.589549			−0.294775	+	0.955567i	$0.595245\pi$
$440$		0			0
$441$	−2121.09			−4.80972
$442$		0			0
$443$	298.428			0.673653			0.336826	−	0.941567i	$-0.390646\pi$
$443$	298.428			0.673653			0.336826	+	0.941567i	$0.390646\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		−	976.308i		−	2.18413i
$448$		0			0
$449$	137.787			0.306875			0.153437	−	0.988158i	$-0.450966\pi$
$449$	137.787			0.306875			0.153437	+	0.988158i	$0.450966\pi$
$450$		0			0
$451$		−	8.74542i		−	0.0193912i
$452$		0			0
$453$	−419.772			−0.926648
$454$		0			0
$455$		0			0
$456$		0			0
$457$			88.3845i			0.193401i	0.995314	+	0.0967007i	$0.0308290\pi$
$457$			88.3845i			0.193401i	−0.995314	+	0.0967007i	$0.969171\pi$
$458$		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
$463$	720.784			1.55677			0.778384	−	0.627789i	$-0.216040\pi$
$463$	720.784			1.55677			0.778384	+	0.627789i	$0.216040\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	821.497i		−	1.75909i	−0.475812	−	0.879547i	$-0.657846\pi$
$467$		−	821.497i		−	1.75909i	0.475812	−	0.879547i	$-0.342154\pi$
$468$		0			0
$469$	579.379			1.23535
$470$		0			0
$471$		−	579.921i		−	1.23125i
$472$		0			0
$473$	−2.89166			−0.00611344
$474$		0			0
$475$		0			0
$476$		0			0
$477$		−	415.910i		−	0.871929i
$478$		0			0
$479$		−	431.638i		−	0.901124i	−0.892745	−	0.450562i	$-0.851224\pi$
$479$		−	431.638i		−	0.901124i	0.892745	−	0.450562i	$-0.148776\pi$
$480$		0			0
$481$			875.003i			1.81913i
$482$		0			0
$483$	−1490.46	−	397.380i	−3.08584	−	0.822733i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	493.280			1.01289			0.506447	−	0.862271i	$-0.330959\pi$
$487$	493.280			1.01289			0.506447	+	0.862271i	$0.330959\pi$
$488$		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
$493$			914.399i			1.85476i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	1366.47i		−	2.74945i
$498$		0			0
$499$	−393.734			−0.789045			−0.394523	−	0.918886i	$-0.629090\pi$
$499$	−393.734			−0.789045			−0.394523	+	0.918886i	$0.629090\pi$
$500$		0			0
$501$	70.2284			0.140177
$502$		0			0
$503$			519.084i			1.03198i	0.856596	+	0.515988i	$0.172575\pi$
$503$			519.084i			1.03198i	−0.856596	+	0.515988i	$0.827425\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−883.820			−1.74323
$508$		0			0
$509$	527.334			1.03602			0.518010	−	0.855375i	$-0.326673\pi$
$509$	527.334			1.03602			0.518010	+	0.855375i	$0.326673\pi$
$510$		0			0
$511$			1265.91i			2.47732i
$512$		0			0
$513$		−	354.998i		−	0.692003i
$514$		0			0
$515$		0			0
$516$		0			0
$517$			4.76023i			0.00920740i
$518$		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
$523$			0.882342i			0.00168708i	1.00000		0.000843540i	$0.000268507\pi$
$523$			0.882342i			0.00168708i	−1.00000		0.000843540i	$0.999731\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			341.507i			0.648021i
$528$		0			0
$529$	−458.784	−	263.359i	−0.867267	−	0.497843i
$530$		0			0
$531$	−249.218			−0.469336
$532$		0			0
$533$	562.715			1.05575
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−1512.02			−2.81567
$538$		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
$543$			50.3212i			0.0926726i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	856.872			1.56649			0.783247	−	0.621710i	$-0.213562\pi$
$547$	856.872			1.56649			0.783247	+	0.621710i	$0.213562\pi$
$548$		0			0
$549$		−	2257.27i		−	4.11160i
$550$		0			0
$551$		−	209.564i		−	0.380334i
$552$		0			0
$553$	−1098.44			−1.98633
$554$		0			0
$555$		0			0
$556$		0			0
$557$			618.036i			1.10958i	0.831991	+	0.554790i	$0.187201\pi$
$557$			618.036i			1.10958i	−0.831991	+	0.554790i	$0.812799\pi$
$558$		0			0
$559$		−	186.061i		−	0.332846i
$560$		0			0
$561$	−38.0464			−0.0678189
$562$		0			0
$563$			170.840i			0.303446i	0.988423	+	0.151723i	$0.0484822\pi$
$563$			170.840i			0.303446i	−0.988423	+	0.151723i	$0.951518\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		−	1919.23i		−	3.38488i
$568$		0			0
$569$		−	652.044i		−	1.14595i	−0.819574	−	0.572973i	$-0.805790\pi$
$569$		−	652.044i		−	1.14595i	0.819574	−	0.572973i	$-0.194210\pi$
$570$		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
$573$		−	1449.94i		−	2.53043i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−30.8641			−0.0534907			−0.0267453	−	0.999642i	$-0.508514\pi$
$577$	−30.8641			−0.0534907			−0.0267453	+	0.999642i	$0.508514\pi$
$578$		0			0
$579$	1012.54			1.74878
$580$		0			0
$581$	726.891			1.25110
$582$		0			0
$583$	5.73581			0.00983844
$584$		0			0
$585$		0			0
$586$		0			0
$587$	494.110			0.841755			0.420877	−	0.907118i	$-0.361722\pi$
$587$	494.110			0.841755			0.420877	+	0.907118i	$0.361722\pi$
$588$		0			0
$589$		−	78.2673i		−	0.132882i
$590$		0			0
$591$	−45.9528			−0.0777544
$592$		0			0
$593$	446.646			0.753198			0.376599	−	0.926376i	$-0.377093\pi$
$593$	446.646			0.753198			0.376599	+	0.926376i	$0.377093\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		−	1737.24i		−	2.90994i
$598$		0			0
$599$	−744.158			−1.24233			−0.621167	−	0.783678i	$-0.713341\pi$
$599$	−744.158			−1.24233			−0.621167	+	0.783678i	$0.713341\pi$
$600$		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
$603$			963.349i			1.59759i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	438.471			0.722358			0.361179	−	0.932496i	$-0.382374\pi$
$607$	438.471			0.722358			0.361179	+	0.932496i	$0.382374\pi$
$608$		0			0
$609$		−	2478.83i		−	4.07032i
$610$		0			0
$611$	−306.292			−0.501296
$612$		0			0
$613$			1051.54i			1.71541i	0.514144	+	0.857704i	$0.328110\pi$
$613$			1051.54i			1.71541i	−0.514144	+	0.857704i	$0.671890\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			567.429i			0.919658i	0.888007	+	0.459829i	$0.152089\pi$
$617$			567.429i			0.919658i	−0.888007	+	0.459829i	$0.847911\pi$
$618$		0			0
$619$			185.256i			0.299282i	0.988740	+	0.149641i	$0.0478118\pi$
$619$			185.256i			0.299282i	−0.988740	+	0.149641i	$0.952188\pi$
$620$		0			0
$621$	370.983	−	1391.45i	0.597396	−	2.24066i
$622$		0			0
$623$	−1860.89			−2.98698
$624$		0			0
$625$		0			0
$626$		0			0
$627$	8.71955			0.0139068
$628$		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
$633$	729.426			1.15233
$634$		0			0
$635$		0			0
$636$		0			0
$637$	1882.18			2.95476
$638$		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
$643$			190.393i			0.296102i	0.988980	+	0.148051i	$0.0472999\pi$
$643$			190.393i			0.296102i	−0.988980	+	0.148051i	$0.952700\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−992.764			−1.53441			−0.767206	−	0.641401i	$-0.778353\pi$
$647$	−992.764			−1.53441			−0.767206	+	0.641401i	$0.778353\pi$
$648$		0			0
$649$		−	3.43696i		−	0.00529577i
$650$		0			0
$651$		−	925.785i		−	1.42210i
$652$		0			0
$653$	−130.683			−0.200127			−0.100063	−	0.994981i	$-0.531905\pi$
$653$	−130.683			−0.200127			−0.100063	+	0.994981i	$0.531905\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−2104.86			−3.20375
$658$		0			0
$659$		−	307.482i		−	0.466589i	−0.972406	−	0.233295i	$-0.925049\pi$
$659$		−	307.482i		−	0.466589i	0.972406	−	0.233295i	$-0.0749507\pi$
$660$		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
$663$		−	2448.06i		−	3.69239i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	219.000	−	821.407i	0.328336	−	1.23150i
$668$		0			0
$669$	−1654.96			−2.47379
$670$		0			0
$671$	31.1300			0.0463934
$672$		0			0
$673$	−921.119			−1.36868			−0.684338	−	0.729165i	$-0.739909\pi$
$673$	−921.119			−1.36868			−0.684338	+	0.729165i	$0.739909\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	98.4249i		−	0.145384i	−0.997354	−	0.0726919i	$-0.976841\pi$
$677$		−	98.4249i		−	0.145384i	0.997354	−	0.0726919i	$-0.0231590\pi$
$678$		0			0
$679$	−459.340			−0.676494
$680$		0			0
$681$		−	2080.12i		−	3.05451i
$682$		0			0
$683$	−13.3472			−0.0195420			−0.00977100	−	0.999952i	$-0.503110\pi$
$683$	−13.3472			−0.0195420			−0.00977100	+	0.999952i	$0.503110\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$			413.636i			0.602090i
$688$		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
$693$	71.6977			0.103460
$694$		0			0
$695$		0			0
$696$		0			0
$697$			764.425i			1.09674i
$698$		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
$703$	272.418			0.387508
$704$		0			0
$705$		0			0
$706$		0			0
$707$			1270.23i			1.79664i
$708$		0			0
$709$		−	123.075i		−	0.173590i	−0.996226	−	0.0867949i	$-0.972338\pi$
$709$		−	123.075i		−	0.173590i	0.996226	−	0.0867949i	$-0.0276625\pi$
$710$		0			0
$711$		−	1826.40i		−	2.56878i
$712$		0			0
$713$	81.7916	−	306.777i	0.114715	−	0.430262i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−1986.08			−2.76999
$718$		0			0
$719$	−754.620			−1.04954			−0.524771	−	0.851244i	$-0.675849\pi$
$719$	−754.620			−1.04954			−0.524771	+	0.851244i	$0.675849\pi$
$720$		0			0
$721$	−1751.60			−2.42941
$722$		0			0
$723$			1336.00i			1.84786i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	509.926i		−	0.701411i	−0.936486	−	0.350706i	$-0.885942\pi$
$727$		−	509.926i		−	0.701411i	0.936486	−	0.350706i	$-0.114058\pi$
$728$		0			0
$729$	129.365			0.177455
$730$		0			0
$731$	252.756			0.345767
$732$		0			0
$733$			1327.62i			1.81122i	0.424116	+	0.905608i	$0.360585\pi$
$733$			1327.62i			1.81122i	−0.424116	+	0.905608i	$0.639415\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−13.2855			−0.0180265
$738$		0			0
$739$	738.809			0.999741			0.499870	−	0.866100i	$-0.333381\pi$
$739$	738.809			0.999741			0.499870	+	0.866100i	$0.333381\pi$
$740$		0			0
$741$			561.051i			0.757154i
$742$		0			0
$743$		−	506.757i		−	0.682041i	−0.940056	−	0.341021i	$-0.889227\pi$
$743$		−	506.757i		−	0.682041i	0.940056	−	0.341021i	$-0.110773\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$			1208.62i			1.61797i
$748$		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
$753$			714.867i			0.949358i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	554.856i		−	0.732967i	−0.930425	−	0.366483i	$-0.880562\pi$
$757$		−	554.856i		−	0.732967i	0.930425	−	0.366483i	$-0.119438\pi$
$758$		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
$763$	2342.07			3.06956
$764$		0			0
$765$		0			0
$766$		0			0
$767$	221.147			0.288328
$768$		0			0
$769$			356.084i			0.463048i	0.972829	+	0.231524i	$0.0743711\pi$
$769$			356.084i			0.463048i	−0.972829	+	0.231524i	$0.925629\pi$
$770$		0			0
$771$	−1931.80			−2.50557
$772$		0			0
$773$		−	940.527i		−	1.21672i	−0.793660	−	0.608362i	$-0.791827\pi$
$773$		−	940.527i		−	1.21672i	0.793660	−	0.608362i	$-0.208173\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	3222.30			4.14711
$778$		0			0
$779$		−	175.193i		−	0.224894i
$780$		0			0
$781$			31.3341i			0.0401205i
$782$		0			0
$783$	2314.16			2.95551
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	325.692i		−	0.413839i	−0.978358	−	0.206920i	$-0.933656\pi$
$787$		−	325.692i		−	0.413839i	0.978358	−	0.206920i	$-0.0663439\pi$
$788$		0			0
$789$			1382.85i			1.75266i
$790$		0			0
$791$	−652.790			−0.825272
$792$		0			0
$793$			2003.03i			2.52589i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			503.556i			0.631814i	0.948790	+	0.315907i	$0.102309\pi$
$797$			503.556i			0.631814i	−0.948790	+	0.315907i	$0.897691\pi$
$798$		0			0
$799$		−	416.085i		−	0.520757i
$800$		0			0
$801$		−	3094.15i		−	3.86285i
$802$		0			0
$803$		−	29.0281i		−	0.0361496i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−832.043			−1.03103
$808$		0			0
$809$	−1210.56			−1.49636			−0.748181	−	0.663495i	$-0.769072\pi$
$809$	−1210.56			−1.49636			−0.748181	+	0.663495i	$0.769072\pi$
$810$		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
$813$	−953.707			−1.17307
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−57.9271			−0.0709022
$818$		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
$823$	51.2070			0.0622199			0.0311100	−	0.999516i	$-0.490096\pi$
$823$	51.2070			0.0622199			0.0311100	+	0.999516i	$0.490096\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			464.159i			0.561256i	0.959817	+	0.280628i	$0.0905428\pi$
$827$			464.159i			0.561256i	−0.959817	+	0.280628i	$0.909457\pi$
$828$		0			0
$829$	1416.57			1.70877			0.854386	−	0.519639i	$-0.173934\pi$
$829$	1416.57			1.70877			0.854386	+	0.519639i	$0.173934\pi$
$830$		0			0
$831$	2463.05			2.96396
$832$		0			0
$833$			2556.86i			3.06947i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	864.287			1.03260
$838$		0			0
$839$		−	174.721i		−	0.208249i	−0.994564	−	0.104125i	$-0.966796\pi$
$839$		−	174.721i		−	0.208249i	0.994564	−	0.104125i	$-0.0332041\pi$
$840$		0			0
$841$	525.108			0.624386
$842$		0			0
$843$			2176.17i			2.58146i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		−	1492.52i		−	1.76213i
$848$		0			0
$849$		−	2262.65i		−	2.66508i
$850$		0			0
$851$	1067.77	+	284.685i	1.25473	+	0.334530i
$852$		0			0
$853$	1224.84			1.43592			0.717961	−	0.696083i	$-0.245076\pi$
$853$	1224.84			1.43592			0.717961	+	0.696083i	$0.245076\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−83.8220			−0.0978086			−0.0489043	−	0.998803i	$-0.515573\pi$
$857$	−83.8220			−0.0978086			−0.0489043	+	0.998803i	$0.515573\pi$
$858$		0			0
$859$	5.35668			0.00623595			0.00311797	−	0.999995i	$-0.499008\pi$
$859$	5.35668			0.00623595			0.00311797	+	0.999995i	$0.499008\pi$
$860$		0			0
$861$		−	2072.27i		−	2.40681i
$862$		0			0
$863$	730.205			0.846124			0.423062	−	0.906101i	$-0.360955\pi$
$863$	730.205			0.846124			0.423062	+	0.906101i	$0.360955\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	1755.29			2.02456
$868$		0			0
$869$	25.1879			0.0289849
$870$		0			0
$871$		−	854.844i		−	0.981451i
$872$		0			0
$873$		−	763.756i		−	0.874864i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	197.939			0.225701			0.112850	−	0.993612i	$-0.464002\pi$
$877$	197.939			0.225701			0.112850	+	0.993612i	$0.464002\pi$
$878$		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
$883$	−294.830			−0.333895			−0.166948	−	0.985966i	$-0.553391\pi$
$883$	−294.830			−0.333895			−0.166948	+	0.985966i	$0.553391\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−509.245			−0.574121			−0.287060	−	0.957912i	$-0.592678\pi$
$887$	−509.245			−0.574121			−0.287060	+	0.957912i	$0.592678\pi$
$888$		0			0
$889$			1783.49i			2.00617i
$890$		0			0
$891$			44.0092i			0.0493930i
$892$		0			0
$893$			95.3592i			0.106785i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−586.314	+	2199.10i	−0.653639	+	2.45161i
$898$		0			0
$899$	510.210			0.567530
$900$		0			0
$901$	−501.359			−0.556447
$902$		0			0
$903$	−685.191			−0.758794
$904$		0			0
$905$		0			0
$906$		0			0
$907$		−	852.611i		−	0.940034i	−0.882657	−	0.470017i	$-0.844248\pi$
$907$		−	852.611i		−	0.940034i	0.882657	−	0.470017i	$-0.155752\pi$
$908$		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
$913$	−16.6681			−0.0182564
$914$		0			0
$915$		0			0
$916$		0			0
$917$			1212.14i			1.32186i
$918$		0			0
$919$			1437.90i			1.56463i	0.622880	+	0.782317i	$0.285962\pi$
$919$			1437.90i			1.56463i	−0.622880	+	0.782317i	$0.714038\pi$
$920$		0			0
$921$	927.728			1.00731
$922$		0			0
$923$	−2016.16			−2.18436
$924$		0			0
$925$		0			0
$926$		0			0
$927$		−	2912.44i		−	3.14179i
$928$		0			0
$929$	−128.712			−0.138549			−0.0692746	−	0.997598i	$-0.522068\pi$
$929$	−128.712			−0.138549			−0.0692746	+	0.997598i	$0.522068\pi$
$930$		0			0
$931$		−	585.988i		−	0.629418i
$932$		0			0
$933$	3219.28			3.45046
$934$		0			0
$935$		0			0
$936$		0			0
$937$			376.018i			0.401300i	0.979663	+	0.200650i	$0.0643054\pi$
$937$			376.018i			0.401300i	−0.979663	+	0.200650i	$0.935695\pi$
$938$		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
$943$	183.081	−	686.685i	0.194148	−	0.728192i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−602.245			−0.635950			−0.317975	−	0.948099i	$-0.603003\pi$
$947$	−602.245			−0.635950			−0.317975	+	0.948099i	$0.603003\pi$
$948$		0			0
$949$	1867.79			1.96816
$950$		0			0
$951$	−1514.17			−1.59219
$952$		0			0
$953$		−	96.0530i		−	0.100790i	−0.998729	−	0.0503951i	$-0.983952\pi$
$953$		−	96.0530i		−	0.100790i	0.998729	−	0.0503951i	$-0.0160480\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$			56.8411i			0.0593951i
$958$		0			0
$959$	1459.67			1.52207
$960$		0			0
$961$	−770.448			−0.801715
$962$		0			0
$963$		−	1652.66i		−	1.71616i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	1635.70			1.69152			0.845760	−	0.533564i	$-0.179148\pi$
$967$	1635.70			1.69152			0.845760	+	0.533564i	$0.179148\pi$
$968$		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
$973$		−	242.742i		−	0.249478i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			775.876i			0.794141i	0.917788	+	0.397071i	$0.129973\pi$
$977$			775.876i			0.794141i	−0.917788	+	0.397071i	$0.870027\pi$
$978$		0			0
$979$	42.6713			0.0435866
$980$		0			0
$981$			3894.22i			3.96965i
$982$		0			0
$983$			1601.98i			1.62969i	0.579680	+	0.814844i	$0.303177\pi$
$983$			1601.98i			1.62969i	−0.579680	+	0.814844i	$0.696823\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$			1127.96i			1.14281i
$988$		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
$993$	−1371.85			−1.38152
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−338.386			−0.339404			−0.169702	−	0.985495i	$-0.554281\pi$
$997$	−338.386			−0.339404			−0.169702	+	0.985495i	$0.554281\pi$
$998$		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.f.c.1701.1	✓	16
5.2	odd	4		2300.3.d.c.1149.2		32
5.3	odd	4		2300.3.d.c.1149.31		32
5.4	even	2		2300.3.f.d.1701.16	yes	16
23.22	odd	2	inner	2300.3.f.c.1701.2	yes	16
115.22	even	4		2300.3.d.c.1149.32		32
115.68	even	4		2300.3.d.c.1149.1		32
115.114	odd	2		2300.3.f.d.1701.15	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2300.3.d.c.1149.1		32	115.68	even	4
2300.3.d.c.1149.2		32	5.2	odd	4
2300.3.d.c.1149.31		32	5.3	odd	4
2300.3.d.c.1149.32		32	115.22	even	4
2300.3.f.c.1701.1	✓	16	1.1	even	1	trivial
2300.3.f.c.1701.2	yes	16	23.22	odd	2	inner
2300.3.f.d.1701.15	yes	16	115.114	odd	2
2300.3.f.d.1701.16	yes	16	5.4	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:	\( N \)	\(=\)	\( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2300.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.43352 q^{3} -12.3431i q^{7} +20.5231 q^{9} +O(q^{10})\)	\(q-5.43352 q^{3} -12.3431i q^{7} +20.5231 q^{9} +0.283034i q^{11} -18.2116 q^{13} -24.7396i q^{17} +5.66988i q^{19} +67.0662i q^{21} +(-5.92519 + 22.2237i) q^{23} -62.6111 q^{27} -36.9609 q^{29} -13.8040 q^{31} -1.53787i q^{33} -48.0466i q^{37} +98.9528 q^{39} -30.8988 q^{41} +10.2166i q^{43} +16.8186 q^{47} -103.351 q^{49} +134.423i q^{51} -20.2654i q^{53} -30.8074i q^{57} -12.1433 q^{59} -109.987i q^{61} -253.318i q^{63} +46.9397i q^{67} +(32.1946 - 120.753i) q^{69} +110.708 q^{71} -102.561 q^{73} +3.49351 q^{77} -88.9924i q^{79} +155.491 q^{81} +58.8907i q^{83} +200.828 q^{87} -150.764i q^{89} +224.786i q^{91} +75.0045 q^{93} -37.2144i q^{97} +5.80875i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 64 q^{9}+O(q^{10}) \) 16 * q + 64 * q^9	\( 16 q + 64 q^{9} - 6 q^{13} + 8 q^{23} - 66 q^{27} - 6 q^{29} + 28 q^{31} + 74 q^{39} - 90 q^{41} - 40 q^{47} - 190 q^{49} - 174 q^{59} + 50 q^{69} + 116 q^{71} - 110 q^{73} - 198 q^{77} + 56 q^{81} + 362 q^{87} + 50 q^{93}+O(q^{100}) \) 16 * q + 64 * q^9 - 6 * q^13 + 8 * q^23 - 66 * q^27 - 6 * q^29 + 28 * q^31 + 74 * q^39 - 90 * q^41 - 40 * q^47 - 190 * q^49 - 174 * q^59 + 50 * q^69 + 116 * q^71 - 110 * q^73 - 198 * q^77 + 56 * q^81 + 362 * q^87 + 50 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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