LMFDB - Embedded newform 928.2.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [928,2,Mod(1,928)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(928, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("928.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 928 = 2^{5} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	928.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:	$7.41011730757$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	928.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.414214 q^{3} +1.00000 q^{5} -2.00000 q^{7} -2.82843 q^{9} +O(q^{10})$	$q+0.414214 q^{3} +1.00000 q^{5} -2.00000 q^{7} -2.82843 q^{9} -2.41421 q^{11} -1.82843 q^{13} +0.414214 q^{15} +2.82843 q^{17} -7.65685 q^{19} -0.828427 q^{21} -1.17157 q^{23} -4.00000 q^{25} -2.41421 q^{27} +1.00000 q^{29} -2.41421 q^{31} -1.00000 q^{33} -2.00000 q^{35} +5.65685 q^{37} -0.757359 q^{39} -4.82843 q^{41} +3.24264 q^{43} -2.82843 q^{45} -4.41421 q^{47} -3.00000 q^{49} +1.17157 q^{51} +11.4853 q^{53} -2.41421 q^{55} -3.17157 q^{57} +4.48528 q^{59} +12.4853 q^{61} +5.65685 q^{63} -1.82843 q^{65} -4.00000 q^{67} -0.485281 q^{69} -8.82843 q^{71} -8.00000 q^{73} -1.65685 q^{75} +4.82843 q^{77} -10.8995 q^{79} +7.48528 q^{81} -12.4853 q^{83} +2.82843 q^{85} +0.414214 q^{87} -3.17157 q^{89} +3.65685 q^{91} -1.00000 q^{93} -7.65685 q^{95} -3.17157 q^{97} +6.82843 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{5} - 4 q^{7}+O(q^{10}) $ 2 * q - 2 * q^3 + 2 * q^5 - 4 * q^7	$ 2 q - 2 q^{3} + 2 q^{5} - 4 q^{7} - 2 q^{11} + 2 q^{13} - 2 q^{15} - 4 q^{19} + 4 q^{21} - 8 q^{23} - 8 q^{25} - 2 q^{27} + 2 q^{29} - 2 q^{31} - 2 q^{33} - 4 q^{35} - 10 q^{39} - 4 q^{41} - 2 q^{43} - 6 q^{47} - 6 q^{49} + 8 q^{51} + 6 q^{53} - 2 q^{55} - 12 q^{57} - 8 q^{59} + 8 q^{61} + 2 q^{65} - 8 q^{67} + 16 q^{69} - 12 q^{71} - 16 q^{73} + 8 q^{75} + 4 q^{77} - 2 q^{79} - 2 q^{81} - 8 q^{83} - 2 q^{87} - 12 q^{89} - 4 q^{91} - 2 q^{93} - 4 q^{95} - 12 q^{97} + 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^5 - 4 * q^7 - 2 * q^11 + 2 * q^13 - 2 * q^15 - 4 * q^19 + 4 * q^21 - 8 * q^23 - 8 * q^25 - 2 * q^27 + 2 * q^29 - 2 * q^31 - 2 * q^33 - 4 * q^35 - 10 * q^39 - 4 * q^41 - 2 * q^43 - 6 * q^47 - 6 * q^49 + 8 * q^51 + 6 * q^53 - 2 * q^55 - 12 * q^57 - 8 * q^59 + 8 * q^61 + 2 * q^65 - 8 * q^67 + 16 * q^69 - 12 * q^71 - 16 * q^73 + 8 * q^75 + 4 * q^77 - 2 * q^79 - 2 * q^81 - 8 * q^83 - 2 * q^87 - 12 * q^89 - 4 * q^91 - 2 * q^93 - 4 * q^95 - 12 * q^97 + 8 * 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
$3$	0.414214	0.239146	0.119573	−	0.992825i	$-0.461847\pi$
$3$	0.414214	0.239146	0.119573	+	0.992825i	$0.461847\pi$
$4$	0	0
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	0	0
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	0	0
$9$	−2.82843	−0.942809
$10$	0	0
$11$	−2.41421	−0.727913	−0.363956	−	0.931416i	$-0.618574\pi$
$11$	−2.41421	−0.727913	−0.363956	+	0.931416i	$0.618574\pi$
$12$	0	0
$13$	−1.82843	−0.507114	−0.253557	−	0.967320i	$-0.581601\pi$
$13$	−1.82843	−0.507114	−0.253557	+	0.967320i	$0.581601\pi$
$14$	0	0
$15$	0.414214	0.106949
$16$	0	0
$17$	2.82843	0.685994	0.342997	−	0.939336i	$-0.388558\pi$
$17$	2.82843	0.685994	0.342997	+	0.939336i	$0.388558\pi$
$18$	0	0
$19$	−7.65685	−1.75660	−0.878301	−	0.478107i	$-0.841323\pi$
$19$	−7.65685	−1.75660	−0.878301	+	0.478107i	$0.841323\pi$
$20$	0	0
$21$	−0.828427	−0.180778
$22$	0	0
$23$	−1.17157	−0.244290	−0.122145	−	0.992512i	$-0.538977\pi$
$23$	−1.17157	−0.244290	−0.122145	+	0.992512i	$0.538977\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	−2.41421	−0.464616
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−2.41421	−0.433606	−0.216803	−	0.976215i	$-0.569563\pi$
$31$	−2.41421	−0.433606	−0.216803	+	0.976215i	$0.569563\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	−2.00000	−0.338062
$36$	0	0
$37$	5.65685	0.929981	0.464991	−	0.885316i	$-0.346058\pi$
$37$	5.65685	0.929981	0.464991	+	0.885316i	$0.346058\pi$
$38$	0	0
$39$	−0.757359	−0.121275
$40$	0	0
$41$	−4.82843	−0.754074	−0.377037	−	0.926198i	$-0.623057\pi$
$41$	−4.82843	−0.754074	−0.377037	+	0.926198i	$0.623057\pi$
$42$	0	0
$43$	3.24264	0.494498	0.247249	−	0.968952i	$-0.420473\pi$
$43$	3.24264	0.494498	0.247249	+	0.968952i	$0.420473\pi$
$44$	0	0
$45$	−2.82843	−0.421637
$46$	0	0
$47$	−4.41421	−0.643879	−0.321940	−	0.946760i	$-0.604335\pi$
$47$	−4.41421	−0.643879	−0.321940	+	0.946760i	$0.604335\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	1.17157	0.164053
$52$	0	0
$53$	11.4853	1.57762	0.788812	−	0.614634i	$-0.210696\pi$
$53$	11.4853	1.57762	0.788812	+	0.614634i	$0.210696\pi$
$54$	0	0
$55$	−2.41421	−0.325532
$56$	0	0
$57$	−3.17157	−0.420085
$58$	0	0
$59$	4.48528	0.583934	0.291967	−	0.956428i	$-0.405690\pi$
$59$	4.48528	0.583934	0.291967	+	0.956428i	$0.405690\pi$
$60$	0	0
$61$	12.4853	1.59858	0.799288	−	0.600948i	$-0.205210\pi$
$61$	12.4853	1.59858	0.799288	+	0.600948i	$0.205210\pi$
$62$	0	0
$63$	5.65685	0.712697
$64$	0	0
$65$	−1.82843	−0.226788
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	0	0
$69$	−0.485281	−0.0584210
$70$	0	0
$71$	−8.82843	−1.04774	−0.523871	−	0.851798i	$-0.675513\pi$
$71$	−8.82843	−1.04774	−0.523871	+	0.851798i	$0.675513\pi$
$72$	0	0
$73$	−8.00000	−0.936329	−0.468165	−	0.883641i	$-0.655085\pi$
$73$	−8.00000	−0.936329	−0.468165	+	0.883641i	$0.655085\pi$
$74$	0	0
$75$	−1.65685	−0.191317
$76$	0	0
$77$	4.82843	0.550250
$78$	0	0
$79$	−10.8995	−1.22629	−0.613144	−	0.789971i	$-0.710096\pi$
$79$	−10.8995	−1.22629	−0.613144	+	0.789971i	$0.710096\pi$
$80$	0	0
$81$	7.48528	0.831698
$82$	0	0
$83$	−12.4853	−1.37044	−0.685219	−	0.728337i	$-0.740293\pi$
$83$	−12.4853	−1.37044	−0.685219	+	0.728337i	$0.740293\pi$
$84$	0	0
$85$	2.82843	0.306786
$86$	0	0
$87$	0.414214	0.0444084
$88$	0	0
$89$	−3.17157	−0.336186	−0.168093	−	0.985771i	$-0.553761\pi$
$89$	−3.17157	−0.336186	−0.168093	+	0.985771i	$0.553761\pi$
$90$	0	0
$91$	3.65685	0.383342
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	−7.65685	−0.785577
$96$	0	0
$97$	−3.17157	−0.322024	−0.161012	−	0.986952i	$-0.551476\pi$
$97$	−3.17157	−0.322024	−0.161012	+	0.986952i	$0.551476\pi$
$98$	0	0
$99$	6.82843	0.686283
$100$	0	0
$101$	−11.3137	−1.12576	−0.562878	−	0.826540i	$-0.690306\pi$
$101$	−11.3137	−1.12576	−0.562878	+	0.826540i	$0.690306\pi$
$102$	0	0
$103$	20.1421	1.98466	0.992332	−	0.123603i	$-0.0394448\pi$
$103$	20.1421	1.98466	0.992332	+	0.123603i	$0.0394448\pi$
$104$	0	0
$105$	−0.828427	−0.0808462
$106$	0	0
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	0	0
$109$	−0.656854	−0.0629152	−0.0314576	−	0.999505i	$-0.510015\pi$
$109$	−0.656854	−0.0629152	−0.0314576	+	0.999505i	$0.510015\pi$
$110$	0	0
$111$	2.34315	0.222402
$112$	0	0
$113$	−3.65685	−0.344008	−0.172004	−	0.985096i	$-0.555024\pi$
$113$	−3.65685	−0.344008	−0.172004	+	0.985096i	$0.555024\pi$
$114$	0	0
$115$	−1.17157	−0.109250
$116$	0	0
$117$	5.17157	0.478112
$118$	0	0
$119$	−5.65685	−0.518563
$120$	0	0
$121$	−5.17157	−0.470143
$122$	0	0
$123$	−2.00000	−0.180334
$124$	0	0
$125$	−9.00000	−0.804984
$126$	0	0
$127$	8.34315	0.740334	0.370167	−	0.928965i	$-0.379300\pi$
$127$	8.34315	0.740334	0.370167	+	0.928965i	$0.379300\pi$
$128$	0	0
$129$	1.34315	0.118257
$130$	0	0
$131$	17.3137	1.51271	0.756353	−	0.654164i	$-0.226979\pi$
$131$	17.3137	1.51271	0.756353	+	0.654164i	$0.226979\pi$
$132$	0	0
$133$	15.3137	1.32787
$134$	0	0
$135$	−2.41421	−0.207782
$136$	0	0
$137$	7.31371	0.624852	0.312426	−	0.949942i	$-0.398858\pi$
$137$	7.31371	0.624852	0.312426	+	0.949942i	$0.398858\pi$
$138$	0	0
$139$	4.48528	0.380437	0.190218	−	0.981742i	$-0.439080\pi$
$139$	4.48528	0.380437	0.190218	+	0.981742i	$0.439080\pi$
$140$	0	0
$141$	−1.82843	−0.153981
$142$	0	0
$143$	4.41421	0.369135
$144$	0	0
$145$	1.00000	0.0830455
$146$	0	0
$147$	−1.24264	−0.102491
$148$	0	0
$149$	6.17157	0.505595	0.252797	−	0.967519i	$-0.418649\pi$
$149$	6.17157	0.505595	0.252797	+	0.967519i	$0.418649\pi$
$150$	0	0
$151$	11.6569	0.948621	0.474311	−	0.880358i	$-0.342697\pi$
$151$	11.6569	0.948621	0.474311	+	0.880358i	$0.342697\pi$
$152$	0	0
$153$	−8.00000	−0.646762
$154$	0	0
$155$	−2.41421	−0.193914
$156$	0	0
$157$	−6.48528	−0.517582	−0.258791	−	0.965933i	$-0.583324\pi$
$157$	−6.48528	−0.517582	−0.258791	+	0.965933i	$0.583324\pi$
$158$	0	0
$159$	4.75736	0.377283
$160$	0	0
$161$	2.34315	0.184666
$162$	0	0
$163$	7.58579	0.594165	0.297082	−	0.954852i	$-0.403986\pi$
$163$	7.58579	0.594165	0.297082	+	0.954852i	$0.403986\pi$
$164$	0	0
$165$	−1.00000	−0.0778499
$166$	0	0
$167$	−6.48528	−0.501846	−0.250923	−	0.968007i	$-0.580734\pi$
$167$	−6.48528	−0.501846	−0.250923	+	0.968007i	$0.580734\pi$
$168$	0	0
$169$	−9.65685	−0.742835
$170$	0	0
$171$	21.6569	1.65614
$172$	0	0
$173$	−4.34315	−0.330203	−0.165102	−	0.986277i	$-0.552795\pi$
$173$	−4.34315	−0.330203	−0.165102	+	0.986277i	$0.552795\pi$
$174$	0	0
$175$	8.00000	0.604743
$176$	0	0
$177$	1.85786	0.139646
$178$	0	0
$179$	−2.48528	−0.185759	−0.0928793	−	0.995677i	$-0.529607\pi$
$179$	−2.48528	−0.185759	−0.0928793	+	0.995677i	$0.529607\pi$
$180$	0	0
$181$	3.68629	0.274000	0.137000	−	0.990571i	$-0.456254\pi$
$181$	3.68629	0.274000	0.137000	+	0.990571i	$0.456254\pi$
$182$	0	0
$183$	5.17157	0.382294
$184$	0	0
$185$	5.65685	0.415900
$186$	0	0
$187$	−6.82843	−0.499344
$188$	0	0
$189$	4.82843	0.351216
$190$	0	0
$191$	17.3137	1.25278	0.626388	−	0.779511i	$-0.284533\pi$
$191$	17.3137	1.25278	0.626388	+	0.779511i	$0.284533\pi$
$192$	0	0
$193$	−27.4558	−1.97631	−0.988157	−	0.153443i	$-0.950964\pi$
$193$	−27.4558	−1.97631	−0.988157	+	0.153443i	$0.950964\pi$
$194$	0	0
$195$	−0.757359	−0.0542356
$196$	0	0
$197$	5.31371	0.378586	0.189293	−	0.981921i	$-0.439380\pi$
$197$	5.31371	0.378586	0.189293	+	0.981921i	$0.439380\pi$
$198$	0	0
$199$	4.34315	0.307877	0.153939	−	0.988080i	$-0.450804\pi$
$199$	4.34315	0.307877	0.153939	+	0.988080i	$0.450804\pi$
$200$	0	0
$201$	−1.65685	−0.116865
$202$	0	0
$203$	−2.00000	−0.140372
$204$	0	0
$205$	−4.82843	−0.337232
$206$	0	0
$207$	3.31371	0.230319
$208$	0	0
$209$	18.4853	1.27865
$210$	0	0
$211$	19.5858	1.34834	0.674171	−	0.738576i	$-0.264501\pi$
$211$	19.5858	1.34834	0.674171	+	0.738576i	$0.264501\pi$
$212$	0	0
$213$	−3.65685	−0.250564
$214$	0	0
$215$	3.24264	0.221146
$216$	0	0
$217$	4.82843	0.327775
$218$	0	0
$219$	−3.31371	−0.223920
$220$	0	0
$221$	−5.17157	−0.347878
$222$	0	0
$223$	−4.82843	−0.323335	−0.161668	−	0.986845i	$-0.551687\pi$
$223$	−4.82843	−0.323335	−0.161668	+	0.986845i	$0.551687\pi$
$224$	0	0
$225$	11.3137	0.754247
$226$	0	0
$227$	13.7990	0.915871	0.457936	−	0.888985i	$-0.348589\pi$
$227$	13.7990	0.915871	0.457936	+	0.888985i	$0.348589\pi$
$228$	0	0
$229$	10.4853	0.692887	0.346443	−	0.938071i	$-0.387389\pi$
$229$	10.4853	0.692887	0.346443	+	0.938071i	$0.387389\pi$
$230$	0	0
$231$	2.00000	0.131590
$232$	0	0
$233$	−4.31371	−0.282600	−0.141300	−	0.989967i	$-0.545128\pi$
$233$	−4.31371	−0.282600	−0.141300	+	0.989967i	$0.545128\pi$
$234$	0	0
$235$	−4.41421	−0.287952
$236$	0	0
$237$	−4.51472	−0.293262
$238$	0	0
$239$	−18.8284	−1.21791	−0.608955	−	0.793205i	$-0.708411\pi$
$239$	−18.8284	−1.21791	−0.608955	+	0.793205i	$0.708411\pi$
$240$	0	0
$241$	23.6274	1.52198	0.760988	−	0.648766i	$-0.224715\pi$
$241$	23.6274	1.52198	0.760988	+	0.648766i	$0.224715\pi$
$242$	0	0
$243$	10.3431	0.663513
$244$	0	0
$245$	−3.00000	−0.191663
$246$	0	0
$247$	14.0000	0.890799
$248$	0	0
$249$	−5.17157	−0.327735
$250$	0	0
$251$	−9.24264	−0.583390	−0.291695	−	0.956511i	$-0.594219\pi$
$251$	−9.24264	−0.583390	−0.291695	+	0.956511i	$0.594219\pi$
$252$	0	0
$253$	2.82843	0.177822
$254$	0	0
$255$	1.17157	0.0733667
$256$	0	0
$257$	14.7990	0.923136	0.461568	−	0.887105i	$-0.347287\pi$
$257$	14.7990	0.923136	0.461568	+	0.887105i	$0.347287\pi$
$258$	0	0
$259$	−11.3137	−0.703000
$260$	0	0
$261$	−2.82843	−0.175075
$262$	0	0
$263$	−25.7279	−1.58645	−0.793226	−	0.608928i	$-0.791600\pi$
$263$	−25.7279	−1.58645	−0.793226	+	0.608928i	$0.791600\pi$
$264$	0	0
$265$	11.4853	0.705535
$266$	0	0
$267$	−1.31371	−0.0803977
$268$	0	0
$269$	−11.5147	−0.702065	−0.351032	−	0.936363i	$-0.614169\pi$
$269$	−11.5147	−0.702065	−0.351032	+	0.936363i	$0.614169\pi$
$270$	0	0
$271$	−3.24264	−0.196976	−0.0984882	−	0.995138i	$-0.531401\pi$
$271$	−3.24264	−0.196976	−0.0984882	+	0.995138i	$0.531401\pi$
$272$	0	0
$273$	1.51472	0.0916749
$274$	0	0
$275$	9.65685	0.582330
$276$	0	0
$277$	−6.00000	−0.360505	−0.180253	−	0.983620i	$-0.557691\pi$
$277$	−6.00000	−0.360505	−0.180253	+	0.983620i	$0.557691\pi$
$278$	0	0
$279$	6.82843	0.408807
$280$	0	0
$281$	−21.9706	−1.31065	−0.655327	−	0.755345i	$-0.727469\pi$
$281$	−21.9706	−1.31065	−0.655327	+	0.755345i	$0.727469\pi$
$282$	0	0
$283$	20.4853	1.21772	0.608862	−	0.793276i	$-0.291626\pi$
$283$	20.4853	1.21772	0.608862	+	0.793276i	$0.291626\pi$
$284$	0	0
$285$	−3.17157	−0.187868
$286$	0	0
$287$	9.65685	0.570026
$288$	0	0
$289$	−9.00000	−0.529412
$290$	0	0
$291$	−1.31371	−0.0770110
$292$	0	0
$293$	−9.31371	−0.544113	−0.272056	−	0.962281i	$-0.587704\pi$
$293$	−9.31371	−0.544113	−0.272056	+	0.962281i	$0.587704\pi$
$294$	0	0
$295$	4.48528	0.261143
$296$	0	0
$297$	5.82843	0.338200
$298$	0	0
$299$	2.14214	0.123883
$300$	0	0
$301$	−6.48528	−0.373805
$302$	0	0
$303$	−4.68629	−0.269220
$304$	0	0
$305$	12.4853	0.714905
$306$	0	0
$307$	3.92893	0.224236	0.112118	−	0.993695i	$-0.464237\pi$
$307$	3.92893	0.224236	0.112118	+	0.993695i	$0.464237\pi$
$308$	0	0
$309$	8.34315	0.474625
$310$	0	0
$311$	−6.68629	−0.379145	−0.189572	−	0.981867i	$-0.560710\pi$
$311$	−6.68629	−0.379145	−0.189572	+	0.981867i	$0.560710\pi$
$312$	0	0
$313$	−16.5147	−0.933467	−0.466734	−	0.884398i	$-0.654569\pi$
$313$	−16.5147	−0.933467	−0.466734	+	0.884398i	$0.654569\pi$
$314$	0	0
$315$	5.65685	0.318728
$316$	0	0
$317$	−22.8284	−1.28217	−0.641086	−	0.767469i	$-0.721516\pi$
$317$	−22.8284	−1.28217	−0.641086	+	0.767469i	$0.721516\pi$
$318$	0	0
$319$	−2.41421	−0.135170
$320$	0	0
$321$	−2.48528	−0.138715
$322$	0	0
$323$	−21.6569	−1.20502
$324$	0	0
$325$	7.31371	0.405692
$326$	0	0
$327$	−0.272078	−0.0150459
$328$	0	0
$329$	8.82843	0.486727
$330$	0	0
$331$	15.5858	0.856672	0.428336	−	0.903619i	$-0.359100\pi$
$331$	15.5858	0.856672	0.428336	+	0.903619i	$0.359100\pi$
$332$	0	0
$333$	−16.0000	−0.876795
$334$	0	0
$335$	−4.00000	−0.218543
$336$	0	0
$337$	12.4853	0.680117	0.340058	−	0.940404i	$-0.389553\pi$
$337$	12.4853	0.680117	0.340058	+	0.940404i	$0.389553\pi$
$338$	0	0
$339$	−1.51472	−0.0822682
$340$	0	0
$341$	5.82843	0.315627
$342$	0	0
$343$	20.0000	1.07990
$344$	0	0
$345$	−0.485281	−0.0261267
$346$	0	0
$347$	−32.8284	−1.76232	−0.881161	−	0.472816i	$-0.843238\pi$
$347$	−32.8284	−1.76232	−0.881161	+	0.472816i	$0.843238\pi$
$348$	0	0
$349$	−7.14214	−0.382310	−0.191155	−	0.981560i	$-0.561223\pi$
$349$	−7.14214	−0.382310	−0.191155	+	0.981560i	$0.561223\pi$
$350$	0	0
$351$	4.41421	0.235613
$352$	0	0
$353$	7.65685	0.407533	0.203767	−	0.979019i	$-0.434682\pi$
$353$	7.65685	0.407533	0.203767	+	0.979019i	$0.434682\pi$
$354$	0	0
$355$	−8.82843	−0.468564
$356$	0	0
$357$	−2.34315	−0.124012
$358$	0	0
$359$	27.7279	1.46342	0.731712	−	0.681614i	$-0.238722\pi$
$359$	27.7279	1.46342	0.731712	+	0.681614i	$0.238722\pi$
$360$	0	0
$361$	39.6274	2.08565
$362$	0	0
$363$	−2.14214	−0.112433
$364$	0	0
$365$	−8.00000	−0.418739
$366$	0	0
$367$	−1.02944	−0.0537362	−0.0268681	−	0.999639i	$-0.508553\pi$
$367$	−1.02944	−0.0537362	−0.0268681	+	0.999639i	$0.508553\pi$
$368$	0	0
$369$	13.6569	0.710947
$370$	0	0
$371$	−22.9706	−1.19257
$372$	0	0
$373$	11.0000	0.569558	0.284779	−	0.958593i	$-0.408080\pi$
$373$	11.0000	0.569558	0.284779	+	0.958593i	$0.408080\pi$
$374$	0	0
$375$	−3.72792	−0.192509
$376$	0	0
$377$	−1.82843	−0.0941688
$378$	0	0
$379$	16.6274	0.854093	0.427047	−	0.904230i	$-0.359554\pi$
$379$	16.6274	0.854093	0.427047	+	0.904230i	$0.359554\pi$
$380$	0	0
$381$	3.45584	0.177048
$382$	0	0
$383$	−26.9706	−1.37813	−0.689066	−	0.724699i	$-0.741979\pi$
$383$	−26.9706	−1.37813	−0.689066	+	0.724699i	$0.741979\pi$
$384$	0	0
$385$	4.82843	0.246079
$386$	0	0
$387$	−9.17157	−0.466217
$388$	0	0
$389$	−0.970563	−0.0492095	−0.0246047	−	0.999697i	$-0.507833\pi$
$389$	−0.970563	−0.0492095	−0.0246047	+	0.999697i	$0.507833\pi$
$390$	0	0
$391$	−3.31371	−0.167581
$392$	0	0
$393$	7.17157	0.361758
$394$	0	0
$395$	−10.8995	−0.548413
$396$	0	0
$397$	4.65685	0.233721	0.116860	−	0.993148i	$-0.462717\pi$
$397$	4.65685	0.233721	0.116860	+	0.993148i	$0.462717\pi$
$398$	0	0
$399$	6.34315	0.317554
$400$	0	0
$401$	−13.9706	−0.697657	−0.348828	−	0.937187i	$-0.613420\pi$
$401$	−13.9706	−0.697657	−0.348828	+	0.937187i	$0.613420\pi$
$402$	0	0
$403$	4.41421	0.219888
$404$	0	0
$405$	7.48528	0.371947
$406$	0	0
$407$	−13.6569	−0.676945
$408$	0	0
$409$	−28.6274	−1.41553	−0.707767	−	0.706446i	$-0.750297\pi$
$409$	−28.6274	−1.41553	−0.707767	+	0.706446i	$0.750297\pi$
$410$	0	0
$411$	3.02944	0.149431
$412$	0	0
$413$	−8.97056	−0.441413
$414$	0	0
$415$	−12.4853	−0.612878
$416$	0	0
$417$	1.85786	0.0909800
$418$	0	0
$419$	−36.8284	−1.79919	−0.899593	−	0.436729i	$-0.856137\pi$
$419$	−36.8284	−1.79919	−0.899593	+	0.436729i	$0.856137\pi$
$420$	0	0
$421$	−10.1421	−0.494297	−0.247149	−	0.968978i	$-0.579494\pi$
$421$	−10.1421	−0.494297	−0.247149	+	0.968978i	$0.579494\pi$
$422$	0	0
$423$	12.4853	0.607055
$424$	0	0
$425$	−11.3137	−0.548795
$426$	0	0
$427$	−24.9706	−1.20841
$428$	0	0
$429$	1.82843	0.0882773
$430$	0	0
$431$	15.7990	0.761011	0.380505	−	0.924779i	$-0.375750\pi$
$431$	15.7990	0.761011	0.380505	+	0.924779i	$0.375750\pi$
$432$	0	0
$433$	28.0000	1.34559	0.672797	−	0.739827i	$-0.265093\pi$
$433$	28.0000	1.34559	0.672797	+	0.739827i	$0.265093\pi$
$434$	0	0
$435$	0.414214	0.0198600
$436$	0	0
$437$	8.97056	0.429120
$438$	0	0
$439$	−26.1421	−1.24770	−0.623848	−	0.781546i	$-0.714432\pi$
$439$	−26.1421	−1.24770	−0.623848	+	0.781546i	$0.714432\pi$
$440$	0	0
$441$	8.48528	0.404061
$442$	0	0
$443$	−15.6569	−0.743880	−0.371940	−	0.928257i	$-0.621307\pi$
$443$	−15.6569	−0.743880	−0.371940	+	0.928257i	$0.621307\pi$
$444$	0	0
$445$	−3.17157	−0.150347
$446$	0	0
$447$	2.55635	0.120911
$448$	0	0
$449$	14.6863	0.693089	0.346544	−	0.938034i	$-0.387355\pi$
$449$	14.6863	0.693089	0.346544	+	0.938034i	$0.387355\pi$
$450$	0	0
$451$	11.6569	0.548900
$452$	0	0
$453$	4.82843	0.226859
$454$	0	0
$455$	3.65685	0.171436
$456$	0	0
$457$	−26.2843	−1.22953	−0.614763	−	0.788712i	$-0.710748\pi$
$457$	−26.2843	−1.22953	−0.614763	+	0.788712i	$0.710748\pi$
$458$	0	0
$459$	−6.82843	−0.318724
$460$	0	0
$461$	−17.3137	−0.806380	−0.403190	−	0.915116i	$-0.632099\pi$
$461$	−17.3137	−0.806380	−0.403190	+	0.915116i	$0.632099\pi$
$462$	0	0
$463$	−23.7990	−1.10603	−0.553016	−	0.833170i	$-0.686523\pi$
$463$	−23.7990	−1.10603	−0.553016	+	0.833170i	$0.686523\pi$
$464$	0	0
$465$	−1.00000	−0.0463739
$466$	0	0
$467$	−20.7574	−0.960536	−0.480268	−	0.877122i	$-0.659461\pi$
$467$	−20.7574	−0.960536	−0.480268	+	0.877122i	$0.659461\pi$
$468$	0	0
$469$	8.00000	0.369406
$470$	0	0
$471$	−2.68629	−0.123778
$472$	0	0
$473$	−7.82843	−0.359951
$474$	0	0
$475$	30.6274	1.40528
$476$	0	0
$477$	−32.4853	−1.48740
$478$	0	0
$479$	−14.5563	−0.665097	−0.332548	−	0.943086i	$-0.607908\pi$
$479$	−14.5563	−0.665097	−0.332548	+	0.943086i	$0.607908\pi$
$480$	0	0
$481$	−10.3431	−0.471607
$482$	0	0
$483$	0.970563	0.0441621
$484$	0	0
$485$	−3.17157	−0.144014
$486$	0	0
$487$	−37.3137	−1.69085	−0.845423	−	0.534098i	$-0.820651\pi$
$487$	−37.3137	−1.69085	−0.845423	+	0.534098i	$0.820651\pi$
$488$	0	0
$489$	3.14214	0.142092
$490$	0	0
$491$	25.7279	1.16108	0.580542	−	0.814230i	$-0.302841\pi$
$491$	25.7279	1.16108	0.580542	+	0.814230i	$0.302841\pi$
$492$	0	0
$493$	2.82843	0.127386
$494$	0	0
$495$	6.82843	0.306915
$496$	0	0
$497$	17.6569	0.792018
$498$	0	0
$499$	39.1127	1.75092	0.875462	−	0.483286i	$-0.160557\pi$
$499$	39.1127	1.75092	0.875462	+	0.483286i	$0.160557\pi$
$500$	0	0
$501$	−2.68629	−0.120015
$502$	0	0
$503$	−15.9289	−0.710236	−0.355118	−	0.934821i	$-0.615559\pi$
$503$	−15.9289	−0.710236	−0.355118	+	0.934821i	$0.615559\pi$
$504$	0	0
$505$	−11.3137	−0.503453
$506$	0	0
$507$	−4.00000	−0.177646
$508$	0	0
$509$	−18.4558	−0.818041	−0.409020	−	0.912525i	$-0.634130\pi$
$509$	−18.4558	−0.818041	−0.409020	+	0.912525i	$0.634130\pi$
$510$	0	0
$511$	16.0000	0.707798
$512$	0	0
$513$	18.4853	0.816145
$514$	0	0
$515$	20.1421	0.887569
$516$	0	0
$517$	10.6569	0.468688
$518$	0	0
$519$	−1.79899	−0.0789669
$520$	0	0
$521$	27.1421	1.18912	0.594559	−	0.804052i	$-0.297327\pi$
$521$	27.1421	1.18912	0.594559	+	0.804052i	$0.297327\pi$
$522$	0	0
$523$	−28.9706	−1.26679	−0.633397	−	0.773827i	$-0.718340\pi$
$523$	−28.9706	−1.26679	−0.633397	+	0.773827i	$0.718340\pi$
$524$	0	0
$525$	3.31371	0.144622
$526$	0	0
$527$	−6.82843	−0.297451
$528$	0	0
$529$	−21.6274	−0.940322
$530$	0	0
$531$	−12.6863	−0.550538
$532$	0	0
$533$	8.82843	0.382402
$534$	0	0
$535$	−6.00000	−0.259403
$536$	0	0
$537$	−1.02944	−0.0444235
$538$	0	0
$539$	7.24264	0.311963
$540$	0	0
$541$	−2.62742	−0.112961	−0.0564807	−	0.998404i	$-0.517988\pi$
$541$	−2.62742	−0.112961	−0.0564807	+	0.998404i	$0.517988\pi$
$542$	0	0
$543$	1.52691	0.0655261
$544$	0	0
$545$	−0.656854	−0.0281365
$546$	0	0
$547$	1.31371	0.0561701	0.0280851	−	0.999606i	$-0.491059\pi$
$547$	1.31371	0.0561701	0.0280851	+	0.999606i	$0.491059\pi$
$548$	0	0
$549$	−35.3137	−1.50715
$550$	0	0
$551$	−7.65685	−0.326193
$552$	0	0
$553$	21.7990	0.926987
$554$	0	0
$555$	2.34315	0.0994610
$556$	0	0
$557$	−45.3137	−1.92000	−0.960002	−	0.279994i	$-0.909667\pi$
$557$	−45.3137	−1.92000	−0.960002	+	0.279994i	$0.909667\pi$
$558$	0	0
$559$	−5.92893	−0.250767
$560$	0	0
$561$	−2.82843	−0.119416
$562$	0	0
$563$	−16.6985	−0.703757	−0.351879	−	0.936046i	$-0.614457\pi$
$563$	−16.6985	−0.703757	−0.351879	+	0.936046i	$0.614457\pi$
$564$	0	0
$565$	−3.65685	−0.153845
$566$	0	0
$567$	−14.9706	−0.628705
$568$	0	0
$569$	−3.65685	−0.153303	−0.0766517	−	0.997058i	$-0.524423\pi$
$569$	−3.65685	−0.153303	−0.0766517	+	0.997058i	$0.524423\pi$
$570$	0	0
$571$	−20.2843	−0.848870	−0.424435	−	0.905458i	$-0.639527\pi$
$571$	−20.2843	−0.848870	−0.424435	+	0.905458i	$0.639527\pi$
$572$	0	0
$573$	7.17157	0.299597
$574$	0	0
$575$	4.68629	0.195432
$576$	0	0
$577$	17.4558	0.726696	0.363348	−	0.931653i	$-0.381634\pi$
$577$	17.4558	0.726696	0.363348	+	0.931653i	$0.381634\pi$
$578$	0	0
$579$	−11.3726	−0.472628
$580$	0	0
$581$	24.9706	1.03595
$582$	0	0
$583$	−27.7279	−1.14837
$584$	0	0
$585$	5.17157	0.213818
$586$	0	0
$587$	−34.8284	−1.43752	−0.718762	−	0.695257i	$-0.755291\pi$
$587$	−34.8284	−1.43752	−0.718762	+	0.695257i	$0.755291\pi$
$588$	0	0
$589$	18.4853	0.761673
$590$	0	0
$591$	2.20101	0.0905375
$592$	0	0
$593$	−8.45584	−0.347240	−0.173620	−	0.984813i	$-0.555546\pi$
$593$	−8.45584	−0.347240	−0.173620	+	0.984813i	$0.555546\pi$
$594$	0	0
$595$	−5.65685	−0.231908
$596$	0	0
$597$	1.79899	0.0736278
$598$	0	0
$599$	−38.8995	−1.58939	−0.794695	−	0.607009i	$-0.792369\pi$
$599$	−38.8995	−1.58939	−0.794695	+	0.607009i	$0.792369\pi$
$600$	0	0
$601$	−44.1421	−1.80060	−0.900298	−	0.435275i	$-0.856651\pi$
$601$	−44.1421	−1.80060	−0.900298	+	0.435275i	$0.856651\pi$
$602$	0	0
$603$	11.3137	0.460730
$604$	0	0
$605$	−5.17157	−0.210254
$606$	0	0
$607$	42.0122	1.70522	0.852611	−	0.522546i	$-0.175018\pi$
$607$	42.0122	1.70522	0.852611	+	0.522546i	$0.175018\pi$
$608$	0	0
$609$	−0.828427	−0.0335696
$610$	0	0
$611$	8.07107	0.326520
$612$	0	0
$613$	−29.6274	−1.19664	−0.598320	−	0.801257i	$-0.704165\pi$
$613$	−29.6274	−1.19664	−0.598320	+	0.801257i	$0.704165\pi$
$614$	0	0
$615$	−2.00000	−0.0806478
$616$	0	0
$617$	31.3137	1.26064	0.630321	−	0.776334i	$-0.282923\pi$
$617$	31.3137	1.26064	0.630321	+	0.776334i	$0.282923\pi$
$618$	0	0
$619$	−15.7279	−0.632159	−0.316079	−	0.948733i	$-0.602367\pi$
$619$	−15.7279	−0.632159	−0.316079	+	0.948733i	$0.602367\pi$
$620$	0	0
$621$	2.82843	0.113501
$622$	0	0
$623$	6.34315	0.254133
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	7.65685	0.305785
$628$	0	0
$629$	16.0000	0.637962
$630$	0	0
$631$	44.8284	1.78459	0.892296	−	0.451451i	$-0.149094\pi$
$631$	44.8284	1.78459	0.892296	+	0.451451i	$0.149094\pi$
$632$	0	0
$633$	8.11270	0.322451
$634$	0	0
$635$	8.34315	0.331088
$636$	0	0
$637$	5.48528	0.217335
$638$	0	0
$639$	24.9706	0.987820
$640$	0	0
$641$	32.7696	1.29432	0.647160	−	0.762354i	$-0.275957\pi$
$641$	32.7696	1.29432	0.647160	+	0.762354i	$0.275957\pi$
$642$	0	0
$643$	19.6569	0.775191	0.387595	−	0.921830i	$-0.373306\pi$
$643$	19.6569	0.775191	0.387595	+	0.921830i	$0.373306\pi$
$644$	0	0
$645$	1.34315	0.0528863
$646$	0	0
$647$	−38.4264	−1.51070	−0.755349	−	0.655323i	$-0.772533\pi$
$647$	−38.4264	−1.51070	−0.755349	+	0.655323i	$0.772533\pi$
$648$	0	0
$649$	−10.8284	−0.425053
$650$	0	0
$651$	2.00000	0.0783862
$652$	0	0
$653$	26.4853	1.03645	0.518225	−	0.855245i	$-0.326593\pi$
$653$	26.4853	1.03645	0.518225	+	0.855245i	$0.326593\pi$
$654$	0	0
$655$	17.3137	0.676503
$656$	0	0
$657$	22.6274	0.882780
$658$	0	0
$659$	−29.2426	−1.13913	−0.569566	−	0.821946i	$-0.692889\pi$
$659$	−29.2426	−1.13913	−0.569566	+	0.821946i	$0.692889\pi$
$660$	0	0
$661$	10.0000	0.388955	0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000	0.388955	0.194477	+	0.980907i	$0.437699\pi$
$662$	0	0
$663$	−2.14214	−0.0831937
$664$	0	0
$665$	15.3137	0.593840
$666$	0	0
$667$	−1.17157	−0.0453635
$668$	0	0
$669$	−2.00000	−0.0773245
$670$	0	0
$671$	−30.1421	−1.16362
$672$	0	0
$673$	−3.68629	−0.142096	−0.0710480	−	0.997473i	$-0.522634\pi$
$673$	−3.68629	−0.142096	−0.0710480	+	0.997473i	$0.522634\pi$
$674$	0	0
$675$	9.65685	0.371692
$676$	0	0
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	0	0
$679$	6.34315	0.243428
$680$	0	0
$681$	5.71573	0.219027
$682$	0	0
$683$	−8.97056	−0.343249	−0.171625	−	0.985162i	$-0.554902\pi$
$683$	−8.97056	−0.343249	−0.171625	+	0.985162i	$0.554902\pi$
$684$	0	0
$685$	7.31371	0.279442
$686$	0	0
$687$	4.34315	0.165701
$688$	0	0
$689$	−21.0000	−0.800036
$690$	0	0
$691$	34.3431	1.30647	0.653237	−	0.757153i	$-0.273410\pi$
$691$	34.3431	1.30647	0.653237	+	0.757153i	$0.273410\pi$
$692$	0	0
$693$	−13.6569	−0.518781
$694$	0	0
$695$	4.48528	0.170136
$696$	0	0
$697$	−13.6569	−0.517290
$698$	0	0
$699$	−1.78680	−0.0675829
$700$	0	0
$701$	33.7696	1.27546	0.637729	−	0.770261i	$-0.279874\pi$
$701$	33.7696	1.27546	0.637729	+	0.770261i	$0.279874\pi$
$702$	0	0
$703$	−43.3137	−1.63361
$704$	0	0
$705$	−1.82843	−0.0688625
$706$	0	0
$707$	22.6274	0.850992
$708$	0	0
$709$	−4.17157	−0.156667	−0.0783334	−	0.996927i	$-0.524960\pi$
$709$	−4.17157	−0.156667	−0.0783334	+	0.996927i	$0.524960\pi$
$710$	0	0
$711$	30.8284	1.15616
$712$	0	0
$713$	2.82843	0.105925
$714$	0	0
$715$	4.41421	0.165082
$716$	0	0
$717$	−7.79899	−0.291259
$718$	0	0
$719$	24.1421	0.900350	0.450175	−	0.892940i	$-0.351362\pi$
$719$	24.1421	0.900350	0.450175	+	0.892940i	$0.351362\pi$
$720$	0	0
$721$	−40.2843	−1.50026
$722$	0	0
$723$	9.78680	0.363975
$724$	0	0
$725$	−4.00000	−0.148556
$726$	0	0
$727$	1.31371	0.0487228	0.0243614	−	0.999703i	$-0.492245\pi$
$727$	1.31371	0.0487228	0.0243614	+	0.999703i	$0.492245\pi$
$728$	0	0
$729$	−18.1716	−0.673021
$730$	0	0
$731$	9.17157	0.339223
$732$	0	0
$733$	32.9706	1.21780	0.608898	−	0.793249i	$-0.291612\pi$
$733$	32.9706	1.21780	0.608898	+	0.793249i	$0.291612\pi$
$734$	0	0
$735$	−1.24264	−0.0458355
$736$	0	0
$737$	9.65685	0.355715
$738$	0	0
$739$	−5.92893	−0.218099	−0.109050	−	0.994036i	$-0.534781\pi$
$739$	−5.92893	−0.218099	−0.109050	+	0.994036i	$0.534781\pi$
$740$	0	0
$741$	5.79899	0.213031
$742$	0	0
$743$	51.2548	1.88036	0.940179	−	0.340682i	$-0.110658\pi$
$743$	51.2548	1.88036	0.940179	+	0.340682i	$0.110658\pi$
$744$	0	0
$745$	6.17157	0.226109
$746$	0	0
$747$	35.3137	1.29206
$748$	0	0
$749$	12.0000	0.438470
$750$	0	0
$751$	20.3431	0.742332	0.371166	−	0.928567i	$-0.378958\pi$
$751$	20.3431	0.742332	0.371166	+	0.928567i	$0.378958\pi$
$752$	0	0
$753$	−3.82843	−0.139516
$754$	0	0
$755$	11.6569	0.424236
$756$	0	0
$757$	−31.7990	−1.15575	−0.577877	−	0.816124i	$-0.696119\pi$
$757$	−31.7990	−1.15575	−0.577877	+	0.816124i	$0.696119\pi$
$758$	0	0
$759$	1.17157	0.0425254
$760$	0	0
$761$	−42.9706	−1.55768	−0.778841	−	0.627222i	$-0.784192\pi$
$761$	−42.9706	−1.55768	−0.778841	+	0.627222i	$0.784192\pi$
$762$	0	0
$763$	1.31371	0.0475594
$764$	0	0
$765$	−8.00000	−0.289241
$766$	0	0
$767$	−8.20101	−0.296121
$768$	0	0
$769$	−45.4558	−1.63918	−0.819590	−	0.572951i	$-0.805799\pi$
$769$	−45.4558	−1.63918	−0.819590	+	0.572951i	$0.805799\pi$
$770$	0	0
$771$	6.12994	0.220764
$772$	0	0
$773$	44.8284	1.61237	0.806183	−	0.591666i	$-0.201530\pi$
$773$	44.8284	1.61237	0.806183	+	0.591666i	$0.201530\pi$
$774$	0	0
$775$	9.65685	0.346884
$776$	0	0
$777$	−4.68629	−0.168120
$778$	0	0
$779$	36.9706	1.32461
$780$	0	0
$781$	21.3137	0.762664
$782$	0	0
$783$	−2.41421	−0.0862770
$784$	0	0
$785$	−6.48528	−0.231470
$786$	0	0
$787$	−41.7990	−1.48997	−0.744987	−	0.667079i	$-0.767544\pi$
$787$	−41.7990	−1.48997	−0.744987	+	0.667079i	$0.767544\pi$
$788$	0	0
$789$	−10.6569	−0.379394
$790$	0	0
$791$	7.31371	0.260046
$792$	0	0
$793$	−22.8284	−0.810661
$794$	0	0
$795$	4.75736	0.168726
$796$	0	0
$797$	−18.1421	−0.642627	−0.321314	−	0.946973i	$-0.604124\pi$
$797$	−18.1421	−0.642627	−0.321314	+	0.946973i	$0.604124\pi$
$798$	0	0
$799$	−12.4853	−0.441698
$800$	0	0
$801$	8.97056	0.316959
$802$	0	0
$803$	19.3137	0.681566
$804$	0	0
$805$	2.34315	0.0825850
$806$	0	0
$807$	−4.76955	−0.167896
$808$	0	0
$809$	19.3137	0.679034	0.339517	−	0.940600i	$-0.389736\pi$
$809$	19.3137	0.679034	0.339517	+	0.940600i	$0.389736\pi$
$810$	0	0
$811$	−24.3431	−0.854803	−0.427402	−	0.904062i	$-0.640571\pi$
$811$	−24.3431	−0.854803	−0.427402	+	0.904062i	$0.640571\pi$
$812$	0	0
$813$	−1.34315	−0.0471062
$814$	0	0
$815$	7.58579	0.265719
$816$	0	0
$817$	−24.8284	−0.868637
$818$	0	0
$819$	−10.3431	−0.361419
$820$	0	0
$821$	−34.7990	−1.21449	−0.607247	−	0.794513i	$-0.707726\pi$
$821$	−34.7990	−1.21449	−0.607247	+	0.794513i	$0.707726\pi$
$822$	0	0
$823$	35.6569	1.24292	0.621460	−	0.783446i	$-0.286540\pi$
$823$	35.6569	1.24292	0.621460	+	0.783446i	$0.286540\pi$
$824$	0	0
$825$	4.00000	0.139262
$826$	0	0
$827$	25.2426	0.877773	0.438886	−	0.898543i	$-0.355373\pi$
$827$	25.2426	0.877773	0.438886	+	0.898543i	$0.355373\pi$
$828$	0	0
$829$	28.4853	0.989335	0.494667	−	0.869082i	$-0.335290\pi$
$829$	28.4853	0.989335	0.494667	+	0.869082i	$0.335290\pi$
$830$	0	0
$831$	−2.48528	−0.0862135
$832$	0	0
$833$	−8.48528	−0.293998
$834$	0	0
$835$	−6.48528	−0.224432
$836$	0	0
$837$	5.82843	0.201460
$838$	0	0
$839$	34.2132	1.18117	0.590585	−	0.806975i	$-0.298897\pi$
$839$	34.2132	1.18117	0.590585	+	0.806975i	$0.298897\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−9.10051	−0.313438
$844$	0	0
$845$	−9.65685	−0.332206
$846$	0	0
$847$	10.3431	0.355395
$848$	0	0
$849$	8.48528	0.291214
$850$	0	0
$851$	−6.62742	−0.227185
$852$	0	0
$853$	22.0000	0.753266	0.376633	−	0.926363i	$-0.377082\pi$
$853$	22.0000	0.753266	0.376633	+	0.926363i	$0.377082\pi$
$854$	0	0
$855$	21.6569	0.740649
$856$	0	0
$857$	−22.4558	−0.767077	−0.383539	−	0.923525i	$-0.625295\pi$
$857$	−22.4558	−0.767077	−0.383539	+	0.923525i	$0.625295\pi$
$858$	0	0
$859$	21.4437	0.731648	0.365824	−	0.930684i	$-0.380787\pi$
$859$	21.4437	0.731648	0.365824	+	0.930684i	$0.380787\pi$
$860$	0	0
$861$	4.00000	0.136320
$862$	0	0
$863$	34.4853	1.17389	0.586946	−	0.809626i	$-0.300330\pi$
$863$	34.4853	1.17389	0.586946	+	0.809626i	$0.300330\pi$
$864$	0	0
$865$	−4.34315	−0.147671
$866$	0	0
$867$	−3.72792	−0.126607
$868$	0	0
$869$	26.3137	0.892631
$870$	0	0
$871$	7.31371	0.247816
$872$	0	0
$873$	8.97056	0.303608
$874$	0	0
$875$	18.0000	0.608511
$876$	0	0
$877$	35.4853	1.19825	0.599126	−	0.800654i	$-0.295515\pi$
$877$	35.4853	1.19825	0.599126	+	0.800654i	$0.295515\pi$
$878$	0	0
$879$	−3.85786	−0.130123
$880$	0	0
$881$	−30.2843	−1.02030	−0.510152	−	0.860085i	$-0.670411\pi$
$881$	−30.2843	−1.02030	−0.510152	+	0.860085i	$0.670411\pi$
$882$	0	0
$883$	−12.6274	−0.424946	−0.212473	−	0.977167i	$-0.568152\pi$
$883$	−12.6274	−0.424946	−0.212473	+	0.977167i	$0.568152\pi$
$884$	0	0
$885$	1.85786	0.0624514
$886$	0	0
$887$	−37.0416	−1.24374	−0.621868	−	0.783122i	$-0.713626\pi$
$887$	−37.0416	−1.24374	−0.621868	+	0.783122i	$0.713626\pi$
$888$	0	0
$889$	−16.6863	−0.559640
$890$	0	0
$891$	−18.0711	−0.605404
$892$	0	0
$893$	33.7990	1.13104
$894$	0	0
$895$	−2.48528	−0.0830738
$896$	0	0
$897$	0.887302	0.0296261
$898$	0	0
$899$	−2.41421	−0.0805185
$900$	0	0
$901$	32.4853	1.08224
$902$	0	0
$903$	−2.68629	−0.0893942
$904$	0	0
$905$	3.68629	0.122536
$906$	0	0
$907$	−46.2843	−1.53684	−0.768422	−	0.639943i	$-0.778958\pi$
$907$	−46.2843	−1.53684	−0.768422	+	0.639943i	$0.778958\pi$
$908$	0	0
$909$	32.0000	1.06137
$910$	0	0
$911$	32.0711	1.06256	0.531281	−	0.847196i	$-0.321711\pi$
$911$	32.0711	1.06256	0.531281	+	0.847196i	$0.321711\pi$
$912$	0	0
$913$	30.1421	0.997559
$914$	0	0
$915$	5.17157	0.170967
$916$	0	0
$917$	−34.6274	−1.14350
$918$	0	0
$919$	−14.2010	−0.468448	−0.234224	−	0.972183i	$-0.575255\pi$
$919$	−14.2010	−0.468448	−0.234224	+	0.972183i	$0.575255\pi$
$920$	0	0
$921$	1.62742	0.0536252
$922$	0	0
$923$	16.1421	0.531325
$924$	0	0
$925$	−22.6274	−0.743985
$926$	0	0
$927$	−56.9706	−1.87116
$928$	0	0
$929$	−34.0000	−1.11550	−0.557752	−	0.830008i	$-0.688336\pi$
$929$	−34.0000	−1.11550	−0.557752	+	0.830008i	$0.688336\pi$
$930$	0	0
$931$	22.9706	0.752830
$932$	0	0
$933$	−2.76955	−0.0906711
$934$	0	0
$935$	−6.82843	−0.223313
$936$	0	0
$937$	30.6863	1.00248	0.501239	−	0.865309i	$-0.332878\pi$
$937$	30.6863	1.00248	0.501239	+	0.865309i	$0.332878\pi$
$938$	0	0
$939$	−6.84062	−0.223235
$940$	0	0
$941$	−24.6569	−0.803790	−0.401895	−	0.915686i	$-0.631648\pi$
$941$	−24.6569	−0.803790	−0.401895	+	0.915686i	$0.631648\pi$
$942$	0	0
$943$	5.65685	0.184213
$944$	0	0
$945$	4.82843	0.157069
$946$	0	0
$947$	−4.07107	−0.132292	−0.0661460	−	0.997810i	$-0.521070\pi$
$947$	−4.07107	−0.132292	−0.0661460	+	0.997810i	$0.521070\pi$
$948$	0	0
$949$	14.6274	0.474826
$950$	0	0
$951$	−9.45584	−0.306627
$952$	0	0
$953$	−28.3137	−0.917171	−0.458585	−	0.888650i	$-0.651644\pi$
$953$	−28.3137	−0.917171	−0.458585	+	0.888650i	$0.651644\pi$
$954$	0	0
$955$	17.3137	0.560258
$956$	0	0
$957$	−1.00000	−0.0323254
$958$	0	0
$959$	−14.6274	−0.472344
$960$	0	0
$961$	−25.1716	−0.811986
$962$	0	0
$963$	16.9706	0.546869
$964$	0	0
$965$	−27.4558	−0.883835
$966$	0	0
$967$	29.5269	0.949521	0.474761	−	0.880115i	$-0.342535\pi$
$967$	29.5269	0.949521	0.474761	+	0.880115i	$0.342535\pi$
$968$	0	0
$969$	−8.97056	−0.288176
$970$	0	0
$971$	42.9706	1.37899	0.689495	−	0.724290i	$-0.257832\pi$
$971$	42.9706	1.37899	0.689495	+	0.724290i	$0.257832\pi$
$972$	0	0
$973$	−8.97056	−0.287583
$974$	0	0
$975$	3.02944	0.0970196
$976$	0	0
$977$	36.4558	1.16633	0.583163	−	0.812355i	$-0.301815\pi$
$977$	36.4558	1.16633	0.583163	+	0.812355i	$0.301815\pi$
$978$	0	0
$979$	7.65685	0.244714
$980$	0	0
$981$	1.85786	0.0593170
$982$	0	0
$983$	38.2132	1.21881	0.609406	−	0.792858i	$-0.291408\pi$
$983$	38.2132	1.21881	0.609406	+	0.792858i	$0.291408\pi$
$984$	0	0
$985$	5.31371	0.169309
$986$	0	0
$987$	3.65685	0.116399
$988$	0	0
$989$	−3.79899	−0.120801
$990$	0	0
$991$	18.7696	0.596234	0.298117	−	0.954529i	$-0.403641\pi$
$991$	18.7696	0.596234	0.298117	+	0.954529i	$0.403641\pi$
$992$	0	0
$993$	6.45584	0.204870
$994$	0	0
$995$	4.34315	0.137687
$996$	0	0
$997$	−0.686292	−0.0217351	−0.0108675	−	0.999941i	$-0.503459\pi$
$997$	−0.686292	−0.0217351	−0.0108675	+	0.999941i	$0.503459\pi$
$998$	0	0
$999$	−13.6569	−0.432084

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	928.2.a.c.1.2	✓	2
3.2	odd	2		8352.2.a.l.1.2		2
4.3	odd	2		928.2.a.e.1.1	yes	2
8.3	odd	2		1856.2.a.q.1.2		2
8.5	even	2		1856.2.a.u.1.1		2
12.11	even	2		8352.2.a.o.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
928.2.a.c.1.2	✓	2	1.1	even	1	trivial
928.2.a.e.1.1	yes	2	4.3	odd	2
1856.2.a.q.1.2		2	8.3	odd	2
1856.2.a.u.1.1		2	8.5	even	2
8352.2.a.l.1.2		2	3.2	odd	2
8352.2.a.o.1.1		2	12.11	even	2

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:	\( N \)	\(=\)	\( 928 = 2^{5} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	928.a (trivial)

\(f(q)\)	\(=\)	\(q+0.414214 q^{3} +1.00000 q^{5} -2.00000 q^{7} -2.82843 q^{9} +O(q^{10})\)	\(q+0.414214 q^{3} +1.00000 q^{5} -2.00000 q^{7} -2.82843 q^{9} -2.41421 q^{11} -1.82843 q^{13} +0.414214 q^{15} +2.82843 q^{17} -7.65685 q^{19} -0.828427 q^{21} -1.17157 q^{23} -4.00000 q^{25} -2.41421 q^{27} +1.00000 q^{29} -2.41421 q^{31} -1.00000 q^{33} -2.00000 q^{35} +5.65685 q^{37} -0.757359 q^{39} -4.82843 q^{41} +3.24264 q^{43} -2.82843 q^{45} -4.41421 q^{47} -3.00000 q^{49} +1.17157 q^{51} +11.4853 q^{53} -2.41421 q^{55} -3.17157 q^{57} +4.48528 q^{59} +12.4853 q^{61} +5.65685 q^{63} -1.82843 q^{65} -4.00000 q^{67} -0.485281 q^{69} -8.82843 q^{71} -8.00000 q^{73} -1.65685 q^{75} +4.82843 q^{77} -10.8995 q^{79} +7.48528 q^{81} -12.4853 q^{83} +2.82843 q^{85} +0.414214 q^{87} -3.17157 q^{89} +3.65685 q^{91} -1.00000 q^{93} -7.65685 q^{95} -3.17157 q^{97} +6.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{5} - 4 q^{7}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^5 - 4 * q^7	\( 2 q - 2 q^{3} + 2 q^{5} - 4 q^{7} - 2 q^{11} + 2 q^{13} - 2 q^{15} - 4 q^{19} + 4 q^{21} - 8 q^{23} - 8 q^{25} - 2 q^{27} + 2 q^{29} - 2 q^{31} - 2 q^{33} - 4 q^{35} - 10 q^{39} - 4 q^{41} - 2 q^{43} - 6 q^{47} - 6 q^{49} + 8 q^{51} + 6 q^{53} - 2 q^{55} - 12 q^{57} - 8 q^{59} + 8 q^{61} + 2 q^{65} - 8 q^{67} + 16 q^{69} - 12 q^{71} - 16 q^{73} + 8 q^{75} + 4 q^{77} - 2 q^{79} - 2 q^{81} - 8 q^{83} - 2 q^{87} - 12 q^{89} - 4 q^{91} - 2 q^{93} - 4 q^{95} - 12 q^{97} + 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^5 - 4 * q^7 - 2 * q^11 + 2 * q^13 - 2 * q^15 - 4 * q^19 + 4 * q^21 - 8 * q^23 - 8 * q^25 - 2 * q^27 + 2 * q^29 - 2 * q^31 - 2 * q^33 - 4 * q^35 - 10 * q^39 - 4 * q^41 - 2 * q^43 - 6 * q^47 - 6 * q^49 + 8 * q^51 + 6 * q^53 - 2 * q^55 - 12 * q^57 - 8 * q^59 + 8 * q^61 + 2 * q^65 - 8 * q^67 + 16 * q^69 - 12 * q^71 - 16 * q^73 + 8 * q^75 + 4 * q^77 - 2 * q^79 - 2 * q^81 - 8 * q^83 - 2 * q^87 - 12 * q^89 - 4 * q^91 - 2 * q^93 - 4 * q^95 - 12 * q^97 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more