LMFDB - Embedded newform 9405.2.a.bm.1.1

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			1.1
Root			$2.69746$ 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)

$f(q)$	$=$	$q-2.69746 q^{2} +5.27631 q^{4} +1.00000 q^{5} -1.40612 q^{7} -8.83771 q^{8} +O(q^{10})$	\(q-2.69746 q^{2} +5.27631 q^{4} +1.00000 q^{5} -1.40612 q^{7} -8.83771 q^{8} -2.69746 q^{10} -1.00000 q^{11} +5.01699 q^{13} +3.79295 q^{14} +13.2868 q^{16} +4.66205 q^{17} +1.00000 q^{19} +5.27631 q^{20} +2.69746 q^{22} -1.13206 q^{23} +1.00000 q^{25} -13.5331 q^{26} -7.41911 q^{28} +0.808221 q^{29} +5.01699 q^{31} -18.1652 q^{32} -12.5757 q^{34} -1.40612 q^{35} -0.673156 q^{37} -2.69746 q^{38} -8.83771 q^{40} -1.91557 q^{41} +9.74641 q^{43} -5.27631 q^{44} +3.05368 q^{46} +11.6882 q^{47} -5.02283 q^{49} -2.69746 q^{50} +26.4712 q^{52} +2.09021 q^{53} -1.00000 q^{55} +12.4269 q^{56} -2.18015 q^{58} +2.94694 q^{59} -2.16107 q^{61} -13.5331 q^{62} +22.4264 q^{64} +5.01699 q^{65} +12.0264 q^{67} +24.5984 q^{68} +3.79295 q^{70} -7.24526 q^{71} +0.115071 q^{73} +1.81581 q^{74} +5.27631 q^{76} +1.40612 q^{77} -11.2787 q^{79} +13.2868 q^{80} +5.16717 q^{82} +10.0518 q^{83} +4.66205 q^{85} -26.2906 q^{86} +8.83771 q^{88} +12.3812 q^{89} -7.05447 q^{91} -5.97308 q^{92} -31.5285 q^{94} +1.00000 q^{95} +2.28517 q^{97} +13.5489 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.69746	−1.90739	−0.953697	−	0.300769i	$-0.902757\pi$
$2$	−2.69746	−1.90739	−0.953697	+	0.300769i	$0.902757\pi$
$3$	0	0
$3$	0	0
$4$	5.27631	2.63815
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.40612	−0.531463	−0.265731	−	0.964047i	$-0.585613\pi$
$7$	−1.40612	−0.531463	−0.265731	+	0.964047i	$0.585613\pi$
$8$	−8.83771	−3.12460
$9$	0	0
$10$	−2.69746	−0.853013
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	5.01699	1.39146	0.695731	−	0.718302i	$-0.255081\pi$
$13$	5.01699	1.39146	0.695731	+	0.718302i	$0.255081\pi$
$14$	3.79295	1.01371
$15$	0	0
$16$	13.2868	3.32170
$17$	4.66205	1.13071	0.565356	−	0.824847i	$-0.308739\pi$
$17$	4.66205	1.13071	0.565356	+	0.824847i	$0.308739\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	5.27631	1.17982
$21$	0	0
$22$	2.69746	0.575101
$23$	−1.13206	−0.236050	−0.118025	−	0.993011i	$-0.537656\pi$
$23$	−1.13206	−0.236050	−0.118025	+	0.993011i	$0.537656\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−13.5331	−2.65407
$27$	0	0
$28$	−7.41911	−1.40208
$29$	0.808221	0.150083	0.0750415	−	0.997180i	$-0.476091\pi$
$29$	0.808221	0.150083	0.0750415	+	0.997180i	$0.476091\pi$
$30$	0	0
$31$	5.01699	0.901077	0.450539	−	0.892757i	$-0.351232\pi$
$31$	5.01699	0.901077	0.450539	+	0.892757i	$0.351232\pi$
$32$	−18.1652	−3.21118
$33$	0	0
$34$	−12.5757	−2.15672
$35$	−1.40612	−0.237677
$36$	0	0
$37$	−0.673156	−0.110666	−0.0553331	−	0.998468i	$-0.517622\pi$
$37$	−0.673156	−0.110666	−0.0553331	+	0.998468i	$0.517622\pi$
$38$	−2.69746	−0.437586
$39$	0	0
$40$	−8.83771	−1.39737
$41$	−1.91557	−0.299161	−0.149581	−	0.988750i	$-0.547792\pi$
$41$	−1.91557	−0.299161	−0.149581	+	0.988750i	$0.547792\pi$
$42$	0	0
$43$	9.74641	1.48631	0.743157	−	0.669117i	$-0.233328\pi$
$43$	9.74641	1.48631	0.743157	+	0.669117i	$0.233328\pi$
$44$	−5.27631	−0.795433
$45$	0	0
$46$	3.05368	0.450241
$47$	11.6882	1.70490	0.852451	−	0.522807i	$-0.175115\pi$
$47$	11.6882	1.70490	0.852451	+	0.522807i	$0.175115\pi$
$48$	0	0
$49$	−5.02283	−0.717548
$50$	−2.69746	−0.381479
$51$	0	0
$52$	26.4712	3.67089
$53$	2.09021	0.287112	0.143556	−	0.989642i	$-0.454146\pi$
$53$	2.09021	0.287112	0.143556	+	0.989642i	$0.454146\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	12.4269	1.66061
$57$	0	0
$58$	−2.18015	−0.286267
$59$	2.94694	0.383659	0.191830	−	0.981428i	$-0.438558\pi$
$59$	2.94694	0.383659	0.191830	+	0.981428i	$0.438558\pi$
$60$	0	0
$61$	−2.16107	−0.276697	−0.138348	−	0.990384i	$-0.544179\pi$
$61$	−2.16107	−0.276697	−0.138348	+	0.990384i	$0.544179\pi$
$62$	−13.5331	−1.71871
$63$	0	0
$64$	22.4264	2.80330
$65$	5.01699	0.622281
$66$	0	0
$67$	12.0264	1.46926	0.734628	−	0.678470i	$-0.237356\pi$
$67$	12.0264	1.46926	0.734628	+	0.678470i	$0.237356\pi$
$68$	24.5984	2.98299
$69$	0	0
$70$	3.79295	0.453344
$71$	−7.24526	−0.859855	−0.429927	−	0.902863i	$-0.641461\pi$
$71$	−7.24526	−0.859855	−0.429927	+	0.902863i	$0.641461\pi$
$72$	0	0
$73$	0.115071	0.0134680	0.00673400	−	0.999977i	$-0.497856\pi$
$73$	0.115071	0.0134680	0.00673400	+	0.999977i	$0.497856\pi$
$74$	1.81581	0.211084
$75$	0	0
$76$	5.27631	0.605234
$77$	1.40612	0.160242
$78$	0	0
$79$	−11.2787	−1.26895	−0.634475	−	0.772944i	$-0.718784\pi$
$79$	−11.2787	−1.26895	−0.634475	+	0.772944i	$0.718784\pi$
$80$	13.2868	1.48551
$81$	0	0
$82$	5.16717	0.570619
$83$	10.0518	1.10333	0.551663	−	0.834067i	$-0.313994\pi$
$83$	10.0518	1.10333	0.551663	+	0.834067i	$0.313994\pi$
$84$	0	0
$85$	4.66205	0.505670
$86$	−26.2906	−2.83499
$87$	0	0
$88$	8.83771	0.942103
$89$	12.3812	1.31240	0.656201	−	0.754586i	$-0.272162\pi$
$89$	12.3812	1.31240	0.656201	+	0.754586i	$0.272162\pi$
$90$	0	0
$91$	−7.05447	−0.739510
$92$	−5.97308	−0.622737
$93$	0	0
$94$	−31.5285	−3.25192
$95$	1.00000	0.102598
$96$	0	0
$97$	2.28517	0.232024	0.116012	−	0.993248i	$-0.462989\pi$
$97$	2.28517	0.232024	0.116012	+	0.993248i	$0.462989\pi$
$98$	13.5489	1.36865
$99$	0	0
$100$	5.27631	0.527631
$101$	−1.59838	−0.159045	−0.0795225	−	0.996833i	$-0.525340\pi$
$101$	−1.59838	−0.159045	−0.0795225	+	0.996833i	$0.525340\pi$
$102$	0	0
$103$	−0.0308290	−0.00303767	−0.00151884	−	0.999999i	$-0.500483\pi$
$103$	−0.0308290	−0.00303767	−0.00151884	+	0.999999i	$0.500483\pi$
$104$	−44.3387	−4.34777
$105$	0	0
$106$	−5.63826	−0.547636
$107$	3.58927	0.346988	0.173494	−	0.984835i	$-0.444494\pi$
$107$	3.58927	0.346988	0.173494	+	0.984835i	$0.444494\pi$
$108$	0	0
$109$	−13.9057	−1.33193	−0.665963	−	0.745985i	$-0.731979\pi$
$109$	−13.9057	−1.33193	−0.665963	+	0.745985i	$0.731979\pi$
$110$	2.69746	0.257193
$111$	0	0
$112$	−18.6828	−1.76536
$113$	2.29670	0.216055	0.108028	−	0.994148i	$-0.465546\pi$
$113$	2.29670	0.216055	0.108028	+	0.994148i	$0.465546\pi$
$114$	0	0
$115$	−1.13206	−0.105565
$116$	4.26442	0.395942
$117$	0	0
$118$	−7.94927	−0.731790
$119$	−6.55539	−0.600932
$120$	0	0
$121$	1.00000	0.0909091
$122$	5.82940	0.527769
$123$	0	0
$124$	26.4712	2.37718
$125$	1.00000	0.0894427
$126$	0	0
$127$	−8.62068	−0.764962	−0.382481	−	0.923963i	$-0.624930\pi$
$127$	−8.62068	−0.764962	−0.382481	+	0.923963i	$0.624930\pi$
$128$	−24.1639	−2.13581
$129$	0	0
$130$	−13.5331	−1.18693
$131$	−1.78441	−0.155905	−0.0779525	−	0.996957i	$-0.524838\pi$
$131$	−1.78441	−0.155905	−0.0779525	+	0.996957i	$0.524838\pi$
$132$	0	0
$133$	−1.40612	−0.121926
$134$	−32.4407	−2.80245
$135$	0	0
$136$	−41.2018	−3.53303
$137$	14.1021	1.20482	0.602410	−	0.798187i	$-0.294207\pi$
$137$	14.1021	1.20482	0.602410	+	0.798187i	$0.294207\pi$
$138$	0	0
$139$	−8.41218	−0.713512	−0.356756	−	0.934198i	$-0.616117\pi$
$139$	−8.41218	−0.713512	−0.356756	+	0.934198i	$0.616117\pi$
$140$	−7.41911	−0.627029
$141$	0	0
$142$	19.5438	1.64008
$143$	−5.01699	−0.419542
$144$	0	0
$145$	0.808221	0.0671191
$146$	−0.310399	−0.0256888
$147$	0	0
$148$	−3.55178	−0.291954
$149$	−9.99865	−0.819121	−0.409561	−	0.912283i	$-0.634318\pi$
$149$	−9.99865	−0.819121	−0.409561	+	0.912283i	$0.634318\pi$
$150$	0	0
$151$	−10.3224	−0.840023	−0.420011	−	0.907519i	$-0.637974\pi$
$151$	−10.3224	−0.840023	−0.420011	+	0.907519i	$0.637974\pi$
$152$	−8.83771	−0.716833
$153$	0	0
$154$	−3.79295	−0.305645
$155$	5.01699	0.402974
$156$	0	0
$157$	−23.8910	−1.90671	−0.953353	−	0.301857i	$-0.902393\pi$
$157$	−23.8910	−1.90671	−0.953353	+	0.301857i	$0.902393\pi$
$158$	30.4238	2.42039
$159$	0	0
$160$	−18.1652	−1.43608
$161$	1.59181	0.125452
$162$	0	0
$163$	22.4639	1.75951	0.879755	−	0.475427i	$-0.157706\pi$
$163$	22.4639	1.75951	0.879755	+	0.475427i	$0.157706\pi$
$164$	−10.1071	−0.789234
$165$	0	0
$166$	−27.1143	−2.10448
$167$	−19.0243	−1.47214	−0.736071	−	0.676905i	$-0.763321\pi$
$167$	−19.0243	−1.47214	−0.736071	+	0.676905i	$0.763321\pi$
$168$	0	0
$169$	12.1702	0.936166
$170$	−12.5757	−0.964512
$171$	0	0
$172$	51.4250	3.92112
$173$	−10.0883	−0.766997	−0.383498	−	0.923542i	$-0.625281\pi$
$173$	−10.0883	−0.766997	−0.383498	+	0.923542i	$0.625281\pi$
$174$	0	0
$175$	−1.40612	−0.106293
$176$	−13.2868	−1.00153
$177$	0	0
$178$	−33.3977	−2.50327
$179$	−9.64997	−0.721272	−0.360636	−	0.932707i	$-0.617440\pi$
$179$	−9.64997	−0.721272	−0.360636	+	0.932707i	$0.617440\pi$
$180$	0	0
$181$	8.35091	0.620718	0.310359	−	0.950619i	$-0.399551\pi$
$181$	8.35091	0.620718	0.310359	+	0.950619i	$0.399551\pi$
$182$	19.0292	1.41054
$183$	0	0
$184$	10.0048	0.737564
$185$	−0.673156	−0.0494914
$186$	0	0
$187$	−4.66205	−0.340923
$188$	61.6706	4.49779
$189$	0	0
$190$	−2.69746	−0.195695
$191$	26.2081	1.89635	0.948175	−	0.317750i	$-0.102927\pi$
$191$	26.2081	1.89635	0.948175	+	0.317750i	$0.102927\pi$
$192$	0	0
$193$	−13.9918	−1.00715	−0.503576	−	0.863951i	$-0.667982\pi$
$193$	−13.9918	−1.00715	−0.503576	+	0.863951i	$0.667982\pi$
$194$	−6.16415	−0.442560
$195$	0	0
$196$	−26.5020	−1.89300
$197$	−23.7263	−1.69043	−0.845215	−	0.534427i	$-0.820528\pi$
$197$	−23.7263	−1.69043	−0.845215	+	0.534427i	$0.820528\pi$
$198$	0	0
$199$	15.6795	1.11149	0.555746	−	0.831352i	$-0.312433\pi$
$199$	15.6795	1.11149	0.555746	+	0.831352i	$0.312433\pi$
$200$	−8.83771	−0.624921
$201$	0	0
$202$	4.31158	0.303361
$203$	−1.13645	−0.0797634
$204$	0	0
$205$	−1.91557	−0.133789
$206$	0.0831601	0.00579404
$207$	0	0
$208$	66.6597	4.62202
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	19.1010	1.31497	0.657485	−	0.753468i	$-0.271620\pi$
$211$	19.1010	1.31497	0.657485	+	0.753468i	$0.271620\pi$
$212$	11.0286	0.757445
$213$	0	0
$214$	−9.68192	−0.661843
$215$	9.74641	0.664700
$216$	0	0
$217$	−7.05447	−0.478889
$218$	37.5101	2.54051
$219$	0	0
$220$	−5.27631	−0.355728
$221$	23.3894	1.57334
$222$	0	0
$223$	−9.15846	−0.613296	−0.306648	−	0.951823i	$-0.599207\pi$
$223$	−9.15846	−0.613296	−0.306648	+	0.951823i	$0.599207\pi$
$224$	25.5424	1.70662
$225$	0	0
$226$	−6.19526	−0.412103
$227$	27.2511	1.80872	0.904359	−	0.426772i	$-0.140349\pi$
$227$	27.2511	1.80872	0.904359	+	0.426772i	$0.140349\pi$
$228$	0	0
$229$	15.1334	1.00004	0.500021	−	0.866013i	$-0.333326\pi$
$229$	15.1334	1.00004	0.500021	+	0.866013i	$0.333326\pi$
$230$	3.05368	0.201354
$231$	0	0
$232$	−7.14283	−0.468950
$233$	−16.3510	−1.07119	−0.535595	−	0.844475i	$-0.679913\pi$
$233$	−16.3510	−1.07119	−0.535595	+	0.844475i	$0.679913\pi$
$234$	0	0
$235$	11.6882	0.762456
$236$	15.5490	1.01215
$237$	0	0
$238$	17.6829	1.14621
$239$	19.6404	1.27043	0.635216	−	0.772335i	$-0.280911\pi$
$239$	19.6404	1.27043	0.635216	+	0.772335i	$0.280911\pi$
$240$	0	0
$241$	5.71386	0.368062	0.184031	−	0.982920i	$-0.441085\pi$
$241$	5.71386	0.368062	0.184031	+	0.982920i	$0.441085\pi$
$242$	−2.69746	−0.173399
$243$	0	0
$244$	−11.4025	−0.729968
$245$	−5.02283	−0.320897
$246$	0	0
$247$	5.01699	0.319223
$248$	−44.3387	−2.81551
$249$	0	0
$250$	−2.69746	−0.170603
$251$	15.9340	1.00575	0.502874	−	0.864360i	$-0.332276\pi$
$251$	15.9340	1.00575	0.502874	+	0.864360i	$0.332276\pi$
$252$	0	0
$253$	1.13206	0.0711719
$254$	23.2540	1.45908
$255$	0	0
$256$	20.3285	1.27053
$257$	16.6133	1.03631	0.518155	−	0.855287i	$-0.326619\pi$
$257$	16.6133	1.03631	0.518155	+	0.855287i	$0.326619\pi$
$258$	0	0
$259$	0.946536	0.0588149
$260$	26.4712	1.64167
$261$	0	0
$262$	4.81339	0.297372
$263$	8.26598	0.509702	0.254851	−	0.966980i	$-0.417974\pi$
$263$	8.26598	0.509702	0.254851	+	0.966980i	$0.417974\pi$
$264$	0	0
$265$	2.09021	0.128400
$266$	3.79295	0.232561
$267$	0	0
$268$	63.4549	3.87612
$269$	23.7962	1.45088	0.725439	−	0.688287i	$-0.241637\pi$
$269$	23.7962	1.45088	0.725439	+	0.688287i	$0.241637\pi$
$270$	0	0
$271$	−27.3012	−1.65843	−0.829215	−	0.558930i	$-0.811212\pi$
$271$	−27.3012	−1.65843	−0.829215	+	0.558930i	$0.811212\pi$
$272$	61.9437	3.75589
$273$	0	0
$274$	−38.0398	−2.29807
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	7.43984	0.447017	0.223508	−	0.974702i	$-0.428249\pi$
$277$	7.43984	0.447017	0.223508	+	0.974702i	$0.428249\pi$
$278$	22.6915	1.36095
$279$	0	0
$280$	12.4269	0.742647
$281$	0.966594	0.0576622	0.0288311	−	0.999584i	$-0.490822\pi$
$281$	0.966594	0.0576622	0.0288311	+	0.999584i	$0.490822\pi$
$282$	0	0
$283$	−8.91259	−0.529799	−0.264899	−	0.964276i	$-0.585339\pi$
$283$	−8.91259	−0.529799	−0.264899	+	0.964276i	$0.585339\pi$
$284$	−38.2282	−2.26843
$285$	0	0
$286$	13.5331	0.800231
$287$	2.69351	0.158993
$288$	0	0
$289$	4.73470	0.278512
$290$	−2.18015	−0.128023
$291$	0	0
$292$	0.607148	0.0355306
$293$	−12.3764	−0.723039	−0.361519	−	0.932365i	$-0.617742\pi$
$293$	−12.3764	−0.723039	−0.361519	+	0.932365i	$0.617742\pi$
$294$	0	0
$295$	2.94694	0.171578
$296$	5.94916	0.345788
$297$	0	0
$298$	26.9710	1.56239
$299$	−5.67952	−0.328455
$300$	0	0
$301$	−13.7046	−0.789920
$302$	27.8442	1.60225
$303$	0	0
$304$	13.2868	0.762050
$305$	−2.16107	−0.123742
$306$	0	0
$307$	1.81161	0.103394	0.0516971	−	0.998663i	$-0.483537\pi$
$307$	1.81161	0.103394	0.0516971	+	0.998663i	$0.483537\pi$
$308$	7.41911	0.422743
$309$	0	0
$310$	−13.5331	−0.768630
$311$	−15.5874	−0.883880	−0.441940	−	0.897045i	$-0.645710\pi$
$311$	−15.5874	−0.883880	−0.441940	+	0.897045i	$0.645710\pi$
$312$	0	0
$313$	22.4976	1.27164	0.635821	−	0.771837i	$-0.280662\pi$
$313$	22.4976	1.27164	0.635821	+	0.771837i	$0.280662\pi$
$314$	64.4450	3.63684
$315$	0	0
$316$	−59.5097	−3.34768
$317$	−28.1537	−1.58127	−0.790635	−	0.612287i	$-0.790250\pi$
$317$	−28.1537	−1.58127	−0.790635	+	0.612287i	$0.790250\pi$
$318$	0	0
$319$	−0.808221	−0.0452517
$320$	22.4264	1.25367
$321$	0	0
$322$	−4.29384	−0.239286
$323$	4.66205	0.259403
$324$	0	0
$325$	5.01699	0.278292
$326$	−60.5956	−3.35608
$327$	0	0
$328$	16.9292	0.934761
$329$	−16.4350	−0.906092
$330$	0	0
$331$	−18.7770	−1.03208	−0.516039	−	0.856565i	$-0.672594\pi$
$331$	−18.7770	−1.03208	−0.516039	+	0.856565i	$0.672594\pi$
$332$	53.0362	2.91074
$333$	0	0
$334$	51.3172	2.80795
$335$	12.0264	0.657072
$336$	0	0
$337$	−5.42312	−0.295416	−0.147708	−	0.989031i	$-0.547190\pi$
$337$	−5.42312	−0.295416	−0.147708	+	0.989031i	$0.547190\pi$
$338$	−32.8286	−1.78564
$339$	0	0
$340$	24.5984	1.33404
$341$	−5.01699	−0.271685
$342$	0	0
$343$	16.9055	0.912812
$344$	−86.1360	−4.64414
$345$	0	0
$346$	27.2127	1.46296
$347$	−19.6791	−1.05643	−0.528214	−	0.849111i	$-0.677138\pi$
$347$	−19.6791	−1.05643	−0.528214	+	0.849111i	$0.677138\pi$
$348$	0	0
$349$	22.4859	1.20364	0.601822	−	0.798630i	$-0.294442\pi$
$349$	22.4859	1.20364	0.601822	+	0.798630i	$0.294442\pi$
$350$	3.79295	0.202742
$351$	0	0
$352$	18.1652	0.968208
$353$	32.7869	1.74507	0.872537	−	0.488549i	$-0.162474\pi$
$353$	32.7869	1.74507	0.872537	+	0.488549i	$0.162474\pi$
$354$	0	0
$355$	−7.24526	−0.384539
$356$	65.3268	3.46232
$357$	0	0
$358$	26.0304	1.37575
$359$	23.9266	1.26280	0.631399	−	0.775458i	$-0.282481\pi$
$359$	23.9266	1.26280	0.631399	+	0.775458i	$0.282481\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−22.5263	−1.18395
$363$	0	0
$364$	−37.2216	−1.95094
$365$	0.115071	0.00602307
$366$	0	0
$367$	−12.9529	−0.676137	−0.338069	−	0.941121i	$-0.609774\pi$
$367$	−12.9529	−0.676137	−0.338069	+	0.941121i	$0.609774\pi$
$368$	−15.0414	−0.784088
$369$	0	0
$370$	1.81581	0.0943996
$371$	−2.93908	−0.152589
$372$	0	0
$373$	−4.89069	−0.253231	−0.126615	−	0.991952i	$-0.540411\pi$
$373$	−4.89069	−0.253231	−0.126615	+	0.991952i	$0.540411\pi$
$374$	12.5757	0.650274
$375$	0	0
$376$	−103.297	−5.32714
$377$	4.05484	0.208835
$378$	0	0
$379$	−33.5136	−1.72148	−0.860738	−	0.509049i	$-0.829997\pi$
$379$	−33.5136	−1.72148	−0.860738	+	0.509049i	$0.829997\pi$
$380$	5.27631	0.270669
$381$	0	0
$382$	−70.6953	−3.61709
$383$	−7.19995	−0.367900	−0.183950	−	0.982936i	$-0.558888\pi$
$383$	−7.19995	−0.367900	−0.183950	+	0.982936i	$0.558888\pi$
$384$	0	0
$385$	1.40612	0.0716624
$386$	37.7423	1.92103
$387$	0	0
$388$	12.0572	0.612114
$389$	−4.71089	−0.238851	−0.119426	−	0.992843i	$-0.538105\pi$
$389$	−4.71089	−0.238851	−0.119426	+	0.992843i	$0.538105\pi$
$390$	0	0
$391$	−5.27771	−0.266905
$392$	44.3904	2.24205
$393$	0	0
$394$	64.0008	3.22432
$395$	−11.2787	−0.567491
$396$	0	0
$397$	−9.32680	−0.468098	−0.234049	−	0.972225i	$-0.575198\pi$
$397$	−9.32680	−0.468098	−0.234049	+	0.972225i	$0.575198\pi$
$398$	−42.2950	−2.12006
$399$	0	0
$400$	13.2868	0.664340
$401$	0.176644	0.00882117	0.00441058	−	0.999990i	$-0.498596\pi$
$401$	0.176644	0.00882117	0.00441058	+	0.999990i	$0.498596\pi$
$402$	0	0
$403$	25.1702	1.25381
$404$	−8.43355	−0.419585
$405$	0	0
$406$	3.06554	0.152140
$407$	0.673156	0.0333671
$408$	0	0
$409$	−7.01295	−0.346768	−0.173384	−	0.984854i	$-0.555470\pi$
$409$	−7.01295	−0.346768	−0.173384	+	0.984854i	$0.555470\pi$
$410$	5.16717	0.255189
$411$	0	0
$412$	−0.162663	−0.00801384
$413$	−4.14375	−0.203901
$414$	0	0
$415$	10.0518	0.493422
$416$	−91.1346	−4.46824
$417$	0	0
$418$	2.69746	0.131937
$419$	−23.5858	−1.15224	−0.576121	−	0.817365i	$-0.695434\pi$
$419$	−23.5858	−1.15224	−0.576121	+	0.817365i	$0.695434\pi$
$420$	0	0
$421$	−21.9892	−1.07169	−0.535844	−	0.844317i	$-0.680006\pi$
$421$	−21.9892	−1.07169	−0.535844	+	0.844317i	$0.680006\pi$
$422$	−51.5243	−2.50817
$423$	0	0
$424$	−18.4726	−0.897111
$425$	4.66205	0.226143
$426$	0	0
$427$	3.03872	0.147054
$428$	18.9381	0.915407
$429$	0	0
$430$	−26.2906	−1.26784
$431$	6.65872	0.320739	0.160370	−	0.987057i	$-0.448731\pi$
$431$	6.65872	0.320739	0.160370	+	0.987057i	$0.448731\pi$
$432$	0	0
$433$	27.7058	1.33146	0.665728	−	0.746195i	$-0.268121\pi$
$433$	27.7058	1.33146	0.665728	+	0.746195i	$0.268121\pi$
$434$	19.0292	0.913430
$435$	0	0
$436$	−73.3708	−3.51382
$437$	−1.13206	−0.0541537
$438$	0	0
$439$	−7.10884	−0.339286	−0.169643	−	0.985506i	$-0.554261\pi$
$439$	−7.10884	−0.339286	−0.169643	+	0.985506i	$0.554261\pi$
$440$	8.83771	0.421321
$441$	0	0
$442$	−63.0921	−3.00099
$443$	19.5733	0.929954	0.464977	−	0.885323i	$-0.346063\pi$
$443$	19.5733	0.929954	0.464977	+	0.885323i	$0.346063\pi$
$444$	0	0
$445$	12.3812	0.586924
$446$	24.7046	1.16980
$447$	0	0
$448$	−31.5341	−1.48985
$449$	−15.8026	−0.745773	−0.372886	−	0.927877i	$-0.621632\pi$
$449$	−15.8026	−0.745773	−0.372886	+	0.927877i	$0.621632\pi$
$450$	0	0
$451$	1.91557	0.0902006
$452$	12.1181	0.569987
$453$	0	0
$454$	−73.5088	−3.44994
$455$	−7.05447	−0.330719
$456$	0	0
$457$	17.2740	0.808045	0.404022	−	0.914749i	$-0.367612\pi$
$457$	17.2740	0.808045	0.404022	+	0.914749i	$0.367612\pi$
$458$	−40.8218	−1.90748
$459$	0	0
$460$	−5.97308	−0.278496
$461$	−6.91319	−0.321979	−0.160990	−	0.986956i	$-0.551469\pi$
$461$	−6.91319	−0.321979	−0.160990	+	0.986956i	$0.551469\pi$
$462$	0	0
$463$	33.5631	1.55981	0.779906	−	0.625897i	$-0.215267\pi$
$463$	33.5631	1.55981	0.779906	+	0.625897i	$0.215267\pi$
$464$	10.7387	0.498530
$465$	0	0
$466$	44.1063	2.04318
$467$	−0.0247568	−0.00114561	−0.000572805	−	1.00000i	$-0.500182\pi$
$467$	−0.0247568	−0.00114561	−0.000572805		1.00000i	$0.500182\pi$
$468$	0	0
$469$	−16.9105	−0.780855
$470$	−31.5285	−1.45430
$471$	0	0
$472$	−26.0442	−1.19878
$473$	−9.74641	−0.448140
$474$	0	0
$475$	1.00000	0.0458831
$476$	−34.5882	−1.58535
$477$	0	0
$478$	−52.9793	−2.42321
$479$	−11.7875	−0.538585	−0.269293	−	0.963058i	$-0.586790\pi$
$479$	−11.7875	−0.538585	−0.269293	+	0.963058i	$0.586790\pi$
$480$	0	0
$481$	−3.37721	−0.153988
$482$	−15.4129	−0.702039
$483$	0	0
$484$	5.27631	0.239832
$485$	2.28517	0.103764
$486$	0	0
$487$	−10.4020	−0.471362	−0.235681	−	0.971831i	$-0.575732\pi$
$487$	−10.4020	−0.471362	−0.235681	+	0.971831i	$0.575732\pi$
$488$	19.0989	0.864567
$489$	0	0
$490$	13.5489	0.612077
$491$	−24.9081	−1.12409	−0.562043	−	0.827108i	$-0.689985\pi$
$491$	−24.9081	−1.12409	−0.562043	+	0.827108i	$0.689985\pi$
$492$	0	0
$493$	3.76797	0.169701
$494$	−13.5331	−0.608885
$495$	0	0
$496$	66.6597	2.99311
$497$	10.1877	0.456981
$498$	0	0
$499$	14.4290	0.645931	0.322965	−	0.946411i	$-0.395320\pi$
$499$	14.4290	0.645931	0.322965	+	0.946411i	$0.395320\pi$
$500$	5.27631	0.235964
$501$	0	0
$502$	−42.9815	−1.91836
$503$	−7.09937	−0.316545	−0.158273	−	0.987395i	$-0.550592\pi$
$503$	−7.09937	−0.316545	−0.158273	+	0.987395i	$0.550592\pi$
$504$	0	0
$505$	−1.59838	−0.0711271
$506$	−3.05368	−0.135753
$507$	0	0
$508$	−45.4853	−2.01809
$509$	−24.1458	−1.07025	−0.535123	−	0.844774i	$-0.679735\pi$
$509$	−24.1458	−1.07025	−0.535123	+	0.844774i	$0.679735\pi$
$510$	0	0
$511$	−0.161803	−0.00715774
$512$	−6.50750	−0.287594
$513$	0	0
$514$	−44.8138	−1.97665
$515$	−0.0308290	−0.00135849
$516$	0	0
$517$	−11.6882	−0.514047
$518$	−2.55325	−0.112183
$519$	0	0
$520$	−44.3387	−1.94438
$521$	−25.1347	−1.10117	−0.550586	−	0.834778i	$-0.685596\pi$
$521$	−25.1347	−1.10117	−0.550586	+	0.834778i	$0.685596\pi$
$522$	0	0
$523$	33.5952	1.46901	0.734507	−	0.678601i	$-0.237413\pi$
$523$	33.5952	1.46901	0.734507	+	0.678601i	$0.237413\pi$
$524$	−9.41511	−0.411301
$525$	0	0
$526$	−22.2972	−0.972203
$527$	23.3894	1.01886
$528$	0	0
$529$	−21.7184	−0.944280
$530$	−5.63826	−0.244910
$531$	0	0
$532$	−7.41911	−0.321659
$533$	−9.61038	−0.416272
$534$	0	0
$535$	3.58927	0.155178
$536$	−106.286	−4.59085
$537$	0	0
$538$	−64.1893	−2.76740
$539$	5.02283	0.216349
$540$	0	0
$541$	21.3488	0.917855	0.458928	−	0.888474i	$-0.348234\pi$
$541$	21.3488	0.917855	0.458928	+	0.888474i	$0.348234\pi$
$542$	73.6440	3.16328
$543$	0	0
$544$	−84.6870	−3.63093
$545$	−13.9057	−0.595655
$546$	0	0
$547$	45.7018	1.95407	0.977033	−	0.213088i	$-0.0683521\pi$
$547$	45.7018	1.95407	0.977033	+	0.213088i	$0.0683521\pi$
$548$	74.4068	3.17850
$549$	0	0
$550$	2.69746	0.115020
$551$	0.808221	0.0344314
$552$	0	0
$553$	15.8591	0.674399
$554$	−20.0687	−0.852638
$555$	0	0
$556$	−44.3852	−1.88235
$557$	−31.8893	−1.35119	−0.675597	−	0.737271i	$-0.736114\pi$
$557$	−31.8893	−1.35119	−0.675597	+	0.737271i	$0.736114\pi$
$558$	0	0
$559$	48.8976	2.06815
$560$	−18.6828	−0.789492
$561$	0	0
$562$	−2.60735	−0.109985
$563$	0.348296	0.0146789	0.00733947	−	0.999973i	$-0.497664\pi$
$563$	0.348296	0.0146789	0.00733947	+	0.999973i	$0.497664\pi$
$564$	0	0
$565$	2.29670	0.0966229
$566$	24.0414	1.01054
$567$	0	0
$568$	64.0316	2.68671
$569$	−21.3120	−0.893448	−0.446724	−	0.894672i	$-0.647409\pi$
$569$	−21.3120	−0.893448	−0.446724	+	0.894672i	$0.647409\pi$
$570$	0	0
$571$	38.1340	1.59586	0.797929	−	0.602752i	$-0.205929\pi$
$571$	38.1340	1.59586	0.797929	+	0.602752i	$0.205929\pi$
$572$	−26.4712	−1.10681
$573$	0	0
$574$	−7.26565	−0.303263
$575$	−1.13206	−0.0472101
$576$	0	0
$577$	−17.3041	−0.720379	−0.360189	−	0.932879i	$-0.617288\pi$
$577$	−17.3041	−0.720379	−0.360189	+	0.932879i	$0.617288\pi$
$578$	−12.7717	−0.531232
$579$	0	0
$580$	4.26442	0.177070
$581$	−14.1340	−0.586376
$582$	0	0
$583$	−2.09021	−0.0865675
$584$	−1.01696	−0.0420822
$585$	0	0
$586$	33.3850	1.37912
$587$	−12.9968	−0.536436	−0.268218	−	0.963358i	$-0.586435\pi$
$587$	−12.9968	−0.536436	−0.268218	+	0.963358i	$0.586435\pi$
$588$	0	0
$589$	5.01699	0.206721
$590$	−7.94927	−0.327266
$591$	0	0
$592$	−8.94408	−0.367599
$593$	−2.38938	−0.0981200	−0.0490600	−	0.998796i	$-0.515623\pi$
$593$	−2.38938	−0.0981200	−0.0490600	+	0.998796i	$0.515623\pi$
$594$	0	0
$595$	−6.55539	−0.268745
$596$	−52.7559	−2.16097
$597$	0	0
$598$	15.3203	0.626493
$599$	−7.18538	−0.293587	−0.146793	−	0.989167i	$-0.546895\pi$
$599$	−7.18538	−0.293587	−0.146793	+	0.989167i	$0.546895\pi$
$600$	0	0
$601$	−5.32419	−0.217178	−0.108589	−	0.994087i	$-0.534633\pi$
$601$	−5.32419	−0.217178	−0.108589	+	0.994087i	$0.534633\pi$
$602$	36.9676	1.50669
$603$	0	0
$604$	−54.4640	−2.21611
$605$	1.00000	0.0406558
$606$	0	0
$607$	−27.9968	−1.13635	−0.568177	−	0.822906i	$-0.692351\pi$
$607$	−27.9968	−1.13635	−0.568177	+	0.822906i	$0.692351\pi$
$608$	−18.1652	−0.736696
$609$	0	0
$610$	5.82940	0.236026
$611$	58.6397	2.37231
$612$	0	0
$613$	15.9169	0.642877	0.321439	−	0.946930i	$-0.395834\pi$
$613$	15.9169	0.642877	0.321439	+	0.946930i	$0.395834\pi$
$614$	−4.88676	−0.197214
$615$	0	0
$616$	−12.4269	−0.500693
$617$	3.91208	0.157494	0.0787471	−	0.996895i	$-0.474908\pi$
$617$	3.91208	0.157494	0.0787471	+	0.996895i	$0.474908\pi$
$618$	0	0
$619$	46.2645	1.85953	0.929764	−	0.368157i	$-0.120011\pi$
$619$	46.2645	1.85953	0.929764	+	0.368157i	$0.120011\pi$
$620$	26.4712	1.06311
$621$	0	0
$622$	42.0464	1.68591
$623$	−17.4094	−0.697492
$624$	0	0
$625$	1.00000	0.0400000
$626$	−60.6865	−2.42552
$627$	0	0
$628$	−126.056	−5.03018
$629$	−3.13829	−0.125132
$630$	0	0
$631$	41.9303	1.66922	0.834610	−	0.550841i	$-0.185693\pi$
$631$	41.9303	1.66922	0.834610	+	0.550841i	$0.185693\pi$
$632$	99.6776	3.96496
$633$	0	0
$634$	75.9436	3.01611
$635$	−8.62068	−0.342101
$636$	0	0
$637$	−25.1995	−0.998440
$638$	2.18015	0.0863128
$639$	0	0
$640$	−24.1639	−0.955162
$641$	−22.3702	−0.883568	−0.441784	−	0.897121i	$-0.645654\pi$
$641$	−22.3702	−0.883568	−0.441784	+	0.897121i	$0.645654\pi$
$642$	0	0
$643$	24.1399	0.951986	0.475993	−	0.879449i	$-0.342089\pi$
$643$	24.1399	0.951986	0.475993	+	0.879449i	$0.342089\pi$
$644$	8.39886	0.330961
$645$	0	0
$646$	−12.5757	−0.494784
$647$	40.1149	1.57708	0.788540	−	0.614984i	$-0.210838\pi$
$647$	40.1149	1.57708	0.788540	+	0.614984i	$0.210838\pi$
$648$	0	0
$649$	−2.94694	−0.115678
$650$	−13.5331	−0.530813
$651$	0	0
$652$	118.527	4.64186
$653$	−28.1355	−1.10103	−0.550514	−	0.834826i	$-0.685568\pi$
$653$	−28.1355	−1.10103	−0.550514	+	0.834826i	$0.685568\pi$
$654$	0	0
$655$	−1.78441	−0.0697228
$656$	−25.4518	−0.993724
$657$	0	0
$658$	44.3328	1.72827
$659$	30.0783	1.17169	0.585843	−	0.810425i	$-0.300764\pi$
$659$	30.0783	1.17169	0.585843	+	0.810425i	$0.300764\pi$
$660$	0	0
$661$	−39.0094	−1.51729	−0.758645	−	0.651505i	$-0.774138\pi$
$661$	−39.0094	−1.51729	−0.758645	+	0.651505i	$0.774138\pi$
$662$	50.6503	1.96858
$663$	0	0
$664$	−88.8347	−3.44745
$665$	−1.40612	−0.0545269
$666$	0	0
$667$	−0.914953	−0.0354271
$668$	−100.378	−3.88373
$669$	0	0
$670$	−32.4407	−1.25329
$671$	2.16107	0.0834272
$672$	0	0
$673$	11.8069	0.455124	0.227562	−	0.973764i	$-0.426925\pi$
$673$	11.8069	0.455124	0.227562	+	0.973764i	$0.426925\pi$
$674$	14.6287	0.563476
$675$	0	0
$676$	64.2135	2.46975
$677$	2.23798	0.0860126	0.0430063	−	0.999075i	$-0.486306\pi$
$677$	2.23798	0.0860126	0.0430063	+	0.999075i	$0.486306\pi$
$678$	0	0
$679$	−3.21321	−0.123312
$680$	−41.2018	−1.58002
$681$	0	0
$682$	13.5331	0.518211
$683$	−1.35333	−0.0517836	−0.0258918	−	0.999665i	$-0.508243\pi$
$683$	−1.35333	−0.0517836	−0.0258918	+	0.999665i	$0.508243\pi$
$684$	0	0
$685$	14.1021	0.538812
$686$	−45.6020	−1.74109
$687$	0	0
$688$	129.499	4.93708
$689$	10.4865	0.399505
$690$	0	0
$691$	−46.8171	−1.78100	−0.890502	−	0.454979i	$-0.849647\pi$
$691$	−46.8171	−1.78100	−0.890502	+	0.454979i	$0.849647\pi$
$692$	−53.2288	−2.02345
$693$	0	0
$694$	53.0835	2.01502
$695$	−8.41218	−0.319092
$696$	0	0
$697$	−8.93047	−0.338266
$698$	−60.6550	−2.29582
$699$	0	0
$700$	−7.41911	−0.280416
$701$	−11.1371	−0.420641	−0.210321	−	0.977632i	$-0.567451\pi$
$701$	−11.1371	−0.420641	−0.210321	+	0.977632i	$0.567451\pi$
$702$	0	0
$703$	−0.673156	−0.0253886
$704$	−22.4264	−0.845225
$705$	0	0
$706$	−88.4416	−3.32854
$707$	2.24751	0.0845264
$708$	0	0
$709$	4.94349	0.185657	0.0928284	−	0.995682i	$-0.470409\pi$
$709$	4.94349	0.185657	0.0928284	+	0.995682i	$0.470409\pi$
$710$	19.5438	0.733467
$711$	0	0
$712$	−109.421	−4.10073
$713$	−5.67952	−0.212700
$714$	0	0
$715$	−5.01699	−0.187625
$716$	−50.9162	−1.90283
$717$	0	0
$718$	−64.5411	−2.40865
$719$	35.5266	1.32492	0.662459	−	0.749098i	$-0.269513\pi$
$719$	35.5266	1.32492	0.662459	+	0.749098i	$0.269513\pi$
$720$	0	0
$721$	0.0433492	0.00161441
$722$	−2.69746	−0.100389
$723$	0	0
$724$	44.0620	1.63755
$725$	0.808221	0.0300166
$726$	0	0
$727$	−4.88967	−0.181348	−0.0906739	−	0.995881i	$-0.528902\pi$
$727$	−4.88967	−0.181348	−0.0906739	+	0.995881i	$0.528902\pi$
$728$	62.3454	2.31067
$729$	0	0
$730$	−0.310399	−0.0114884
$731$	45.4382	1.68059
$732$	0	0
$733$	7.10138	0.262295	0.131148	−	0.991363i	$-0.458134\pi$
$733$	7.10138	0.262295	0.131148	+	0.991363i	$0.458134\pi$
$734$	34.9400	1.28966
$735$	0	0
$736$	20.5641	0.758001
$737$	−12.0264	−0.442998
$738$	0	0
$739$	28.3115	1.04145	0.520727	−	0.853723i	$-0.325661\pi$
$739$	28.3115	1.04145	0.520727	+	0.853723i	$0.325661\pi$
$740$	−3.55178	−0.130566
$741$	0	0
$742$	7.92805	0.291048
$743$	34.7657	1.27543	0.637714	−	0.770273i	$-0.279880\pi$
$743$	34.7657	1.27543	0.637714	+	0.770273i	$0.279880\pi$
$744$	0	0
$745$	−9.99865	−0.366322
$746$	13.1925	0.483010
$747$	0	0
$748$	−24.5984	−0.899406
$749$	−5.04694	−0.184411
$750$	0	0
$751$	30.9171	1.12818	0.564091	−	0.825713i	$-0.309227\pi$
$751$	30.9171	1.12818	0.564091	+	0.825713i	$0.309227\pi$
$752$	155.299	5.66317
$753$	0	0
$754$	−10.9378	−0.398330
$755$	−10.3224	−0.375670
$756$	0	0
$757$	−7.25068	−0.263530	−0.131765	−	0.991281i	$-0.542064\pi$
$757$	−7.25068	−0.263530	−0.131765	+	0.991281i	$0.542064\pi$
$758$	90.4016	3.28353
$759$	0	0
$760$	−8.83771	−0.320578
$761$	31.0893	1.12699	0.563494	−	0.826120i	$-0.309457\pi$
$761$	31.0893	1.12699	0.563494	+	0.826120i	$0.309457\pi$
$762$	0	0
$763$	19.5531	0.707868
$764$	138.282	5.00286
$765$	0	0
$766$	19.4216	0.701731
$767$	14.7848	0.533847
$768$	0	0
$769$	−4.18138	−0.150784	−0.0753922	−	0.997154i	$-0.524021\pi$
$769$	−4.18138	−0.150784	−0.0753922	+	0.997154i	$0.524021\pi$
$770$	−3.79295	−0.136688
$771$	0	0
$772$	−73.8250	−2.65702
$773$	29.7300	1.06931	0.534656	−	0.845070i	$-0.320441\pi$
$773$	29.7300	1.06931	0.534656	+	0.845070i	$0.320441\pi$
$774$	0	0
$775$	5.01699	0.180215
$776$	−20.1957	−0.724982
$777$	0	0
$778$	12.7074	0.455584
$779$	−1.91557	−0.0686323
$780$	0	0
$781$	7.24526	0.259256
$782$	14.2364	0.509093
$783$	0	0
$784$	−66.7373	−2.38348
$785$	−23.8910	−0.852705
$786$	0	0
$787$	−30.6245	−1.09164	−0.545822	−	0.837901i	$-0.683783\pi$
$787$	−30.6245	−1.09164	−0.545822	+	0.837901i	$0.683783\pi$
$788$	−125.187	−4.45961
$789$	0	0
$790$	30.4238	1.08243
$791$	−3.22943	−0.114825
$792$	0	0
$793$	−10.8421	−0.385013
$794$	25.1587	0.892848
$795$	0	0
$796$	82.7300	2.93229
$797$	−9.33845	−0.330785	−0.165392	−	0.986228i	$-0.552889\pi$
$797$	−9.33845	−0.330785	−0.165392	+	0.986228i	$0.552889\pi$
$798$	0	0
$799$	54.4911	1.92776
$800$	−18.1652	−0.642237
$801$	0	0
$802$	−0.476490	−0.0168254
$803$	−0.115071	−0.00406075
$804$	0	0
$805$	1.59181	0.0561038
$806$	−67.8956	−2.39152
$807$	0	0
$808$	14.1260	0.496952
$809$	−13.3621	−0.469787	−0.234894	−	0.972021i	$-0.575474\pi$
$809$	−13.3621	−0.469787	−0.234894	+	0.972021i	$0.575474\pi$
$810$	0	0
$811$	−15.2884	−0.536849	−0.268425	−	0.963301i	$-0.586503\pi$
$811$	−15.2884	−0.536849	−0.268425	+	0.963301i	$0.586503\pi$
$812$	−5.99628	−0.210428
$813$	0	0
$814$	−1.81581	−0.0636442
$815$	22.4639	0.786877
$816$	0	0
$817$	9.74641	0.340984
$818$	18.9172	0.661424
$819$	0	0
$820$	−10.1071	−0.352956
$821$	50.8035	1.77306	0.886528	−	0.462676i	$-0.153111\pi$
$821$	50.8035	1.77306	0.886528	+	0.462676i	$0.153111\pi$
$822$	0	0
$823$	−41.1037	−1.43279	−0.716393	−	0.697697i	$-0.754208\pi$
$823$	−41.1037	−1.43279	−0.716393	+	0.697697i	$0.754208\pi$
$824$	0.272458	0.00949152
$825$	0	0
$826$	11.1776	0.388919
$827$	−15.3631	−0.534228	−0.267114	−	0.963665i	$-0.586070\pi$
$827$	−15.3631	−0.534228	−0.267114	+	0.963665i	$0.586070\pi$
$828$	0	0
$829$	24.3478	0.845635	0.422818	−	0.906215i	$-0.361041\pi$
$829$	24.3478	0.845635	0.422818	+	0.906215i	$0.361041\pi$
$830$	−27.1143	−0.941151
$831$	0	0
$832$	112.513	3.90068
$833$	−23.4167	−0.811340
$834$	0	0
$835$	−19.0243	−0.658362
$836$	−5.27631	−0.182485
$837$	0	0
$838$	63.6218	2.19778
$839$	41.3971	1.42919	0.714593	−	0.699541i	$-0.246612\pi$
$839$	41.3971	1.42919	0.714593	+	0.699541i	$0.246612\pi$
$840$	0	0
$841$	−28.3468	−0.977475
$842$	59.3150	2.04413
$843$	0	0
$844$	100.783	3.46909
$845$	12.1702	0.418666
$846$	0	0
$847$	−1.40612	−0.0483148
$848$	27.7721	0.953699
$849$	0	0
$850$	−12.5757	−0.431343
$851$	0.762051	0.0261228
$852$	0	0
$853$	−5.51540	−0.188844	−0.0944218	−	0.995532i	$-0.530100\pi$
$853$	−5.51540	−0.188844	−0.0944218	+	0.995532i	$0.530100\pi$
$854$	−8.19683	−0.280490
$855$	0	0
$856$	−31.7209	−1.08420
$857$	−15.1585	−0.517804	−0.258902	−	0.965904i	$-0.583361\pi$
$857$	−15.1585	−0.517804	−0.258902	+	0.965904i	$0.583361\pi$
$858$	0	0
$859$	−7.51823	−0.256519	−0.128259	−	0.991741i	$-0.540939\pi$
$859$	−7.51823	−0.256519	−0.128259	+	0.991741i	$0.540939\pi$
$860$	51.4250	1.75358
$861$	0	0
$862$	−17.9616	−0.611776
$863$	25.2622	0.859936	0.429968	−	0.902844i	$-0.358525\pi$
$863$	25.2622	0.859936	0.429968	+	0.902844i	$0.358525\pi$
$864$	0	0
$865$	−10.0883	−0.343011
$866$	−74.7353	−2.53961
$867$	0	0
$868$	−37.2216	−1.26338
$869$	11.2787	0.382603
$870$	0	0
$871$	60.3362	2.04442
$872$	122.895	4.16174
$873$	0	0
$874$	3.05368	0.103292
$875$	−1.40612	−0.0475355
$876$	0	0
$877$	−14.0647	−0.474932	−0.237466	−	0.971396i	$-0.576317\pi$
$877$	−14.0647	−0.474932	−0.237466	+	0.971396i	$0.576317\pi$
$878$	19.1758	0.647153
$879$	0	0
$880$	−13.2868	−0.447898
$881$	−41.7826	−1.40769	−0.703846	−	0.710352i	$-0.748536\pi$
$881$	−41.7826	−1.40769	−0.703846	+	0.710352i	$0.748536\pi$
$882$	0	0
$883$	34.7074	1.16800	0.583999	−	0.811754i	$-0.301487\pi$
$883$	34.7074	1.16800	0.583999	+	0.811754i	$0.301487\pi$
$884$	123.410	4.15072
$885$	0	0
$886$	−52.7982	−1.77379
$887$	−0.659136	−0.0221316	−0.0110658	−	0.999939i	$-0.503522\pi$
$887$	−0.659136	−0.0221316	−0.0110658	+	0.999939i	$0.503522\pi$
$888$	0	0
$889$	12.1217	0.406548
$890$	−33.3977	−1.11949
$891$	0	0
$892$	−48.3228	−1.61797
$893$	11.6882	0.391131
$894$	0	0
$895$	−9.64997	−0.322563
$896$	33.9773	1.13510
$897$	0	0
$898$	42.6271	1.42248
$899$	4.05484	0.135236
$900$	0	0
$901$	9.74465	0.324641
$902$	−5.16717	−0.172048
$903$	0	0
$904$	−20.2976	−0.675087
$905$	8.35091	0.277594
$906$	0	0
$907$	12.6324	0.419451	0.209726	−	0.977760i	$-0.432743\pi$
$907$	12.6324	0.419451	0.209726	+	0.977760i	$0.432743\pi$
$908$	143.785	4.77167
$909$	0	0
$910$	19.0292	0.630811
$911$	−38.3845	−1.27174	−0.635868	−	0.771798i	$-0.719358\pi$
$911$	−38.3845	−1.27174	−0.635868	+	0.771798i	$0.719358\pi$
$912$	0	0
$913$	−10.0518	−0.332665
$914$	−46.5960	−1.54126
$915$	0	0
$916$	79.8484	2.63827
$917$	2.50910	0.0828577
$918$	0	0
$919$	28.5593	0.942084	0.471042	−	0.882111i	$-0.343878\pi$
$919$	28.5593	0.942084	0.471042	+	0.882111i	$0.343878\pi$
$920$	10.0048	0.329849
$921$	0	0
$922$	18.6481	0.614142
$923$	−36.3494	−1.19646
$924$	0	0
$925$	−0.673156	−0.0221332
$926$	−90.5353	−2.97518
$927$	0	0
$928$	−14.6815	−0.481944
$929$	14.6340	0.480125	0.240063	−	0.970757i	$-0.422832\pi$
$929$	14.6340	0.480125	0.240063	+	0.970757i	$0.422832\pi$
$930$	0	0
$931$	−5.02283	−0.164617
$932$	−86.2730	−2.82597
$933$	0	0
$934$	0.0667807	0.00218513
$935$	−4.66205	−0.152465
$936$	0	0
$937$	1.73736	0.0567570	0.0283785	−	0.999597i	$-0.490966\pi$
$937$	1.73736	0.0567570	0.0283785	+	0.999597i	$0.490966\pi$
$938$	45.6155	1.48940
$939$	0	0
$940$	61.6706	2.01147
$941$	−4.73828	−0.154463	−0.0772317	−	0.997013i	$-0.524608\pi$
$941$	−4.73828	−0.154463	−0.0772317	+	0.997013i	$0.524608\pi$
$942$	0	0
$943$	2.16853	0.0706172
$944$	39.1554	1.27440
$945$	0	0
$946$	26.2906	0.854780
$947$	49.0444	1.59373	0.796864	−	0.604159i	$-0.206491\pi$
$947$	49.0444	1.59373	0.796864	+	0.604159i	$0.206491\pi$
$948$	0	0
$949$	0.577308	0.0187402
$950$	−2.69746	−0.0875173
$951$	0	0
$952$	57.9346	1.87767
$953$	30.5961	0.991104	0.495552	−	0.868578i	$-0.334966\pi$
$953$	30.5961	0.991104	0.495552	+	0.868578i	$0.334966\pi$
$954$	0	0
$955$	26.2081	0.848073
$956$	103.629	3.35159
$957$	0	0
$958$	31.7964	1.02729
$959$	−19.8291	−0.640317
$960$	0	0
$961$	−5.82984	−0.188059
$962$	9.10991	0.293715
$963$	0	0
$964$	30.1481	0.971004
$965$	−13.9918	−0.450412
$966$	0	0
$967$	50.1971	1.61423	0.807115	−	0.590395i	$-0.201028\pi$
$967$	50.1971	1.61423	0.807115	+	0.590395i	$0.201028\pi$
$968$	−8.83771	−0.284055
$969$	0	0
$970$	−6.16415	−0.197919
$971$	−27.0879	−0.869294	−0.434647	−	0.900601i	$-0.643127\pi$
$971$	−27.0879	−0.869294	−0.434647	+	0.900601i	$0.643127\pi$
$972$	0	0
$973$	11.8285	0.379205
$974$	28.0591	0.899072
$975$	0	0
$976$	−28.7137	−0.919102
$977$	10.3603	0.331455	0.165728	−	0.986172i	$-0.447003\pi$
$977$	10.3603	0.331455	0.165728	+	0.986172i	$0.447003\pi$
$978$	0	0
$979$	−12.3812	−0.395704
$980$	−26.5020	−0.846575
$981$	0	0
$982$	67.1887	2.14408
$983$	23.8319	0.760119	0.380059	−	0.924962i	$-0.375904\pi$
$983$	23.8319	0.760119	0.380059	+	0.924962i	$0.375904\pi$
$984$	0	0
$985$	−23.7263	−0.755983
$986$	−10.1639	−0.323686
$987$	0	0
$988$	26.4712	0.842160
$989$	−11.0335	−0.350845
$990$	0	0
$991$	3.72047	0.118185	0.0590923	−	0.998253i	$-0.481179\pi$
$991$	3.72047	0.118185	0.0590923	+	0.998253i	$0.481179\pi$
$992$	−91.1346	−2.89353
$993$	0	0
$994$	−27.4809	−0.871642
$995$	15.6795	0.497075
$996$	0	0
$997$	53.3247	1.68881	0.844406	−	0.535704i	$-0.179954\pi$
$997$	53.3247	1.68881	0.844406	+	0.535704i	$0.179954\pi$
$998$	−38.9217	−1.23204
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9405.2.a.bm.1.1		10
3.2	odd	2		3135.2.a.x.1.10	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3135.2.a.x.1.10	✓	10	3.2	odd	2
9405.2.a.bm.1.1		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)

\(f(q)\)	\(=\)	\(q-2.69746 q^{2} +5.27631 q^{4} +1.00000 q^{5} -1.40612 q^{7} -8.83771 q^{8} +O(q^{10})\)	\(q-2.69746 q^{2} +5.27631 q^{4} +1.00000 q^{5} -1.40612 q^{7} -8.83771 q^{8} -2.69746 q^{10} -1.00000 q^{11} +5.01699 q^{13} +3.79295 q^{14} +13.2868 q^{16} +4.66205 q^{17} +1.00000 q^{19} +5.27631 q^{20} +2.69746 q^{22} -1.13206 q^{23} +1.00000 q^{25} -13.5331 q^{26} -7.41911 q^{28} +0.808221 q^{29} +5.01699 q^{31} -18.1652 q^{32} -12.5757 q^{34} -1.40612 q^{35} -0.673156 q^{37} -2.69746 q^{38} -8.83771 q^{40} -1.91557 q^{41} +9.74641 q^{43} -5.27631 q^{44} +3.05368 q^{46} +11.6882 q^{47} -5.02283 q^{49} -2.69746 q^{50} +26.4712 q^{52} +2.09021 q^{53} -1.00000 q^{55} +12.4269 q^{56} -2.18015 q^{58} +2.94694 q^{59} -2.16107 q^{61} -13.5331 q^{62} +22.4264 q^{64} +5.01699 q^{65} +12.0264 q^{67} +24.5984 q^{68} +3.79295 q^{70} -7.24526 q^{71} +0.115071 q^{73} +1.81581 q^{74} +5.27631 q^{76} +1.40612 q^{77} -11.2787 q^{79} +13.2868 q^{80} +5.16717 q^{82} +10.0518 q^{83} +4.66205 q^{85} -26.2906 q^{86} +8.83771 q^{88} +12.3812 q^{89} -7.05447 q^{91} -5.97308 q^{92} -31.5285 q^{94} +1.00000 q^{95} +2.28517 q^{97} +13.5489 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 4 q^{2} + 16 q^{4} + 10 q^{5} + 5 q^{7} - 3 q^{8}+O(q^{10}) \) 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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