LMFDB - Embedded newform 7800.2.a.bk.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7800,2,Mod(1,7800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7800.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:	$62.2833135766$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.2700.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 15x - 20 $ x^3 - 15*x - 20
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.80560$ of defining polynomial
Character	$\chi$	$=$	7800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -2.80560 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -2.80560 q^{7} +1.00000 q^{9} +2.80560 q^{11} -1.00000 q^{13} +1.48261 q^{17} -2.48261 q^{19} +2.80560 q^{21} +6.96523 q^{23} -1.00000 q^{27} +1.00000 q^{29} +2.80560 q^{31} -2.80560 q^{33} -6.48261 q^{37} +1.00000 q^{39} +4.48261 q^{41} +2.00000 q^{43} -7.28822 q^{47} +0.871407 q^{49} -1.48261 q^{51} -7.96523 q^{53} +2.48261 q^{57} +11.7708 q^{59} -12.4478 q^{61} -2.80560 q^{63} +1.67701 q^{67} -6.96523 q^{69} +6.48261 q^{71} -1.61121 q^{73} -7.87141 q^{77} +2.87141 q^{79} +1.00000 q^{81} -7.77083 q^{83} -1.00000 q^{87} +8.96523 q^{89} +2.80560 q^{91} -2.80560 q^{93} -10.9652 q^{97} +2.80560 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + 3 * q^9	$ 3 q - 3 q^{3} + 3 q^{9} - 3 q^{13} - 6 q^{17} + 3 q^{19} - 3 q^{27} + 3 q^{29} - 9 q^{37} + 3 q^{39} + 3 q^{41} + 6 q^{43} - 3 q^{47} + 9 q^{49} + 6 q^{51} - 3 q^{53} - 3 q^{57} + 6 q^{59} - 6 q^{61} + 3 q^{67} + 9 q^{71} + 12 q^{73} - 30 q^{77} + 15 q^{79} + 3 q^{81} + 6 q^{83} - 3 q^{87} + 6 q^{89} - 12 q^{97}+O(q^{100}) $ 3 * q - 3 * q^3 + 3 * q^9 - 3 * q^13 - 6 * q^17 + 3 * q^19 - 3 * q^27 + 3 * q^29 - 9 * q^37 + 3 * q^39 + 3 * q^41 + 6 * q^43 - 3 * q^47 + 9 * q^49 + 6 * q^51 - 3 * q^53 - 3 * q^57 + 6 * q^59 - 6 * q^61 + 3 * q^67 + 9 * q^71 + 12 * q^73 - 30 * q^77 + 15 * q^79 + 3 * q^81 + 6 * q^83 - 3 * q^87 + 6 * q^89 - 12 * q^97

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
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.80560	−1.06042	−0.530209	−	0.847867i	$-0.677887\pi$
$7$	−2.80560	−1.06042	−0.530209	+	0.847867i	$0.677887\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.80560	0.845921	0.422961	−	0.906148i	$-0.360991\pi$
$11$	2.80560	0.845921	0.422961	+	0.906148i	$0.360991\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.48261	0.359586	0.179793	−	0.983704i	$-0.442457\pi$
$17$	1.48261	0.359586	0.179793	+	0.983704i	$0.442457\pi$
$18$	0	0
$19$	−2.48261	−0.569550	−0.284775	−	0.958594i	$-0.591919\pi$
$19$	−2.48261	−0.569550	−0.284775	+	0.958594i	$0.591919\pi$
$20$	0	0
$21$	2.80560	0.612233
$22$	0	0
$23$	6.96523	1.45235	0.726175	−	0.687510i	$-0.241296\pi$
$23$	6.96523	1.45235	0.726175	+	0.687510i	$0.241296\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	1.00000	0.185695	0.0928477	−	0.995680i	$-0.470403\pi$
$29$	1.00000	0.185695	0.0928477	+	0.995680i	$0.470403\pi$
$30$	0	0
$31$	2.80560	0.503901	0.251951	−	0.967740i	$-0.418928\pi$
$31$	2.80560	0.503901	0.251951	+	0.967740i	$0.418928\pi$
$32$	0	0
$33$	−2.80560	−0.488393
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−6.48261	−1.06573	−0.532867	−	0.846199i	$-0.678886\pi$
$37$	−6.48261	−1.06573	−0.532867	+	0.846199i	$0.678886\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	4.48261	0.700067	0.350033	−	0.936737i	$-0.386170\pi$
$41$	4.48261	0.700067	0.350033	+	0.936737i	$0.386170\pi$
$42$	0	0
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−7.28822	−1.06310	−0.531548	−	0.847028i	$-0.678389\pi$
$47$	−7.28822	−1.06310	−0.531548	+	0.847028i	$0.678389\pi$
$48$	0	0
$49$	0.871407	0.124487
$50$	0	0
$51$	−1.48261	−0.207607
$52$	0	0
$53$	−7.96523	−1.09411	−0.547054	−	0.837097i	$-0.684251\pi$
$53$	−7.96523	−1.09411	−0.547054	+	0.837097i	$0.684251\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.48261	0.328830
$58$	0	0
$59$	11.7708	1.53243	0.766216	−	0.642583i	$-0.222137\pi$
$59$	11.7708	1.53243	0.766216	+	0.642583i	$0.222137\pi$
$60$	0	0
$61$	−12.4478	−1.59378	−0.796891	−	0.604123i	$-0.793524\pi$
$61$	−12.4478	−1.59378	−0.796891	+	0.604123i	$0.793524\pi$
$62$	0	0
$63$	−2.80560	−0.353473
$64$	0	0
$65$	0	0
$66$	0	0
$67$	1.67701	0.204879	0.102440	−	0.994739i	$-0.467335\pi$
$67$	1.67701	0.204879	0.102440	+	0.994739i	$0.467335\pi$
$68$	0	0
$69$	−6.96523	−0.838515
$70$	0	0
$71$	6.48261	0.769345	0.384672	−	0.923053i	$-0.374314\pi$
$71$	6.48261	0.769345	0.384672	+	0.923053i	$0.374314\pi$
$72$	0	0
$73$	−1.61121	−0.188577	−0.0942887	−	0.995545i	$-0.530058\pi$
$73$	−1.61121	−0.188577	−0.0942887	+	0.995545i	$0.530058\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−7.87141	−0.897030
$78$	0	0
$79$	2.87141	0.323059	0.161529	−	0.986868i	$-0.448357\pi$
$79$	2.87141	0.323059	0.161529	+	0.986868i	$0.448357\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−7.77083	−0.852959	−0.426480	−	0.904497i	$-0.640246\pi$
$83$	−7.77083	−0.852959	−0.426480	+	0.904497i	$0.640246\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	8.96523	0.950312	0.475156	−	0.879902i	$-0.342392\pi$
$89$	8.96523	0.950312	0.475156	+	0.879902i	$0.342392\pi$
$90$	0	0
$91$	2.80560	0.294107
$92$	0	0
$93$	−2.80560	−0.290927
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−10.9652	−1.11335	−0.556675	−	0.830730i	$-0.687923\pi$
$97$	−10.9652	−1.11335	−0.556675	+	0.830730i	$0.687923\pi$
$98$	0	0
$99$	2.80560	0.281974
$100$	0	0
$101$	8.83663	0.879278	0.439639	−	0.898175i	$-0.355106\pi$
$101$	8.83663	0.879278	0.439639	+	0.898175i	$0.355106\pi$
$102$	0	0
$103$	2.64598	0.260716	0.130358	−	0.991467i	$-0.458387\pi$
$103$	2.64598	0.260716	0.130358	+	0.991467i	$0.458387\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.48261	−0.240003	−0.120002	−	0.992774i	$-0.538290\pi$
$107$	−2.48261	−0.240003	−0.120002	+	0.992774i	$0.538290\pi$
$108$	0	0
$109$	−11.7050	−1.12114	−0.560569	−	0.828108i	$-0.689418\pi$
$109$	−11.7050	−1.12114	−0.560569	+	0.828108i	$0.689418\pi$
$110$	0	0
$111$	6.48261	0.615302
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−4.15962	−0.381312
$120$	0	0
$121$	−3.12859	−0.284418
$122$	0	0
$123$	−4.48261	−0.404184
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−10.0938	−0.895682	−0.447841	−	0.894113i	$-0.647807\pi$
$127$	−10.0938	−0.895682	−0.447841	+	0.894113i	$0.647807\pi$
$128$	0	0
$129$	−2.00000	−0.176090
$130$	0	0
$131$	3.51739	0.307316	0.153658	−	0.988124i	$-0.450895\pi$
$131$	3.51739	0.307316	0.153658	+	0.988124i	$0.450895\pi$
$132$	0	0
$133$	6.96523	0.603962
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.05904	0.432223	0.216112	−	0.976369i	$-0.430662\pi$
$137$	5.05904	0.432223	0.216112	+	0.976369i	$0.430662\pi$
$138$	0	0
$139$	9.61121	0.815212	0.407606	−	0.913158i	$-0.366364\pi$
$139$	9.61121	0.815212	0.407606	+	0.913158i	$0.366364\pi$
$140$	0	0
$141$	7.28822	0.613778
$142$	0	0
$143$	−2.80560	−0.234616
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−0.871407	−0.0718725
$148$	0	0
$149$	4.25719	0.348762	0.174381	−	0.984678i	$-0.444208\pi$
$149$	4.25719	0.348762	0.174381	+	0.984678i	$0.444208\pi$
$150$	0	0
$151$	19.1248	1.55636	0.778179	−	0.628042i	$-0.216144\pi$
$151$	19.1248	1.55636	0.778179	+	0.628042i	$0.216144\pi$
$152$	0	0
$153$	1.48261	0.119862
$154$	0	0
$155$	0	0
$156$	0	0
$157$	6.44784	0.514594	0.257297	−	0.966332i	$-0.417168\pi$
$157$	6.44784	0.514594	0.257297	+	0.966332i	$0.417168\pi$
$158$	0	0
$159$	7.96523	0.631683
$160$	0	0
$161$	−19.5417	−1.54010
$162$	0	0
$163$	13.6112	1.06611	0.533056	−	0.846080i	$-0.321044\pi$
$163$	13.6112	1.06611	0.533056	+	0.846080i	$0.321044\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	7.44784	0.576331	0.288166	−	0.957581i	$-0.406955\pi$
$167$	7.44784	0.576331	0.288166	+	0.957581i	$0.406955\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−2.48261	−0.189850
$172$	0	0
$173$	24.1529	1.83631	0.918154	−	0.396224i	$-0.129679\pi$
$173$	24.1529	1.83631	0.918154	+	0.396224i	$0.129679\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−11.7708	−0.884750
$178$	0	0
$179$	−1.35402	−0.101204	−0.0506021	−	0.998719i	$-0.516114\pi$
$179$	−1.35402	−0.101204	−0.0506021	+	0.998719i	$0.516114\pi$
$180$	0	0
$181$	−12.7050	−0.944357	−0.472179	−	0.881503i	$-0.656532\pi$
$181$	−12.7050	−0.944357	−0.472179	+	0.881503i	$0.656532\pi$
$182$	0	0
$183$	12.4478	0.920171
$184$	0	0
$185$	0	0
$186$	0	0
$187$	4.15962	0.304182
$188$	0	0
$189$	2.80560	0.204078
$190$	0	0
$191$	4.64598	0.336171	0.168086	−	0.985772i	$-0.446241\pi$
$191$	4.64598	0.336171	0.168086	+	0.985772i	$0.446241\pi$
$192$	0	0
$193$	−4.57643	−0.329419	−0.164709	−	0.986342i	$-0.552669\pi$
$193$	−4.57643	−0.329419	−0.164709	+	0.986342i	$0.552669\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−13.3540	−0.951435	−0.475717	−	0.879598i	$-0.657811\pi$
$197$	−13.3540	−0.951435	−0.475717	+	0.879598i	$0.657811\pi$
$198$	0	0
$199$	−10.6703	−0.756394	−0.378197	−	0.925725i	$-0.623456\pi$
$199$	−10.6703	−0.756394	−0.378197	+	0.925725i	$0.623456\pi$
$200$	0	0
$201$	−1.67701	−0.118287
$202$	0	0
$203$	−2.80560	−0.196915
$204$	0	0
$205$	0	0
$206$	0	0
$207$	6.96523	0.484117
$208$	0	0
$209$	−6.96523	−0.481795
$210$	0	0
$211$	−2.96523	−0.204135	−0.102067	−	0.994777i	$-0.532546\pi$
$211$	−2.96523	−0.204135	−0.102067	+	0.994777i	$0.532546\pi$
$212$	0	0
$213$	−6.48261	−0.444181
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−7.87141	−0.534346
$218$	0	0
$219$	1.61121	0.108875
$220$	0	0
$221$	−1.48261	−0.0997313
$222$	0	0
$223$	26.5764	1.77969	0.889845	−	0.456263i	$-0.150813\pi$
$223$	26.5764	1.77969	0.889845	+	0.456263i	$0.150813\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	25.3820	1.68466	0.842332	−	0.538958i	$-0.181182\pi$
$227$	25.3820	1.68466	0.842332	+	0.538958i	$0.181182\pi$
$228$	0	0
$229$	20.8019	1.37463	0.687313	−	0.726362i	$-0.258790\pi$
$229$	20.8019	1.37463	0.687313	+	0.726362i	$0.258790\pi$
$230$	0	0
$231$	7.87141	0.517901
$232$	0	0
$233$	8.25719	0.540946	0.270473	−	0.962728i	$-0.412820\pi$
$233$	8.25719	0.540946	0.270473	+	0.962728i	$0.412820\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−2.87141	−0.186518
$238$	0	0
$239$	−7.06279	−0.456854	−0.228427	−	0.973561i	$-0.573358\pi$
$239$	−7.06279	−0.456854	−0.228427	+	0.973561i	$0.573358\pi$
$240$	0	0
$241$	22.5069	1.44980	0.724898	−	0.688856i	$-0.241887\pi$
$241$	22.5069	1.44980	0.724898	+	0.688856i	$0.241887\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.48261	0.157965
$248$	0	0
$249$	7.77083	0.492456
$250$	0	0
$251$	−8.67025	−0.547261	−0.273631	−	0.961835i	$-0.588225\pi$
$251$	−8.67025	−0.547261	−0.273631	+	0.961835i	$0.588225\pi$
$252$	0	0
$253$	19.5417	1.22857
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−24.4478	−1.52501	−0.762507	−	0.646980i	$-0.776032\pi$
$257$	−24.4478	−1.52501	−0.762507	+	0.646980i	$0.776032\pi$
$258$	0	0
$259$	18.1876	1.13012
$260$	0	0
$261$	1.00000	0.0618984
$262$	0	0
$263$	−14.3192	−0.882963	−0.441481	−	0.897270i	$-0.645547\pi$
$263$	−14.3192	−0.882963	−0.441481	+	0.897270i	$0.645547\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−8.96523	−0.548663
$268$	0	0
$269$	20.2224	1.23298	0.616491	−	0.787362i	$-0.288554\pi$
$269$	20.2224	1.23298	0.616491	+	0.787362i	$0.288554\pi$
$270$	0	0
$271$	9.19440	0.558520	0.279260	−	0.960216i	$-0.409911\pi$
$271$	9.19440	0.558520	0.279260	+	0.960216i	$0.409911\pi$
$272$	0	0
$273$	−2.80560	−0.169803
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−21.2224	−1.27513	−0.637566	−	0.770396i	$-0.720058\pi$
$277$	−21.2224	−1.27513	−0.637566	+	0.770396i	$0.720058\pi$
$278$	0	0
$279$	2.80560	0.167967
$280$	0	0
$281$	13.1286	0.783186	0.391593	−	0.920138i	$-0.371924\pi$
$281$	13.1286	0.783186	0.391593	+	0.920138i	$0.371924\pi$
$282$	0	0
$283$	24.8336	1.47621	0.738103	−	0.674688i	$-0.235722\pi$
$283$	24.8336	1.47621	0.738103	+	0.674688i	$0.235722\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−12.5764	−0.742363
$288$	0	0
$289$	−14.8019	−0.870698
$290$	0	0
$291$	10.9652	0.642793
$292$	0	0
$293$	−4.25719	−0.248707	−0.124354	−	0.992238i	$-0.539686\pi$
$293$	−4.25719	−0.248707	−0.124354	+	0.992238i	$0.539686\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−2.80560	−0.162798
$298$	0	0
$299$	−6.96523	−0.402809
$300$	0	0
$301$	−5.61121	−0.323425
$302$	0	0
$303$	−8.83663	−0.507651
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−1.19065	−0.0679542	−0.0339771	−	0.999423i	$-0.510817\pi$
$307$	−1.19065	−0.0679542	−0.0339771	+	0.999423i	$0.510817\pi$
$308$	0	0
$309$	−2.64598	−0.150525
$310$	0	0
$311$	12.6460	0.717088	0.358544	−	0.933513i	$-0.383273\pi$
$311$	12.6460	0.717088	0.358544	+	0.933513i	$0.383273\pi$
$312$	0	0
$313$	15.0000	0.847850	0.423925	−	0.905697i	$-0.360652\pi$
$313$	15.0000	0.847850	0.423925	+	0.905697i	$0.360652\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−20.8336	−1.17013	−0.585066	−	0.810985i	$-0.698932\pi$
$317$	−20.8336	−1.17013	−0.585066	+	0.810985i	$0.698932\pi$
$318$	0	0
$319$	2.80560	0.157084
$320$	0	0
$321$	2.48261	0.138566
$322$	0	0
$323$	−3.68075	−0.204803
$324$	0	0
$325$	0	0
$326$	0	0
$327$	11.7050	0.647289
$328$	0	0
$329$	20.4478	1.12733
$330$	0	0
$331$	−1.61121	−0.0885599	−0.0442799	−	0.999019i	$-0.514099\pi$
$331$	−1.61121	−0.0885599	−0.0442799	+	0.999019i	$0.514099\pi$
$332$	0	0
$333$	−6.48261	−0.355245
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−0.198141	−0.0107934	−0.00539672	−	0.999985i	$-0.501718\pi$
$337$	−0.198141	−0.0107934	−0.00539672	+	0.999985i	$0.501718\pi$
$338$	0	0
$339$	−6.00000	−0.325875
$340$	0	0
$341$	7.87141	0.426261
$342$	0	0
$343$	17.1944	0.928410
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7.77457	0.417361	0.208680	−	0.977984i	$-0.433083\pi$
$347$	7.77457	0.417361	0.208680	+	0.977984i	$0.433083\pi$
$348$	0	0
$349$	0.645980	0.0345785	0.0172893	−	0.999851i	$-0.494496\pi$
$349$	0.645980	0.0345785	0.0172893	+	0.999851i	$0.494496\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	22.7398	1.21032	0.605159	−	0.796105i	$-0.293110\pi$
$353$	22.7398	1.21032	0.605159	+	0.796105i	$0.293110\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	4.15962	0.220151
$358$	0	0
$359$	−14.5106	−0.765842	−0.382921	−	0.923781i	$-0.625082\pi$
$359$	−14.5106	−0.765842	−0.382921	+	0.923781i	$0.625082\pi$
$360$	0	0
$361$	−12.8366	−0.675612
$362$	0	0
$363$	3.12859	0.164209
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−9.05904	−0.472878	−0.236439	−	0.971646i	$-0.575980\pi$
$367$	−9.05904	−0.472878	−0.236439	+	0.971646i	$0.575980\pi$
$368$	0	0
$369$	4.48261	0.233356
$370$	0	0
$371$	22.3473	1.16021
$372$	0	0
$373$	−1.41306	−0.0731657	−0.0365829	−	0.999331i	$-0.511647\pi$
$373$	−1.41306	−0.0731657	−0.0365829	+	0.999331i	$0.511647\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1.00000	−0.0515026
$378$	0	0
$379$	32.9932	1.69475	0.847374	−	0.530996i	$-0.178182\pi$
$379$	32.9932	1.69475	0.847374	+	0.530996i	$0.178182\pi$
$380$	0	0
$381$	10.0938	0.517122
$382$	0	0
$383$	0.225427	0.0115188	0.00575940	−	0.999983i	$-0.498167\pi$
$383$	0.225427	0.0115188	0.00575940	+	0.999983i	$0.498167\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2.00000	0.101666
$388$	0	0
$389$	18.2815	0.926907	0.463453	−	0.886121i	$-0.346610\pi$
$389$	18.2815	0.926907	0.463453	+	0.886121i	$0.346610\pi$
$390$	0	0
$391$	10.3267	0.522245
$392$	0	0
$393$	−3.51739	−0.177429
$394$	0	0
$395$	0	0
$396$	0	0
$397$	2.09382	0.105086	0.0525429	−	0.998619i	$-0.483267\pi$
$397$	2.09382	0.105086	0.0525429	+	0.998619i	$0.483267\pi$
$398$	0	0
$399$	−6.96523	−0.348697
$400$	0	0
$401$	4.64598	0.232009	0.116005	−	0.993249i	$-0.462991\pi$
$401$	4.64598	0.232009	0.116005	+	0.993249i	$0.462991\pi$
$402$	0	0
$403$	−2.80560	−0.139757
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−18.1876	−0.901528
$408$	0	0
$409$	32.5764	1.61080	0.805400	−	0.592731i	$-0.201950\pi$
$409$	32.5764	1.61080	0.805400	+	0.592731i	$0.201950\pi$
$410$	0	0
$411$	−5.05904	−0.249544
$412$	0	0
$413$	−33.0243	−1.62502
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−9.61121	−0.470663
$418$	0	0
$419$	−14.7398	−0.720086	−0.360043	−	0.932936i	$-0.617238\pi$
$419$	−14.7398	−0.720086	−0.360043	+	0.932936i	$0.617238\pi$
$420$	0	0
$421$	−7.03477	−0.342854	−0.171427	−	0.985197i	$-0.554838\pi$
$421$	−7.03477	−0.342854	−0.171427	+	0.985197i	$0.554838\pi$
$422$	0	0
$423$	−7.28822	−0.354365
$424$	0	0
$425$	0	0
$426$	0	0
$427$	34.9237	1.69008
$428$	0	0
$429$	2.80560	0.135456
$430$	0	0
$431$	33.3783	1.60778	0.803888	−	0.594781i	$-0.202761\pi$
$431$	33.3783	1.60778	0.803888	+	0.594781i	$0.202761\pi$
$432$	0	0
$433$	19.3858	0.931621	0.465811	−	0.884884i	$-0.345763\pi$
$433$	19.3858	0.931621	0.465811	+	0.884884i	$0.345763\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−17.2920	−0.827187
$438$	0	0
$439$	7.83663	0.374022	0.187011	−	0.982358i	$-0.440120\pi$
$439$	7.83663	0.374022	0.187011	+	0.982358i	$0.440120\pi$
$440$	0	0
$441$	0.871407	0.0414956
$442$	0	0
$443$	−18.2815	−0.868578	−0.434289	−	0.900774i	$-0.643000\pi$
$443$	−18.2815	−0.868578	−0.434289	+	0.900774i	$0.643000\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−4.25719	−0.201358
$448$	0	0
$449$	−5.05904	−0.238751	−0.119376	−	0.992849i	$-0.538089\pi$
$449$	−5.05904	−0.238751	−0.119376	+	0.992849i	$0.538089\pi$
$450$	0	0
$451$	12.5764	0.592201
$452$	0	0
$453$	−19.1248	−0.898564
$454$	0	0
$455$	0	0
$456$	0	0
$457$	17.3540	0.811787	0.405893	−	0.913920i	$-0.366960\pi$
$457$	17.3540	0.811787	0.405893	+	0.913920i	$0.366960\pi$
$458$	0	0
$459$	−1.48261	−0.0692024
$460$	0	0
$461$	27.4796	1.27985	0.639926	−	0.768436i	$-0.278965\pi$
$461$	27.4796	1.27985	0.639926	+	0.768436i	$0.278965\pi$
$462$	0	0
$463$	30.1596	1.40164	0.700818	−	0.713340i	$-0.252818\pi$
$463$	30.1596	1.40164	0.700818	+	0.713340i	$0.252818\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	24.0938	1.11493	0.557464	−	0.830201i	$-0.311774\pi$
$467$	24.0938	1.11493	0.557464	+	0.830201i	$0.311774\pi$
$468$	0	0
$469$	−4.70502	−0.217258
$470$	0	0
$471$	−6.44784	−0.297101
$472$	0	0
$473$	5.61121	0.258004
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−7.96523	−0.364703
$478$	0	0
$479$	−18.3230	−0.837199	−0.418599	−	0.908171i	$-0.637479\pi$
$479$	−18.3230	−0.837199	−0.418599	+	0.908171i	$0.637479\pi$
$480$	0	0
$481$	6.48261	0.295582
$482$	0	0
$483$	19.5417	0.889176
$484$	0	0
$485$	0	0
$486$	0	0
$487$	25.0628	1.13570	0.567852	−	0.823131i	$-0.307775\pi$
$487$	25.0628	1.13570	0.567852	+	0.823131i	$0.307775\pi$
$488$	0	0
$489$	−13.6112	−0.615520
$490$	0	0
$491$	27.5417	1.24294	0.621469	−	0.783439i	$-0.286536\pi$
$491$	27.5417	1.24294	0.621469	+	0.783439i	$0.286536\pi$
$492$	0	0
$493$	1.48261	0.0667735
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−18.1876	−0.815827
$498$	0	0
$499$	−29.7330	−1.33103	−0.665517	−	0.746383i	$-0.731789\pi$
$499$	−29.7330	−1.33103	−0.665517	+	0.746383i	$0.731789\pi$
$500$	0	0
$501$	−7.44784	−0.332745
$502$	0	0
$503$	−1.86839	−0.0833074	−0.0416537	−	0.999132i	$-0.513263\pi$
$503$	−1.86839	−0.0833074	−0.0416537	+	0.999132i	$0.513263\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	−15.7988	−0.700271	−0.350136	−	0.936699i	$-0.613865\pi$
$509$	−15.7988	−0.700271	−0.350136	+	0.936699i	$0.613865\pi$
$510$	0	0
$511$	4.52040	0.199971
$512$	0	0
$513$	2.48261	0.109610
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−20.4478	−0.899295
$518$	0	0
$519$	−24.1529	−1.06019
$520$	0	0
$521$	25.6733	1.12477	0.562383	−	0.826877i	$-0.309885\pi$
$521$	25.6733	1.12477	0.562383	+	0.826877i	$0.309885\pi$
$522$	0	0
$523$	3.35402	0.146661	0.0733305	−	0.997308i	$-0.476637\pi$
$523$	3.35402	0.146661	0.0733305	+	0.997308i	$0.476637\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.15962	0.181196
$528$	0	0
$529$	25.5144	1.10932
$530$	0	0
$531$	11.7708	0.510810
$532$	0	0
$533$	−4.48261	−0.194164
$534$	0	0
$535$	0	0
$536$	0	0
$537$	1.35402	0.0584303
$538$	0	0
$539$	2.44482	0.105306
$540$	0	0
$541$	27.0908	1.16472	0.582362	−	0.812929i	$-0.302129\pi$
$541$	27.0908	1.16472	0.582362	+	0.812929i	$0.302129\pi$
$542$	0	0
$543$	12.7050	0.545225
$544$	0	0
$545$	0	0
$546$	0	0
$547$	4.38879	0.187651	0.0938256	−	0.995589i	$-0.470090\pi$
$547$	4.38879	0.187651	0.0938256	+	0.995589i	$0.470090\pi$
$548$	0	0
$549$	−12.4478	−0.531261
$550$	0	0
$551$	−2.48261	−0.105763
$552$	0	0
$553$	−8.05603	−0.342577
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−18.3192	−0.776211	−0.388106	−	0.921615i	$-0.626870\pi$
$557$	−18.3192	−0.776211	−0.388106	+	0.921615i	$0.626870\pi$
$558$	0	0
$559$	−2.00000	−0.0845910
$560$	0	0
$561$	−4.15962	−0.175619
$562$	0	0
$563$	28.9895	1.22176	0.610881	−	0.791723i	$-0.290815\pi$
$563$	28.9895	1.22176	0.610881	+	0.791723i	$0.290815\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−2.80560	−0.117824
$568$	0	0
$569$	−22.8927	−0.959710	−0.479855	−	0.877348i	$-0.659311\pi$
$569$	−22.8927	−0.959710	−0.479855	+	0.877348i	$0.659311\pi$
$570$	0	0
$571$	28.8957	1.20925	0.604623	−	0.796512i	$-0.293324\pi$
$571$	28.8957	1.20925	0.604623	+	0.796512i	$0.293324\pi$
$572$	0	0
$573$	−4.64598	−0.194089
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−8.89568	−0.370332	−0.185166	−	0.982707i	$-0.559282\pi$
$577$	−8.89568	−0.370332	−0.185166	+	0.982707i	$0.559282\pi$
$578$	0	0
$579$	4.57643	0.190190
$580$	0	0
$581$	21.8019	0.904494
$582$	0	0
$583$	−22.3473	−0.925529
$584$	0	0
$585$	0	0
$586$	0	0
$587$	13.1944	0.544591	0.272296	−	0.962214i	$-0.412217\pi$
$587$	13.1944	0.544591	0.272296	+	0.962214i	$0.412217\pi$
$588$	0	0
$589$	−6.96523	−0.286997
$590$	0	0
$591$	13.3540	0.549311
$592$	0	0
$593$	34.7323	1.42629	0.713143	−	0.701019i	$-0.247271\pi$
$593$	34.7323	1.42629	0.713143	+	0.701019i	$0.247271\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	10.6703	0.436704
$598$	0	0
$599$	−25.8684	−1.05695	−0.528477	−	0.848948i	$-0.677237\pi$
$599$	−25.8684	−1.05695	−0.528477	+	0.848948i	$0.677237\pi$
$600$	0	0
$601$	−21.9652	−0.895980	−0.447990	−	0.894039i	$-0.647860\pi$
$601$	−21.9652	−0.895980	−0.447990	+	0.894039i	$0.647860\pi$
$602$	0	0
$603$	1.67701	0.0682931
$604$	0	0
$605$	0	0
$606$	0	0
$607$	23.4478	0.951718	0.475859	−	0.879521i	$-0.342137\pi$
$607$	23.4478	0.951718	0.475859	+	0.879521i	$0.342137\pi$
$608$	0	0
$609$	2.80560	0.113689
$610$	0	0
$611$	7.28822	0.294850
$612$	0	0
$613$	2.77759	0.112186	0.0560929	−	0.998426i	$-0.482136\pi$
$613$	2.77759	0.112186	0.0560929	+	0.998426i	$0.482136\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6.28146	0.252882	0.126441	−	0.991974i	$-0.459645\pi$
$617$	6.28146	0.252882	0.126441	+	0.991974i	$0.459645\pi$
$618$	0	0
$619$	2.90317	0.116688	0.0583440	−	0.998297i	$-0.481418\pi$
$619$	2.90317	0.116688	0.0583440	+	0.998297i	$0.481418\pi$
$620$	0	0
$621$	−6.96523	−0.279505
$622$	0	0
$623$	−25.1529	−1.00773
$624$	0	0
$625$	0	0
$626$	0	0
$627$	6.96523	0.278164
$628$	0	0
$629$	−9.61121	−0.383224
$630$	0	0
$631$	−44.1801	−1.75878	−0.879392	−	0.476098i	$-0.842051\pi$
$631$	−44.1801	−1.75878	−0.879392	+	0.476098i	$0.842051\pi$
$632$	0	0
$633$	2.96523	0.117857
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−0.871407	−0.0345264
$638$	0	0
$639$	6.48261	0.256448
$640$	0	0
$641$	9.03176	0.356733	0.178366	−	0.983964i	$-0.442919\pi$
$641$	9.03176	0.356733	0.178366	+	0.983964i	$0.442919\pi$
$642$	0	0
$643$	−35.8231	−1.41273	−0.706363	−	0.707850i	$-0.749665\pi$
$643$	−35.8231	−1.41273	−0.706363	+	0.707850i	$0.749665\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	31.8609	1.25258	0.626291	−	0.779590i	$-0.284572\pi$
$647$	31.8609	1.25258	0.626291	+	0.779590i	$0.284572\pi$
$648$	0	0
$649$	33.0243	1.29632
$650$	0	0
$651$	7.87141	0.308505
$652$	0	0
$653$	−18.9062	−0.739856	−0.369928	−	0.929060i	$-0.620618\pi$
$653$	−18.9062	−0.739856	−0.369928	+	0.929060i	$0.620618\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−1.61121	−0.0628591
$658$	0	0
$659$	15.0590	0.586617	0.293309	−	0.956018i	$-0.405244\pi$
$659$	15.0590	0.586617	0.293309	+	0.956018i	$0.405244\pi$
$660$	0	0
$661$	6.73980	0.262148	0.131074	−	0.991373i	$-0.458157\pi$
$661$	6.73980	0.262148	0.131074	+	0.991373i	$0.458157\pi$
$662$	0	0
$663$	1.48261	0.0575799
$664$	0	0
$665$	0	0
$666$	0	0
$667$	6.96523	0.269695
$668$	0	0
$669$	−26.5764	−1.02750
$670$	0	0
$671$	−34.9237	−1.34821
$672$	0	0
$673$	−6.22241	−0.239856	−0.119928	−	0.992783i	$-0.538266\pi$
$673$	−6.22241	−0.239856	−0.119928	+	0.992783i	$0.538266\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−35.1529	−1.35103	−0.675517	−	0.737344i	$-0.736080\pi$
$677$	−35.1529	−1.35103	−0.675517	+	0.737344i	$0.736080\pi$
$678$	0	0
$679$	30.7641	1.18062
$680$	0	0
$681$	−25.3820	−0.972642
$682$	0	0
$683$	−26.3473	−1.00815	−0.504075	−	0.863660i	$-0.668166\pi$
$683$	−26.3473	−1.00815	−0.504075	+	0.863660i	$0.668166\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−20.8019	−0.793640
$688$	0	0
$689$	7.96523	0.303451
$690$	0	0
$691$	−27.4759	−1.04523	−0.522615	−	0.852569i	$-0.675044\pi$
$691$	−27.4759	−1.04523	−0.522615	+	0.852569i	$0.675044\pi$
$692$	0	0
$693$	−7.87141	−0.299010
$694$	0	0
$695$	0	0
$696$	0	0
$697$	6.64598	0.251734
$698$	0	0
$699$	−8.25719	−0.312315
$700$	0	0
$701$	3.68377	0.139134	0.0695670	−	0.997577i	$-0.477838\pi$
$701$	3.68377	0.139134	0.0695670	+	0.997577i	$0.477838\pi$
$702$	0	0
$703$	16.0938	0.606990
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−24.7921	−0.932402
$708$	0	0
$709$	7.09683	0.266527	0.133264	−	0.991081i	$-0.457454\pi$
$709$	7.09683	0.266527	0.133264	+	0.991081i	$0.457454\pi$
$710$	0	0
$711$	2.87141	0.107686
$712$	0	0
$713$	19.5417	0.731841
$714$	0	0
$715$	0	0
$716$	0	0
$717$	7.06279	0.263765
$718$	0	0
$719$	−29.0908	−1.08490	−0.542452	−	0.840087i	$-0.682504\pi$
$719$	−29.0908	−1.08490	−0.542452	+	0.840087i	$0.682504\pi$
$720$	0	0
$721$	−7.42357	−0.276468
$722$	0	0
$723$	−22.5069	−0.837040
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−51.6597	−1.91595	−0.957977	−	0.286845i	$-0.907393\pi$
$727$	−51.6597	−1.91595	−0.957977	+	0.286845i	$0.907393\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	2.96523	0.109673
$732$	0	0
$733$	45.4343	1.67815	0.839077	−	0.544012i	$-0.183095\pi$
$733$	45.4343	1.67815	0.839077	+	0.544012i	$0.183095\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.70502	0.173312
$738$	0	0
$739$	6.51063	0.239497	0.119749	−	0.992804i	$-0.461791\pi$
$739$	6.51063	0.239497	0.119749	+	0.992804i	$0.461791\pi$
$740$	0	0
$741$	−2.48261	−0.0912011
$742$	0	0
$743$	−7.09309	−0.260220	−0.130110	−	0.991500i	$-0.541533\pi$
$743$	−7.09309	−0.260220	−0.130110	+	0.991500i	$0.541533\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−7.77083	−0.284320
$748$	0	0
$749$	6.96523	0.254504
$750$	0	0
$751$	−7.24668	−0.264435	−0.132218	−	0.991221i	$-0.542210\pi$
$751$	−7.24668	−0.264435	−0.132218	+	0.991221i	$0.542210\pi$
$752$	0	0
$753$	8.67025	0.315961
$754$	0	0
$755$	0	0
$756$	0	0
$757$	44.5659	1.61978	0.809888	−	0.586584i	$-0.199528\pi$
$757$	44.5659	1.61978	0.809888	+	0.586584i	$0.199528\pi$
$758$	0	0
$759$	−19.5417	−0.709317
$760$	0	0
$761$	1.51739	0.0550052	0.0275026	−	0.999622i	$-0.491245\pi$
$761$	1.51739	0.0550052	0.0275026	+	0.999622i	$0.491245\pi$
$762$	0	0
$763$	32.8396	1.18888
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−11.7708	−0.425020
$768$	0	0
$769$	49.6672	1.79105	0.895523	−	0.445015i	$-0.146802\pi$
$769$	49.6672	1.79105	0.895523	+	0.445015i	$0.146802\pi$
$770$	0	0
$771$	24.4478	0.880467
$772$	0	0
$773$	−52.6325	−1.89306	−0.946529	−	0.322619i	$-0.895437\pi$
$773$	−52.6325	−1.89306	−0.946529	+	0.322619i	$0.895437\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−18.1876	−0.652478
$778$	0	0
$779$	−11.1286	−0.398723
$780$	0	0
$781$	18.1876	0.650805
$782$	0	0
$783$	−1.00000	−0.0357371
$784$	0	0
$785$	0	0
$786$	0	0
$787$	17.9660	0.640417	0.320209	−	0.947347i	$-0.396247\pi$
$787$	17.9660	0.640417	0.320209	+	0.947347i	$0.396247\pi$
$788$	0	0
$789$	14.3192	0.509779
$790$	0	0
$791$	−16.8336	−0.598535
$792$	0	0
$793$	12.4478	0.442036
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−3.94096	−0.139596	−0.0697979	−	0.997561i	$-0.522235\pi$
$797$	−3.94096	−0.139596	−0.0697979	+	0.997561i	$0.522235\pi$
$798$	0	0
$799$	−10.8056	−0.382275
$800$	0	0
$801$	8.96523	0.316771
$802$	0	0
$803$	−4.52040	−0.159522
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−20.2224	−0.711863
$808$	0	0
$809$	−39.6672	−1.39463	−0.697313	−	0.716767i	$-0.745621\pi$
$809$	−39.6672	−1.39463	−0.697313	+	0.716767i	$0.745621\pi$
$810$	0	0
$811$	14.9237	0.524042	0.262021	−	0.965062i	$-0.415611\pi$
$811$	14.9237	0.524042	0.262021	+	0.965062i	$0.415611\pi$
$812$	0	0
$813$	−9.19440	−0.322462
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−4.96523	−0.173711
$818$	0	0
$819$	2.80560	0.0980357
$820$	0	0
$821$	−25.7988	−0.900386	−0.450193	−	0.892931i	$-0.648645\pi$
$821$	−25.7988	−0.900386	−0.450193	+	0.892931i	$0.648645\pi$
$822$	0	0
$823$	19.9062	0.693886	0.346943	−	0.937886i	$-0.387220\pi$
$823$	19.9062	0.693886	0.346943	+	0.937886i	$0.387220\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	22.6665	0.788192	0.394096	−	0.919069i	$-0.371058\pi$
$827$	22.6665	0.788192	0.394096	+	0.919069i	$0.371058\pi$
$828$	0	0
$829$	−21.6703	−0.752639	−0.376319	−	0.926490i	$-0.622810\pi$
$829$	−21.6703	−0.752639	−0.376319	+	0.926490i	$0.622810\pi$
$830$	0	0
$831$	21.2224	0.736197
$832$	0	0
$833$	1.29196	0.0447637
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−2.80560	−0.0969758
$838$	0	0
$839$	20.0560	0.692411	0.346205	−	0.938159i	$-0.387470\pi$
$839$	20.0560	0.692411	0.346205	+	0.938159i	$0.387470\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	0	0
$843$	−13.1286	−0.452173
$844$	0	0
$845$	0	0
$846$	0	0
$847$	8.77759	0.301602
$848$	0	0
$849$	−24.8336	−0.852288
$850$	0	0
$851$	−45.1529	−1.54782
$852$	0	0
$853$	−30.9895	−1.06106	−0.530530	−	0.847666i	$-0.678007\pi$
$853$	−30.9895	−1.06106	−0.530530	+	0.847666i	$0.678007\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2.51437	0.0858893	0.0429446	−	0.999077i	$-0.486326\pi$
$857$	2.51437	0.0858893	0.0429446	+	0.999077i	$0.486326\pi$
$858$	0	0
$859$	−33.0833	−1.12879	−0.564394	−	0.825506i	$-0.690890\pi$
$859$	−33.0833	−1.12879	−0.564394	+	0.825506i	$0.690890\pi$
$860$	0	0
$861$	12.5764	0.428604
$862$	0	0
$863$	32.2534	1.09792	0.548960	−	0.835849i	$-0.315024\pi$
$863$	32.2534	1.09792	0.548960	+	0.835849i	$0.315024\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	14.8019	0.502697
$868$	0	0
$869$	8.05603	0.273282
$870$	0	0
$871$	−1.67701	−0.0568233
$872$	0	0
$873$	−10.9652	−0.371117
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−9.05904	−0.305902	−0.152951	−	0.988234i	$-0.548878\pi$
$877$	−9.05904	−0.305902	−0.152951	+	0.988234i	$0.548878\pi$
$878$	0	0
$879$	4.25719	0.143591
$880$	0	0
$881$	−16.4478	−0.554142	−0.277071	−	0.960849i	$-0.589364\pi$
$881$	−16.4478	−0.554142	−0.277071	+	0.960849i	$0.589364\pi$
$882$	0	0
$883$	38.5689	1.29795	0.648974	−	0.760810i	$-0.275198\pi$
$883$	38.5689	1.29795	0.648974	+	0.760810i	$0.275198\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.18764	0.0734537	0.0367268	−	0.999325i	$-0.488307\pi$
$887$	2.18764	0.0734537	0.0367268	+	0.999325i	$0.488307\pi$
$888$	0	0
$889$	28.3192	0.949797
$890$	0	0
$891$	2.80560	0.0939912
$892$	0	0
$893$	18.0938	0.605487
$894$	0	0
$895$	0	0
$896$	0	0
$897$	6.96523	0.232562
$898$	0	0
$899$	2.80560	0.0935721
$900$	0	0
$901$	−11.8093	−0.393426
$902$	0	0
$903$	5.61121	0.186729
$904$	0	0
$905$	0	0
$906$	0	0
$907$	11.6733	0.387604	0.193802	−	0.981041i	$-0.437918\pi$
$907$	11.6733	0.387604	0.193802	+	0.981041i	$0.437918\pi$
$908$	0	0
$909$	8.83663	0.293093
$910$	0	0
$911$	50.3753	1.66901	0.834504	−	0.551002i	$-0.185754\pi$
$911$	50.3753	1.66901	0.834504	+	0.551002i	$0.185754\pi$
$912$	0	0
$913$	−21.8019	−0.721536
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−9.86839	−0.325883
$918$	0	0
$919$	28.4131	0.937261	0.468630	−	0.883394i	$-0.344748\pi$
$919$	28.4131	0.937261	0.468630	+	0.883394i	$0.344748\pi$
$920$	0	0
$921$	1.19065	0.0392334
$922$	0	0
$923$	−6.48261	−0.213378
$924$	0	0
$925$	0	0
$926$	0	0
$927$	2.64598	0.0869054
$928$	0	0
$929$	29.3162	0.961834	0.480917	−	0.876766i	$-0.340304\pi$
$929$	29.3162	0.961834	0.480917	+	0.876766i	$0.340304\pi$
$930$	0	0
$931$	−2.16337	−0.0709015
$932$	0	0
$933$	−12.6460	−0.414011
$934$	0	0
$935$	0	0
$936$	0	0
$937$	42.5039	1.38854	0.694270	−	0.719714i	$-0.255727\pi$
$937$	42.5039	1.38854	0.694270	+	0.719714i	$0.255727\pi$
$938$	0	0
$939$	−15.0000	−0.489506
$940$	0	0
$941$	−3.68075	−0.119989	−0.0599946	−	0.998199i	$-0.519108\pi$
$941$	−3.68075	−0.119989	−0.0599946	+	0.998199i	$0.519108\pi$
$942$	0	0
$943$	31.2224	1.01674
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−2.34726	−0.0762757	−0.0381379	−	0.999272i	$-0.512143\pi$
$947$	−2.34726	−0.0762757	−0.0381379	+	0.999272i	$0.512143\pi$
$948$	0	0
$949$	1.61121	0.0523019
$950$	0	0
$951$	20.8336	0.675576
$952$	0	0
$953$	−15.1559	−0.490947	−0.245474	−	0.969403i	$-0.578943\pi$
$953$	−15.1559	−0.490947	−0.245474	+	0.969403i	$0.578943\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−2.80560	−0.0906923
$958$	0	0
$959$	−14.1937	−0.458337
$960$	0	0
$961$	−23.1286	−0.746084
$962$	0	0
$963$	−2.48261	−0.0800011
$964$	0	0
$965$	0	0
$966$	0	0
$967$	41.8889	1.34706	0.673528	−	0.739161i	$-0.264778\pi$
$967$	41.8889	1.34706	0.673528	+	0.739161i	$0.264778\pi$
$968$	0	0
$969$	3.68075	0.118243
$970$	0	0
$971$	−51.1211	−1.64055	−0.820277	−	0.571966i	$-0.806181\pi$
$971$	−51.1211	−1.64055	−0.820277	+	0.571966i	$0.806181\pi$
$972$	0	0
$973$	−26.9652	−0.864465
$974$	0	0
$975$	0	0
$976$	0	0
$977$	26.1181	0.835592	0.417796	−	0.908541i	$-0.362803\pi$
$977$	26.1181	0.835592	0.417796	+	0.908541i	$0.362803\pi$
$978$	0	0
$979$	25.1529	0.803889
$980$	0	0
$981$	−11.7050	−0.373713
$982$	0	0
$983$	29.5076	0.941147	0.470573	−	0.882361i	$-0.344047\pi$
$983$	29.5076	0.941147	0.470573	+	0.882361i	$0.344047\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−20.4478	−0.650862
$988$	0	0
$989$	13.9305	0.442963
$990$	0	0
$991$	34.9199	1.10927	0.554634	−	0.832094i	$-0.312858\pi$
$991$	34.9199	1.10927	0.554634	+	0.832094i	$0.312858\pi$
$992$	0	0
$993$	1.61121	0.0511301
$994$	0	0
$995$	0	0
$996$	0	0
$997$	13.6703	0.432941	0.216471	−	0.976289i	$-0.430545\pi$
$997$	13.6703	0.432941	0.216471	+	0.976289i	$0.430545\pi$
$998$	0	0
$999$	6.48261	0.205101

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7800.2.a.bk.1.1	✓	3
5.4	even	2		7800.2.a.bq.1.3	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7800.2.a.bk.1.1	✓	3	1.1	even	1	trivial
7800.2.a.bq.1.3	yes	3	5.4	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:	\( N \)	\(=\)	\( 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7800.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -2.80560 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -2.80560 q^{7} +1.00000 q^{9} +2.80560 q^{11} -1.00000 q^{13} +1.48261 q^{17} -2.48261 q^{19} +2.80560 q^{21} +6.96523 q^{23} -1.00000 q^{27} +1.00000 q^{29} +2.80560 q^{31} -2.80560 q^{33} -6.48261 q^{37} +1.00000 q^{39} +4.48261 q^{41} +2.00000 q^{43} -7.28822 q^{47} +0.871407 q^{49} -1.48261 q^{51} -7.96523 q^{53} +2.48261 q^{57} +11.7708 q^{59} -12.4478 q^{61} -2.80560 q^{63} +1.67701 q^{67} -6.96523 q^{69} +6.48261 q^{71} -1.61121 q^{73} -7.87141 q^{77} +2.87141 q^{79} +1.00000 q^{81} -7.77083 q^{83} -1.00000 q^{87} +8.96523 q^{89} +2.80560 q^{91} -2.80560 q^{93} -10.9652 q^{97} +2.80560 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + 3 * q^9	\( 3 q - 3 q^{3} + 3 q^{9} - 3 q^{13} - 6 q^{17} + 3 q^{19} - 3 q^{27} + 3 q^{29} - 9 q^{37} + 3 q^{39} + 3 q^{41} + 6 q^{43} - 3 q^{47} + 9 q^{49} + 6 q^{51} - 3 q^{53} - 3 q^{57} + 6 q^{59} - 6 q^{61} + 3 q^{67} + 9 q^{71} + 12 q^{73} - 30 q^{77} + 15 q^{79} + 3 q^{81} + 6 q^{83} - 3 q^{87} + 6 q^{89} - 12 q^{97}+O(q^{100}) \) 3 * q - 3 * q^3 + 3 * q^9 - 3 * q^13 - 6 * q^17 + 3 * q^19 - 3 * q^27 + 3 * q^29 - 9 * q^37 + 3 * q^39 + 3 * q^41 + 6 * q^43 - 3 * q^47 + 9 * q^49 + 6 * q^51 - 3 * q^53 - 3 * q^57 + 6 * q^59 - 6 * q^61 + 3 * q^67 + 9 * q^71 + 12 * q^73 - 30 * q^77 + 15 * q^79 + 3 * q^81 + 6 * q^83 - 3 * q^87 + 6 * q^89 - 12 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more