LMFDB - Embedded newform 9300.2.g.t.3349.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9300,2,Mod(3349,9300)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9300.3349"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9300, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9300.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,0,0,-12,0,-6,0,0,0,0,0,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$74.2608738798$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} + 8 x^{10} + 6 x^{9} + 44 x^{8} - 164 x^{7} + 322 x^{6} + 216 x^{5} + 304 x^{4} + \cdots + 576 $ x^12 - 4x^11 + 8x^10 + 6x^9 + 44x^8 - 164x^7 + 322x^6 + 216x^5 + 304x^4 - 976x^3 + 2048x^2 + 1536*x + 576
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3349.10
Root			$-0.324975 + 0.324975i$ of defining polynomial
Character	$\chi$	$=$	9300.3349
Dual form			9300.2.g.t.3349.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +0.649950i q^{7} -1.00000 q^{9} +5.35424 q^{11} +0.343092i q^{13} -5.13187i q^{17} +1.53116 q^{19} -0.649950 q^{21} +6.81578i q^{23} -1.00000i q^{27} -1.68424 q^{29} -1.00000 q^{31} +5.35424i q^{33} +4.23545i q^{37} -0.343092 q^{39} +4.30686 q^{41} -4.66109i q^{43} +3.25762i q^{47} +6.57757 q^{49} +5.13187 q^{51} +5.24274i q^{53} +1.53116i q^{57} -13.9245 q^{59} +4.53845 q^{61} -0.649950i q^{63} -0.342756i q^{67} -6.81578 q^{69} +3.50382 q^{71} +2.00000i q^{73} +3.47999i q^{77} -1.64961 q^{79} +1.00000 q^{81} -8.10177i q^{83} -1.68424i q^{87} +8.61186 q^{89} -0.222992 q^{91} -1.00000i q^{93} +7.92762i q^{97} -5.35424 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{9} - 6 q^{11} + 6 q^{19} + 8 q^{21} - 12 q^{31} - 4 q^{39} + 36 q^{41} - 28 q^{49} + 8 q^{51} + 24 q^{59} + 56 q^{61} - 10 q^{69} + 50 q^{71} - 6 q^{79} + 12 q^{81} - 8 q^{89} + 32 q^{91} + 6 q^{99}+O(q^{100}) $ 12 * q - 12 * q^9 - 6 * q^11 + 6 * q^19 + 8 * q^21 - 12 * q^31 - 4 * q^39 + 36 * q^41 - 28 * q^49 + 8 * q^51 + 24 * q^59 + 56 * q^61 - 10 * q^69 + 50 * q^71 - 6 * q^79 + 12 * q^81 - 8 * q^89 + 32 * q^91 + 6 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/9300\mathbb{Z}\right)^\times$.

$n$	$1801$	$2977$	$3101$	$4651$
$\chi(n)$	$1$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.649950i	0.245658i	0.992428	+	0.122829i	$0.0391967\pi$
$7$	0.649950i	0.245658i	−0.992428	+	0.122829i	$0.960803\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	5.35424	1.61436	0.807182	−	0.590303i	$-0.200992\pi$
$11$	5.35424	1.61436	0.807182	+	0.590303i	$0.200992\pi$
$12$	0	0
$13$	0.343092i	0.0951565i	0.998868	+	0.0475783i	$0.0151503\pi$
$13$	0.343092i	0.0951565i	−0.998868	+	0.0475783i	$0.984850\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 5.13187i	− 1.24466i	−0.782754	−	0.622331i	$-0.786186\pi$
$17$	− 5.13187i	− 1.24466i	0.782754	−	0.622331i	$-0.213814\pi$
$18$	0	0
$19$	1.53116	0.351272	0.175636	−	0.984455i	$-0.443802\pi$
$19$	1.53116	0.351272	0.175636	+	0.984455i	$0.443802\pi$
$20$	0	0
$21$	−0.649950	−0.141831
$22$	0	0
$23$	6.81578i	1.42119i	0.703602	+	0.710594i	$0.251574\pi$
$23$	6.81578i	1.42119i	−0.703602	+	0.710594i	$0.748426\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	−1.68424	−0.312756	−0.156378	−	0.987697i	$-0.549982\pi$
$29$	−1.68424	−0.312756	−0.156378	+	0.987697i	$0.549982\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	5.35424i	0.932053i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.23545i	0.696303i	0.937438	+	0.348152i	$0.113191\pi$
$37$	4.23545i	0.696303i	−0.937438	+	0.348152i	$0.886809\pi$
$38$	0	0
$39$	−0.343092	−0.0549386
$40$	0	0
$41$	4.30686	0.672618	0.336309	−	0.941752i	$-0.390821\pi$
$41$	4.30686	0.672618	0.336309	+	0.941752i	$0.390821\pi$
$42$	0	0
$43$	− 4.66109i	− 0.710810i	−0.934712	−	0.355405i	$-0.884343\pi$
$43$	− 4.66109i	− 0.710810i	0.934712	−	0.355405i	$-0.115657\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.25762i	0.475173i	0.971366	+	0.237587i	$0.0763563\pi$
$47$	3.25762i	0.475173i	−0.971366	+	0.237587i	$0.923644\pi$
$48$	0	0
$49$	6.57757	0.939652
$50$	0	0
$51$	5.13187	0.718606
$52$	0	0
$53$	5.24274i	0.720146i	0.932924	+	0.360073i	$0.117248\pi$
$53$	5.24274i	0.720146i	−0.932924	+	0.360073i	$0.882752\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.53116i	0.202807i
$58$	0	0
$59$	−13.9245	−1.81282	−0.906408	−	0.422403i	$-0.861187\pi$
$59$	−13.9245	−1.81282	−0.906408	+	0.422403i	$0.861187\pi$
$60$	0	0
$61$	4.53845	0.581089	0.290545	−	0.956861i	$-0.406163\pi$
$61$	4.53845	0.581089	0.290545	+	0.956861i	$0.406163\pi$
$62$	0	0
$63$	− 0.649950i	− 0.0818860i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 0.342756i	− 0.0418743i	−0.999781	−	0.0209371i	$-0.993335\pi$
$67$	− 0.342756i	− 0.0418743i	0.999781	−	0.0209371i	$-0.00666498\pi$
$68$	0	0
$69$	−6.81578	−0.820524
$70$	0	0
$71$	3.50382	0.415827	0.207914	−	0.978147i	$-0.433333\pi$
$71$	3.50382	0.415827	0.207914	+	0.978147i	$0.433333\pi$
$72$	0	0
$73$	2.00000i	0.234082i	0.993127	+	0.117041i	$0.0373409\pi$
$73$	2.00000i	0.234082i	−0.993127	+	0.117041i	$0.962659\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.47999i	0.396581i
$78$	0	0
$79$	−1.64961	−0.185596	−0.0927980	−	0.995685i	$-0.529581\pi$
$79$	−1.64961	−0.185596	−0.0927980	+	0.995685i	$0.529581\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 8.10177i	− 0.889284i	−0.895708	−	0.444642i	$-0.853331\pi$
$83$	− 8.10177i	− 0.889284i	0.895708	−	0.444642i	$-0.146669\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 1.68424i	− 0.180570i
$88$	0	0
$89$	8.61186	0.912855	0.456428	−	0.889761i	$-0.349129\pi$
$89$	8.61186	0.912855	0.456428	+	0.889761i	$0.349129\pi$
$90$	0	0
$91$	−0.222992	−0.0233760
$92$	0	0
$93$	− 1.00000i	− 0.103695i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	7.92762i	0.804927i	0.915436	+	0.402464i	$0.131846\pi$
$97$	7.92762i	0.804927i	−0.915436	+	0.402464i	$0.868154\pi$
$98$	0	0
$99$	−5.35424	−0.538121
$100$	0	0
$101$	18.8775	1.87838	0.939189	−	0.343401i	$-0.111579\pi$
$101$	18.8775	1.87838	0.939189	+	0.343401i	$0.111579\pi$
$102$	0	0
$103$	0.952622i	0.0938646i	0.998898	+	0.0469323i	$0.0149445\pi$
$103$	0.952622i	0.0938646i	−0.998898	+	0.0469323i	$0.985055\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0.658511i	0.0636607i	0.999493	+	0.0318303i	$0.0101336\pi$
$107$	0.658511i	0.0636607i	−0.999493	+	0.0318303i	$0.989866\pi$
$108$	0	0
$109$	11.5887	1.11000	0.554999	−	0.831851i	$-0.312719\pi$
$109$	11.5887	1.11000	0.554999	+	0.831851i	$0.312719\pi$
$110$	0	0
$111$	−4.23545	−0.402011
$112$	0	0
$113$	− 6.81612i	− 0.641207i	−0.947214	−	0.320603i	$-0.896114\pi$
$113$	− 6.81612i	− 0.641207i	0.947214	−	0.320603i	$-0.103886\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 0.343092i	− 0.0317188i
$118$	0	0
$119$	3.33546	0.305761
$120$	0	0
$121$	17.6679	1.60617
$122$	0	0
$123$	4.30686i	0.388336i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 0.545076i	− 0.0483677i	−0.999708	−	0.0241838i	$-0.992301\pi$
$127$	− 0.545076i	− 0.0483677i	0.999708	−	0.0241838i	$-0.00769871\pi$
$128$	0	0
$129$	4.66109	0.410387
$130$	0	0
$131$	−13.3353	−1.16511	−0.582554	−	0.812792i	$-0.697947\pi$
$131$	−13.3353	−1.16511	−0.582554	+	0.812792i	$0.697947\pi$
$132$	0	0
$133$	0.995177i	0.0862928i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 2.74656i	− 0.234655i	−0.993093	−	0.117327i	$-0.962567\pi$
$137$	− 2.74656i	− 0.234655i	0.993093	−	0.117327i	$-0.0374327\pi$
$138$	0	0
$139$	6.85104	0.581098	0.290549	−	0.956860i	$-0.406162\pi$
$139$	6.85104	0.581098	0.290549	+	0.956860i	$0.406162\pi$
$140$	0	0
$141$	−3.25762	−0.274341
$142$	0	0
$143$	1.83699i	0.153617i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	6.57757i	0.542508i
$148$	0	0
$149$	−8.79360	−0.720400	−0.360200	−	0.932875i	$-0.617292\pi$
$149$	−8.79360	−0.720400	−0.360200	+	0.932875i	$0.617292\pi$
$150$	0	0
$151$	5.91357	0.481239	0.240620	−	0.970620i	$-0.422649\pi$
$151$	5.91357	0.481239	0.240620	+	0.970620i	$0.422649\pi$
$152$	0	0
$153$	5.13187i	0.414887i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	9.57854i	0.764451i	0.924069	+	0.382225i	$0.124842\pi$
$157$	9.57854i	0.764451i	−0.924069	+	0.382225i	$0.875158\pi$
$158$	0	0
$159$	−5.24274	−0.415776
$160$	0	0
$161$	−4.42992	−0.349126
$162$	0	0
$163$	2.02886i	0.158912i	0.996838	+	0.0794562i	$0.0253184\pi$
$163$	2.02886i	0.158912i	−0.996838	+	0.0794562i	$0.974682\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 4.35293i	− 0.336840i	−0.985715	−	0.168420i	$-0.946134\pi$
$167$	− 4.35293i	− 0.336840i	0.985715	−	0.168420i	$-0.0538664\pi$
$168$	0	0
$169$	12.8823	0.990945
$170$	0	0
$171$	−1.53116	−0.117091
$172$	0	0
$173$	− 4.06066i	− 0.308726i	−0.988014	−	0.154363i	$-0.950667\pi$
$173$	− 4.06066i	− 0.308726i	0.988014	−	0.154363i	$-0.0493325\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 13.9245i	− 1.04663i
$178$	0	0
$179$	11.3762	0.850298	0.425149	−	0.905123i	$-0.360222\pi$
$179$	11.3762	0.850298	0.425149	+	0.905123i	$0.360222\pi$
$180$	0	0
$181$	13.2132	0.982130	0.491065	−	0.871123i	$-0.336608\pi$
$181$	13.2132	0.982130	0.491065	+	0.871123i	$0.336608\pi$
$182$	0	0
$183$	4.53845i	0.335492i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 27.4773i	− 2.00934i
$188$	0	0
$189$	0.649950	0.0472769
$190$	0	0
$191$	−3.90544	−0.282588	−0.141294	−	0.989968i	$-0.545126\pi$
$191$	−3.90544	−0.282588	−0.141294	+	0.989968i	$0.545126\pi$
$192$	0	0
$193$	− 11.1272i	− 0.800951i	−0.916308	−	0.400475i	$-0.868845\pi$
$193$	− 11.1272i	− 0.800951i	0.916308	−	0.400475i	$-0.131155\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 13.2660i	− 0.945163i	−0.881287	−	0.472582i	$-0.843322\pi$
$197$	− 13.2660i	− 0.945163i	0.881287	−	0.472582i	$-0.156678\pi$
$198$	0	0
$199$	−11.6685	−0.827157	−0.413579	−	0.910468i	$-0.635721\pi$
$199$	−11.6685	−0.827157	−0.413579	+	0.910468i	$0.635721\pi$
$200$	0	0
$201$	0.342756	0.0241761
$202$	0	0
$203$	− 1.09467i	− 0.0768311i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 6.81578i	− 0.473730i
$208$	0	0
$209$	8.19819	0.567081
$210$	0	0
$211$	5.28429	0.363785	0.181893	−	0.983318i	$-0.441778\pi$
$211$	5.28429	0.363785	0.181893	+	0.983318i	$0.441778\pi$
$212$	0	0
$213$	3.50382i	0.240078i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 0.649950i	− 0.0441215i
$218$	0	0
$219$	−2.00000	−0.135147
$220$	0	0
$221$	1.76070	0.118438
$222$	0	0
$223$	17.5849i	1.17757i	0.808290	+	0.588785i	$0.200393\pi$
$223$	17.5849i	1.17757i	−0.808290	+	0.588785i	$0.799607\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	21.7285i	1.44217i	0.692845	+	0.721086i	$0.256357\pi$
$227$	21.7285i	1.44217i	−0.692845	+	0.721086i	$0.743643\pi$
$228$	0	0
$229$	−25.8445	−1.70785	−0.853926	−	0.520395i	$-0.825785\pi$
$229$	−25.8445	−1.70785	−0.853926	+	0.520395i	$0.825785\pi$
$230$	0	0
$231$	−3.47999	−0.228966
$232$	0	0
$233$	11.4699i	0.751419i	0.926737	+	0.375710i	$0.122601\pi$
$233$	11.4699i	0.751419i	−0.926737	+	0.375710i	$0.877399\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 1.64961i	− 0.107154i
$238$	0	0
$239$	−4.54264	−0.293839	−0.146919	−	0.989148i	$-0.546936\pi$
$239$	−4.54264	−0.293839	−0.146919	+	0.989148i	$0.546936\pi$
$240$	0	0
$241$	−13.2216	−0.851677	−0.425839	−	0.904799i	$-0.640021\pi$
$241$	−13.2216	−0.851677	−0.425839	+	0.904799i	$0.640021\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0.525328i	0.0334258i
$248$	0	0
$249$	8.10177	0.513429
$250$	0	0
$251$	21.5605	1.36089	0.680443	−	0.732801i	$-0.261787\pi$
$251$	21.5605	1.36089	0.680443	+	0.732801i	$0.261787\pi$
$252$	0	0
$253$	36.4933i	2.29432i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 11.8169i	− 0.737116i	−0.929605	−	0.368558i	$-0.879852\pi$
$257$	− 11.8169i	− 0.737116i	0.929605	−	0.368558i	$-0.120148\pi$
$258$	0	0
$259$	−2.75283	−0.171052
$260$	0	0
$261$	1.68424	0.104252
$262$	0	0
$263$	15.8413i	0.976817i	0.872615	+	0.488409i	$0.162422\pi$
$263$	15.8413i	0.976817i	−0.872615	+	0.488409i	$0.837578\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	8.61186i	0.527037i
$268$	0	0
$269$	19.1761	1.16919	0.584595	−	0.811325i	$-0.301253\pi$
$269$	19.1761	1.16919	0.584595	+	0.811325i	$0.301253\pi$
$270$	0	0
$271$	−0.842540	−0.0511807	−0.0255903	−	0.999673i	$-0.508147\pi$
$271$	−0.842540	−0.0511807	−0.0255903	+	0.999673i	$0.508147\pi$
$272$	0	0
$273$	− 0.222992i	− 0.0134961i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	31.9176i	1.91774i	0.283841	+	0.958871i	$0.408391\pi$
$277$	31.9176i	1.91774i	−0.283841	+	0.958871i	$0.591609\pi$
$278$	0	0
$279$	1.00000	0.0598684
$280$	0	0
$281$	15.5023	0.924790	0.462395	−	0.886674i	$-0.346990\pi$
$281$	15.5023	0.924790	0.462395	+	0.886674i	$0.346990\pi$
$282$	0	0
$283$	− 10.9026i	− 0.648093i	−0.946041	−	0.324046i	$-0.894957\pi$
$283$	− 10.9026i	− 0.648093i	0.946041	−	0.324046i	$-0.105043\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.79924i	0.165234i
$288$	0	0
$289$	−9.33613	−0.549184
$290$	0	0
$291$	−7.92762	−0.464725
$292$	0	0
$293$	− 7.74714i	− 0.452593i	−0.974058	−	0.226296i	$-0.927338\pi$
$293$	− 7.74714i	− 0.452593i	0.974058	−	0.226296i	$-0.0726618\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 5.35424i	− 0.310684i
$298$	0	0
$299$	−2.33844	−0.135235
$300$	0	0
$301$	3.02948	0.174616
$302$	0	0
$303$	18.8775i	1.08448i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 1.46995i	− 0.0838943i	−0.999120	−	0.0419471i	$-0.986644\pi$
$307$	− 1.46995i	− 0.0838943i	0.999120	−	0.0419471i	$-0.0133561\pi$
$308$	0	0
$309$	−0.952622	−0.0541927
$310$	0	0
$311$	19.8469	1.12541	0.562707	−	0.826657i	$-0.309760\pi$
$311$	19.8469	1.12541	0.562707	+	0.826657i	$0.309760\pi$
$312$	0	0
$313$	1.89449i	0.107083i	0.998566	+	0.0535414i	$0.0170509\pi$
$313$	1.89449i	0.107083i	−0.998566	+	0.0535414i	$0.982949\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	27.7231i	1.55709i	0.627591	+	0.778543i	$0.284041\pi$
$317$	27.7231i	1.55709i	−0.627591	+	0.778543i	$0.715959\pi$
$318$	0	0
$319$	−9.01784	−0.504902
$320$	0	0
$321$	−0.658511	−0.0367545
$322$	0	0
$323$	− 7.85772i	− 0.437215i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	11.5887i	0.640857i
$328$	0	0
$329$	−2.11729	−0.116730
$330$	0	0
$331$	5.93845	0.326407	0.163203	−	0.986592i	$-0.447817\pi$
$331$	5.93845	0.326407	0.163203	+	0.986592i	$0.447817\pi$
$332$	0	0
$333$	− 4.23545i	− 0.232101i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	7.17313i	0.390745i	0.980729	+	0.195373i	$0.0625916\pi$
$337$	7.17313i	0.390745i	−0.980729	+	0.195373i	$0.937408\pi$
$338$	0	0
$339$	6.81612	0.370201
$340$	0	0
$341$	−5.35424	−0.289948
$342$	0	0
$343$	8.82474i	0.476491i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	18.3362i	0.984338i	0.870500	+	0.492169i	$0.163796\pi$
$347$	18.3362i	0.984338i	−0.870500	+	0.492169i	$0.836204\pi$
$348$	0	0
$349$	−15.7837	−0.844884	−0.422442	−	0.906390i	$-0.638827\pi$
$349$	−15.7837	−0.844884	−0.422442	+	0.906390i	$0.638827\pi$
$350$	0	0
$351$	0.343092	0.0183129
$352$	0	0
$353$	6.23635i	0.331928i	0.986132	+	0.165964i	$0.0530735\pi$
$353$	6.23635i	0.331928i	−0.986132	+	0.165964i	$0.946926\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	3.33546i	0.176531i
$358$	0	0
$359$	−7.03911	−0.371510	−0.185755	−	0.982596i	$-0.559473\pi$
$359$	−7.03911	−0.371510	−0.185755	+	0.982596i	$0.559473\pi$
$360$	0	0
$361$	−16.6556	−0.876608
$362$	0	0
$363$	17.6679i	0.927322i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	4.29037i	0.223956i	0.993711	+	0.111978i	$0.0357186\pi$
$367$	4.29037i	0.223956i	−0.993711	+	0.111978i	$0.964281\pi$
$368$	0	0
$369$	−4.30686	−0.224206
$370$	0	0
$371$	−3.40752	−0.176910
$372$	0	0
$373$	33.1822i	1.71811i	0.511882	+	0.859056i	$0.328948\pi$
$373$	33.1822i	1.71811i	−0.511882	+	0.859056i	$0.671052\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 0.577850i	− 0.0297608i
$378$	0	0
$379$	−0.194003	−0.00996524	−0.00498262	−	0.999988i	$-0.501586\pi$
$379$	−0.194003	−0.00996524	−0.00498262	+	0.999988i	$0.501586\pi$
$380$	0	0
$381$	0.545076	0.0279251
$382$	0	0
$383$	0.304031i	0.0155353i	0.999970	+	0.00776763i	$0.00247254\pi$
$383$	0.304031i	0.0155353i	−0.999970	+	0.00776763i	$0.997527\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4.66109i	0.236937i
$388$	0	0
$389$	−10.0759	−0.510870	−0.255435	−	0.966826i	$-0.582219\pi$
$389$	−10.0759	−0.510870	−0.255435	+	0.966826i	$0.582219\pi$
$390$	0	0
$391$	34.9777	1.76890
$392$	0	0
$393$	− 13.3353i	− 0.672676i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 6.12089i	− 0.307199i	−0.988133	−	0.153599i	$-0.950913\pi$
$397$	− 6.12089i	− 0.307199i	0.988133	−	0.153599i	$-0.0490865\pi$
$398$	0	0
$399$	−0.995177	−0.0498212
$400$	0	0
$401$	32.6625	1.63109	0.815544	−	0.578695i	$-0.196438\pi$
$401$	32.6625	1.63109	0.815544	+	0.578695i	$0.196438\pi$
$402$	0	0
$403$	− 0.343092i	− 0.0170906i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	22.6776i	1.12409i
$408$	0	0
$409$	−36.4199	−1.80085	−0.900423	−	0.435016i	$-0.856743\pi$
$409$	−36.4199	−1.80085	−0.900423	+	0.435016i	$0.856743\pi$
$410$	0	0
$411$	2.74656	0.135478
$412$	0	0
$413$	− 9.05023i	− 0.445333i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	6.85104i	0.335497i
$418$	0	0
$419$	3.50449	0.171206	0.0856029	−	0.996329i	$-0.472718\pi$
$419$	3.50449	0.171206	0.0856029	+	0.996329i	$0.472718\pi$
$420$	0	0
$421$	30.8953	1.50574	0.752872	−	0.658167i	$-0.228668\pi$
$421$	30.8953	1.50574	0.752872	+	0.658167i	$0.228668\pi$
$422$	0	0
$423$	− 3.25762i	− 0.158391i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	2.94977i	0.142749i
$428$	0	0
$429$	−1.83699	−0.0886909
$430$	0	0
$431$	7.44637	0.358679	0.179339	−	0.983787i	$-0.442604\pi$
$431$	7.44637	0.358679	0.179339	+	0.983787i	$0.442604\pi$
$432$	0	0
$433$	− 20.7559i	− 0.997465i	−0.866756	−	0.498733i	$-0.833799\pi$
$433$	− 20.7559i	− 0.997465i	0.866756	−	0.498733i	$-0.166201\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	10.4360i	0.499224i
$438$	0	0
$439$	−2.48163	−0.118442	−0.0592209	−	0.998245i	$-0.518862\pi$
$439$	−2.48163	−0.118442	−0.0592209	+	0.998245i	$0.518862\pi$
$440$	0	0
$441$	−6.57757	−0.313217
$442$	0	0
$443$	0.757538i	0.0359917i	0.999838	+	0.0179959i	$0.00572857\pi$
$443$	0.757538i	0.0359917i	−0.999838	+	0.0179959i	$0.994271\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 8.79360i	− 0.415923i
$448$	0	0
$449$	−21.0354	−0.992723	−0.496362	−	0.868116i	$-0.665331\pi$
$449$	−21.0354	−0.992723	−0.496362	+	0.868116i	$0.665331\pi$
$450$	0	0
$451$	23.0599	1.08585
$452$	0	0
$453$	5.91357i	0.277844i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 35.0835i	− 1.64114i	−0.571549	−	0.820568i	$-0.693657\pi$
$457$	− 35.0835i	− 1.64114i	0.571549	−	0.820568i	$-0.306343\pi$
$458$	0	0
$459$	−5.13187	−0.239535
$460$	0	0
$461$	41.0666	1.91266	0.956332	−	0.292282i	$-0.0944147\pi$
$461$	41.0666	1.91266	0.956332	+	0.292282i	$0.0944147\pi$
$462$	0	0
$463$	17.3619i	0.806874i	0.915007	+	0.403437i	$0.132185\pi$
$463$	17.3619i	0.806874i	−0.915007	+	0.403437i	$0.867815\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	17.6429i	0.816418i	0.912889	+	0.408209i	$0.133846\pi$
$467$	17.6429i	0.816418i	−0.912889	+	0.408209i	$0.866154\pi$
$468$	0	0
$469$	0.222774	0.0102867
$470$	0	0
$471$	−9.57854	−0.441356
$472$	0	0
$473$	− 24.9566i	− 1.14751i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 5.24274i	− 0.240049i
$478$	0	0
$479$	3.00292	0.137207	0.0686035	−	0.997644i	$-0.478146\pi$
$479$	3.00292	0.137207	0.0686035	+	0.997644i	$0.478146\pi$
$480$	0	0
$481$	−1.45315	−0.0662578
$482$	0	0
$483$	− 4.42992i	− 0.201568i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	15.3888i	0.697334i	0.937247	+	0.348667i	$0.113366\pi$
$487$	15.3888i	0.697334i	−0.937247	+	0.348667i	$0.886634\pi$
$488$	0	0
$489$	−2.02886	−0.0917481
$490$	0	0
$491$	16.5958	0.748956	0.374478	−	0.927236i	$-0.377822\pi$
$491$	16.5958	0.748956	0.374478	+	0.927236i	$0.377822\pi$
$492$	0	0
$493$	8.64333i	0.389276i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.27731i	0.102151i
$498$	0	0
$499$	12.2579	0.548740	0.274370	−	0.961624i	$-0.411531\pi$
$499$	12.2579	0.548740	0.274370	+	0.961624i	$0.411531\pi$
$500$	0	0
$501$	4.35293	0.194474
$502$	0	0
$503$	− 2.64670i	− 0.118011i	−0.998258	−	0.0590053i	$-0.981207\pi$
$503$	− 2.64670i	− 0.118011i	0.998258	−	0.0590053i	$-0.0187929\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	12.8823i	0.572122i
$508$	0	0
$509$	−21.6152	−0.958075	−0.479038	−	0.877794i	$-0.659014\pi$
$509$	−21.6152	−0.958075	−0.479038	+	0.877794i	$0.659014\pi$
$510$	0	0
$511$	−1.29990	−0.0575042
$512$	0	0
$513$	− 1.53116i	− 0.0676023i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	17.4421i	0.767102i
$518$	0	0
$519$	4.06066	0.178243
$520$	0	0
$521$	−6.59610	−0.288980	−0.144490	−	0.989506i	$-0.546154\pi$
$521$	−6.59610	−0.288980	−0.144490	+	0.989506i	$0.546154\pi$
$522$	0	0
$523$	42.9488i	1.87802i	0.343888	+	0.939011i	$0.388256\pi$
$523$	42.9488i	1.87802i	−0.343888	+	0.939011i	$0.611744\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.13187i	0.223548i
$528$	0	0
$529$	−23.4549	−1.01978
$530$	0	0
$531$	13.9245	0.604272
$532$	0	0
$533$	1.47765i	0.0640040i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	11.3762i	0.490920i
$538$	0	0
$539$	35.2178	1.51694
$540$	0	0
$541$	−14.0430	−0.603754	−0.301877	−	0.953347i	$-0.597613\pi$
$541$	−14.0430	−0.603754	−0.301877	+	0.953347i	$0.597613\pi$
$542$	0	0
$543$	13.2132i	0.567033i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 29.5256i	− 1.26242i	−0.775611	−	0.631212i	$-0.782558\pi$
$547$	− 29.5256i	− 1.26242i	0.775611	−	0.631212i	$-0.217442\pi$
$548$	0	0
$549$	−4.53845	−0.193696
$550$	0	0
$551$	−2.57885	−0.109863
$552$	0	0
$553$	− 1.07217i	− 0.0455932i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 19.1673i	− 0.812145i	−0.913841	−	0.406072i	$-0.866898\pi$
$557$	− 19.1673i	− 0.812145i	0.913841	−	0.406072i	$-0.133102\pi$
$558$	0	0
$559$	1.59918	0.0676382
$560$	0	0
$561$	27.4773	1.16009
$562$	0	0
$563$	− 11.2021i	− 0.472114i	−0.971739	−	0.236057i	$-0.924145\pi$
$563$	− 11.2021i	− 0.472114i	0.971739	−	0.236057i	$-0.0758551\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0.649950i	0.0272953i
$568$	0	0
$569$	6.30788	0.264440	0.132220	−	0.991220i	$-0.457789\pi$
$569$	6.30788	0.264440	0.132220	+	0.991220i	$0.457789\pi$
$570$	0	0
$571$	−32.4125	−1.35642	−0.678211	−	0.734868i	$-0.737244\pi$
$571$	−32.4125	−1.35642	−0.678211	+	0.734868i	$0.737244\pi$
$572$	0	0
$573$	− 3.90544i	− 0.163152i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	34.7319i	1.44591i	0.690897	+	0.722953i	$0.257216\pi$
$577$	34.7319i	1.44591i	−0.690897	+	0.722953i	$0.742784\pi$
$578$	0	0
$579$	11.1272	0.462429
$580$	0	0
$581$	5.26574	0.218460
$582$	0	0
$583$	28.0709i	1.16258i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	43.2265i	1.78415i	0.451890	+	0.892073i	$0.350750\pi$
$587$	43.2265i	1.78415i	−0.451890	+	0.892073i	$0.649250\pi$
$588$	0	0
$589$	−1.53116	−0.0630903
$590$	0	0
$591$	13.2660	0.545690
$592$	0	0
$593$	− 45.1582i	− 1.85442i	−0.374536	−	0.927212i	$-0.622198\pi$
$593$	− 45.1582i	− 1.85442i	0.374536	−	0.927212i	$-0.377802\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 11.6685i	− 0.477559i
$598$	0	0
$599$	8.13772	0.332498	0.166249	−	0.986084i	$-0.446834\pi$
$599$	8.13772	0.332498	0.166249	+	0.986084i	$0.446834\pi$
$600$	0	0
$601$	−2.08104	−0.0848875	−0.0424438	−	0.999099i	$-0.513514\pi$
$601$	−2.08104	−0.0848875	−0.0424438	+	0.999099i	$0.513514\pi$
$602$	0	0
$603$	0.342756i	0.0139581i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	3.68238i	0.149463i	0.997204	+	0.0747317i	$0.0238100\pi$
$607$	3.68238i	0.149463i	−0.997204	+	0.0747317i	$0.976190\pi$
$608$	0	0
$609$	1.09467	0.0443585
$610$	0	0
$611$	−1.11766	−0.0452158
$612$	0	0
$613$	− 37.6568i	− 1.52094i	−0.649370	−	0.760472i	$-0.724968\pi$
$613$	− 37.6568i	− 1.52094i	0.649370	−	0.760472i	$-0.275032\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	34.0450i	1.37060i	0.728260	+	0.685300i	$0.240329\pi$
$617$	34.0450i	1.37060i	−0.728260	+	0.685300i	$0.759671\pi$
$618$	0	0
$619$	2.97114	0.119420	0.0597102	−	0.998216i	$-0.480982\pi$
$619$	2.97114	0.119420	0.0597102	+	0.998216i	$0.480982\pi$
$620$	0	0
$621$	6.81578	0.273508
$622$	0	0
$623$	5.59728i	0.224250i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	8.19819i	0.327404i
$628$	0	0
$629$	21.7358	0.866662
$630$	0	0
$631$	5.40274	0.215080	0.107540	−	0.994201i	$-0.465703\pi$
$631$	5.40274	0.215080	0.107540	+	0.994201i	$0.465703\pi$
$632$	0	0
$633$	5.28429i	0.210032i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	2.25671i	0.0894140i
$638$	0	0
$639$	−3.50382	−0.138609
$640$	0	0
$641$	30.8883	1.22002	0.610008	−	0.792395i	$-0.291166\pi$
$641$	30.8883	1.22002	0.610008	+	0.792395i	$0.291166\pi$
$642$	0	0
$643$	36.5110i	1.43985i	0.694050	+	0.719926i	$0.255824\pi$
$643$	36.5110i	1.43985i	−0.694050	+	0.719926i	$0.744176\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 3.19555i	− 0.125630i	−0.998025	−	0.0628150i	$-0.979992\pi$
$647$	− 3.19555i	− 0.125630i	0.998025	−	0.0628150i	$-0.0200078\pi$
$648$	0	0
$649$	−74.5551	−2.92654
$650$	0	0
$651$	0.649950	0.0254735
$652$	0	0
$653$	− 0.885526i	− 0.0346533i	−0.999850	−	0.0173267i	$-0.994484\pi$
$653$	− 0.885526i	− 0.0346533i	0.999850	−	0.0173267i	$-0.00551552\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 2.00000i	− 0.0780274i
$658$	0	0
$659$	36.7476	1.43148	0.715742	−	0.698364i	$-0.246088\pi$
$659$	36.7476	1.43148	0.715742	+	0.698364i	$0.246088\pi$
$660$	0	0
$661$	−37.4321	−1.45594	−0.727969	−	0.685610i	$-0.759536\pi$
$661$	−37.4321	−1.45594	−0.727969	+	0.685610i	$0.759536\pi$
$662$	0	0
$663$	1.76070i	0.0683801i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 11.4794i	− 0.444486i
$668$	0	0
$669$	−17.5849	−0.679870
$670$	0	0
$671$	24.3000	0.938089
$672$	0	0
$673$	14.3238i	0.552140i	0.961137	+	0.276070i	$0.0890322\pi$
$673$	14.3238i	0.552140i	−0.961137	+	0.276070i	$0.910968\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	12.5132i	0.480921i	0.970659	+	0.240461i	$0.0772985\pi$
$677$	12.5132i	0.480921i	−0.970659	+	0.240461i	$0.922702\pi$
$678$	0	0
$679$	−5.15255	−0.197737
$680$	0	0
$681$	−21.7285	−0.832639
$682$	0	0
$683$	− 17.2965i	− 0.661834i	−0.943660	−	0.330917i	$-0.892642\pi$
$683$	− 17.2965i	− 0.661834i	0.943660	−	0.330917i	$-0.107358\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 25.8445i	− 0.986028i
$688$	0	0
$689$	−1.79874	−0.0685266
$690$	0	0
$691$	−45.5203	−1.73167	−0.865837	−	0.500326i	$-0.833214\pi$
$691$	−45.5203	−1.73167	−0.865837	+	0.500326i	$0.833214\pi$
$692$	0	0
$693$	− 3.47999i	− 0.132194i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 22.1023i	− 0.837183i
$698$	0	0
$699$	−11.4699	−0.433832
$700$	0	0
$701$	39.6022	1.49576	0.747878	−	0.663836i	$-0.231073\pi$
$701$	39.6022	1.49576	0.747878	+	0.663836i	$0.231073\pi$
$702$	0	0
$703$	6.48514i	0.244592i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	12.2694i	0.461439i
$708$	0	0
$709$	8.12306	0.305068	0.152534	−	0.988298i	$-0.451257\pi$
$709$	8.12306	0.305068	0.152534	+	0.988298i	$0.451257\pi$
$710$	0	0
$711$	1.64961	0.0618654
$712$	0	0
$713$	− 6.81578i	− 0.255253i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 4.54264i	− 0.169648i
$718$	0	0
$719$	−4.93397	−0.184006	−0.0920030	−	0.995759i	$-0.529327\pi$
$719$	−4.93397	−0.184006	−0.0920030	+	0.995759i	$0.529327\pi$
$720$	0	0
$721$	−0.619156	−0.0230586
$722$	0	0
$723$	− 13.2216i	− 0.491716i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	6.66944i	0.247356i	0.992322	+	0.123678i	$0.0394690\pi$
$727$	6.66944i	0.247356i	−0.992322	+	0.123678i	$0.960531\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−23.9202	−0.884719
$732$	0	0
$733$	− 20.2657i	− 0.748533i	−0.927321	−	0.374266i	$-0.877895\pi$
$733$	− 20.2657i	− 0.748533i	0.927321	−	0.374266i	$-0.122105\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 1.83519i	− 0.0676003i
$738$	0	0
$739$	18.9049	0.695429	0.347714	−	0.937601i	$-0.386958\pi$
$739$	18.9049	0.695429	0.347714	+	0.937601i	$0.386958\pi$
$740$	0	0
$741$	−0.525328	−0.0192984
$742$	0	0
$743$	− 49.8029i	− 1.82709i	−0.406734	−	0.913547i	$-0.633333\pi$
$743$	− 49.8029i	− 1.82709i	0.406734	−	0.913547i	$-0.366667\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	8.10177i	0.296428i
$748$	0	0
$749$	−0.427999	−0.0156387
$750$	0	0
$751$	−30.7190	−1.12095	−0.560476	−	0.828171i	$-0.689382\pi$
$751$	−30.7190	−1.12095	−0.560476	+	0.828171i	$0.689382\pi$
$752$	0	0
$753$	21.5605i	0.785708i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	29.9090i	1.08706i	0.839389	+	0.543531i	$0.182913\pi$
$757$	29.9090i	1.08706i	−0.839389	+	0.543531i	$0.817087\pi$
$758$	0	0
$759$	−36.4933	−1.32462
$760$	0	0
$761$	−13.2632	−0.480790	−0.240395	−	0.970675i	$-0.577277\pi$
$761$	−13.2632	−0.480790	−0.240395	+	0.970675i	$0.577277\pi$
$762$	0	0
$763$	7.53208i	0.272680i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 4.77738i	− 0.172501i
$768$	0	0
$769$	17.7608	0.640470	0.320235	−	0.947338i	$-0.396238\pi$
$769$	17.7608	0.640470	0.320235	+	0.947338i	$0.396238\pi$
$770$	0	0
$771$	11.8169	0.425574
$772$	0	0
$773$	29.6130i	1.06511i	0.846396	+	0.532554i	$0.178768\pi$
$773$	29.6130i	1.06511i	−0.846396	+	0.532554i	$0.821232\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 2.75283i	− 0.0987571i
$778$	0	0
$779$	6.59449	0.236272
$780$	0	0
$781$	18.7603	0.671296
$782$	0	0
$783$	1.68424i	0.0601900i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 1.41312i	− 0.0503724i	−0.999683	−	0.0251862i	$-0.991982\pi$
$787$	− 1.41312i	− 0.0503724i	0.999683	−	0.0251862i	$-0.00801786\pi$
$788$	0	0
$789$	−15.8413	−0.563966
$790$	0	0
$791$	4.43014	0.157518
$792$	0	0
$793$	1.55711i	0.0552944i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 23.2353i	− 0.823037i	−0.911401	−	0.411519i	$-0.864998\pi$
$797$	− 23.2353i	− 0.823037i	0.911401	−	0.411519i	$-0.135002\pi$
$798$	0	0
$799$	16.7177	0.591430
$800$	0	0
$801$	−8.61186	−0.304285
$802$	0	0
$803$	10.7085i	0.377894i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	19.1761i	0.675032i
$808$	0	0
$809$	28.7950	1.01238	0.506190	−	0.862422i	$-0.331054\pi$
$809$	28.7950	1.01238	0.506190	+	0.862422i	$0.331054\pi$
$810$	0	0
$811$	−39.2861	−1.37952	−0.689761	−	0.724037i	$-0.742284\pi$
$811$	−39.2861	−1.37952	−0.689761	+	0.724037i	$0.742284\pi$
$812$	0	0
$813$	− 0.842540i	− 0.0295492i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 7.13688i	− 0.249688i
$818$	0	0
$819$	0.222992	0.00779199
$820$	0	0
$821$	35.6506	1.24421	0.622107	−	0.782932i	$-0.286277\pi$
$821$	35.6506	1.24421	0.622107	+	0.782932i	$0.286277\pi$
$822$	0	0
$823$	12.7389i	0.444051i	0.975041	+	0.222025i	$0.0712667\pi$
$823$	12.7389i	0.444051i	−0.975041	+	0.222025i	$0.928733\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 47.7979i	− 1.66210i	−0.556201	−	0.831048i	$-0.687742\pi$
$827$	− 47.7979i	− 1.66210i	0.556201	−	0.831048i	$-0.312258\pi$
$828$	0	0
$829$	−23.0923	−0.802029	−0.401015	−	0.916072i	$-0.631342\pi$
$829$	−23.0923	−0.802029	−0.401015	+	0.916072i	$0.631342\pi$
$830$	0	0
$831$	−31.9176	−1.10721
$832$	0	0
$833$	− 33.7552i	− 1.16955i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.00000i	0.0345651i
$838$	0	0
$839$	−14.5270	−0.501527	−0.250763	−	0.968048i	$-0.580682\pi$
$839$	−14.5270	−0.501527	−0.250763	+	0.968048i	$0.580682\pi$
$840$	0	0
$841$	−26.1633	−0.902183
$842$	0	0
$843$	15.5023i	0.533928i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	11.4832i	0.394568i
$848$	0	0
$849$	10.9026	0.374177
$850$	0	0
$851$	−28.8679	−0.989578
$852$	0	0
$853$	− 45.1017i	− 1.54425i	−0.635470	−	0.772126i	$-0.719193\pi$
$853$	− 45.1017i	− 1.54425i	0.635470	−	0.772126i	$-0.280807\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 5.22869i	− 0.178609i	−0.996004	−	0.0893043i	$-0.971536\pi$
$857$	− 5.22869i	− 0.178609i	0.996004	−	0.0893043i	$-0.0284644\pi$
$858$	0	0
$859$	−17.4749	−0.596235	−0.298118	−	0.954529i	$-0.596359\pi$
$859$	−17.4749	−0.596235	−0.298118	+	0.954529i	$0.596359\pi$
$860$	0	0
$861$	−2.79924	−0.0953979
$862$	0	0
$863$	16.6956i	0.568324i	0.958776	+	0.284162i	$0.0917154\pi$
$863$	16.6956i	0.568324i	−0.958776	+	0.284162i	$0.908285\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 9.33613i	− 0.317072i
$868$	0	0
$869$	−8.83242	−0.299619
$870$	0	0
$871$	0.117597	0.00398461
$872$	0	0
$873$	− 7.92762i	− 0.268309i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 13.7028i	− 0.462711i	−0.972869	−	0.231356i	$-0.925684\pi$
$877$	− 13.7028i	− 0.462711i	0.972869	−	0.231356i	$-0.0743161\pi$
$878$	0	0
$879$	7.74714	0.261305
$880$	0	0
$881$	−8.95675	−0.301761	−0.150880	−	0.988552i	$-0.548211\pi$
$881$	−8.95675	−0.301761	−0.150880	+	0.988552i	$0.548211\pi$
$882$	0	0
$883$	40.8511i	1.37475i	0.726303	+	0.687375i	$0.241237\pi$
$883$	40.8511i	1.37475i	−0.726303	+	0.687375i	$0.758763\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	46.2905i	1.55428i	0.629326	+	0.777141i	$0.283331\pi$
$887$	46.2905i	1.55428i	−0.629326	+	0.777141i	$0.716669\pi$
$888$	0	0
$889$	0.354272	0.0118819
$890$	0	0
$891$	5.35424	0.179374
$892$	0	0
$893$	4.98794i	0.166915i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 2.33844i	− 0.0780782i
$898$	0	0
$899$	1.68424	0.0561727
$900$	0	0
$901$	26.9051	0.896338
$902$	0	0
$903$	3.02948i	0.100815i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	29.9467i	0.994365i	0.867646	+	0.497182i	$0.165632\pi$
$907$	29.9467i	0.994365i	−0.867646	+	0.497182i	$0.834368\pi$
$908$	0	0
$909$	−18.8775	−0.626126
$910$	0	0
$911$	−47.8940	−1.58680	−0.793399	−	0.608702i	$-0.791690\pi$
$911$	−47.8940	−1.58680	−0.793399	+	0.608702i	$0.791690\pi$
$912$	0	0
$913$	− 43.3788i	− 1.43563i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 8.66726i	− 0.286218i
$918$	0	0
$919$	47.0420	1.55177	0.775886	−	0.630873i	$-0.217303\pi$
$919$	47.0420	1.55177	0.775886	+	0.630873i	$0.217303\pi$
$920$	0	0
$921$	1.46995	0.0484364
$922$	0	0
$923$	1.20213i	0.0395687i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 0.952622i	− 0.0312882i
$928$	0	0
$929$	−2.38689	−0.0783114	−0.0391557	−	0.999233i	$-0.512467\pi$
$929$	−2.38689	−0.0783114	−0.0391557	+	0.999233i	$0.512467\pi$
$930$	0	0
$931$	10.0713	0.330074
$932$	0	0
$933$	19.8469i	0.649758i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 19.7649i	− 0.645691i	−0.946452	−	0.322846i	$-0.895361\pi$
$937$	− 19.7649i	− 0.645691i	0.946452	−	0.322846i	$-0.104639\pi$
$938$	0	0
$939$	−1.89449	−0.0618243
$940$	0	0
$941$	1.81125	0.0590450	0.0295225	−	0.999564i	$-0.490601\pi$
$941$	1.81125	0.0590450	0.0295225	+	0.999564i	$0.490601\pi$
$942$	0	0
$943$	29.3546i	0.955918i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	24.6290i	0.800335i	0.916442	+	0.400167i	$0.131048\pi$
$947$	24.6290i	0.800335i	−0.916442	+	0.400167i	$0.868952\pi$
$948$	0	0
$949$	−0.686183	−0.0222745
$950$	0	0
$951$	−27.7231	−0.898984
$952$	0	0
$953$	− 32.1879i	− 1.04267i	−0.853352	−	0.521335i	$-0.825434\pi$
$953$	− 32.1879i	− 1.04267i	0.853352	−	0.521335i	$-0.174566\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 9.01784i	− 0.291506i
$958$	0	0
$959$	1.78513	0.0576448
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	− 0.658511i	− 0.0212202i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 19.3094i	− 0.620948i	−0.950582	−	0.310474i	$-0.899512\pi$
$967$	− 19.3094i	− 0.620948i	0.950582	−	0.310474i	$-0.100488\pi$
$968$	0	0
$969$	7.85772	0.252426
$970$	0	0
$971$	44.4190	1.42547	0.712736	−	0.701432i	$-0.247456\pi$
$971$	44.4190	1.42547	0.712736	+	0.701432i	$0.247456\pi$
$972$	0	0
$973$	4.45284i	0.142751i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	15.4034i	0.492800i	0.969168	+	0.246400i	$0.0792476\pi$
$977$	15.4034i	0.492800i	−0.969168	+	0.246400i	$0.920752\pi$
$978$	0	0
$979$	46.1099	1.47368
$980$	0	0
$981$	−11.5887	−0.369999
$982$	0	0
$983$	25.9914i	0.828997i	0.910050	+	0.414498i	$0.136043\pi$
$983$	25.9914i	0.828997i	−0.910050	+	0.414498i	$0.863957\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 2.11729i	− 0.0673941i
$988$	0	0
$989$	31.7690	1.01020
$990$	0	0
$991$	−7.40420	−0.235202	−0.117601	−	0.993061i	$-0.537520\pi$
$991$	−7.40420	−0.235202	−0.117601	+	0.993061i	$0.537520\pi$
$992$	0	0
$993$	5.93845i	0.188451i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	2.57638i	0.0815949i	0.999167	+	0.0407974i	$0.0129898\pi$
$997$	2.57638i	0.0815949i	−0.999167	+	0.0407974i	$0.987010\pi$
$998$	0	0
$999$	4.23545	0.134004

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9300.2.g.t.3349.10		12
5.2	odd	4		9300.2.a.ba.1.3	yes	6
5.3	odd	4		9300.2.a.y.1.4	✓	6
5.4	even	2	inner	9300.2.g.t.3349.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9300.2.a.y.1.4	✓	6	5.3	odd	4
9300.2.a.ba.1.3	yes	6	5.2	odd	4
9300.2.g.t.3349.3		12	5.4	even	2	inner
9300.2.g.t.3349.10		12	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 9300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9300.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +0.649950i q^{7} -1.00000 q^{9} +5.35424 q^{11} +0.343092i q^{13} -5.13187i q^{17} +1.53116 q^{19} -0.649950 q^{21} +6.81578i q^{23} -1.00000i q^{27} -1.68424 q^{29} -1.00000 q^{31} +5.35424i q^{33} +4.23545i q^{37} -0.343092 q^{39} +4.30686 q^{41} -4.66109i q^{43} +3.25762i q^{47} +6.57757 q^{49} +5.13187 q^{51} +5.24274i q^{53} +1.53116i q^{57} -13.9245 q^{59} +4.53845 q^{61} -0.649950i q^{63} -0.342756i q^{67} -6.81578 q^{69} +3.50382 q^{71} +2.00000i q^{73} +3.47999i q^{77} -1.64961 q^{79} +1.00000 q^{81} -8.10177i q^{83} -1.68424i q^{87} +8.61186 q^{89} -0.222992 q^{91} -1.00000i q^{93} +7.92762i q^{97} -5.35424 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{9} - 6 q^{11} + 6 q^{19} + 8 q^{21} - 12 q^{31} - 4 q^{39} + 36 q^{41} - 28 q^{49} + 8 q^{51} + 24 q^{59} + 56 q^{61} - 10 q^{69} + 50 q^{71} - 6 q^{79} + 12 q^{81} - 8 q^{89} + 32 q^{91} + 6 q^{99}+O(q^{100}) \) 12 * q - 12 * q^9 - 6 * q^11 + 6 * q^19 + 8 * q^21 - 12 * q^31 - 4 * q^39 + 36 * q^41 - 28 * q^49 + 8 * q^51 + 24 * q^59 + 56 * q^61 - 10 * q^69 + 50 * q^71 - 6 * q^79 + 12 * q^81 - 8 * q^89 + 32 * q^91 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more