LMFDB - Embedded newform 9405.2.a.w.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9405,2,Mod(1,9405)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [6,0,0,8,-6,0,5,0,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$75.0993031010$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.131947641.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 10x^{4} + 25x^{2} - 3x - 9 $ x^6 - 10x^4 + 25x^2 - 3*x - 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1045)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.65636$ 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)

$f(q)$	$=$	\(q-1.65636 q^{2} +0.743534 q^{4} -1.00000 q^{5} +2.81120 q^{7} +2.08116 q^{8} +1.65636 q^{10} +1.00000 q^{11} -5.98258 q^{13} -4.65636 q^{14} -4.93423 q^{16} +6.49551 q^{17} -1.00000 q^{19} -0.743534 q^{20} -1.65636 q^{22} -2.77149 q^{23} +1.00000 q^{25} +9.90932 q^{26} +2.09022 q^{28} +1.78540 q^{29} -3.15789 q^{31} +4.01054 q^{32} -10.7589 q^{34} -2.81120 q^{35} +3.90101 q^{37} +1.65636 q^{38} -2.08116 q^{40} +4.63245 q^{41} -2.40738 q^{43} +0.743534 q^{44} +4.59059 q^{46} +4.10602 q^{47} +0.902838 q^{49} -1.65636 q^{50} -4.44825 q^{52} -11.5083 q^{53} -1.00000 q^{55} +5.85056 q^{56} -2.95726 q^{58} -5.33014 q^{59} +7.18233 q^{61} +5.23060 q^{62} +3.22555 q^{64} +5.98258 q^{65} -13.2370 q^{67} +4.82964 q^{68} +4.65636 q^{70} +0.437851 q^{71} +11.2457 q^{73} -6.46148 q^{74} -0.743534 q^{76} +2.81120 q^{77} +4.44176 q^{79} +4.93423 q^{80} -7.67301 q^{82} -17.7115 q^{83} -6.49551 q^{85} +3.98750 q^{86} +2.08116 q^{88} -17.5897 q^{89} -16.8182 q^{91} -2.06070 q^{92} -6.80106 q^{94} +1.00000 q^{95} +4.99386 q^{97} -1.49543 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 8 q^{4} - 6 q^{5} + 5 q^{7} + 6 q^{11} - 9 q^{13} - 18 q^{14} + 4 q^{16} + 5 q^{17} - 6 q^{19} - 8 q^{20} - 8 q^{23} + 6 q^{25} + 22 q^{26} + 10 q^{28} + 5 q^{29} - q^{31} - 15 q^{32} - 22 q^{34}+ \cdots - 33 q^{98}+O(q^{100}) $ 6 * q + 8 * q^4 - 6 * q^5 + 5 * q^7 + 6 * q^11 - 9 * q^13 - 18 * q^14 + 4 * q^16 + 5 * q^17 - 6 * q^19 - 8 * q^20 - 8 * q^23 + 6 * q^25 + 22 * q^26 + 10 * q^28 + 5 * q^29 - q^31 - 15 * q^32 - 22 * q^34 - 5 * q^35 + 9 * q^37 - 25 * q^41 + 15 * q^43 + 8 * q^44 - 16 * q^46 - 24 * q^47 + 13 * q^49 - 27 * q^52 - 5 * q^53 - 6 * q^55 + 12 * q^56 + 13 * q^58 - 39 * q^59 - 11 * q^61 + 42 * q^62 - 14 * q^64 + 9 * q^65 + 24 * q^67 - 45 * q^68 + 18 * q^70 + 24 * q^71 - 26 * q^73 - q^74 - 8 * q^76 + 5 * q^77 + 11 * q^79 - 4 * q^80 + 8 * q^82 - 39 * q^83 - 5 * q^85 - 18 * q^86 - 22 * q^89 - 26 * q^91 + 11 * q^92 - 30 * q^94 + 6 * q^95 + 22 * q^97 - 33 * q^98

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 $

$2$	−1.65636	−1.17122	−0.585612	−	0.810591i	$-0.699146\pi$
$2$	−1.65636	−1.17122	−0.585612	+	0.810591i	$0.699146\pi$
$3$	0	0
$3$	0	0
$4$	0.743534	0.371767
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.81120	1.06253	0.531267	−	0.847205i	$-0.321716\pi$
$7$	2.81120	1.06253	0.531267	+	0.847205i	$0.321716\pi$
$8$	2.08116	0.735802
$9$	0	0
$10$	1.65636	0.523788
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−5.98258	−1.65927	−0.829635	−	0.558306i	$-0.811451\pi$
$13$	−5.98258	−1.65927	−0.829635	+	0.558306i	$0.811451\pi$
$14$	−4.65636	−1.24446
$15$	0	0
$16$	−4.93423	−1.23356
$17$	6.49551	1.57539	0.787697	−	0.616063i	$-0.211273\pi$
$17$	6.49551	1.57539	0.787697	+	0.616063i	$0.211273\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−0.743534	−0.166259
$21$	0	0
$22$	−1.65636	−0.353137
$23$	−2.77149	−0.577895	−0.288948	−	0.957345i	$-0.593305\pi$
$23$	−2.77149	−0.577895	−0.288948	+	0.957345i	$0.593305\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	9.90932	1.94338
$27$	0	0
$28$	2.09022	0.395015
$29$	1.78540	0.331540	0.165770	−	0.986164i	$-0.446989\pi$
$29$	1.78540	0.331540	0.165770	+	0.986164i	$0.446989\pi$
$30$	0	0
$31$	−3.15789	−0.567173	−0.283587	−	0.958947i	$-0.591524\pi$
$31$	−3.15789	−0.567173	−0.283587	+	0.958947i	$0.591524\pi$
$32$	4.01054	0.708969
$33$	0	0
$34$	−10.7589	−1.84514
$35$	−2.81120	−0.475179
$36$	0	0
$37$	3.90101	0.641322	0.320661	−	0.947194i	$-0.396095\pi$
$37$	3.90101	0.641322	0.320661	+	0.947194i	$0.396095\pi$
$38$	1.65636	0.268697
$39$	0	0
$40$	−2.08116	−0.329061
$41$	4.63245	0.723467	0.361734	−	0.932281i	$-0.382185\pi$
$41$	4.63245	0.723467	0.361734	+	0.932281i	$0.382185\pi$
$42$	0	0
$43$	−2.40738	−0.367122	−0.183561	−	0.983008i	$-0.558763\pi$
$43$	−2.40738	−0.367122	−0.183561	+	0.983008i	$0.558763\pi$
$44$	0.743534	0.112092
$45$	0	0
$46$	4.59059	0.676845
$47$	4.10602	0.598925	0.299462	−	0.954108i	$-0.403193\pi$
$47$	4.10602	0.598925	0.299462	+	0.954108i	$0.403193\pi$
$48$	0	0
$49$	0.902838	0.128977
$50$	−1.65636	−0.234245
$51$	0	0
$52$	−4.44825	−0.616862
$53$	−11.5083	−1.58078	−0.790391	−	0.612603i	$-0.790123\pi$
$53$	−11.5083	−1.58078	−0.790391	+	0.612603i	$0.790123\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	5.85056	0.781814
$57$	0	0
$58$	−2.95726	−0.388308
$59$	−5.33014	−0.693925	−0.346963	−	0.937879i	$-0.612787\pi$
$59$	−5.33014	−0.693925	−0.346963	+	0.937879i	$0.612787\pi$
$60$	0	0
$61$	7.18233	0.919603	0.459802	−	0.888022i	$-0.347920\pi$
$61$	7.18233	0.919603	0.459802	+	0.888022i	$0.347920\pi$
$62$	5.23060	0.664287
$63$	0	0
$64$	3.22555	0.403194
$65$	5.98258	0.742048
$66$	0	0
$67$	−13.2370	−1.61716	−0.808580	−	0.588387i	$-0.799763\pi$
$67$	−13.2370	−1.61716	−0.808580	+	0.588387i	$0.799763\pi$
$68$	4.82964	0.585679
$69$	0	0
$70$	4.65636	0.556542
$71$	0.437851	0.0519633	0.0259817	−	0.999662i	$-0.491729\pi$
$71$	0.437851	0.0519633	0.0259817	+	0.999662i	$0.491729\pi$
$72$	0	0
$73$	11.2457	1.31621	0.658103	−	0.752928i	$-0.271359\pi$
$73$	11.2457	1.31621	0.658103	+	0.752928i	$0.271359\pi$
$74$	−6.46148	−0.751132
$75$	0	0
$76$	−0.743534	−0.0852892
$77$	2.81120	0.320366
$78$	0	0
$79$	4.44176	0.499737	0.249869	−	0.968280i	$-0.419613\pi$
$79$	4.44176	0.499737	0.249869	+	0.968280i	$0.419613\pi$
$80$	4.93423	0.551663
$81$	0	0
$82$	−7.67301	−0.847343
$83$	−17.7115	−1.94409	−0.972046	−	0.234790i	$-0.924560\pi$
$83$	−17.7115	−1.94409	−0.972046	+	0.234790i	$0.924560\pi$
$84$	0	0
$85$	−6.49551	−0.704537
$86$	3.98750	0.429983
$87$	0	0
$88$	2.08116	0.221853
$89$	−17.5897	−1.86451	−0.932253	−	0.361807i	$-0.882160\pi$
$89$	−17.5897	−1.86451	−0.932253	+	0.361807i	$0.882160\pi$
$90$	0	0
$91$	−16.8182	−1.76303
$92$	−2.06070	−0.214842
$93$	0	0
$94$	−6.80106	−0.701475
$95$	1.00000	0.102598
$96$	0	0
$97$	4.99386	0.507049	0.253525	−	0.967329i	$-0.418410\pi$
$97$	4.99386	0.507049	0.253525	+	0.967329i	$0.418410\pi$
$98$	−1.49543	−0.151061
$99$	0	0
$100$	0.743534	0.0743534
$101$	16.7267	1.66437	0.832186	−	0.554497i	$-0.187089\pi$
$101$	16.7267	1.66437	0.832186	+	0.554497i	$0.187089\pi$
$102$	0	0
$103$	4.48754	0.442171	0.221085	−	0.975254i	$-0.429040\pi$
$103$	4.48754	0.442171	0.221085	+	0.975254i	$0.429040\pi$
$104$	−12.4507	−1.22089
$105$	0	0
$106$	19.0618	1.85145
$107$	−1.05672	−0.102157	−0.0510783	−	0.998695i	$-0.516266\pi$
$107$	−1.05672	−0.102157	−0.0510783	+	0.998695i	$0.516266\pi$
$108$	0	0
$109$	−3.48106	−0.333425	−0.166712	−	0.986006i	$-0.553315\pi$
$109$	−3.48106	−0.333425	−0.166712	+	0.986006i	$0.553315\pi$
$110$	1.65636	0.157928
$111$	0	0
$112$	−13.8711	−1.31069
$113$	8.94084	0.841084	0.420542	−	0.907273i	$-0.361840\pi$
$113$	8.94084	0.841084	0.420542	+	0.907273i	$0.361840\pi$
$114$	0	0
$115$	2.77149	0.258443
$116$	1.32750	0.123256
$117$	0	0
$118$	8.82864	0.812742
$119$	18.2602	1.67391
$120$	0	0
$121$	1.00000	0.0909091
$122$	−11.8965	−1.07706
$123$	0	0
$124$	−2.34800	−0.210856
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	2.14707	0.190522	0.0952609	−	0.995452i	$-0.469631\pi$
$127$	2.14707	0.190522	0.0952609	+	0.995452i	$0.469631\pi$
$128$	−13.3638	−1.18120
$129$	0	0
$130$	−9.90932	−0.869105
$131$	−13.1523	−1.14912	−0.574562	−	0.818461i	$-0.694828\pi$
$131$	−13.1523	−1.14912	−0.574562	+	0.818461i	$0.694828\pi$
$132$	0	0
$133$	−2.81120	−0.243762
$134$	21.9253	1.89406
$135$	0	0
$136$	13.5182	1.15918
$137$	0.316167	0.0270120	0.0135060	−	0.999909i	$-0.495701\pi$
$137$	0.316167	0.0270120	0.0135060	+	0.999909i	$0.495701\pi$
$138$	0	0
$139$	−4.07415	−0.345565	−0.172782	−	0.984960i	$-0.555276\pi$
$139$	−4.07415	−0.345565	−0.172782	+	0.984960i	$0.555276\pi$
$140$	−2.09022	−0.176656
$141$	0	0
$142$	−0.725239	−0.0608607
$143$	−5.98258	−0.500289
$144$	0	0
$145$	−1.78540	−0.148269
$146$	−18.6269	−1.54157
$147$	0	0
$148$	2.90053	0.238422
$149$	−4.80464	−0.393612	−0.196806	−	0.980442i	$-0.563057\pi$
$149$	−4.80464	−0.393612	−0.196806	+	0.980442i	$0.563057\pi$
$150$	0	0
$151$	−18.2399	−1.48434	−0.742169	−	0.670213i	$-0.766203\pi$
$151$	−18.2399	−1.48434	−0.742169	+	0.670213i	$0.766203\pi$
$152$	−2.08116	−0.168805
$153$	0	0
$154$	−4.65636	−0.375220
$155$	3.15789	0.253648
$156$	0	0
$157$	−18.1161	−1.44582	−0.722910	−	0.690942i	$-0.757196\pi$
$157$	−18.1161	−1.44582	−0.722910	+	0.690942i	$0.757196\pi$
$158$	−7.35716	−0.585304
$159$	0	0
$160$	−4.01054	−0.317061
$161$	−7.79120	−0.614033
$162$	0	0
$163$	−23.9946	−1.87940	−0.939699	−	0.342002i	$-0.888895\pi$
$163$	−23.9946	−1.87940	−0.939699	+	0.342002i	$0.888895\pi$
$164$	3.44438	0.268961
$165$	0	0
$166$	29.3367	2.27697
$167$	−3.10562	−0.240320	−0.120160	−	0.992755i	$-0.538341\pi$
$167$	−3.10562	−0.240320	−0.120160	+	0.992755i	$0.538341\pi$
$168$	0	0
$169$	22.7913	1.75318
$170$	10.7589	0.825172
$171$	0	0
$172$	−1.78997	−0.136484
$173$	−10.9411	−0.831839	−0.415919	−	0.909401i	$-0.636540\pi$
$173$	−10.9411	−0.831839	−0.415919	+	0.909401i	$0.636540\pi$
$174$	0	0
$175$	2.81120	0.212507
$176$	−4.93423	−0.371931
$177$	0	0
$178$	29.1349	2.18376
$179$	−1.88648	−0.141002	−0.0705011	−	0.997512i	$-0.522460\pi$
$179$	−1.88648	−0.141002	−0.0705011	+	0.997512i	$0.522460\pi$
$180$	0	0
$181$	8.50254	0.631989	0.315995	−	0.948761i	$-0.397662\pi$
$181$	8.50254	0.631989	0.315995	+	0.948761i	$0.397662\pi$
$182$	27.8571	2.06490
$183$	0	0
$184$	−5.76792	−0.425216
$185$	−3.90101	−0.286808
$186$	0	0
$187$	6.49551	0.474999
$188$	3.05297	0.222660
$189$	0	0
$190$	−1.65636	−0.120165
$191$	23.2282	1.68073	0.840366	−	0.542019i	$-0.182340\pi$
$191$	23.2282	1.68073	0.840366	+	0.542019i	$0.182340\pi$
$192$	0	0
$193$	15.5701	1.12076	0.560380	−	0.828235i	$-0.310655\pi$
$193$	15.5701	1.12076	0.560380	+	0.828235i	$0.310655\pi$
$194$	−8.27163	−0.593868
$195$	0	0
$196$	0.671291	0.0479493
$197$	22.8694	1.62938	0.814689	−	0.579898i	$-0.196908\pi$
$197$	22.8694	1.62938	0.814689	+	0.579898i	$0.196908\pi$
$198$	0	0
$199$	−23.7525	−1.68377	−0.841885	−	0.539657i	$-0.818554\pi$
$199$	−23.7525	−1.68377	−0.841885	+	0.539657i	$0.818554\pi$
$200$	2.08116	0.147160
$201$	0	0
$202$	−27.7055	−1.94935
$203$	5.01911	0.352272
$204$	0	0
$205$	−4.63245	−0.323544
$206$	−7.43300	−0.517881
$207$	0	0
$208$	29.5194	2.04680
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	7.48728	0.515446	0.257723	−	0.966219i	$-0.417028\pi$
$211$	7.48728	0.515446	0.257723	+	0.966219i	$0.417028\pi$
$212$	−8.55679	−0.587682
$213$	0	0
$214$	1.75030	0.119648
$215$	2.40738	0.164182
$216$	0	0
$217$	−8.87745	−0.602640
$218$	5.76589	0.390515
$219$	0	0
$220$	−0.743534	−0.0501290
$221$	−38.8600	−2.61400
$222$	0	0
$223$	−8.80258	−0.589464	−0.294732	−	0.955580i	$-0.595230\pi$
$223$	−8.80258	−0.589464	−0.294732	+	0.955580i	$0.595230\pi$
$224$	11.2744	0.753304
$225$	0	0
$226$	−14.8093	−0.985098
$227$	12.5157	0.830695	0.415347	−	0.909663i	$-0.363660\pi$
$227$	12.5157	0.830695	0.415347	+	0.909663i	$0.363660\pi$
$228$	0	0
$229$	−2.31366	−0.152891	−0.0764454	−	0.997074i	$-0.524357\pi$
$229$	−2.31366	−0.152891	−0.0764454	+	0.997074i	$0.524357\pi$
$230$	−4.59059	−0.302694
$231$	0	0
$232$	3.71570	0.243948
$233$	13.0851	0.857234	0.428617	−	0.903486i	$-0.359001\pi$
$233$	13.0851	0.857234	0.428617	+	0.903486i	$0.359001\pi$
$234$	0	0
$235$	−4.10602	−0.267847
$236$	−3.96314	−0.257978
$237$	0	0
$238$	−30.2455	−1.96052
$239$	−20.9482	−1.35503	−0.677514	−	0.735510i	$-0.736943\pi$
$239$	−20.9482	−1.35503	−0.677514	+	0.735510i	$0.736943\pi$
$240$	0	0
$241$	−14.2896	−0.920477	−0.460238	−	0.887795i	$-0.652236\pi$
$241$	−14.2896	−0.920477	−0.460238	+	0.887795i	$0.652236\pi$
$242$	−1.65636	−0.106475
$243$	0	0
$244$	5.34031	0.341878
$245$	−0.902838	−0.0576802
$246$	0	0
$247$	5.98258	0.380663
$248$	−6.57207	−0.417327
$249$	0	0
$250$	1.65636	0.104758
$251$	−11.1296	−0.702495	−0.351247	−	0.936283i	$-0.614242\pi$
$251$	−11.1296	−0.702495	−0.351247	+	0.936283i	$0.614242\pi$
$252$	0	0
$253$	−2.77149	−0.174242
$254$	−3.55633	−0.223144
$255$	0	0
$256$	15.6841	0.980257
$257$	−7.27753	−0.453960	−0.226980	−	0.973899i	$-0.572885\pi$
$257$	−7.27753	−0.453960	−0.226980	+	0.973899i	$0.572885\pi$
$258$	0	0
$259$	10.9665	0.681426
$260$	4.44825	0.275869
$261$	0	0
$262$	21.7850	1.34588
$263$	16.6526	1.02684	0.513421	−	0.858137i	$-0.328378\pi$
$263$	16.6526	1.02684	0.513421	+	0.858137i	$0.328378\pi$
$264$	0	0
$265$	11.5083	0.706947
$266$	4.65636	0.285500
$267$	0	0
$268$	−9.84217	−0.601206
$269$	7.11358	0.433723	0.216861	−	0.976202i	$-0.430418\pi$
$269$	7.11358	0.433723	0.216861	+	0.976202i	$0.430418\pi$
$270$	0	0
$271$	11.2711	0.684669	0.342334	−	0.939578i	$-0.388782\pi$
$271$	11.2711	0.684669	0.342334	+	0.939578i	$0.388782\pi$
$272$	−32.0503	−1.94334
$273$	0	0
$274$	−0.523687	−0.0316371
$275$	1.00000	0.0603023
$276$	0	0
$277$	−4.24890	−0.255292	−0.127646	−	0.991820i	$-0.540742\pi$
$277$	−4.24890	−0.255292	−0.127646	+	0.991820i	$0.540742\pi$
$278$	6.74827	0.404734
$279$	0	0
$280$	−5.85056	−0.349638
$281$	−0.222832	−0.0132931	−0.00664653	−	0.999978i	$-0.502116\pi$
$281$	−0.222832	−0.0132931	−0.00664653	+	0.999978i	$0.502116\pi$
$282$	0	0
$283$	12.1767	0.723828	0.361914	−	0.932211i	$-0.382123\pi$
$283$	12.1767	0.723828	0.361914	+	0.932211i	$0.382123\pi$
$284$	0.325557	0.0193182
$285$	0	0
$286$	9.90932	0.585950
$287$	13.0227	0.768708
$288$	0	0
$289$	25.1917	1.48187
$290$	2.95726	0.173657
$291$	0	0
$292$	8.36153	0.489322
$293$	−24.1175	−1.40896	−0.704480	−	0.709724i	$-0.748820\pi$
$293$	−24.1175	−1.40896	−0.704480	+	0.709724i	$0.748820\pi$
$294$	0	0
$295$	5.33014	0.310333
$296$	8.11863	0.471886
$297$	0	0
$298$	7.95822	0.461008
$299$	16.5807	0.958884
$300$	0	0
$301$	−6.76763	−0.390080
$302$	30.2118	1.73849
$303$	0	0
$304$	4.93423	0.282997
$305$	−7.18233	−0.411259
$306$	0	0
$307$	19.7231	1.12566	0.562829	−	0.826573i	$-0.309713\pi$
$307$	19.7231	1.12566	0.562829	+	0.826573i	$0.309713\pi$
$308$	2.09022	0.119101
$309$	0	0
$310$	−5.23060	−0.297078
$311$	26.2571	1.48890	0.744451	−	0.667677i	$-0.232711\pi$
$311$	26.2571	1.48890	0.744451	+	0.667677i	$0.232711\pi$
$312$	0	0
$313$	2.77164	0.156662	0.0783311	−	0.996927i	$-0.475041\pi$
$313$	2.77164	0.156662	0.0783311	+	0.996927i	$0.475041\pi$
$314$	30.0068	1.69338
$315$	0	0
$316$	3.30260	0.185786
$317$	−4.00082	−0.224709	−0.112354	−	0.993668i	$-0.535839\pi$
$317$	−4.00082	−0.224709	−0.112354	+	0.993668i	$0.535839\pi$
$318$	0	0
$319$	1.78540	0.0999631
$320$	−3.22555	−0.180314
$321$	0	0
$322$	12.9051	0.719170
$323$	−6.49551	−0.361420
$324$	0	0
$325$	−5.98258	−0.331854
$326$	39.7437	2.20120
$327$	0	0
$328$	9.64088	0.532329
$329$	11.5428	0.636377
$330$	0	0
$331$	−17.5839	−0.966500	−0.483250	−	0.875483i	$-0.660544\pi$
$331$	−17.5839	−0.966500	−0.483250	+	0.875483i	$0.660544\pi$
$332$	−13.1691	−0.722749
$333$	0	0
$334$	5.14403	0.281469
$335$	13.2370	0.723216
$336$	0	0
$337$	32.7681	1.78499	0.892497	−	0.451054i	$-0.148952\pi$
$337$	32.7681	1.78499	0.892497	+	0.451054i	$0.148952\pi$
$338$	−37.7506	−2.05336
$339$	0	0
$340$	−4.82964	−0.261924
$341$	−3.15789	−0.171009
$342$	0	0
$343$	−17.1403	−0.925491
$344$	−5.01015	−0.270129
$345$	0	0
$346$	18.1225	0.974270
$347$	19.6495	1.05484	0.527421	−	0.849604i	$-0.323159\pi$
$347$	19.6495	1.05484	0.527421	+	0.849604i	$0.323159\pi$
$348$	0	0
$349$	19.9610	1.06849	0.534243	−	0.845331i	$-0.320597\pi$
$349$	19.9610	1.06849	0.534243	+	0.845331i	$0.320597\pi$
$350$	−4.65636	−0.248893
$351$	0	0
$352$	4.01054	0.213762
$353$	23.2030	1.23497	0.617486	−	0.786582i	$-0.288151\pi$
$353$	23.2030	1.23497	0.617486	+	0.786582i	$0.288151\pi$
$354$	0	0
$355$	−0.437851	−0.0232387
$356$	−13.0786	−0.693162
$357$	0	0
$358$	3.12469	0.165145
$359$	9.83399	0.519018	0.259509	−	0.965741i	$-0.416439\pi$
$359$	9.83399	0.519018	0.259509	+	0.965741i	$0.416439\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−14.0833	−0.740201
$363$	0	0
$364$	−12.5049	−0.655436
$365$	−11.2457	−0.588625
$366$	0	0
$367$	1.73835	0.0907411	0.0453705	−	0.998970i	$-0.485553\pi$
$367$	1.73835	0.0907411	0.0453705	+	0.998970i	$0.485553\pi$
$368$	13.6751	0.712866
$369$	0	0
$370$	6.46148	0.335916
$371$	−32.3520	−1.67963
$372$	0	0
$373$	−6.58428	−0.340921	−0.170460	−	0.985365i	$-0.554526\pi$
$373$	−6.58428	−0.340921	−0.170460	+	0.985365i	$0.554526\pi$
$374$	−10.7589	−0.556331
$375$	0	0
$376$	8.54530	0.440690
$377$	−10.6813	−0.550114
$378$	0	0
$379$	−3.74050	−0.192137	−0.0960684	−	0.995375i	$-0.530627\pi$
$379$	−3.74050	−0.192137	−0.0960684	+	0.995375i	$0.530627\pi$
$380$	0.743534	0.0381425
$381$	0	0
$382$	−38.4743	−1.96852
$383$	−27.5711	−1.40882	−0.704409	−	0.709795i	$-0.748788\pi$
$383$	−27.5711	−1.40882	−0.704409	+	0.709795i	$0.748788\pi$
$384$	0	0
$385$	−2.81120	−0.143272
$386$	−25.7897	−1.31266
$387$	0	0
$388$	3.71310	0.188504
$389$	17.1560	0.869844	0.434922	−	0.900468i	$-0.356776\pi$
$389$	17.1560	0.869844	0.434922	+	0.900468i	$0.356776\pi$
$390$	0	0
$391$	−18.0022	−0.910413
$392$	1.87895	0.0949014
$393$	0	0
$394$	−37.8800	−1.90837
$395$	−4.44176	−0.223489
$396$	0	0
$397$	−14.1506	−0.710201	−0.355100	−	0.934828i	$-0.615553\pi$
$397$	−14.1506	−0.710201	−0.355100	+	0.934828i	$0.615553\pi$
$398$	39.3427	1.97207
$399$	0	0
$400$	−4.93423	−0.246711
$401$	−10.3891	−0.518807	−0.259403	−	0.965769i	$-0.583526\pi$
$401$	−10.3891	−0.518807	−0.259403	+	0.965769i	$0.583526\pi$
$402$	0	0
$403$	18.8923	0.941093
$404$	12.4369	0.618758
$405$	0	0
$406$	−8.31346	−0.412590
$407$	3.90101	0.193366
$408$	0	0
$409$	4.11163	0.203307	0.101653	−	0.994820i	$-0.467587\pi$
$409$	4.11163	0.203307	0.101653	+	0.994820i	$0.467587\pi$
$410$	7.67301	0.378943
$411$	0	0
$412$	3.33664	0.164385
$413$	−14.9841	−0.737319
$414$	0	0
$415$	17.7115	0.869424
$416$	−23.9934	−1.17637
$417$	0	0
$418$	1.65636	0.0810153
$419$	−16.4494	−0.803604	−0.401802	−	0.915726i	$-0.631616\pi$
$419$	−16.4494	−0.803604	−0.401802	+	0.915726i	$0.631616\pi$
$420$	0	0
$421$	−38.6187	−1.88216	−0.941079	−	0.338187i	$-0.890186\pi$
$421$	−38.6187	−1.88216	−0.941079	+	0.338187i	$0.890186\pi$
$422$	−12.4016	−0.603703
$423$	0	0
$424$	−23.9506	−1.16314
$425$	6.49551	0.315079
$426$	0	0
$427$	20.1910	0.977109
$428$	−0.785704	−0.0379784
$429$	0	0
$430$	−3.98750	−0.192294
$431$	24.2965	1.17032	0.585161	−	0.810917i	$-0.301031\pi$
$431$	24.2965	1.17032	0.585161	+	0.810917i	$0.301031\pi$
$432$	0	0
$433$	22.8348	1.09737	0.548686	−	0.836028i	$-0.315128\pi$
$433$	22.8348	1.09737	0.548686	+	0.836028i	$0.315128\pi$
$434$	14.7043	0.705827
$435$	0	0
$436$	−2.58828	−0.123956
$437$	2.77149	0.132578
$438$	0	0
$439$	−23.3798	−1.11586	−0.557929	−	0.829889i	$-0.688404\pi$
$439$	−23.3798	−1.11586	−0.557929	+	0.829889i	$0.688404\pi$
$440$	−2.08116	−0.0992155
$441$	0	0
$442$	64.3661	3.06158
$443$	−14.6539	−0.696228	−0.348114	−	0.937452i	$-0.613178\pi$
$443$	−14.6539	−0.696228	−0.348114	+	0.937452i	$0.613178\pi$
$444$	0	0
$445$	17.5897	0.833833
$446$	14.5803	0.690395
$447$	0	0
$448$	9.06766	0.428407
$449$	−19.5934	−0.924672	−0.462336	−	0.886705i	$-0.652989\pi$
$449$	−19.5934	−0.924672	−0.462336	+	0.886705i	$0.652989\pi$
$450$	0	0
$451$	4.63245	0.218134
$452$	6.64782	0.312687
$453$	0	0
$454$	−20.7305	−0.972930
$455$	16.8182	0.788451
$456$	0	0
$457$	−32.4549	−1.51817	−0.759087	−	0.650989i	$-0.774354\pi$
$457$	−32.4549	−1.51817	−0.759087	+	0.650989i	$0.774354\pi$
$458$	3.83225	0.179069
$459$	0	0
$460$	2.06070	0.0960804
$461$	−9.88505	−0.460393	−0.230196	−	0.973144i	$-0.573937\pi$
$461$	−9.88505	−0.460393	−0.230196	+	0.973144i	$0.573937\pi$
$462$	0	0
$463$	9.99709	0.464604	0.232302	−	0.972644i	$-0.425374\pi$
$463$	9.99709	0.464604	0.232302	+	0.972644i	$0.425374\pi$
$464$	−8.80956	−0.408973
$465$	0	0
$466$	−21.6737	−1.00401
$467$	−26.3637	−1.21997	−0.609984	−	0.792414i	$-0.708824\pi$
$467$	−26.3637	−1.21997	−0.609984	+	0.792414i	$0.708824\pi$
$468$	0	0
$469$	−37.2119	−1.71829
$470$	6.80106	0.313709
$471$	0	0
$472$	−11.0929	−0.510592
$473$	−2.40738	−0.110692
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	13.5771	0.622304
$477$	0	0
$478$	34.6978	1.58704
$479$	−19.3573	−0.884459	−0.442230	−	0.896902i	$-0.645812\pi$
$479$	−19.3573	−0.884459	−0.442230	+	0.896902i	$0.645812\pi$
$480$	0	0
$481$	−23.3381	−1.06413
$482$	23.6688	1.07809
$483$	0	0
$484$	0.743534	0.0337970
$485$	−4.99386	−0.226759
$486$	0	0
$487$	15.4579	0.700462	0.350231	−	0.936663i	$-0.386103\pi$
$487$	15.4579	0.700462	0.350231	+	0.936663i	$0.386103\pi$
$488$	14.9476	0.676646
$489$	0	0
$490$	1.49543	0.0675565
$491$	−33.7098	−1.52130	−0.760650	−	0.649162i	$-0.775120\pi$
$491$	−33.7098	−1.52130	−0.760650	+	0.649162i	$0.775120\pi$
$492$	0	0
$493$	11.5971	0.522306
$494$	−9.90932	−0.445841
$495$	0	0
$496$	15.5817	0.699640
$497$	1.23089	0.0552127
$498$	0	0
$499$	32.9388	1.47454	0.737272	−	0.675596i	$-0.236114\pi$
$499$	32.9388	1.47454	0.737272	+	0.675596i	$0.236114\pi$
$500$	−0.743534	−0.0332518
$501$	0	0
$502$	18.4347	0.822779
$503$	−8.51652	−0.379733	−0.189866	−	0.981810i	$-0.560805\pi$
$503$	−8.51652	−0.379733	−0.189866	+	0.981810i	$0.560805\pi$
$504$	0	0
$505$	−16.7267	−0.744329
$506$	4.59059	0.204076
$507$	0	0
$508$	1.59642	0.0708297
$509$	−19.8390	−0.879350	−0.439675	−	0.898157i	$-0.644906\pi$
$509$	−19.8390	−0.879350	−0.439675	+	0.898157i	$0.644906\pi$
$510$	0	0
$511$	31.6138	1.39851
$512$	0.748955	0.0330994
$513$	0	0
$514$	12.0542	0.531689
$515$	−4.48754	−0.197745
$516$	0	0
$517$	4.10602	0.180583
$518$	−18.1645	−0.798102
$519$	0	0
$520$	12.4507	0.546000
$521$	−42.2657	−1.85169	−0.925846	−	0.377900i	$-0.876646\pi$
$521$	−42.2657	−1.85169	−0.925846	+	0.377900i	$0.876646\pi$
$522$	0	0
$523$	−28.4461	−1.24386	−0.621931	−	0.783072i	$-0.713652\pi$
$523$	−28.4461	−1.24386	−0.621931	+	0.783072i	$0.713652\pi$
$524$	−9.77920	−0.427206
$525$	0	0
$526$	−27.5827	−1.20266
$527$	−20.5121	−0.893521
$528$	0	0
$529$	−15.3189	−0.666037
$530$	−19.0618	−0.827994
$531$	0	0
$532$	−2.09022	−0.0906226
$533$	−27.7140	−1.20043
$534$	0	0
$535$	1.05672	0.0456858
$536$	−27.5484	−1.18991
$537$	0	0
$538$	−11.7827	−0.507987
$539$	0.902838	0.0388880
$540$	0	0
$541$	18.7126	0.804515	0.402258	−	0.915526i	$-0.368226\pi$
$541$	18.7126	0.804515	0.402258	+	0.915526i	$0.368226\pi$
$542$	−18.6690	−0.801901
$543$	0	0
$544$	26.0505	1.11691
$545$	3.48106	0.149112
$546$	0	0
$547$	22.8994	0.979109	0.489555	−	0.871973i	$-0.337159\pi$
$547$	22.8994	0.979109	0.489555	+	0.871973i	$0.337159\pi$
$548$	0.235081	0.0100422
$549$	0	0
$550$	−1.65636	−0.0706275
$551$	−1.78540	−0.0760605
$552$	0	0
$553$	12.4867	0.530987
$554$	7.03772	0.299004
$555$	0	0
$556$	−3.02927	−0.128470
$557$	−13.6065	−0.576524	−0.288262	−	0.957552i	$-0.593077\pi$
$557$	−13.6065	−0.576524	−0.288262	+	0.957552i	$0.593077\pi$
$558$	0	0
$559$	14.4024	0.609155
$560$	13.8711	0.586160
$561$	0	0
$562$	0.369091	0.0155692
$563$	38.8581	1.63767	0.818836	−	0.574028i	$-0.194620\pi$
$563$	38.8581	1.63767	0.818836	+	0.574028i	$0.194620\pi$
$564$	0	0
$565$	−8.94084	−0.376144
$566$	−20.1690	−0.847765
$567$	0	0
$568$	0.911238	0.0382347
$569$	27.9139	1.17021	0.585107	−	0.810956i	$-0.301053\pi$
$569$	27.9139	1.17021	0.585107	+	0.810956i	$0.301053\pi$
$570$	0	0
$571$	−39.5183	−1.65379	−0.826895	−	0.562357i	$-0.809895\pi$
$571$	−39.5183	−1.65379	−0.826895	+	0.562357i	$0.809895\pi$
$572$	−4.44825	−0.185991
$573$	0	0
$574$	−21.5704	−0.900330
$575$	−2.77149	−0.115579
$576$	0	0
$577$	12.3632	0.514685	0.257343	−	0.966320i	$-0.417153\pi$
$577$	12.3632	0.514685	0.257343	+	0.966320i	$0.417153\pi$
$578$	−41.7266	−1.73560
$579$	0	0
$580$	−1.32750	−0.0551216
$581$	−49.7906	−2.06566
$582$	0	0
$583$	−11.5083	−0.476624
$584$	23.4041	0.968467
$585$	0	0
$586$	39.9473	1.65021
$587$	−6.03747	−0.249193	−0.124597	−	0.992207i	$-0.539764\pi$
$587$	−6.03747	−0.249193	−0.124597	+	0.992207i	$0.539764\pi$
$588$	0	0
$589$	3.15789	0.130118
$590$	−8.82864	−0.363469
$591$	0	0
$592$	−19.2485	−0.791106
$593$	1.44341	0.0592739	0.0296369	−	0.999561i	$-0.490565\pi$
$593$	1.44341	0.0592739	0.0296369	+	0.999561i	$0.490565\pi$
$594$	0	0
$595$	−18.2602	−0.748594
$596$	−3.57241	−0.146332
$597$	0	0
$598$	−27.4636	−1.12307
$599$	−20.4093	−0.833902	−0.416951	−	0.908929i	$-0.636901\pi$
$599$	−20.4093	−0.833902	−0.416951	+	0.908929i	$0.636901\pi$
$600$	0	0
$601$	−33.1258	−1.35123	−0.675616	−	0.737254i	$-0.736122\pi$
$601$	−33.1258	−1.35123	−0.675616	+	0.737254i	$0.736122\pi$
$602$	11.2096	0.456871
$603$	0	0
$604$	−13.5620	−0.551828
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	7.79823	0.316520	0.158260	−	0.987397i	$-0.449412\pi$
$607$	7.79823	0.316520	0.158260	+	0.987397i	$0.449412\pi$
$608$	−4.01054	−0.162649
$609$	0	0
$610$	11.8965	0.481677
$611$	−24.5646	−0.993778
$612$	0	0
$613$	28.4672	1.14978	0.574890	−	0.818231i	$-0.305045\pi$
$613$	28.4672	1.14978	0.574890	+	0.818231i	$0.305045\pi$
$614$	−32.6686	−1.31840
$615$	0	0
$616$	5.85056	0.235726
$617$	−1.26620	−0.0509751	−0.0254876	−	0.999675i	$-0.508114\pi$
$617$	−1.26620	−0.0509751	−0.0254876	+	0.999675i	$0.508114\pi$
$618$	0	0
$619$	2.57985	0.103693	0.0518464	−	0.998655i	$-0.483489\pi$
$619$	2.57985	0.103693	0.0518464	+	0.998655i	$0.483489\pi$
$620$	2.34800	0.0942978
$621$	0	0
$622$	−43.4912	−1.74384
$623$	−49.4482	−1.98110
$624$	0	0
$625$	1.00000	0.0400000
$626$	−4.59083	−0.183487
$627$	0	0
$628$	−13.4699	−0.537508
$629$	25.3391	1.01033
$630$	0	0
$631$	−18.3838	−0.731848	−0.365924	−	0.930645i	$-0.619247\pi$
$631$	−18.3838	−0.731848	−0.365924	+	0.930645i	$0.619247\pi$
$632$	9.24402	0.367707
$633$	0	0
$634$	6.62681	0.263184
$635$	−2.14707	−0.0852039
$636$	0	0
$637$	−5.40130	−0.214007
$638$	−2.95726	−0.117079
$639$	0	0
$640$	13.3638	0.528249
$641$	−20.9430	−0.827199	−0.413600	−	0.910459i	$-0.635729\pi$
$641$	−20.9430	−0.827199	−0.413600	+	0.910459i	$0.635729\pi$
$642$	0	0
$643$	9.37012	0.369522	0.184761	−	0.982784i	$-0.440849\pi$
$643$	9.37012	0.369522	0.184761	+	0.982784i	$0.440849\pi$
$644$	−5.79302	−0.228277
$645$	0	0
$646$	10.7589	0.423304
$647$	15.3143	0.602069	0.301034	−	0.953613i	$-0.402668\pi$
$647$	15.3143	0.602069	0.301034	+	0.953613i	$0.402668\pi$
$648$	0	0
$649$	−5.33014	−0.209226
$650$	9.90932	0.388676
$651$	0	0
$652$	−17.8408	−0.698698
$653$	−7.61281	−0.297912	−0.148956	−	0.988844i	$-0.547591\pi$
$653$	−7.61281	−0.297912	−0.148956	+	0.988844i	$0.547591\pi$
$654$	0	0
$655$	13.1523	0.513904
$656$	−22.8576	−0.892438
$657$	0	0
$658$	−19.1191	−0.745341
$659$	−27.8742	−1.08582	−0.542912	−	0.839789i	$-0.682678\pi$
$659$	−27.8742	−1.08582	−0.542912	+	0.839789i	$0.682678\pi$
$660$	0	0
$661$	−40.0185	−1.55654	−0.778269	−	0.627931i	$-0.783902\pi$
$661$	−40.0185	−1.55654	−0.778269	+	0.627931i	$0.783902\pi$
$662$	29.1253	1.13199
$663$	0	0
$664$	−36.8606	−1.43047
$665$	2.81120	0.109014
$666$	0	0
$667$	−4.94821	−0.191595
$668$	−2.30913	−0.0893431
$669$	0	0
$670$	−21.9253	−0.847048
$671$	7.18233	0.277271
$672$	0	0
$673$	−41.4748	−1.59874	−0.799369	−	0.600841i	$-0.794832\pi$
$673$	−41.4748	−1.59874	−0.799369	+	0.600841i	$0.794832\pi$
$674$	−54.2759	−2.09063
$675$	0	0
$676$	16.9461	0.651773
$677$	0.266167	0.0102296	0.00511482	−	0.999987i	$-0.498372\pi$
$677$	0.266167	0.0102296	0.00511482	+	0.999987i	$0.498372\pi$
$678$	0	0
$679$	14.0387	0.538757
$680$	−13.5182	−0.518400
$681$	0	0
$682$	5.23060	0.200290
$683$	−29.3614	−1.12348	−0.561742	−	0.827312i	$-0.689869\pi$
$683$	−29.3614	−1.12348	−0.561742	+	0.827312i	$0.689869\pi$
$684$	0	0
$685$	−0.316167	−0.0120801
$686$	28.3906	1.08396
$687$	0	0
$688$	11.8786	0.452866
$689$	68.8491	2.62294
$690$	0	0
$691$	−22.7603	−0.865844	−0.432922	−	0.901431i	$-0.642517\pi$
$691$	−22.7603	−0.865844	−0.432922	+	0.901431i	$0.642517\pi$
$692$	−8.13510	−0.309250
$693$	0	0
$694$	−32.5468	−1.23546
$695$	4.07415	0.154541
$696$	0	0
$697$	30.0902	1.13975
$698$	−33.0626	−1.25144
$699$	0	0
$700$	2.09022	0.0790029
$701$	13.6101	0.514046	0.257023	−	0.966405i	$-0.417258\pi$
$701$	13.6101	0.514046	0.257023	+	0.966405i	$0.417258\pi$
$702$	0	0
$703$	−3.90101	−0.147129
$704$	3.22555	0.121568
$705$	0	0
$706$	−38.4326	−1.44643
$707$	47.0221	1.76845
$708$	0	0
$709$	−32.9904	−1.23898	−0.619491	−	0.785004i	$-0.712661\pi$
$709$	−32.9904	−1.23898	−0.619491	+	0.785004i	$0.712661\pi$
$710$	0.725239	0.0272177
$711$	0	0
$712$	−36.6071	−1.37191
$713$	8.75204	0.327767
$714$	0	0
$715$	5.98258	0.223736
$716$	−1.40266	−0.0524199
$717$	0	0
$718$	−16.2886	−0.607886
$719$	3.85276	0.143684	0.0718418	−	0.997416i	$-0.477112\pi$
$719$	3.85276	0.143684	0.0718418	+	0.997416i	$0.477112\pi$
$720$	0	0
$721$	12.6154	0.469821
$722$	−1.65636	−0.0616434
$723$	0	0
$724$	6.32193	0.234953
$725$	1.78540	0.0663080
$726$	0	0
$727$	−35.5951	−1.32015	−0.660074	−	0.751200i	$-0.729475\pi$
$727$	−35.5951	−1.32015	−0.660074	+	0.751200i	$0.729475\pi$
$728$	−35.0015	−1.29724
$729$	0	0
$730$	18.6269	0.689412
$731$	−15.6372	−0.578362
$732$	0	0
$733$	−22.8053	−0.842332	−0.421166	−	0.906984i	$-0.638379\pi$
$733$	−22.8053	−0.842332	−0.421166	+	0.906984i	$0.638379\pi$
$734$	−2.87933	−0.106278
$735$	0	0
$736$	−11.1152	−0.409710
$737$	−13.2370	−0.487592
$738$	0	0
$739$	−37.2882	−1.37167	−0.685835	−	0.727757i	$-0.740563\pi$
$739$	−37.2882	−1.37167	−0.685835	+	0.727757i	$0.740563\pi$
$740$	−2.90053	−0.106626
$741$	0	0
$742$	53.5866	1.96723
$743$	−24.9220	−0.914301	−0.457150	−	0.889389i	$-0.651130\pi$
$743$	−24.9220	−0.914301	−0.457150	+	0.889389i	$0.651130\pi$
$744$	0	0
$745$	4.80464	0.176028
$746$	10.9059	0.399295
$747$	0	0
$748$	4.82964	0.176589
$749$	−2.97064	−0.108545
$750$	0	0
$751$	−1.46203	−0.0533501	−0.0266751	−	0.999644i	$-0.508492\pi$
$751$	−1.46203	−0.0533501	−0.0266751	+	0.999644i	$0.508492\pi$
$752$	−20.2600	−0.738807
$753$	0	0
$754$	17.6921	0.644308
$755$	18.2399	0.663816
$756$	0	0
$757$	4.66986	0.169729	0.0848645	−	0.996393i	$-0.472954\pi$
$757$	4.66986	0.169729	0.0848645	+	0.996393i	$0.472954\pi$
$758$	6.19562	0.225035
$759$	0	0
$760$	2.08116	0.0754917
$761$	−18.4084	−0.667303	−0.333652	−	0.942696i	$-0.608281\pi$
$761$	−18.4084	−0.667303	−0.333652	+	0.942696i	$0.608281\pi$
$762$	0	0
$763$	−9.78595	−0.354275
$764$	17.2709	0.624841
$765$	0	0
$766$	45.6677	1.65004
$767$	31.8880	1.15141
$768$	0	0
$769$	−3.57354	−0.128865	−0.0644326	−	0.997922i	$-0.520524\pi$
$769$	−3.57354	−0.128865	−0.0644326	+	0.997922i	$0.520524\pi$
$770$	4.65636	0.167804
$771$	0	0
$772$	11.5769	0.416662
$773$	−34.6561	−1.24649	−0.623246	−	0.782026i	$-0.714186\pi$
$773$	−34.6561	−1.24649	−0.623246	+	0.782026i	$0.714186\pi$
$774$	0	0
$775$	−3.15789	−0.113435
$776$	10.3930	0.373088
$777$	0	0
$778$	−28.4166	−1.01878
$779$	−4.63245	−0.165975
$780$	0	0
$781$	0.437851	0.0156675
$782$	29.8182	1.06630
$783$	0	0
$784$	−4.45481	−0.159100
$785$	18.1161	0.646591
$786$	0	0
$787$	30.3693	1.08255	0.541274	−	0.840846i	$-0.317942\pi$
$787$	30.3693	1.08255	0.541274	+	0.840846i	$0.317942\pi$
$788$	17.0042	0.605749
$789$	0	0
$790$	7.35716	0.261756
$791$	25.1345	0.893679
$792$	0	0
$793$	−42.9689	−1.52587
$794$	23.4386	0.831804
$795$	0	0
$796$	−17.6608	−0.625970
$797$	34.4392	1.21990	0.609950	−	0.792440i	$-0.291190\pi$
$797$	34.4392	1.21990	0.609950	+	0.792440i	$0.291190\pi$
$798$	0	0
$799$	26.6707	0.943542
$800$	4.01054	0.141794
$801$	0	0
$802$	17.2081	0.607639
$803$	11.2457	0.396851
$804$	0	0
$805$	7.79120	0.274604
$806$	−31.2925	−1.10223
$807$	0	0
$808$	34.8110	1.22465
$809$	8.23292	0.289454	0.144727	−	0.989472i	$-0.453770\pi$
$809$	8.23292	0.289454	0.144727	+	0.989472i	$0.453770\pi$
$810$	0	0
$811$	35.9943	1.26393	0.631966	−	0.774996i	$-0.282248\pi$
$811$	35.9943	1.26393	0.631966	+	0.774996i	$0.282248\pi$
$812$	3.73188	0.130963
$813$	0	0
$814$	−6.46148	−0.226475
$815$	23.9946	0.840492
$816$	0	0
$817$	2.40738	0.0842237
$818$	−6.81034	−0.238118
$819$	0	0
$820$	−3.44438	−0.120283
$821$	−31.2285	−1.08988	−0.544941	−	0.838474i	$-0.683448\pi$
$821$	−31.2285	−1.08988	−0.544941	+	0.838474i	$0.683448\pi$
$822$	0	0
$823$	−8.20565	−0.286031	−0.143015	−	0.989720i	$-0.545680\pi$
$823$	−8.20565	−0.286031	−0.143015	+	0.989720i	$0.545680\pi$
$824$	9.33931	0.325350
$825$	0	0
$826$	24.8191	0.863566
$827$	15.5275	0.539943	0.269972	−	0.962868i	$-0.412986\pi$
$827$	15.5275	0.539943	0.269972	+	0.962868i	$0.412986\pi$
$828$	0	0
$829$	3.31730	0.115215	0.0576074	−	0.998339i	$-0.481653\pi$
$829$	3.31730	0.115215	0.0576074	+	0.998339i	$0.481653\pi$
$830$	−29.3367	−1.01829
$831$	0	0
$832$	−19.2971	−0.669007
$833$	5.86440	0.203189
$834$	0	0
$835$	3.10562	0.107474
$836$	−0.743534	−0.0257157
$837$	0	0
$838$	27.2461	0.941201
$839$	−7.02051	−0.242375	−0.121188	−	0.992630i	$-0.538670\pi$
$839$	−7.02051	−0.242375	−0.121188	+	0.992630i	$0.538670\pi$
$840$	0	0
$841$	−25.8124	−0.890081
$842$	63.9665	2.20443
$843$	0	0
$844$	5.56705	0.191626
$845$	−22.7913	−0.784044
$846$	0	0
$847$	2.81120	0.0965939
$848$	56.7844	1.94998
$849$	0	0
$850$	−10.7589	−0.369028
$851$	−10.8116	−0.370617
$852$	0	0
$853$	3.62821	0.124227	0.0621137	−	0.998069i	$-0.480216\pi$
$853$	3.62821	0.124227	0.0621137	+	0.998069i	$0.480216\pi$
$854$	−33.4435	−1.14441
$855$	0	0
$856$	−2.19920	−0.0751670
$857$	−16.1864	−0.552918	−0.276459	−	0.961026i	$-0.589161\pi$
$857$	−16.1864	−0.552918	−0.276459	+	0.961026i	$0.589161\pi$
$858$	0	0
$859$	25.5004	0.870063	0.435032	−	0.900415i	$-0.356737\pi$
$859$	25.5004	0.870063	0.435032	+	0.900415i	$0.356737\pi$
$860$	1.78997	0.0610375
$861$	0	0
$862$	−40.2438	−1.37071
$863$	39.9012	1.35825	0.679126	−	0.734022i	$-0.262359\pi$
$863$	39.9012	1.35825	0.679126	+	0.734022i	$0.262359\pi$
$864$	0	0
$865$	10.9411	0.372010
$866$	−37.8228	−1.28527
$867$	0	0
$868$	−6.60068	−0.224042
$869$	4.44176	0.150676
$870$	0	0
$871$	79.1916	2.68330
$872$	−7.24465	−0.245335
$873$	0	0
$874$	−4.59059	−0.155279
$875$	−2.81120	−0.0950359
$876$	0	0
$877$	11.1804	0.377537	0.188768	−	0.982022i	$-0.439550\pi$
$877$	11.1804	0.377537	0.188768	+	0.982022i	$0.439550\pi$
$878$	38.7254	1.30692
$879$	0	0
$880$	4.93423	0.166333
$881$	−2.82033	−0.0950195	−0.0475097	−	0.998871i	$-0.515129\pi$
$881$	−2.82033	−0.0950195	−0.0475097	+	0.998871i	$0.515129\pi$
$882$	0	0
$883$	−17.0119	−0.572495	−0.286247	−	0.958156i	$-0.592408\pi$
$883$	−17.0119	−0.572495	−0.286247	+	0.958156i	$0.592408\pi$
$884$	−28.8937	−0.971800
$885$	0	0
$886$	24.2722	0.815439
$887$	−10.8354	−0.363817	−0.181908	−	0.983316i	$-0.558227\pi$
$887$	−10.8354	−0.363817	−0.181908	+	0.983316i	$0.558227\pi$
$888$	0	0
$889$	6.03584	0.202436
$890$	−29.1349	−0.976605
$891$	0	0
$892$	−6.54501	−0.219143
$893$	−4.10602	−0.137403
$894$	0	0
$895$	1.88648	0.0630581
$896$	−37.5682	−1.25506
$897$	0	0
$898$	32.4538	1.08300
$899$	−5.63808	−0.188041
$900$	0	0
$901$	−74.7521	−2.49035
$902$	−7.67301	−0.255483
$903$	0	0
$904$	18.6073	0.618871
$905$	−8.50254	−0.282634
$906$	0	0
$907$	38.8963	1.29153	0.645766	−	0.763536i	$-0.276538\pi$
$907$	38.8963	1.29153	0.645766	+	0.763536i	$0.276538\pi$
$908$	9.30583	0.308825
$909$	0	0
$910$	−27.8571	−0.923453
$911$	−11.2431	−0.372502	−0.186251	−	0.982502i	$-0.559634\pi$
$911$	−11.2431	−0.372502	−0.186251	+	0.982502i	$0.559634\pi$
$912$	0	0
$913$	−17.7115	−0.586166
$914$	53.7570	1.77812
$915$	0	0
$916$	−1.72028	−0.0568398
$917$	−36.9738	−1.22098
$918$	0	0
$919$	−9.48069	−0.312739	−0.156370	−	0.987699i	$-0.549979\pi$
$919$	−9.48069	−0.312739	−0.156370	+	0.987699i	$0.549979\pi$
$920$	5.76792	0.190163
$921$	0	0
$922$	16.3732	0.539223
$923$	−2.61948	−0.0862211
$924$	0	0
$925$	3.90101	0.128264
$926$	−16.5588	−0.544156
$927$	0	0
$928$	7.16040	0.235052
$929$	39.8543	1.30758	0.653789	−	0.756677i	$-0.273178\pi$
$929$	39.8543	1.30758	0.653789	+	0.756677i	$0.273178\pi$
$930$	0	0
$931$	−0.902838	−0.0295893
$932$	9.72922	0.318691
$933$	0	0
$934$	43.6679	1.42886
$935$	−6.49551	−0.212426
$936$	0	0
$937$	−56.3053	−1.83942	−0.919708	−	0.392604i	$-0.871574\pi$
$937$	−56.3053	−1.83942	−0.919708	+	0.392604i	$0.871574\pi$
$938$	61.6363	2.01250
$939$	0	0
$940$	−3.05297	−0.0995768
$941$	21.4276	0.698519	0.349259	−	0.937026i	$-0.386433\pi$
$941$	21.4276	0.698519	0.349259	+	0.937026i	$0.386433\pi$
$942$	0	0
$943$	−12.8388	−0.418088
$944$	26.3001	0.855996
$945$	0	0
$946$	3.98750	0.129645
$947$	−16.2072	−0.526662	−0.263331	−	0.964706i	$-0.584821\pi$
$947$	−16.2072	−0.526662	−0.263331	+	0.964706i	$0.584821\pi$
$948$	0	0
$949$	−67.2781	−2.18394
$950$	1.65636	0.0537395
$951$	0	0
$952$	38.0024	1.23166
$953$	−58.8844	−1.90745	−0.953726	−	0.300676i	$-0.902788\pi$
$953$	−58.8844	−1.90745	−0.953726	+	0.300676i	$0.902788\pi$
$954$	0	0
$955$	−23.2282	−0.751646
$956$	−15.5757	−0.503755
$957$	0	0
$958$	32.0628	1.03590
$959$	0.888809	0.0287011
$960$	0	0
$961$	−21.0278	−0.678315
$962$	38.6563	1.24633
$963$	0	0
$964$	−10.6248	−0.342203
$965$	−15.5701	−0.501220
$966$	0	0
$967$	−24.4443	−0.786074	−0.393037	−	0.919523i	$-0.628576\pi$
$967$	−24.4443	−0.786074	−0.393037	+	0.919523i	$0.628576\pi$
$968$	2.08116	0.0668911
$969$	0	0
$970$	8.27163	0.265586
$971$	−59.3586	−1.90491	−0.952454	−	0.304683i	$-0.901449\pi$
$971$	−59.3586	−1.90491	−0.952454	+	0.304683i	$0.901449\pi$
$972$	0	0
$973$	−11.4532	−0.367174
$974$	−25.6038	−0.820399
$975$	0	0
$976$	−35.4392	−1.13438
$977$	5.33048	0.170537	0.0852686	−	0.996358i	$-0.472825\pi$
$977$	5.33048	0.170537	0.0852686	+	0.996358i	$0.472825\pi$
$978$	0	0
$979$	−17.5897	−0.562170
$980$	−0.671291	−0.0214436
$981$	0	0
$982$	55.8356	1.78178
$983$	31.3817	1.00092	0.500460	−	0.865760i	$-0.333164\pi$
$983$	31.3817	1.00092	0.500460	+	0.865760i	$0.333164\pi$
$984$	0	0
$985$	−22.8694	−0.728680
$986$	−19.2090	−0.611738
$987$	0	0
$988$	4.44825	0.141518
$989$	6.67203	0.212158
$990$	0	0
$991$	−25.0162	−0.794667	−0.397334	−	0.917674i	$-0.630064\pi$
$991$	−25.0162	−0.794667	−0.397334	+	0.917674i	$0.630064\pi$
$992$	−12.6648	−0.402108
$993$	0	0
$994$	−2.03879	−0.0646665
$995$	23.7525	0.753005
$996$	0	0
$997$	−16.2641	−0.515091	−0.257545	−	0.966266i	$-0.582914\pi$
$997$	−16.2641	−0.515091	−0.257545	+	0.966266i	$0.582914\pi$
$998$	−54.5586	−1.72702
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9405.2.a.w.1.2		6
3.2	odd	2		1045.2.a.g.1.5	✓	6
15.14	odd	2		5225.2.a.k.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.a.g.1.5	✓	6	3.2	odd	2
5225.2.a.k.1.2		6	15.14	odd	2
9405.2.a.w.1.2		6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9405 = 3^{2} \cdot 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9405.a (trivial)

\(f(q)\)	\(=\)	\(q-1.65636 q^{2} +0.743534 q^{4} -1.00000 q^{5} +2.81120 q^{7} +2.08116 q^{8} +1.65636 q^{10} +1.00000 q^{11} -5.98258 q^{13} -4.65636 q^{14} -4.93423 q^{16} +6.49551 q^{17} -1.00000 q^{19} -0.743534 q^{20} -1.65636 q^{22} -2.77149 q^{23} +1.00000 q^{25} +9.90932 q^{26} +2.09022 q^{28} +1.78540 q^{29} -3.15789 q^{31} +4.01054 q^{32} -10.7589 q^{34} -2.81120 q^{35} +3.90101 q^{37} +1.65636 q^{38} -2.08116 q^{40} +4.63245 q^{41} -2.40738 q^{43} +0.743534 q^{44} +4.59059 q^{46} +4.10602 q^{47} +0.902838 q^{49} -1.65636 q^{50} -4.44825 q^{52} -11.5083 q^{53} -1.00000 q^{55} +5.85056 q^{56} -2.95726 q^{58} -5.33014 q^{59} +7.18233 q^{61} +5.23060 q^{62} +3.22555 q^{64} +5.98258 q^{65} -13.2370 q^{67} +4.82964 q^{68} +4.65636 q^{70} +0.437851 q^{71} +11.2457 q^{73} -6.46148 q^{74} -0.743534 q^{76} +2.81120 q^{77} +4.44176 q^{79} +4.93423 q^{80} -7.67301 q^{82} -17.7115 q^{83} -6.49551 q^{85} +3.98750 q^{86} +2.08116 q^{88} -17.5897 q^{89} -16.8182 q^{91} -2.06070 q^{92} -6.80106 q^{94} +1.00000 q^{95} +4.99386 q^{97} -1.49543 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 8 q^{4} - 6 q^{5} + 5 q^{7} + 6 q^{11} - 9 q^{13} - 18 q^{14} + 4 q^{16} + 5 q^{17} - 6 q^{19} - 8 q^{20} - 8 q^{23} + 6 q^{25} + 22 q^{26} + 10 q^{28} + 5 q^{29} - q^{31} - 15 q^{32} - 22 q^{34}+ \cdots - 33 q^{98}+O(q^{100}) \) 6 * q + 8 * q^4 - 6 * q^5 + 5 * q^7 + 6 * q^11 - 9 * q^13 - 18 * q^14 + 4 * q^16 + 5 * q^17 - 6 * q^19 - 8 * q^20 - 8 * q^23 + 6 * q^25 + 22 * q^26 + 10 * q^28 + 5 * q^29 - q^31 - 15 * q^32 - 22 * q^34 - 5 * q^35 + 9 * q^37 - 25 * q^41 + 15 * q^43 + 8 * q^44 - 16 * q^46 - 24 * q^47 + 13 * q^49 - 27 * q^52 - 5 * q^53 - 6 * q^55 + 12 * q^56 + 13 * q^58 - 39 * q^59 - 11 * q^61 + 42 * q^62 - 14 * q^64 + 9 * q^65 + 24 * q^67 - 45 * q^68 + 18 * q^70 + 24 * q^71 - 26 * q^73 - q^74 - 8 * q^76 + 5 * q^77 + 11 * q^79 - 4 * q^80 + 8 * q^82 - 39 * q^83 - 5 * q^85 - 18 * q^86 - 22 * q^89 - 26 * q^91 + 11 * q^92 - 30 * q^94 + 6 * q^95 + 22 * q^97 - 33 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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