LMFDB - Embedded newform 6900.2.f.r.6349.2

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			6349.2
Root			$2.79911 - 2.79911i$ of defining polynomial
Character	$\chi$	$=$	6900.6349
Dual form			6900.2.f.r.6349.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +1.52644i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +1.52644i q^{7} -1.00000 q^{9} -3.59821 q^{11} -5.59821i q^{13} +4.07177i q^{17} +1.59821 q^{19} +1.52644 q^{21} -1.00000i q^{23} +1.00000i q^{27} +4.07177 q^{29} +2.47356 q^{31} +3.59821i q^{33} +9.66998i q^{37} -5.59821 q^{39} +5.01889 q^{41} -7.05288i q^{43} -4.54533i q^{47} +4.66998 q^{49} +4.07177 q^{51} -5.01889i q^{53} -1.59821i q^{57} -4.07177 q^{59} +6.54533 q^{61} -1.52644i q^{63} +2.47356i q^{67} -1.00000 q^{69} -5.01889 q^{71} +8.79463i q^{73} -5.49245i q^{77} -2.94712 q^{79} +1.00000 q^{81} -9.12465i q^{83} -4.07177i q^{87} -12.2493 q^{89} +8.54533 q^{91} -2.47356i q^{93} -13.0907i q^{97} +3.59821 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	1.52644i	0.576940i	0.957489	+	0.288470i	$0.0931465\pi$
$7$	1.52644i	0.576940i	−0.957489	+	0.288470i	$0.906853\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−3.59821	−1.08490	−0.542451	−	0.840088i	$-0.682503\pi$
$11$	−3.59821	−1.08490	−0.542451	+	0.840088i	$0.682503\pi$
$12$	0	0
$13$	− 5.59821i	− 1.55266i	−0.630324	−	0.776332i	$-0.717078\pi$
$13$	− 5.59821i	− 1.55266i	0.630324	−	0.776332i	$-0.282922\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.07177i	0.987550i	0.869590	+	0.493775i	$0.164383\pi$
$17$	4.07177i	0.987550i	−0.869590	+	0.493775i	$0.835617\pi$
$18$	0	0
$19$	1.59821	0.366655	0.183327	−	0.983052i	$-0.441313\pi$
$19$	1.59821	0.366655	0.183327	+	0.983052i	$0.441313\pi$
$20$	0	0
$21$	1.52644	0.333096
$22$	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
$29$	4.07177	0.756109	0.378054	−	0.925783i	$-0.376593\pi$
$29$	4.07177	0.756109	0.378054	+	0.925783i	$0.376593\pi$
$30$	0	0
$31$	2.47356	0.444265	0.222132	−	0.975017i	$-0.428698\pi$
$31$	2.47356	0.444265	0.222132	+	0.975017i	$0.428698\pi$
$32$	0	0
$33$	3.59821i	0.626368i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	9.66998i	1.58974i	0.606783	+	0.794868i	$0.292460\pi$
$37$	9.66998i	1.58974i	−0.606783	+	0.794868i	$0.707540\pi$
$38$	0	0
$39$	−5.59821	−0.896431
$40$	0	0
$41$	5.01889	0.783819	0.391910	−	0.920004i	$-0.371815\pi$
$41$	5.01889	0.783819	0.391910	+	0.920004i	$0.371815\pi$
$42$	0	0
$43$	− 7.05288i	− 1.07555i	−0.843087	−	0.537777i	$-0.819264\pi$
$43$	− 7.05288i	− 1.07555i	0.843087	−	0.537777i	$-0.180736\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 4.54533i	− 0.663005i	−0.943454	−	0.331502i	$-0.892445\pi$
$47$	− 4.54533i	− 0.663005i	0.943454	−	0.331502i	$-0.107555\pi$
$48$	0	0
$49$	4.66998	0.667140
$50$	0	0
$51$	4.07177	0.570162
$52$	0	0
$53$	− 5.01889i	− 0.689398i	−0.938713	−	0.344699i	$-0.887981\pi$
$53$	− 5.01889i	− 0.689398i	0.938713	−	0.344699i	$-0.112019\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 1.59821i	− 0.211688i
$58$	0	0
$59$	−4.07177	−0.530099	−0.265050	−	0.964235i	$-0.585388\pi$
$59$	−4.07177	−0.530099	−0.265050	+	0.964235i	$0.585388\pi$
$60$	0	0
$61$	6.54533	0.838044	0.419022	−	0.907976i	$-0.362373\pi$
$61$	6.54533	0.838044	0.419022	+	0.907976i	$0.362373\pi$
$62$	0	0
$63$	− 1.52644i	− 0.192313i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	2.47356i	0.302193i	0.988519	+	0.151097i	$0.0482805\pi$
$67$	2.47356i	0.302193i	−0.988519	+	0.151097i	$0.951720\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−5.01889	−0.595633	−0.297816	−	0.954623i	$-0.596258\pi$
$71$	−5.01889	−0.595633	−0.297816	+	0.954623i	$0.596258\pi$
$72$	0	0
$73$	8.79463i	1.02933i	0.857390	+	0.514667i	$0.172084\pi$
$73$	8.79463i	1.02933i	−0.857390	+	0.514667i	$0.827916\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 5.49245i	− 0.625923i
$78$	0	0
$79$	−2.94712	−0.331577	−0.165788	−	0.986161i	$-0.553017\pi$
$79$	−2.94712	−0.331577	−0.165788	+	0.986161i	$0.553017\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 9.12465i	− 1.00156i	−0.865574	−	0.500780i	$-0.833046\pi$
$83$	− 9.12465i	− 1.00156i	0.865574	−	0.500780i	$-0.166954\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 4.07177i	− 0.436540i
$88$	0	0
$89$	−12.2493	−1.29842	−0.649212	−	0.760608i	$-0.724901\pi$
$89$	−12.2493	−1.29842	−0.649212	+	0.760608i	$0.724901\pi$
$90$	0	0
$91$	8.54533	0.895794
$92$	0	0
$93$	− 2.47356i	− 0.256496i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 13.0907i	− 1.32916i	−0.747219	−	0.664578i	$-0.768611\pi$
$97$	− 13.0907i	− 1.32916i	0.747219	−	0.664578i	$-0.231389\pi$
$98$	0	0
$99$	3.59821	0.361634
$100$	0	0
$101$	−12.2153	−1.21547	−0.607735	−	0.794140i	$-0.707922\pi$
$101$	−12.2153	−1.21547	−0.607735	+	0.794140i	$0.707922\pi$
$102$	0	0
$103$	0.143542i	0.0141436i	0.999975	+	0.00707181i	$0.00225105\pi$
$103$	0.143542i	0.0141436i	−0.999975	+	0.00707181i	$0.997749\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.26819i	0.509295i	0.967034	+	0.254648i	$0.0819595\pi$
$107$	5.26819i	0.509295i	−0.967034	+	0.254648i	$0.918040\pi$
$108$	0	0
$109$	−12.7946	−1.22550	−0.612752	−	0.790275i	$-0.709937\pi$
$109$	−12.7946	−1.22550	−0.612752	+	0.790275i	$0.709937\pi$
$110$	0	0
$111$	9.66998	0.917834
$112$	0	0
$113$	− 5.01889i	− 0.472138i	−0.971736	−	0.236069i	$-0.924141\pi$
$113$	− 5.01889i	− 0.472138i	0.971736	−	0.236069i	$-0.0758591\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	5.59821i	0.517555i
$118$	0	0
$119$	−6.21531	−0.569757
$120$	0	0
$121$	1.94712	0.177011
$122$	0	0
$123$	− 5.01889i	− 0.452538i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 17.7040i	− 1.57097i	−0.618879	−	0.785487i	$-0.712413\pi$
$127$	− 17.7040i	− 1.57097i	0.618879	−	0.785487i	$-0.287587\pi$
$128$	0	0
$129$	−7.05288	−0.620971
$130$	0	0
$131$	−6.94712	−0.606973	−0.303486	−	0.952836i	$-0.598151\pi$
$131$	−6.94712	−0.606973	−0.303486	+	0.952836i	$0.598151\pi$
$132$	0	0
$133$	2.43957i	0.211538i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 12.2493i	− 1.04653i	−0.852171	−	0.523264i	$-0.824714\pi$
$137$	− 12.2493i	− 1.04653i	0.852171	−	0.523264i	$-0.175286\pi$
$138$	0	0
$139$	6.61710	0.561255	0.280628	−	0.959817i	$-0.409457\pi$
$139$	6.61710	0.561255	0.280628	+	0.959817i	$0.409457\pi$
$140$	0	0
$141$	−4.54533	−0.382786
$142$	0	0
$143$	20.1435i	1.68449i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 4.66998i	− 0.385174i
$148$	0	0
$149$	−9.59821	−0.786316	−0.393158	−	0.919471i	$-0.628617\pi$
$149$	−9.59821	−0.786316	−0.393158	+	0.919471i	$0.628617\pi$
$150$	0	0
$151$	−4.94712	−0.402591	−0.201295	−	0.979531i	$-0.564515\pi$
$151$	−4.94712	−0.402591	−0.201295	+	0.979531i	$0.564515\pi$
$152$	0	0
$153$	− 4.07177i	− 0.329183i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 7.63220i	− 0.609116i	−0.952494	−	0.304558i	$-0.901491\pi$
$157$	− 7.63220i	− 0.609116i	0.952494	−	0.304558i	$-0.0985087\pi$
$158$	0	0
$159$	−5.01889	−0.398024
$160$	0	0
$161$	1.52644	0.120300
$162$	0	0
$163$	− 23.3400i	− 1.82813i	−0.405571	−	0.914064i	$-0.632927\pi$
$163$	− 23.3400i	− 1.82813i	0.405571	−	0.914064i	$-0.367073\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 7.45467i	− 0.576860i	−0.957501	−	0.288430i	$-0.906867\pi$
$167$	− 7.45467i	− 0.576860i	0.957501	−	0.288430i	$-0.0931332\pi$
$168$	0	0
$169$	−18.3400	−1.41077
$170$	0	0
$171$	−1.59821	−0.122218
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	4.07177i	0.306053i
$178$	0	0
$179$	22.2871	1.66581	0.832907	−	0.553412i	$-0.186675\pi$
$179$	22.2871	1.66581	0.832907	+	0.553412i	$0.186675\pi$
$180$	0	0
$181$	20.2493	1.50512	0.752559	−	0.658524i	$-0.228819\pi$
$181$	20.2493	1.50512	0.752559	+	0.658524i	$0.228819\pi$
$182$	0	0
$183$	− 6.54533i	− 0.483845i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 14.6511i	− 1.07139i
$188$	0	0
$189$	−1.52644	−0.111032
$190$	0	0
$191$	−21.8475	−1.58083	−0.790415	−	0.612571i	$-0.790135\pi$
$191$	−21.8475	−1.58083	−0.790415	+	0.612571i	$0.790135\pi$
$192$	0	0
$193$	− 12.0378i	− 0.866499i	−0.901274	−	0.433249i	$-0.857367\pi$
$193$	− 12.0378i	− 0.866499i	0.901274	−	0.433249i	$-0.142633\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 18.2493i	− 1.30021i	−0.759845	−	0.650104i	$-0.774725\pi$
$197$	− 18.2493i	− 1.30021i	0.759845	−	0.650104i	$-0.225275\pi$
$198$	0	0
$199$	17.1964	1.21902	0.609511	−	0.792778i	$-0.291366\pi$
$199$	17.1964	1.21902	0.609511	+	0.792778i	$0.291366\pi$
$200$	0	0
$201$	2.47356	0.174471
$202$	0	0
$203$	6.21531i	0.436229i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.00000i	0.0695048i
$208$	0	0
$209$	−5.75070	−0.397784
$210$	0	0
$211$	−5.66998	−0.390338	−0.195169	−	0.980770i	$-0.562525\pi$
$211$	−5.66998	−0.390338	−0.195169	+	0.980770i	$0.562525\pi$
$212$	0	0
$213$	5.01889i	0.343889i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	3.77574i	0.256314i
$218$	0	0
$219$	8.79463	0.594286
$220$	0	0
$221$	22.7946	1.53333
$222$	0	0
$223$	− 11.3400i	− 0.759380i	−0.925114	−	0.379690i	$-0.876031\pi$
$223$	− 11.3400i	− 0.759380i	0.925114	−	0.379690i	$-0.123969\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 26.3928i	− 1.75175i	−0.482534	−	0.875877i	$-0.660284\pi$
$227$	− 26.3928i	− 1.75175i	0.482534	−	0.875877i	$-0.339716\pi$
$228$	0	0
$229$	4.24930	0.280802	0.140401	−	0.990095i	$-0.455161\pi$
$229$	4.24930	0.280802	0.140401	+	0.990095i	$0.455161\pi$
$230$	0	0
$231$	−5.49245	−0.361377
$232$	0	0
$233$	1.89424i	0.124096i	0.998073	+	0.0620479i	$0.0197632\pi$
$233$	1.89424i	0.124096i	−0.998073	+	0.0620479i	$0.980237\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	2.94712i	0.191436i
$238$	0	0
$239$	6.98111	0.451570	0.225785	−	0.974177i	$-0.427505\pi$
$239$	6.98111	0.451570	0.225785	+	0.974177i	$0.427505\pi$
$240$	0	0
$241$	3.20537	0.206476	0.103238	−	0.994657i	$-0.467080\pi$
$241$	3.20537	0.206476	0.103238	+	0.994657i	$0.467080\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 8.94712i	− 0.569292i
$248$	0	0
$249$	−9.12465	−0.578251
$250$	0	0
$251$	13.4457	0.848686	0.424343	−	0.905501i	$-0.360505\pi$
$251$	13.4457	0.848686	0.424343	+	0.905501i	$0.360505\pi$
$252$	0	0
$253$	3.59821i	0.226218i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	18.9382i	1.18133i	0.806917	+	0.590665i	$0.201135\pi$
$257$	18.9382i	1.18133i	−0.806917	+	0.590665i	$0.798865\pi$
$258$	0	0
$259$	−14.7606	−0.917182
$260$	0	0
$261$	−4.07177	−0.252036
$262$	0	0
$263$	0.981108i	0.0604977i	0.999542	+	0.0302489i	$0.00962998\pi$
$263$	0.981108i	0.0604977i	−0.999542	+	0.0302489i	$0.990370\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	12.2493i	0.749645i
$268$	0	0
$269$	3.12465	0.190513	0.0952567	−	0.995453i	$-0.469633\pi$
$269$	3.12465	0.190513	0.0952567	+	0.995453i	$0.469633\pi$
$270$	0	0
$271$	−9.52644	−0.578690	−0.289345	−	0.957225i	$-0.593437\pi$
$271$	−9.52644	−0.578690	−0.289345	+	0.957225i	$0.593437\pi$
$272$	0	0
$273$	− 8.54533i	− 0.517187i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	4.39284i	0.263940i	0.991254	+	0.131970i	$0.0421303\pi$
$277$	4.39284i	0.263940i	−0.991254	+	0.131970i	$0.957870\pi$
$278$	0	0
$279$	−2.47356	−0.148088
$280$	0	0
$281$	12.9382	0.771827	0.385913	−	0.922535i	$-0.373886\pi$
$281$	12.9382	0.771827	0.385913	+	0.922535i	$0.373886\pi$
$282$	0	0
$283$	− 19.7757i	− 1.17555i	−0.809026	−	0.587773i	$-0.800005\pi$
$283$	− 19.7757i	− 1.17555i	0.809026	−	0.587773i	$-0.199995\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	7.66104i	0.452217i
$288$	0	0
$289$	0.420681	0.0247459
$290$	0	0
$291$	−13.0907	−0.767388
$292$	0	0
$293$	11.2682i	0.658295i	0.944279	+	0.329147i	$0.106761\pi$
$293$	11.2682i	0.658295i	−0.944279	+	0.329147i	$0.893239\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 3.59821i	− 0.208789i
$298$	0	0
$299$	−5.59821	−0.323753
$300$	0	0
$301$	10.7658	0.620530
$302$	0	0
$303$	12.2153i	0.701751i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 25.8475i	− 1.47520i	−0.675240	−	0.737598i	$-0.735960\pi$
$307$	− 25.8475i	− 1.47520i	0.675240	−	0.737598i	$-0.264040\pi$
$308$	0	0
$309$	0.143542	0.00816582
$310$	0	0
$311$	−29.0529	−1.64744	−0.823719	−	0.566998i	$-0.808105\pi$
$311$	−29.0529	−1.64744	−0.823719	+	0.566998i	$0.808105\pi$
$312$	0	0
$313$	21.9571i	1.24109i	0.784172	+	0.620543i	$0.213088\pi$
$313$	21.9571i	1.24109i	−0.784172	+	0.620543i	$0.786912\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.65109i	0.148900i	0.997225	+	0.0744500i	$0.0237201\pi$
$317$	2.65109i	0.148900i	−0.997225	+	0.0744500i	$0.976280\pi$
$318$	0	0
$319$	−14.6511	−0.820304
$320$	0	0
$321$	5.26819	0.294042
$322$	0	0
$323$	6.50755i	0.362090i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	12.7946i	0.707545i
$328$	0	0
$329$	6.93817	0.382514
$330$	0	0
$331$	7.77574	0.427393	0.213697	−	0.976900i	$-0.431450\pi$
$331$	7.77574	0.427393	0.213697	+	0.976900i	$0.431450\pi$
$332$	0	0
$333$	− 9.66998i	− 0.529912i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 16.9471i	− 0.923168i	−0.887096	−	0.461584i	$-0.847281\pi$
$337$	− 16.9471i	− 0.923168i	0.887096	−	0.461584i	$-0.152719\pi$
$338$	0	0
$339$	−5.01889	−0.272589
$340$	0	0
$341$	−8.90039	−0.481983
$342$	0	0
$343$	17.8135i	0.961840i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.90934i	0.156181i	0.996946	+	0.0780907i	$0.0248824\pi$
$347$	2.90934i	0.156181i	−0.996946	+	0.0780907i	$0.975118\pi$
$348$	0	0
$349$	18.8664	1.00990	0.504948	−	0.863150i	$-0.331512\pi$
$349$	18.8664	1.00990	0.504948	+	0.863150i	$0.331512\pi$
$350$	0	0
$351$	5.59821	0.298810
$352$	0	0
$353$	− 18.9382i	− 1.00798i	−0.863710	−	0.503989i	$-0.831865\pi$
$353$	− 18.9382i	− 1.00798i	0.863710	−	0.503989i	$-0.168135\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	6.21531i	0.328949i
$358$	0	0
$359$	4.54533	0.239893	0.119947	−	0.992780i	$-0.461728\pi$
$359$	4.54533	0.239893	0.119947	+	0.992780i	$0.461728\pi$
$360$	0	0
$361$	−16.4457	−0.865564
$362$	0	0
$363$	− 1.94712i	− 0.102197i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 33.9571i	− 1.77254i	−0.463165	−	0.886272i	$-0.653286\pi$
$367$	− 33.9571i	− 1.77254i	0.463165	−	0.886272i	$-0.346714\pi$
$368$	0	0
$369$	−5.01889	−0.261273
$370$	0	0
$371$	7.66104	0.397741
$372$	0	0
$373$	12.2115i	0.632288i	0.948711	+	0.316144i	$0.102388\pi$
$373$	12.2115i	0.632288i	−0.948711	+	0.316144i	$0.897612\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 22.7946i	− 1.17398i
$378$	0	0
$379$	−13.0529	−0.670481	−0.335241	−	0.942133i	$-0.608818\pi$
$379$	−13.0529	−0.670481	−0.335241	+	0.942133i	$0.608818\pi$
$380$	0	0
$381$	−17.7040	−0.907002
$382$	0	0
$383$	26.4268i	1.35035i	0.737659	+	0.675174i	$0.235931\pi$
$383$	26.4268i	1.35035i	−0.737659	+	0.675174i	$0.764069\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	7.05288i	0.358518i
$388$	0	0
$389$	15.5893	0.790407	0.395204	−	0.918594i	$-0.370674\pi$
$389$	15.5893	0.790407	0.395204	+	0.918594i	$0.370674\pi$
$390$	0	0
$391$	4.07177	0.205918
$392$	0	0
$393$	6.94712i	0.350436i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 35.4457i	− 1.77897i	−0.456965	−	0.889485i	$-0.651063\pi$
$397$	− 35.4457i	− 1.77897i	0.456965	−	0.889485i	$-0.348937\pi$
$398$	0	0
$399$	2.43957	0.122131
$400$	0	0
$401$	27.5893	1.37774	0.688871	−	0.724884i	$-0.258107\pi$
$401$	27.5893	1.37774	0.688871	+	0.724884i	$0.258107\pi$
$402$	0	0
$403$	− 13.8475i	− 0.689794i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 34.7946i	− 1.72471i
$408$	0	0
$409$	−12.7606	−0.630973	−0.315487	−	0.948930i	$-0.602168\pi$
$409$	−12.7606	−0.630973	−0.315487	+	0.948930i	$0.602168\pi$
$410$	0	0
$411$	−12.2493	−0.604213
$412$	0	0
$413$	− 6.21531i	− 0.305836i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 6.61710i	− 0.324041i
$418$	0	0
$419$	22.7946	1.11359	0.556795	−	0.830650i	$-0.312031\pi$
$419$	22.7946	1.11359	0.556795	+	0.830650i	$0.312031\pi$
$420$	0	0
$421$	26.6889	1.30074	0.650368	−	0.759619i	$-0.274615\pi$
$421$	26.6889	1.30074	0.650368	+	0.759619i	$0.274615\pi$
$422$	0	0
$423$	4.54533i	0.221002i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	9.99105i	0.483501i
$428$	0	0
$429$	20.1435	0.972539
$430$	0	0
$431$	−27.4079	−1.32019	−0.660097	−	0.751180i	$-0.729485\pi$
$431$	−27.4079	−1.32019	−0.660097	+	0.751180i	$0.729485\pi$
$432$	0	0
$433$	8.57932i	0.412296i	0.978521	+	0.206148i	$0.0660928\pi$
$433$	8.57932i	0.412296i	−0.978521	+	0.206148i	$0.933907\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 1.59821i	− 0.0764528i
$438$	0	0
$439$	−9.96222	−0.475471	−0.237735	−	0.971330i	$-0.576405\pi$
$439$	−9.96222	−0.475471	−0.237735	+	0.971330i	$0.576405\pi$
$440$	0	0
$441$	−4.66998	−0.222380
$442$	0	0
$443$	1.63599i	0.0777284i	0.999245	+	0.0388642i	$0.0123740\pi$
$443$	1.63599i	0.0777284i	−0.999245	+	0.0388642i	$0.987626\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	9.59821i	0.453980i
$448$	0	0
$449$	−38.6082	−1.82203	−0.911016	−	0.412372i	$-0.864701\pi$
$449$	−38.6082	−1.82203	−0.911016	+	0.412372i	$0.864701\pi$
$450$	0	0
$451$	−18.0590	−0.850366
$452$	0	0
$453$	4.94712i	0.232436i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 17.6700i	− 0.826567i	−0.910602	−	0.413283i	$-0.864382\pi$
$457$	− 17.6700i	− 0.826567i	0.910602	−	0.413283i	$-0.135618\pi$
$458$	0	0
$459$	−4.07177	−0.190054
$460$	0	0
$461$	−15.0907	−0.702842	−0.351421	−	0.936217i	$-0.614301\pi$
$461$	−15.0907	−0.702842	−0.351421	+	0.936217i	$0.614301\pi$
$462$	0	0
$463$	− 22.1346i	− 1.02868i	−0.857586	−	0.514341i	$-0.828037\pi$
$463$	− 22.1346i	− 1.02868i	0.857586	−	0.514341i	$-0.171963\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	29.2682i	1.35437i	0.735813	+	0.677185i	$0.236800\pi$
$467$	29.2682i	1.35437i	−0.735813	+	0.677185i	$0.763200\pi$
$468$	0	0
$469$	−3.77574	−0.174348
$470$	0	0
$471$	−7.63220	−0.351673
$472$	0	0
$473$	25.3777i	1.16687i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	5.01889i	0.229799i
$478$	0	0
$479$	−18.9382	−0.865307	−0.432654	−	0.901560i	$-0.642423\pi$
$479$	−18.9382	−0.865307	−0.432654	+	0.901560i	$0.642423\pi$
$480$	0	0
$481$	54.1346	2.46833
$482$	0	0
$483$	− 1.52644i	− 0.0694554i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 5.20537i	− 0.235878i	−0.993021	−	0.117939i	$-0.962371\pi$
$487$	− 5.20537i	− 0.235878i	0.993021	−	0.117939i	$-0.0376287\pi$
$488$	0	0
$489$	−23.3400	−1.05547
$490$	0	0
$491$	−12.2153	−0.551269	−0.275635	−	0.961262i	$-0.588888\pi$
$491$	−12.2153	−0.551269	−0.275635	+	0.961262i	$0.588888\pi$
$492$	0	0
$493$	16.5793i	0.746695i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 7.66104i	− 0.343644i
$498$	0	0
$499$	−35.1157	−1.57199	−0.785997	−	0.618230i	$-0.787850\pi$
$499$	−35.1157	−1.57199	−0.785997	+	0.618230i	$0.787850\pi$
$500$	0	0
$501$	−7.45467	−0.333050
$502$	0	0
$503$	6.21531i	0.277127i	0.990354	+	0.138564i	$0.0442485\pi$
$503$	6.21531i	0.277127i	−0.990354	+	0.138564i	$0.955751\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	18.3400i	0.814506i
$508$	0	0
$509$	−9.85646	−0.436880	−0.218440	−	0.975850i	$-0.570097\pi$
$509$	−9.85646	−0.436880	−0.218440	+	0.975850i	$0.570097\pi$
$510$	0	0
$511$	−13.4245	−0.593864
$512$	0	0
$513$	1.59821i	0.0705627i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	16.3551i	0.719295i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−41.2253	−1.80611	−0.903056	−	0.429524i	$-0.858682\pi$
$521$	−41.2253	−1.80611	−0.903056	+	0.429524i	$0.858682\pi$
$522$	0	0
$523$	22.6799i	0.991724i	0.868401	+	0.495862i	$0.165148\pi$
$523$	22.6799i	0.991724i	−0.868401	+	0.495862i	$0.834852\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	10.0718i	0.438733i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	4.07177	0.176700
$532$	0	0
$533$	− 28.0968i	− 1.21701i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 22.2871i	− 0.961759i
$538$	0	0
$539$	−16.8036	−0.723781
$540$	0	0
$541$	22.1435	0.952025	0.476013	−	0.879438i	$-0.342082\pi$
$541$	22.1435	0.952025	0.476013	+	0.879438i	$0.342082\pi$
$542$	0	0
$543$	− 20.2493i	− 0.868981i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 28.0000i	− 1.19719i	−0.801050	−	0.598597i	$-0.795725\pi$
$547$	− 28.0000i	− 1.19719i	0.801050	−	0.598597i	$-0.204275\pi$
$548$	0	0
$549$	−6.54533	−0.279348
$550$	0	0
$551$	6.50755	0.277231
$552$	0	0
$553$	− 4.49860i	− 0.191300i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 9.37395i	− 0.397187i	−0.980082	−	0.198594i	$-0.936363\pi$
$557$	− 9.37395i	− 0.397187i	0.980082	−	0.198594i	$-0.0636374\pi$
$558$	0	0
$559$	−39.4835	−1.66997
$560$	0	0
$561$	−14.6511	−0.618570
$562$	0	0
$563$	− 27.3060i	− 1.15081i	−0.817869	−	0.575405i	$-0.804845\pi$
$563$	− 27.3060i	− 1.15081i	0.817869	−	0.575405i	$-0.195155\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.52644i	0.0641044i
$568$	0	0
$569$	−23.3022	−0.976878	−0.488439	−	0.872598i	$-0.662434\pi$
$569$	−23.3022	−0.976878	−0.488439	+	0.872598i	$0.662434\pi$
$570$	0	0
$571$	32.9382	1.37842	0.689210	−	0.724562i	$-0.257958\pi$
$571$	32.9382	1.37842	0.689210	+	0.724562i	$0.257958\pi$
$572$	0	0
$573$	21.8475i	0.912693i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	41.3400i	1.72101i	0.509446	+	0.860503i	$0.329850\pi$
$577$	41.3400i	1.72101i	−0.509446	+	0.860503i	$0.670150\pi$
$578$	0	0
$579$	−12.0378	−0.500273
$580$	0	0
$581$	13.9282	0.577840
$582$	0	0
$583$	18.0590i	0.747929i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	23.0529i	0.951494i	0.879582	+	0.475747i	$0.157822\pi$
$587$	23.0529i	0.951494i	−0.879582	+	0.475747i	$0.842178\pi$
$588$	0	0
$589$	3.95327	0.162892
$590$	0	0
$591$	−18.2493	−0.750676
$592$	0	0
$593$	− 31.9533i	− 1.31216i	−0.754690	−	0.656082i	$-0.772213\pi$
$593$	− 31.9533i	− 1.31216i	0.754690	−	0.656082i	$-0.227787\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 17.1964i	− 0.703803i
$598$	0	0
$599$	−26.6421	−1.08857	−0.544284	−	0.838901i	$-0.683199\pi$
$599$	−26.6421	−1.08857	−0.544284	+	0.838901i	$0.683199\pi$
$600$	0	0
$601$	7.45846	0.304237	0.152119	−	0.988362i	$-0.451390\pi$
$601$	7.45846	0.304237	0.152119	+	0.988362i	$0.451390\pi$
$602$	0	0
$603$	− 2.47356i	− 0.100731i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	13.4925i	0.547642i	0.961781	+	0.273821i	$0.0882875\pi$
$607$	13.4925i	0.547642i	−0.961781	+	0.273821i	$0.911712\pi$
$608$	0	0
$609$	6.21531	0.251857
$610$	0	0
$611$	−25.4457	−1.02942
$612$	0	0
$613$	41.4457i	1.67398i	0.547221	+	0.836988i	$0.315686\pi$
$613$	41.4457i	1.67398i	−0.547221	+	0.836988i	$0.684314\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	25.2304i	1.01574i	0.861434	+	0.507869i	$0.169567\pi$
$617$	25.2304i	1.01574i	−0.861434	+	0.507869i	$0.830433\pi$
$618$	0	0
$619$	5.12845	0.206130	0.103065	−	0.994675i	$-0.467135\pi$
$619$	5.12845	0.206130	0.103065	+	0.994675i	$0.467135\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	− 18.6978i	− 0.749112i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	5.75070i	0.229661i
$628$	0	0
$629$	−39.3740	−1.56994
$630$	0	0
$631$	−11.6360	−0.463222	−0.231611	−	0.972809i	$-0.574400\pi$
$631$	−11.6360	−0.463222	−0.231611	+	0.972809i	$0.574400\pi$
$632$	0	0
$633$	5.66998i	0.225362i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 26.1435i	− 1.03584i
$638$	0	0
$639$	5.01889	0.198544
$640$	0	0
$641$	17.7418	0.700757	0.350379	−	0.936608i	$-0.386053\pi$
$641$	17.7418	0.700757	0.350379	+	0.936608i	$0.386053\pi$
$642$	0	0
$643$	− 31.7757i	− 1.25311i	−0.779376	−	0.626556i	$-0.784464\pi$
$643$	− 31.7757i	− 1.25311i	0.779376	−	0.626556i	$-0.215536\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 46.7946i	− 1.83969i	−0.392286	−	0.919843i	$-0.628316\pi$
$647$	− 46.7946i	− 1.83969i	0.392286	−	0.919843i	$-0.371684\pi$
$648$	0	0
$649$	14.6511	0.575106
$650$	0	0
$651$	3.77574	0.147983
$652$	0	0
$653$	− 3.59821i	− 0.140809i	−0.997519	−	0.0704044i	$-0.977571\pi$
$653$	− 3.59821i	− 0.140809i	0.997519	−	0.0704044i	$-0.0224290\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 8.79463i	− 0.343111i
$658$	0	0
$659$	5.49245	0.213956	0.106978	−	0.994261i	$-0.465883\pi$
$659$	5.49245	0.213956	0.106978	+	0.994261i	$0.465883\pi$
$660$	0	0
$661$	4.90934	0.190951	0.0954755	−	0.995432i	$-0.469563\pi$
$661$	4.90934	0.190951	0.0954755	+	0.995432i	$0.469563\pi$
$662$	0	0
$663$	− 22.7946i	− 0.885270i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 4.07177i	− 0.157660i
$668$	0	0
$669$	−11.3400	−0.438428
$670$	0	0
$671$	−23.5515	−0.909195
$672$	0	0
$673$	− 25.2253i	− 0.972362i	−0.873858	−	0.486181i	$-0.838390\pi$
$673$	− 25.2253i	− 0.972362i	0.873858	−	0.486181i	$-0.161610\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	49.6610i	1.90863i	0.298805	+	0.954314i	$0.403412\pi$
$677$	49.6610i	1.90863i	−0.298805	+	0.954314i	$0.596588\pi$
$678$	0	0
$679$	19.9821	0.766843
$680$	0	0
$681$	−26.3928	−1.01138
$682$	0	0
$683$	− 9.84751i	− 0.376805i	−0.982092	−	0.188402i	$-0.939669\pi$
$683$	− 9.84751i	− 0.376805i	0.982092	−	0.188402i	$-0.0603309\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 4.24930i	− 0.162121i
$688$	0	0
$689$	−28.0968	−1.07040
$690$	0	0
$691$	−17.4457	−0.663667	−0.331833	−	0.943338i	$-0.607667\pi$
$691$	−17.4457	−0.663667	−0.331833	+	0.943338i	$0.607667\pi$
$692$	0	0
$693$	5.49245i	0.208641i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	20.4358i	0.774060i
$698$	0	0
$699$	1.89424	0.0716468
$700$	0	0
$701$	24.5075	0.925637	0.462819	−	0.886453i	$-0.346838\pi$
$701$	24.5075	0.925637	0.462819	+	0.886453i	$0.346838\pi$
$702$	0	0
$703$	15.4547i	0.582884i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 18.6459i	− 0.701253i
$708$	0	0
$709$	41.4547	1.55686	0.778431	−	0.627730i	$-0.216016\pi$
$709$	41.4547	1.55686	0.778431	+	0.627730i	$0.216016\pi$
$710$	0	0
$711$	2.94712	0.110526
$712$	0	0
$713$	− 2.47356i	− 0.0926356i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 6.98111i	− 0.260714i
$718$	0	0
$719$	−30.9811	−1.15540	−0.577700	−	0.816249i	$-0.696050\pi$
$719$	−30.9811	−1.15540	−0.577700	+	0.816249i	$0.696050\pi$
$720$	0	0
$721$	−0.219108	−0.00816002
$722$	0	0
$723$	− 3.20537i	− 0.119209i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	15.8513i	0.587892i	0.955822	+	0.293946i	$0.0949687\pi$
$727$	15.8513i	0.587892i	−0.955822	+	0.293946i	$0.905031\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	28.7177	1.06216
$732$	0	0
$733$	− 0.0807173i	− 0.00298136i	−0.999999	−	0.00149068i	$-0.999526\pi$
$733$	− 0.0807173i	− 0.00298136i	0.999999	−	0.00149068i	$-0.000474499\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 8.90039i	− 0.327850i
$738$	0	0
$739$	41.2215	1.51636	0.758178	−	0.652048i	$-0.226090\pi$
$739$	41.2215	1.51636	0.758178	+	0.652048i	$0.226090\pi$
$740$	0	0
$741$	−8.94712	−0.328681
$742$	0	0
$743$	− 7.71292i	− 0.282959i	−0.989941	−	0.141480i	$-0.954814\pi$
$743$	− 7.71292i	− 0.282959i	0.989941	−	0.141480i	$-0.0451860\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	9.12465i	0.333854i
$748$	0	0
$749$	−8.04158	−0.293833
$750$	0	0
$751$	24.7946	0.904769	0.452384	−	0.891823i	$-0.350573\pi$
$751$	24.7946	0.904769	0.452384	+	0.891823i	$0.350573\pi$
$752$	0	0
$753$	− 13.4457i	− 0.489989i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 30.6171i	− 1.11280i	−0.830915	−	0.556399i	$-0.812183\pi$
$757$	− 30.6171i	− 1.11280i	0.830915	−	0.556399i	$-0.187817\pi$
$758$	0	0
$759$	3.59821	0.130607
$760$	0	0
$761$	37.1624	1.34714	0.673569	−	0.739125i	$-0.264761\pi$
$761$	37.1624	1.34714	0.673569	+	0.739125i	$0.264761\pi$
$762$	0	0
$763$	− 19.5302i	− 0.707042i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	22.7946i	0.823066i
$768$	0	0
$769$	40.4217	1.45764	0.728822	−	0.684704i	$-0.240068\pi$
$769$	40.4217	1.45764	0.728822	+	0.684704i	$0.240068\pi$
$770$	0	0
$771$	18.9382	0.682042
$772$	0	0
$773$	− 25.1964i	− 0.906252i	−0.891446	−	0.453126i	$-0.850309\pi$
$773$	− 25.1964i	− 0.906252i	0.891446	−	0.453126i	$-0.149691\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	14.7606i	0.529535i
$778$	0	0
$779$	8.02125	0.287391
$780$	0	0
$781$	18.0590	0.646203
$782$	0	0
$783$	4.07177i	0.145513i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	9.66998i	0.344698i	0.985036	+	0.172349i	$0.0551356\pi$
$787$	9.66998i	0.344698i	−0.985036	+	0.172349i	$0.944864\pi$
$788$	0	0
$789$	0.981108	0.0349284
$790$	0	0
$791$	7.66104	0.272395
$792$	0	0
$793$	− 36.6421i	− 1.30120i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	37.6610i	1.33402i	0.745048	+	0.667011i	$0.232427\pi$
$797$	37.6610i	1.33402i	−0.745048	+	0.667011i	$0.767573\pi$
$798$	0	0
$799$	18.5075	0.654750
$800$	0	0
$801$	12.2493	0.432808
$802$	0	0
$803$	− 31.6449i	− 1.11673i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 3.12465i	− 0.109993i
$808$	0	0
$809$	11.2002	0.393779	0.196889	−	0.980426i	$-0.436916\pi$
$809$	11.2002	0.393779	0.196889	+	0.980426i	$0.436916\pi$
$810$	0	0
$811$	37.0099	1.29959	0.649797	−	0.760107i	$-0.274854\pi$
$811$	37.0099	1.29959	0.649797	+	0.760107i	$0.274854\pi$
$812$	0	0
$813$	9.52644i	0.334107i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 11.2720i	− 0.394357i
$818$	0	0
$819$	−8.54533	−0.298598
$820$	0	0
$821$	−12.2493	−0.427504	−0.213752	−	0.976888i	$-0.568568\pi$
$821$	−12.2493	−0.427504	−0.213752	+	0.976888i	$0.568568\pi$
$822$	0	0
$823$	48.5742i	1.69319i	0.532238	+	0.846595i	$0.321351\pi$
$823$	48.5742i	1.69319i	−0.532238	+	0.846595i	$0.678649\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 10.5704i	− 0.367568i	−0.982967	−	0.183784i	$-0.941165\pi$
$827$	− 10.5704i	− 0.367568i	0.982967	−	0.183784i	$-0.0588347\pi$
$828$	0	0
$829$	12.1865	0.423254	0.211627	−	0.977351i	$-0.432124\pi$
$829$	12.1865	0.423254	0.211627	+	0.977351i	$0.432124\pi$
$830$	0	0
$831$	4.39284	0.152386
$832$	0	0
$833$	19.0151i	0.658834i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	2.47356i	0.0854988i
$838$	0	0
$839$	25.4457	0.878484	0.439242	−	0.898369i	$-0.355247\pi$
$839$	25.4457	0.878484	0.439242	+	0.898369i	$0.355247\pi$
$840$	0	0
$841$	−12.4207	−0.428299
$842$	0	0
$843$	− 12.9382i	− 0.445614i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2.97216i	0.102125i
$848$	0	0
$849$	−19.7757	−0.678702
$850$	0	0
$851$	9.66998	0.331483
$852$	0	0
$853$	− 17.2720i	− 0.591382i	−0.955284	−	0.295691i	$-0.904450\pi$
$853$	− 17.2720i	− 0.591382i	0.955284	−	0.295691i	$-0.0955498\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	41.7328i	1.42557i	0.701385	+	0.712783i	$0.252565\pi$
$857$	41.7328i	1.42557i	−0.701385	+	0.712783i	$0.747435\pi$
$858$	0	0
$859$	−34.6171	−1.18112	−0.590560	−	0.806994i	$-0.701093\pi$
$859$	−34.6171	−1.18112	−0.590560	+	0.806994i	$0.701093\pi$
$860$	0	0
$861$	7.66104	0.261087
$862$	0	0
$863$	− 38.4608i	− 1.30922i	−0.755966	−	0.654611i	$-0.772833\pi$
$863$	− 38.4608i	− 1.30922i	0.755966	−	0.654611i	$-0.227167\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 0.420681i	− 0.0142871i
$868$	0	0
$869$	10.6044	0.359728
$870$	0	0
$871$	13.8475	0.469205
$872$	0	0
$873$	13.0907i	0.443052i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 27.3022i	− 0.921929i	−0.887419	−	0.460965i	$-0.847504\pi$
$877$	− 27.3022i	− 0.921929i	0.887419	−	0.460965i	$-0.152496\pi$
$878$	0	0
$879$	11.2682	0.380067
$880$	0	0
$881$	−4.79463	−0.161535	−0.0807676	−	0.996733i	$-0.525737\pi$
$881$	−4.79463	−0.161535	−0.0807676	+	0.996733i	$0.525737\pi$
$882$	0	0
$883$	52.6710i	1.77252i	0.463188	+	0.886260i	$0.346705\pi$
$883$	52.6710i	1.77252i	−0.463188	+	0.886260i	$0.653295\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	48.9292i	1.64288i	0.570293	+	0.821441i	$0.306830\pi$
$887$	48.9292i	1.64288i	−0.570293	+	0.821441i	$0.693170\pi$
$888$	0	0
$889$	27.0240	0.906357
$890$	0	0
$891$	−3.59821	−0.120545
$892$	0	0
$893$	− 7.26440i	− 0.243094i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	5.59821i	0.186919i
$898$	0	0
$899$	10.0718	0.335912
$900$	0	0
$901$	20.4358	0.680814
$902$	0	0
$903$	− 10.7658i	− 0.358263i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 5.73796i	− 0.190526i	−0.995452	−	0.0952629i	$-0.969631\pi$
$907$	− 5.73796i	− 0.190526i	0.995452	−	0.0952629i	$-0.0303692\pi$
$908$	0	0
$909$	12.2153	0.405156
$910$	0	0
$911$	−42.6799	−1.41405	−0.707025	−	0.707189i	$-0.749963\pi$
$911$	−42.6799	−1.41405	−0.707025	+	0.707189i	$0.749963\pi$
$912$	0	0
$913$	32.8324i	1.08659i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 10.6044i	− 0.350187i
$918$	0	0
$919$	−25.5515	−0.842866	−0.421433	−	0.906860i	$-0.638473\pi$
$919$	−25.5515	−0.842866	−0.421433	+	0.906860i	$0.638473\pi$
$920$	0	0
$921$	−25.8475	−0.851704
$922$	0	0
$923$	28.0968i	0.924818i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 0.143542i	− 0.00471454i
$928$	0	0
$929$	53.9481	1.76998	0.884990	−	0.465610i	$-0.154165\pi$
$929$	53.9481	1.76998	0.884990	+	0.465610i	$0.154165\pi$
$930$	0	0
$931$	7.46362	0.244610
$932$	0	0
$933$	29.0529i	0.951149i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	7.48351i	0.244475i	0.992501	+	0.122238i	$0.0390070\pi$
$937$	7.48351i	0.244475i	−0.992501	+	0.122238i	$0.960993\pi$
$938$	0	0
$939$	21.9571	0.716542
$940$	0	0
$941$	22.6133	0.737173	0.368586	−	0.929594i	$-0.379842\pi$
$941$	22.6133	0.737173	0.368586	+	0.929594i	$0.379842\pi$
$942$	0	0
$943$	− 5.01889i	− 0.163438i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 7.19642i	− 0.233852i	−0.993141	−	0.116926i	$-0.962696\pi$
$947$	− 7.19642i	− 0.233852i	0.993141	−	0.116926i	$-0.0373041\pi$
$948$	0	0
$949$	49.2342	1.59821
$950$	0	0
$951$	2.65109	0.0859675
$952$	0	0
$953$	60.6799i	1.96562i	0.184631	+	0.982808i	$0.440891\pi$
$953$	60.6799i	1.96562i	−0.184631	+	0.982808i	$0.559109\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	14.6511i	0.473602i
$958$	0	0
$959$	18.6978	0.603784
$960$	0	0
$961$	−24.8815	−0.802629
$962$	0	0
$963$	− 5.26819i	− 0.169765i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	4.40179i	0.141552i	0.997492	+	0.0707760i	$0.0225476\pi$
$967$	4.40179i	0.141552i	−0.997492	+	0.0707760i	$0.977452\pi$
$968$	0	0
$969$	6.50755	0.209053
$970$	0	0
$971$	1.01510	0.0325760	0.0162880	−	0.999867i	$-0.494815\pi$
$971$	1.01510	0.0325760	0.0162880	+	0.999867i	$0.494815\pi$
$972$	0	0
$973$	10.1006i	0.323811i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 18.4646i	− 0.590735i	−0.955384	−	0.295368i	$-0.904558\pi$
$977$	− 18.4646i	− 0.590735i	0.955384	−	0.295368i	$-0.0954421\pi$
$978$	0	0
$979$	44.0756	1.40866
$980$	0	0
$981$	12.7946	0.408501
$982$	0	0
$983$	− 44.0917i	− 1.40631i	−0.711039	−	0.703153i	$-0.751775\pi$
$983$	− 44.0917i	− 1.40631i	0.711039	−	0.703153i	$-0.248225\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 6.93817i	− 0.220845i
$988$	0	0
$989$	−7.05288	−0.224269
$990$	0	0
$991$	8.22426	0.261252	0.130626	−	0.991432i	$-0.458301\pi$
$991$	8.22426	0.261252	0.130626	+	0.991432i	$0.458301\pi$
$992$	0	0
$993$	− 7.77574i	− 0.246756i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	14.4306i	0.457023i	0.973541	+	0.228511i	$0.0733858\pi$
$997$	14.4306i	0.457023i	−0.973541	+	0.228511i	$0.926614\pi$
$998$	0	0
$999$	−9.66998	−0.305945

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6900.2.f.r.6349.2		6
5.2	odd	4		6900.2.a.x.1.2		3
5.3	odd	4		1380.2.a.j.1.2	✓	3
5.4	even	2	inner	6900.2.f.r.6349.5		6
15.8	even	4		4140.2.a.s.1.2		3
20.3	even	4		5520.2.a.bv.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.2.a.j.1.2	✓	3	5.3	odd	4
4140.2.a.s.1.2		3	15.8	even	4
5520.2.a.bv.1.2		3	20.3	even	4
6900.2.a.x.1.2		3	5.2	odd	4
6900.2.f.r.6349.2		6	1.1	even	1	trivial
6900.2.f.r.6349.5		6	5.4	even	2	inner

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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +1.52644i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +1.52644i q^{7} -1.00000 q^{9} -3.59821 q^{11} -5.59821i q^{13} +4.07177i q^{17} +1.59821 q^{19} +1.52644 q^{21} -1.00000i q^{23} +1.00000i q^{27} +4.07177 q^{29} +2.47356 q^{31} +3.59821i q^{33} +9.66998i q^{37} -5.59821 q^{39} +5.01889 q^{41} -7.05288i q^{43} -4.54533i q^{47} +4.66998 q^{49} +4.07177 q^{51} -5.01889i q^{53} -1.59821i q^{57} -4.07177 q^{59} +6.54533 q^{61} -1.52644i q^{63} +2.47356i q^{67} -1.00000 q^{69} -5.01889 q^{71} +8.79463i q^{73} -5.49245i q^{77} -2.94712 q^{79} +1.00000 q^{81} -9.12465i q^{83} -4.07177i q^{87} -12.2493 q^{89} +8.54533 q^{91} -2.47356i q^{93} -13.0907i q^{97} +3.59821 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{9}+O(q^{10}) \) 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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Database

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