LMFDB - Embedded newform 6468.2.a.bf.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6468,2,Mod(1,6468)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6468, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6468.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6468 = 2^{2} \cdot 3 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6468.a (trivial)

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:	yes
Analytic conductor:	$51.6472400274$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.152932864.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 15x^{4} + 20x^{3} + 58x^{2} - 16x - 28 $ x^6 - 2x^5 - 15x^4 + 20x^3 + 58x^2 - 16*x - 28
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.64899$ of defining polynomial
Character	$\chi$	$=$	6468.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +2.64899 q^{5} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +2.64899 q^{5} +1.00000 q^{9} +1.00000 q^{11} -3.86981 q^{13} +2.64899 q^{15} +4.11439 q^{17} +3.53779 q^{19} +4.77256 q^{23} +2.01714 q^{25} +1.00000 q^{27} +1.99082 q^{29} +9.43660 q^{31} +1.00000 q^{33} +1.76019 q^{37} -3.86981 q^{39} -1.72200 q^{41} -1.16972 q^{43} +2.64899 q^{45} -12.2000 q^{47} +4.11439 q^{51} -5.70336 q^{53} +2.64899 q^{55} +3.53779 q^{57} +0.282686 q^{59} +2.17025 q^{61} -10.2511 q^{65} +6.36360 q^{67} +4.77256 q^{69} +3.91062 q^{71} -9.72271 q^{73} +2.01714 q^{75} -3.56473 q^{79} +1.00000 q^{81} -13.5630 q^{83} +10.8990 q^{85} +1.99082 q^{87} +1.86247 q^{89} +9.43660 q^{93} +9.37155 q^{95} +11.5871 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{3} + 4 q^{5} + 6 q^{9}+O(q^{10}) $ 6 * q + 6 * q^3 + 4 * q^5 + 6 * q^9	$ 6 q + 6 q^{3} + 4 q^{5} + 6 q^{9} + 6 q^{11} + 8 q^{13} + 4 q^{15} + 8 q^{17} + 4 q^{19} + 8 q^{23} + 6 q^{25} + 6 q^{27} + 4 q^{29} + 8 q^{31} + 6 q^{33} + 4 q^{37} + 8 q^{39} + 4 q^{41} + 8 q^{43} + 4 q^{45} + 4 q^{47} + 8 q^{51} - 4 q^{53} + 4 q^{55} + 4 q^{57} + 16 q^{59} + 20 q^{65} - 8 q^{67} + 8 q^{69} + 12 q^{71} + 20 q^{73} + 6 q^{75} - 16 q^{79} + 6 q^{81} + 8 q^{83} + 4 q^{85} + 4 q^{87} + 4 q^{89} + 8 q^{93} - 4 q^{95} + 24 q^{97} + 6 q^{99}+O(q^{100}) $ 6 * q + 6 * q^3 + 4 * q^5 + 6 * q^9 + 6 * q^11 + 8 * q^13 + 4 * q^15 + 8 * q^17 + 4 * q^19 + 8 * q^23 + 6 * q^25 + 6 * q^27 + 4 * q^29 + 8 * q^31 + 6 * q^33 + 4 * q^37 + 8 * q^39 + 4 * q^41 + 8 * q^43 + 4 * q^45 + 4 * q^47 + 8 * q^51 - 4 * q^53 + 4 * q^55 + 4 * q^57 + 16 * q^59 + 20 * q^65 - 8 * q^67 + 8 * q^69 + 12 * q^71 + 20 * q^73 + 6 * q^75 - 16 * q^79 + 6 * q^81 + 8 * q^83 + 4 * q^85 + 4 * q^87 + 4 * q^89 + 8 * q^93 - 4 * q^95 + 24 * q^97 + 6 * q^99

Display 100 coefficients 10 coefficients

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	2.64899	1.18466	0.592332	−	0.805694i	$-0.298207\pi$
$5$	2.64899	1.18466	0.592332	+	0.805694i	$0.298207\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−3.86981	−1.07329	−0.536646	−	0.843808i	$-0.680309\pi$
$13$	−3.86981	−1.07329	−0.536646	+	0.843808i	$0.680309\pi$
$14$	0	0
$15$	2.64899	0.683966
$16$	0	0
$17$	4.11439	0.997885	0.498943	−	0.866635i	$-0.333722\pi$
$17$	4.11439	0.997885	0.498943	+	0.866635i	$0.333722\pi$
$18$	0	0
$19$	3.53779	0.811624	0.405812	−	0.913957i	$-0.366989\pi$
$19$	3.53779	0.811624	0.405812	+	0.913957i	$0.366989\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.77256	0.995148	0.497574	−	0.867422i	$-0.334224\pi$
$23$	4.77256	0.995148	0.497574	+	0.867422i	$0.334224\pi$
$24$	0	0
$25$	2.01714	0.403428
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.99082	0.369685	0.184843	−	0.982768i	$-0.440822\pi$
$29$	1.99082	0.369685	0.184843	+	0.982768i	$0.440822\pi$
$30$	0	0
$31$	9.43660	1.69486	0.847432	−	0.530904i	$-0.178147\pi$
$31$	9.43660	1.69486	0.847432	+	0.530904i	$0.178147\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.76019	0.289374	0.144687	−	0.989477i	$-0.453783\pi$
$37$	1.76019	0.289374	0.144687	+	0.989477i	$0.453783\pi$
$38$	0	0
$39$	−3.86981	−0.619665
$40$	0	0
$41$	−1.72200	−0.268931	−0.134465	−	0.990918i	$-0.542932\pi$
$41$	−1.72200	−0.268931	−0.134465	+	0.990918i	$0.542932\pi$
$42$	0	0
$43$	−1.16972	−0.178381	−0.0891906	−	0.996015i	$-0.528428\pi$
$43$	−1.16972	−0.178381	−0.0891906	+	0.996015i	$0.528428\pi$
$44$	0	0
$45$	2.64899	0.394888
$46$	0	0
$47$	−12.2000	−1.77955	−0.889775	−	0.456399i	$-0.849139\pi$
$47$	−12.2000	−1.77955	−0.889775	+	0.456399i	$0.849139\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	4.11439	0.576129
$52$	0	0
$53$	−5.70336	−0.783416	−0.391708	−	0.920089i	$-0.628116\pi$
$53$	−5.70336	−0.783416	−0.391708	+	0.920089i	$0.628116\pi$
$54$	0	0
$55$	2.64899	0.357190
$56$	0	0
$57$	3.53779	0.468591
$58$	0	0
$59$	0.282686	0.0368026	0.0184013	−	0.999831i	$-0.494142\pi$
$59$	0.282686	0.0368026	0.0184013	+	0.999831i	$0.494142\pi$
$60$	0	0
$61$	2.17025	0.277873	0.138936	−	0.990301i	$-0.455632\pi$
$61$	2.17025	0.277873	0.138936	+	0.990301i	$0.455632\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−10.2511	−1.27149
$66$	0	0
$67$	6.36360	0.777437	0.388719	−	0.921357i	$-0.372918\pi$
$67$	6.36360	0.777437	0.388719	+	0.921357i	$0.372918\pi$
$68$	0	0
$69$	4.77256	0.574549
$70$	0	0
$71$	3.91062	0.464105	0.232053	−	0.972703i	$-0.425456\pi$
$71$	3.91062	0.464105	0.232053	+	0.972703i	$0.425456\pi$
$72$	0	0
$73$	−9.72271	−1.13796	−0.568979	−	0.822352i	$-0.692661\pi$
$73$	−9.72271	−1.13796	−0.568979	+	0.822352i	$0.692661\pi$
$74$	0	0
$75$	2.01714	0.232919
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−3.56473	−0.401064	−0.200532	−	0.979687i	$-0.564267\pi$
$79$	−3.56473	−0.401064	−0.200532	+	0.979687i	$0.564267\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−13.5630	−1.48873	−0.744367	−	0.667771i	$-0.767249\pi$
$83$	−13.5630	−1.48873	−0.744367	+	0.667771i	$0.767249\pi$
$84$	0	0
$85$	10.8990	1.18216
$86$	0	0
$87$	1.99082	0.213438
$88$	0	0
$89$	1.86247	0.197422	0.0987108	−	0.995116i	$-0.468528\pi$
$89$	1.86247	0.197422	0.0987108	+	0.995116i	$0.468528\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	9.43660	0.978530
$94$	0	0
$95$	9.37155	0.961501
$96$	0	0
$97$	11.5871	1.17649	0.588247	−	0.808681i	$-0.299818\pi$
$97$	11.5871	1.17649	0.588247	+	0.808681i	$0.299818\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	7.37808	0.734147	0.367073	−	0.930192i	$-0.380360\pi$
$101$	7.37808	0.734147	0.367073	+	0.930192i	$0.380360\pi$
$102$	0	0
$103$	10.8628	1.07035	0.535173	−	0.844742i	$-0.320246\pi$
$103$	10.8628	1.07035	0.535173	+	0.844742i	$0.320246\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.10175	0.299857	0.149929	−	0.988697i	$-0.452096\pi$
$107$	3.10175	0.299857	0.149929	+	0.988697i	$0.452096\pi$
$108$	0	0
$109$	−6.86468	−0.657517	−0.328759	−	0.944414i	$-0.606630\pi$
$109$	−6.86468	−0.657517	−0.328759	+	0.944414i	$0.606630\pi$
$110$	0	0
$111$	1.76019	0.167070
$112$	0	0
$113$	9.83836	0.925515	0.462757	−	0.886485i	$-0.346860\pi$
$113$	9.83836	0.925515	0.462757	+	0.886485i	$0.346860\pi$
$114$	0	0
$115$	12.6425	1.17892
$116$	0	0
$117$	−3.86981	−0.357764
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−1.72200	−0.155267
$124$	0	0
$125$	−7.90156	−0.706737
$126$	0	0
$127$	14.0139	1.24353	0.621765	−	0.783203i	$-0.286416\pi$
$127$	14.0139	1.24353	0.621765	+	0.783203i	$0.286416\pi$
$128$	0	0
$129$	−1.16972	−0.102988
$130$	0	0
$131$	−9.00993	−0.787201	−0.393601	−	0.919282i	$-0.628771\pi$
$131$	−9.00993	−0.787201	−0.393601	+	0.919282i	$0.628771\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	2.64899	0.227989
$136$	0	0
$137$	11.1958	0.956524	0.478262	−	0.878217i	$-0.341267\pi$
$137$	11.1958	0.956524	0.478262	+	0.878217i	$0.341267\pi$
$138$	0	0
$139$	19.5582	1.65891	0.829454	−	0.558575i	$-0.188652\pi$
$139$	19.5582	1.65891	0.829454	+	0.558575i	$0.188652\pi$
$140$	0	0
$141$	−12.2000	−1.02742
$142$	0	0
$143$	−3.86981	−0.323610
$144$	0	0
$145$	5.27365	0.437953
$146$	0	0
$147$	0	0
$148$	0	0
$149$	3.87908	0.317787	0.158893	−	0.987296i	$-0.449207\pi$
$149$	3.87908	0.317787	0.158893	+	0.987296i	$0.449207\pi$
$150$	0	0
$151$	−19.7575	−1.60784	−0.803921	−	0.594736i	$-0.797257\pi$
$151$	−19.7575	−1.60784	−0.803921	+	0.594736i	$0.797257\pi$
$152$	0	0
$153$	4.11439	0.332628
$154$	0	0
$155$	24.9975	2.00784
$156$	0	0
$157$	11.4772	0.915983	0.457992	−	0.888957i	$-0.348569\pi$
$157$	11.4772	0.915983	0.457992	+	0.888957i	$0.348569\pi$
$158$	0	0
$159$	−5.70336	−0.452306
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−8.17847	−0.640587	−0.320293	−	0.947318i	$-0.603782\pi$
$163$	−8.17847	−0.640587	−0.320293	+	0.947318i	$0.603782\pi$
$164$	0	0
$165$	2.64899	0.206223
$166$	0	0
$167$	4.55553	0.352517	0.176259	−	0.984344i	$-0.443600\pi$
$167$	4.55553	0.352517	0.176259	+	0.984344i	$0.443600\pi$
$168$	0	0
$169$	1.97541	0.151954
$170$	0	0
$171$	3.53779	0.270541
$172$	0	0
$173$	16.8590	1.28177	0.640884	−	0.767637i	$-0.278568\pi$
$173$	16.8590	1.28177	0.640884	+	0.767637i	$0.278568\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0.282686	0.0212480
$178$	0	0
$179$	−8.25281	−0.616844	−0.308422	−	0.951250i	$-0.599801\pi$
$179$	−8.25281	−0.616844	−0.308422	+	0.951250i	$0.599801\pi$
$180$	0	0
$181$	−4.04837	−0.300913	−0.150456	−	0.988617i	$-0.548074\pi$
$181$	−4.04837	−0.300913	−0.150456	+	0.988617i	$0.548074\pi$
$182$	0	0
$183$	2.17025	0.160430
$184$	0	0
$185$	4.66273	0.342811
$186$	0	0
$187$	4.11439	0.300874
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−18.4872	−1.33769	−0.668843	−	0.743404i	$-0.733210\pi$
$191$	−18.4872	−1.33769	−0.668843	+	0.743404i	$0.733210\pi$
$192$	0	0
$193$	4.75295	0.342125	0.171062	−	0.985260i	$-0.445280\pi$
$193$	4.75295	0.342125	0.171062	+	0.985260i	$0.445280\pi$
$194$	0	0
$195$	−10.2511	−0.734095
$196$	0	0
$197$	21.4635	1.52921	0.764605	−	0.644500i	$-0.222934\pi$
$197$	21.4635	1.52921	0.764605	+	0.644500i	$0.222934\pi$
$198$	0	0
$199$	−11.7471	−0.832732	−0.416366	−	0.909197i	$-0.636696\pi$
$199$	−11.7471	−0.832732	−0.416366	+	0.909197i	$0.636696\pi$
$200$	0	0
$201$	6.36360	0.448854
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−4.56155	−0.318592
$206$	0	0
$207$	4.77256	0.331716
$208$	0	0
$209$	3.53779	0.244714
$210$	0	0
$211$	−7.84392	−0.539998	−0.269999	−	0.962861i	$-0.587023\pi$
$211$	−7.84392	−0.539998	−0.269999	+	0.962861i	$0.587023\pi$
$212$	0	0
$213$	3.91062	0.267951
$214$	0	0
$215$	−3.09858	−0.211322
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−9.72271	−0.657000
$220$	0	0
$221$	−15.9219	−1.07102
$222$	0	0
$223$	21.0729	1.41115	0.705574	−	0.708636i	$-0.250689\pi$
$223$	21.0729	1.41115	0.705574	+	0.708636i	$0.250689\pi$
$224$	0	0
$225$	2.01714	0.134476
$226$	0	0
$227$	21.0506	1.39718	0.698588	−	0.715524i	$-0.253812\pi$
$227$	21.0506	1.39718	0.698588	+	0.715524i	$0.253812\pi$
$228$	0	0
$229$	13.0812	0.864431	0.432216	−	0.901770i	$-0.357732\pi$
$229$	13.0812	0.864431	0.432216	+	0.901770i	$0.357732\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−12.7972	−0.838372	−0.419186	−	0.907900i	$-0.637685\pi$
$233$	−12.7972	−0.838372	−0.419186	+	0.907900i	$0.637685\pi$
$234$	0	0
$235$	−32.3176	−2.10817
$236$	0	0
$237$	−3.56473	−0.231554
$238$	0	0
$239$	−9.56760	−0.618877	−0.309438	−	0.950920i	$-0.600141\pi$
$239$	−9.56760	−0.618877	−0.309438	+	0.950920i	$0.600141\pi$
$240$	0	0
$241$	−22.2314	−1.43205	−0.716024	−	0.698076i	$-0.754040\pi$
$241$	−22.2314	−1.43205	−0.716024	+	0.698076i	$0.754040\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−13.6905	−0.871109
$248$	0	0
$249$	−13.5630	−0.859521
$250$	0	0
$251$	−1.69770	−0.107158	−0.0535791	−	0.998564i	$-0.517063\pi$
$251$	−1.69770	−0.107158	−0.0535791	+	0.998564i	$0.517063\pi$
$252$	0	0
$253$	4.77256	0.300048
$254$	0	0
$255$	10.8990	0.682520
$256$	0	0
$257$	15.7299	0.981206	0.490603	−	0.871383i	$-0.336776\pi$
$257$	15.7299	0.981206	0.490603	+	0.871383i	$0.336776\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	1.99082	0.123228
$262$	0	0
$263$	−13.2913	−0.819576	−0.409788	−	0.912181i	$-0.634397\pi$
$263$	−13.2913	−0.819576	−0.409788	+	0.912181i	$0.634397\pi$
$264$	0	0
$265$	−15.1081	−0.928085
$266$	0	0
$267$	1.86247	0.113981
$268$	0	0
$269$	15.0513	0.917696	0.458848	−	0.888515i	$-0.348262\pi$
$269$	15.0513	0.917696	0.458848	+	0.888515i	$0.348262\pi$
$270$	0	0
$271$	−24.4520	−1.48535	−0.742677	−	0.669649i	$-0.766444\pi$
$271$	−24.4520	−1.48535	−0.742677	+	0.669649i	$0.766444\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.01714	0.121638
$276$	0	0
$277$	−23.7635	−1.42781	−0.713906	−	0.700242i	$-0.753076\pi$
$277$	−23.7635	−1.42781	−0.713906	+	0.700242i	$0.753076\pi$
$278$	0	0
$279$	9.43660	0.564955
$280$	0	0
$281$	24.5318	1.46344	0.731722	−	0.681603i	$-0.238717\pi$
$281$	24.5318	1.46344	0.731722	+	0.681603i	$0.238717\pi$
$282$	0	0
$283$	−1.85194	−0.110086	−0.0550431	−	0.998484i	$-0.517530\pi$
$283$	−1.85194	−0.110086	−0.0550431	+	0.998484i	$0.517530\pi$
$284$	0	0
$285$	9.37155	0.555123
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−0.0718180	−0.00422459
$290$	0	0
$291$	11.5871	0.679249
$292$	0	0
$293$	7.41822	0.433377	0.216688	−	0.976241i	$-0.430474\pi$
$293$	7.41822	0.433377	0.216688	+	0.976241i	$0.430474\pi$
$294$	0	0
$295$	0.748832	0.0435987
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	−18.4689	−1.06808
$300$	0	0
$301$	0	0
$302$	0	0
$303$	7.37808	0.423860
$304$	0	0
$305$	5.74898	0.329186
$306$	0	0
$307$	17.0470	0.972921	0.486461	−	0.873703i	$-0.338288\pi$
$307$	17.0470	0.972921	0.486461	+	0.873703i	$0.338288\pi$
$308$	0	0
$309$	10.8628	0.617965
$310$	0	0
$311$	−10.1579	−0.576001	−0.288001	−	0.957630i	$-0.592990\pi$
$311$	−10.1579	−0.576001	−0.288001	+	0.957630i	$0.592990\pi$
$312$	0	0
$313$	27.7592	1.56905	0.784523	−	0.620100i	$-0.212908\pi$
$313$	27.7592	1.56905	0.784523	+	0.620100i	$0.212908\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−25.4315	−1.42838	−0.714188	−	0.699954i	$-0.753204\pi$
$317$	−25.4315	−1.42838	−0.714188	+	0.699954i	$0.753204\pi$
$318$	0	0
$319$	1.99082	0.111464
$320$	0	0
$321$	3.10175	0.173123
$322$	0	0
$323$	14.5558	0.809907
$324$	0	0
$325$	−7.80594	−0.432996
$326$	0	0
$327$	−6.86468	−0.379618
$328$	0	0
$329$	0	0
$330$	0	0
$331$	8.20653	0.451072	0.225536	−	0.974235i	$-0.427587\pi$
$331$	8.20653	0.451072	0.225536	+	0.974235i	$0.427587\pi$
$332$	0	0
$333$	1.76019	0.0964579
$334$	0	0
$335$	16.8571	0.921002
$336$	0	0
$337$	−19.6406	−1.06989	−0.534947	−	0.844886i	$-0.679668\pi$
$337$	−19.6406	−1.06989	−0.534947	+	0.844886i	$0.679668\pi$
$338$	0	0
$339$	9.83836	0.534346
$340$	0	0
$341$	9.43660	0.511021
$342$	0	0
$343$	0	0
$344$	0	0
$345$	12.6425	0.680647
$346$	0	0
$347$	−5.12596	−0.275176	−0.137588	−	0.990490i	$-0.543935\pi$
$347$	−5.12596	−0.275176	−0.137588	+	0.990490i	$0.543935\pi$
$348$	0	0
$349$	21.7043	1.16181	0.580903	−	0.813973i	$-0.302699\pi$
$349$	21.7043	1.16181	0.580903	+	0.813973i	$0.302699\pi$
$350$	0	0
$351$	−3.86981	−0.206555
$352$	0	0
$353$	37.4092	1.99109	0.995545	−	0.0942833i	$-0.0300559\pi$
$353$	37.4092	1.99109	0.995545	+	0.0942833i	$0.0300559\pi$
$354$	0	0
$355$	10.3592	0.549808
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−31.9281	−1.68510	−0.842550	−	0.538618i	$-0.818947\pi$
$359$	−31.9281	−1.68510	−0.842550	+	0.538618i	$0.818947\pi$
$360$	0	0
$361$	−6.48407	−0.341267
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	−25.7553	−1.34810
$366$	0	0
$367$	−29.3205	−1.53052	−0.765259	−	0.643722i	$-0.777389\pi$
$367$	−29.3205	−1.53052	−0.765259	+	0.643722i	$0.777389\pi$
$368$	0	0
$369$	−1.72200	−0.0896435
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−24.3190	−1.25919	−0.629596	−	0.776922i	$-0.716780\pi$
$373$	−24.3190	−1.25919	−0.629596	+	0.776922i	$0.716780\pi$
$374$	0	0
$375$	−7.90156	−0.408035
$376$	0	0
$377$	−7.70407	−0.396780
$378$	0	0
$379$	10.9095	0.560384	0.280192	−	0.959944i	$-0.409602\pi$
$379$	10.9095	0.560384	0.280192	+	0.959944i	$0.409602\pi$
$380$	0	0
$381$	14.0139	0.717953
$382$	0	0
$383$	−18.2476	−0.932407	−0.466203	−	0.884678i	$-0.654378\pi$
$383$	−18.2476	−0.932407	−0.466203	+	0.884678i	$0.654378\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.16972	−0.0594604
$388$	0	0
$389$	16.8365	0.853643	0.426822	−	0.904336i	$-0.359633\pi$
$389$	16.8365	0.853643	0.426822	+	0.904336i	$0.359633\pi$
$390$	0	0
$391$	19.6362	0.993043
$392$	0	0
$393$	−9.00993	−0.454491
$394$	0	0
$395$	−9.44293	−0.475125
$396$	0	0
$397$	15.2294	0.764343	0.382172	−	0.924091i	$-0.375176\pi$
$397$	15.2294	0.764343	0.382172	+	0.924091i	$0.375176\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−3.75362	−0.187447	−0.0937235	−	0.995598i	$-0.529877\pi$
$401$	−3.75362	−0.187447	−0.0937235	+	0.995598i	$0.529877\pi$
$402$	0	0
$403$	−36.5178	−1.81908
$404$	0	0
$405$	2.64899	0.131629
$406$	0	0
$407$	1.76019	0.0872495
$408$	0	0
$409$	13.7324	0.679022	0.339511	−	0.940602i	$-0.389738\pi$
$409$	13.7324	0.679022	0.339511	+	0.940602i	$0.389738\pi$
$410$	0	0
$411$	11.1958	0.552250
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−35.9283	−1.76365
$416$	0	0
$417$	19.5582	0.957771
$418$	0	0
$419$	12.6495	0.617968	0.308984	−	0.951067i	$-0.400011\pi$
$419$	12.6495	0.617968	0.308984	+	0.951067i	$0.400011\pi$
$420$	0	0
$421$	3.36927	0.164208	0.0821041	−	0.996624i	$-0.473836\pi$
$421$	3.36927	0.164208	0.0821041	+	0.996624i	$0.473836\pi$
$422$	0	0
$423$	−12.2000	−0.593183
$424$	0	0
$425$	8.29930	0.402575
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−3.86981	−0.186836
$430$	0	0
$431$	11.4032	0.549274	0.274637	−	0.961548i	$-0.411442\pi$
$431$	11.4032	0.549274	0.274637	+	0.961548i	$0.411442\pi$
$432$	0	0
$433$	−13.0528	−0.627279	−0.313639	−	0.949542i	$-0.601548\pi$
$433$	−13.0528	−0.627279	−0.313639	+	0.949542i	$0.601548\pi$
$434$	0	0
$435$	5.27365	0.252852
$436$	0	0
$437$	16.8843	0.807685
$438$	0	0
$439$	−34.1481	−1.62980	−0.814900	−	0.579602i	$-0.803208\pi$
$439$	−34.1481	−1.62980	−0.814900	+	0.579602i	$0.803208\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	16.4444	0.781296	0.390648	−	0.920540i	$-0.372251\pi$
$443$	16.4444	0.781296	0.390648	+	0.920540i	$0.372251\pi$
$444$	0	0
$445$	4.93367	0.233878
$446$	0	0
$447$	3.87908	0.183474
$448$	0	0
$449$	−5.86627	−0.276846	−0.138423	−	0.990373i	$-0.544203\pi$
$449$	−5.86627	−0.276846	−0.138423	+	0.990373i	$0.544203\pi$
$450$	0	0
$451$	−1.72200	−0.0810856
$452$	0	0
$453$	−19.7575	−0.928288
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−7.85307	−0.367351	−0.183676	−	0.982987i	$-0.558800\pi$
$457$	−7.85307	−0.367351	−0.183676	+	0.982987i	$0.558800\pi$
$458$	0	0
$459$	4.11439	0.192043
$460$	0	0
$461$	33.9153	1.57959	0.789796	−	0.613370i	$-0.210186\pi$
$461$	33.9153	1.57959	0.789796	+	0.613370i	$0.210186\pi$
$462$	0	0
$463$	−19.8452	−0.922282	−0.461141	−	0.887327i	$-0.652560\pi$
$463$	−19.8452	−0.922282	−0.461141	+	0.887327i	$0.652560\pi$
$464$	0	0
$465$	24.9975	1.15923
$466$	0	0
$467$	−29.0337	−1.34352	−0.671760	−	0.740769i	$-0.734461\pi$
$467$	−29.0337	−1.34352	−0.671760	+	0.740769i	$0.734461\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	11.4772	0.528843
$472$	0	0
$473$	−1.16972	−0.0537839
$474$	0	0
$475$	7.13621	0.327432
$476$	0	0
$477$	−5.70336	−0.261139
$478$	0	0
$479$	−19.0043	−0.868327	−0.434163	−	0.900834i	$-0.642956\pi$
$479$	−19.0043	−0.868327	−0.434163	+	0.900834i	$0.642956\pi$
$480$	0	0
$481$	−6.81160	−0.310582
$482$	0	0
$483$	0	0
$484$	0	0
$485$	30.6942	1.39375
$486$	0	0
$487$	22.5288	1.02088	0.510439	−	0.859914i	$-0.329483\pi$
$487$	22.5288	1.02088	0.510439	+	0.859914i	$0.329483\pi$
$488$	0	0
$489$	−8.17847	−0.369843
$490$	0	0
$491$	−4.32527	−0.195197	−0.0975983	−	0.995226i	$-0.531116\pi$
$491$	−4.32527	−0.195197	−0.0975983	+	0.995226i	$0.531116\pi$
$492$	0	0
$493$	8.19099	0.368903
$494$	0	0
$495$	2.64899	0.119063
$496$	0	0
$497$	0	0
$498$	0	0
$499$	5.15821	0.230913	0.115457	−	0.993313i	$-0.463167\pi$
$499$	5.15821	0.230913	0.115457	+	0.993313i	$0.463167\pi$
$500$	0	0
$501$	4.55553	0.203526
$502$	0	0
$503$	2.07830	0.0926666	0.0463333	−	0.998926i	$-0.485246\pi$
$503$	2.07830	0.0926666	0.0463333	+	0.998926i	$0.485246\pi$
$504$	0	0
$505$	19.5445	0.869717
$506$	0	0
$507$	1.97541	0.0877310
$508$	0	0
$509$	−20.0678	−0.889488	−0.444744	−	0.895658i	$-0.646705\pi$
$509$	−20.0678	−0.889488	−0.444744	+	0.895658i	$0.646705\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	3.53779	0.156197
$514$	0	0
$515$	28.7755	1.26800
$516$	0	0
$517$	−12.2000	−0.536555
$518$	0	0
$519$	16.8590	0.740030
$520$	0	0
$521$	22.3458	0.978987	0.489493	−	0.872007i	$-0.337182\pi$
$521$	22.3458	0.978987	0.489493	+	0.872007i	$0.337182\pi$
$522$	0	0
$523$	12.9923	0.568113	0.284056	−	0.958808i	$-0.408320\pi$
$523$	12.9923	0.568113	0.284056	+	0.958808i	$0.408320\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	38.8258	1.69128
$528$	0	0
$529$	−0.222668	−0.00968122
$530$	0	0
$531$	0.282686	0.0122675
$532$	0	0
$533$	6.66379	0.288641
$534$	0	0
$535$	8.21650	0.355230
$536$	0	0
$537$	−8.25281	−0.356135
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−32.5186	−1.39809	−0.699043	−	0.715080i	$-0.746390\pi$
$541$	−32.5186	−1.39809	−0.699043	+	0.715080i	$0.746390\pi$
$542$	0	0
$543$	−4.04837	−0.173732
$544$	0	0
$545$	−18.1845	−0.778937
$546$	0	0
$547$	27.3373	1.16886	0.584428	−	0.811445i	$-0.301319\pi$
$547$	27.3373	1.16886	0.584428	+	0.811445i	$0.301319\pi$
$548$	0	0
$549$	2.17025	0.0926242
$550$	0	0
$551$	7.04308	0.300045
$552$	0	0
$553$	0	0
$554$	0	0
$555$	4.66273	0.197922
$556$	0	0
$557$	−24.2142	−1.02599	−0.512994	−	0.858392i	$-0.671464\pi$
$557$	−24.2142	−1.02599	−0.512994	+	0.858392i	$0.671464\pi$
$558$	0	0
$559$	4.52660	0.191455
$560$	0	0
$561$	4.11439	0.173710
$562$	0	0
$563$	15.5474	0.655244	0.327622	−	0.944809i	$-0.393753\pi$
$563$	15.5474	0.655244	0.327622	+	0.944809i	$0.393753\pi$
$564$	0	0
$565$	26.0617	1.09642
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−2.21160	−0.0927151	−0.0463575	−	0.998925i	$-0.514761\pi$
$569$	−2.21160	−0.0927151	−0.0463575	+	0.998925i	$0.514761\pi$
$570$	0	0
$571$	−30.3567	−1.27039	−0.635193	−	0.772353i	$-0.719080\pi$
$571$	−30.3567	−1.27039	−0.635193	+	0.772353i	$0.719080\pi$
$572$	0	0
$573$	−18.4872	−0.772313
$574$	0	0
$575$	9.62692	0.401470
$576$	0	0
$577$	27.5252	1.14589	0.572945	−	0.819594i	$-0.305801\pi$
$577$	27.5252	1.14589	0.572945	+	0.819594i	$0.305801\pi$
$578$	0	0
$579$	4.75295	0.197526
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−5.70336	−0.236209
$584$	0	0
$585$	−10.2511	−0.423830
$586$	0	0
$587$	−23.8933	−0.986183	−0.493091	−	0.869978i	$-0.664133\pi$
$587$	−23.8933	−0.986183	−0.493091	+	0.869978i	$0.664133\pi$
$588$	0	0
$589$	33.3847	1.37559
$590$	0	0
$591$	21.4635	0.882889
$592$	0	0
$593$	−36.0691	−1.48118	−0.740591	−	0.671956i	$-0.765454\pi$
$593$	−36.0691	−1.48118	−0.740591	+	0.671956i	$0.765454\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−11.7471	−0.480778
$598$	0	0
$599$	1.78889	0.0730921	0.0365461	−	0.999332i	$-0.488364\pi$
$599$	1.78889	0.0730921	0.0365461	+	0.999332i	$0.488364\pi$
$600$	0	0
$601$	−26.3017	−1.07287	−0.536434	−	0.843942i	$-0.680229\pi$
$601$	−26.3017	−1.07287	−0.536434	+	0.843942i	$0.680229\pi$
$602$	0	0
$603$	6.36360	0.259146
$604$	0	0
$605$	2.64899	0.107697
$606$	0	0
$607$	−31.8034	−1.29086	−0.645430	−	0.763819i	$-0.723322\pi$
$607$	−31.8034	−1.29086	−0.645430	+	0.763819i	$0.723322\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	47.2116	1.90998
$612$	0	0
$613$	41.8145	1.68887	0.844435	−	0.535657i	$-0.179936\pi$
$613$	41.8145	1.68887	0.844435	+	0.535657i	$0.179936\pi$
$614$	0	0
$615$	−4.56155	−0.183939
$616$	0	0
$617$	−12.5090	−0.503592	−0.251796	−	0.967780i	$-0.581021\pi$
$617$	−12.5090	−0.503592	−0.251796	+	0.967780i	$0.581021\pi$
$618$	0	0
$619$	−29.4329	−1.18301	−0.591504	−	0.806302i	$-0.701466\pi$
$619$	−29.4329	−1.18301	−0.591504	+	0.806302i	$0.701466\pi$
$620$	0	0
$621$	4.77256	0.191516
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−31.0168	−1.24067
$626$	0	0
$627$	3.53779	0.141286
$628$	0	0
$629$	7.24211	0.288762
$630$	0	0
$631$	40.4201	1.60910	0.804550	−	0.593885i	$-0.202407\pi$
$631$	40.4201	1.60910	0.804550	+	0.593885i	$0.202407\pi$
$632$	0	0
$633$	−7.84392	−0.311768
$634$	0	0
$635$	37.1226	1.47317
$636$	0	0
$637$	0	0
$638$	0	0
$639$	3.91062	0.154702
$640$	0	0
$641$	18.0258	0.711975	0.355987	−	0.934491i	$-0.384145\pi$
$641$	18.0258	0.711975	0.355987	+	0.934491i	$0.384145\pi$
$642$	0	0
$643$	16.8425	0.664203	0.332101	−	0.943244i	$-0.392242\pi$
$643$	16.8425	0.664203	0.332101	+	0.943244i	$0.392242\pi$
$644$	0	0
$645$	−3.09858	−0.122007
$646$	0	0
$647$	20.4565	0.804227	0.402114	−	0.915590i	$-0.368276\pi$
$647$	20.4565	0.804227	0.402114	+	0.915590i	$0.368276\pi$
$648$	0	0
$649$	0.282686	0.0110964
$650$	0	0
$651$	0	0
$652$	0	0
$653$	6.70769	0.262492	0.131246	−	0.991350i	$-0.458102\pi$
$653$	6.70769	0.262492	0.131246	+	0.991350i	$0.458102\pi$
$654$	0	0
$655$	−23.8672	−0.932569
$656$	0	0
$657$	−9.72271	−0.379319
$658$	0	0
$659$	−34.6131	−1.34833	−0.674167	−	0.738579i	$-0.735497\pi$
$659$	−34.6131	−1.34833	−0.674167	+	0.738579i	$0.735497\pi$
$660$	0	0
$661$	−44.1492	−1.71721	−0.858603	−	0.512642i	$-0.828667\pi$
$661$	−44.1492	−1.71721	−0.858603	+	0.512642i	$0.828667\pi$
$662$	0	0
$663$	−15.9219	−0.618355
$664$	0	0
$665$	0	0
$666$	0	0
$667$	9.50129	0.367891
$668$	0	0
$669$	21.0729	0.814727
$670$	0	0
$671$	2.17025	0.0837817
$672$	0	0
$673$	−14.0748	−0.542545	−0.271272	−	0.962503i	$-0.587444\pi$
$673$	−14.0748	−0.542545	−0.271272	+	0.962503i	$0.587444\pi$
$674$	0	0
$675$	2.01714	0.0776398
$676$	0	0
$677$	8.79757	0.338118	0.169059	−	0.985606i	$-0.445927\pi$
$677$	8.79757	0.338118	0.169059	+	0.985606i	$0.445927\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	21.0506	0.806660
$682$	0	0
$683$	−43.0225	−1.64621	−0.823105	−	0.567890i	$-0.807760\pi$
$683$	−43.0225	−1.64621	−0.823105	+	0.567890i	$0.807760\pi$
$684$	0	0
$685$	29.6576	1.13316
$686$	0	0
$687$	13.0812	0.499080
$688$	0	0
$689$	22.0709	0.840834
$690$	0	0
$691$	−3.70463	−0.140931	−0.0704654	−	0.997514i	$-0.522448\pi$
$691$	−3.70463	−0.140931	−0.0704654	+	0.997514i	$0.522448\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	51.8095	1.96525
$696$	0	0
$697$	−7.08496	−0.268362
$698$	0	0
$699$	−12.7972	−0.484034
$700$	0	0
$701$	−22.8505	−0.863052	−0.431526	−	0.902100i	$-0.642025\pi$
$701$	−22.8505	−0.863052	−0.431526	+	0.902100i	$0.642025\pi$
$702$	0	0
$703$	6.22718	0.234863
$704$	0	0
$705$	−32.3176	−1.21715
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−24.5752	−0.922941	−0.461470	−	0.887156i	$-0.652678\pi$
$709$	−24.5752	−0.922941	−0.461470	+	0.887156i	$0.652678\pi$
$710$	0	0
$711$	−3.56473	−0.133688
$712$	0	0
$713$	45.0368	1.68664
$714$	0	0
$715$	−10.2511	−0.383368
$716$	0	0
$717$	−9.56760	−0.357309
$718$	0	0
$719$	29.4412	1.09797	0.548986	−	0.835831i	$-0.315014\pi$
$719$	29.4412	1.09797	0.548986	+	0.835831i	$0.315014\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−22.2314	−0.826793
$724$	0	0
$725$	4.01575	0.149141
$726$	0	0
$727$	0.337111	0.0125028	0.00625138	−	0.999980i	$-0.498010\pi$
$727$	0.337111	0.0125028	0.00625138	+	0.999980i	$0.498010\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−4.81270	−0.178004
$732$	0	0
$733$	40.2743	1.48757	0.743783	−	0.668421i	$-0.233029\pi$
$733$	40.2743	1.48757	0.743783	+	0.668421i	$0.233029\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6.36360	0.234406
$738$	0	0
$739$	−45.6393	−1.67887	−0.839434	−	0.543462i	$-0.817113\pi$
$739$	−45.6393	−1.67887	−0.839434	+	0.543462i	$0.817113\pi$
$740$	0	0
$741$	−13.6905	−0.502935
$742$	0	0
$743$	9.44143	0.346373	0.173186	−	0.984889i	$-0.444594\pi$
$743$	9.44143	0.346373	0.173186	+	0.984889i	$0.444594\pi$
$744$	0	0
$745$	10.2756	0.376470
$746$	0	0
$747$	−13.5630	−0.496245
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−27.1300	−0.989988	−0.494994	−	0.868896i	$-0.664830\pi$
$751$	−27.1300	−0.989988	−0.494994	+	0.868896i	$0.664830\pi$
$752$	0	0
$753$	−1.69770	−0.0618678
$754$	0	0
$755$	−52.3374	−1.90475
$756$	0	0
$757$	−33.3687	−1.21281	−0.606403	−	0.795158i	$-0.707388\pi$
$757$	−33.3687	−1.21281	−0.606403	+	0.795158i	$0.707388\pi$
$758$	0	0
$759$	4.77256	0.173233
$760$	0	0
$761$	−32.4111	−1.17490	−0.587450	−	0.809260i	$-0.699868\pi$
$761$	−32.4111	−1.17490	−0.587450	+	0.809260i	$0.699868\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	10.8990	0.394053
$766$	0	0
$767$	−1.09394	−0.0394999
$768$	0	0
$769$	18.5669	0.669538	0.334769	−	0.942300i	$-0.391342\pi$
$769$	18.5669	0.669538	0.334769	+	0.942300i	$0.391342\pi$
$770$	0	0
$771$	15.7299	0.566500
$772$	0	0
$773$	8.05784	0.289820	0.144910	−	0.989445i	$-0.453711\pi$
$773$	8.05784	0.289820	0.144910	+	0.989445i	$0.453711\pi$
$774$	0	0
$775$	19.0350	0.683756
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−6.09205	−0.218270
$780$	0	0
$781$	3.91062	0.139933
$782$	0	0
$783$	1.99082	0.0711459
$784$	0	0
$785$	30.4031	1.08513
$786$	0	0
$787$	28.8535	1.02852	0.514258	−	0.857636i	$-0.328067\pi$
$787$	28.8535	1.02852	0.514258	+	0.857636i	$0.328067\pi$
$788$	0	0
$789$	−13.2913	−0.473182
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−8.39846	−0.298238
$794$	0	0
$795$	−15.1081	−0.535830
$796$	0	0
$797$	−40.5149	−1.43511	−0.717555	−	0.696501i	$-0.754739\pi$
$797$	−40.5149	−1.43511	−0.717555	+	0.696501i	$0.754739\pi$
$798$	0	0
$799$	−50.1954	−1.77579
$800$	0	0
$801$	1.86247	0.0658072
$802$	0	0
$803$	−9.72271	−0.343107
$804$	0	0
$805$	0	0
$806$	0	0
$807$	15.0513	0.529832
$808$	0	0
$809$	18.8223	0.661757	0.330878	−	0.943673i	$-0.392655\pi$
$809$	18.8223	0.661757	0.330878	+	0.943673i	$0.392655\pi$
$810$	0	0
$811$	−30.6428	−1.07601	−0.538007	−	0.842940i	$-0.680823\pi$
$811$	−30.6428	−1.07601	−0.538007	+	0.842940i	$0.680823\pi$
$812$	0	0
$813$	−24.4520	−0.857570
$814$	0	0
$815$	−21.6647	−0.758880
$816$	0	0
$817$	−4.13823	−0.144778
$818$	0	0
$819$	0	0
$820$	0	0
$821$	24.7958	0.865378	0.432689	−	0.901543i	$-0.357565\pi$
$821$	24.7958	0.865378	0.432689	+	0.901543i	$0.357565\pi$
$822$	0	0
$823$	48.3523	1.68546	0.842728	−	0.538340i	$-0.180948\pi$
$823$	48.3523	1.68546	0.842728	+	0.538340i	$0.180948\pi$
$824$	0	0
$825$	2.01714	0.0702278
$826$	0	0
$827$	−30.7512	−1.06932	−0.534661	−	0.845067i	$-0.679561\pi$
$827$	−30.7512	−1.06932	−0.534661	+	0.845067i	$0.679561\pi$
$828$	0	0
$829$	−20.7531	−0.720784	−0.360392	−	0.932801i	$-0.617357\pi$
$829$	−20.7531	−0.720784	−0.360392	+	0.932801i	$0.617357\pi$
$830$	0	0
$831$	−23.7635	−0.824347
$832$	0	0
$833$	0	0
$834$	0	0
$835$	12.0675	0.417614
$836$	0	0
$837$	9.43660	0.326177
$838$	0	0
$839$	−7.80753	−0.269546	−0.134773	−	0.990877i	$-0.543030\pi$
$839$	−7.80753	−0.269546	−0.134773	+	0.990877i	$0.543030\pi$
$840$	0	0
$841$	−25.0367	−0.863333
$842$	0	0
$843$	24.5318	0.844920
$844$	0	0
$845$	5.23283	0.180015
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−1.85194	−0.0635583
$850$	0	0
$851$	8.40062	0.287970
$852$	0	0
$853$	12.4816	0.427362	0.213681	−	0.976903i	$-0.431455\pi$
$853$	12.4816	0.427362	0.213681	+	0.976903i	$0.431455\pi$
$854$	0	0
$855$	9.37155	0.320500
$856$	0	0
$857$	34.8041	1.18889	0.594443	−	0.804138i	$-0.297373\pi$
$857$	34.8041	1.18889	0.594443	+	0.804138i	$0.297373\pi$
$858$	0	0
$859$	−4.49799	−0.153469	−0.0767346	−	0.997052i	$-0.524449\pi$
$859$	−4.49799	−0.153469	−0.0767346	+	0.997052i	$0.524449\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	8.44971	0.287632	0.143816	−	0.989604i	$-0.454063\pi$
$863$	8.44971	0.287632	0.143816	+	0.989604i	$0.454063\pi$
$864$	0	0
$865$	44.6594	1.51846
$866$	0	0
$867$	−0.0718180	−0.00243907
$868$	0	0
$869$	−3.56473	−0.120925
$870$	0	0
$871$	−24.6259	−0.834417
$872$	0	0
$873$	11.5871	0.392165
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−22.8194	−0.770557	−0.385279	−	0.922800i	$-0.625895\pi$
$877$	−22.8194	−0.770557	−0.385279	+	0.922800i	$0.625895\pi$
$878$	0	0
$879$	7.41822	0.250210
$880$	0	0
$881$	38.7166	1.30440	0.652198	−	0.758049i	$-0.273847\pi$
$881$	38.7166	1.30440	0.652198	+	0.758049i	$0.273847\pi$
$882$	0	0
$883$	−47.9657	−1.61417	−0.807087	−	0.590433i	$-0.798957\pi$
$883$	−47.9657	−1.61417	−0.807087	+	0.590433i	$0.798957\pi$
$884$	0	0
$885$	0.748832	0.0251717
$886$	0	0
$887$	−12.5723	−0.422136	−0.211068	−	0.977471i	$-0.567694\pi$
$887$	−12.5723	−0.422136	−0.211068	+	0.977471i	$0.567694\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	−43.1609	−1.44432
$894$	0	0
$895$	−21.8616	−0.730753
$896$	0	0
$897$	−18.4689	−0.616658
$898$	0	0
$899$	18.7865	0.626566
$900$	0	0
$901$	−23.4658	−0.781760
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−10.7241	−0.356481
$906$	0	0
$907$	−3.76568	−0.125037	−0.0625187	−	0.998044i	$-0.519913\pi$
$907$	−3.76568	−0.125037	−0.0625187	+	0.998044i	$0.519913\pi$
$908$	0	0
$909$	7.37808	0.244716
$910$	0	0
$911$	12.8068	0.424307	0.212154	−	0.977236i	$-0.431952\pi$
$911$	12.8068	0.424307	0.212154	+	0.977236i	$0.431952\pi$
$912$	0	0
$913$	−13.5630	−0.448870
$914$	0	0
$915$	5.74898	0.190055
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−12.9369	−0.426749	−0.213375	−	0.976970i	$-0.568445\pi$
$919$	−12.9369	−0.426749	−0.213375	+	0.976970i	$0.568445\pi$
$920$	0	0
$921$	17.0470	0.561716
$922$	0	0
$923$	−15.1333	−0.498120
$924$	0	0
$925$	3.55055	0.116741
$926$	0	0
$927$	10.8628	0.356782
$928$	0	0
$929$	23.6785	0.776866	0.388433	−	0.921477i	$-0.373017\pi$
$929$	23.6785	0.776866	0.388433	+	0.921477i	$0.373017\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−10.1579	−0.332554
$934$	0	0
$935$	10.8990	0.356434
$936$	0	0
$937$	2.96818	0.0969661	0.0484830	−	0.998824i	$-0.484561\pi$
$937$	2.96818	0.0969661	0.0484830	+	0.998824i	$0.484561\pi$
$938$	0	0
$939$	27.7592	0.905889
$940$	0	0
$941$	−54.5847	−1.77941	−0.889705	−	0.456535i	$-0.849090\pi$
$941$	−54.5847	−1.77941	−0.889705	+	0.456535i	$0.849090\pi$
$942$	0	0
$943$	−8.21833	−0.267626
$944$	0	0
$945$	0	0
$946$	0	0
$947$	16.2641	0.528513	0.264256	−	0.964452i	$-0.414873\pi$
$947$	16.2641	0.528513	0.264256	+	0.964452i	$0.414873\pi$
$948$	0	0
$949$	37.6250	1.22136
$950$	0	0
$951$	−25.4315	−0.824673
$952$	0	0
$953$	17.3398	0.561691	0.280845	−	0.959753i	$-0.409385\pi$
$953$	17.3398	0.561691	0.280845	+	0.959753i	$0.409385\pi$
$954$	0	0
$955$	−48.9723	−1.58471
$956$	0	0
$957$	1.99082	0.0643539
$958$	0	0
$959$	0	0
$960$	0	0
$961$	58.0495	1.87256
$962$	0	0
$963$	3.10175	0.0999525
$964$	0	0
$965$	12.5905	0.405303
$966$	0	0
$967$	25.4705	0.819077	0.409539	−	0.912293i	$-0.365690\pi$
$967$	25.4705	0.819077	0.409539	+	0.912293i	$0.365690\pi$
$968$	0	0
$969$	14.5558	0.467600
$970$	0	0
$971$	35.5072	1.13948	0.569740	−	0.821825i	$-0.307044\pi$
$971$	35.5072	1.13948	0.569740	+	0.821825i	$0.307044\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−7.80594	−0.249990
$976$	0	0
$977$	49.5921	1.58659	0.793296	−	0.608836i	$-0.208363\pi$
$977$	49.5921	1.58659	0.793296	+	0.608836i	$0.208363\pi$
$978$	0	0
$979$	1.86247	0.0595249
$980$	0	0
$981$	−6.86468	−0.219172
$982$	0	0
$983$	−32.0033	−1.02075	−0.510373	−	0.859953i	$-0.670493\pi$
$983$	−32.0033	−1.02075	−0.510373	+	0.859953i	$0.670493\pi$
$984$	0	0
$985$	56.8565	1.81160
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−5.58258	−0.177516
$990$	0	0
$991$	13.5690	0.431035	0.215517	−	0.976500i	$-0.430856\pi$
$991$	13.5690	0.431035	0.215517	+	0.976500i	$0.430856\pi$
$992$	0	0
$993$	8.20653	0.260426
$994$	0	0
$995$	−31.1180	−0.986507
$996$	0	0
$997$	−30.2659	−0.958531	−0.479265	−	0.877670i	$-0.659097\pi$
$997$	−30.2659	−0.958531	−0.479265	+	0.877670i	$0.659097\pi$
$998$	0	0
$999$	1.76019	0.0556900

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6468.2.a.bf.1.5	yes	6
7.6	odd	2		6468.2.a.bc.1.2	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6468.2.a.bc.1.2	✓	6	7.6	odd	2
6468.2.a.bf.1.5	yes	6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6468 = 2^{2} \cdot 3 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6468.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.64899 q^{5} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +2.64899 q^{5} +1.00000 q^{9} +1.00000 q^{11} -3.86981 q^{13} +2.64899 q^{15} +4.11439 q^{17} +3.53779 q^{19} +4.77256 q^{23} +2.01714 q^{25} +1.00000 q^{27} +1.99082 q^{29} +9.43660 q^{31} +1.00000 q^{33} +1.76019 q^{37} -3.86981 q^{39} -1.72200 q^{41} -1.16972 q^{43} +2.64899 q^{45} -12.2000 q^{47} +4.11439 q^{51} -5.70336 q^{53} +2.64899 q^{55} +3.53779 q^{57} +0.282686 q^{59} +2.17025 q^{61} -10.2511 q^{65} +6.36360 q^{67} +4.77256 q^{69} +3.91062 q^{71} -9.72271 q^{73} +2.01714 q^{75} -3.56473 q^{79} +1.00000 q^{81} -13.5630 q^{83} +10.8990 q^{85} +1.99082 q^{87} +1.86247 q^{89} +9.43660 q^{93} +9.37155 q^{95} +11.5871 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} + 4 q^{5} + 6 q^{9}+O(q^{10}) \) 6 * q + 6 * q^3 + 4 * q^5 + 6 * q^9	\( 6 q + 6 q^{3} + 4 q^{5} + 6 q^{9} + 6 q^{11} + 8 q^{13} + 4 q^{15} + 8 q^{17} + 4 q^{19} + 8 q^{23} + 6 q^{25} + 6 q^{27} + 4 q^{29} + 8 q^{31} + 6 q^{33} + 4 q^{37} + 8 q^{39} + 4 q^{41} + 8 q^{43} + 4 q^{45} + 4 q^{47} + 8 q^{51} - 4 q^{53} + 4 q^{55} + 4 q^{57} + 16 q^{59} + 20 q^{65} - 8 q^{67} + 8 q^{69} + 12 q^{71} + 20 q^{73} + 6 q^{75} - 16 q^{79} + 6 q^{81} + 8 q^{83} + 4 q^{85} + 4 q^{87} + 4 q^{89} + 8 q^{93} - 4 q^{95} + 24 q^{97} + 6 q^{99}+O(q^{100}) \) 6 * q + 6 * q^3 + 4 * q^5 + 6 * q^9 + 6 * q^11 + 8 * q^13 + 4 * q^15 + 8 * q^17 + 4 * q^19 + 8 * q^23 + 6 * q^25 + 6 * q^27 + 4 * q^29 + 8 * q^31 + 6 * q^33 + 4 * q^37 + 8 * q^39 + 4 * q^41 + 8 * q^43 + 4 * q^45 + 4 * q^47 + 8 * q^51 - 4 * q^53 + 4 * q^55 + 4 * q^57 + 16 * q^59 + 20 * q^65 - 8 * q^67 + 8 * q^69 + 12 * q^71 + 20 * q^73 + 6 * q^75 - 16 * q^79 + 6 * q^81 + 8 * q^83 + 4 * q^85 + 4 * q^87 + 4 * q^89 + 8 * q^93 - 4 * q^95 + 24 * q^97 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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