LMFDB - Embedded newform 6900.2.f.r.6349.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6900,2,Mod(6349,6900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6900, 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("6900.6349");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6900.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:	$55.0967773947$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.158155776.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 24x^{3} + 81x^{2} + 54x + 18 $ x^6 - 2x^5 + 2x^4 + 24x^3 + 81x^2 + 54*x + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1380)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			6349.6
Root			$-1.42234 - 1.42234i$ of defining polynomial
Character	$\chi$	$=$	6900.6349
Dual form			6900.2.f.r.6349.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +3.73549i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +3.73549i q^{7} -1.00000 q^{9} +4.84469 q^{11} -2.84469i q^{13} -0.890804i q^{17} -6.84469 q^{19} -3.73549 q^{21} +1.00000i q^{23} -1.00000i q^{27} +0.890804 q^{29} +7.73549 q^{31} +4.84469i q^{33} +1.95388i q^{37} +2.84469 q^{39} +12.3618 q^{41} -3.47098i q^{43} +6.62629i q^{47} -6.95388 q^{49} +0.890804 q^{51} +12.3618i q^{53} -6.84469i q^{57} -0.890804 q^{59} +8.62629 q^{61} -3.73549i q^{63} -7.73549i q^{67} -1.00000 q^{69} -12.3618 q^{71} +16.5341i q^{73} +18.0973i q^{77} -13.4710 q^{79} +1.00000 q^{81} -4.58018i q^{83} +0.890804i q^{87} +15.1604 q^{89} +10.6263 q^{91} +7.73549i q^{93} +17.2526i q^{97} -4.84469 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{9}+O(q^{10}) $ 6 * q - 6 * q^9	$ 6 q - 6 q^{9} + 8 q^{11} - 20 q^{19} + 4 q^{21} + 20 q^{31} - 4 q^{39} + 16 q^{41} - 26 q^{49} + 20 q^{61} - 6 q^{69} - 16 q^{71} - 28 q^{79} + 6 q^{81} - 4 q^{89} + 32 q^{91} - 8 q^{99}+O(q^{100}) $ 6 * q - 6 * q^9 + 8 * q^11 - 20 * q^19 + 4 * q^21 + 20 * q^31 - 4 * q^39 + 16 * q^41 - 26 * q^49 + 20 * q^61 - 6 * q^69 - 16 * q^71 - 28 * q^79 + 6 * q^81 - 4 * q^89 + 32 * q^91 - 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6900\mathbb{Z}\right)^\times$.

$n$	$277$	$1201$	$3451$	$4601$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$7$	3.73549i	1.41188i	0.708270	+	0.705941i	$0.249476\pi$
$7$	3.73549i	1.41188i	−0.708270	+	0.705941i	$0.750524\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	4.84469	1.46073	0.730364	−	0.683058i	$-0.239351\pi$
$11$	4.84469	1.46073	0.730364	+	0.683058i	$0.239351\pi$
$12$	0	0
$13$	− 2.84469i	− 0.788974i	−0.918902	−	0.394487i	$-0.870922\pi$
$13$	− 2.84469i	− 0.788974i	0.918902	−	0.394487i	$-0.129078\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 0.890804i	− 0.216052i	−0.994148	−	0.108026i	$-0.965547\pi$
$17$	− 0.890804i	− 0.216052i	0.994148	−	0.108026i	$-0.0344529\pi$
$18$	0	0
$19$	−6.84469	−1.57028	−0.785139	−	0.619319i	$-0.787409\pi$
$19$	−6.84469	−1.57028	−0.785139	+	0.619319i	$0.787409\pi$
$20$	0	0
$21$	−3.73549	−0.815151
$22$	0	0
$23$	1.00000i	0.208514i
$23$	1.00000i	0.208514i
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	0.890804	0.165418	0.0827091	−	0.996574i	$-0.473643\pi$
$29$	0.890804	0.165418	0.0827091	+	0.996574i	$0.473643\pi$
$30$	0	0
$31$	7.73549	1.38933	0.694667	−	0.719331i	$-0.255552\pi$
$31$	7.73549	1.38933	0.694667	+	0.719331i	$0.255552\pi$
$32$	0	0
$33$	4.84469i	0.843352i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.95388i	0.321216i	0.987018	+	0.160608i	$0.0513455\pi$
$37$	1.95388i	0.321216i	−0.987018	+	0.160608i	$0.948654\pi$
$38$	0	0
$39$	2.84469	0.455514
$40$	0	0
$41$	12.3618	1.93059	0.965293	−	0.261169i	$-0.0841080\pi$
$41$	12.3618	1.93059	0.965293	+	0.261169i	$0.0841080\pi$
$42$	0	0
$43$	− 3.47098i	− 0.529319i	−0.964342	−	0.264660i	$-0.914740\pi$
$43$	− 3.47098i	− 0.529319i	0.964342	−	0.264660i	$-0.0852596\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.62629i	0.966544i	0.875470	+	0.483272i	$0.160552\pi$
$47$	6.62629i	0.966544i	−0.875470	+	0.483272i	$0.839448\pi$
$48$	0	0
$49$	−6.95388	−0.993412
$50$	0	0
$51$	0.890804	0.124737
$52$	0	0
$53$	12.3618i	1.69802i	0.528376	+	0.849011i	$0.322801\pi$
$53$	12.3618i	1.69802i	−0.528376	+	0.849011i	$0.677199\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 6.84469i	− 0.906601i
$58$	0	0
$59$	−0.890804	−0.115973	−0.0579864	−	0.998317i	$-0.518468\pi$
$59$	−0.890804	−0.115973	−0.0579864	+	0.998317i	$0.518468\pi$
$60$	0	0
$61$	8.62629	1.10448	0.552242	−	0.833684i	$-0.313773\pi$
$61$	8.62629	1.10448	0.552242	+	0.833684i	$0.313773\pi$
$62$	0	0
$63$	− 3.73549i	− 0.470627i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 7.73549i	− 0.945040i	−0.881320	−	0.472520i	$-0.843344\pi$
$67$	− 7.73549i	− 0.945040i	0.881320	−	0.472520i	$-0.156656\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−12.3618	−1.46707	−0.733537	−	0.679650i	$-0.762132\pi$
$71$	−12.3618	−1.46707	−0.733537	+	0.679650i	$0.762132\pi$
$72$	0	0
$73$	16.5341i	1.93516i	0.252555	+	0.967582i	$0.418729\pi$
$73$	16.5341i	1.93516i	−0.252555	+	0.967582i	$0.581271\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	18.0973i	2.06238i
$78$	0	0
$79$	−13.4710	−1.51560	−0.757802	−	0.652485i	$-0.773727\pi$
$79$	−13.4710	−1.51560	−0.757802	+	0.652485i	$0.773727\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 4.58018i	− 0.502740i	−0.967891	−	0.251370i	$-0.919119\pi$
$83$	− 4.58018i	− 0.502740i	0.967891	−	0.251370i	$-0.0808810\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0.890804i	0.0955042i
$88$	0	0
$89$	15.1604	1.60699	0.803497	−	0.595309i	$-0.202970\pi$
$89$	15.1604	1.60699	0.803497	+	0.595309i	$0.202970\pi$
$90$	0	0
$91$	10.6263	1.11394
$92$	0	0
$93$	7.73549i	0.802133i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	17.2526i	1.75173i	0.482552	+	0.875867i	$0.339710\pi$
$97$	17.2526i	1.75173i	−0.482552	+	0.875867i	$0.660290\pi$
$98$	0	0
$99$	−4.84469	−0.486909
$100$	0	0
$101$	−2.67241	−0.265915	−0.132957	−	0.991122i	$-0.542447\pi$
$101$	−2.67241	−0.265915	−0.132957	+	0.991122i	$0.542447\pi$
$102$	0	0
$103$	6.21839i	0.612716i	0.951916	+	0.306358i	$0.0991105\pi$
$103$	6.21839i	0.612716i	−0.951916	+	0.306358i	$0.900889\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.7986i	1.43063i	0.698801	+	0.715316i	$0.253717\pi$
$107$	14.7986i	1.43063i	−0.698801	+	0.715316i	$0.746283\pi$
$108$	0	0
$109$	12.5341	1.20054	0.600272	−	0.799796i	$-0.295059\pi$
$109$	12.5341	1.20054	0.600272	+	0.799796i	$0.295059\pi$
$110$	0	0
$111$	−1.95388	−0.185454
$112$	0	0
$113$	12.3618i	1.16290i	0.813583	+	0.581449i	$0.197514\pi$
$113$	12.3618i	1.16290i	−0.813583	+	0.581449i	$0.802486\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.84469i	0.262991i
$118$	0	0
$119$	3.32759	0.305040
$120$	0	0
$121$	12.4710	1.13373
$122$	0	0
$123$	12.3618i	1.11462i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 11.7866i	− 1.04590i	−0.852365	−	0.522948i	$-0.824832\pi$
$127$	− 11.7866i	− 1.04590i	0.852365	−	0.522948i	$-0.175168\pi$
$128$	0	0
$129$	3.47098	0.305603
$130$	0	0
$131$	−17.4710	−1.52645	−0.763223	−	0.646135i	$-0.776384\pi$
$131$	−17.4710	−1.52645	−0.763223	+	0.646135i	$0.776384\pi$
$132$	0	0
$133$	− 25.5683i	− 2.21705i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 15.1604i	− 1.29524i	−0.761965	−	0.647618i	$-0.775765\pi$
$137$	− 15.1604i	− 1.29524i	0.761965	−	0.647618i	$-0.224235\pi$
$138$	0	0
$139$	5.51710	0.467954	0.233977	−	0.972242i	$-0.424826\pi$
$139$	5.51710	0.467954	0.233977	+	0.972242i	$0.424826\pi$
$140$	0	0
$141$	−6.62629	−0.558035
$142$	0	0
$143$	− 13.7816i	− 1.15248i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 6.95388i	− 0.573547i
$148$	0	0
$149$	−1.15531	−0.0946470	−0.0473235	−	0.998880i	$-0.515069\pi$
$149$	−1.15531	−0.0946470	−0.0473235	+	0.998880i	$0.515069\pi$
$150$	0	0
$151$	−15.4710	−1.25901	−0.629505	−	0.776996i	$-0.716742\pi$
$151$	−15.4710	−1.25901	−0.629505	+	0.776996i	$0.716742\pi$
$152$	0	0
$153$	0.890804i	0.0720172i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 18.6774i	− 1.49062i	−0.666717	−	0.745311i	$-0.732301\pi$
$157$	− 18.6774i	− 1.49062i	0.666717	−	0.745311i	$-0.267699\pi$
$158$	0	0
$159$	−12.3618	−0.980353
$160$	0	0
$161$	−3.73549	−0.294398
$162$	0	0
$163$	0.0922364i	0.00722451i	0.999993	+	0.00361226i	$0.00114982\pi$
$163$	0.0922364i	0.00722451i	−0.999993	+	0.00361226i	$0.998850\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	5.37371i	0.415830i	0.978147	+	0.207915i	$0.0666677\pi$
$167$	5.37371i	0.415830i	−0.978147	+	0.207915i	$0.933332\pi$
$168$	0	0
$169$	4.90776	0.377520
$170$	0	0
$171$	6.84469	0.523426
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 0.890804i	− 0.0669569i
$178$	0	0
$179$	9.56322	0.714788	0.357394	−	0.933954i	$-0.383665\pi$
$179$	9.56322	0.714788	0.357394	+	0.933954i	$0.383665\pi$
$180$	0	0
$181$	−7.16035	−0.532225	−0.266112	−	0.963942i	$-0.585739\pi$
$181$	−7.16035	−0.532225	−0.266112	+	0.963942i	$0.585739\pi$
$182$	0	0
$183$	8.62629i	0.637674i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 4.31566i	− 0.315593i
$188$	0	0
$189$	3.73549	0.271717
$190$	0	0
$191$	14.0050	1.01337	0.506684	−	0.862132i	$-0.330871\pi$
$191$	14.0050	1.01337	0.506684	+	0.862132i	$0.330871\pi$
$192$	0	0
$193$	26.7236i	1.92360i	0.273745	+	0.961802i	$0.411737\pi$
$193$	26.7236i	1.92360i	−0.273745	+	0.961802i	$0.588263\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 9.16035i	− 0.652648i	−0.945258	−	0.326324i	$-0.894190\pi$
$197$	− 9.16035i	− 0.652648i	0.945258	−	0.326324i	$-0.105810\pi$
$198$	0	0
$199$	0.310629	0.0220199	0.0110099	−	0.999939i	$-0.496495\pi$
$199$	0.310629	0.0220199	0.0110099	+	0.999939i	$0.496495\pi$
$200$	0	0
$201$	7.73549	0.545619
$202$	0	0
$203$	3.32759i	0.233551i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 1.00000i	− 0.0695048i
$208$	0	0
$209$	−33.1604	−2.29375
$210$	0	0
$211$	5.95388	0.409882	0.204941	−	0.978774i	$-0.434300\pi$
$211$	5.95388	0.409882	0.204941	+	0.978774i	$0.434300\pi$
$212$	0	0
$213$	− 12.3618i	− 0.847015i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	28.8958i	1.96158i
$218$	0	0
$219$	−16.5341	−1.11727
$220$	0	0
$221$	−2.53406	−0.170459
$222$	0	0
$223$	− 11.9078i	− 0.797403i	−0.917081	−	0.398701i	$-0.869461\pi$
$223$	− 11.9078i	− 0.797403i	0.917081	−	0.398701i	$-0.130539\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 7.37874i	− 0.489744i	−0.969555	−	0.244872i	$-0.921254\pi$
$227$	− 7.37874i	− 0.489744i	0.969555	−	0.244872i	$-0.0787460\pi$
$228$	0	0
$229$	−23.1604	−1.53048	−0.765240	−	0.643746i	$-0.777379\pi$
$229$	−23.1604	−1.53048	−0.765240	+	0.643746i	$0.777379\pi$
$230$	0	0
$231$	−18.0973	−1.19071
$232$	0	0
$233$	− 22.9420i	− 1.50298i	−0.659746	−	0.751489i	$-0.729336\pi$
$233$	− 22.9420i	− 1.50298i	0.659746	−	0.751489i	$-0.270664\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 13.4710i	− 0.875034i
$238$	0	0
$239$	−0.361783	−0.0234018	−0.0117009	−	0.999932i	$-0.503725\pi$
$239$	−0.361783	−0.0234018	−0.0117009	+	0.999932i	$0.503725\pi$
$240$	0	0
$241$	28.5341	1.83804	0.919020	−	0.394211i	$-0.128982\pi$
$241$	28.5341	1.83804	0.919020	+	0.394211i	$0.128982\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	19.4710i	1.23891i
$248$	0	0
$249$	4.58018	0.290257
$250$	0	0
$251$	−30.8497	−1.94722	−0.973609	−	0.228224i	$-0.926708\pi$
$251$	−30.8497	−1.94722	−0.973609	+	0.228224i	$0.926708\pi$
$252$	0	0
$253$	4.84469i	0.304583i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	12.7524i	0.795476i	0.917499	+	0.397738i	$0.130205\pi$
$257$	12.7524i	0.795476i	−0.917499	+	0.397738i	$0.869795\pi$
$258$	0	0
$259$	−7.29870	−0.453519
$260$	0	0
$261$	−0.890804	−0.0551394
$262$	0	0
$263$	6.36178i	0.392284i	0.980575	+	0.196142i	$0.0628414\pi$
$263$	6.36178i	0.392284i	−0.980575	+	0.196142i	$0.937159\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	15.1604i	0.927798i
$268$	0	0
$269$	−10.5802	−0.645085	−0.322542	−	0.946555i	$-0.604537\pi$
$269$	−10.5802	−0.645085	−0.322542	+	0.946555i	$0.604537\pi$
$270$	0	0
$271$	−4.26451	−0.259051	−0.129525	−	0.991576i	$-0.541345\pi$
$271$	−4.26451	−0.259051	−0.129525	+	0.991576i	$0.541345\pi$
$272$	0	0
$273$	10.6263i	0.643133i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	29.3787i	1.76520i	0.470127	+	0.882599i	$0.344208\pi$
$277$	29.3787i	1.76520i	−0.470127	+	0.882599i	$0.655792\pi$
$278$	0	0
$279$	−7.73549	−0.463112
$280$	0	0
$281$	−18.7524	−1.11868	−0.559339	−	0.828939i	$-0.688945\pi$
$281$	−18.7524	−1.11868	−0.559339	+	0.828939i	$0.688945\pi$
$282$	0	0
$283$	− 12.8958i	− 0.766578i	−0.923628	−	0.383289i	$-0.874791\pi$
$283$	− 12.8958i	− 0.766578i	0.923628	−	0.383289i	$-0.125209\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	46.1773i	2.72576i
$288$	0	0
$289$	16.2065	0.953322
$290$	0	0
$291$	−17.2526	−1.01136
$292$	0	0
$293$	8.79857i	0.514018i	0.966409	+	0.257009i	$0.0827370\pi$
$293$	8.79857i	0.514018i	−0.966409	+	0.257009i	$0.917263\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 4.84469i	− 0.281117i
$298$	0	0
$299$	2.84469	0.164512
$300$	0	0
$301$	12.9658	0.747337
$302$	0	0
$303$	− 2.67241i	− 0.153526i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 10.0050i	− 0.571018i	−0.958376	−	0.285509i	$-0.907837\pi$
$307$	− 10.0050i	− 0.571018i	0.958376	−	0.285509i	$-0.0921626\pi$
$308$	0	0
$309$	−6.21839	−0.353752
$310$	0	0
$311$	−18.5290	−1.05068	−0.525342	−	0.850891i	$-0.676063\pi$
$311$	−18.5290	−1.05068	−0.525342	+	0.850891i	$0.676063\pi$
$312$	0	0
$313$	2.39067i	0.135128i	0.997715	+	0.0675642i	$0.0215228\pi$
$313$	2.39067i	0.135128i	−0.997715	+	0.0675642i	$0.978477\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	16.3157i	0.916379i	0.888855	+	0.458190i	$0.151502\pi$
$317$	16.3157i	0.916379i	−0.888855	+	0.458190i	$0.848498\pi$
$318$	0	0
$319$	4.31566	0.241631
$320$	0	0
$321$	−14.7986	−0.825975
$322$	0	0
$323$	6.09727i	0.339261i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	12.5341i	0.693135i
$328$	0	0
$329$	−24.7524	−1.36465
$330$	0	0
$331$	−24.8958	−1.36840	−0.684200	−	0.729295i	$-0.739848\pi$
$331$	−24.8958	−1.36840	−0.684200	+	0.729295i	$0.739848\pi$
$332$	0	0
$333$	− 1.95388i	− 0.107072i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	27.4710i	1.49644i	0.663451	+	0.748220i	$0.269091\pi$
$337$	27.4710i	1.49644i	−0.663451	+	0.748220i	$0.730909\pi$
$338$	0	0
$339$	−12.3618	−0.671400
$340$	0	0
$341$	37.4760	2.02944
$342$	0	0
$343$	0.172274i	0.00930193i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.25259i	0.0672424i	0.999435	+	0.0336212i	$0.0107040\pi$
$347$	1.25259i	0.0672424i	−0.999435	+	0.0336212i	$0.989296\pi$
$348$	0	0
$349$	−9.64325	−0.516192	−0.258096	−	0.966119i	$-0.583095\pi$
$349$	−9.64325	−0.516192	−0.258096	+	0.966119i	$0.583095\pi$
$350$	0	0
$351$	−2.84469	−0.151838
$352$	0	0
$353$	− 12.7524i	− 0.678744i	−0.940652	−	0.339372i	$-0.889785\pi$
$353$	− 12.7524i	− 0.678744i	0.940652	−	0.339372i	$-0.110215\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	3.32759i	0.176115i
$358$	0	0
$359$	6.62629	0.349722	0.174861	−	0.984593i	$-0.444052\pi$
$359$	6.62629	0.349722	0.174861	+	0.984593i	$0.444052\pi$
$360$	0	0
$361$	27.8497	1.46577
$362$	0	0
$363$	12.4710i	0.654557i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	9.60933i	0.501603i	0.968039	+	0.250802i	$0.0806942\pi$
$367$	9.60933i	0.501603i	−0.968039	+	0.250802i	$0.919306\pi$
$368$	0	0
$369$	−12.3618	−0.643529
$370$	0	0
$371$	−46.1773	−2.39741
$372$	0	0
$373$	29.8839i	1.54733i	0.633595	+	0.773665i	$0.281579\pi$
$373$	29.8839i	1.54733i	−0.633595	+	0.773665i	$0.718421\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 2.53406i	− 0.130511i
$378$	0	0
$379$	−2.52902	−0.129907	−0.0649535	−	0.997888i	$-0.520690\pi$
$379$	−2.52902	−0.129907	−0.0649535	+	0.997888i	$0.520690\pi$
$380$	0	0
$381$	11.7866	0.603848
$382$	0	0
$383$	25.2115i	1.28825i	0.764921	+	0.644124i	$0.222778\pi$
$383$	25.2115i	1.28825i	−0.764921	+	0.644124i	$0.777222\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.47098i	0.176440i
$388$	0	0
$389$	−35.0681	−1.77802	−0.889012	−	0.457884i	$-0.848608\pi$
$389$	−35.0681	−1.77802	−0.889012	+	0.457884i	$0.848608\pi$
$390$	0	0
$391$	0.890804	0.0450499
$392$	0	0
$393$	− 17.4710i	− 0.881294i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 8.84972i	− 0.444155i	−0.975029	−	0.222077i	$-0.928716\pi$
$397$	− 8.84972i	− 0.444155i	0.975029	−	0.222077i	$-0.0712838\pi$
$398$	0	0
$399$	25.5683	1.28001
$400$	0	0
$401$	−23.0681	−1.15197	−0.575983	−	0.817461i	$-0.695381\pi$
$401$	−23.0681	−1.15197	−0.575983	+	0.817461i	$0.695381\pi$
$402$	0	0
$403$	− 22.0050i	− 1.09615i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	9.46594i	0.469209i
$408$	0	0
$409$	−5.29870	−0.262004	−0.131002	−	0.991382i	$-0.541819\pi$
$409$	−5.29870	−0.262004	−0.131002	+	0.991382i	$0.541819\pi$
$410$	0	0
$411$	15.1604	0.747805
$412$	0	0
$413$	− 3.32759i	− 0.163740i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	5.51710i	0.270173i
$418$	0	0
$419$	−2.53406	−0.123797	−0.0618984	−	0.998082i	$-0.519715\pi$
$419$	−2.53406	−0.123797	−0.0618984	+	0.998082i	$0.519715\pi$
$420$	0	0
$421$	22.4079	1.09209	0.546047	−	0.837754i	$-0.316132\pi$
$421$	22.4079	1.09209	0.546047	+	0.837754i	$0.316132\pi$
$422$	0	0
$423$	− 6.62629i	− 0.322181i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	32.2234i	1.55940i
$428$	0	0
$429$	13.7816	0.665382
$430$	0	0
$431$	31.5733	1.52083	0.760416	−	0.649436i	$-0.224995\pi$
$431$	31.5733	1.52083	0.760416	+	0.649436i	$0.224995\pi$
$432$	0	0
$433$	7.20647i	0.346321i	0.984894	+	0.173160i	$0.0553979\pi$
$433$	7.20647i	0.346321i	−0.984894	+	0.173160i	$0.944602\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 6.84469i	− 0.327426i
$438$	0	0
$439$	4.72357	0.225443	0.112722	−	0.993627i	$-0.464043\pi$
$439$	4.72357	0.225443	0.112722	+	0.993627i	$0.464043\pi$
$440$	0	0
$441$	6.95388	0.331137
$442$	0	0
$443$	− 7.87888i	− 0.374337i	−0.982328	−	0.187168i	$-0.940069\pi$
$443$	− 7.87888i	− 0.374337i	0.982328	−	0.187168i	$-0.0599310\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 1.15531i	− 0.0546445i
$448$	0	0
$449$	4.70633	0.222105	0.111053	−	0.993815i	$-0.464578\pi$
$449$	4.70633	0.222105	0.111053	+	0.993815i	$0.464578\pi$
$450$	0	0
$451$	59.8890	2.82006
$452$	0	0
$453$	− 15.4710i	− 0.726890i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6.04612i	0.282825i	0.989951	+	0.141413i	$0.0451645\pi$
$457$	6.04612i	0.282825i	−0.989951	+	0.141413i	$0.954836\pi$
$458$	0	0
$459$	−0.890804	−0.0415792
$460$	0	0
$461$	−19.2526	−0.896682	−0.448341	−	0.893863i	$-0.647985\pi$
$461$	−19.2526	−0.896682	−0.448341	+	0.893863i	$0.647985\pi$
$462$	0	0
$463$	− 26.4418i	− 1.22886i	−0.788973	−	0.614428i	$-0.789387\pi$
$463$	− 26.4418i	− 1.22886i	0.788973	−	0.614428i	$-0.210613\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 9.20143i	− 0.425792i	−0.977075	−	0.212896i	$-0.931711\pi$
$467$	− 9.20143i	− 0.425792i	0.977075	−	0.212896i	$-0.0682895\pi$
$468$	0	0
$469$	28.8958	1.33429
$470$	0	0
$471$	18.6774	0.860611
$472$	0	0
$473$	− 16.8158i	− 0.773191i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 12.3618i	− 0.566007i
$478$	0	0
$479$	12.7524	0.582674	0.291337	−	0.956620i	$-0.405900\pi$
$479$	12.7524	0.582674	0.291337	+	0.956620i	$0.405900\pi$
$480$	0	0
$481$	5.55818	0.253431
$482$	0	0
$483$	− 3.73549i	− 0.169971i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	30.5341i	1.38363i	0.722075	+	0.691815i	$0.243189\pi$
$487$	30.5341i	1.38363i	−0.722075	+	0.691815i	$0.756811\pi$
$488$	0	0
$489$	−0.0922364	−0.00417107
$490$	0	0
$491$	−2.67241	−0.120604	−0.0603021	−	0.998180i	$-0.519206\pi$
$491$	−2.67241	−0.120604	−0.0603021	+	0.998180i	$0.519206\pi$
$492$	0	0
$493$	− 0.793532i	− 0.0357389i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 46.1773i	− 2.07134i
$498$	0	0
$499$	20.8036	0.931297	0.465649	−	0.884970i	$-0.345821\pi$
$499$	20.8036	0.931297	0.465649	+	0.884970i	$0.345821\pi$
$500$	0	0
$501$	−5.37371	−0.240080
$502$	0	0
$503$	3.32759i	0.148370i	0.997245	+	0.0741849i	$0.0236355\pi$
$503$	3.32759i	0.148370i	−0.997245	+	0.0741849i	$0.976364\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	4.90776i	0.217961i
$508$	0	0
$509$	−16.2184	−0.718868	−0.359434	−	0.933171i	$-0.617030\pi$
$509$	−16.2184	−0.718868	−0.359434	+	0.933171i	$0.617030\pi$
$510$	0	0
$511$	−61.7628	−2.73223
$512$	0	0
$513$	6.84469i	0.302200i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	32.1023i	1.41186i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	3.18923	0.139723	0.0698614	−	0.997557i	$-0.477744\pi$
$521$	3.18923	0.139723	0.0698614	+	0.997557i	$0.477744\pi$
$522$	0	0
$523$	23.8155i	1.04138i	0.853746	+	0.520690i	$0.174325\pi$
$523$	23.8155i	1.04138i	−0.853746	+	0.520690i	$0.825675\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 6.89080i	− 0.300168i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	0.890804	0.0386576
$532$	0	0
$533$	− 35.1654i	− 1.52318i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	9.56322i	0.412683i
$538$	0	0
$539$	−33.6894	−1.45110
$540$	0	0
$541$	15.7816	0.678504	0.339252	−	0.940695i	$-0.389826\pi$
$541$	15.7816	0.678504	0.339252	+	0.940695i	$0.389826\pi$
$542$	0	0
$543$	− 7.16035i	− 0.307280i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	28.0000i	1.19719i	0.801050	+	0.598597i	$0.204275\pi$
$547$	28.0000i	1.19719i	−0.801050	+	0.598597i	$0.795725\pi$
$548$	0	0
$549$	−8.62629	−0.368161
$550$	0	0
$551$	−6.09727	−0.259753
$552$	0	0
$553$	− 50.3207i	− 2.13985i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 31.7405i	− 1.34489i	−0.740147	−	0.672445i	$-0.765244\pi$
$557$	− 31.7405i	− 1.34489i	0.740147	−	0.672445i	$-0.234756\pi$
$558$	0	0
$559$	−9.87384	−0.417619
$560$	0	0
$561$	4.31566	0.182208
$562$	0	0
$563$	21.9250i	0.924029i	0.886872	+	0.462014i	$0.152873\pi$
$563$	21.9250i	0.924029i	−0.886872	+	0.462014i	$0.847127\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	3.73549i	0.156876i
$568$	0	0
$569$	14.6313	0.613377	0.306689	−	0.951810i	$-0.400779\pi$
$569$	14.6313	0.613377	0.306689	+	0.951810i	$0.400779\pi$
$570$	0	0
$571$	1.24755	0.0522084	0.0261042	−	0.999659i	$-0.491690\pi$
$571$	1.24755	0.0522084	0.0261042	+	0.999659i	$0.491690\pi$
$572$	0	0
$573$	14.0050i	0.585069i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 18.0922i	− 0.753190i	−0.926378	−	0.376595i	$-0.877095\pi$
$577$	− 18.0922i	− 0.753190i	0.926378	−	0.376595i	$-0.122905\pi$
$578$	0	0
$579$	−26.7236	−1.11059
$580$	0	0
$581$	17.1092	0.709809
$582$	0	0
$583$	59.8890i	2.48035i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 12.5290i	− 0.517128i	−0.965994	−	0.258564i	$-0.916751\pi$
$587$	− 12.5290i	− 0.517128i	0.965994	−	0.258564i	$-0.0832493\pi$
$588$	0	0
$589$	−52.9470	−2.18164
$590$	0	0
$591$	9.16035	0.376806
$592$	0	0
$593$	− 24.9470i	− 1.02445i	−0.858851	−	0.512225i	$-0.828821\pi$
$593$	− 24.9470i	− 1.02445i	0.858851	−	0.512225i	$-0.171179\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0.310629i	0.0127132i
$598$	0	0
$599$	34.5391	1.41123	0.705615	−	0.708596i	$-0.250671\pi$
$599$	34.5391	1.41123	0.705615	+	0.708596i	$0.250671\pi$
$600$	0	0
$601$	37.9300	1.54720	0.773599	−	0.633675i	$-0.218454\pi$
$601$	37.9300	1.54720	0.773599	+	0.633675i	$0.218454\pi$
$602$	0	0
$603$	7.73549i	0.315013i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 26.0973i	− 1.05926i	−0.848230	−	0.529628i	$-0.822332\pi$
$607$	− 26.0973i	− 1.05926i	0.848230	−	0.529628i	$-0.177668\pi$
$608$	0	0
$609$	−3.32759	−0.134841
$610$	0	0
$611$	18.8497	0.762578
$612$	0	0
$613$	2.84972i	0.115099i	0.998343	+	0.0575496i	$0.0183287\pi$
$613$	2.84972i	0.115099i	−0.998343	+	0.0575496i	$0.981671\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	9.52213i	0.383347i	0.981459	+	0.191673i	$0.0613914\pi$
$617$	9.52213i	0.383347i	−0.981459	+	0.191673i	$0.938609\pi$
$618$	0	0
$619$	23.9762	0.963683	0.481841	−	0.876258i	$-0.339968\pi$
$619$	23.9762	0.963683	0.481841	+	0.876258i	$0.339968\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	56.6313i	2.26889i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 33.1604i	− 1.32430i
$628$	0	0
$629$	1.74053	0.0693993
$630$	0	0
$631$	−17.8789	−0.711747	−0.355873	−	0.934534i	$-0.615817\pi$
$631$	−17.8789	−0.711747	−0.355873	+	0.934534i	$0.615817\pi$
$632$	0	0
$633$	5.95388i	0.236646i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	19.7816i	0.783776i
$638$	0	0
$639$	12.3618	0.489025
$640$	0	0
$641$	2.93692	0.116001	0.0580007	−	0.998317i	$-0.481527\pi$
$641$	2.93692	0.116001	0.0580007	+	0.998317i	$0.481527\pi$
$642$	0	0
$643$	− 0.895840i	− 0.0353285i	−0.999844	−	0.0176642i	$-0.994377\pi$
$643$	− 0.895840i	− 0.0353285i	0.999844	−	0.0176642i	$-0.00562299\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	21.4659i	0.843913i	0.906616	+	0.421957i	$0.138657\pi$
$647$	21.4659i	0.843913i	−0.906616	+	0.421957i	$0.861343\pi$
$648$	0	0
$649$	−4.31566	−0.169405
$650$	0	0
$651$	−28.8958	−1.13252
$652$	0	0
$653$	− 4.84469i	− 0.189587i	−0.995497	−	0.0947936i	$-0.969781\pi$
$653$	− 4.84469i	− 0.189587i	0.995497	−	0.0947936i	$-0.0302191\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 16.5341i	− 0.645055i
$658$	0	0
$659$	18.0973	0.704970	0.352485	−	0.935818i	$-0.385337\pi$
$659$	18.0973	0.704970	0.352485	+	0.935818i	$0.385337\pi$
$660$	0	0
$661$	0.747413	0.0290710	0.0145355	−	0.999894i	$-0.495373\pi$
$661$	0.747413	0.0290710	0.0145355	+	0.999894i	$0.495373\pi$
$662$	0	0
$663$	− 2.53406i	− 0.0984146i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0.890804i	0.0344921i
$668$	0	0
$669$	11.9078	0.460381
$670$	0	0
$671$	41.7917	1.61335
$672$	0	0
$673$	− 19.1892i	− 0.739691i	−0.929093	−	0.369845i	$-0.879411\pi$
$673$	− 19.1892i	− 0.739691i	0.929093	−	0.369845i	$-0.120589\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	4.17731i	0.160547i	0.996773	+	0.0802735i	$0.0255794\pi$
$677$	4.17731i	0.160547i	−0.996773	+	0.0802735i	$0.974421\pi$
$678$	0	0
$679$	−64.4469	−2.47324
$680$	0	0
$681$	7.37874	0.282754
$682$	0	0
$683$	− 26.0050i	− 0.995055i	−0.867448	−	0.497528i	$-0.834241\pi$
$683$	− 26.0050i	− 0.995055i	0.867448	−	0.497528i	$-0.165759\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 23.1604i	− 0.883622i
$688$	0	0
$689$	35.1654	1.33969
$690$	0	0
$691$	26.8497	1.02141	0.510706	−	0.859756i	$-0.329384\pi$
$691$	26.8497	1.02141	0.510706	+	0.859756i	$0.329384\pi$
$692$	0	0
$693$	− 18.0973i	− 0.687459i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 11.0119i	− 0.417106i
$698$	0	0
$699$	22.9420	0.867745
$700$	0	0
$701$	11.9027	0.449560	0.224780	−	0.974410i	$-0.427834\pi$
$701$	11.9027	0.449560	0.224780	+	0.974410i	$0.427834\pi$
$702$	0	0
$703$	− 13.3737i	− 0.504399i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 9.98277i	− 0.375441i
$708$	0	0
$709$	39.3737	1.47871	0.739355	−	0.673315i	$-0.235130\pi$
$709$	39.3737	1.47871	0.739355	+	0.673315i	$0.235130\pi$
$710$	0	0
$711$	13.4710	0.505201
$712$	0	0
$713$	7.73549i	0.289696i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 0.361783i	− 0.0135110i
$718$	0	0
$719$	−23.6382	−0.881557	−0.440778	−	0.897616i	$-0.645298\pi$
$719$	−23.6382	−0.881557	−0.440778	+	0.897616i	$0.645298\pi$
$720$	0	0
$721$	−23.2287	−0.865083
$722$	0	0
$723$	28.5341i	1.06119i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 12.5513i	− 0.465502i	−0.972536	−	0.232751i	$-0.925227\pi$
$727$	− 12.5513i	− 0.465502i	0.972536	−	0.232751i	$-0.0747727\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−3.09196	−0.114360
$732$	0	0
$733$	39.1142i	1.44472i	0.691519	+	0.722359i	$0.256942\pi$
$733$	39.1142i	1.44472i	−0.691519	+	0.722359i	$0.743058\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 37.4760i	− 1.38045i
$738$	0	0
$739$	−35.7456	−1.31492	−0.657461	−	0.753489i	$-0.728370\pi$
$739$	−35.7456	−1.31492	−0.657461	+	0.753489i	$0.728370\pi$
$740$	0	0
$741$	−19.4710	−0.715284
$742$	0	0
$743$	20.4368i	0.749753i	0.927075	+	0.374876i	$0.122315\pi$
$743$	20.4368i	0.749753i	−0.927075	+	0.374876i	$0.877685\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	4.58018i	0.167580i
$748$	0	0
$749$	−55.2799	−2.01988
$750$	0	0
$751$	−0.534057	−0.0194880	−0.00974401	−	0.999953i	$-0.503102\pi$
$751$	−0.534057	−0.0194880	−0.00974401	+	0.999953i	$0.503102\pi$
$752$	0	0
$753$	− 30.8497i	− 1.12423i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	29.5171i	1.07282i	0.843958	+	0.536409i	$0.180219\pi$
$757$	29.5171i	1.07282i	−0.843958	+	0.536409i	$0.819781\pi$
$758$	0	0
$759$	−4.84469	−0.175851
$760$	0	0
$761$	38.1434	1.38270	0.691348	−	0.722522i	$-0.257017\pi$
$761$	38.1434	1.38270	0.691348	+	0.722522i	$0.257017\pi$
$762$	0	0
$763$	46.8208i	1.69503i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.53406i	0.0914995i
$768$	0	0
$769$	−20.8786	−0.752902	−0.376451	−	0.926437i	$-0.622856\pi$
$769$	−20.8786	−0.752902	−0.376451	+	0.926437i	$0.622856\pi$
$770$	0	0
$771$	−12.7524	−0.459268
$772$	0	0
$773$	8.31063i	0.298913i	0.988768	+	0.149456i	$0.0477523\pi$
$773$	8.31063i	0.298913i	−0.988768	+	0.149456i	$0.952248\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 7.29870i	− 0.261840i
$778$	0	0
$779$	−84.6125	−3.03156
$780$	0	0
$781$	−59.8890	−2.14300
$782$	0	0
$783$	− 0.890804i	− 0.0318347i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1.95388i	0.0696484i	0.999393	+	0.0348242i	$0.0110871\pi$
$787$	1.95388i	0.0696484i	−0.999393	+	0.0348242i	$0.988913\pi$
$788$	0	0
$789$	−6.36178	−0.226485
$790$	0	0
$791$	−46.1773	−1.64188
$792$	0	0
$793$	− 24.5391i	− 0.871409i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	16.1773i	0.573030i	0.958076	+	0.286515i	$0.0924968\pi$
$797$	16.1773i	0.573030i	−0.958076	+	0.286515i	$0.907503\pi$
$798$	0	0
$799$	5.90273	0.208823
$800$	0	0
$801$	−15.1604	−0.535665
$802$	0	0
$803$	80.1023i	2.82675i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 10.5802i	− 0.372440i
$808$	0	0
$809$	26.8670	0.944592	0.472296	−	0.881440i	$-0.343425\pi$
$809$	26.8670	0.944592	0.472296	+	0.881440i	$0.343425\pi$
$810$	0	0
$811$	2.13835	0.0750878	0.0375439	−	0.999295i	$-0.488047\pi$
$811$	2.13835	0.0750878	0.0375439	+	0.999295i	$0.488047\pi$
$812$	0	0
$813$	− 4.26451i	− 0.149563i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	23.7578i	0.831179i
$818$	0	0
$819$	−10.6263	−0.371313
$820$	0	0
$821$	15.1604	0.529100	0.264550	−	0.964372i	$-0.414777\pi$
$821$	15.1604	0.529100	0.264550	+	0.964372i	$0.414777\pi$
$822$	0	0
$823$	− 23.1264i	− 0.806137i	−0.915170	−	0.403068i	$-0.867944\pi$
$823$	− 23.1264i	− 0.806137i	0.915170	−	0.403068i	$-0.132056\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 47.4299i	− 1.64930i	−0.565644	−	0.824650i	$-0.691372\pi$
$827$	− 47.4299i	− 1.64930i	0.565644	−	0.824650i	$-0.308628\pi$
$828$	0	0
$829$	30.1723	1.04793	0.523963	−	0.851741i	$-0.324453\pi$
$829$	30.1723	1.04793	0.523963	+	0.851741i	$0.324453\pi$
$830$	0	0
$831$	−29.3787	−1.01914
$832$	0	0
$833$	6.19454i	0.214628i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 7.73549i	− 0.267378i
$838$	0	0
$839$	−18.8497	−0.650765	−0.325382	−	0.945583i	$-0.605493\pi$
$839$	−18.8497	−0.650765	−0.325382	+	0.945583i	$0.605493\pi$
$840$	0	0
$841$	−28.2065	−0.972637
$842$	0	0
$843$	− 18.7524i	− 0.645869i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	46.5852i	1.60069i
$848$	0	0
$849$	12.8958	0.442584
$850$	0	0
$851$	−1.95388	−0.0669782
$852$	0	0
$853$	29.7578i	1.01889i	0.860504	+	0.509443i	$0.170149\pi$
$853$	29.7578i	1.01889i	−0.860504	+	0.509443i	$0.829851\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	15.2865i	0.522177i	0.965315	+	0.261089i	$0.0840815\pi$
$857$	15.2865i	0.522177i	−0.965315	+	0.261089i	$0.915919\pi$
$858$	0	0
$859$	−33.5171	−1.14359	−0.571794	−	0.820397i	$-0.693752\pi$
$859$	−33.5171	−1.14359	−0.571794	+	0.820397i	$0.693752\pi$
$860$	0	0
$861$	−46.1773	−1.57372
$862$	0	0
$863$	− 31.0443i	− 1.05676i	−0.849008	−	0.528380i	$-0.822800\pi$
$863$	− 31.0443i	− 1.05676i	0.849008	−	0.528380i	$-0.177200\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	16.2065i	0.550401i
$868$	0	0
$869$	−65.2627	−2.21388
$870$	0	0
$871$	−22.0050	−0.745612
$872$	0	0
$873$	− 17.2526i	− 0.583912i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 10.6313i	− 0.358994i	−0.983758	−	0.179497i	$-0.942553\pi$
$877$	− 10.6313i	− 0.358994i	0.983758	−	0.179497i	$-0.0574471\pi$
$878$	0	0
$879$	−8.79857	−0.296768
$880$	0	0
$881$	20.5341	0.691810	0.345905	−	0.938270i	$-0.387572\pi$
$881$	20.5341	0.691810	0.345905	+	0.938270i	$0.387572\pi$
$882$	0	0
$883$	36.0390i	1.21281i	0.795157	+	0.606404i	$0.207388\pi$
$883$	36.0390i	1.21281i	−0.795157	+	0.606404i	$0.792612\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	24.9759i	0.838608i	0.907846	+	0.419304i	$0.137726\pi$
$887$	24.9759i	0.838608i	−0.907846	+	0.419304i	$0.862274\pi$
$888$	0	0
$889$	44.0289	1.47668
$890$	0	0
$891$	4.84469	0.162303
$892$	0	0
$893$	− 45.3549i	− 1.51774i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	2.84469i	0.0949813i
$898$	0	0
$899$	6.89080	0.229821
$900$	0	0
$901$	11.0119	0.366860
$902$	0	0
$903$	12.9658i	0.431475i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 41.6194i	− 1.38195i	−0.722879	−	0.690975i	$-0.757182\pi$
$907$	− 41.6194i	− 1.38195i	0.722879	−	0.690975i	$-0.242818\pi$
$908$	0	0
$909$	2.67241	0.0886383
$910$	0	0
$911$	3.81553	0.126414	0.0632070	−	0.998000i	$-0.479867\pi$
$911$	3.81553	0.126414	0.0632070	+	0.998000i	$0.479867\pi$
$912$	0	0
$913$	− 22.1895i	− 0.734366i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 65.2627i	− 2.15516i
$918$	0	0
$919$	39.7917	1.31261	0.656303	−	0.754497i	$-0.272119\pi$
$919$	39.7917	1.31261	0.656303	+	0.754497i	$0.272119\pi$
$920$	0	0
$921$	10.0050	0.329677
$922$	0	0
$923$	35.1654i	1.15748i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 6.21839i	− 0.204239i
$928$	0	0
$929$	−12.6141	−0.413855	−0.206928	−	0.978356i	$-0.566346\pi$
$929$	−12.6141	−0.413855	−0.206928	+	0.978356i	$0.566346\pi$
$930$	0	0
$931$	47.5971	1.55993
$932$	0	0
$933$	− 18.5290i	− 0.606613i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	22.1262i	0.722830i	0.932405	+	0.361415i	$0.117706\pi$
$937$	22.1262i	0.722830i	−0.932405	+	0.361415i	$0.882294\pi$
$938$	0	0
$939$	−2.39067	−0.0780164
$940$	0	0
$941$	−11.0392	−0.359869	−0.179934	−	0.983679i	$-0.557589\pi$
$941$	−11.0392	−0.359869	−0.179934	+	0.983679i	$0.557589\pi$
$942$	0	0
$943$	12.3618i	0.402555i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 9.68937i	− 0.314862i	−0.987530	−	0.157431i	$-0.949679\pi$
$947$	− 9.68937i	− 0.314862i	0.987530	−	0.157431i	$-0.0503212\pi$
$948$	0	0
$949$	47.0342	1.52679
$950$	0	0
$951$	−16.3157	−0.529072
$952$	0	0
$953$	− 14.1845i	− 0.459480i	−0.973252	−	0.229740i	$-0.926212\pi$
$953$	− 14.1845i	− 0.459480i	0.973252	−	0.229740i	$-0.0737876\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	4.31566i	0.139506i
$958$	0	0
$959$	56.6313	1.82872
$960$	0	0
$961$	28.8378	0.930252
$962$	0	0
$963$	− 14.7986i	− 0.476877i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 12.8447i	− 0.413057i	−0.978441	−	0.206529i	$-0.933783\pi$
$967$	− 12.8447i	− 0.413057i	0.978441	−	0.206529i	$-0.0662167\pi$
$968$	0	0
$969$	−6.09727	−0.195873
$970$	0	0
$971$	−24.1945	−0.776440	−0.388220	−	0.921567i	$-0.626910\pi$
$971$	−24.1945	−0.776440	−0.388220	+	0.921567i	$0.626910\pi$
$972$	0	0
$973$	20.6091i	0.660696i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 18.4879i	− 0.591482i	−0.955268	−	0.295741i	$-0.904434\pi$
$977$	− 18.4879i	− 0.591482i	0.955268	−	0.295741i	$-0.0955665\pi$
$978$	0	0
$979$	73.4471	2.34738
$980$	0	0
$981$	−12.5341	−0.400182
$982$	0	0
$983$	− 28.8325i	− 0.919614i	−0.888019	−	0.459807i	$-0.847919\pi$
$983$	− 28.8325i	− 0.919614i	0.888019	−	0.459807i	$-0.152081\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 24.7524i	− 0.787879i
$988$	0	0
$989$	3.47098	0.110371
$990$	0	0
$991$	40.8958	1.29910	0.649550	−	0.760319i	$-0.274957\pi$
$991$	40.8958	1.29910	0.649550	+	0.760319i	$0.274957\pi$
$992$	0	0
$993$	− 24.8958i	− 0.790046i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	4.65518i	0.147431i	0.997279	+	0.0737155i	$0.0234857\pi$
$997$	4.65518i	0.147431i	−0.997279	+	0.0737155i	$0.976514\pi$
$998$	0	0
$999$	1.95388	0.0618181

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6900.2.f.r.6349.6		6
5.2	odd	4		1380.2.a.j.1.1	✓	3
5.3	odd	4		6900.2.a.x.1.3		3
5.4	even	2	inner	6900.2.f.r.6349.1		6
15.2	even	4		4140.2.a.s.1.1		3
20.7	even	4		5520.2.a.bv.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.2.a.j.1.1	✓	3	5.2	odd	4
4140.2.a.s.1.1		3	15.2	even	4
5520.2.a.bv.1.3		3	20.7	even	4
6900.2.a.x.1.3		3	5.3	odd	4
6900.2.f.r.6349.1		6	5.4	even	2	inner
6900.2.f.r.6349.6		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:	\( N \)	\(=\)	\( 6900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6900.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +3.73549i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +3.73549i q^{7} -1.00000 q^{9} +4.84469 q^{11} -2.84469i q^{13} -0.890804i q^{17} -6.84469 q^{19} -3.73549 q^{21} +1.00000i q^{23} -1.00000i q^{27} +0.890804 q^{29} +7.73549 q^{31} +4.84469i q^{33} +1.95388i q^{37} +2.84469 q^{39} +12.3618 q^{41} -3.47098i q^{43} +6.62629i q^{47} -6.95388 q^{49} +0.890804 q^{51} +12.3618i q^{53} -6.84469i q^{57} -0.890804 q^{59} +8.62629 q^{61} -3.73549i q^{63} -7.73549i q^{67} -1.00000 q^{69} -12.3618 q^{71} +16.5341i q^{73} +18.0973i q^{77} -13.4710 q^{79} +1.00000 q^{81} -4.58018i q^{83} +0.890804i q^{87} +15.1604 q^{89} +10.6263 q^{91} +7.73549i q^{93} +17.2526i q^{97} -4.84469 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^9	\( 6 q - 6 q^{9} + 8 q^{11} - 20 q^{19} + 4 q^{21} + 20 q^{31} - 4 q^{39} + 16 q^{41} - 26 q^{49} + 20 q^{61} - 6 q^{69} - 16 q^{71} - 28 q^{79} + 6 q^{81} - 4 q^{89} + 32 q^{91} - 8 q^{99}+O(q^{100}) \) 6 * q - 6 * q^9 + 8 * q^11 - 20 * q^19 + 4 * q^21 + 20 * q^31 - 4 * q^39 + 16 * q^41 - 26 * q^49 + 20 * q^61 - 6 * q^69 - 16 * q^71 - 28 * q^79 + 6 * q^81 - 4 * q^89 + 32 * q^91 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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