LMFDB - Embedded newform 9405.2.a.bm.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9405,2,Mod(1,9405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9405, 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("9405.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9405 = 3^{2} \cdot 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9405.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:	$75.0993031010$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 4x^{9} - 10x^{8} + 55x^{7} + 5x^{6} - 232x^{5} + 166x^{4} + 276x^{3} - 337x^{2} + 63x + 9 $ x^10 - 4x^9 - 10x^8 + 55x^7 + 5x^6 - 232x^5 + 166x^4 + 276x^3 - 337x^2 + 63*x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 3135)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.91754$ of defining polynomial
Character	$\chi$	$=$	9405.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.91754 q^{2} +1.67697 q^{4} +1.00000 q^{5} +4.35488 q^{7} +0.619418 q^{8} +O(q^{10})$	\(q-1.91754 q^{2} +1.67697 q^{4} +1.00000 q^{5} +4.35488 q^{7} +0.619418 q^{8} -1.91754 q^{10} -1.00000 q^{11} +4.48588 q^{13} -8.35066 q^{14} -4.54171 q^{16} -2.68689 q^{17} +1.00000 q^{19} +1.67697 q^{20} +1.91754 q^{22} +2.22008 q^{23} +1.00000 q^{25} -8.60187 q^{26} +7.30301 q^{28} +8.34690 q^{29} +4.48588 q^{31} +7.47009 q^{32} +5.15222 q^{34} +4.35488 q^{35} -10.6488 q^{37} -1.91754 q^{38} +0.619418 q^{40} -0.241461 q^{41} +12.4091 q^{43} -1.67697 q^{44} -4.25709 q^{46} -6.41516 q^{47} +11.9649 q^{49} -1.91754 q^{50} +7.52270 q^{52} -4.35482 q^{53} -1.00000 q^{55} +2.69749 q^{56} -16.0055 q^{58} -7.32266 q^{59} -7.45506 q^{61} -8.60187 q^{62} -5.24080 q^{64} +4.48588 q^{65} -12.5140 q^{67} -4.50584 q^{68} -8.35066 q^{70} +10.3984 q^{71} -2.70595 q^{73} +20.4195 q^{74} +1.67697 q^{76} -4.35488 q^{77} +8.67975 q^{79} -4.54171 q^{80} +0.463013 q^{82} -1.01594 q^{83} -2.68689 q^{85} -23.7950 q^{86} -0.619418 q^{88} -11.1078 q^{89} +19.5354 q^{91} +3.72301 q^{92} +12.3013 q^{94} +1.00000 q^{95} +17.2009 q^{97} -22.9433 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 4 q^{2} + 16 q^{4} + 10 q^{5} + 5 q^{7} - 3 q^{8}+O(q^{10}) $ 10 * q - 4 * q^2 + 16 * q^4 + 10 * q^5 + 5 * q^7 - 3 * q^8	$ 10 q - 4 q^{2} + 16 q^{4} + 10 q^{5} + 5 q^{7} - 3 q^{8} - 4 q^{10} - 10 q^{11} + 4 q^{13} - 6 q^{14} + 20 q^{16} - 10 q^{17} + 10 q^{19} + 16 q^{20} + 4 q^{22} + 6 q^{23} + 10 q^{25} + 3 q^{26} + 34 q^{28} - 5 q^{29} + 4 q^{31} - 30 q^{32} + 5 q^{34} + 5 q^{35} + 13 q^{37} - 4 q^{38} - 3 q^{40} - 9 q^{41} + 40 q^{43} - 16 q^{44} - 12 q^{46} + 24 q^{47} + 29 q^{49} - 4 q^{50} + 13 q^{52} - 13 q^{53} - 10 q^{55} - 8 q^{56} + 27 q^{58} - 5 q^{61} + 3 q^{62} + 27 q^{64} + 4 q^{65} + 39 q^{67} - 16 q^{68} - 6 q^{70} - 11 q^{71} + 30 q^{73} + 30 q^{74} + 16 q^{76} - 5 q^{77} + 4 q^{79} + 20 q^{80} + 24 q^{82} - 19 q^{83} - 10 q^{85} - 12 q^{86} + 3 q^{88} + 15 q^{89} - 17 q^{91} + 19 q^{92} + 2 q^{94} + 10 q^{95} + 58 q^{97} - 30 q^{98}+O(q^{100}) $ 10 * q - 4 * q^2 + 16 * q^4 + 10 * q^5 + 5 * q^7 - 3 * q^8 - 4 * q^10 - 10 * q^11 + 4 * q^13 - 6 * q^14 + 20 * q^16 - 10 * q^17 + 10 * q^19 + 16 * q^20 + 4 * q^22 + 6 * q^23 + 10 * q^25 + 3 * q^26 + 34 * q^28 - 5 * q^29 + 4 * q^31 - 30 * q^32 + 5 * q^34 + 5 * q^35 + 13 * q^37 - 4 * q^38 - 3 * q^40 - 9 * q^41 + 40 * q^43 - 16 * q^44 - 12 * q^46 + 24 * q^47 + 29 * q^49 - 4 * q^50 + 13 * q^52 - 13 * q^53 - 10 * q^55 - 8 * q^56 + 27 * q^58 - 5 * q^61 + 3 * q^62 + 27 * q^64 + 4 * q^65 + 39 * q^67 - 16 * q^68 - 6 * q^70 - 11 * q^71 + 30 * q^73 + 30 * q^74 + 16 * q^76 - 5 * q^77 + 4 * q^79 + 20 * q^80 + 24 * q^82 - 19 * q^83 - 10 * q^85 - 12 * q^86 + 3 * q^88 + 15 * q^89 - 17 * q^91 + 19 * q^92 + 2 * q^94 + 10 * q^95 + 58 * q^97 - 30 * q^98

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 $

$2$	−1.91754	−1.35591	−0.677954	−	0.735104i	$-0.737133\pi$
$2$	−1.91754	−1.35591	−0.677954	+	0.735104i	$0.737133\pi$
$3$	0	0
$3$	0	0
$4$	1.67697	0.838486
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.35488	1.64599	0.822994	−	0.568050i	$-0.192302\pi$
$7$	4.35488	1.64599	0.822994	+	0.568050i	$0.192302\pi$
$8$	0.619418	0.218997
$9$	0	0
$10$	−1.91754	−0.606380
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	4.48588	1.24416	0.622079	−	0.782954i	$-0.286288\pi$
$13$	4.48588	1.24416	0.622079	+	0.782954i	$0.286288\pi$
$14$	−8.35066	−2.23181
$15$	0	0
$16$	−4.54171	−1.13543
$17$	−2.68689	−0.651666	−0.325833	−	0.945427i	$-0.605645\pi$
$17$	−2.68689	−0.651666	−0.325833	+	0.945427i	$0.605645\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	1.67697	0.374983
$21$	0	0
$22$	1.91754	0.408822
$23$	2.22008	0.462918	0.231459	−	0.972845i	$-0.425650\pi$
$23$	2.22008	0.462918	0.231459	+	0.972845i	$0.425650\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−8.60187	−1.68696
$27$	0	0
$28$	7.30301	1.38014
$29$	8.34690	1.54998	0.774990	−	0.631973i	$-0.217755\pi$
$29$	8.34690	1.54998	0.774990	+	0.631973i	$0.217755\pi$
$30$	0	0
$31$	4.48588	0.805687	0.402844	−	0.915269i	$-0.368022\pi$
$31$	4.48588	0.805687	0.402844	+	0.915269i	$0.368022\pi$
$32$	7.47009	1.32054
$33$	0	0
$34$	5.15222	0.883599
$35$	4.35488	0.736108
$36$	0	0
$37$	−10.6488	−1.75065	−0.875326	−	0.483534i	$-0.839353\pi$
$37$	−10.6488	−1.75065	−0.875326	+	0.483534i	$0.839353\pi$
$38$	−1.91754	−0.311067
$39$	0	0
$40$	0.619418	0.0979386
$41$	−0.241461	−0.0377099	−0.0188550	−	0.999822i	$-0.506002\pi$
$41$	−0.241461	−0.0377099	−0.0188550	+	0.999822i	$0.506002\pi$
$42$	0	0
$43$	12.4091	1.89237	0.946186	−	0.323623i	$-0.104901\pi$
$43$	12.4091	1.89237	0.946186	+	0.323623i	$0.104901\pi$
$44$	−1.67697	−0.252813
$45$	0	0
$46$	−4.25709	−0.627674
$47$	−6.41516	−0.935747	−0.467873	−	0.883796i	$-0.654980\pi$
$47$	−6.41516	−0.935747	−0.467873	+	0.883796i	$0.654980\pi$
$48$	0	0
$49$	11.9649	1.70928
$50$	−1.91754	−0.271182
$51$	0	0
$52$	7.52270	1.04321
$53$	−4.35482	−0.598180	−0.299090	−	0.954225i	$-0.596683\pi$
$53$	−4.35482	−0.598180	−0.299090	+	0.954225i	$0.596683\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	2.69749	0.360467
$57$	0	0
$58$	−16.0055	−2.10163
$59$	−7.32266	−0.953330	−0.476665	−	0.879085i	$-0.658154\pi$
$59$	−7.32266	−0.953330	−0.476665	+	0.879085i	$0.658154\pi$
$60$	0	0
$61$	−7.45506	−0.954522	−0.477261	−	0.878762i	$-0.658370\pi$
$61$	−7.45506	−0.954522	−0.477261	+	0.878762i	$0.658370\pi$
$62$	−8.60187	−1.09244
$63$	0	0
$64$	−5.24080	−0.655100
$65$	4.48588	0.556405
$66$	0	0
$67$	−12.5140	−1.52882	−0.764412	−	0.644728i	$-0.776971\pi$
$67$	−12.5140	−1.52882	−0.764412	+	0.644728i	$0.776971\pi$
$68$	−4.50584	−0.546413
$69$	0	0
$70$	−8.35066	−0.998095
$71$	10.3984	1.23407	0.617034	−	0.786936i	$-0.288334\pi$
$71$	10.3984	1.23407	0.617034	+	0.786936i	$0.288334\pi$
$72$	0	0
$73$	−2.70595	−0.316708	−0.158354	−	0.987382i	$-0.550619\pi$
$73$	−2.70595	−0.316708	−0.158354	+	0.987382i	$0.550619\pi$
$74$	20.4195	2.37372
$75$	0	0
$76$	1.67697	0.192362
$77$	−4.35488	−0.496284
$78$	0	0
$79$	8.67975	0.976549	0.488274	−	0.872690i	$-0.337627\pi$
$79$	8.67975	0.976549	0.488274	+	0.872690i	$0.337627\pi$
$80$	−4.54171	−0.507778
$81$	0	0
$82$	0.463013	0.0511312
$83$	−1.01594	−0.111514	−0.0557572	−	0.998444i	$-0.517757\pi$
$83$	−1.01594	−0.111514	−0.0557572	+	0.998444i	$0.517757\pi$
$84$	0	0
$85$	−2.68689	−0.291434
$86$	−23.7950	−2.56588
$87$	0	0
$88$	−0.619418	−0.0660302
$89$	−11.1078	−1.17742	−0.588710	−	0.808344i	$-0.700364\pi$
$89$	−11.1078	−1.17742	−0.588710	+	0.808344i	$0.700364\pi$
$90$	0	0
$91$	19.5354	2.04787
$92$	3.72301	0.388150
$93$	0	0
$94$	12.3013	1.26879
$95$	1.00000	0.102598
$96$	0	0
$97$	17.2009	1.74648	0.873242	−	0.487286i	$-0.162013\pi$
$97$	17.2009	1.74648	0.873242	+	0.487286i	$0.162013\pi$
$98$	−22.9433	−2.31762
$99$	0	0
$100$	1.67697	0.167697
$101$	7.65081	0.761284	0.380642	−	0.924722i	$-0.375703\pi$
$101$	7.65081	0.761284	0.380642	+	0.924722i	$0.375703\pi$
$102$	0	0
$103$	0.0969205	0.00954986	0.00477493	−	0.999989i	$-0.498480\pi$
$103$	0.0969205	0.00954986	0.00477493	+	0.999989i	$0.498480\pi$
$104$	2.77863	0.272468
$105$	0	0
$106$	8.35055	0.811077
$107$	3.42305	0.330919	0.165459	−	0.986217i	$-0.447089\pi$
$107$	3.42305	0.330919	0.165459	+	0.986217i	$0.447089\pi$
$108$	0	0
$109$	16.0990	1.54200	0.771001	−	0.636834i	$-0.219756\pi$
$109$	16.0990	1.54200	0.771001	+	0.636834i	$0.219756\pi$
$110$	1.91754	0.182831
$111$	0	0
$112$	−19.7786	−1.86890
$113$	8.47515	0.797275	0.398638	−	0.917108i	$-0.369483\pi$
$113$	8.47515	0.797275	0.398638	+	0.917108i	$0.369483\pi$
$114$	0	0
$115$	2.22008	0.207023
$116$	13.9975	1.29964
$117$	0	0
$118$	14.0415	1.29263
$119$	−11.7011	−1.07263
$120$	0	0
$121$	1.00000	0.0909091
$122$	14.2954	1.29424
$123$	0	0
$124$	7.52270	0.675558
$125$	1.00000	0.0894427
$126$	0	0
$127$	1.54006	0.136658	0.0683289	−	0.997663i	$-0.478233\pi$
$127$	1.54006	0.136658	0.0683289	+	0.997663i	$0.478233\pi$
$128$	−4.89071	−0.432282
$129$	0	0
$130$	−8.60187	−0.754434
$131$	8.69465	0.759655	0.379828	−	0.925057i	$-0.375983\pi$
$131$	8.69465	0.759655	0.379828	+	0.925057i	$0.375983\pi$
$132$	0	0
$133$	4.35488	0.377616
$134$	23.9961	2.07295
$135$	0	0
$136$	−1.66431	−0.142713
$137$	21.4556	1.83308	0.916539	−	0.399944i	$-0.130971\pi$
$137$	21.4556	1.83308	0.916539	+	0.399944i	$0.130971\pi$
$138$	0	0
$139$	20.3942	1.72981	0.864907	−	0.501932i	$-0.167377\pi$
$139$	20.3942	1.72981	0.864907	+	0.501932i	$0.167377\pi$
$140$	7.30301	0.617217
$141$	0	0
$142$	−19.9395	−1.67328
$143$	−4.48588	−0.375128
$144$	0	0
$145$	8.34690	0.693172
$146$	5.18878	0.429427
$147$	0	0
$148$	−17.8577	−1.46790
$149$	20.3622	1.66813	0.834067	−	0.551662i	$-0.186006\pi$
$149$	20.3622	1.66813	0.834067	+	0.551662i	$0.186006\pi$
$150$	0	0
$151$	−18.9136	−1.53917	−0.769584	−	0.638546i	$-0.779536\pi$
$151$	−18.9136	−1.53917	−0.769584	+	0.638546i	$0.779536\pi$
$152$	0.619418	0.0502415
$153$	0	0
$154$	8.35066	0.672916
$155$	4.48588	0.360314
$156$	0	0
$157$	−18.3845	−1.46725	−0.733623	−	0.679557i	$-0.762172\pi$
$157$	−18.3845	−1.46725	−0.733623	+	0.679557i	$0.762172\pi$
$158$	−16.6438	−1.32411
$159$	0	0
$160$	7.47009	0.590562
$161$	9.66816	0.761957
$162$	0	0
$163$	4.38591	0.343531	0.171766	−	0.985138i	$-0.445053\pi$
$163$	4.38591	0.343531	0.171766	+	0.985138i	$0.445053\pi$
$164$	−0.404924	−0.0316193
$165$	0	0
$166$	1.94812	0.151203
$167$	0.271973	0.0210459	0.0105229	−	0.999945i	$-0.496650\pi$
$167$	0.271973	0.0210459	0.0105229	+	0.999945i	$0.496650\pi$
$168$	0	0
$169$	7.12310	0.547931
$170$	5.15222	0.395157
$171$	0	0
$172$	20.8098	1.58673
$173$	10.4563	0.794981	0.397491	−	0.917606i	$-0.369881\pi$
$173$	10.4563	0.794981	0.397491	+	0.917606i	$0.369881\pi$
$174$	0	0
$175$	4.35488	0.329198
$176$	4.54171	0.342344
$177$	0	0
$178$	21.2996	1.59647
$179$	−12.8392	−0.959647	−0.479824	−	0.877365i	$-0.659299\pi$
$179$	−12.8392	−0.959647	−0.479824	+	0.877365i	$0.659299\pi$
$180$	0	0
$181$	20.3421	1.51202	0.756008	−	0.654563i	$-0.227147\pi$
$181$	20.3421	1.51202	0.756008	+	0.654563i	$0.227147\pi$
$182$	−37.4601	−2.77672
$183$	0	0
$184$	1.37516	0.101378
$185$	−10.6488	−0.782915
$186$	0	0
$187$	2.68689	0.196485
$188$	−10.7580	−0.784611
$189$	0	0
$190$	−1.91754	−0.139113
$191$	−9.29643	−0.672666	−0.336333	−	0.941743i	$-0.609187\pi$
$191$	−9.29643	−0.672666	−0.336333	+	0.941743i	$0.609187\pi$
$192$	0	0
$193$	−0.652103	−0.0469394	−0.0234697	−	0.999725i	$-0.507471\pi$
$193$	−0.652103	−0.0469394	−0.0234697	+	0.999725i	$0.507471\pi$
$194$	−32.9834	−2.36807
$195$	0	0
$196$	20.0649	1.43321
$197$	24.6478	1.75608	0.878042	−	0.478584i	$-0.158850\pi$
$197$	24.6478	1.75608	0.878042	+	0.478584i	$0.158850\pi$
$198$	0	0
$199$	−20.9127	−1.48246	−0.741230	−	0.671251i	$-0.765757\pi$
$199$	−20.9127	−1.48246	−0.741230	+	0.671251i	$0.765757\pi$
$200$	0.619418	0.0437995
$201$	0	0
$202$	−14.6708	−1.03223
$203$	36.3497	2.55125
$204$	0	0
$205$	−0.241461	−0.0168644
$206$	−0.185849	−0.0129487
$207$	0	0
$208$	−20.3735	−1.41265
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	−25.9949	−1.78956	−0.894781	−	0.446504i	$-0.852669\pi$
$211$	−25.9949	−1.78956	−0.894781	+	0.446504i	$0.852669\pi$
$212$	−7.30291	−0.501566
$213$	0	0
$214$	−6.56385	−0.448696
$215$	12.4091	0.846295
$216$	0	0
$217$	19.5354	1.32615
$218$	−30.8705	−2.09081
$219$	0	0
$220$	−1.67697	−0.113061
$221$	−12.0530	−0.810775
$222$	0	0
$223$	7.35616	0.492605	0.246302	−	0.969193i	$-0.420784\pi$
$223$	7.35616	0.492605	0.246302	+	0.969193i	$0.420784\pi$
$224$	32.5313	2.17359
$225$	0	0
$226$	−16.2515	−1.08103
$227$	−1.59761	−0.106037	−0.0530184	−	0.998594i	$-0.516884\pi$
$227$	−1.59761	−0.106037	−0.0530184	+	0.998594i	$0.516884\pi$
$228$	0	0
$229$	−20.7505	−1.37123	−0.685615	−	0.727964i	$-0.740467\pi$
$229$	−20.7505	−1.37123	−0.685615	+	0.727964i	$0.740467\pi$
$230$	−4.25709	−0.280704
$231$	0	0
$232$	5.17022	0.339442
$233$	−8.60488	−0.563724	−0.281862	−	0.959455i	$-0.590952\pi$
$233$	−8.60488	−0.563724	−0.281862	+	0.959455i	$0.590952\pi$
$234$	0	0
$235$	−6.41516	−0.418479
$236$	−12.2799	−0.799354
$237$	0	0
$238$	22.4373	1.45439
$239$	−0.604538	−0.0391043	−0.0195522	−	0.999809i	$-0.506224\pi$
$239$	−0.604538	−0.0391043	−0.0195522	+	0.999809i	$0.506224\pi$
$240$	0	0
$241$	−8.57771	−0.552539	−0.276269	−	0.961080i	$-0.589098\pi$
$241$	−8.57771	−0.552539	−0.276269	+	0.961080i	$0.589098\pi$
$242$	−1.91754	−0.123264
$243$	0	0
$244$	−12.5019	−0.800354
$245$	11.9649	0.764412
$246$	0	0
$247$	4.48588	0.285430
$248$	2.77863	0.176443
$249$	0	0
$250$	−1.91754	−0.121276
$251$	−8.70635	−0.549540	−0.274770	−	0.961510i	$-0.588602\pi$
$251$	−8.70635	−0.549540	−0.274770	+	0.961510i	$0.588602\pi$
$252$	0	0
$253$	−2.22008	−0.139575
$254$	−2.95312	−0.185295
$255$	0	0
$256$	19.8597	1.24123
$257$	9.55966	0.596315	0.298158	−	0.954517i	$-0.403628\pi$
$257$	9.55966	0.596315	0.298158	+	0.954517i	$0.403628\pi$
$258$	0	0
$259$	−46.3742	−2.88155
$260$	7.52270	0.466538
$261$	0	0
$262$	−16.6724	−1.03002
$263$	10.5228	0.648865	0.324433	−	0.945909i	$-0.394827\pi$
$263$	10.5228	0.648865	0.324433	+	0.945909i	$0.394827\pi$
$264$	0	0
$265$	−4.35482	−0.267514
$266$	−8.35066	−0.512012
$267$	0	0
$268$	−20.9856	−1.28190
$269$	−2.00890	−0.122485	−0.0612423	−	0.998123i	$-0.519506\pi$
$269$	−2.00890	−0.122485	−0.0612423	+	0.998123i	$0.519506\pi$
$270$	0	0
$271$	13.1748	0.800309	0.400155	−	0.916448i	$-0.368956\pi$
$271$	13.1748	0.800309	0.400155	+	0.916448i	$0.368956\pi$
$272$	12.2031	0.739919
$273$	0	0
$274$	−41.1421	−2.48549
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	−19.8281	−1.19136	−0.595678	−	0.803224i	$-0.703116\pi$
$277$	−19.8281	−1.19136	−0.595678	+	0.803224i	$0.703116\pi$
$278$	−39.1068	−2.34547
$279$	0	0
$280$	2.69749	0.161206
$281$	6.14826	0.366775	0.183387	−	0.983041i	$-0.441294\pi$
$281$	6.14826	0.366775	0.183387	+	0.983041i	$0.441294\pi$
$282$	0	0
$283$	−4.92430	−0.292719	−0.146360	−	0.989231i	$-0.546756\pi$
$283$	−4.92430	−0.292719	−0.146360	+	0.989231i	$0.546756\pi$
$284$	17.4379	1.03475
$285$	0	0
$286$	8.60187	0.508639
$287$	−1.05153	−0.0620701
$288$	0	0
$289$	−9.78064	−0.575332
$290$	−16.0055	−0.939878
$291$	0	0
$292$	−4.53781	−0.265555
$293$	3.81220	0.222711	0.111355	−	0.993781i	$-0.464481\pi$
$293$	3.81220	0.222711	0.111355	+	0.993781i	$0.464481\pi$
$294$	0	0
$295$	−7.32266	−0.426342
$296$	−6.59606	−0.383388
$297$	0	0
$298$	−39.0454	−2.26184
$299$	9.95899	0.575943
$300$	0	0
$301$	54.0402	3.11482
$302$	36.2677	2.08697
$303$	0	0
$304$	−4.54171	−0.260485
$305$	−7.45506	−0.426875
$306$	0	0
$307$	−4.54309	−0.259288	−0.129644	−	0.991561i	$-0.541383\pi$
$307$	−4.54309	−0.259288	−0.129644	+	0.991561i	$0.541383\pi$
$308$	−7.30301	−0.416128
$309$	0	0
$310$	−8.60187	−0.488553
$311$	2.14788	0.121795	0.0608974	−	0.998144i	$-0.480604\pi$
$311$	2.14788	0.121795	0.0608974	+	0.998144i	$0.480604\pi$
$312$	0	0
$313$	−3.06876	−0.173457	−0.0867283	−	0.996232i	$-0.527641\pi$
$313$	−3.06876	−0.173457	−0.0867283	+	0.996232i	$0.527641\pi$
$314$	35.2531	1.98945
$315$	0	0
$316$	14.5557	0.818823
$317$	−28.4889	−1.60010	−0.800049	−	0.599934i	$-0.795193\pi$
$317$	−28.4889	−1.60010	−0.800049	+	0.599934i	$0.795193\pi$
$318$	0	0
$319$	−8.34690	−0.467337
$320$	−5.24080	−0.292970
$321$	0	0
$322$	−18.5391	−1.03314
$323$	−2.68689	−0.149502
$324$	0	0
$325$	4.48588	0.248832
$326$	−8.41018	−0.465797
$327$	0	0
$328$	−0.149566	−0.00825838
$329$	−27.9372	−1.54023
$330$	0	0
$331$	2.62990	0.144552	0.0722762	−	0.997385i	$-0.476974\pi$
$331$	2.62990	0.144552	0.0722762	+	0.997385i	$0.476974\pi$
$332$	−1.70371	−0.0935033
$333$	0	0
$334$	−0.521519	−0.0285363
$335$	−12.5140	−0.683711
$336$	0	0
$337$	−10.9202	−0.594861	−0.297431	−	0.954743i	$-0.596130\pi$
$337$	−10.9202	−0.594861	−0.297431	+	0.954743i	$0.596130\pi$
$338$	−13.6589	−0.742944
$339$	0	0
$340$	−4.50584	−0.244363
$341$	−4.48588	−0.242924
$342$	0	0
$343$	21.6217	1.16746
$344$	7.68643	0.414425
$345$	0	0
$346$	−20.0505	−1.07792
$347$	34.1518	1.83337	0.916683	−	0.399615i	$-0.130856\pi$
$347$	34.1518	1.83337	0.916683	+	0.399615i	$0.130856\pi$
$348$	0	0
$349$	−20.2909	−1.08615	−0.543073	−	0.839686i	$-0.682739\pi$
$349$	−20.2909	−1.08615	−0.543073	+	0.839686i	$0.682739\pi$
$350$	−8.35066	−0.446362
$351$	0	0
$352$	−7.47009	−0.398157
$353$	8.33866	0.443822	0.221911	−	0.975067i	$-0.428771\pi$
$353$	8.33866	0.443822	0.221911	+	0.975067i	$0.428771\pi$
$354$	0	0
$355$	10.3984	0.551892
$356$	−18.6274	−0.987251
$357$	0	0
$358$	24.6197	1.30119
$359$	0.876631	0.0462668	0.0231334	−	0.999732i	$-0.492636\pi$
$359$	0.876631	0.0462668	0.0231334	+	0.999732i	$0.492636\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−39.0068	−2.05015
$363$	0	0
$364$	32.7604	1.71711
$365$	−2.70595	−0.141636
$366$	0	0
$367$	−21.1494	−1.10399	−0.551995	−	0.833847i	$-0.686133\pi$
$367$	−21.1494	−1.10399	−0.551995	+	0.833847i	$0.686133\pi$
$368$	−10.0829	−0.525609
$369$	0	0
$370$	20.4195	1.06156
$371$	−18.9647	−0.984598
$372$	0	0
$373$	−23.6208	−1.22304	−0.611518	−	0.791230i	$-0.709441\pi$
$373$	−23.6208	−1.22304	−0.611518	+	0.791230i	$0.709441\pi$
$374$	−5.15222	−0.266415
$375$	0	0
$376$	−3.97366	−0.204926
$377$	37.4432	1.92842
$378$	0	0
$379$	3.01645	0.154945	0.0774724	−	0.996995i	$-0.475315\pi$
$379$	3.01645	0.154945	0.0774724	+	0.996995i	$0.475315\pi$
$380$	1.67697	0.0860269
$381$	0	0
$382$	17.8263	0.912073
$383$	−7.89623	−0.403479	−0.201739	−	0.979439i	$-0.564659\pi$
$383$	−7.89623	−0.403479	−0.201739	+	0.979439i	$0.564659\pi$
$384$	0	0
$385$	−4.35488	−0.221945
$386$	1.25044	0.0636455
$387$	0	0
$388$	28.8454	1.46440
$389$	−7.51649	−0.381101	−0.190551	−	0.981677i	$-0.561027\pi$
$389$	−7.51649	−0.381101	−0.190551	+	0.981677i	$0.561027\pi$
$390$	0	0
$391$	−5.96509	−0.301668
$392$	7.41130	0.374327
$393$	0	0
$394$	−47.2632	−2.38109
$395$	8.67975	0.436726
$396$	0	0
$397$	30.0391	1.50762	0.753809	−	0.657094i	$-0.228215\pi$
$397$	30.0391	1.50762	0.753809	+	0.657094i	$0.228215\pi$
$398$	40.1010	2.01008
$399$	0	0
$400$	−4.54171	−0.227085
$401$	−9.63569	−0.481184	−0.240592	−	0.970626i	$-0.577342\pi$
$401$	−9.63569	−0.481184	−0.240592	+	0.970626i	$0.577342\pi$
$402$	0	0
$403$	20.1231	1.00240
$404$	12.8302	0.638327
$405$	0	0
$406$	−69.7021	−3.45926
$407$	10.6488	0.527841
$408$	0	0
$409$	24.5489	1.21386	0.606932	−	0.794754i	$-0.292400\pi$
$409$	24.5489	1.21386	0.606932	+	0.794754i	$0.292400\pi$
$410$	0.463013	0.0228666
$411$	0	0
$412$	0.162533	0.00800743
$413$	−31.8893	−1.56917
$414$	0	0
$415$	−1.01594	−0.0498707
$416$	33.5099	1.64296
$417$	0	0
$418$	1.91754	0.0937901
$419$	−13.5830	−0.663573	−0.331787	−	0.943354i	$-0.607651\pi$
$419$	−13.5830	−0.663573	−0.331787	+	0.943354i	$0.607651\pi$
$420$	0	0
$421$	−18.5925	−0.906145	−0.453073	−	0.891474i	$-0.649672\pi$
$421$	−18.5925	−0.906145	−0.453073	+	0.891474i	$0.649672\pi$
$422$	49.8464	2.42648
$423$	0	0
$424$	−2.69745	−0.131000
$425$	−2.68689	−0.130333
$426$	0	0
$427$	−32.4658	−1.57113
$428$	5.74037	0.277471
$429$	0	0
$430$	−23.7950	−1.14750
$431$	−30.7873	−1.48297	−0.741487	−	0.670968i	$-0.765879\pi$
$431$	−30.7873	−1.48297	−0.741487	+	0.670968i	$0.765879\pi$
$432$	0	0
$433$	−0.795465	−0.0382276	−0.0191138	−	0.999817i	$-0.506084\pi$
$433$	−0.795465	−0.0382276	−0.0191138	+	0.999817i	$0.506084\pi$
$434$	−37.4601	−1.79814
$435$	0	0
$436$	26.9976	1.29295
$437$	2.22008	0.106201
$438$	0	0
$439$	39.9329	1.90589	0.952946	−	0.303139i	$-0.0980348\pi$
$439$	39.9329	1.90589	0.952946	+	0.303139i	$0.0980348\pi$
$440$	−0.619418	−0.0295296
$441$	0	0
$442$	23.1122	1.09934
$443$	12.5915	0.598241	0.299120	−	0.954215i	$-0.403307\pi$
$443$	12.5915	0.598241	0.299120	+	0.954215i	$0.403307\pi$
$444$	0	0
$445$	−11.1078	−0.526558
$446$	−14.1058	−0.667927
$447$	0	0
$448$	−22.8230	−1.07829
$449$	0.837131	0.0395067	0.0197533	−	0.999805i	$-0.493712\pi$
$449$	0.837131	0.0395067	0.0197533	+	0.999805i	$0.493712\pi$
$450$	0	0
$451$	0.241461	0.0113700
$452$	14.2126	0.668505
$453$	0	0
$454$	3.06348	0.143776
$455$	19.5354	0.915836
$456$	0	0
$457$	−5.06499	−0.236930	−0.118465	−	0.992958i	$-0.537797\pi$
$457$	−5.06499	−0.236930	−0.118465	+	0.992958i	$0.537797\pi$
$458$	39.7899	1.85926
$459$	0	0
$460$	3.72301	0.173586
$461$	35.5471	1.65559	0.827797	−	0.561027i	$-0.189594\pi$
$461$	35.5471	1.65559	0.827797	+	0.561027i	$0.189594\pi$
$462$	0	0
$463$	14.0904	0.654838	0.327419	−	0.944879i	$-0.393821\pi$
$463$	14.0904	0.654838	0.327419	+	0.944879i	$0.393821\pi$
$464$	−37.9092	−1.75989
$465$	0	0
$466$	16.5002	0.764358
$467$	0.945675	0.0437606	0.0218803	−	0.999761i	$-0.493035\pi$
$467$	0.945675	0.0437606	0.0218803	+	0.999761i	$0.493035\pi$
$468$	0	0
$469$	−54.4968	−2.51643
$470$	12.3013	0.567418
$471$	0	0
$472$	−4.53579	−0.208777
$473$	−12.4091	−0.570572
$474$	0	0
$475$	1.00000	0.0458831
$476$	−19.6224	−0.899389
$477$	0	0
$478$	1.15923	0.0530219
$479$	−18.2687	−0.834716	−0.417358	−	0.908742i	$-0.637044\pi$
$479$	−18.2687	−0.834716	−0.417358	+	0.908742i	$0.637044\pi$
$480$	0	0
$481$	−47.7692	−2.17809
$482$	16.4481	0.749192
$483$	0	0
$484$	1.67697	0.0762260
$485$	17.2009	0.781052
$486$	0	0
$487$	3.35944	0.152231	0.0761153	−	0.997099i	$-0.475748\pi$
$487$	3.35944	0.152231	0.0761153	+	0.997099i	$0.475748\pi$
$488$	−4.61780	−0.209038
$489$	0	0
$490$	−22.9433	−1.03647
$491$	24.1485	1.08981	0.544904	−	0.838498i	$-0.316566\pi$
$491$	24.1485	1.08981	0.544904	+	0.838498i	$0.316566\pi$
$492$	0	0
$493$	−22.4272	−1.01007
$494$	−8.60187	−0.387016
$495$	0	0
$496$	−20.3735	−0.914799
$497$	45.2840	2.03126
$498$	0	0
$499$	26.2251	1.17400	0.586999	−	0.809587i	$-0.300309\pi$
$499$	26.2251	1.17400	0.586999	+	0.809587i	$0.300309\pi$
$500$	1.67697	0.0749965
$501$	0	0
$502$	16.6948	0.745126
$503$	−33.1869	−1.47973	−0.739865	−	0.672756i	$-0.765111\pi$
$503$	−33.1869	−1.47973	−0.739865	+	0.672756i	$0.765111\pi$
$504$	0	0
$505$	7.65081	0.340457
$506$	4.25709	0.189251
$507$	0	0
$508$	2.58263	0.114586
$509$	17.1477	0.760059	0.380030	−	0.924974i	$-0.375914\pi$
$509$	17.1477	0.760059	0.380030	+	0.924974i	$0.375914\pi$
$510$	0	0
$511$	−11.7841	−0.521298
$512$	−28.3005	−1.25072
$513$	0	0
$514$	−18.3311	−0.808548
$515$	0.0969205	0.00427083
$516$	0	0
$517$	6.41516	0.282138
$518$	88.9245	3.90712
$519$	0	0
$520$	2.77863	0.121851
$521$	6.24480	0.273590	0.136795	−	0.990599i	$-0.456320\pi$
$521$	6.24480	0.273590	0.136795	+	0.990599i	$0.456320\pi$
$522$	0	0
$523$	27.2098	1.18980	0.594900	−	0.803800i	$-0.297192\pi$
$523$	27.2098	1.18980	0.594900	+	0.803800i	$0.297192\pi$
$524$	14.5807	0.636961
$525$	0	0
$526$	−20.1780	−0.879801
$527$	−12.0530	−0.525039
$528$	0	0
$529$	−18.0713	−0.785707
$530$	8.35055	0.362725
$531$	0	0
$532$	7.30301	0.316626
$533$	−1.08317	−0.0469171
$534$	0	0
$535$	3.42305	0.147991
$536$	−7.75138	−0.334809
$537$	0	0
$538$	3.85214	0.166078
$539$	−11.9649	−0.515367
$540$	0	0
$541$	−8.83478	−0.379837	−0.189918	−	0.981800i	$-0.560822\pi$
$541$	−8.83478	−0.379837	−0.189918	+	0.981800i	$0.560822\pi$
$542$	−25.2632	−1.08515
$543$	0	0
$544$	−20.0713	−0.860549
$545$	16.0990	0.689605
$546$	0	0
$547$	31.3950	1.34235	0.671176	−	0.741298i	$-0.265789\pi$
$547$	31.3950	1.34235	0.671176	+	0.741298i	$0.265789\pi$
$548$	35.9805	1.53701
$549$	0	0
$550$	1.91754	0.0817643
$551$	8.34690	0.355590
$552$	0	0
$553$	37.7992	1.60739
$554$	38.0212	1.61537
$555$	0	0
$556$	34.2005	1.45043
$557$	47.1474	1.99770	0.998849	−	0.0479717i	$-0.0152757\pi$
$557$	47.1474	1.99770	0.998849	+	0.0479717i	$0.0152757\pi$
$558$	0	0
$559$	55.6658	2.35441
$560$	−19.7786	−0.835797
$561$	0	0
$562$	−11.7896	−0.497313
$563$	20.6669	0.871007	0.435503	−	0.900187i	$-0.356570\pi$
$563$	20.6669	0.871007	0.435503	+	0.900187i	$0.356570\pi$
$564$	0	0
$565$	8.47515	0.356552
$566$	9.44256	0.396900
$567$	0	0
$568$	6.44099	0.270258
$569$	−23.3464	−0.978731	−0.489365	−	0.872079i	$-0.662772\pi$
$569$	−23.3464	−0.978731	−0.489365	+	0.872079i	$0.662772\pi$
$570$	0	0
$571$	0.905142	0.0378790	0.0189395	−	0.999821i	$-0.493971\pi$
$571$	0.905142	0.0378790	0.0189395	+	0.999821i	$0.493971\pi$
$572$	−7.52270	−0.314540
$573$	0	0
$574$	2.01636	0.0841614
$575$	2.22008	0.0925836
$576$	0	0
$577$	1.42069	0.0591442	0.0295721	−	0.999563i	$-0.490586\pi$
$577$	1.42069	0.0591442	0.0295721	+	0.999563i	$0.490586\pi$
$578$	18.7548	0.780097
$579$	0	0
$580$	13.9975	0.581215
$581$	−4.42431	−0.183551
$582$	0	0
$583$	4.35482	0.180358
$584$	−1.67612	−0.0693582
$585$	0	0
$586$	−7.31005	−0.301975
$587$	44.8959	1.85305	0.926526	−	0.376230i	$-0.122780\pi$
$587$	44.8959	1.85305	0.926526	+	0.376230i	$0.122780\pi$
$588$	0	0
$589$	4.48588	0.184837
$590$	14.0415	0.578080
$591$	0	0
$592$	48.3637	1.98774
$593$	−9.05007	−0.371642	−0.185821	−	0.982584i	$-0.559494\pi$
$593$	−9.05007	−0.371642	−0.185821	+	0.982584i	$0.559494\pi$
$594$	0	0
$595$	−11.7011	−0.479697
$596$	34.1468	1.39871
$597$	0	0
$598$	−19.0968	−0.780926
$599$	33.9418	1.38682	0.693411	−	0.720542i	$-0.256107\pi$
$599$	33.9418	1.38682	0.693411	+	0.720542i	$0.256107\pi$
$600$	0	0
$601$	7.60296	0.310131	0.155066	−	0.987904i	$-0.450441\pi$
$601$	7.60296	0.310131	0.155066	+	0.987904i	$0.450441\pi$
$602$	−103.624	−4.22341
$603$	0	0
$604$	−31.7176	−1.29057
$605$	1.00000	0.0406558
$606$	0	0
$607$	−33.0809	−1.34271	−0.671356	−	0.741135i	$-0.734288\pi$
$607$	−33.0809	−1.34271	−0.671356	+	0.741135i	$0.734288\pi$
$608$	7.47009	0.302952
$609$	0	0
$610$	14.2954	0.578804
$611$	−28.7776	−1.16422
$612$	0	0
$613$	−22.0583	−0.890928	−0.445464	−	0.895300i	$-0.646961\pi$
$613$	−22.0583	−0.890928	−0.445464	+	0.895300i	$0.646961\pi$
$614$	8.71157	0.351570
$615$	0	0
$616$	−2.69749	−0.108685
$617$	−19.9256	−0.802175	−0.401088	−	0.916040i	$-0.631368\pi$
$617$	−19.9256	−0.802175	−0.401088	+	0.916040i	$0.631368\pi$
$618$	0	0
$619$	−9.14804	−0.367691	−0.183845	−	0.982955i	$-0.558855\pi$
$619$	−9.14804	−0.367691	−0.183845	+	0.982955i	$0.558855\pi$
$620$	7.52270	0.302119
$621$	0	0
$622$	−4.11864	−0.165143
$623$	−48.3729	−1.93802
$624$	0	0
$625$	1.00000	0.0400000
$626$	5.88448	0.235191
$627$	0	0
$628$	−30.8304	−1.23027
$629$	28.6121	1.14084
$630$	0	0
$631$	14.1578	0.563614	0.281807	−	0.959471i	$-0.409066\pi$
$631$	14.1578	0.563614	0.281807	+	0.959471i	$0.409066\pi$
$632$	5.37640	0.213862
$633$	0	0
$634$	54.6288	2.16959
$635$	1.54006	0.0611152
$636$	0	0
$637$	53.6733	2.12661
$638$	16.0055	0.633665
$639$	0	0
$640$	−4.89071	−0.193322
$641$	−37.5553	−1.48335	−0.741673	−	0.670761i	$-0.765967\pi$
$641$	−37.5553	−1.48335	−0.741673	+	0.670761i	$0.765967\pi$
$642$	0	0
$643$	20.0444	0.790472	0.395236	−	0.918580i	$-0.370663\pi$
$643$	20.0444	0.790472	0.395236	+	0.918580i	$0.370663\pi$
$644$	16.2132	0.638891
$645$	0	0
$646$	5.15222	0.202711
$647$	14.4284	0.567240	0.283620	−	0.958937i	$-0.408465\pi$
$647$	14.4284	0.567240	0.283620	+	0.958937i	$0.408465\pi$
$648$	0	0
$649$	7.32266	0.287440
$650$	−8.60187	−0.337393
$651$	0	0
$652$	7.35506	0.288046
$653$	15.3712	0.601520	0.300760	−	0.953700i	$-0.402760\pi$
$653$	15.3712	0.601520	0.300760	+	0.953700i	$0.402760\pi$
$654$	0	0
$655$	8.69465	0.339728
$656$	1.09665	0.0428169
$657$	0	0
$658$	53.5708	2.08841
$659$	−34.3302	−1.33731	−0.668657	−	0.743571i	$-0.733131\pi$
$659$	−34.3302	−1.33731	−0.668657	+	0.743571i	$0.733131\pi$
$660$	0	0
$661$	1.40733	0.0547387	0.0273693	−	0.999625i	$-0.491287\pi$
$661$	1.40733	0.0547387	0.0273693	+	0.999625i	$0.491287\pi$
$662$	−5.04295	−0.196000
$663$	0	0
$664$	−0.629294	−0.0244214
$665$	4.35488	0.168875
$666$	0	0
$667$	18.5307	0.717513
$668$	0.456091	0.0176467
$669$	0	0
$670$	23.9961	0.927049
$671$	7.45506	0.287799
$672$	0	0
$673$	−7.11989	−0.274452	−0.137226	−	0.990540i	$-0.543819\pi$
$673$	−7.11989	−0.274452	−0.137226	+	0.990540i	$0.543819\pi$
$674$	20.9400	0.806577
$675$	0	0
$676$	11.9452	0.459432
$677$	−29.6437	−1.13930	−0.569650	−	0.821888i	$-0.692921\pi$
$677$	−29.6437	−1.13930	−0.569650	+	0.821888i	$0.692921\pi$
$678$	0	0
$679$	74.9077	2.87469
$680$	−1.66431	−0.0638232
$681$	0	0
$682$	8.60187	0.329382
$683$	−15.7884	−0.604125	−0.302062	−	0.953288i	$-0.597675\pi$
$683$	−15.7884	−0.604125	−0.302062	+	0.953288i	$0.597675\pi$
$684$	0	0
$685$	21.4556	0.819778
$686$	−41.4606	−1.58297
$687$	0	0
$688$	−56.3586	−2.14865
$689$	−19.5352	−0.744231
$690$	0	0
$691$	19.6816	0.748722	0.374361	−	0.927283i	$-0.377862\pi$
$691$	19.6816	0.748722	0.374361	+	0.927283i	$0.377862\pi$
$692$	17.5350	0.666581
$693$	0	0
$694$	−65.4876	−2.48588
$695$	20.3942	0.773597
$696$	0	0
$697$	0.648779	0.0245743
$698$	38.9086	1.47271
$699$	0	0
$700$	7.30301	0.276028
$701$	−35.9057	−1.35614	−0.678069	−	0.734998i	$-0.737183\pi$
$701$	−35.9057	−1.35614	−0.678069	+	0.734998i	$0.737183\pi$
$702$	0	0
$703$	−10.6488	−0.401627
$704$	5.24080	0.197520
$705$	0	0
$706$	−15.9897	−0.601782
$707$	33.3183	1.25307
$708$	0	0
$709$	1.98234	0.0744482	0.0372241	−	0.999307i	$-0.488148\pi$
$709$	1.98234	0.0744482	0.0372241	+	0.999307i	$0.488148\pi$
$710$	−19.9395	−0.748315
$711$	0	0
$712$	−6.88035	−0.257852
$713$	9.95899	0.372967
$714$	0	0
$715$	−4.48588	−0.167762
$716$	−21.5310	−0.804651
$717$	0	0
$718$	−1.68098	−0.0627336
$719$	−29.8026	−1.11145	−0.555725	−	0.831366i	$-0.687559\pi$
$719$	−29.8026	−1.11145	−0.555725	+	0.831366i	$0.687559\pi$
$720$	0	0
$721$	0.422077	0.0157190
$722$	−1.91754	−0.0713636
$723$	0	0
$724$	34.1131	1.26780
$725$	8.34690	0.309996
$726$	0	0
$727$	7.46607	0.276901	0.138451	−	0.990369i	$-0.455788\pi$
$727$	7.46607	0.276901	0.138451	+	0.990369i	$0.455788\pi$
$728$	12.1006	0.448478
$729$	0	0
$730$	5.18878	0.192046
$731$	−33.3419	−1.23319
$732$	0	0
$733$	27.5634	1.01808	0.509038	−	0.860744i	$-0.330001\pi$
$733$	27.5634	1.01808	0.509038	+	0.860744i	$0.330001\pi$
$734$	40.5549	1.49691
$735$	0	0
$736$	16.5842	0.611300
$737$	12.5140	0.460958
$738$	0	0
$739$	−26.8209	−0.986624	−0.493312	−	0.869852i	$-0.664214\pi$
$739$	−26.8209	−0.986624	−0.493312	+	0.869852i	$0.664214\pi$
$740$	−17.8577	−0.656464
$741$	0	0
$742$	36.3656	1.33502
$743$	−40.5866	−1.48898	−0.744489	−	0.667635i	$-0.767307\pi$
$743$	−40.5866	−1.48898	−0.744489	+	0.667635i	$0.767307\pi$
$744$	0	0
$745$	20.3622	0.746013
$746$	45.2938	1.65832
$747$	0	0
$748$	4.50584	0.164750
$749$	14.9070	0.544689
$750$	0	0
$751$	−38.6226	−1.40936	−0.704679	−	0.709527i	$-0.748909\pi$
$751$	−38.6226	−1.40936	−0.704679	+	0.709527i	$0.748909\pi$
$752$	29.1358	1.06247
$753$	0	0
$754$	−71.7989	−2.61476
$755$	−18.9136	−0.688337
$756$	0	0
$757$	51.7109	1.87947	0.939733	−	0.341910i	$-0.111074\pi$
$757$	51.7109	1.87947	0.939733	+	0.341910i	$0.111074\pi$
$758$	−5.78418	−0.210091
$759$	0	0
$760$	0.619418	0.0224687
$761$	−15.5534	−0.563809	−0.281905	−	0.959442i	$-0.590966\pi$
$761$	−15.5534	−0.563809	−0.281905	+	0.959442i	$0.590966\pi$
$762$	0	0
$763$	70.1091	2.53812
$764$	−15.5899	−0.564021
$765$	0	0
$766$	15.1414	0.547080
$767$	−32.8486	−1.18609
$768$	0	0
$769$	−33.7422	−1.21677	−0.608387	−	0.793640i	$-0.708183\pi$
$769$	−33.7422	−1.21677	−0.608387	+	0.793640i	$0.708183\pi$
$770$	8.35066	0.300937
$771$	0	0
$772$	−1.09356	−0.0393581
$773$	11.3900	0.409671	0.204836	−	0.978796i	$-0.434334\pi$
$773$	11.3900	0.409671	0.204836	+	0.978796i	$0.434334\pi$
$774$	0	0
$775$	4.48588	0.161137
$776$	10.6545	0.382476
$777$	0	0
$778$	14.4132	0.516738
$779$	−0.241461	−0.00865125
$780$	0	0
$781$	−10.3984	−0.372086
$782$	11.4383	0.409034
$783$	0	0
$784$	−54.3413	−1.94076
$785$	−18.3845	−0.656172
$786$	0	0
$787$	−3.78061	−0.134764	−0.0673822	−	0.997727i	$-0.521465\pi$
$787$	−3.78061	−0.134764	−0.0673822	+	0.997727i	$0.521465\pi$
$788$	41.3337	1.47245
$789$	0	0
$790$	−16.6438	−0.592160
$791$	36.9082	1.31231
$792$	0	0
$793$	−33.4425	−1.18758
$794$	−57.6012	−2.04419
$795$	0	0
$796$	−35.0700	−1.24302
$797$	−50.2367	−1.77947	−0.889737	−	0.456474i	$-0.849112\pi$
$797$	−50.2367	−1.77947	−0.889737	+	0.456474i	$0.849112\pi$
$798$	0	0
$799$	17.2368	0.609794
$800$	7.47009	0.264107
$801$	0	0
$802$	18.4769	0.652441
$803$	2.70595	0.0954910
$804$	0	0
$805$	9.66816	0.340758
$806$	−38.5869	−1.35917
$807$	0	0
$808$	4.73905	0.166719
$809$	55.4633	1.94998	0.974992	−	0.222238i	$-0.0713363\pi$
$809$	55.4633	1.94998	0.974992	+	0.222238i	$0.0713363\pi$
$810$	0	0
$811$	−14.1011	−0.495155	−0.247578	−	0.968868i	$-0.579635\pi$
$811$	−14.1011	−0.495155	−0.247578	+	0.968868i	$0.579635\pi$
$812$	60.9575	2.13919
$813$	0	0
$814$	−20.4195	−0.715704
$815$	4.38591	0.153632
$816$	0	0
$817$	12.4091	0.434140
$818$	−47.0735	−1.64589
$819$	0	0
$820$	−0.404924	−0.0141406
$821$	−51.6548	−1.80276	−0.901382	−	0.433025i	$-0.857446\pi$
$821$	−51.6548	−1.80276	−0.901382	+	0.433025i	$0.857446\pi$
$822$	0	0
$823$	15.5348	0.541511	0.270755	−	0.962648i	$-0.412727\pi$
$823$	15.5348	0.541511	0.270755	+	0.962648i	$0.412727\pi$
$824$	0.0600343	0.00209139
$825$	0	0
$826$	61.1491	2.12765
$827$	−0.501542	−0.0174403	−0.00872017	−	0.999962i	$-0.502776\pi$
$827$	−0.501542	−0.0174403	−0.00872017	+	0.999962i	$0.502776\pi$
$828$	0	0
$829$	−17.5982	−0.611211	−0.305605	−	0.952158i	$-0.598859\pi$
$829$	−17.5982	−0.611211	−0.305605	+	0.952158i	$0.598859\pi$
$830$	1.94812	0.0676201
$831$	0	0
$832$	−23.5096	−0.815048
$833$	−32.1484	−1.11388
$834$	0	0
$835$	0.271973	0.00941200
$836$	−1.67697	−0.0579993
$837$	0	0
$838$	26.0460	0.899744
$839$	7.77520	0.268430	0.134215	−	0.990952i	$-0.457149\pi$
$839$	7.77520	0.268430	0.134215	+	0.990952i	$0.457149\pi$
$840$	0	0
$841$	40.6707	1.40244
$842$	35.6520	1.22865
$843$	0	0
$844$	−43.5927	−1.50052
$845$	7.12310	0.245042
$846$	0	0
$847$	4.35488	0.149635
$848$	19.7783	0.679190
$849$	0	0
$850$	5.15222	0.176720
$851$	−23.6411	−0.810408
$852$	0	0
$853$	28.5328	0.976945	0.488472	−	0.872579i	$-0.337554\pi$
$853$	28.5328	0.976945	0.488472	+	0.872579i	$0.337554\pi$
$854$	62.2547	2.13031
$855$	0	0
$856$	2.12030	0.0724704
$857$	−1.57508	−0.0538037	−0.0269019	−	0.999638i	$-0.508564\pi$
$857$	−1.57508	−0.0538037	−0.0269019	+	0.999638i	$0.508564\pi$
$858$	0	0
$859$	13.9593	0.476286	0.238143	−	0.971230i	$-0.423461\pi$
$859$	13.9593	0.476286	0.238143	+	0.971230i	$0.423461\pi$
$860$	20.8098	0.709607
$861$	0	0
$862$	59.0360	2.01078
$863$	26.7704	0.911276	0.455638	−	0.890165i	$-0.349411\pi$
$863$	26.7704	0.911276	0.455638	+	0.890165i	$0.349411\pi$
$864$	0	0
$865$	10.4563	0.355526
$866$	1.52534	0.0518331
$867$	0	0
$868$	32.7604	1.11196
$869$	−8.67975	−0.294440
$870$	0	0
$871$	−56.1361	−1.90210
$872$	9.97200	0.337695
$873$	0	0
$874$	−4.25709	−0.143998
$875$	4.35488	0.147222
$876$	0	0
$877$	0.0151551	0.000511751 0	0.000255875	−	1.00000i	$-0.499919\pi$
$877$	0.0151551	0.000511751 0	0.000255875		1.00000i	$0.499919\pi$
$878$	−76.5730	−2.58421
$879$	0	0
$880$	4.54171	0.153101
$881$	34.8001	1.17245	0.586223	−	0.810150i	$-0.300614\pi$
$881$	34.8001	1.17245	0.586223	+	0.810150i	$0.300614\pi$
$882$	0	0
$883$	6.21108	0.209019	0.104510	−	0.994524i	$-0.466673\pi$
$883$	6.21108	0.209019	0.104510	+	0.994524i	$0.466673\pi$
$884$	−20.2126	−0.679824
$885$	0	0
$886$	−24.1448	−0.811160
$887$	10.2615	0.344548	0.172274	−	0.985049i	$-0.444889\pi$
$887$	10.2615	0.344548	0.172274	+	0.985049i	$0.444889\pi$
$888$	0	0
$889$	6.70675	0.224937
$890$	21.2996	0.713965
$891$	0	0
$892$	12.3361	0.413043
$893$	−6.41516	−0.214675
$894$	0	0
$895$	−12.8392	−0.429167
$896$	−21.2984	−0.711531
$897$	0	0
$898$	−1.60523	−0.0535674
$899$	37.4432	1.24880
$900$	0	0
$901$	11.7009	0.389814
$902$	−0.463013	−0.0154166
$903$	0	0
$904$	5.24967	0.174601
$905$	20.3421	0.676194
$906$	0	0
$907$	−15.4443	−0.512820	−0.256410	−	0.966568i	$-0.582540\pi$
$907$	−15.4443	−0.512820	−0.256410	+	0.966568i	$0.582540\pi$
$908$	−2.67914	−0.0889105
$909$	0	0
$910$	−37.4601	−1.24179
$911$	−46.1833	−1.53012	−0.765061	−	0.643958i	$-0.777291\pi$
$911$	−46.1833	−1.53012	−0.765061	+	0.643958i	$0.777291\pi$
$912$	0	0
$913$	1.01594	0.0336228
$914$	9.71233	0.321255
$915$	0	0
$916$	−34.7980	−1.14976
$917$	37.8641	1.25038
$918$	0	0
$919$	53.5240	1.76559	0.882797	−	0.469755i	$-0.155658\pi$
$919$	53.5240	1.76559	0.882797	+	0.469755i	$0.155658\pi$
$920$	1.37516	0.0453375
$921$	0	0
$922$	−68.1632	−2.24483
$923$	46.6462	1.53538
$924$	0	0
$925$	−10.6488	−0.350130
$926$	−27.0190	−0.887900
$927$	0	0
$928$	62.3520	2.04681
$929$	55.6828	1.82689	0.913446	−	0.406960i	$-0.133411\pi$
$929$	55.6828	1.82689	0.913446	+	0.406960i	$0.133411\pi$
$930$	0	0
$931$	11.9649	0.392135
$932$	−14.4301	−0.472675
$933$	0	0
$934$	−1.81337	−0.0593354
$935$	2.68689	0.0878706
$936$	0	0
$937$	−50.6135	−1.65347	−0.826736	−	0.562590i	$-0.809805\pi$
$937$	−50.6135	−1.65347	−0.826736	+	0.562590i	$0.809805\pi$
$938$	104.500	3.41204
$939$	0	0
$940$	−10.7580	−0.350889
$941$	24.5207	0.799352	0.399676	−	0.916656i	$-0.369123\pi$
$941$	24.5207	0.799352	0.399676	+	0.916656i	$0.369123\pi$
$942$	0	0
$943$	−0.536063	−0.0174566
$944$	33.2574	1.08244
$945$	0	0
$946$	23.7950	0.773643
$947$	−12.3764	−0.402179	−0.201089	−	0.979573i	$-0.564448\pi$
$947$	−12.3764	−0.402179	−0.201089	+	0.979573i	$0.564448\pi$
$948$	0	0
$949$	−12.1386	−0.394035
$950$	−1.91754	−0.0622133
$951$	0	0
$952$	−7.24785	−0.234904
$953$	24.7785	0.802656	0.401328	−	0.915934i	$-0.368549\pi$
$953$	24.7785	0.802656	0.401328	+	0.915934i	$0.368549\pi$
$954$	0	0
$955$	−9.29643	−0.300825
$956$	−1.01379	−0.0327884
$957$	0	0
$958$	35.0309	1.13180
$959$	93.4366	3.01723
$960$	0	0
$961$	−10.8769	−0.350868
$962$	91.5995	2.95329
$963$	0	0
$964$	−14.3846	−0.463296
$965$	−0.652103	−0.0209919
$966$	0	0
$967$	−14.7702	−0.474978	−0.237489	−	0.971390i	$-0.576324\pi$
$967$	−14.7702	−0.474978	−0.237489	+	0.971390i	$0.576324\pi$
$968$	0.619418	0.0199089
$969$	0	0
$970$	−32.9834	−1.05903
$971$	−4.28321	−0.137455	−0.0687273	−	0.997635i	$-0.521894\pi$
$971$	−4.28321	−0.137455	−0.0687273	+	0.997635i	$0.521894\pi$
$972$	0	0
$973$	88.8143	2.84725
$974$	−6.44186	−0.206411
$975$	0	0
$976$	33.8587	1.08379
$977$	−21.8380	−0.698660	−0.349330	−	0.937000i	$-0.613591\pi$
$977$	−21.8380	−0.698660	−0.349330	+	0.937000i	$0.613591\pi$
$978$	0	0
$979$	11.1078	0.355006
$980$	20.0649	0.640949
$981$	0	0
$982$	−46.3059	−1.47768
$983$	17.1859	0.548146	0.274073	−	0.961709i	$-0.411629\pi$
$983$	17.1859	0.548146	0.274073	+	0.961709i	$0.411629\pi$
$984$	0	0
$985$	24.6478	0.785345
$986$	43.0051	1.36956
$987$	0	0
$988$	7.52270	0.239329
$989$	27.5492	0.876013
$990$	0	0
$991$	−33.7365	−1.07167	−0.535837	−	0.844321i	$-0.680004\pi$
$991$	−33.7365	−1.07167	−0.535837	+	0.844321i	$0.680004\pi$
$992$	33.5099	1.06394
$993$	0	0
$994$	−86.8339	−2.75421
$995$	−20.9127	−0.662977
$996$	0	0
$997$	55.0169	1.74240	0.871201	−	0.490926i	$-0.163342\pi$
$997$	55.0169	1.74240	0.871201	+	0.490926i	$0.163342\pi$
$998$	−50.2879	−1.59183
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9405.2.a.bm.1.3		10
3.2	odd	2		3135.2.a.x.1.8	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3135.2.a.x.1.8	✓	10	3.2	odd	2
9405.2.a.bm.1.3		10	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:	\( N \)	\(=\)	\( 9405 = 3^{2} \cdot 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9405.a (trivial)

\(f(q)\)	\(=\)	\(q-1.91754 q^{2} +1.67697 q^{4} +1.00000 q^{5} +4.35488 q^{7} +0.619418 q^{8} +O(q^{10})\)	\(q-1.91754 q^{2} +1.67697 q^{4} +1.00000 q^{5} +4.35488 q^{7} +0.619418 q^{8} -1.91754 q^{10} -1.00000 q^{11} +4.48588 q^{13} -8.35066 q^{14} -4.54171 q^{16} -2.68689 q^{17} +1.00000 q^{19} +1.67697 q^{20} +1.91754 q^{22} +2.22008 q^{23} +1.00000 q^{25} -8.60187 q^{26} +7.30301 q^{28} +8.34690 q^{29} +4.48588 q^{31} +7.47009 q^{32} +5.15222 q^{34} +4.35488 q^{35} -10.6488 q^{37} -1.91754 q^{38} +0.619418 q^{40} -0.241461 q^{41} +12.4091 q^{43} -1.67697 q^{44} -4.25709 q^{46} -6.41516 q^{47} +11.9649 q^{49} -1.91754 q^{50} +7.52270 q^{52} -4.35482 q^{53} -1.00000 q^{55} +2.69749 q^{56} -16.0055 q^{58} -7.32266 q^{59} -7.45506 q^{61} -8.60187 q^{62} -5.24080 q^{64} +4.48588 q^{65} -12.5140 q^{67} -4.50584 q^{68} -8.35066 q^{70} +10.3984 q^{71} -2.70595 q^{73} +20.4195 q^{74} +1.67697 q^{76} -4.35488 q^{77} +8.67975 q^{79} -4.54171 q^{80} +0.463013 q^{82} -1.01594 q^{83} -2.68689 q^{85} -23.7950 q^{86} -0.619418 q^{88} -11.1078 q^{89} +19.5354 q^{91} +3.72301 q^{92} +12.3013 q^{94} +1.00000 q^{95} +17.2009 q^{97} -22.9433 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 4 q^{2} + 16 q^{4} + 10 q^{5} + 5 q^{7} - 3 q^{8}+O(q^{10}) \) 10 * q - 4 * q^2 + 16 * q^4 + 10 * q^5 + 5 * q^7 - 3 * q^8	\( 10 q - 4 q^{2} + 16 q^{4} + 10 q^{5} + 5 q^{7} - 3 q^{8} - 4 q^{10} - 10 q^{11} + 4 q^{13} - 6 q^{14} + 20 q^{16} - 10 q^{17} + 10 q^{19} + 16 q^{20} + 4 q^{22} + 6 q^{23} + 10 q^{25} + 3 q^{26} + 34 q^{28} - 5 q^{29} + 4 q^{31} - 30 q^{32} + 5 q^{34} + 5 q^{35} + 13 q^{37} - 4 q^{38} - 3 q^{40} - 9 q^{41} + 40 q^{43} - 16 q^{44} - 12 q^{46} + 24 q^{47} + 29 q^{49} - 4 q^{50} + 13 q^{52} - 13 q^{53} - 10 q^{55} - 8 q^{56} + 27 q^{58} - 5 q^{61} + 3 q^{62} + 27 q^{64} + 4 q^{65} + 39 q^{67} - 16 q^{68} - 6 q^{70} - 11 q^{71} + 30 q^{73} + 30 q^{74} + 16 q^{76} - 5 q^{77} + 4 q^{79} + 20 q^{80} + 24 q^{82} - 19 q^{83} - 10 q^{85} - 12 q^{86} + 3 q^{88} + 15 q^{89} - 17 q^{91} + 19 q^{92} + 2 q^{94} + 10 q^{95} + 58 q^{97} - 30 q^{98}+O(q^{100}) \) 10 * q - 4 * q^2 + 16 * q^4 + 10 * q^5 + 5 * q^7 - 3 * q^8 - 4 * q^10 - 10 * q^11 + 4 * q^13 - 6 * q^14 + 20 * q^16 - 10 * q^17 + 10 * q^19 + 16 * q^20 + 4 * q^22 + 6 * q^23 + 10 * q^25 + 3 * q^26 + 34 * q^28 - 5 * q^29 + 4 * q^31 - 30 * q^32 + 5 * q^34 + 5 * q^35 + 13 * q^37 - 4 * q^38 - 3 * q^40 - 9 * q^41 + 40 * q^43 - 16 * q^44 - 12 * q^46 + 24 * q^47 + 29 * q^49 - 4 * q^50 + 13 * q^52 - 13 * q^53 - 10 * q^55 - 8 * q^56 + 27 * q^58 - 5 * q^61 + 3 * q^62 + 27 * q^64 + 4 * q^65 + 39 * q^67 - 16 * q^68 - 6 * q^70 - 11 * q^71 + 30 * q^73 + 30 * q^74 + 16 * q^76 - 5 * q^77 + 4 * q^79 + 20 * q^80 + 24 * q^82 - 19 * q^83 - 10 * q^85 - 12 * q^86 + 3 * q^88 + 15 * q^89 - 17 * q^91 + 19 * q^92 + 2 * q^94 + 10 * q^95 + 58 * q^97 - 30 * q^98

Introduction

L-functions

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