LMFDB - Embedded newform 5700.2.f.q.3649.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5700,2,Mod(3649,5700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5700, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5700.3649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5700 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5700.f (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:	$45.5147291521$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.5089536.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 16x^{2} - 24x + 18 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 16x^2 - 24*x + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 1140)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3649.4
Root			$-1.33641 - 1.33641i$ of defining polynomial
Character	$\chi$	$=$	5700.3649
Dual form			5700.2.f.q.3649.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -4.67282i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} -4.67282i q^{7} -1.00000 q^{9} +5.81681 q^{11} -2.67282i q^{13} +6.48963i q^{17} +1.00000 q^{19} +4.67282 q^{21} +8.20166i q^{23} -1.00000i q^{27} +3.81681 q^{29} -6.20166 q^{31} +5.81681i q^{33} +12.0185i q^{37} +2.67282 q^{39} +0.183190 q^{41} -6.96080i q^{43} +4.20166i q^{47} -14.8353 q^{49} -6.48963 q^{51} +10.7776i q^{53} +1.00000i q^{57} +7.05767 q^{59} +5.14399 q^{61} +4.67282i q^{63} -10.6913i q^{67} -8.20166 q^{69} +11.0577 q^{71} -4.28797i q^{73} -27.1809i q^{77} +6.28797 q^{79} +1.00000 q^{81} +1.14399i q^{83} +3.81681i q^{87} +3.81681 q^{89} -12.4896 q^{91} -6.20166i q^{93} +6.67282i q^{97} -5.81681 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{9}+O(q^{10}) $ 6 * q - 6 * q^9	$ 6 q - 6 q^{9} + 12 q^{11} + 6 q^{19} + 8 q^{21} - 4 q^{39} + 24 q^{41} - 6 q^{49} + 4 q^{51} + 8 q^{59} + 28 q^{61} - 12 q^{69} + 32 q^{71} + 32 q^{79} + 6 q^{81} - 32 q^{91} - 12 q^{99}+O(q^{100}) $ 6 * q - 6 * q^9 + 12 * q^11 + 6 * q^19 + 8 * q^21 - 4 * q^39 + 24 * q^41 - 6 * q^49 + 4 * q^51 + 8 * q^59 + 28 * q^61 - 12 * q^69 + 32 * q^71 + 32 * q^79 + 6 * q^81 - 32 * q^91 - 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5700\mathbb{Z}\right)^\times$.

$n$	$1901$	$2851$	$3877$	$4201$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 4.67282i	− 1.76616i	−0.469221	−	0.883081i	$-0.655465\pi$
$7$	− 4.67282i	− 1.76616i	0.469221	−	0.883081i	$-0.344535\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	5.81681	1.75383	0.876917	−	0.480642i	$-0.159596\pi$
$11$	5.81681	1.75383	0.876917	+	0.480642i	$0.159596\pi$
$12$	0	0
$13$	− 2.67282i	− 0.741308i	−0.928771	−	0.370654i	$-0.879134\pi$
$13$	− 2.67282i	− 0.741308i	0.928771	−	0.370654i	$-0.120866\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.48963i	1.57397i	0.616974	+	0.786984i	$0.288358\pi$
$17$	6.48963i	1.57397i	−0.616974	+	0.786984i	$0.711642\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	4.67282	1.01969
$22$	0	0
$23$	8.20166i	1.71016i	0.518492	+	0.855082i	$0.326493\pi$
$23$	8.20166i	1.71016i	−0.518492	+	0.855082i	$0.673507\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	3.81681	0.708764	0.354382	−	0.935101i	$-0.384691\pi$
$29$	3.81681	0.708764	0.354382	+	0.935101i	$0.384691\pi$
$30$	0	0
$31$	−6.20166	−1.11385	−0.556926	−	0.830562i	$-0.688019\pi$
$31$	−6.20166	−1.11385	−0.556926	+	0.830562i	$0.688019\pi$
$32$	0	0
$33$	5.81681i	1.01258i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	12.0185i	1.97582i	0.155014	+	0.987912i	$0.450458\pi$
$37$	12.0185i	1.97582i	−0.155014	+	0.987912i	$0.549542\pi$
$38$	0	0
$39$	2.67282	0.427994
$40$	0	0
$41$	0.183190	0.0286094	0.0143047	−	0.999898i	$-0.495447\pi$
$41$	0.183190	0.0286094	0.0143047	+	0.999898i	$0.495447\pi$
$42$	0	0
$43$	− 6.96080i	− 1.06151i	−0.847525	−	0.530756i	$-0.821908\pi$
$43$	− 6.96080i	− 1.06151i	0.847525	−	0.530756i	$-0.178092\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.20166i	0.612875i	0.951891	+	0.306438i	$0.0991371\pi$
$47$	4.20166i	0.612875i	−0.951891	+	0.306438i	$0.900863\pi$
$48$	0	0
$49$	−14.8353	−2.11933
$50$	0	0
$51$	−6.48963	−0.908731
$52$	0	0
$53$	10.7776i	1.48042i	0.672377	+	0.740209i	$0.265273\pi$
$53$	10.7776i	1.48042i	−0.672377	+	0.740209i	$0.734727\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.00000i	0.132453i
$58$	0	0
$59$	7.05767	0.918831	0.459415	−	0.888221i	$-0.348059\pi$
$59$	7.05767	0.918831	0.459415	+	0.888221i	$0.348059\pi$
$60$	0	0
$61$	5.14399	0.658620	0.329310	−	0.944222i	$-0.393184\pi$
$61$	5.14399	0.658620	0.329310	+	0.944222i	$0.393184\pi$
$62$	0	0
$63$	4.67282i	0.588720i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 10.6913i	− 1.30615i	−0.757293	−	0.653075i	$-0.773479\pi$
$67$	− 10.6913i	− 1.30615i	0.757293	−	0.653075i	$-0.226521\pi$
$68$	0	0
$69$	−8.20166	−0.987364
$70$	0	0
$71$	11.0577	1.31230	0.656152	−	0.754629i	$-0.272183\pi$
$71$	11.0577	1.31230	0.656152	+	0.754629i	$0.272183\pi$
$72$	0	0
$73$	− 4.28797i	− 0.501869i	−0.968004	−	0.250935i	$-0.919262\pi$
$73$	− 4.28797i	− 0.501869i	0.968004	−	0.250935i	$-0.0807379\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 27.1809i	− 3.09755i
$78$	0	0
$79$	6.28797	0.707452	0.353726	−	0.935349i	$-0.384914\pi$
$79$	6.28797	0.707452	0.353726	+	0.935349i	$0.384914\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	1.14399i	0.125569i	0.998027	+	0.0627844i	$0.0199981\pi$
$83$	1.14399i	0.125569i	−0.998027	+	0.0627844i	$0.980002\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	3.81681i	0.409205i
$88$	0	0
$89$	3.81681	0.404581	0.202291	−	0.979326i	$-0.435161\pi$
$89$	3.81681	0.404581	0.202291	+	0.979326i	$0.435161\pi$
$90$	0	0
$91$	−12.4896	−1.30927
$92$	0	0
$93$	− 6.20166i	− 0.643082i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	6.67282i	0.677523i	0.940872	+	0.338761i	$0.110008\pi$
$97$	6.67282i	0.677523i	−0.940872	+	0.338761i	$0.889992\pi$
$98$	0	0
$99$	−5.81681	−0.584611
$100$	0	0
$101$	0.654353	0.0651105	0.0325553	−	0.999470i	$-0.489636\pi$
$101$	0.654353	0.0651105	0.0325553	+	0.999470i	$0.489636\pi$
$102$	0	0
$103$	− 1.71203i	− 0.168691i	−0.996437	−	0.0843455i	$-0.973120\pi$
$103$	− 1.71203i	− 0.168691i	0.996437	−	0.0843455i	$-0.0268799\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	9.05767	0.867568	0.433784	−	0.901017i	$-0.357178\pi$
$109$	9.05767	0.867568	0.433784	+	0.901017i	$0.357178\pi$
$110$	0	0
$111$	−12.0185	−1.14074
$112$	0	0
$113$	11.5473i	1.08628i	0.839642	+	0.543140i	$0.182765\pi$
$113$	11.5473i	1.08628i	−0.839642	+	0.543140i	$0.817235\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.67282i	0.247103i
$118$	0	0
$119$	30.3249	2.77988
$120$	0	0
$121$	22.8353	2.07593
$122$	0	0
$123$	0.183190i	0.0165177i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	4.00000i	0.354943i	0.984126	+	0.177471i	$0.0567917\pi$
$127$	4.00000i	0.354943i	−0.984126	+	0.177471i	$0.943208\pi$
$128$	0	0
$129$	6.96080	0.612864
$130$	0	0
$131$	2.18319	0.190746	0.0953731	−	0.995442i	$-0.469596\pi$
$131$	2.18319	0.190746	0.0953731	+	0.995442i	$0.469596\pi$
$132$	0	0
$133$	− 4.67282i	− 0.405185i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 2.00000i	− 0.170872i	−0.996344	−	0.0854358i	$-0.972772\pi$
$137$	− 2.00000i	− 0.170872i	0.996344	−	0.0854358i	$-0.0272282\pi$
$138$	0	0
$139$	4.36638	0.370351	0.185176	−	0.982705i	$-0.440715\pi$
$139$	4.36638	0.370351	0.185176	+	0.982705i	$0.440715\pi$
$140$	0	0
$141$	−4.20166	−0.353844
$142$	0	0
$143$	− 15.5473i	− 1.30013i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 14.8353i	− 1.22359i
$148$	0	0
$149$	−4.28797	−0.351284	−0.175642	−	0.984454i	$-0.556200\pi$
$149$	−4.28797	−0.351284	−0.175642	+	0.984454i	$0.556200\pi$
$150$	0	0
$151$	16.8930	1.37473	0.687365	−	0.726313i	$-0.258767\pi$
$151$	16.8930	1.37473	0.687365	+	0.726313i	$0.258767\pi$
$152$	0	0
$153$	− 6.48963i	− 0.524656i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 16.3249i	− 1.30287i	−0.758704	−	0.651435i	$-0.774167\pi$
$157$	− 16.3249i	− 1.30287i	0.758704	−	0.651435i	$-0.225833\pi$
$158$	0	0
$159$	−10.7776	−0.854720
$160$	0	0
$161$	38.3249	3.02043
$162$	0	0
$163$	− 18.0185i	− 1.41132i	−0.708553	−	0.705658i	$-0.750652\pi$
$163$	− 18.0185i	− 1.41132i	0.708553	−	0.705658i	$-0.249348\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	7.14399i	0.552818i	0.961040	+	0.276409i	$0.0891445\pi$
$167$	7.14399i	0.552818i	−0.961040	+	0.276409i	$0.910856\pi$
$168$	0	0
$169$	5.85601	0.450463
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	− 22.4112i	− 1.70389i	−0.523628	−	0.851947i	$-0.675422\pi$
$173$	− 22.4112i	− 1.70389i	0.523628	−	0.851947i	$-0.324578\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	7.05767i	0.530487i
$178$	0	0
$179$	−16.0369	−1.19866	−0.599329	−	0.800503i	$-0.704566\pi$
$179$	−16.0369	−1.19866	−0.599329	+	0.800503i	$0.704566\pi$
$180$	0	0
$181$	−11.9216	−0.886125	−0.443063	−	0.896491i	$-0.646108\pi$
$181$	−11.9216	−0.886125	−0.443063	+	0.896491i	$0.646108\pi$
$182$	0	0
$183$	5.14399i	0.380254i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	37.7490i	2.76048i
$188$	0	0
$189$	−4.67282	−0.339898
$190$	0	0
$191$	8.47116	0.612952	0.306476	−	0.951878i	$-0.400850\pi$
$191$	8.47116	0.612952	0.306476	+	0.951878i	$0.400850\pi$
$192$	0	0
$193$	− 13.7305i	− 0.988343i	−0.869364	−	0.494171i	$-0.835472\pi$
$193$	− 13.7305i	− 0.988343i	0.869364	−	0.494171i	$-0.164528\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	19.4689i	1.38710i	0.720408	+	0.693551i	$0.243955\pi$
$197$	19.4689i	1.38710i	−0.720408	+	0.693551i	$0.756045\pi$
$198$	0	0
$199$	0.942326	0.0667997	0.0333998	−	0.999442i	$-0.489367\pi$
$199$	0.942326	0.0667997	0.0333998	+	0.999442i	$0.489367\pi$
$200$	0	0
$201$	10.6913	0.754106
$202$	0	0
$203$	− 17.8353i	− 1.25179i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 8.20166i	− 0.570055i
$208$	0	0
$209$	5.81681	0.402357
$210$	0	0
$211$	−5.71203	−0.393232	−0.196616	−	0.980481i	$-0.562995\pi$
$211$	−5.71203	−0.393232	−0.196616	+	0.980481i	$0.562995\pi$
$212$	0	0
$213$	11.0577i	0.757659i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	28.9793i	1.96724i
$218$	0	0
$219$	4.28797	0.289754
$220$	0	0
$221$	17.3456	1.16679
$222$	0	0
$223$	− 1.71203i	− 0.114646i	−0.998356	−	0.0573229i	$-0.981744\pi$
$223$	− 1.71203i	− 0.114646i	0.998356	−	0.0573229i	$-0.0182565\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0.452692i	0.0300462i	0.999887	+	0.0150231i	$0.00478218\pi$
$227$	0.452692i	0.0300462i	−0.999887	+	0.0150231i	$0.995218\pi$
$228$	0	0
$229$	14.1233	0.933291	0.466645	−	0.884444i	$-0.345462\pi$
$229$	14.1233	0.933291	0.466645	+	0.884444i	$0.345462\pi$
$230$	0	0
$231$	27.1809	1.78837
$232$	0	0
$233$	− 11.7120i	− 0.767280i	−0.923483	−	0.383640i	$-0.874670\pi$
$233$	− 11.7120i	− 0.767280i	0.923483	−	0.383640i	$-0.125330\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	6.28797i	0.408448i
$238$	0	0
$239$	−25.4874	−1.64864	−0.824321	−	0.566123i	$-0.808443\pi$
$239$	−25.4874	−1.64864	−0.824321	+	0.566123i	$0.808443\pi$
$240$	0	0
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 2.67282i	− 0.170068i
$248$	0	0
$249$	−1.14399	−0.0724972
$250$	0	0
$251$	3.12552	0.197281	0.0986404	−	0.995123i	$-0.468551\pi$
$251$	3.12552	0.197281	0.0986404	+	0.995123i	$0.468551\pi$
$252$	0	0
$253$	47.7075i	2.99935i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	14.5680i	0.908729i	0.890816	+	0.454365i	$0.150134\pi$
$257$	14.5680i	0.908729i	−0.890816	+	0.454365i	$0.849866\pi$
$258$	0	0
$259$	56.1602	3.48962
$260$	0	0
$261$	−3.81681	−0.236255
$262$	0	0
$263$	5.54731i	0.342062i	0.985266	+	0.171031i	$0.0547098\pi$
$263$	5.54731i	0.342062i	−0.985266	+	0.171031i	$0.945290\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	3.81681i	0.233585i
$268$	0	0
$269$	−29.1625	−1.77807	−0.889033	−	0.457843i	$-0.848622\pi$
$269$	−29.1625	−1.77807	−0.889033	+	0.457843i	$0.848622\pi$
$270$	0	0
$271$	1.92159	0.116728	0.0583642	−	0.998295i	$-0.481412\pi$
$271$	1.92159	0.116728	0.0583642	+	0.998295i	$0.481412\pi$
$272$	0	0
$273$	− 12.4896i	− 0.755907i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	26.9793i	1.62103i	0.585720	+	0.810514i	$0.300812\pi$
$277$	26.9793i	1.62103i	−0.585720	+	0.810514i	$0.699188\pi$
$278$	0	0
$279$	6.20166	0.371284
$280$	0	0
$281$	4.39276	0.262050	0.131025	−	0.991379i	$-0.458173\pi$
$281$	4.39276	0.262050	0.131025	+	0.991379i	$0.458173\pi$
$282$	0	0
$283$	− 24.8824i	− 1.47910i	−0.673099	−	0.739552i	$-0.735037\pi$
$283$	− 24.8824i	− 1.47910i	0.673099	−	0.739552i	$-0.264963\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 0.856013i	− 0.0505289i
$288$	0	0
$289$	−25.1153	−1.47737
$290$	0	0
$291$	−6.67282	−0.391168
$292$	0	0
$293$	1.83528i	0.107218i	0.998562	+	0.0536091i	$0.0170725\pi$
$293$	1.83528i	0.107218i	−0.998562	+	0.0536091i	$0.982927\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 5.81681i	− 0.337526i
$298$	0	0
$299$	21.9216	1.26776
$300$	0	0
$301$	−32.5266	−1.87480
$302$	0	0
$303$	0.654353i	0.0375916i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	12.3664i	0.705787i	0.935664	+	0.352893i	$0.114802\pi$
$307$	12.3664i	0.705787i	−0.935664	+	0.352893i	$0.885198\pi$
$308$	0	0
$309$	1.71203	0.0973938
$310$	0	0
$311$	−6.79608	−0.385370	−0.192685	−	0.981261i	$-0.561720\pi$
$311$	−6.79608	−0.385370	−0.192685	+	0.981261i	$0.561720\pi$
$312$	0	0
$313$	5.23030i	0.295634i	0.989015	+	0.147817i	$0.0472247\pi$
$313$	5.23030i	0.295634i	−0.989015	+	0.147817i	$0.952775\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 31.7569i	− 1.78364i	−0.452387	−	0.891822i	$-0.649427\pi$
$317$	− 31.7569i	− 1.78364i	0.452387	−	0.891822i	$-0.350573\pi$
$318$	0	0
$319$	22.2017	1.24305
$320$	0	0
$321$	0	0
$322$	0	0
$323$	6.48963i	0.361093i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	9.05767i	0.500891i
$328$	0	0
$329$	19.6336	1.08244
$330$	0	0
$331$	12.8930	0.708661	0.354330	−	0.935120i	$-0.384709\pi$
$331$	12.8930	0.708661	0.354330	+	0.935120i	$0.384709\pi$
$332$	0	0
$333$	− 12.0185i	− 0.658608i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	24.2280i	1.31979i	0.751360	+	0.659893i	$0.229398\pi$
$337$	24.2280i	1.31979i	−0.751360	+	0.659893i	$0.770602\pi$
$338$	0	0
$339$	−11.5473	−0.627164
$340$	0	0
$341$	−36.0739	−1.95351
$342$	0	0
$343$	36.6129i	1.97691i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	6.45269i	0.346399i	0.984887	+	0.173199i	$0.0554105\pi$
$347$	6.45269i	0.346399i	−0.984887	+	0.173199i	$0.944590\pi$
$348$	0	0
$349$	−11.3087	−0.605341	−0.302671	−	0.953095i	$-0.597878\pi$
$349$	−11.3087	−0.605341	−0.302671	+	0.953095i	$0.597878\pi$
$350$	0	0
$351$	−2.67282	−0.142665
$352$	0	0
$353$	− 32.6913i	− 1.73998i	−0.493068	−	0.869991i	$-0.664124\pi$
$353$	− 32.6913i	− 1.73998i	0.493068	−	0.869991i	$-0.335876\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	30.3249i	1.60496i
$358$	0	0
$359$	−5.04711	−0.266376	−0.133188	−	0.991091i	$-0.542521\pi$
$359$	−5.04711	−0.266376	−0.133188	+	0.991091i	$0.542521\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	22.8353i	1.19854i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	34.2280i	1.78669i	0.449372	+	0.893345i	$0.351648\pi$
$367$	34.2280i	1.78669i	−0.449372	+	0.893345i	$0.648352\pi$
$368$	0	0
$369$	−0.183190	−0.00953648
$370$	0	0
$371$	50.3619	2.61466
$372$	0	0
$373$	26.3064i	1.36210i	0.732239	+	0.681048i	$0.238476\pi$
$373$	26.3064i	1.36210i	−0.732239	+	0.681048i	$0.761524\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 10.2017i	− 0.525412i
$378$	0	0
$379$	38.6050	1.98300	0.991502	−	0.130089i	$-0.0415262\pi$
$379$	38.6050	1.98300	0.991502	+	0.130089i	$0.0415262\pi$
$380$	0	0
$381$	−4.00000	−0.204926
$382$	0	0
$383$	8.40332i	0.429390i	0.976681	+	0.214695i	$0.0688757\pi$
$383$	8.40332i	0.429390i	−0.976681	+	0.214695i	$0.931124\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	6.96080i	0.353837i
$388$	0	0
$389$	−12.2880	−0.623025	−0.311512	−	0.950242i	$-0.600836\pi$
$389$	−12.2880	−0.623025	−0.311512	+	0.950242i	$0.600836\pi$
$390$	0	0
$391$	−53.2258	−2.69174
$392$	0	0
$393$	2.18319i	0.110127i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 34.6498i	− 1.73903i	−0.493911	−	0.869513i	$-0.664433\pi$
$397$	− 34.6498i	− 1.73903i	0.493911	−	0.869513i	$-0.335567\pi$
$398$	0	0
$399$	4.67282	0.233934
$400$	0	0
$401$	−5.73840	−0.286562	−0.143281	−	0.989682i	$-0.545765\pi$
$401$	−5.73840	−0.286562	−0.143281	+	0.989682i	$0.545765\pi$
$402$	0	0
$403$	16.5759i	0.825707i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	69.9092i	3.46527i
$408$	0	0
$409$	−28.9009	−1.42906	−0.714528	−	0.699607i	$-0.753358\pi$
$409$	−28.9009	−1.42906	−0.714528	+	0.699607i	$0.753358\pi$
$410$	0	0
$411$	2.00000	0.0986527
$412$	0	0
$413$	− 32.9793i	− 1.62280i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	4.36638i	0.213823i
$418$	0	0
$419$	4.87448	0.238134	0.119067	−	0.992886i	$-0.462010\pi$
$419$	4.87448	0.238134	0.119067	+	0.992886i	$0.462010\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	0	0
$423$	− 4.20166i	− 0.204292i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 24.0369i	− 1.16323i
$428$	0	0
$429$	15.5473	0.750631
$430$	0	0
$431$	10.6544	0.513202	0.256601	−	0.966517i	$-0.417397\pi$
$431$	10.6544	0.513202	0.256601	+	0.966517i	$0.417397\pi$
$432$	0	0
$433$	− 6.88239i	− 0.330747i	−0.986231	−	0.165373i	$-0.947117\pi$
$433$	− 6.88239i	− 0.330747i	0.986231	−	0.165373i	$-0.0528829\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	8.20166i	0.392339i
$438$	0	0
$439$	−40.0739	−1.91262	−0.956311	−	0.292351i	$-0.905562\pi$
$439$	−40.0739	−1.91262	−0.956311	+	0.292351i	$0.905562\pi$
$440$	0	0
$441$	14.8353	0.706442
$442$	0	0
$443$	− 30.8930i	− 1.46777i	−0.679274	−	0.733884i	$-0.737705\pi$
$443$	− 30.8930i	− 1.46777i	0.679274	−	0.733884i	$-0.262295\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 4.28797i	− 0.202814i
$448$	0	0
$449$	12.7961	0.603884	0.301942	−	0.953326i	$-0.402365\pi$
$449$	12.7961	0.603884	0.301942	+	0.953326i	$0.402365\pi$
$450$	0	0
$451$	1.06558	0.0501762
$452$	0	0
$453$	16.8930i	0.793700i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 28.3249i	− 1.32498i	−0.749069	−	0.662492i	$-0.769499\pi$
$457$	− 28.3249i	− 1.32498i	0.749069	−	0.662492i	$-0.230501\pi$
$458$	0	0
$459$	6.48963	0.302910
$460$	0	0
$461$	1.42405	0.0663248	0.0331624	−	0.999450i	$-0.489442\pi$
$461$	1.42405	0.0663248	0.0331624	+	0.999450i	$0.489442\pi$
$462$	0	0
$463$	31.7305i	1.47464i	0.675543	+	0.737321i	$0.263909\pi$
$463$	31.7305i	1.47464i	−0.675543	+	0.737321i	$0.736091\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	14.4896i	0.670500i	0.942129	+	0.335250i	$0.108821\pi$
$467$	14.4896i	0.670500i	−0.942129	+	0.335250i	$0.891179\pi$
$468$	0	0
$469$	−49.9585	−2.30687
$470$	0	0
$471$	16.3249	0.752212
$472$	0	0
$473$	− 40.4896i	− 1.86172i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 10.7776i	− 0.493473i
$478$	0	0
$479$	31.1994	1.42554	0.712768	−	0.701399i	$-0.247441\pi$
$479$	31.1994	1.42554	0.712768	+	0.701399i	$0.247441\pi$
$480$	0	0
$481$	32.1233	1.46469
$482$	0	0
$483$	38.3249i	1.74384i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 38.3249i	− 1.73667i	−0.495980	−	0.868334i	$-0.665191\pi$
$487$	− 38.3249i	− 1.73667i	0.495980	−	0.868334i	$-0.334809\pi$
$488$	0	0
$489$	18.0185	0.814823
$490$	0	0
$491$	15.1625	0.684272	0.342136	−	0.939650i	$-0.388850\pi$
$491$	15.1625	0.684272	0.342136	+	0.939650i	$0.388850\pi$
$492$	0	0
$493$	24.7697i	1.11557i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 51.6706i	− 2.31774i
$498$	0	0
$499$	26.9009	1.20425	0.602124	−	0.798403i	$-0.294321\pi$
$499$	26.9009	1.20425	0.602124	+	0.798403i	$0.294321\pi$
$500$	0	0
$501$	−7.14399	−0.319170
$502$	0	0
$503$	34.8560i	1.55415i	0.629406	+	0.777076i	$0.283298\pi$
$503$	34.8560i	1.55415i	−0.629406	+	0.777076i	$0.716702\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	5.85601i	0.260075i
$508$	0	0
$509$	12.7591	0.565539	0.282769	−	0.959188i	$-0.408747\pi$
$509$	12.7591	0.565539	0.282769	+	0.959188i	$0.408747\pi$
$510$	0	0
$511$	−20.0369	−0.886382
$512$	0	0
$513$	− 1.00000i	− 0.0441511i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	24.4403i	1.07488i
$518$	0	0
$519$	22.4112	0.983744
$520$	0	0
$521$	5.52884	0.242223	0.121111	−	0.992639i	$-0.461354\pi$
$521$	5.52884	0.242223	0.121111	+	0.992639i	$0.461354\pi$
$522$	0	0
$523$	11.0577i	0.483518i	0.970336	+	0.241759i	$0.0777244\pi$
$523$	11.0577i	0.483518i	−0.970336	+	0.241759i	$0.922276\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 40.2465i	− 1.75317i
$528$	0	0
$529$	−44.2672	−1.92466
$530$	0	0
$531$	−7.05767	−0.306277
$532$	0	0
$533$	− 0.489634i	− 0.0212084i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 16.0369i	− 0.692045i
$538$	0	0
$539$	−86.2940	−3.71695
$540$	0	0
$541$	−23.9585	−1.03006	−0.515029	−	0.857173i	$-0.672219\pi$
$541$	−23.9585	−1.03006	−0.515029	+	0.857173i	$0.672219\pi$
$542$	0	0
$543$	− 11.9216i	− 0.511605i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 0.209567i	− 0.00896042i	−0.999990	−	0.00448021i	$-0.998574\pi$
$547$	− 0.209567i	− 0.00896042i	0.999990	−	0.00448021i	$-0.00142610\pi$
$548$	0	0
$549$	−5.14399	−0.219540
$550$	0	0
$551$	3.81681	0.162602
$552$	0	0
$553$	− 29.3826i	− 1.24947i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	14.9793i	0.634692i	0.948310	+	0.317346i	$0.102792\pi$
$557$	14.9793i	0.634692i	−0.948310	+	0.317346i	$0.897208\pi$
$558$	0	0
$559$	−18.6050	−0.786907
$560$	0	0
$561$	−37.7490	−1.59376
$562$	0	0
$563$	16.6992i	0.703787i	0.936040	+	0.351894i	$0.114462\pi$
$563$	16.6992i	0.703787i	−0.936040	+	0.351894i	$0.885538\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 4.67282i	− 0.196240i
$568$	0	0
$569$	−34.5450	−1.44820	−0.724102	−	0.689693i	$-0.757745\pi$
$569$	−34.5450	−1.44820	−0.724102	+	0.689693i	$0.757745\pi$
$570$	0	0
$571$	−5.92159	−0.247811	−0.123905	−	0.992294i	$-0.539542\pi$
$571$	−5.92159	−0.247811	−0.123905	+	0.992294i	$0.539542\pi$
$572$	0	0
$573$	8.47116i	0.353888i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 24.6544i	− 1.02637i	−0.858277	−	0.513187i	$-0.828465\pi$
$577$	− 24.6544i	− 1.02637i	0.858277	−	0.513187i	$-0.171535\pi$
$578$	0	0
$579$	13.7305	0.570620
$580$	0	0
$581$	5.34565	0.221775
$582$	0	0
$583$	62.6913i	2.59641i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 12.2017i	− 0.503616i	−0.967777	−	0.251808i	$-0.918975\pi$
$587$	− 12.2017i	− 0.503616i	0.967777	−	0.251808i	$-0.0810252\pi$
$588$	0	0
$589$	−6.20166	−0.255535
$590$	0	0
$591$	−19.4689	−0.800844
$592$	0	0
$593$	− 1.02073i	− 0.0419164i	−0.999780	−	0.0209582i	$-0.993328\pi$
$593$	− 1.02073i	− 0.0419164i	0.999780	−	0.0209582i	$-0.00667170\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0.942326i	0.0385668i
$598$	0	0
$599$	−7.42405	−0.303339	−0.151669	−	0.988431i	$-0.548465\pi$
$599$	−7.42405	−0.303339	−0.151669	+	0.988431i	$0.548465\pi$
$600$	0	0
$601$	19.3456	0.789125	0.394563	−	0.918869i	$-0.370896\pi$
$601$	19.3456	0.789125	0.394563	+	0.918869i	$0.370896\pi$
$602$	0	0
$603$	10.6913i	0.435383i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	16.0369i	0.650919i	0.945556	+	0.325460i	$0.105519\pi$
$607$	16.0369i	0.650919i	−0.945556	+	0.325460i	$0.894481\pi$
$608$	0	0
$609$	17.8353	0.722722
$610$	0	0
$611$	11.2303	0.454329
$612$	0	0
$613$	− 36.1523i	− 1.46018i	−0.683352	−	0.730089i	$-0.739479\pi$
$613$	− 36.1523i	− 1.46018i	0.683352	−	0.730089i	$-0.260521\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	40.4482i	1.62838i	0.580597	+	0.814191i	$0.302819\pi$
$617$	40.4482i	1.62838i	−0.580597	+	0.814191i	$0.697181\pi$
$618$	0	0
$619$	4.19376	0.168561	0.0842806	−	0.996442i	$-0.473141\pi$
$619$	4.19376	0.168561	0.0842806	+	0.996442i	$0.473141\pi$
$620$	0	0
$621$	8.20166	0.329121
$622$	0	0
$623$	− 17.8353i	− 0.714555i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	5.81681i	0.232301i
$628$	0	0
$629$	−77.9955	−3.10988
$630$	0	0
$631$	5.30871	0.211336	0.105668	−	0.994401i	$-0.466302\pi$
$631$	5.30871	0.211336	0.105668	+	0.994401i	$0.466302\pi$
$632$	0	0
$633$	− 5.71203i	− 0.227033i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	39.6521i	1.57107i
$638$	0	0
$639$	−11.0577	−0.437435
$640$	0	0
$641$	−7.24086	−0.285997	−0.142998	−	0.989723i	$-0.545674\pi$
$641$	−7.24086	−0.285997	−0.142998	+	0.989723i	$0.545674\pi$
$642$	0	0
$643$	2.38485i	0.0940493i	0.998894	+	0.0470247i	$0.0149739\pi$
$643$	2.38485i	0.0940493i	−0.998894	+	0.0470247i	$0.985026\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 1.47342i	− 0.0579263i	−0.999580	−	0.0289631i	$-0.990779\pi$
$647$	− 1.47342i	− 0.0579263i	0.999580	−	0.0289631i	$-0.00922054\pi$
$648$	0	0
$649$	41.0532	1.61148
$650$	0	0
$651$	−28.9793	−1.13579
$652$	0	0
$653$	− 25.9137i	− 1.01408i	−0.861922	−	0.507040i	$-0.830739\pi$
$653$	− 25.9137i	− 1.01408i	0.861922	−	0.507040i	$-0.169261\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	4.28797i	0.167290i
$658$	0	0
$659$	24.9793	0.973054	0.486527	−	0.873666i	$-0.338264\pi$
$659$	24.9793	0.973054	0.486527	+	0.873666i	$0.338264\pi$
$660$	0	0
$661$	39.4195	1.53324	0.766621	−	0.642100i	$-0.221937\pi$
$661$	39.4195	1.53324	0.766621	+	0.642100i	$0.221937\pi$
$662$	0	0
$663$	17.3456i	0.673649i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	31.3042i	1.21210i
$668$	0	0
$669$	1.71203	0.0661908
$670$	0	0
$671$	29.9216	1.15511
$672$	0	0
$673$	12.0185i	0.463278i	0.972802	+	0.231639i	$0.0744088\pi$
$673$	12.0185i	0.463278i	−0.972802	+	0.231639i	$0.925591\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 22.2017i	− 0.853279i	−0.904422	−	0.426640i	$-0.859697\pi$
$677$	− 22.2017i	− 0.853279i	0.904422	−	0.426640i	$-0.140303\pi$
$678$	0	0
$679$	31.1809	1.19661
$680$	0	0
$681$	−0.452692	−0.0173472
$682$	0	0
$683$	− 32.0739i	− 1.22727i	−0.789589	−	0.613637i	$-0.789706\pi$
$683$	− 32.0739i	− 1.22727i	0.789589	−	0.613637i	$-0.210294\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	14.1233i	0.538836i
$688$	0	0
$689$	28.8066	1.09745
$690$	0	0
$691$	44.0369	1.67524	0.837622	−	0.546250i	$-0.183945\pi$
$691$	44.0369	1.67524	0.837622	+	0.546250i	$0.183945\pi$
$692$	0	0
$693$	27.1809i	1.03252i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1.18883i	0.0450303i
$698$	0	0
$699$	11.7120	0.442990
$700$	0	0
$701$	−18.9793	−0.716837	−0.358419	−	0.933561i	$-0.616684\pi$
$701$	−18.9793	−0.716837	−0.358419	+	0.933561i	$0.616684\pi$
$702$	0	0
$703$	12.0185i	0.453285i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 3.05767i	− 0.114996i
$708$	0	0
$709$	2.12325	0.0797405	0.0398702	−	0.999205i	$-0.487306\pi$
$709$	2.12325	0.0797405	0.0398702	+	0.999205i	$0.487306\pi$
$710$	0	0
$711$	−6.28797	−0.235817
$712$	0	0
$713$	− 50.8639i	− 1.90487i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 25.4874i	− 0.951843i
$718$	0	0
$719$	11.5658	0.431331	0.215665	−	0.976467i	$-0.430808\pi$
$719$	11.5658	0.431331	0.215665	+	0.976467i	$0.430808\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	0	0
$723$	14.0000i	0.520666i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 32.3064i	− 1.19818i	−0.800682	−	0.599090i	$-0.795529\pi$
$727$	− 32.3064i	− 1.19818i	0.800682	−	0.599090i	$-0.204471\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	45.1730	1.67078
$732$	0	0
$733$	18.4033i	0.679742i	0.940472	+	0.339871i	$0.110383\pi$
$733$	18.4033i	0.679742i	−0.940472	+	0.339871i	$0.889617\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 62.1892i	− 2.29077i
$738$	0	0
$739$	28.0000	1.03000	0.514998	−	0.857191i	$-0.327793\pi$
$739$	28.0000	1.03000	0.514998	+	0.857191i	$0.327793\pi$
$740$	0	0
$741$	2.67282	0.0981886
$742$	0	0
$743$	33.5552i	1.23102i	0.788129	+	0.615511i	$0.211050\pi$
$743$	33.5552i	1.23102i	−0.788129	+	0.615511i	$0.788950\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 1.14399i	− 0.0418563i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	9.06558	0.330808	0.165404	−	0.986226i	$-0.447107\pi$
$751$	9.06558	0.330808	0.165404	+	0.986226i	$0.447107\pi$
$752$	0	0
$753$	3.12552i	0.113900i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 49.1316i	− 1.78572i	−0.450337	−	0.892858i	$-0.648696\pi$
$757$	− 49.1316i	− 1.78572i	0.450337	−	0.892858i	$-0.351304\pi$
$758$	0	0
$759$	−47.7075	−1.73167
$760$	0	0
$761$	−26.4033	−0.957120	−0.478560	−	0.878055i	$-0.658841\pi$
$761$	−26.4033	−0.957120	−0.478560	+	0.878055i	$0.658841\pi$
$762$	0	0
$763$	− 42.3249i	− 1.53226i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 18.8639i	− 0.681137i
$768$	0	0
$769$	−16.6913	−0.601903	−0.300952	−	0.953639i	$-0.597304\pi$
$769$	−16.6913	−0.601903	−0.300952	+	0.953639i	$0.597304\pi$
$770$	0	0
$771$	−14.5680	−0.524655
$772$	0	0
$773$	37.8722i	1.36217i	0.732205	+	0.681085i	$0.238491\pi$
$773$	37.8722i	1.36217i	−0.732205	+	0.681085i	$0.761509\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	56.1602i	2.01474i
$778$	0	0
$779$	0.183190	0.00656345
$780$	0	0
$781$	64.3204	2.30156
$782$	0	0
$783$	− 3.81681i	− 0.136402i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	14.4817i	0.516218i	0.966116	+	0.258109i	$0.0830993\pi$
$787$	14.4817i	0.516218i	−0.966116	+	0.258109i	$0.916901\pi$
$788$	0	0
$789$	−5.54731	−0.197489
$790$	0	0
$791$	53.9585	1.91854
$792$	0	0
$793$	− 13.7490i	− 0.488240i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	42.6419i	1.51045i	0.655463	+	0.755227i	$0.272473\pi$
$797$	42.6419i	1.51045i	−0.655463	+	0.755227i	$0.727527\pi$
$798$	0	0
$799$	−27.2672	−0.964646
$800$	0	0
$801$	−3.81681	−0.134860
$802$	0	0
$803$	− 24.9423i	− 0.880196i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 29.1625i	− 1.02657i
$808$	0	0
$809$	41.6706	1.46506	0.732529	−	0.680735i	$-0.238340\pi$
$809$	41.6706	1.46506	0.732529	+	0.680735i	$0.238340\pi$
$810$	0	0
$811$	53.9585	1.89474	0.947370	−	0.320140i	$-0.103730\pi$
$811$	53.9585	1.89474	0.947370	+	0.320140i	$0.103730\pi$
$812$	0	0
$813$	1.92159i	0.0673932i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 6.96080i	− 0.243527i
$818$	0	0
$819$	12.4896	0.436423
$820$	0	0
$821$	7.74897	0.270441	0.135220	−	0.990816i	$-0.456826\pi$
$821$	7.74897	0.270441	0.135220	+	0.990816i	$0.456826\pi$
$822$	0	0
$823$	− 38.0554i	− 1.32653i	−0.748385	−	0.663264i	$-0.769171\pi$
$823$	− 38.0554i	− 1.32653i	0.748385	−	0.663264i	$-0.230829\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	48.9299i	1.70146i	0.525604	+	0.850730i	$0.323840\pi$
$827$	48.9299i	1.70146i	−0.525604	+	0.850730i	$0.676160\pi$
$828$	0	0
$829$	18.0369	0.626449	0.313224	−	0.949679i	$-0.398591\pi$
$829$	18.0369	0.626449	0.313224	+	0.949679i	$0.398591\pi$
$830$	0	0
$831$	−26.9793	−0.935900
$832$	0	0
$833$	− 96.2755i	− 3.33575i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	6.20166i	0.214361i
$838$	0	0
$839$	−27.0577	−0.934135	−0.467067	−	0.884222i	$-0.654689\pi$
$839$	−27.0577	−0.934135	−0.467067	+	0.884222i	$0.654689\pi$
$840$	0	0
$841$	−14.4320	−0.497654
$842$	0	0
$843$	4.39276i	0.151295i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 106.705i	− 3.66644i
$848$	0	0
$849$	24.8824	0.853961
$850$	0	0
$851$	−98.5714	−3.37898
$852$	0	0
$853$	− 41.6336i	− 1.42551i	−0.701414	−	0.712754i	$-0.747448\pi$
$853$	− 41.6336i	− 1.42551i	0.701414	−	0.712754i	$-0.252552\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 45.8722i	− 1.56697i	−0.621414	−	0.783483i	$-0.713441\pi$
$857$	− 45.8722i	− 1.56697i	0.621414	−	0.783483i	$-0.286559\pi$
$858$	0	0
$859$	−30.5187	−1.04128	−0.520642	−	0.853775i	$-0.674307\pi$
$859$	−30.5187	−1.04128	−0.520642	+	0.853775i	$0.674307\pi$
$860$	0	0
$861$	0.856013	0.0291729
$862$	0	0
$863$	7.42405i	0.252718i	0.991985	+	0.126359i	$0.0403291\pi$
$863$	7.42405i	0.252718i	−0.991985	+	0.126359i	$0.959671\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 25.1153i	− 0.852962i
$868$	0	0
$869$	36.5759	1.24075
$870$	0	0
$871$	−28.5759	−0.968259
$872$	0	0
$873$	− 6.67282i	− 0.225841i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	24.0554i	0.812294i	0.913808	+	0.406147i	$0.133128\pi$
$877$	24.0554i	0.812294i	−0.913808	+	0.406147i	$0.866872\pi$
$878$	0	0
$879$	−1.83528	−0.0619025
$880$	0	0
$881$	−0.481728	−0.0162298	−0.00811492	−	0.999967i	$-0.502583\pi$
$881$	−0.481728	−0.0162298	−0.00811492	+	0.999967i	$0.502583\pi$
$882$	0	0
$883$	25.4056i	0.854966i	0.904023	+	0.427483i	$0.140600\pi$
$883$	25.4056i	0.854966i	−0.904023	+	0.427483i	$0.859400\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 28.2307i	− 0.947894i	−0.880553	−	0.473947i	$-0.842829\pi$
$887$	− 28.2307i	− 0.947894i	0.880553	−	0.473947i	$-0.157171\pi$
$888$	0	0
$889$	18.6913	0.626886
$890$	0	0
$891$	5.81681	0.194870
$892$	0	0
$893$	4.20166i	0.140603i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	21.9216i	0.731941i
$898$	0	0
$899$	−23.6706	−0.789457
$900$	0	0
$901$	−69.9427	−2.33013
$902$	0	0
$903$	− 32.5266i	− 1.08242i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	37.1888i	1.23483i	0.786636	+	0.617417i	$0.211821\pi$
$907$	37.1888i	1.23483i	−0.786636	+	0.617417i	$0.788179\pi$
$908$	0	0
$909$	−0.654353	−0.0217035
$910$	0	0
$911$	18.5187	0.613551	0.306775	−	0.951782i	$-0.400750\pi$
$911$	18.5187	0.613551	0.306775	+	0.951782i	$0.400750\pi$
$912$	0	0
$913$	6.65435i	0.220227i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 10.2017i	− 0.336889i
$918$	0	0
$919$	−12.3294	−0.406711	−0.203355	−	0.979105i	$-0.565185\pi$
$919$	−12.3294	−0.406711	−0.203355	+	0.979105i	$0.565185\pi$
$920$	0	0
$921$	−12.3664	−0.407486
$922$	0	0
$923$	− 29.5552i	− 0.972822i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	1.71203i	0.0562303i
$928$	0	0
$929$	7.67508	0.251811	0.125906	−	0.992042i	$-0.459816\pi$
$929$	7.67508	0.251811	0.125906	+	0.992042i	$0.459816\pi$
$930$	0	0
$931$	−14.8353	−0.486207
$932$	0	0
$933$	− 6.79608i	− 0.222494i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	9.05767i	0.295901i	0.988995	+	0.147951i	$0.0472677\pi$
$937$	9.05767i	0.295901i	−0.988995	+	0.147951i	$0.952732\pi$
$938$	0	0
$939$	−5.23030	−0.170684
$940$	0	0
$941$	−56.2201	−1.83272	−0.916362	−	0.400351i	$-0.868888\pi$
$941$	−56.2201	−1.83272	−0.916362	+	0.400351i	$0.868888\pi$
$942$	0	0
$943$	1.50246i	0.0489268i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	33.2179i	1.07944i	0.841846	+	0.539718i	$0.181469\pi$
$947$	33.2179i	1.07944i	−0.841846	+	0.539718i	$0.818531\pi$
$948$	0	0
$949$	−11.4610	−0.372040
$950$	0	0
$951$	31.7569	1.02979
$952$	0	0
$953$	6.81455i	0.220745i	0.993890	+	0.110372i	$0.0352044\pi$
$953$	6.81455i	0.220745i	−0.993890	+	0.110372i	$0.964796\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	22.2017i	0.717678i
$958$	0	0
$959$	−9.34565	−0.301787
$960$	0	0
$961$	7.46060	0.240664
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 45.2857i	− 1.45629i	−0.685423	−	0.728145i	$-0.740383\pi$
$967$	− 45.2857i	− 1.45629i	0.685423	−	0.728145i	$-0.259617\pi$
$968$	0	0
$969$	−6.48963	−0.208477
$970$	0	0
$971$	15.0207	0.482038	0.241019	−	0.970520i	$-0.422518\pi$
$971$	15.0207	0.482038	0.241019	+	0.970520i	$0.422518\pi$
$972$	0	0
$973$	− 20.4033i	− 0.654100i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 6.77761i	− 0.216835i	−0.994105	−	0.108417i	$-0.965422\pi$
$977$	− 6.77761i	− 0.216835i	0.994105	−	0.108417i	$-0.0345783\pi$
$978$	0	0
$979$	22.2017	0.709568
$980$	0	0
$981$	−9.05767	−0.289189
$982$	0	0
$983$	33.1025i	1.05581i	0.849305	+	0.527903i	$0.177022\pi$
$983$	33.1025i	1.05581i	−0.849305	+	0.527903i	$0.822978\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	19.6336i	0.624945i
$988$	0	0
$989$	57.0901	1.81536
$990$	0	0
$991$	−47.2672	−1.50149	−0.750747	−	0.660590i	$-0.770306\pi$
$991$	−47.2672	−1.50149	−0.750747	+	0.660590i	$0.770306\pi$
$992$	0	0
$993$	12.8930i	0.409146i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 4.09422i	− 0.129665i	−0.997896	−	0.0648326i	$-0.979349\pi$
$997$	− 4.09422i	− 0.129665i	0.997896	−	0.0648326i	$-0.0206513\pi$
$998$	0	0
$999$	12.0185	0.380248

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5700.2.f.q.3649.4		6
5.2	odd	4		1140.2.a.g.1.3	✓	3
5.3	odd	4		5700.2.a.w.1.1		3
5.4	even	2	inner	5700.2.f.q.3649.3		6
15.2	even	4		3420.2.a.m.1.3		3
20.7	even	4		4560.2.a.br.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1140.2.a.g.1.3	✓	3	5.2	odd	4
3420.2.a.m.1.3		3	15.2	even	4
4560.2.a.br.1.1		3	20.7	even	4
5700.2.a.w.1.1		3	5.3	odd	4
5700.2.f.q.3649.3		6	5.4	even	2	inner
5700.2.f.q.3649.4		6	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 5700 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5700.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -4.67282i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} -4.67282i q^{7} -1.00000 q^{9} +5.81681 q^{11} -2.67282i q^{13} +6.48963i q^{17} +1.00000 q^{19} +4.67282 q^{21} +8.20166i q^{23} -1.00000i q^{27} +3.81681 q^{29} -6.20166 q^{31} +5.81681i q^{33} +12.0185i q^{37} +2.67282 q^{39} +0.183190 q^{41} -6.96080i q^{43} +4.20166i q^{47} -14.8353 q^{49} -6.48963 q^{51} +10.7776i q^{53} +1.00000i q^{57} +7.05767 q^{59} +5.14399 q^{61} +4.67282i q^{63} -10.6913i q^{67} -8.20166 q^{69} +11.0577 q^{71} -4.28797i q^{73} -27.1809i q^{77} +6.28797 q^{79} +1.00000 q^{81} +1.14399i q^{83} +3.81681i q^{87} +3.81681 q^{89} -12.4896 q^{91} -6.20166i q^{93} +6.67282i q^{97} -5.81681 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^9	\( 6 q - 6 q^{9} + 12 q^{11} + 6 q^{19} + 8 q^{21} - 4 q^{39} + 24 q^{41} - 6 q^{49} + 4 q^{51} + 8 q^{59} + 28 q^{61} - 12 q^{69} + 32 q^{71} + 32 q^{79} + 6 q^{81} - 32 q^{91} - 12 q^{99}+O(q^{100}) \) 6 * q - 6 * q^9 + 12 * q^11 + 6 * q^19 + 8 * q^21 - 4 * q^39 + 24 * q^41 - 6 * q^49 + 4 * q^51 + 8 * q^59 + 28 * q^61 - 12 * q^69 + 32 * q^71 + 32 * q^79 + 6 * q^81 - 32 * q^91 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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