LMFDB - Embedded newform 5700.2.f.q.3649.5

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			3649.5
Root			$0.675970 + 0.675970i$ of defining polynomial
Character	$\chi$	$=$	5700.3649
Dual form			5700.2.f.q.3649.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -0.648061i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} -0.648061i q^{7} -1.00000 q^{9} -3.52420 q^{11} +1.35194i q^{13} -6.87614i q^{17} +1.00000 q^{19} +0.648061 q^{21} +5.46838i q^{23} -1.00000i q^{27} -5.52420 q^{29} -3.46838 q^{31} -3.52420i q^{33} -0.0558176i q^{37} -1.35194 q^{39} +9.52420 q^{41} +7.69646i q^{43} +1.46838i q^{47} +6.58002 q^{49} +6.87614 q^{51} -13.2207i q^{53} +1.00000i q^{57} +9.64064 q^{59} -0.172260 q^{61} +0.648061i q^{63} +5.40776i q^{67} -5.46838 q^{69} +13.6406 q^{71} +6.34452i q^{73} +2.28390i q^{77} -4.34452 q^{79} +1.00000 q^{81} -4.17226i q^{83} -5.52420i q^{87} -5.52420 q^{89} +0.876139 q^{91} -3.46838i q^{93} +2.64806i q^{97} +3.52420 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	− 0.648061i	− 0.244944i	−0.992472	−	0.122472i	$-0.960918\pi$
$7$	− 0.648061i	− 0.244944i	0.992472	−	0.122472i	$-0.0390822\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−3.52420	−1.06259	−0.531293	−	0.847188i	$-0.678294\pi$
$11$	−3.52420	−1.06259	−0.531293	+	0.847188i	$0.678294\pi$
$12$	0	0
$13$	1.35194i	0.374960i	0.982268	+	0.187480i	$0.0600321\pi$
$13$	1.35194i	0.374960i	−0.982268	+	0.187480i	$0.939968\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 6.87614i	− 1.66771i	−0.551985	−	0.833854i	$-0.686129\pi$
$17$	− 6.87614i	− 1.66771i	0.551985	−	0.833854i	$-0.313871\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0.648061	0.141418
$22$	0	0
$23$	5.46838i	1.14024i	0.821563	+	0.570118i	$0.193103\pi$
$23$	5.46838i	1.14024i	−0.821563	+	0.570118i	$0.806897\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	−5.52420	−1.02582	−0.512909	−	0.858443i	$-0.671432\pi$
$29$	−5.52420	−1.02582	−0.512909	+	0.858443i	$0.671432\pi$
$30$	0	0
$31$	−3.46838	−0.622940	−0.311470	−	0.950256i	$-0.600821\pi$
$31$	−3.46838	−0.622940	−0.311470	+	0.950256i	$0.600821\pi$
$32$	0	0
$33$	− 3.52420i	− 0.613484i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 0.0558176i	− 0.00917636i	−0.999989	−	0.00458818i	$-0.998540\pi$
$37$	− 0.0558176i	− 0.00917636i	0.999989	−	0.00458818i	$-0.00146047\pi$
$38$	0	0
$39$	−1.35194	−0.216484
$40$	0	0
$41$	9.52420	1.48743	0.743715	−	0.668497i	$-0.233062\pi$
$41$	9.52420	1.48743	0.743715	+	0.668497i	$0.233062\pi$
$42$	0	0
$43$	7.69646i	1.17370i	0.809696	+	0.586850i	$0.199632\pi$
$43$	7.69646i	1.17370i	−0.809696	+	0.586850i	$0.800368\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.46838i	0.214186i	0.994249	+	0.107093i	$0.0341542\pi$
$47$	1.46838i	0.214186i	−0.994249	+	0.107093i	$0.965846\pi$
$48$	0	0
$49$	6.58002	0.940002
$50$	0	0
$51$	6.87614	0.962852
$52$	0	0
$53$	− 13.2207i	− 1.81600i	−0.418973	−	0.907999i	$-0.637610\pi$
$53$	− 13.2207i	− 1.81600i	0.418973	−	0.907999i	$-0.362390\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.00000i	0.132453i
$58$	0	0
$59$	9.64064	1.25510	0.627552	−	0.778574i	$-0.284057\pi$
$59$	9.64064	1.25510	0.627552	+	0.778574i	$0.284057\pi$
$60$	0	0
$61$	−0.172260	−0.0220557	−0.0110278	−	0.999939i	$-0.503510\pi$
$61$	−0.172260	−0.0220557	−0.0110278	+	0.999939i	$0.503510\pi$
$62$	0	0
$63$	0.648061i	0.0816480i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	5.40776i	0.660663i	0.943865	+	0.330331i	$0.107160\pi$
$67$	5.40776i	0.660663i	−0.943865	+	0.330331i	$0.892840\pi$
$68$	0	0
$69$	−5.46838	−0.658316
$70$	0	0
$71$	13.6406	1.61885	0.809423	−	0.587226i	$-0.199780\pi$
$71$	13.6406	1.61885	0.809423	+	0.587226i	$0.199780\pi$
$72$	0	0
$73$	6.34452i	0.742570i	0.928519	+	0.371285i	$0.121083\pi$
$73$	6.34452i	0.742570i	−0.928519	+	0.371285i	$0.878917\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.28390i	0.260274i
$78$	0	0
$79$	−4.34452	−0.488797	−0.244398	−	0.969675i	$-0.578590\pi$
$79$	−4.34452	−0.488797	−0.244398	+	0.969675i	$0.578590\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 4.17226i	− 0.457965i	−0.973431	−	0.228983i	$-0.926460\pi$
$83$	− 4.17226i	− 0.457965i	0.973431	−	0.228983i	$-0.0735399\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 5.52420i	− 0.592256i
$88$	0	0
$89$	−5.52420	−0.585564	−0.292782	−	0.956179i	$-0.594581\pi$
$89$	−5.52420	−0.585564	−0.292782	+	0.956179i	$0.594581\pi$
$90$	0	0
$91$	0.876139	0.0918443
$92$	0	0
$93$	− 3.46838i	− 0.359654i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	2.64806i	0.268870i	0.990922	+	0.134435i	$0.0429219\pi$
$97$	2.64806i	0.268870i	−0.990922	+	0.134435i	$0.957078\pi$
$98$	0	0
$99$	3.52420	0.354195
$100$	0	0
$101$	8.70388	0.866068	0.433034	−	0.901378i	$-0.357443\pi$
$101$	8.70388	0.866068	0.433034	+	0.901378i	$0.357443\pi$
$102$	0	0
$103$	− 12.3445i	− 1.21634i	−0.793806	−	0.608171i	$-0.791904\pi$
$103$	− 12.3445i	− 1.21634i	0.793806	−	0.608171i	$-0.208096\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	11.6406	1.11497	0.557486	−	0.830187i	$-0.311766\pi$
$109$	11.6406	1.11497	0.557486	+	0.830187i	$0.311766\pi$
$110$	0	0
$111$	0.0558176	0.00529797
$112$	0	0
$113$	0.764504i	0.0719184i	0.999353	+	0.0359592i	$0.0114486\pi$
$113$	0.764504i	0.0719184i	−0.999353	+	0.0359592i	$0.988551\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 1.35194i	− 0.124987i
$118$	0	0
$119$	−4.45616	−0.408495
$120$	0	0
$121$	1.41998	0.129089
$122$	0	0
$123$	9.52420i	0.858768i
$124$	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
$129$	−7.69646	−0.677636
$130$	0	0
$131$	11.5242	1.00687	0.503437	−	0.864032i	$-0.332069\pi$
$131$	11.5242	1.00687	0.503437	+	0.864032i	$0.332069\pi$
$132$	0	0
$133$	− 0.648061i	− 0.0561940i
$134$	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
$139$	23.0484	1.95494	0.977470	−	0.211075i	$-0.0676965\pi$
$139$	23.0484	1.95494	0.977470	+	0.211075i	$0.0676965\pi$
$140$	0	0
$141$	−1.46838	−0.123660
$142$	0	0
$143$	− 4.76450i	− 0.398428i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	6.58002i	0.542711i
$148$	0	0
$149$	6.34452	0.519763	0.259882	−	0.965640i	$-0.416316\pi$
$149$	6.34452	0.519763	0.259882	+	0.965640i	$0.416316\pi$
$150$	0	0
$151$	−1.93937	−0.157824	−0.0789120	−	0.996882i	$-0.525145\pi$
$151$	−1.93937	−0.157824	−0.0789120	+	0.996882i	$0.525145\pi$
$152$	0	0
$153$	6.87614i	0.555903i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	18.4562i	1.47296i	0.676458	+	0.736481i	$0.263514\pi$
$157$	18.4562i	1.47296i	−0.676458	+	0.736481i	$0.736486\pi$
$158$	0	0
$159$	13.2207	1.04847
$160$	0	0
$161$	3.54384	0.279294
$162$	0	0
$163$	− 5.94418i	− 0.465584i	−0.972527	−	0.232792i	$-0.925214\pi$
$163$	− 5.94418i	− 0.465584i	0.972527	−	0.232792i	$-0.0747862\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.82774i	0.141435i	0.997496	+	0.0707174i	$0.0225288\pi$
$167$	1.82774i	0.141435i	−0.997496	+	0.0707174i	$0.977471\pi$
$168$	0	0
$169$	11.1723	0.859405
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	20.2691i	1.54103i	0.637423	+	0.770514i	$0.280000\pi$
$173$	20.2691i	1.54103i	−0.637423	+	0.770514i	$0.720000\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	9.64064i	0.724635i
$178$	0	0
$179$	8.11164	0.606292	0.303146	−	0.952944i	$-0.401963\pi$
$179$	8.11164	0.606292	0.303146	+	0.952944i	$0.401963\pi$
$180$	0	0
$181$	17.3929	1.29281	0.646403	−	0.762996i	$-0.276273\pi$
$181$	17.3929	1.29281	0.646403	+	0.762996i	$0.276273\pi$
$182$	0	0
$183$	− 0.172260i	− 0.0127339i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	24.2329i	1.77208i
$188$	0	0
$189$	−0.648061	−0.0471395
$190$	0	0
$191$	7.17968	0.519503	0.259752	−	0.965675i	$-0.416359\pi$
$191$	7.17968	0.519503	0.259752	+	0.965675i	$0.416359\pi$
$192$	0	0
$193$	− 12.2887i	− 0.884560i	−0.896877	−	0.442280i	$-0.854170\pi$
$193$	− 12.2887i	− 0.884560i	0.896877	−	0.442280i	$-0.145830\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 20.6284i	− 1.46971i	−0.678222	−	0.734857i	$-0.737249\pi$
$197$	− 20.6284i	− 1.46971i	0.678222	−	0.734857i	$-0.262751\pi$
$198$	0	0
$199$	−1.64064	−0.116302	−0.0581510	−	0.998308i	$-0.518520\pi$
$199$	−1.64064	−0.116302	−0.0581510	+	0.998308i	$0.518520\pi$
$200$	0	0
$201$	−5.40776	−0.381434
$202$	0	0
$203$	3.58002i	0.251268i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 5.46838i	− 0.380079i
$208$	0	0
$209$	−3.52420	−0.243774
$210$	0	0
$211$	−16.3445	−1.12520	−0.562602	−	0.826728i	$-0.690199\pi$
$211$	−16.3445	−1.12520	−0.562602	+	0.826728i	$0.690199\pi$
$212$	0	0
$213$	13.6406i	0.934641i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	2.24772i	0.152585i
$218$	0	0
$219$	−6.34452	−0.428723
$220$	0	0
$221$	9.29612	0.625325
$222$	0	0
$223$	− 12.3445i	− 0.826650i	−0.910584	−	0.413325i	$-0.864367\pi$
$223$	− 12.3445i	− 0.826650i	0.910584	−	0.413325i	$-0.135633\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	11.2355i	0.745726i	0.927886	+	0.372863i	$0.121624\pi$
$227$	11.2355i	0.745726i	−0.927886	+	0.372863i	$0.878376\pi$
$228$	0	0
$229$	−17.9245	−1.18449	−0.592243	−	0.805759i	$-0.701758\pi$
$229$	−17.9245	−1.18449	−0.592243	+	0.805759i	$0.701758\pi$
$230$	0	0
$231$	−2.28390	−0.150269
$232$	0	0
$233$	− 22.3445i	− 1.46384i	−0.681392	−	0.731919i	$-0.738625\pi$
$233$	− 22.3445i	− 1.46384i	0.681392	−	0.731919i	$-0.261375\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 4.34452i	− 0.282207i
$238$	0	0
$239$	26.6842	1.72606	0.863030	−	0.505153i	$-0.168564\pi$
$239$	26.6842	1.72606	0.863030	+	0.505153i	$0.168564\pi$
$240$	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
$247$	1.35194i	0.0860218i
$248$	0	0
$249$	4.17226	0.264406
$250$	0	0
$251$	9.88356	0.623845	0.311922	−	0.950108i	$-0.399027\pi$
$251$	9.88356	0.623845	0.311922	+	0.950108i	$0.399027\pi$
$252$	0	0
$253$	− 19.2717i	− 1.21160i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	30.5168i	1.90358i	0.306743	+	0.951792i	$0.400761\pi$
$257$	30.5168i	1.90358i	−0.306743	+	0.951792i	$0.599239\pi$
$258$	0	0
$259$	−0.0361732	−0.00224769
$260$	0	0
$261$	5.52420	0.341939
$262$	0	0
$263$	− 5.23550i	− 0.322835i	−0.986886	−	0.161417i	$-0.948394\pi$
$263$	− 5.23550i	− 0.322835i	0.986886	−	0.161417i	$-0.0516065\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 5.52420i	− 0.338076i
$268$	0	0
$269$	−11.7719	−0.717747	−0.358873	−	0.933386i	$-0.616839\pi$
$269$	−11.7719	−0.717747	−0.358873	+	0.933386i	$0.616839\pi$
$270$	0	0
$271$	−27.3929	−1.66400	−0.832001	−	0.554775i	$-0.812804\pi$
$271$	−27.3929	−1.66400	−0.832001	+	0.554775i	$0.812804\pi$
$272$	0	0
$273$	0.876139i	0.0530263i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	0.247722i	0.0148842i	0.999972	+	0.00744210i	$0.00236892\pi$
$277$	0.247722i	0.0148842i	−0.999972	+	0.00744210i	$0.997631\pi$
$278$	0	0
$279$	3.46838	0.207647
$280$	0	0
$281$	−26.2132	−1.56375	−0.781875	−	0.623435i	$-0.785737\pi$
$281$	−26.2132	−1.56375	−0.781875	+	0.623435i	$0.785737\pi$
$282$	0	0
$283$	19.0894i	1.13475i	0.823461	+	0.567373i	$0.192040\pi$
$283$	19.0894i	1.13475i	−0.823461	+	0.567373i	$0.807960\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 6.17226i	− 0.364337i
$288$	0	0
$289$	−30.2813	−1.78125
$290$	0	0
$291$	−2.64806	−0.155232
$292$	0	0
$293$	− 19.5800i	− 1.14388i	−0.820297	−	0.571938i	$-0.806192\pi$
$293$	− 19.5800i	− 1.14388i	0.820297	−	0.571938i	$-0.193808\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	3.52420i	0.204495i
$298$	0	0
$299$	−7.39292	−0.427544
$300$	0	0
$301$	4.98777	0.287491
$302$	0	0
$303$	8.70388i	0.500025i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	31.0484i	1.77203i	0.463661	+	0.886013i	$0.346536\pi$
$307$	31.0484i	1.77203i	−0.463661	+	0.886013i	$0.653464\pi$
$308$	0	0
$309$	12.3445	0.702255
$310$	0	0
$311$	29.2765	1.66012	0.830058	−	0.557677i	$-0.188307\pi$
$311$	29.2765	1.66012	0.830058	+	0.557677i	$0.188307\pi$
$312$	0	0
$313$	− 7.98516i	− 0.451348i	−0.974203	−	0.225674i	$-0.927542\pi$
$313$	− 7.98516i	− 0.451348i	0.974203	−	0.225674i	$-0.0724584\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	18.9729i	1.06563i	0.846233	+	0.532813i	$0.178865\pi$
$317$	18.9729i	1.06563i	−0.846233	+	0.532813i	$0.821135\pi$
$318$	0	0
$319$	19.4684	1.09002
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 6.87614i	− 0.382599i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	11.6406i	0.643729i
$328$	0	0
$329$	0.951601	0.0524635
$330$	0	0
$331$	−5.93937	−0.326458	−0.163229	−	0.986588i	$-0.552191\pi$
$331$	−5.93937	−0.326458	−0.163229	+	0.986588i	$0.552191\pi$
$332$	0	0
$333$	0.0558176i	0.00305879i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 27.7933i	− 1.51400i	−0.653418	−	0.756998i	$-0.726665\pi$
$337$	− 27.7933i	− 1.51400i	0.653418	−	0.756998i	$-0.273335\pi$
$338$	0	0
$339$	−0.764504	−0.0415221
$340$	0	0
$341$	12.2233	0.661927
$342$	0	0
$343$	− 8.80068i	− 0.475192i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	17.2355i	0.925250i	0.886554	+	0.462625i	$0.153092\pi$
$347$	17.2355i	0.925250i	−0.886554	+	0.462625i	$0.846908\pi$
$348$	0	0
$349$	−27.4078	−1.46710	−0.733552	−	0.679634i	$-0.762139\pi$
$349$	−27.4078	−1.46710	−0.733552	+	0.679634i	$0.762139\pi$
$350$	0	0
$351$	1.35194	0.0721612
$352$	0	0
$353$	− 16.5922i	− 0.883116i	−0.897233	−	0.441558i	$-0.854426\pi$
$353$	− 16.5922i	− 0.883116i	0.897233	−	0.441558i	$-0.145574\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 4.45616i	− 0.235845i
$358$	0	0
$359$	17.5094	0.924109	0.462054	−	0.886852i	$-0.347112\pi$
$359$	17.5094	0.924109	0.462054	+	0.886852i	$0.347112\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	1.41998i	0.0745298i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 17.7933i	− 0.928801i	−0.885625	−	0.464400i	$-0.846270\pi$
$367$	− 17.7933i	− 0.928801i	0.885625	−	0.464400i	$-0.153730\pi$
$368$	0	0
$369$	−9.52420	−0.495810
$370$	0	0
$371$	−8.56779	−0.444818
$372$	0	0
$373$	3.59966i	0.186383i	0.995648	+	0.0931917i	$0.0297069\pi$
$373$	3.59966i	0.186383i	−0.995648	+	0.0931917i	$0.970293\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 7.46838i	− 0.384641i
$378$	0	0
$379$	30.4051	1.56181	0.780904	−	0.624651i	$-0.214759\pi$
$379$	30.4051	1.56181	0.780904	+	0.624651i	$0.214759\pi$
$380$	0	0
$381$	−4.00000	−0.204926
$382$	0	0
$383$	2.93676i	0.150062i	0.997181	+	0.0750308i	$0.0239055\pi$
$383$	2.93676i	0.150062i	−0.997181	+	0.0750308i	$0.976094\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 7.69646i	− 0.391233i
$388$	0	0
$389$	−1.65548	−0.0839361	−0.0419681	−	0.999119i	$-0.513363\pi$
$389$	−1.65548	−0.0839361	−0.0419681	+	0.999119i	$0.513363\pi$
$390$	0	0
$391$	37.6014	1.90158
$392$	0	0
$393$	11.5242i	0.581319i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	34.9123i	1.75220i	0.482131	+	0.876099i	$0.339863\pi$
$397$	34.9123i	1.75220i	−0.482131	+	0.876099i	$0.660137\pi$
$398$	0	0
$399$	0.648061	0.0324436
$400$	0	0
$401$	32.9171	1.64380	0.821901	−	0.569630i	$-0.192913\pi$
$401$	32.9171	1.64380	0.821901	+	0.569630i	$0.192913\pi$
$402$	0	0
$403$	− 4.68904i	− 0.233578i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.196712i	0.00975067i
$408$	0	0
$409$	27.1452	1.34224	0.671122	−	0.741347i	$-0.265813\pi$
$409$	27.1452	1.34224	0.671122	+	0.741347i	$0.265813\pi$
$410$	0	0
$411$	2.00000	0.0986527
$412$	0	0
$413$	− 6.24772i	− 0.307430i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	23.0484i	1.12868i
$418$	0	0
$419$	−1.88356	−0.0920178	−0.0460089	−	0.998941i	$-0.514650\pi$
$419$	−1.88356	−0.0920178	−0.0460089	+	0.998941i	$0.514650\pi$
$420$	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
$423$	− 1.46838i	− 0.0713952i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0.111635i	0.00540241i
$428$	0	0
$429$	4.76450	0.230032
$430$	0	0
$431$	18.7039	0.900934	0.450467	−	0.892793i	$-0.351257\pi$
$431$	18.7039	0.900934	0.450467	+	0.892793i	$0.351257\pi$
$432$	0	0
$433$	37.0894i	1.78240i	0.453609	+	0.891201i	$0.350136\pi$
$433$	37.0894i	1.78240i	−0.453609	+	0.891201i	$0.649864\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.46838i	0.261588i
$438$	0	0
$439$	8.22327	0.392475	0.196238	−	0.980556i	$-0.437128\pi$
$439$	8.22327	0.392475	0.196238	+	0.980556i	$0.437128\pi$
$440$	0	0
$441$	−6.58002	−0.313334
$442$	0	0
$443$	− 12.0606i	− 0.573018i	−0.958078	−	0.286509i	$-0.907505\pi$
$443$	− 12.0606i	− 0.573018i	0.958078	−	0.286509i	$-0.0924948\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	6.34452i	0.300086i
$448$	0	0
$449$	−23.2765	−1.09848	−0.549242	−	0.835663i	$-0.685084\pi$
$449$	−23.2765	−1.09848	−0.549242	+	0.835663i	$0.685084\pi$
$450$	0	0
$451$	−33.5652	−1.58052
$452$	0	0
$453$	− 1.93937i	− 0.0911198i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6.45616i	0.302006i	0.988533	+	0.151003i	$0.0482504\pi$
$457$	6.45616i	0.302006i	−0.988533	+	0.151003i	$0.951750\pi$
$458$	0	0
$459$	−6.87614	−0.320951
$460$	0	0
$461$	22.6890	1.05673	0.528367	−	0.849016i	$-0.322805\pi$
$461$	22.6890	1.05673	0.528367	+	0.849016i	$0.322805\pi$
$462$	0	0
$463$	30.2887i	1.40764i	0.710381	+	0.703818i	$0.248523\pi$
$463$	30.2887i	1.40764i	−0.710381	+	0.703818i	$0.751477\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.12386i	0.0520061i	0.999662	+	0.0260030i	$0.00827796\pi$
$467$	1.12386i	0.0520061i	−0.999662	+	0.0260030i	$0.991722\pi$
$468$	0	0
$469$	3.50456	0.161825
$470$	0	0
$471$	−18.4562	−0.850415
$472$	0	0
$473$	− 27.1239i	− 1.24716i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	13.2207i	0.605332i
$478$	0	0
$479$	−10.3397	−0.472434	−0.236217	−	0.971700i	$-0.575908\pi$
$479$	−10.3397	−0.472434	−0.236217	+	0.971700i	$0.575908\pi$
$480$	0	0
$481$	0.0754620	0.00344077
$482$	0	0
$483$	3.54384i	0.161250i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 3.54384i	− 0.160587i	−0.996771	−	0.0802935i	$-0.974414\pi$
$487$	− 3.54384i	− 0.160587i	0.996771	−	0.0802935i	$-0.0255857\pi$
$488$	0	0
$489$	5.94418	0.268805
$490$	0	0
$491$	−2.22808	−0.100552	−0.0502759	−	0.998735i	$-0.516010\pi$
$491$	−2.22808	−0.100552	−0.0502759	+	0.998735i	$0.516010\pi$
$492$	0	0
$493$	37.9852i	1.71077i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 8.83997i	− 0.396527i
$498$	0	0
$499$	−29.1452	−1.30472	−0.652359	−	0.757910i	$-0.726221\pi$
$499$	−29.1452	−1.30472	−0.652359	+	0.757910i	$0.726221\pi$
$500$	0	0
$501$	−1.82774	−0.0816574
$502$	0	0
$503$	40.1723i	1.79119i	0.444868	+	0.895596i	$0.353251\pi$
$503$	40.1723i	1.79119i	−0.444868	+	0.895596i	$0.646749\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	11.1723i	0.496178i
$508$	0	0
$509$	0.835158	0.0370177	0.0185089	−	0.999829i	$-0.494108\pi$
$509$	0.835158	0.0370177	0.0185089	+	0.999829i	$0.494108\pi$
$510$	0	0
$511$	4.11164	0.181888
$512$	0	0
$513$	− 1.00000i	− 0.0441511i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 5.17487i	− 0.227591i
$518$	0	0
$519$	−20.2691	−0.889713
$520$	0	0
$521$	6.82032	0.298804	0.149402	−	0.988777i	$-0.452265\pi$
$521$	6.82032	0.298804	0.149402	+	0.988777i	$0.452265\pi$
$522$	0	0
$523$	13.6406i	0.596464i	0.954493	+	0.298232i	$0.0963969\pi$
$523$	13.6406i	0.596464i	−0.954493	+	0.298232i	$0.903603\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	23.8491i	1.03888i
$528$	0	0
$529$	−6.90320	−0.300139
$530$	0	0
$531$	−9.64064	−0.418368
$532$	0	0
$533$	12.8761i	0.557727i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	8.11164i	0.350043i
$538$	0	0
$539$	−23.1893	−0.998834
$540$	0	0
$541$	29.5046	1.26850	0.634250	−	0.773128i	$-0.281309\pi$
$541$	29.5046	1.26850	0.634250	+	0.773128i	$0.281309\pi$
$542$	0	0
$543$	17.3929i	0.746402i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	39.7374i	1.69905i	0.527548	+	0.849525i	$0.323111\pi$
$547$	39.7374i	1.69905i	−0.527548	+	0.849525i	$0.676889\pi$
$548$	0	0
$549$	0.172260	0.00735189
$550$	0	0
$551$	−5.52420	−0.235339
$552$	0	0
$553$	2.81551i	0.119728i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 11.7523i	− 0.497960i	−0.968509	−	0.248980i	$-0.919905\pi$
$557$	− 11.7523i	− 0.497960i	0.968509	−	0.248980i	$-0.0800953\pi$
$558$	0	0
$559$	−10.4051	−0.440091
$560$	0	0
$561$	−24.2329	−1.02311
$562$	0	0
$563$	− 36.6136i	− 1.54308i	−0.636182	−	0.771539i	$-0.719487\pi$
$563$	− 36.6136i	− 1.54308i	0.636182	−	0.771539i	$-0.280513\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 0.648061i	− 0.0272160i
$568$	0	0
$569$	15.0436	0.630660	0.315330	−	0.948982i	$-0.397885\pi$
$569$	15.0436	0.630660	0.315330	+	0.948982i	$0.397885\pi$
$570$	0	0
$571$	23.3929	0.978963	0.489482	−	0.872014i	$-0.337186\pi$
$571$	23.3929	0.978963	0.489482	+	0.872014i	$0.337186\pi$
$572$	0	0
$573$	7.17968i	0.299935i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 32.7039i	− 1.36148i	−0.732525	−	0.680740i	$-0.761658\pi$
$577$	− 32.7039i	− 1.36148i	0.732525	−	0.680740i	$-0.238342\pi$
$578$	0	0
$579$	12.2887	0.510701
$580$	0	0
$581$	−2.70388	−0.112176
$582$	0	0
$583$	46.5922i	1.92965i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 9.46838i	− 0.390802i	−0.980723	−	0.195401i	$-0.937399\pi$
$587$	− 9.46838i	− 0.390802i	0.980723	−	0.195401i	$-0.0626008\pi$
$588$	0	0
$589$	−3.46838	−0.142912
$590$	0	0
$591$	20.6284	0.848540
$592$	0	0
$593$	− 27.7523i	− 1.13965i	−0.821766	−	0.569825i	$-0.807011\pi$
$593$	− 27.7523i	− 1.13965i	0.821766	−	0.569825i	$-0.192989\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 1.64064i	− 0.0671470i
$598$	0	0
$599$	−28.6890	−1.17220	−0.586101	−	0.810238i	$-0.699338\pi$
$599$	−28.6890	−1.17220	−0.586101	+	0.810238i	$0.699338\pi$
$600$	0	0
$601$	11.2961	0.460778	0.230389	−	0.973099i	$-0.426000\pi$
$601$	11.2961	0.460778	0.230389	+	0.973099i	$0.426000\pi$
$602$	0	0
$603$	− 5.40776i	− 0.220221i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 8.11164i	− 0.329241i	−0.986357	−	0.164621i	$-0.947360\pi$
$607$	− 8.11164i	− 0.329241i	0.986357	−	0.164621i	$-0.0526400\pi$
$608$	0	0
$609$	−3.58002	−0.145070
$610$	0	0
$611$	−1.98516	−0.0803111
$612$	0	0
$613$	− 17.1696i	− 0.693476i	−0.937962	−	0.346738i	$-0.887289\pi$
$613$	− 17.1696i	− 0.693476i	0.937962	−	0.346738i	$-0.112711\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 26.3807i	− 1.06205i	−0.847357	−	0.531023i	$-0.821808\pi$
$617$	− 26.3807i	− 1.06205i	0.847357	−	0.531023i	$-0.178192\pi$
$618$	0	0
$619$	38.6742	1.55445	0.777224	−	0.629224i	$-0.216627\pi$
$619$	38.6742	1.55445	0.777224	+	0.629224i	$0.216627\pi$
$620$	0	0
$621$	5.46838	0.219439
$622$	0	0
$623$	3.58002i	0.143430i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 3.52420i	− 0.140743i
$628$	0	0
$629$	−0.383810	−0.0153035
$630$	0	0
$631$	21.4078	0.852229	0.426115	−	0.904669i	$-0.359882\pi$
$631$	21.4078	0.852229	0.426115	+	0.904669i	$0.359882\pi$
$632$	0	0
$633$	− 16.3445i	− 0.649636i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	8.89578i	0.352464i
$638$	0	0
$639$	−13.6406	−0.539615
$640$	0	0
$641$	−19.1648	−0.756966	−0.378483	−	0.925608i	$-0.623554\pi$
$641$	−19.1648	−0.756966	−0.378483	+	0.925608i	$0.623554\pi$
$642$	0	0
$643$	8.99258i	0.354633i	0.984154	+	0.177316i	$0.0567416\pi$
$643$	8.99258i	0.354633i	−0.984154	+	0.177316i	$0.943258\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 38.9878i	− 1.53277i	−0.642383	−	0.766384i	$-0.722054\pi$
$647$	− 38.9878i	− 1.53277i	0.642383	−	0.766384i	$-0.277946\pi$
$648$	0	0
$649$	−33.9755	−1.33366
$650$	0	0
$651$	−2.24772	−0.0880952
$652$	0	0
$653$	− 33.8129i	− 1.32320i	−0.749856	−	0.661601i	$-0.769877\pi$
$653$	− 33.8129i	− 1.32320i	0.749856	−	0.661601i	$-0.230123\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 6.34452i	− 0.247523i
$658$	0	0
$659$	−1.75228	−0.0682590	−0.0341295	−	0.999417i	$-0.510866\pi$
$659$	−1.75228	−0.0682590	−0.0341295	+	0.999417i	$0.510866\pi$
$660$	0	0
$661$	−16.9271	−0.658390	−0.329195	−	0.944262i	$-0.606777\pi$
$661$	−16.9271	−0.658390	−0.329195	+	0.944262i	$0.606777\pi$
$662$	0	0
$663$	9.29612i	0.361031i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 30.2084i	− 1.16968i
$668$	0	0
$669$	12.3445	0.477267
$670$	0	0
$671$	0.607080	0.0234361
$672$	0	0
$673$	− 0.0558176i	− 0.00215161i	−0.999999	−	0.00107581i	$-0.999658\pi$
$673$	− 0.0558176i	− 0.00215161i	0.999999	−	0.00107581i	$-0.000342440\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 19.4684i	− 0.748231i	−0.927382	−	0.374115i	$-0.877946\pi$
$677$	− 19.4684i	− 0.748231i	0.927382	−	0.374115i	$-0.122054\pi$
$678$	0	0
$679$	1.71610	0.0658580
$680$	0	0
$681$	−11.2355	−0.430545
$682$	0	0
$683$	16.2233i	0.620766i	0.950612	+	0.310383i	$0.100457\pi$
$683$	16.2233i	0.620766i	−0.950612	+	0.310383i	$0.899543\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 17.9245i	− 0.683864i
$688$	0	0
$689$	17.8735	0.680927
$690$	0	0
$691$	19.8884	0.756589	0.378295	−	0.925685i	$-0.376511\pi$
$691$	19.8884	0.756589	0.378295	+	0.925685i	$0.376511\pi$
$692$	0	0
$693$	− 2.28390i	− 0.0867580i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 65.4897i	− 2.48060i
$698$	0	0
$699$	22.3445	0.845147
$700$	0	0
$701$	7.75228	0.292799	0.146400	−	0.989226i	$-0.453231\pi$
$701$	7.75228	0.292799	0.146400	+	0.989226i	$0.453231\pi$
$702$	0	0
$703$	− 0.0558176i	− 0.00210520i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 5.64064i	− 0.212138i
$708$	0	0
$709$	−29.9245	−1.12384	−0.561920	−	0.827192i	$-0.689937\pi$
$709$	−29.9245	−1.12384	−0.561920	+	0.827192i	$0.689937\pi$
$710$	0	0
$711$	4.34452	0.162932
$712$	0	0
$713$	− 18.9664i	− 0.710299i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	26.6842i	0.996541i
$718$	0	0
$719$	−11.2913	−0.421095	−0.210547	−	0.977584i	$-0.567525\pi$
$719$	−11.2913	−0.421095	−0.210547	+	0.977584i	$0.567525\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	0	0
$723$	14.0000i	0.520666i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 9.59966i	− 0.356032i	−0.984028	−	0.178016i	$-0.943032\pi$
$727$	− 9.59966i	− 0.356032i	0.984028	−	0.178016i	$-0.0569678\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	52.9219	1.95739
$732$	0	0
$733$	12.9368i	0.477830i	0.971040	+	0.238915i	$0.0767918\pi$
$733$	12.9368i	0.477830i	−0.971040	+	0.238915i	$0.923208\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 19.0580i	− 0.702011i
$738$	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
$741$	−1.35194	−0.0496647
$742$	0	0
$743$	− 14.4413i	− 0.529801i	−0.964276	−	0.264900i	$-0.914661\pi$
$743$	− 14.4413i	− 0.529801i	0.964276	−	0.264900i	$-0.0853391\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	4.17226i	0.152655i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−25.5652	−0.932887	−0.466443	−	0.884551i	$-0.654465\pi$
$751$	−25.5652	−0.932887	−0.466443	+	0.884551i	$0.654465\pi$
$752$	0	0
$753$	9.88356i	0.360177i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 3.41737i	− 0.124206i	−0.998070	−	0.0621032i	$-0.980219\pi$
$757$	− 3.41737i	− 0.124206i	0.998070	−	0.0621032i	$-0.0197808\pi$
$758$	0	0
$759$	19.2717	0.699517
$760$	0	0
$761$	−20.9368	−0.758957	−0.379479	−	0.925201i	$-0.623897\pi$
$761$	−20.9368	−0.758957	−0.379479	+	0.925201i	$0.623897\pi$
$762$	0	0
$763$	− 7.54384i	− 0.273105i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	13.0336i	0.470615i
$768$	0	0
$769$	−0.592243	−0.0213568	−0.0106784	−	0.999943i	$-0.503399\pi$
$769$	−0.592243	−0.0213568	−0.0106784	+	0.999943i	$0.503399\pi$
$770$	0	0
$771$	−30.5168	−1.09904
$772$	0	0
$773$	− 7.69165i	− 0.276650i	−0.990387	−	0.138325i	$-0.955828\pi$
$773$	− 7.69165i	− 0.276650i	0.990387	−	0.138325i	$-0.0441718\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 0.0361732i	− 0.00129771i
$778$	0	0
$779$	9.52420	0.341240
$780$	0	0
$781$	−48.0723	−1.72016
$782$	0	0
$783$	5.52420i	0.197419i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	38.3297i	1.36631i	0.730275	+	0.683153i	$0.239392\pi$
$787$	38.3297i	1.36631i	−0.730275	+	0.683153i	$0.760608\pi$
$788$	0	0
$789$	5.23550	0.186389
$790$	0	0
$791$	0.495445	0.0176160
$792$	0	0
$793$	− 0.232886i	− 0.00827001i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	10.2935i	0.364615i	0.983242	+	0.182307i	$0.0583566\pi$
$797$	10.2935i	0.364615i	−0.983242	+	0.182307i	$0.941643\pi$
$798$	0	0
$799$	10.0968	0.357199
$800$	0	0
$801$	5.52420	0.195188
$802$	0	0
$803$	− 22.3594i	− 0.789045i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 11.7719i	− 0.414391i
$808$	0	0
$809$	−1.16003	−0.0407846	−0.0203923	−	0.999792i	$-0.506492\pi$
$809$	−1.16003	−0.0407846	−0.0203923	+	0.999792i	$0.506492\pi$
$810$	0	0
$811$	0.495445	0.0173974	0.00869871	−	0.999962i	$-0.497231\pi$
$811$	0.495445	0.0173974	0.00869871	+	0.999962i	$0.497231\pi$
$812$	0	0
$813$	− 27.3929i	− 0.960712i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	7.69646i	0.269265i
$818$	0	0
$819$	−0.876139	−0.0306148
$820$	0	0
$821$	−5.76711	−0.201274	−0.100637	−	0.994923i	$-0.532088\pi$
$821$	−5.76711	−0.201274	−0.100637	+	0.994923i	$0.532088\pi$
$822$	0	0
$823$	− 1.83255i	− 0.0638786i	−0.999490	−	0.0319393i	$-0.989832\pi$
$823$	− 1.83255i	− 0.0638786i	0.999490	−	0.0319393i	$-0.0101683\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	5.94899i	0.206867i	0.994636	+	0.103433i	$0.0329828\pi$
$827$	5.94899i	0.206867i	−0.994636	+	0.103433i	$0.967017\pi$
$828$	0	0
$829$	−6.11164	−0.212266	−0.106133	−	0.994352i	$-0.533847\pi$
$829$	−6.11164	−0.212266	−0.106133	+	0.994352i	$0.533847\pi$
$830$	0	0
$831$	−0.247722	−0.00859339
$832$	0	0
$833$	− 45.2451i	− 1.56765i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	3.46838i	0.119885i
$838$	0	0
$839$	−29.6406	−1.02331	−0.511654	−	0.859191i	$-0.670967\pi$
$839$	−29.6406	−1.02331	−0.511654	+	0.859191i	$0.670967\pi$
$840$	0	0
$841$	1.51678	0.0523028
$842$	0	0
$843$	− 26.2132i	− 0.902832i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 0.920235i	− 0.0316197i
$848$	0	0
$849$	−19.0894	−0.655146
$850$	0	0
$851$	0.305232	0.0104632
$852$	0	0
$853$	− 22.9516i	− 0.785848i	−0.919571	−	0.392924i	$-0.871464\pi$
$853$	− 22.9516i	− 0.785848i	0.919571	−	0.392924i	$-0.128536\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 0.308348i	− 0.0105330i	−0.999986	−	0.00526648i	$-0.998324\pi$
$857$	− 0.308348i	− 0.0105330i	0.999986	−	0.00526648i	$-0.00167638\pi$
$858$	0	0
$859$	−30.2180	−1.03103	−0.515513	−	0.856882i	$-0.672399\pi$
$859$	−30.2180	−1.03103	−0.515513	+	0.856882i	$0.672399\pi$
$860$	0	0
$861$	6.17226	0.210350
$862$	0	0
$863$	28.6890i	0.976586i	0.872680	+	0.488293i	$0.162380\pi$
$863$	28.6890i	0.976586i	−0.872680	+	0.488293i	$0.837620\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 30.2813i	− 1.02841i
$868$	0	0
$869$	15.3110	0.519389
$870$	0	0
$871$	−7.31096	−0.247722
$872$	0	0
$873$	− 2.64806i	− 0.0896233i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 12.1675i	− 0.410866i	−0.978671	−	0.205433i	$-0.934140\pi$
$877$	− 12.1675i	− 0.410866i	0.978671	−	0.205433i	$-0.0658602\pi$
$878$	0	0
$879$	19.5800	0.660418
$880$	0	0
$881$	−24.3297	−0.819688	−0.409844	−	0.912156i	$-0.634417\pi$
$881$	−24.3297	−0.819688	−0.409844	+	0.912156i	$0.634417\pi$
$882$	0	0
$883$	58.7449i	1.97692i	0.151476	+	0.988461i	$0.451597\pi$
$883$	58.7449i	1.97692i	−0.151476	+	0.988461i	$0.548403\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 38.5626i	− 1.29480i	−0.762149	−	0.647402i	$-0.775855\pi$
$887$	− 38.5626i	− 1.29480i	0.762149	−	0.647402i	$-0.224145\pi$
$888$	0	0
$889$	2.59224	0.0869410
$890$	0	0
$891$	−3.52420	−0.118065
$892$	0	0
$893$	1.46838i	0.0491375i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 7.39292i	− 0.246842i
$898$	0	0
$899$	19.1600	0.639023
$900$	0	0
$901$	−90.9071	−3.02855
$902$	0	0
$903$	4.98777i	0.165983i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 29.4897i	− 0.979190i	−0.871950	−	0.489595i	$-0.837145\pi$
$907$	− 29.4897i	− 0.979190i	0.871950	−	0.489595i	$-0.162855\pi$
$908$	0	0
$909$	−8.70388	−0.288689
$910$	0	0
$911$	18.2180	0.603591	0.301795	−	0.953373i	$-0.402414\pi$
$911$	18.2180	0.603591	0.301795	+	0.953373i	$0.402414\pi$
$912$	0	0
$913$	14.7039i	0.486627i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 7.46838i	− 0.246628i
$918$	0	0
$919$	−55.1600	−1.81956	−0.909781	−	0.415089i	$-0.863750\pi$
$919$	−55.1600	−1.81956	−0.909781	+	0.415089i	$0.863750\pi$
$920$	0	0
$921$	−31.0484	−1.02308
$922$	0	0
$923$	18.4413i	0.607003i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	12.3445i	0.405447i
$928$	0	0
$929$	42.4562	1.39294	0.696471	−	0.717585i	$-0.254753\pi$
$929$	42.4562	1.39294	0.696471	+	0.717585i	$0.254753\pi$
$930$	0	0
$931$	6.58002	0.215651
$932$	0	0
$933$	29.2765i	0.958469i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	11.6406i	0.380283i	0.981757	+	0.190142i	$0.0608947\pi$
$937$	11.6406i	0.380283i	−0.981757	+	0.190142i	$0.939105\pi$
$938$	0	0
$939$	7.98516	0.260586
$940$	0	0
$941$	−41.4126	−1.35001	−0.675006	−	0.737813i	$-0.735859\pi$
$941$	−41.4126	−1.35001	−0.675006	+	0.737813i	$0.735859\pi$
$942$	0	0
$943$	52.0820i	1.69602i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 20.3955i	− 0.662766i	−0.943496	−	0.331383i	$-0.892485\pi$
$947$	− 20.3955i	− 0.662766i	0.943496	−	0.331383i	$-0.107515\pi$
$948$	0	0
$949$	−8.57741	−0.278434
$950$	0	0
$951$	−18.9729	−0.615240
$952$	0	0
$953$	− 41.3323i	− 1.33888i	−0.742864	−	0.669442i	$-0.766533\pi$
$953$	− 41.3323i	− 1.33888i	0.742864	−	0.669442i	$-0.233467\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	19.4684i	0.629323i
$958$	0	0
$959$	−1.29612	−0.0418539
$960$	0	0
$961$	−18.9703	−0.611946
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	4.15262i	0.133539i	0.997768	+	0.0667696i	$0.0212692\pi$
$967$	4.15262i	0.133539i	−0.997768	+	0.0667696i	$0.978731\pi$
$968$	0	0
$969$	6.87614	0.220893
$970$	0	0
$971$	41.7523	1.33989	0.669947	−	0.742409i	$-0.266317\pi$
$971$	41.7523	1.33989	0.669947	+	0.742409i	$0.266317\pi$
$972$	0	0
$973$	− 14.9368i	− 0.478851i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	17.2207i	0.550938i	0.961310	+	0.275469i	$0.0888331\pi$
$977$	17.2207i	0.550938i	−0.961310	+	0.275469i	$0.911167\pi$
$978$	0	0
$979$	19.4684	0.622212
$980$	0	0
$981$	−11.6406	−0.371657
$982$	0	0
$983$	− 25.6768i	− 0.818963i	−0.912318	−	0.409482i	$-0.865710\pi$
$983$	− 25.6768i	− 0.818963i	0.912318	−	0.409482i	$-0.134290\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0.951601i	0.0302898i
$988$	0	0
$989$	−42.0872	−1.33829
$990$	0	0
$991$	−9.90320	−0.314586	−0.157293	−	0.987552i	$-0.550277\pi$
$991$	−9.90320	−0.314586	−0.157293	+	0.987552i	$0.550277\pi$
$992$	0	0
$993$	− 5.93937i	− 0.188480i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	41.0187i	1.29908i	0.760329	+	0.649538i	$0.225038\pi$
$997$	41.0187i	1.29908i	−0.760329	+	0.649538i	$0.774962\pi$
$998$	0	0
$999$	−0.0558176	−0.00176599

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5700.2.f.q.3649.5		6
5.2	odd	4		1140.2.a.g.1.2	✓	3
5.3	odd	4		5700.2.a.w.1.2		3
5.4	even	2	inner	5700.2.f.q.3649.2		6
15.2	even	4		3420.2.a.m.1.2		3
20.7	even	4		4560.2.a.br.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1140.2.a.g.1.2	✓	3	5.2	odd	4
3420.2.a.m.1.2		3	15.2	even	4
4560.2.a.br.1.2		3	20.7	even	4
5700.2.a.w.1.2		3	5.3	odd	4
5700.2.f.q.3649.2		6	5.4	even	2	inner
5700.2.f.q.3649.5		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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -0.648061i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} -0.648061i q^{7} -1.00000 q^{9} -3.52420 q^{11} +1.35194i q^{13} -6.87614i q^{17} +1.00000 q^{19} +0.648061 q^{21} +5.46838i q^{23} -1.00000i q^{27} -5.52420 q^{29} -3.46838 q^{31} -3.52420i q^{33} -0.0558176i q^{37} -1.35194 q^{39} +9.52420 q^{41} +7.69646i q^{43} +1.46838i q^{47} +6.58002 q^{49} +6.87614 q^{51} -13.2207i q^{53} +1.00000i q^{57} +9.64064 q^{59} -0.172260 q^{61} +0.648061i q^{63} +5.40776i q^{67} -5.46838 q^{69} +13.6406 q^{71} +6.34452i q^{73} +2.28390i q^{77} -4.34452 q^{79} +1.00000 q^{81} -4.17226i q^{83} -5.52420i q^{87} -5.52420 q^{89} +0.876139 q^{91} -3.46838i q^{93} +2.64806i q^{97} +3.52420 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more