LMFDB - Embedded newform 2900.2.c.f.349.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2900,2,Mod(349,2900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2900, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2900.349");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2900 = 2^{2} \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2900.c (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:	$23.1566165862$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 580)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			349.2
Root			$-1.33641 + 1.33641i$ of defining polynomial
Character	$\chi$	$=$	2900.349
Dual form			2900.2.c.f.349.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.67282i q^{3} +4.67282i q^{7} -4.14399 q^{9} -0.672824 q^{11} -1.14399i q^{13} -3.52884i q^{17} -5.52884 q^{19} +12.4896 q^{21} -3.81681i q^{23} +3.05767i q^{27} +1.00000 q^{29} -1.52884 q^{31} +1.79834i q^{33} -7.16246i q^{37} -3.05767 q^{39} +2.85601 q^{41} +8.96080i q^{43} +6.67282i q^{47} -14.8353 q^{49} -9.43196 q^{51} +10.4896i q^{53} +14.7776i q^{57} -10.7776 q^{59} -14.4896 q^{61} -19.3641i q^{63} +7.81681i q^{67} -10.2017 q^{69} +4.48963 q^{71} -4.96080i q^{73} -3.14399i q^{77} -2.38485 q^{79} -4.25934 q^{81} +14.0185i q^{83} -2.67282i q^{87} +1.63362 q^{89} +5.34565 q^{91} +4.08631i q^{93} +9.32718i q^{97} +2.78817 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 22 q^{9} + 16 q^{11} - 16 q^{19} + 32 q^{21} + 6 q^{29} + 8 q^{31} + 16 q^{39} + 20 q^{41} - 6 q^{49} - 48 q^{51} - 16 q^{59} - 44 q^{61} - 24 q^{69} - 16 q^{71} + 46 q^{81} - 36 q^{89} - 8 q^{91}+ \cdots - 72 q^{99}+O(q^{100}) $ 6 * q - 22 * q^9 + 16 * q^11 - 16 * q^19 + 32 * q^21 + 6 * q^29 + 8 * q^31 + 16 * q^39 + 20 * q^41 - 6 * q^49 - 48 * q^51 - 16 * q^59 - 44 * q^61 - 24 * q^69 - 16 * q^71 + 46 * q^81 - 36 * q^89 - 8 * q^91 - 72 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2900\mathbb{Z}\right)^\times$.

$n$	$901$	$1277$	$1451$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	− 2.67282i	− 1.54316i	−0.636135	−	0.771578i	$-0.719468\pi$
$3$	− 2.67282i	− 1.54316i	0.636135	−	0.771578i	$-0.280532\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.67282i	1.76616i	0.469221	+	0.883081i	$0.344535\pi$
$7$	4.67282i	1.76616i	−0.469221	+	0.883081i	$0.655465\pi$
$8$	0	0
$9$	−4.14399	−1.38133
$10$	0	0
$11$	−0.672824	−0.202864	−0.101432	−	0.994842i	$-0.532342\pi$
$11$	−0.672824	−0.202864	−0.101432	+	0.994842i	$0.532342\pi$
$12$	0	0
$13$	− 1.14399i	− 0.317285i	−0.987336	−	0.158642i	$-0.949288\pi$
$13$	− 1.14399i	− 0.317285i	0.987336	−	0.158642i	$-0.0507117\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 3.52884i	− 0.855869i	−0.903810	−	0.427934i	$-0.859241\pi$
$17$	− 3.52884i	− 0.855869i	0.903810	−	0.427934i	$-0.140759\pi$
$18$	0	0
$19$	−5.52884	−1.26840	−0.634201	−	0.773168i	$-0.718671\pi$
$19$	−5.52884	−1.26840	−0.634201	+	0.773168i	$0.718671\pi$
$20$	0	0
$21$	12.4896	2.72546
$22$	0	0
$23$	− 3.81681i	− 0.795860i	−0.917416	−	0.397930i	$-0.869729\pi$
$23$	− 3.81681i	− 0.795860i	0.917416	−	0.397930i	$-0.130271\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	3.05767i	0.588450i
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−1.52884	−0.274587	−0.137294	−	0.990530i	$-0.543840\pi$
$31$	−1.52884	−0.274587	−0.137294	+	0.990530i	$0.543840\pi$
$32$	0	0
$33$	1.79834i	0.313051i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 7.16246i	− 1.17750i	−0.808315	−	0.588750i	$-0.799620\pi$
$37$	− 7.16246i	− 1.17750i	0.808315	−	0.588750i	$-0.200380\pi$
$38$	0	0
$39$	−3.05767	−0.489620
$40$	0	0
$41$	2.85601	0.446034	0.223017	−	0.974815i	$-0.428409\pi$
$41$	2.85601	0.446034	0.223017	+	0.974815i	$0.428409\pi$
$42$	0	0
$43$	8.96080i	1.36651i	0.730180	+	0.683254i	$0.239436\pi$
$43$	8.96080i	1.36651i	−0.730180	+	0.683254i	$0.760564\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.67282i	0.973331i	0.873588	+	0.486666i	$0.161787\pi$
$47$	6.67282i	0.973331i	−0.873588	+	0.486666i	$0.838213\pi$
$48$	0	0
$49$	−14.8353	−2.11933
$50$	0	0
$51$	−9.43196	−1.32074
$52$	0	0
$53$	10.4896i	1.44086i	0.693527	+	0.720431i	$0.256056\pi$
$53$	10.4896i	1.44086i	−0.693527	+	0.720431i	$0.743944\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	14.7776i	1.95734i
$58$	0	0
$59$	−10.7776	−1.40312	−0.701562	−	0.712608i	$-0.747514\pi$
$59$	−10.7776	−1.40312	−0.701562	+	0.712608i	$0.747514\pi$
$60$	0	0
$61$	−14.4896	−1.85521	−0.927604	−	0.373566i	$-0.878135\pi$
$61$	−14.4896	−1.85521	−0.927604	+	0.373566i	$0.878135\pi$
$62$	0	0
$63$	− 19.3641i	− 2.43965i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	7.81681i	0.954975i	0.878638	+	0.477488i	$0.158452\pi$
$67$	7.81681i	0.954975i	−0.878638	+	0.477488i	$0.841548\pi$
$68$	0	0
$69$	−10.2017	−1.22814
$70$	0	0
$71$	4.48963	0.532822	0.266411	−	0.963860i	$-0.414162\pi$
$71$	4.48963	0.532822	0.266411	+	0.963860i	$0.414162\pi$
$72$	0	0
$73$	− 4.96080i	− 0.580617i	−0.956933	−	0.290309i	$-0.906242\pi$
$73$	− 4.96080i	− 0.580617i	0.956933	−	0.290309i	$-0.0937580\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 3.14399i	− 0.358291i
$78$	0	0
$79$	−2.38485	−0.268317	−0.134158	−	0.990960i	$-0.542833\pi$
$79$	−2.38485	−0.268317	−0.134158	+	0.990960i	$0.542833\pi$
$80$	0	0
$81$	−4.25934	−0.473259
$82$	0	0
$83$	14.0185i	1.53873i	0.638811	+	0.769364i	$0.279427\pi$
$83$	14.0185i	1.53873i	−0.638811	+	0.769364i	$0.720573\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 2.67282i	− 0.286557i
$88$	0	0
$89$	1.63362	0.173163	0.0865817	−	0.996245i	$-0.472406\pi$
$89$	1.63362	0.173163	0.0865817	+	0.996245i	$0.472406\pi$
$90$	0	0
$91$	5.34565	0.560376
$92$	0	0
$93$	4.08631i	0.423731i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	9.32718i	0.947031i	0.880785	+	0.473516i	$0.157015\pi$
$97$	9.32718i	0.947031i	−0.880785	+	0.473516i	$0.842985\pi$
$98$	0	0
$99$	2.78817	0.280222
$100$	0	0
$101$	−16.3249	−1.62439	−0.812195	−	0.583386i	$-0.801727\pi$
$101$	−16.3249	−1.62439	−0.812195	+	0.583386i	$0.801727\pi$
$102$	0	0
$103$	− 13.5288i	− 1.33304i	−0.745489	−	0.666518i	$-0.767784\pi$
$103$	− 13.5288i	− 1.33304i	0.745489	−	0.666518i	$-0.232216\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 0.672824i	− 0.0650443i	−0.999471	−	0.0325222i	$-0.989646\pi$
$107$	− 0.672824i	− 0.0650443i	0.999471	−	0.0325222i	$-0.0103540\pi$
$108$	0	0
$109$	−13.5473	−1.29760	−0.648798	−	0.760960i	$-0.724728\pi$
$109$	−13.5473	−1.29760	−0.648798	+	0.760960i	$0.724728\pi$
$110$	0	0
$111$	−19.1440	−1.81707
$112$	0	0
$113$	20.0185i	1.88318i	0.336762	+	0.941590i	$0.390668\pi$
$113$	20.0185i	1.88318i	−0.336762	+	0.941590i	$0.609332\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4.74066i	0.438275i
$118$	0	0
$119$	16.4896	1.51160
$120$	0	0
$121$	−10.5473	−0.958846
$122$	0	0
$123$	− 7.63362i	− 0.688300i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	3.03920i	0.269686i	0.990867	+	0.134843i	$0.0430530\pi$
$127$	3.03920i	0.269686i	−0.990867	+	0.134843i	$0.956947\pi$
$128$	0	0
$129$	23.9506	2.10874
$130$	0	0
$131$	−3.81681	−0.333476	−0.166738	−	0.986001i	$-0.553323\pi$
$131$	−3.81681	−0.333476	−0.166738	+	0.986001i	$0.553323\pi$
$132$	0	0
$133$	− 25.8353i	− 2.24020i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 16.1048i	− 1.37592i	−0.725746	−	0.687962i	$-0.758505\pi$
$137$	− 16.1048i	− 1.37592i	0.725746	−	0.687962i	$-0.241495\pi$
$138$	0	0
$139$	−3.05767	−0.259349	−0.129674	−	0.991557i	$-0.541393\pi$
$139$	−3.05767	−0.259349	−0.129674	+	0.991557i	$0.541393\pi$
$140$	0	0
$141$	17.8353	1.50200
$142$	0	0
$143$	0.769701i	0.0643657i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	39.6521i	3.27045i
$148$	0	0
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	0	0
$151$	6.77761	0.551554	0.275777	−	0.961222i	$-0.411065\pi$
$151$	6.77761	0.551554	0.275777	+	0.961222i	$0.411065\pi$
$152$	0	0
$153$	14.6235i	1.18224i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 8.96080i	− 0.715149i	−0.933885	−	0.357575i	$-0.883604\pi$
$157$	− 8.96080i	− 0.715149i	0.933885	−	0.357575i	$-0.116396\pi$
$158$	0	0
$159$	28.0369	2.22347
$160$	0	0
$161$	17.8353	1.40562
$162$	0	0
$163$	− 6.30644i	− 0.493959i	−0.969021	−	0.246979i	$-0.920562\pi$
$163$	− 6.30644i	− 0.493959i	0.969021	−	0.246979i	$-0.0794380\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 12.7961i	− 0.990190i	−0.868839	−	0.495095i	$-0.835133\pi$
$167$	− 12.7961i	− 0.990190i	0.868839	−	0.495095i	$-0.164867\pi$
$168$	0	0
$169$	11.6913	0.899330
$170$	0	0
$171$	22.9114	1.75208
$172$	0	0
$173$	18.1233i	1.37789i	0.724816	+	0.688943i	$0.241925\pi$
$173$	18.1233i	1.37789i	−0.724816	+	0.688943i	$0.758075\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	28.8066i	2.16524i
$178$	0	0
$179$	−6.28797	−0.469985	−0.234993	−	0.971997i	$-0.575507\pi$
$179$	−6.28797	−0.469985	−0.234993	+	0.971997i	$0.575507\pi$
$180$	0	0
$181$	4.56804	0.339540	0.169770	−	0.985484i	$-0.445698\pi$
$181$	4.56804	0.339540	0.169770	+	0.985484i	$0.445698\pi$
$182$	0	0
$183$	38.7282i	2.86287i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2.37429i	0.173625i
$188$	0	0
$189$	−14.2880	−1.03930
$190$	0	0
$191$	21.6521	1.56669	0.783345	−	0.621587i	$-0.213512\pi$
$191$	21.6521	1.56669	0.783345	+	0.621587i	$0.213512\pi$
$192$	0	0
$193$	− 13.2409i	− 0.953098i	−0.879148	−	0.476549i	$-0.841887\pi$
$193$	− 13.2409i	− 0.953098i	0.879148	−	0.476549i	$-0.158113\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 7.34565i	− 0.523356i	−0.965155	−	0.261678i	$-0.915724\pi$
$197$	− 7.34565i	− 0.523356i	0.965155	−	0.261678i	$-0.0842758\pi$
$198$	0	0
$199$	−18.7776	−1.33111	−0.665555	−	0.746349i	$-0.731805\pi$
$199$	−18.7776	−1.33111	−0.665555	+	0.746349i	$0.731805\pi$
$200$	0	0
$201$	20.8930	1.47368
$202$	0	0
$203$	4.67282i	0.327968i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	15.8168i	1.09934i
$208$	0	0
$209$	3.71993	0.257313
$210$	0	0
$211$	17.6521	1.21522	0.607610	−	0.794235i	$-0.292128\pi$
$211$	17.6521	1.21522	0.607610	+	0.794235i	$0.292128\pi$
$212$	0	0
$213$	− 12.0000i	− 0.822226i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 7.14399i	− 0.484965i
$218$	0	0
$219$	−13.2593	−0.895983
$220$	0	0
$221$	−4.03694	−0.271554
$222$	0	0
$223$	7.45043i	0.498918i	0.968385	+	0.249459i	$0.0802527\pi$
$223$	7.45043i	0.498918i	−0.968385	+	0.249459i	$0.919747\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	24.2201i	1.60755i	0.594936	+	0.803773i	$0.297178\pi$
$227$	24.2201i	1.60755i	−0.594936	+	0.803773i	$0.702822\pi$
$228$	0	0
$229$	−2.00000	−0.132164	−0.0660819	−	0.997814i	$-0.521050\pi$
$229$	−2.00000	−0.132164	−0.0660819	+	0.997814i	$0.521050\pi$
$230$	0	0
$231$	−8.40332	−0.552898
$232$	0	0
$233$	− 0.287973i	− 0.0188657i	−0.999956	−	0.00943287i	$-0.996997\pi$
$233$	− 0.287973i	− 0.0188657i	0.999956	−	0.00943287i	$-0.00300262\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	6.37429i	0.414054i
$238$	0	0
$239$	−7.51037	−0.485805	−0.242903	−	0.970051i	$-0.578100\pi$
$239$	−7.51037	−0.485805	−0.242903	+	0.970051i	$0.578100\pi$
$240$	0	0
$241$	−21.6706	−1.39592	−0.697962	−	0.716135i	$-0.745909\pi$
$241$	−21.6706	−1.39592	−0.697962	+	0.716135i	$0.745909\pi$
$242$	0	0
$243$	20.5575i	1.31876i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	6.32492i	0.402445i
$248$	0	0
$249$	37.4689	2.37450
$250$	0	0
$251$	−12.6728	−0.799902	−0.399951	−	0.916537i	$-0.630973\pi$
$251$	−12.6728	−0.799902	−0.399951	+	0.916537i	$0.630973\pi$
$252$	0	0
$253$	2.56804i	0.161451i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 16.9714i	− 1.05864i	−0.848421	−	0.529322i	$-0.822446\pi$
$257$	− 16.9714i	− 1.05864i	0.848421	−	0.529322i	$-0.177554\pi$
$258$	0	0
$259$	33.4689	2.07966
$260$	0	0
$261$	−4.14399	−0.256506
$262$	0	0
$263$	5.90312i	0.364002i	0.983298	+	0.182001i	$0.0582574\pi$
$263$	5.90312i	0.364002i	−0.983298	+	0.182001i	$0.941743\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 4.36638i	− 0.267218i
$268$	0	0
$269$	14.8560	0.905787	0.452894	−	0.891565i	$-0.350392\pi$
$269$	14.8560	0.905787	0.452894	+	0.891565i	$0.350392\pi$
$270$	0	0
$271$	22.5944	1.37251	0.686257	−	0.727360i	$-0.259253\pi$
$271$	22.5944	1.37251	0.686257	+	0.727360i	$0.259253\pi$
$272$	0	0
$273$	− 14.2880i	− 0.864747i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 27.4689i	− 1.65045i	−0.564807	−	0.825223i	$-0.691049\pi$
$277$	− 27.4689i	− 1.65045i	0.564807	−	0.825223i	$-0.308951\pi$
$278$	0	0
$279$	6.33548	0.379295
$280$	0	0
$281$	17.5473	1.04678	0.523392	−	0.852092i	$-0.324666\pi$
$281$	17.5473	1.04678	0.523392	+	0.852092i	$0.324666\pi$
$282$	0	0
$283$	26.5081i	1.57574i	0.615839	+	0.787872i	$0.288817\pi$
$283$	26.5081i	1.57574i	−0.615839	+	0.787872i	$0.711183\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	13.3456i	0.787769i
$288$	0	0
$289$	4.54731	0.267489
$290$	0	0
$291$	24.9299	1.46142
$292$	0	0
$293$	− 22.7098i	− 1.32672i	−0.748301	−	0.663359i	$-0.769130\pi$
$293$	− 22.7098i	− 1.32672i	0.748301	−	0.663359i	$-0.230870\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 2.05728i	− 0.119375i
$298$	0	0
$299$	−4.36638	−0.252514
$300$	0	0
$301$	−41.8722	−2.41347
$302$	0	0
$303$	43.6336i	2.50669i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 20.9608i	− 1.19630i	−0.801386	−	0.598148i	$-0.795904\pi$
$307$	− 20.9608i	− 1.19630i	0.801386	−	0.598148i	$-0.204096\pi$
$308$	0	0
$309$	−36.1602	−2.05708
$310$	0	0
$311$	−27.4874	−1.55867	−0.779333	−	0.626610i	$-0.784442\pi$
$311$	−27.4874	−1.55867	−0.779333	+	0.626610i	$0.784442\pi$
$312$	0	0
$313$	− 18.8560i	− 1.06580i	−0.846177	−	0.532902i	$-0.821101\pi$
$313$	− 18.8560i	− 1.06580i	0.846177	−	0.532902i	$-0.178899\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 29.4504i	− 1.65410i	−0.562128	−	0.827050i	$-0.690017\pi$
$317$	− 29.4504i	− 1.65410i	0.562128	−	0.827050i	$-0.309983\pi$
$318$	0	0
$319$	−0.672824	−0.0376709
$320$	0	0
$321$	−1.79834	−0.100374
$322$	0	0
$323$	19.5104i	1.08559i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	36.2096i	2.00239i
$328$	0	0
$329$	−31.1809	−1.71906
$330$	0	0
$331$	3.60724	0.198272	0.0991360	−	0.995074i	$-0.468392\pi$
$331$	3.60724	0.198272	0.0991360	+	0.995074i	$0.468392\pi$
$332$	0	0
$333$	29.6811i	1.62652i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	18.3064i	0.997216i	0.866828	+	0.498608i	$0.166155\pi$
$337$	18.3064i	0.997216i	−0.866828	+	0.498608i	$0.833845\pi$
$338$	0	0
$339$	53.5058	2.90604
$340$	0	0
$341$	1.02864	0.0557039
$342$	0	0
$343$	− 36.6129i	− 1.97691i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 6.38485i	− 0.342757i	−0.985205	−	0.171378i	$-0.945178\pi$
$347$	− 6.38485i	− 0.342757i	0.985205	−	0.171378i	$-0.0548221\pi$
$348$	0	0
$349$	−17.2672	−0.924294	−0.462147	−	0.886803i	$-0.652921\pi$
$349$	−17.2672	−0.924294	−0.462147	+	0.886803i	$0.652921\pi$
$350$	0	0
$351$	3.49794	0.186706
$352$	0	0
$353$	− 9.91369i	− 0.527652i	−0.964570	−	0.263826i	$-0.915015\pi$
$353$	− 9.91369i	− 0.527652i	0.964570	−	0.263826i	$-0.0849845\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 44.0739i	− 2.33264i
$358$	0	0
$359$	3.81681	0.201444	0.100722	−	0.994915i	$-0.467885\pi$
$359$	3.81681	0.201444	0.100722	+	0.994915i	$0.467885\pi$
$360$	0	0
$361$	11.5680	0.608844
$362$	0	0
$363$	28.1911i	1.47965i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	7.98153i	0.416632i	0.978062	+	0.208316i	$0.0667983\pi$
$367$	7.98153i	0.416632i	−0.978062	+	0.208316i	$0.933202\pi$
$368$	0	0
$369$	−11.8353	−0.616120
$370$	0	0
$371$	−49.0162	−2.54479
$372$	0	0
$373$	− 26.0369i	− 1.34814i	−0.738667	−	0.674071i	$-0.764544\pi$
$373$	− 26.0369i	− 1.34814i	0.738667	−	0.674071i	$-0.235456\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 1.14399i	− 0.0589183i
$378$	0	0
$379$	−28.8824	−1.48359	−0.741794	−	0.670627i	$-0.766025\pi$
$379$	−28.8824	−1.48359	−0.741794	+	0.670627i	$0.766025\pi$
$380$	0	0
$381$	8.12325	0.416167
$382$	0	0
$383$	0.0968776i	0.00495021i	0.999997	+	0.00247511i	$0.000787851\pi$
$383$	0.0968776i	0.00495021i	−0.999997	+	0.00247511i	$0.999212\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 37.1334i	− 1.88760i
$388$	0	0
$389$	0.740665	0.0375532	0.0187766	−	0.999824i	$-0.494023\pi$
$389$	0.740665	0.0375532	0.0187766	+	0.999824i	$0.494023\pi$
$390$	0	0
$391$	−13.4689	−0.681152
$392$	0	0
$393$	10.2017i	0.514606i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	23.9216i	1.20059i	0.799779	+	0.600295i	$0.204950\pi$
$397$	23.9216i	1.20059i	−0.799779	+	0.600295i	$0.795050\pi$
$398$	0	0
$399$	−69.0532	−3.45698
$400$	0	0
$401$	−5.54731	−0.277019	−0.138510	−	0.990361i	$-0.544231\pi$
$401$	−5.54731	−0.277019	−0.138510	+	0.990361i	$0.544231\pi$
$402$	0	0
$403$	1.74897i	0.0871224i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.81907i	0.238872i
$408$	0	0
$409$	34.8930	1.72535	0.862673	−	0.505762i	$-0.168789\pi$
$409$	34.8930	1.72535	0.862673	+	0.505762i	$0.168789\pi$
$410$	0	0
$411$	−43.0452	−2.12327
$412$	0	0
$413$	− 50.3619i	− 2.47814i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	8.17262i	0.400215i
$418$	0	0
$419$	−12.1233	−0.592260	−0.296130	−	0.955148i	$-0.595696\pi$
$419$	−12.1233	−0.592260	−0.296130	+	0.955148i	$0.595696\pi$
$420$	0	0
$421$	19.0656	0.929200	0.464600	−	0.885521i	$-0.346198\pi$
$421$	19.0656	0.929200	0.464600	+	0.885521i	$0.346198\pi$
$422$	0	0
$423$	− 27.6521i	− 1.34449i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 67.7075i	− 3.27660i
$428$	0	0
$429$	2.05728	0.0993262
$430$	0	0
$431$	2.69129	0.129635	0.0648176	−	0.997897i	$-0.479353\pi$
$431$	2.69129	0.129635	0.0648176	+	0.997897i	$0.479353\pi$
$432$	0	0
$433$	17.4874i	0.840390i	0.907434	+	0.420195i	$0.138038\pi$
$433$	17.4874i	0.840390i	−0.907434	+	0.420195i	$0.861962\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	21.1025i	1.00947i
$438$	0	0
$439$	4.45269	0.212515	0.106258	−	0.994339i	$-0.466113\pi$
$439$	4.45269	0.212515	0.106258	+	0.994339i	$0.466113\pi$
$440$	0	0
$441$	61.4772	2.92749
$442$	0	0
$443$	11.6151i	0.551852i	0.961179	+	0.275926i	$0.0889845\pi$
$443$	11.6151i	0.551852i	−0.961179	+	0.275926i	$0.911015\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	48.1108i	2.27556i
$448$	0	0
$449$	9.91369	0.467856	0.233928	−	0.972254i	$-0.424842\pi$
$449$	9.91369	0.467856	0.233928	+	0.972254i	$0.424842\pi$
$450$	0	0
$451$	−1.92159	−0.0904843
$452$	0	0
$453$	− 18.1153i	− 0.851133i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 32.6050i	− 1.52520i	−0.646872	−	0.762598i	$-0.723923\pi$
$457$	− 32.6050i	− 1.52520i	0.646872	−	0.762598i	$-0.276077\pi$
$458$	0	0
$459$	10.7900	0.503636
$460$	0	0
$461$	5.23030	0.243599	0.121800	−	0.992555i	$-0.461133\pi$
$461$	5.23030	0.243599	0.121800	+	0.992555i	$0.461133\pi$
$462$	0	0
$463$	− 35.0841i	− 1.63049i	−0.579113	−	0.815247i	$-0.696601\pi$
$463$	− 35.0841i	− 1.63049i	0.579113	−	0.815247i	$-0.303399\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 21.3641i	− 0.988614i	−0.869288	−	0.494307i	$-0.835422\pi$
$467$	− 21.3641i	− 0.988614i	0.869288	−	0.494307i	$-0.164578\pi$
$468$	0	0
$469$	−36.5266	−1.68664
$470$	0	0
$471$	−23.9506	−1.10359
$472$	0	0
$473$	− 6.02904i	− 0.277215i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 43.4689i	− 1.99030i
$478$	0	0
$479$	−5.03920	−0.230247	−0.115124	−	0.993351i	$-0.536726\pi$
$479$	−5.03920	−0.230247	−0.115124	+	0.993351i	$0.536726\pi$
$480$	0	0
$481$	−8.19376	−0.373603
$482$	0	0
$483$	− 47.6706i	− 2.16909i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 28.8824i	− 1.30879i	−0.756155	−	0.654393i	$-0.772924\pi$
$487$	− 28.8824i	− 1.30879i	0.756155	−	0.654393i	$-0.227076\pi$
$488$	0	0
$489$	−16.8560	−0.762255
$490$	0	0
$491$	5.16246	0.232978	0.116489	−	0.993192i	$-0.462836\pi$
$491$	5.16246	0.232978	0.116489	+	0.993192i	$0.462836\pi$
$492$	0	0
$493$	− 3.52884i	− 0.158931i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	20.9793i	0.941049i
$498$	0	0
$499$	27.0946	1.21292	0.606461	−	0.795113i	$-0.292589\pi$
$499$	27.0946	1.21292	0.606461	+	0.795113i	$0.292589\pi$
$500$	0	0
$501$	−34.2017	−1.52802
$502$	0	0
$503$	25.7305i	1.14727i	0.819112	+	0.573633i	$0.194466\pi$
$503$	25.7305i	1.14727i	−0.819112	+	0.573633i	$0.805534\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 31.2488i	− 1.38781i
$508$	0	0
$509$	−11.8353	−0.524590	−0.262295	−	0.964988i	$-0.584479\pi$
$509$	−11.8353	−0.524590	−0.262295	+	0.964988i	$0.584479\pi$
$510$	0	0
$511$	23.1809	1.02546
$512$	0	0
$513$	− 16.9054i	− 0.746391i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 4.48963i	− 0.197454i
$518$	0	0
$519$	48.4403	2.12629
$520$	0	0
$521$	−0.164719	−0.00721646	−0.00360823	−	0.999993i	$-0.501149\pi$
$521$	−0.164719	−0.00721646	−0.00360823	+	0.999993i	$0.501149\pi$
$522$	0	0
$523$	− 11.6072i	− 0.507549i	−0.967263	−	0.253775i	$-0.918328\pi$
$523$	− 11.6072i	− 0.507549i	0.967263	−	0.253775i	$-0.0816722\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.39502i	0.235011i
$528$	0	0
$529$	8.43196	0.366607
$530$	0	0
$531$	44.6623	1.93818
$532$	0	0
$533$	− 3.26724i	− 0.141520i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	16.8066i	0.725260i
$538$	0	0
$539$	9.98153	0.429935
$540$	0	0
$541$	−35.5922	−1.53023	−0.765113	−	0.643896i	$-0.777317\pi$
$541$	−35.5922	−1.53023	−0.765113	+	0.643896i	$0.777317\pi$
$542$	0	0
$543$	− 12.2096i	− 0.523963i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	44.3803i	1.89757i	0.315929	+	0.948783i	$0.397684\pi$
$547$	44.3803i	1.89757i	−0.315929	+	0.948783i	$0.602316\pi$
$548$	0	0
$549$	60.0448	2.56265
$550$	0	0
$551$	−5.52884	−0.235536
$552$	0	0
$553$	− 11.1440i	− 0.473891i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	26.0739i	1.10479i	0.833584	+	0.552393i	$0.186285\pi$
$557$	26.0739i	1.10479i	−0.833584	+	0.552393i	$0.813715\pi$
$558$	0	0
$559$	10.2510	0.433572
$560$	0	0
$561$	6.34605	0.267930
$562$	0	0
$563$	− 16.4218i	− 0.692096i	−0.938217	−	0.346048i	$-0.887523\pi$
$563$	− 16.4218i	− 0.692096i	0.938217	−	0.346048i	$-0.112477\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 19.9031i	− 0.835853i
$568$	0	0
$569$	7.79834	0.326923	0.163462	−	0.986550i	$-0.447734\pi$
$569$	7.79834	0.326923	0.163462	+	0.986550i	$0.447734\pi$
$570$	0	0
$571$	−33.3826	−1.39702	−0.698509	−	0.715601i	$-0.746153\pi$
$571$	−33.3826	−1.39702	−0.698509	+	0.715601i	$0.746153\pi$
$572$	0	0
$573$	− 57.8722i	− 2.41765i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	8.10478i	0.337407i	0.985667	+	0.168703i	$0.0539580\pi$
$577$	8.10478i	0.337407i	−0.985667	+	0.168703i	$0.946042\pi$
$578$	0	0
$579$	−35.3905	−1.47078
$580$	0	0
$581$	−65.5058	−2.71764
$582$	0	0
$583$	− 7.05767i	− 0.292299i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 28.5865i	− 1.17989i	−0.807443	−	0.589946i	$-0.799149\pi$
$587$	− 28.5865i	− 1.17989i	0.807443	−	0.589946i	$-0.200851\pi$
$588$	0	0
$589$	8.45269	0.348287
$590$	0	0
$591$	−19.6336	−0.807619
$592$	0	0
$593$	22.4896i	0.923539i	0.887000	+	0.461769i	$0.152785\pi$
$593$	22.4896i	0.923539i	−0.887000	+	0.461769i	$0.847215\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	50.1892i	2.05411i
$598$	0	0
$599$	31.7305	1.29647	0.648237	−	0.761439i	$-0.275507\pi$
$599$	31.7305	1.29647	0.648237	+	0.761439i	$0.275507\pi$
$600$	0	0
$601$	4.56804	0.186334	0.0931671	−	0.995650i	$-0.470301\pi$
$601$	4.56804	0.186334	0.0931671	+	0.995650i	$0.470301\pi$
$602$	0	0
$603$	− 32.3928i	− 1.31914i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	12.7882i	0.519056i	0.965736	+	0.259528i	$0.0835670\pi$
$607$	12.7882i	0.519056i	−0.965736	+	0.259528i	$0.916433\pi$
$608$	0	0
$609$	12.4896	0.506106
$610$	0	0
$611$	7.63362	0.308823
$612$	0	0
$613$	− 6.97927i	− 0.281890i	−0.990017	−	0.140945i	$-0.954986\pi$
$613$	− 6.97927i	− 0.281890i	0.990017	−	0.140945i	$-0.0450141\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 30.9193i	− 1.24477i	−0.782713	−	0.622383i	$-0.786165\pi$
$617$	− 30.9193i	− 1.24477i	0.782713	−	0.622383i	$-0.213835\pi$
$618$	0	0
$619$	20.7098	0.832396	0.416198	−	0.909274i	$-0.363362\pi$
$619$	20.7098	0.832396	0.416198	+	0.909274i	$0.363362\pi$
$620$	0	0
$621$	11.6706	0.468324
$622$	0	0
$623$	7.63362i	0.305835i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 9.94272i	− 0.397074i
$628$	0	0
$629$	−25.2751	−1.00779
$630$	0	0
$631$	9.67508	0.385159	0.192580	−	0.981281i	$-0.438315\pi$
$631$	9.67508	0.385159	0.192580	+	0.981281i	$0.438315\pi$
$632$	0	0
$633$	− 47.1809i	− 1.87527i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	16.9714i	0.672430i
$638$	0	0
$639$	−18.6050	−0.736002
$640$	0	0
$641$	23.5058	0.928425	0.464213	−	0.885724i	$-0.346337\pi$
$641$	23.5058	0.928425	0.464213	+	0.885724i	$0.346337\pi$
$642$	0	0
$643$	40.6235i	1.60203i	0.598643	+	0.801016i	$0.295707\pi$
$643$	40.6235i	1.60203i	−0.598643	+	0.801016i	$0.704293\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	10.2280i	0.402106i	0.979580	+	0.201053i	$0.0644363\pi$
$647$	10.2280i	0.402106i	−0.979580	+	0.201053i	$0.935564\pi$
$648$	0	0
$649$	7.25143	0.284644
$650$	0	0
$651$	−19.0946	−0.748377
$652$	0	0
$653$	− 28.2280i	− 1.10465i	−0.833629	−	0.552324i	$-0.813741\pi$
$653$	− 28.2280i	− 1.10465i	0.833629	−	0.552324i	$-0.186259\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	20.5575i	0.802023i
$658$	0	0
$659$	15.0471	0.586152	0.293076	−	0.956089i	$-0.405321\pi$
$659$	15.0471	0.586152	0.293076	+	0.956089i	$0.405321\pi$
$660$	0	0
$661$	13.8767	0.539743	0.269871	−	0.962896i	$-0.413019\pi$
$661$	13.8767	0.539743	0.269871	+	0.962896i	$0.413019\pi$
$662$	0	0
$663$	10.7900i	0.419050i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 3.81681i	− 0.147787i
$668$	0	0
$669$	19.9137	0.769908
$670$	0	0
$671$	9.74897	0.376355
$672$	0	0
$673$	15.3456i	0.591531i	0.955261	+	0.295766i	$0.0955747\pi$
$673$	15.3456i	0.591531i	−0.955261	+	0.295766i	$0.904425\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	50.5530i	1.94291i	0.237228	+	0.971454i	$0.423761\pi$
$677$	50.5530i	1.94291i	−0.237228	+	0.971454i	$0.576239\pi$
$678$	0	0
$679$	−43.5843	−1.67261
$680$	0	0
$681$	64.7361	2.48069
$682$	0	0
$683$	41.6027i	1.59188i	0.605373	+	0.795942i	$0.293024\pi$
$683$	41.6027i	1.59188i	−0.605373	+	0.795942i	$0.706976\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	5.34565i	0.203949i
$688$	0	0
$689$	12.0000	0.457164
$690$	0	0
$691$	−41.6627	−1.58492	−0.792461	−	0.609922i	$-0.791201\pi$
$691$	−41.6627	−1.58492	−0.792461	+	0.609922i	$0.791201\pi$
$692$	0	0
$693$	13.0286i	0.494917i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 10.0784i	− 0.381747i
$698$	0	0
$699$	−0.769701	−0.0291128
$700$	0	0
$701$	−44.8145	−1.69262	−0.846311	−	0.532689i	$-0.821182\pi$
$701$	−44.8145	−1.69262	−0.846311	+	0.532689i	$0.821182\pi$
$702$	0	0
$703$	39.6001i	1.49354i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 76.2835i	− 2.86893i
$708$	0	0
$709$	−23.4320	−0.880006	−0.440003	−	0.897996i	$-0.645023\pi$
$709$	−23.4320	−0.880006	−0.440003	+	0.897996i	$0.645023\pi$
$710$	0	0
$711$	9.88279	0.370634
$712$	0	0
$713$	5.83528i	0.218533i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	20.0739i	0.749673i
$718$	0	0
$719$	−32.3328	−1.20581	−0.602905	−	0.797813i	$-0.705990\pi$
$719$	−32.3328	−1.20581	−0.602905	+	0.797813i	$0.705990\pi$
$720$	0	0
$721$	63.2179	2.35436
$722$	0	0
$723$	57.9216i	2.15413i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 4.01847i	− 0.149037i	−0.997220	−	0.0745184i	$-0.976258\pi$
$727$	− 4.01847i	− 0.149037i	0.997220	−	0.0745184i	$-0.0237420\pi$
$728$	0	0
$729$	42.1685	1.56180
$730$	0	0
$731$	31.6212	1.16955
$732$	0	0
$733$	− 3.49189i	− 0.128976i	−0.997918	−	0.0644880i	$-0.979459\pi$
$733$	− 3.49189i	− 0.128976i	0.997918	−	0.0644880i	$-0.0205414\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 5.25934i	− 0.193730i
$738$	0	0
$739$	45.4090	1.67040	0.835198	−	0.549949i	$-0.185353\pi$
$739$	45.4090	1.67040	0.835198	+	0.549949i	$0.185353\pi$
$740$	0	0
$741$	16.9054	0.621035
$742$	0	0
$743$	40.6683i	1.49198i	0.665960	+	0.745988i	$0.268022\pi$
$743$	40.6683i	1.49198i	−0.665960	+	0.745988i	$0.731978\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 58.0924i	− 2.12549i
$748$	0	0
$749$	3.14399	0.114879
$750$	0	0
$751$	49.0268	1.78901	0.894506	−	0.447055i	$-0.147527\pi$
$751$	49.0268	1.78901	0.894506	+	0.447055i	$0.147527\pi$
$752$	0	0
$753$	33.8722i	1.23437i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	51.7260i	1.88001i	0.341157	+	0.940006i	$0.389181\pi$
$757$	51.7260i	1.88001i	−0.341157	+	0.940006i	$0.610819\pi$
$758$	0	0
$759$	6.86392	0.249144
$760$	0	0
$761$	34.2307	1.24086	0.620431	−	0.784261i	$-0.286958\pi$
$761$	34.2307	1.24086	0.620431	+	0.784261i	$0.286958\pi$
$762$	0	0
$763$	− 63.3042i	− 2.29177i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	12.3294i	0.445190i
$768$	0	0
$769$	−22.5971	−0.814871	−0.407436	−	0.913234i	$-0.633577\pi$
$769$	−22.5971	−0.814871	−0.407436	+	0.913234i	$0.633577\pi$
$770$	0	0
$771$	−45.3615	−1.63365
$772$	0	0
$773$	13.8538i	0.498285i	0.968467	+	0.249142i	$0.0801487\pi$
$773$	13.8538i	0.498285i	−0.968467	+	0.249142i	$0.919851\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 89.4565i	− 3.20923i
$778$	0	0
$779$	−15.7904	−0.565751
$780$	0	0
$781$	−3.02073	−0.108090
$782$	0	0
$783$	3.05767i	0.109272i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 20.0969i	− 0.716376i	−0.933649	−	0.358188i	$-0.883395\pi$
$787$	− 20.0969i	− 0.716376i	0.933649	−	0.358188i	$-0.116605\pi$
$788$	0	0
$789$	15.7780	0.561712
$790$	0	0
$791$	−93.5428	−3.32600
$792$	0	0
$793$	16.5759i	0.588629i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 43.6890i	− 1.54754i	−0.633464	−	0.773772i	$-0.718367\pi$
$797$	− 43.6890i	− 1.54754i	0.633464	−	0.773772i	$-0.281633\pi$
$798$	0	0
$799$	23.5473	0.833044
$800$	0	0
$801$	−6.76970	−0.239196
$802$	0	0
$803$	3.33774i	0.117786i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 39.7075i	− 1.39777i
$808$	0	0
$809$	18.4403	0.648325	0.324163	−	0.946001i	$-0.394918\pi$
$809$	18.4403	0.648325	0.324163	+	0.946001i	$0.394918\pi$
$810$	0	0
$811$	15.0577	0.528746	0.264373	−	0.964420i	$-0.414835\pi$
$811$	15.0577	0.528746	0.264373	+	0.964420i	$0.414835\pi$
$812$	0	0
$813$	− 60.3909i	− 2.11800i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 49.5428i	− 1.73328i
$818$	0	0
$819$	−22.1523	−0.774064
$820$	0	0
$821$	50.9793	1.77919	0.889594	−	0.456751i	$-0.150987\pi$
$821$	50.9793	1.77919	0.889594	+	0.456751i	$0.150987\pi$
$822$	0	0
$823$	− 2.84545i	− 0.0991861i	−0.998770	−	0.0495930i	$-0.984208\pi$
$823$	− 2.84545i	− 0.0991861i	0.998770	−	0.0495930i	$-0.0157924\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	15.0761i	0.524249i	0.965034	+	0.262124i	$0.0844231\pi$
$827$	15.0761i	0.524249i	−0.965034	+	0.262124i	$0.915577\pi$
$828$	0	0
$829$	34.6992	1.20515	0.602577	−	0.798061i	$-0.294141\pi$
$829$	34.6992	1.20515	0.602577	+	0.798061i	$0.294141\pi$
$830$	0	0
$831$	−73.4195	−2.54690
$832$	0	0
$833$	52.3513i	1.81386i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 4.67469i	− 0.161581i
$838$	0	0
$839$	14.7882	0.510544	0.255272	−	0.966869i	$-0.417835\pi$
$839$	14.7882	0.510544	0.255272	+	0.966869i	$0.417835\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	− 46.9009i	− 1.61535i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 49.2857i	− 1.69348i
$848$	0	0
$849$	70.8515	2.43162
$850$	0	0
$851$	−27.3377	−0.937126
$852$	0	0
$853$	35.7753i	1.22492i	0.790500	+	0.612462i	$0.209821\pi$
$853$	35.7753i	1.22492i	−0.790500	+	0.612462i	$0.790179\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	12.4112i	0.423959i	0.977274	+	0.211980i	$0.0679911\pi$
$857$	12.4112i	0.423959i	−0.977274	+	0.211980i	$0.932009\pi$
$858$	0	0
$859$	0.0599353	0.00204497	0.00102248	−	0.999999i	$-0.499675\pi$
$859$	0.0599353	0.00204497	0.00102248	+	0.999999i	$0.499675\pi$
$860$	0	0
$861$	35.6706	1.21565
$862$	0	0
$863$	− 44.5530i	− 1.51660i	−0.651906	−	0.758300i	$-0.726030\pi$
$863$	− 44.5530i	− 1.51660i	0.651906	−	0.758300i	$-0.273970\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 12.1542i	− 0.412777i
$868$	0	0
$869$	1.60458	0.0544318
$870$	0	0
$871$	8.94233	0.302999
$872$	0	0
$873$	− 38.6517i	− 1.30816i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 29.9137i	− 1.01011i	−0.863086	−	0.505057i	$-0.831472\pi$
$877$	− 29.9137i	− 1.01011i	0.863086	−	0.505057i	$-0.168528\pi$
$878$	0	0
$879$	−60.6992	−2.04733
$880$	0	0
$881$	−32.2386	−1.08615	−0.543073	−	0.839685i	$-0.682739\pi$
$881$	−32.2386	−1.08615	−0.543073	+	0.839685i	$0.682739\pi$
$882$	0	0
$883$	36.7098i	1.23538i	0.786421	+	0.617691i	$0.211932\pi$
$883$	36.7098i	1.23538i	−0.786421	+	0.617691i	$0.788068\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.63588i	0.0885042i	0.999020	+	0.0442521i	$0.0140905\pi$
$887$	2.63588i	0.0885042i	−0.999020	+	0.0442521i	$0.985910\pi$
$888$	0	0
$889$	−14.2017	−0.476308
$890$	0	0
$891$	2.86578	0.0960073
$892$	0	0
$893$	− 36.8930i	− 1.23458i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	11.6706i	0.389669i
$898$	0	0
$899$	−1.52884	−0.0509896
$900$	0	0
$901$	37.0162	1.23319
$902$	0	0
$903$	111.917i	3.72437i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 16.0185i	− 0.531885i	−0.963989	−	0.265942i	$-0.914317\pi$
$907$	− 16.0185i	− 0.531885i	0.963989	−	0.265942i	$-0.0856831\pi$
$908$	0	0
$909$	67.6502	2.24382
$910$	0	0
$911$	37.9480	1.25727	0.628636	−	0.777700i	$-0.283613\pi$
$911$	37.9480	1.25727	0.628636	+	0.777700i	$0.283613\pi$
$912$	0	0
$913$	− 9.43196i	− 0.312152i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 17.8353i	− 0.588973i
$918$	0	0
$919$	7.58425	0.250181	0.125091	−	0.992145i	$-0.460078\pi$
$919$	7.58425	0.250181	0.125091	+	0.992145i	$0.460078\pi$
$920$	0	0
$921$	−56.0245	−1.84607
$922$	0	0
$923$	− 5.13608i	− 0.169056i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	56.0633i	1.84136i
$928$	0	0
$929$	−26.4033	−0.866265	−0.433132	−	0.901330i	$-0.642592\pi$
$929$	−26.4033	−0.866265	−0.433132	+	0.901330i	$0.642592\pi$
$930$	0	0
$931$	82.0219	2.68816
$932$	0	0
$933$	73.4689i	2.40526i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 44.6050i	− 1.45718i	−0.684949	−	0.728591i	$-0.740176\pi$
$937$	− 44.6050i	− 1.45718i	0.684949	−	0.728591i	$-0.259824\pi$
$938$	0	0
$939$	−50.3988	−1.64470
$940$	0	0
$941$	−9.26724	−0.302103	−0.151052	−	0.988526i	$-0.548266\pi$
$941$	−9.26724	−0.302103	−0.151052	+	0.988526i	$0.548266\pi$
$942$	0	0
$943$	− 10.9009i	− 0.354981i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	28.4218i	0.923584i	0.886988	+	0.461792i	$0.152793\pi$
$947$	28.4218i	0.923584i	−0.886988	+	0.461792i	$0.847207\pi$
$948$	0	0
$949$	−5.67508	−0.184221
$950$	0	0
$951$	−78.7158	−2.55254
$952$	0	0
$953$	55.2100i	1.78843i	0.447642	+	0.894213i	$0.352264\pi$
$953$	55.2100i	1.78843i	−0.447642	+	0.894213i	$0.647736\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	1.79834i	0.0581320i
$958$	0	0
$959$	75.2548	2.43010
$960$	0	0
$961$	−28.6627	−0.924602
$962$	0	0
$963$	2.78817i	0.0898476i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	34.1496i	1.09818i	0.835764	+	0.549089i	$0.185025\pi$
$967$	34.1496i	1.09818i	−0.835764	+	0.549089i	$0.814975\pi$
$968$	0	0
$969$	52.1478	1.67523
$970$	0	0
$971$	−41.4090	−1.32888	−0.664438	−	0.747343i	$-0.731329\pi$
$971$	−41.4090	−1.32888	−0.664438	+	0.747343i	$0.731329\pi$
$972$	0	0
$973$	− 14.2880i	− 0.458051i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	13.9506i	0.446320i	0.974782	+	0.223160i	$0.0716373\pi$
$977$	13.9506i	0.446320i	−0.974782	+	0.223160i	$0.928363\pi$
$978$	0	0
$979$	−1.09914	−0.0351286
$980$	0	0
$981$	56.1399	1.79241
$982$	0	0
$983$	− 39.6521i	− 1.26471i	−0.774681	−	0.632353i	$-0.782089\pi$
$983$	− 39.6521i	− 1.26471i	0.774681	−	0.632353i	$-0.217911\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	83.3411i	2.65278i
$988$	0	0
$989$	34.2017	1.08755
$990$	0	0
$991$	2.93442	0.0932149	0.0466075	−	0.998913i	$-0.485159\pi$
$991$	2.93442	0.0932149	0.0466075	+	0.998913i	$0.485159\pi$
$992$	0	0
$993$	− 9.64153i	− 0.305965i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 12.9977i	− 0.411643i	−0.978590	−	0.205821i	$-0.934013\pi$
$997$	− 12.9977i	− 0.411643i	0.978590	−	0.205821i	$-0.0659866\pi$
$998$	0	0
$999$	21.9005	0.692900

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2900.2.c.f.349.2		6
5.2	odd	4		580.2.a.c.1.1	✓	3
5.3	odd	4		2900.2.a.g.1.3		3
5.4	even	2	inner	2900.2.c.f.349.5		6
15.2	even	4		5220.2.a.x.1.1		3
20.7	even	4		2320.2.a.m.1.3		3
40.27	even	4		9280.2.a.bw.1.1		3
40.37	odd	4		9280.2.a.bk.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
580.2.a.c.1.1	✓	3	5.2	odd	4
2320.2.a.m.1.3		3	20.7	even	4
2900.2.a.g.1.3		3	5.3	odd	4
2900.2.c.f.349.2		6	1.1	even	1	trivial
2900.2.c.f.349.5		6	5.4	even	2	inner
5220.2.a.x.1.1		3	15.2	even	4
9280.2.a.bk.1.3		3	40.37	odd	4
9280.2.a.bw.1.1		3	40.27	even	4

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:	\( N \)	\(=\)	\( 2900 = 2^{2} \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2900.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.67282i q^{3} +4.67282i q^{7} -4.14399 q^{9} -0.672824 q^{11} -1.14399i q^{13} -3.52884i q^{17} -5.52884 q^{19} +12.4896 q^{21} -3.81681i q^{23} +3.05767i q^{27} +1.00000 q^{29} -1.52884 q^{31} +1.79834i q^{33} -7.16246i q^{37} -3.05767 q^{39} +2.85601 q^{41} +8.96080i q^{43} +6.67282i q^{47} -14.8353 q^{49} -9.43196 q^{51} +10.4896i q^{53} +14.7776i q^{57} -10.7776 q^{59} -14.4896 q^{61} -19.3641i q^{63} +7.81681i q^{67} -10.2017 q^{69} +4.48963 q^{71} -4.96080i q^{73} -3.14399i q^{77} -2.38485 q^{79} -4.25934 q^{81} +14.0185i q^{83} -2.67282i q^{87} +1.63362 q^{89} +5.34565 q^{91} +4.08631i q^{93} +9.32718i q^{97} +2.78817 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 22 q^{9} + 16 q^{11} - 16 q^{19} + 32 q^{21} + 6 q^{29} + 8 q^{31} + 16 q^{39} + 20 q^{41} - 6 q^{49} - 48 q^{51} - 16 q^{59} - 44 q^{61} - 24 q^{69} - 16 q^{71} + 46 q^{81} - 36 q^{89} - 8 q^{91}+ \cdots - 72 q^{99}+O(q^{100}) \) 6 * q - 22 * q^9 + 16 * q^11 - 16 * q^19 + 32 * q^21 + 6 * q^29 + 8 * q^31 + 16 * q^39 + 20 * q^41 - 6 * q^49 - 48 * q^51 - 16 * q^59 - 44 * q^61 - 24 * q^69 - 16 * q^71 + 46 * q^81 - 36 * q^89 - 8 * q^91 - 72 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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