LMFDB - Embedded newform 9405.2.a.v.1.4

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 = [5,3,0,5,-5,0,-11,-3,0,-3,5] 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:	$0$
Dimension:	$5$
Coefficient field:	$\Q(\zeta_{22})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1 $ x^5 - x^4 - 4x^3 + 3x^2 + 3*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
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			1.4
Root			$1.30972$ 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)

$f(q)$	$=$	\(q+1.54620 q^{2} +0.390736 q^{4} -1.00000 q^{5} -4.16140 q^{7} -2.48825 q^{8} -1.54620 q^{10} +1.00000 q^{11} +1.82862 q^{13} -6.43436 q^{14} -4.62880 q^{16} -4.80511 q^{17} +1.00000 q^{19} -0.390736 q^{20} +1.54620 q^{22} +5.53843 q^{23} +1.00000 q^{25} +2.82741 q^{26} -1.62601 q^{28} -3.98933 q^{29} -8.51574 q^{31} -2.18056 q^{32} -7.42966 q^{34} +4.16140 q^{35} -7.11059 q^{37} +1.54620 q^{38} +2.48825 q^{40} +3.85750 q^{41} -12.0581 q^{43} +0.390736 q^{44} +8.56352 q^{46} -5.75936 q^{47} +10.3172 q^{49} +1.54620 q^{50} +0.714506 q^{52} +2.82914 q^{53} -1.00000 q^{55} +10.3546 q^{56} -6.16831 q^{58} +10.8553 q^{59} +7.35724 q^{61} -13.1670 q^{62} +5.88602 q^{64} -1.82862 q^{65} -11.7680 q^{67} -1.87753 q^{68} +6.43436 q^{70} +4.55334 q^{71} -6.03445 q^{73} -10.9944 q^{74} +0.390736 q^{76} -4.16140 q^{77} +9.68391 q^{79} +4.62880 q^{80} +5.96447 q^{82} -0.508463 q^{83} +4.80511 q^{85} -18.6442 q^{86} -2.48825 q^{88} -6.86161 q^{89} -7.60961 q^{91} +2.16406 q^{92} -8.90513 q^{94} -1.00000 q^{95} -3.72641 q^{97} +15.9525 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 3 q^{2} + 5 q^{4} - 5 q^{5} - 11 q^{7} - 3 q^{8} - 3 q^{10} + 5 q^{11} + q^{13} - 3 q^{16} + 3 q^{17} + 5 q^{19} - 5 q^{20} + 3 q^{22} + 8 q^{23} + 5 q^{25} + 16 q^{26} - 22 q^{28} - 11 q^{29} - 5 q^{31}+ \cdots - 10 q^{98}+O(q^{100}) $ 5 * q + 3 * q^2 + 5 * q^4 - 5 * q^5 - 11 * q^7 - 3 * q^8 - 3 * q^10 + 5 * q^11 + q^13 - 3 * q^16 + 3 * q^17 + 5 * q^19 - 5 * q^20 + 3 * q^22 + 8 * q^23 + 5 * q^25 + 16 * q^26 - 22 * q^28 - 11 * q^29 - 5 * q^31 + 2 * q^32 + 4 * q^34 + 11 * q^35 - 9 * q^37 + 3 * q^38 + 3 * q^40 - 15 * q^41 - 13 * q^43 + 5 * q^44 + 18 * q^46 + 20 * q^47 + 20 * q^49 + 3 * q^50 + q^52 + 5 * q^53 - 5 * q^55 - 33 * q^58 + 17 * q^59 + 3 * q^61 - 14 * q^62 - 17 * q^64 - q^65 - 28 * q^67 + 25 * q^68 + 6 * q^71 - 16 * q^73 + 21 * q^74 + 5 * q^76 - 11 * q^77 + 3 * q^79 + 3 * q^80 + 2 * q^82 + 33 * q^83 - 3 * q^85 - 10 * q^86 - 3 * q^88 + 16 * q^89 - 22 * q^91 + 19 * q^92 - 10 * q^94 - 5 * q^95 - 14 * q^97 - 10 * 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 $

$2$	1.54620	1.09333	0.546664	−	0.837352i	$-0.315897\pi$
$2$	1.54620	1.09333	0.546664	+	0.837352i	$0.315897\pi$
$3$	0	0
$3$	0	0
$4$	0.390736	0.195368
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−4.16140	−1.57286	−0.786430	−	0.617679i	$-0.788073\pi$
$7$	−4.16140	−1.57286	−0.786430	+	0.617679i	$0.788073\pi$
$8$	−2.48825	−0.879728
$9$	0	0
$10$	−1.54620	−0.488951
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	1.82862	0.507167	0.253584	−	0.967313i	$-0.418391\pi$
$13$	1.82862	0.507167	0.253584	+	0.967313i	$0.418391\pi$
$14$	−6.43436	−1.71965
$15$	0	0
$16$	−4.62880	−1.15720
$17$	−4.80511	−1.16541	−0.582705	−	0.812684i	$-0.698006\pi$
$17$	−4.80511	−1.16541	−0.582705	+	0.812684i	$0.698006\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−0.390736	−0.0873711
$21$	0	0
$22$	1.54620	0.329651
$23$	5.53843	1.15484	0.577421	−	0.816446i	$-0.304059\pi$
$23$	5.53843	1.15484	0.577421	+	0.816446i	$0.304059\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	2.82741	0.554501
$27$	0	0
$28$	−1.62601	−0.307286
$29$	−3.98933	−0.740800	−0.370400	−	0.928872i	$-0.620779\pi$
$29$	−3.98933	−0.740800	−0.370400	+	0.928872i	$0.620779\pi$
$30$	0	0
$31$	−8.51574	−1.52947	−0.764736	−	0.644343i	$-0.777131\pi$
$31$	−8.51574	−1.52947	−0.764736	+	0.644343i	$0.777131\pi$
$32$	−2.18056	−0.385472
$33$	0	0
$34$	−7.42966	−1.27418
$35$	4.16140	0.703405
$36$	0	0
$37$	−7.11059	−1.16897	−0.584487	−	0.811403i	$-0.698704\pi$
$37$	−7.11059	−1.16897	−0.584487	+	0.811403i	$0.698704\pi$
$38$	1.54620	0.250827
$39$	0	0
$40$	2.48825	0.393426
$41$	3.85750	0.602441	0.301220	−	0.953555i	$-0.402606\pi$
$41$	3.85750	0.602441	0.301220	+	0.953555i	$0.402606\pi$
$42$	0	0
$43$	−12.0581	−1.83884	−0.919420	−	0.393277i	$-0.871341\pi$
$43$	−12.0581	−1.83884	−0.919420	+	0.393277i	$0.871341\pi$
$44$	0.390736	0.0589056
$45$	0	0
$46$	8.56352	1.26262
$47$	−5.75936	−0.840090	−0.420045	−	0.907503i	$-0.637986\pi$
$47$	−5.75936	−0.840090	−0.420045	+	0.907503i	$0.637986\pi$
$48$	0	0
$49$	10.3172	1.47389
$50$	1.54620	0.218666
$51$	0	0
$52$	0.714506	0.0990842
$53$	2.82914	0.388612	0.194306	−	0.980941i	$-0.437754\pi$
$53$	2.82914	0.388612	0.194306	+	0.980941i	$0.437754\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	10.3546	1.38369
$57$	0	0
$58$	−6.16831	−0.809938
$59$	10.8553	1.41324	0.706619	−	0.707594i	$-0.250219\pi$
$59$	10.8553	1.41324	0.706619	+	0.707594i	$0.250219\pi$
$60$	0	0
$61$	7.35724	0.941998	0.470999	−	0.882134i	$-0.343893\pi$
$61$	7.35724	0.941998	0.470999	+	0.882134i	$0.343893\pi$
$62$	−13.1670	−1.67222
$63$	0	0
$64$	5.88602	0.735752
$65$	−1.82862	−0.226812
$66$	0	0
$67$	−11.7680	−1.43769	−0.718845	−	0.695170i	$-0.755329\pi$
$67$	−11.7680	−1.43769	−0.718845	+	0.695170i	$0.755329\pi$
$68$	−1.87753	−0.227684
$69$	0	0
$70$	6.43436	0.769053
$71$	4.55334	0.540382	0.270191	−	0.962807i	$-0.412913\pi$
$71$	4.55334	0.540382	0.270191	+	0.962807i	$0.412913\pi$
$72$	0	0
$73$	−6.03445	−0.706278	−0.353139	−	0.935571i	$-0.614886\pi$
$73$	−6.03445	−0.706278	−0.353139	+	0.935571i	$0.614886\pi$
$74$	−10.9944	−1.27807
$75$	0	0
$76$	0.390736	0.0448204
$77$	−4.16140	−0.474235
$78$	0	0
$79$	9.68391	1.08953	0.544763	−	0.838590i	$-0.316620\pi$
$79$	9.68391	1.08953	0.544763	+	0.838590i	$0.316620\pi$
$80$	4.62880	0.517515
$81$	0	0
$82$	5.96447	0.658666
$83$	−0.508463	−0.0558110	−0.0279055	−	0.999611i	$-0.508884\pi$
$83$	−0.508463	−0.0558110	−0.0279055	+	0.999611i	$0.508884\pi$
$84$	0	0
$85$	4.80511	0.521187
$86$	−18.6442	−2.01046
$87$	0	0
$88$	−2.48825	−0.265248
$89$	−6.86161	−0.727329	−0.363665	−	0.931530i	$-0.618475\pi$
$89$	−6.86161	−0.727329	−0.363665	+	0.931530i	$0.618475\pi$
$90$	0	0
$91$	−7.60961	−0.797704
$92$	2.16406	0.225619
$93$	0	0
$94$	−8.90513	−0.918494
$95$	−1.00000	−0.102598
$96$	0	0
$97$	−3.72641	−0.378360	−0.189180	−	0.981942i	$-0.560583\pi$
$97$	−3.72641	−0.378360	−0.189180	+	0.981942i	$0.560583\pi$
$98$	15.9525	1.61145
$99$	0	0
$100$	0.390736	0.0390736
$101$	14.0934	1.40234	0.701171	−	0.712994i	$-0.252661\pi$
$101$	14.0934	1.40234	0.701171	+	0.712994i	$0.252661\pi$
$102$	0	0
$103$	10.8474	1.06883	0.534414	−	0.845223i	$-0.320532\pi$
$103$	10.8474	1.06883	0.534414	+	0.845223i	$0.320532\pi$
$104$	−4.55005	−0.446169
$105$	0	0
$106$	4.37442	0.424881
$107$	9.49711	0.918120	0.459060	−	0.888405i	$-0.348186\pi$
$107$	9.49711	0.918120	0.459060	+	0.888405i	$0.348186\pi$
$108$	0	0
$109$	10.2467	0.981456	0.490728	−	0.871313i	$-0.336731\pi$
$109$	10.2467	0.981456	0.490728	+	0.871313i	$0.336731\pi$
$110$	−1.54620	−0.147424
$111$	0	0
$112$	19.2623	1.82011
$113$	7.19036	0.676412	0.338206	−	0.941072i	$-0.390180\pi$
$113$	7.19036	0.676412	0.338206	+	0.941072i	$0.390180\pi$
$114$	0	0
$115$	−5.53843	−0.516461
$116$	−1.55877	−0.144728
$117$	0	0
$118$	16.7845	1.54513
$119$	19.9960	1.83303
$120$	0	0
$121$	1.00000	0.0909091
$122$	11.3758	1.02991
$123$	0	0
$124$	−3.32740	−0.298810
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−3.35345	−0.297570	−0.148785	−	0.988870i	$-0.547536\pi$
$127$	−3.35345	−0.297570	−0.148785	+	0.988870i	$0.547536\pi$
$128$	13.4621	1.18989
$129$	0	0
$130$	−2.82741	−0.247980
$131$	5.15388	0.450297	0.225148	−	0.974324i	$-0.427713\pi$
$131$	5.15388	0.450297	0.225148	+	0.974324i	$0.427713\pi$
$132$	0	0
$133$	−4.16140	−0.360839
$134$	−18.1957	−1.57187
$135$	0	0
$136$	11.9563	1.02524
$137$	11.3104	0.966316	0.483158	−	0.875533i	$-0.339490\pi$
$137$	11.3104	0.966316	0.483158	+	0.875533i	$0.339490\pi$
$138$	0	0
$139$	5.14614	0.436490	0.218245	−	0.975894i	$-0.429967\pi$
$139$	5.14614	0.436490	0.218245	+	0.975894i	$0.429967\pi$
$140$	1.62601	0.137423
$141$	0	0
$142$	7.04038	0.590816
$143$	1.82862	0.152917
$144$	0	0
$145$	3.98933	0.331296
$146$	−9.33046	−0.772195
$147$	0	0
$148$	−2.77836	−0.228380
$149$	−10.3400	−0.847089	−0.423545	−	0.905875i	$-0.639214\pi$
$149$	−10.3400	−0.847089	−0.423545	+	0.905875i	$0.639214\pi$
$150$	0	0
$151$	−4.86952	−0.396276	−0.198138	−	0.980174i	$-0.563489\pi$
$151$	−4.86952	−0.396276	−0.198138	+	0.980174i	$0.563489\pi$
$152$	−2.48825	−0.201823
$153$	0	0
$154$	−6.43436	−0.518495
$155$	8.51574	0.684001
$156$	0	0
$157$	−17.3831	−1.38732	−0.693661	−	0.720302i	$-0.744003\pi$
$157$	−17.3831	−1.38732	−0.693661	+	0.720302i	$0.744003\pi$
$158$	14.9733	1.19121
$159$	0	0
$160$	2.18056	0.172388
$161$	−23.0476	−1.81641
$162$	0	0
$163$	17.0128	1.33255	0.666274	−	0.745707i	$-0.267888\pi$
$163$	17.0128	1.33255	0.666274	+	0.745707i	$0.267888\pi$
$164$	1.50726	0.117698
$165$	0	0
$166$	−0.786185	−0.0610198
$167$	10.3330	0.799590	0.399795	−	0.916604i	$-0.369081\pi$
$167$	10.3330	0.799590	0.399795	+	0.916604i	$0.369081\pi$
$168$	0	0
$169$	−9.65616	−0.742781
$170$	7.42966	0.569829
$171$	0	0
$172$	−4.71152	−0.359250
$173$	−15.1982	−1.15550	−0.577748	−	0.816215i	$-0.696068\pi$
$173$	−15.1982	−1.15550	−0.577748	+	0.816215i	$0.696068\pi$
$174$	0	0
$175$	−4.16140	−0.314572
$176$	−4.62880	−0.348909
$177$	0	0
$178$	−10.6094	−0.795210
$179$	10.2858	0.768794	0.384397	−	0.923168i	$-0.374409\pi$
$179$	10.2858	0.768794	0.384397	+	0.923168i	$0.374409\pi$
$180$	0	0
$181$	23.0997	1.71699	0.858495	−	0.512823i	$-0.171400\pi$
$181$	23.0997	1.71699	0.858495	+	0.512823i	$0.171400\pi$
$182$	−11.7660	−0.872152
$183$	0	0
$184$	−13.7810	−1.01595
$185$	7.11059	0.522781
$186$	0	0
$187$	−4.80511	−0.351384
$188$	−2.25039	−0.164126
$189$	0	0
$190$	−1.54620	−0.112173
$191$	−13.6788	−0.989766	−0.494883	−	0.868960i	$-0.664789\pi$
$191$	−13.6788	−0.989766	−0.494883	+	0.868960i	$0.664789\pi$
$192$	0	0
$193$	9.48835	0.682987	0.341493	−	0.939884i	$-0.389067\pi$
$193$	9.48835	0.682987	0.341493	+	0.939884i	$0.389067\pi$
$194$	−5.76178	−0.413672
$195$	0	0
$196$	4.03131	0.287951
$197$	17.2237	1.22713	0.613567	−	0.789643i	$-0.289734\pi$
$197$	17.2237	1.22713	0.613567	+	0.789643i	$0.289734\pi$
$198$	0	0
$199$	14.4608	1.02510	0.512549	−	0.858658i	$-0.328701\pi$
$199$	14.4608	1.02510	0.512549	+	0.858658i	$0.328701\pi$
$200$	−2.48825	−0.175946
$201$	0	0
$202$	21.7911	1.53322
$203$	16.6012	1.16518
$204$	0	0
$205$	−3.85750	−0.269420
$206$	16.7723	1.16858
$207$	0	0
$208$	−8.46430	−0.586894
$209$	1.00000	0.0691714
$210$	0	0
$211$	1.14231	0.0786401	0.0393200	−	0.999227i	$-0.487481\pi$
$211$	1.14231	0.0786401	0.0393200	+	0.999227i	$0.487481\pi$
$212$	1.10545	0.0759223
$213$	0	0
$214$	14.6844	1.00381
$215$	12.0581	0.822354
$216$	0	0
$217$	35.4374	2.40565
$218$	15.8434	1.07305
$219$	0	0
$220$	−0.390736	−0.0263434
$221$	−8.78671	−0.591058
$222$	0	0
$223$	6.86812	0.459923	0.229962	−	0.973200i	$-0.426140\pi$
$223$	6.86812	0.459923	0.229962	+	0.973200i	$0.426140\pi$
$224$	9.07417	0.606293
$225$	0	0
$226$	11.1177	0.739541
$227$	0.104970	0.00696713	0.00348356	−	0.999994i	$-0.498891\pi$
$227$	0.104970	0.00696713	0.00348356	+	0.999994i	$0.498891\pi$
$228$	0	0
$229$	14.9679	0.989106	0.494553	−	0.869147i	$-0.335332\pi$
$229$	14.9679	0.989106	0.494553	+	0.869147i	$0.335332\pi$
$230$	−8.56352	−0.564662
$231$	0	0
$232$	9.92644	0.651702
$233$	16.8004	1.10063	0.550316	−	0.834956i	$-0.314507\pi$
$233$	16.8004	1.10063	0.550316	+	0.834956i	$0.314507\pi$
$234$	0	0
$235$	5.75936	0.375700
$236$	4.24155	0.276101
$237$	0	0
$238$	30.9178	2.00410
$239$	3.44343	0.222737	0.111369	−	0.993779i	$-0.464477\pi$
$239$	3.44343	0.222737	0.111369	+	0.993779i	$0.464477\pi$
$240$	0	0
$241$	−14.8135	−0.954222	−0.477111	−	0.878843i	$-0.658316\pi$
$241$	−14.8135	−0.954222	−0.477111	+	0.878843i	$0.658316\pi$
$242$	1.54620	0.0993935
$243$	0	0
$244$	2.87474	0.184036
$245$	−10.3172	−0.659144
$246$	0	0
$247$	1.82862	0.116352
$248$	21.1893	1.34552
$249$	0	0
$250$	−1.54620	−0.0977903
$251$	−8.76253	−0.553086	−0.276543	−	0.961002i	$-0.589189\pi$
$251$	−8.76253	−0.553086	−0.276543	+	0.961002i	$0.589189\pi$
$252$	0	0
$253$	5.53843	0.348198
$254$	−5.18510	−0.325342
$255$	0	0
$256$	9.04303	0.565189
$257$	−2.52960	−0.157792	−0.0788960	−	0.996883i	$-0.525140\pi$
$257$	−2.52960	−0.157792	−0.0788960	+	0.996883i	$0.525140\pi$
$258$	0	0
$259$	29.5900	1.83863
$260$	−0.714506	−0.0443118
$261$	0	0
$262$	7.96893	0.492322
$263$	15.3262	0.945053	0.472527	−	0.881316i	$-0.343342\pi$
$263$	15.3262	0.945053	0.472527	+	0.881316i	$0.343342\pi$
$264$	0	0
$265$	−2.82914	−0.173793
$266$	−6.43436	−0.394516
$267$	0	0
$268$	−4.59818	−0.280878
$269$	−19.1624	−1.16835	−0.584176	−	0.811627i	$-0.698582\pi$
$269$	−19.1624	−1.16835	−0.584176	+	0.811627i	$0.698582\pi$
$270$	0	0
$271$	30.2567	1.83796	0.918982	−	0.394299i	$-0.129012\pi$
$271$	30.2567	1.83796	0.918982	+	0.394299i	$0.129012\pi$
$272$	22.2419	1.34861
$273$	0	0
$274$	17.4882	1.05650
$275$	1.00000	0.0603023
$276$	0	0
$277$	4.07877	0.245069	0.122535	−	0.992464i	$-0.460898\pi$
$277$	4.07877	0.245069	0.122535	+	0.992464i	$0.460898\pi$
$278$	7.95697	0.477227
$279$	0	0
$280$	−10.3546	−0.618805
$281$	−3.04066	−0.181390	−0.0906952	−	0.995879i	$-0.528909\pi$
$281$	−3.04066	−0.181390	−0.0906952	+	0.995879i	$0.528909\pi$
$282$	0	0
$283$	2.24468	0.133432	0.0667162	−	0.997772i	$-0.478748\pi$
$283$	2.24468	0.133432	0.0667162	+	0.997772i	$0.478748\pi$
$284$	1.77915	0.105573
$285$	0	0
$286$	2.82741	0.167188
$287$	−16.0526	−0.947556
$288$	0	0
$289$	6.08907	0.358181
$290$	6.16831	0.362215
$291$	0	0
$292$	−2.35787	−0.137984
$293$	−18.3882	−1.07425	−0.537124	−	0.843503i	$-0.680489\pi$
$293$	−18.3882	−1.07425	−0.537124	+	0.843503i	$0.680489\pi$
$294$	0	0
$295$	−10.8553	−0.632020
$296$	17.6929	1.02838
$297$	0	0
$298$	−15.9878	−0.926147
$299$	10.1277	0.585698
$300$	0	0
$301$	50.1785	2.89224
$302$	−7.52925	−0.433260
$303$	0	0
$304$	−4.62880	−0.265480
$305$	−7.35724	−0.421274
$306$	0	0
$307$	13.5196	0.771607	0.385804	−	0.922581i	$-0.373924\pi$
$307$	13.5196	0.771607	0.385804	+	0.922581i	$0.373924\pi$
$308$	−1.62601	−0.0926503
$309$	0	0
$310$	13.1670	0.747838
$311$	−27.3678	−1.55188	−0.775941	−	0.630805i	$-0.782725\pi$
$311$	−27.3678	−1.55188	−0.775941	+	0.630805i	$0.782725\pi$
$312$	0	0
$313$	−21.1505	−1.19550	−0.597748	−	0.801684i	$-0.703937\pi$
$313$	−21.1505	−1.19550	−0.597748	+	0.801684i	$0.703937\pi$
$314$	−26.8777	−1.51680
$315$	0	0
$316$	3.78385	0.212858
$317$	4.68075	0.262897	0.131448	−	0.991323i	$-0.458037\pi$
$317$	4.68075	0.262897	0.131448	+	0.991323i	$0.458037\pi$
$318$	0	0
$319$	−3.98933	−0.223360
$320$	−5.88602	−0.329038
$321$	0	0
$322$	−35.6362	−1.98593
$323$	−4.80511	−0.267363
$324$	0	0
$325$	1.82862	0.101433
$326$	26.3053	1.45691
$327$	0	0
$328$	−9.59842	−0.529984
$329$	23.9670	1.32134
$330$	0	0
$331$	24.3025	1.33578	0.667892	−	0.744258i	$-0.267197\pi$
$331$	24.3025	1.33578	0.667892	+	0.744258i	$0.267197\pi$
$332$	−0.198674	−0.0109037
$333$	0	0
$334$	15.9769	0.874215
$335$	11.7680	0.642955
$336$	0	0
$337$	11.6901	0.636803	0.318401	−	0.947956i	$-0.396854\pi$
$337$	11.6901	0.636803	0.318401	+	0.947956i	$0.396854\pi$
$338$	−14.9304	−0.812104
$339$	0	0
$340$	1.87753	0.101823
$341$	−8.51574	−0.461153
$342$	0	0
$343$	−13.8043	−0.745365
$344$	30.0035	1.61768
$345$	0	0
$346$	−23.4994	−1.26334
$347$	−9.85130	−0.528845	−0.264423	−	0.964407i	$-0.585181\pi$
$347$	−9.85130	−0.528845	−0.264423	+	0.964407i	$0.585181\pi$
$348$	0	0
$349$	−3.99813	−0.214015	−0.107007	−	0.994258i	$-0.534127\pi$
$349$	−3.99813	−0.214015	−0.107007	+	0.994258i	$0.534127\pi$
$350$	−6.43436	−0.343931
$351$	0	0
$352$	−2.18056	−0.116224
$353$	−7.73116	−0.411488	−0.205744	−	0.978606i	$-0.565961\pi$
$353$	−7.73116	−0.411488	−0.205744	+	0.978606i	$0.565961\pi$
$354$	0	0
$355$	−4.55334	−0.241666
$356$	−2.68107	−0.142097
$357$	0	0
$358$	15.9038	0.840545
$359$	−4.83542	−0.255204	−0.127602	−	0.991825i	$-0.540728\pi$
$359$	−4.83542	−0.255204	−0.127602	+	0.991825i	$0.540728\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	35.7168	1.87723
$363$	0	0
$364$	−2.97334	−0.155846
$365$	6.03445	0.315857
$366$	0	0
$367$	−29.8171	−1.55644	−0.778220	−	0.627992i	$-0.783877\pi$
$367$	−29.8171	−1.55644	−0.778220	+	0.627992i	$0.783877\pi$
$368$	−25.6363	−1.33638
$369$	0	0
$370$	10.9944	0.571571
$371$	−11.7732	−0.611233
$372$	0	0
$373$	−28.2081	−1.46056	−0.730280	−	0.683148i	$-0.760610\pi$
$373$	−28.2081	−1.46056	−0.730280	+	0.683148i	$0.760610\pi$
$374$	−7.42966	−0.384179
$375$	0	0
$376$	14.3307	0.739050
$377$	−7.29496	−0.375710
$378$	0	0
$379$	−33.7427	−1.73325	−0.866624	−	0.498962i	$-0.833715\pi$
$379$	−33.7427	−1.73325	−0.866624	+	0.498962i	$0.833715\pi$
$380$	−0.390736	−0.0200443
$381$	0	0
$382$	−21.1502	−1.08214
$383$	29.4579	1.50523	0.752615	−	0.658461i	$-0.228792\pi$
$383$	29.4579	1.50523	0.752615	+	0.658461i	$0.228792\pi$
$384$	0	0
$385$	4.16140	0.212085
$386$	14.6709	0.746729
$387$	0	0
$388$	−1.45604	−0.0739193
$389$	24.5828	1.24640	0.623199	−	0.782063i	$-0.285833\pi$
$389$	24.5828	1.24640	0.623199	+	0.782063i	$0.285833\pi$
$390$	0	0
$391$	−26.6128	−1.34586
$392$	−25.6718	−1.29662
$393$	0	0
$394$	26.6312	1.34166
$395$	−9.68391	−0.487250
$396$	0	0
$397$	22.5952	1.13402	0.567009	−	0.823711i	$-0.308100\pi$
$397$	22.5952	1.13402	0.567009	+	0.823711i	$0.308100\pi$
$398$	22.3593	1.12077
$399$	0	0
$400$	−4.62880	−0.231440
$401$	10.9449	0.546562	0.273281	−	0.961934i	$-0.411891\pi$
$401$	10.9449	0.546562	0.273281	+	0.961934i	$0.411891\pi$
$402$	0	0
$403$	−15.5720	−0.775699
$404$	5.50677	0.273972
$405$	0	0
$406$	25.6688	1.27392
$407$	−7.11059	−0.352459
$408$	0	0
$409$	30.7338	1.51969	0.759844	−	0.650106i	$-0.225275\pi$
$409$	30.7338	1.51969	0.759844	+	0.650106i	$0.225275\pi$
$410$	−5.96447	−0.294564
$411$	0	0
$412$	4.23847	0.208814
$413$	−45.1732	−2.22283
$414$	0	0
$415$	0.508463	0.0249595
$416$	−3.98741	−0.195499
$417$	0	0
$418$	1.54620	0.0756271
$419$	35.3598	1.72744	0.863720	−	0.503972i	$-0.168129\pi$
$419$	35.3598	1.72744	0.863720	+	0.503972i	$0.168129\pi$
$420$	0	0
$421$	−38.7689	−1.88948	−0.944740	−	0.327820i	$-0.893686\pi$
$421$	−38.7689	−1.88948	−0.944740	+	0.327820i	$0.893686\pi$
$422$	1.76624	0.0859795
$423$	0	0
$424$	−7.03960	−0.341873
$425$	−4.80511	−0.233082
$426$	0	0
$427$	−30.6164	−1.48163
$428$	3.71086	0.179371
$429$	0	0
$430$	18.6442	0.899104
$431$	−29.7370	−1.43238	−0.716191	−	0.697904i	$-0.754116\pi$
$431$	−29.7370	−1.43238	−0.716191	+	0.697904i	$0.754116\pi$
$432$	0	0
$433$	−8.54319	−0.410559	−0.205280	−	0.978703i	$-0.565810\pi$
$433$	−8.54319	−0.410559	−0.205280	+	0.978703i	$0.565810\pi$
$434$	54.7933	2.63016
$435$	0	0
$436$	4.00375	0.191745
$437$	5.53843	0.264939
$438$	0	0
$439$	−2.07249	−0.0989146	−0.0494573	−	0.998776i	$-0.515749\pi$
$439$	−2.07249	−0.0989146	−0.0494573	+	0.998776i	$0.515749\pi$
$440$	2.48825	0.118622
$441$	0	0
$442$	−13.5860	−0.646221
$443$	−23.8457	−1.13295	−0.566473	−	0.824081i	$-0.691692\pi$
$443$	−23.8457	−1.13295	−0.566473	+	0.824081i	$0.691692\pi$
$444$	0	0
$445$	6.86161	0.325271
$446$	10.6195	0.502848
$447$	0	0
$448$	−24.4941	−1.15724
$449$	−5.44413	−0.256925	−0.128462	−	0.991714i	$-0.541004\pi$
$449$	−5.44413	−0.256925	−0.128462	+	0.991714i	$0.541004\pi$
$450$	0	0
$451$	3.85750	0.181643
$452$	2.80953	0.132149
$453$	0	0
$454$	0.162305	0.00761736
$455$	7.60961	0.356744
$456$	0	0
$457$	−23.4484	−1.09687	−0.548435	−	0.836193i	$-0.684776\pi$
$457$	−23.4484	−1.09687	−0.548435	+	0.836193i	$0.684776\pi$
$458$	23.1434	1.08142
$459$	0	0
$460$	−2.16406	−0.100900
$461$	37.5982	1.75112	0.875560	−	0.483109i	$-0.160492\pi$
$461$	37.5982	1.75112	0.875560	+	0.483109i	$0.160492\pi$
$462$	0	0
$463$	27.2719	1.26743	0.633716	−	0.773566i	$-0.281529\pi$
$463$	27.2719	1.26743	0.633716	+	0.773566i	$0.281529\pi$
$464$	18.4658	0.857253
$465$	0	0
$466$	25.9768	1.20335
$467$	−14.2213	−0.658082	−0.329041	−	0.944316i	$-0.606725\pi$
$467$	−14.2213	−0.658082	−0.329041	+	0.944316i	$0.606725\pi$
$468$	0	0
$469$	48.9713	2.26129
$470$	8.90513	0.410763
$471$	0	0
$472$	−27.0106	−1.24327
$473$	−12.0581	−0.554431
$474$	0	0
$475$	1.00000	0.0458831
$476$	7.81314	0.358115
$477$	0	0
$478$	5.32424	0.243525
$479$	−1.56056	−0.0713037	−0.0356518	−	0.999364i	$-0.511351\pi$
$479$	−1.56056	−0.0713037	−0.0356518	+	0.999364i	$0.511351\pi$
$480$	0	0
$481$	−13.0025	−0.592865
$482$	−22.9047	−1.04328
$483$	0	0
$484$	0.390736	0.0177607
$485$	3.72641	0.169208
$486$	0	0
$487$	−17.8046	−0.806803	−0.403402	−	0.915023i	$-0.632172\pi$
$487$	−17.8046	−0.806803	−0.403402	+	0.915023i	$0.632172\pi$
$488$	−18.3066	−0.828702
$489$	0	0
$490$	−15.9525	−0.720661
$491$	−4.97956	−0.224724	−0.112362	−	0.993667i	$-0.535842\pi$
$491$	−4.97956	−0.224724	−0.112362	+	0.993667i	$0.535842\pi$
$492$	0	0
$493$	19.1692	0.863336
$494$	2.82741	0.127211
$495$	0	0
$496$	39.4176	1.76990
$497$	−18.9483	−0.849946
$498$	0	0
$499$	7.14052	0.319654	0.159827	−	0.987145i	$-0.448906\pi$
$499$	7.14052	0.319654	0.159827	+	0.987145i	$0.448906\pi$
$500$	−0.390736	−0.0174742
$501$	0	0
$502$	−13.5486	−0.604705
$503$	−39.9024	−1.77916	−0.889579	−	0.456781i	$-0.849002\pi$
$503$	−39.9024	−1.77916	−0.889579	+	0.456781i	$0.849002\pi$
$504$	0	0
$505$	−14.0934	−0.627146
$506$	8.56352	0.380695
$507$	0	0
$508$	−1.31031	−0.0581357
$509$	−2.18899	−0.0970252	−0.0485126	−	0.998823i	$-0.515448\pi$
$509$	−2.18899	−0.0970252	−0.0485126	+	0.998823i	$0.515448\pi$
$510$	0	0
$511$	25.1117	1.11088
$512$	−12.9418	−0.571953
$513$	0	0
$514$	−3.91127	−0.172519
$515$	−10.8474	−0.477994
$516$	0	0
$517$	−5.75936	−0.253297
$518$	45.7520	2.01023
$519$	0	0
$520$	4.55005	0.199533
$521$	−21.6375	−0.947957	−0.473979	−	0.880536i	$-0.657183\pi$
$521$	−21.6375	−0.947957	−0.473979	+	0.880536i	$0.657183\pi$
$522$	0	0
$523$	26.8411	1.17368	0.586840	−	0.809703i	$-0.300372\pi$
$523$	26.8411	1.17368	0.586840	+	0.809703i	$0.300372\pi$
$524$	2.01380	0.0879735
$525$	0	0
$526$	23.6974	1.03325
$527$	40.9191	1.78246
$528$	0	0
$529$	7.67419	0.333661
$530$	−4.37442	−0.190013
$531$	0	0
$532$	−1.62601	−0.0704963
$533$	7.05390	0.305538
$534$	0	0
$535$	−9.49711	−0.410596
$536$	29.2817	1.26478
$537$	0	0
$538$	−29.6289	−1.27739
$539$	10.3172	0.444395
$540$	0	0
$541$	−44.0968	−1.89587	−0.947935	−	0.318465i	$-0.896833\pi$
$541$	−44.0968	−1.89587	−0.947935	+	0.318465i	$0.896833\pi$
$542$	46.7829	2.00950
$543$	0	0
$544$	10.4778	0.449232
$545$	−10.2467	−0.438920
$546$	0	0
$547$	−21.5021	−0.919362	−0.459681	−	0.888084i	$-0.652036\pi$
$547$	−21.5021	−0.919362	−0.459681	+	0.888084i	$0.652036\pi$
$548$	4.41939	0.188787
$549$	0	0
$550$	1.54620	0.0659302
$551$	−3.98933	−0.169951
$552$	0	0
$553$	−40.2986	−1.71367
$554$	6.30659	0.267941
$555$	0	0
$556$	2.01078	0.0852761
$557$	33.2260	1.40783	0.703915	−	0.710284i	$-0.251434\pi$
$557$	33.2260	1.40783	0.703915	+	0.710284i	$0.251434\pi$
$558$	0	0
$559$	−22.0496	−0.932600
$560$	−19.2623	−0.813979
$561$	0	0
$562$	−4.70146	−0.198319
$563$	7.31538	0.308306	0.154153	−	0.988047i	$-0.450735\pi$
$563$	7.31538	0.308306	0.154153	+	0.988047i	$0.450735\pi$
$564$	0	0
$565$	−7.19036	−0.302501
$566$	3.47073	0.145885
$567$	0	0
$568$	−11.3298	−0.475389
$569$	−7.20515	−0.302056	−0.151028	−	0.988530i	$-0.548258\pi$
$569$	−7.20515	−0.302056	−0.151028	+	0.988530i	$0.548258\pi$
$570$	0	0
$571$	15.0655	0.630472	0.315236	−	0.949013i	$-0.397916\pi$
$571$	15.0655	0.630472	0.315236	+	0.949013i	$0.397916\pi$
$572$	0.714506	0.0298750
$573$	0	0
$574$	−24.8206	−1.03599
$575$	5.53843	0.230968
$576$	0	0
$577$	5.82824	0.242633	0.121316	−	0.992614i	$-0.461288\pi$
$577$	5.82824	0.242633	0.121316	+	0.992614i	$0.461288\pi$
$578$	9.41492	0.391609
$579$	0	0
$580$	1.55877	0.0647245
$581$	2.11592	0.0877830
$582$	0	0
$583$	2.82914	0.117171
$584$	15.0152	0.621333
$585$	0	0
$586$	−28.4318	−1.17451
$587$	40.6364	1.67724	0.838622	−	0.544714i	$-0.183362\pi$
$587$	40.6364	1.67724	0.838622	+	0.544714i	$0.183362\pi$
$588$	0	0
$589$	−8.51574	−0.350885
$590$	−16.7845	−0.691005
$591$	0	0
$592$	32.9135	1.35273
$593$	32.0839	1.31753	0.658764	−	0.752350i	$-0.271080\pi$
$593$	32.0839	1.31753	0.658764	+	0.752350i	$0.271080\pi$
$594$	0	0
$595$	−19.9960	−0.819755
$596$	−4.04022	−0.165494
$597$	0	0
$598$	15.6594	0.640361
$599$	47.1879	1.92805	0.964023	−	0.265820i	$-0.0856428\pi$
$599$	47.1879	1.92805	0.964023	+	0.265820i	$0.0856428\pi$
$600$	0	0
$601$	−6.75179	−0.275411	−0.137706	−	0.990473i	$-0.543973\pi$
$601$	−6.75179	−0.275411	−0.137706	+	0.990473i	$0.543973\pi$
$602$	77.5860	3.16217
$603$	0	0
$604$	−1.90269	−0.0774195
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	37.4849	1.52147	0.760733	−	0.649065i	$-0.224840\pi$
$607$	37.4849	1.52147	0.760733	+	0.649065i	$0.224840\pi$
$608$	−2.18056	−0.0884332
$609$	0	0
$610$	−11.3758	−0.460592
$611$	−10.5317	−0.426066
$612$	0	0
$613$	35.4094	1.43017	0.715085	−	0.699037i	$-0.246388\pi$
$613$	35.4094	1.43017	0.715085	+	0.699037i	$0.246388\pi$
$614$	20.9041	0.843620
$615$	0	0
$616$	10.3546	0.417198
$617$	27.4510	1.10513	0.552567	−	0.833469i	$-0.313648\pi$
$617$	27.4510	1.10513	0.552567	+	0.833469i	$0.313648\pi$
$618$	0	0
$619$	−7.15830	−0.287716	−0.143858	−	0.989598i	$-0.545951\pi$
$619$	−7.15830	−0.287716	−0.143858	+	0.989598i	$0.545951\pi$
$620$	3.32740	0.133632
$621$	0	0
$622$	−42.3160	−1.69672
$623$	28.5539	1.14399
$624$	0	0
$625$	1.00000	0.0400000
$626$	−32.7029	−1.30707
$627$	0	0
$628$	−6.79219	−0.271038
$629$	34.1671	1.36233
$630$	0	0
$631$	−28.7158	−1.14316	−0.571578	−	0.820547i	$-0.693669\pi$
$631$	−28.7158	−1.14316	−0.571578	+	0.820547i	$0.693669\pi$
$632$	−24.0959	−0.958485
$633$	0	0
$634$	7.23737	0.287433
$635$	3.35345	0.133078
$636$	0	0
$637$	18.8663	0.747509
$638$	−6.16831	−0.244206
$639$	0	0
$640$	−13.4621	−0.532135
$641$	42.9018	1.69452	0.847259	−	0.531180i	$-0.178251\pi$
$641$	42.9018	1.69452	0.847259	+	0.531180i	$0.178251\pi$
$642$	0	0
$643$	−26.1604	−1.03167	−0.515834	−	0.856689i	$-0.672518\pi$
$643$	−26.1604	−1.03167	−0.515834	+	0.856689i	$0.672518\pi$
$644$	−9.00552	−0.354867
$645$	0	0
$646$	−7.42966	−0.292316
$647$	−32.5031	−1.27783	−0.638915	−	0.769277i	$-0.720616\pi$
$647$	−32.5031	−1.27783	−0.638915	+	0.769277i	$0.720616\pi$
$648$	0	0
$649$	10.8553	0.426107
$650$	2.82741	0.110900
$651$	0	0
$652$	6.64752	0.260337
$653$	−18.9389	−0.741137	−0.370569	−	0.928805i	$-0.620837\pi$
$653$	−18.9389	−0.741137	−0.370569	+	0.928805i	$0.620837\pi$
$654$	0	0
$655$	−5.15388	−0.201379
$656$	−17.8556	−0.697144
$657$	0	0
$658$	37.0578	1.44466
$659$	−43.5740	−1.69740	−0.848701	−	0.528873i	$-0.822615\pi$
$659$	−43.5740	−1.69740	−0.848701	+	0.528873i	$0.822615\pi$
$660$	0	0
$661$	−34.9291	−1.35858	−0.679292	−	0.733868i	$-0.737713\pi$
$661$	−34.9291	−1.35858	−0.679292	+	0.733868i	$0.737713\pi$
$662$	37.5765	1.46045
$663$	0	0
$664$	1.26518	0.0490985
$665$	4.16140	0.161372
$666$	0	0
$667$	−22.0946	−0.855507
$668$	4.03746	0.156214
$669$	0	0
$670$	18.1957	0.702961
$671$	7.35724	0.284023
$672$	0	0
$673$	39.0030	1.50346	0.751728	−	0.659473i	$-0.229220\pi$
$673$	39.0030	1.50346	0.751728	+	0.659473i	$0.229220\pi$
$674$	18.0753	0.696235
$675$	0	0
$676$	−3.77300	−0.145116
$677$	−27.3864	−1.05255	−0.526273	−	0.850316i	$-0.676411\pi$
$677$	−27.3864	−1.05255	−0.526273	+	0.850316i	$0.676411\pi$
$678$	0	0
$679$	15.5071	0.595107
$680$	−11.9563	−0.458503
$681$	0	0
$682$	−13.1670	−0.504192
$683$	−44.9382	−1.71951	−0.859755	−	0.510706i	$-0.829384\pi$
$683$	−44.9382	−1.71951	−0.859755	+	0.510706i	$0.829384\pi$
$684$	0	0
$685$	−11.3104	−0.432150
$686$	−21.3443	−0.814929
$687$	0	0
$688$	55.8144	2.12790
$689$	5.17342	0.197092
$690$	0	0
$691$	25.4509	0.968199	0.484100	−	0.875013i	$-0.339147\pi$
$691$	25.4509	0.968199	0.484100	+	0.875013i	$0.339147\pi$
$692$	−5.93847	−0.225747
$693$	0	0
$694$	−15.2321	−0.578202
$695$	−5.14614	−0.195204
$696$	0	0
$697$	−18.5357	−0.702091
$698$	−6.18191	−0.233989
$699$	0	0
$700$	−1.62601	−0.0614573
$701$	−50.6146	−1.91169	−0.955844	−	0.293873i	$-0.905056\pi$
$701$	−50.6146	−1.91169	−0.955844	+	0.293873i	$0.905056\pi$
$702$	0	0
$703$	−7.11059	−0.268181
$704$	5.88602	0.221838
$705$	0	0
$706$	−11.9539	−0.449892
$707$	−58.6481	−2.20569
$708$	0	0
$709$	25.8266	0.969938	0.484969	−	0.874531i	$-0.338831\pi$
$709$	25.8266	0.969938	0.484969	+	0.874531i	$0.338831\pi$
$710$	−7.04038	−0.264221
$711$	0	0
$712$	17.0734	0.639852
$713$	−47.1638	−1.76630
$714$	0	0
$715$	−1.82862	−0.0683864
$716$	4.01901	0.150198
$717$	0	0
$718$	−7.47653	−0.279021
$719$	26.1220	0.974187	0.487093	−	0.873350i	$-0.338057\pi$
$719$	26.1220	0.974187	0.487093	+	0.873350i	$0.338057\pi$
$720$	0	0
$721$	−45.1404	−1.68112
$722$	1.54620	0.0575436
$723$	0	0
$724$	9.02588	0.335444
$725$	−3.98933	−0.148160
$726$	0	0
$727$	−32.1853	−1.19369	−0.596844	−	0.802357i	$-0.703579\pi$
$727$	−32.1853	−1.19369	−0.596844	+	0.802357i	$0.703579\pi$
$728$	18.9346	0.701762
$729$	0	0
$730$	9.33046	0.345336
$731$	57.9404	2.14300
$732$	0	0
$733$	−24.9746	−0.922457	−0.461229	−	0.887281i	$-0.652591\pi$
$733$	−24.9746	−0.922457	−0.461229	+	0.887281i	$0.652591\pi$
$734$	−46.1032	−1.70170
$735$	0	0
$736$	−12.0769	−0.445159
$737$	−11.7680	−0.433480
$738$	0	0
$739$	−21.0642	−0.774859	−0.387429	−	0.921899i	$-0.626637\pi$
$739$	−21.0642	−0.774859	−0.387429	+	0.921899i	$0.626637\pi$
$740$	2.77836	0.102134
$741$	0	0
$742$	−18.2037	−0.668279
$743$	19.3902	0.711357	0.355679	−	0.934608i	$-0.384250\pi$
$743$	19.3902	0.711357	0.355679	+	0.934608i	$0.384250\pi$
$744$	0	0
$745$	10.3400	0.378830
$746$	−43.6154	−1.59687
$747$	0	0
$748$	−1.87753	−0.0686492
$749$	−39.5213	−1.44408
$750$	0	0
$751$	−32.5618	−1.18820	−0.594098	−	0.804393i	$-0.702491\pi$
$751$	−32.5618	−1.18820	−0.594098	+	0.804393i	$0.702491\pi$
$752$	26.6589	0.972151
$753$	0	0
$754$	−11.2795	−0.410774
$755$	4.86952	0.177220
$756$	0	0
$757$	40.4497	1.47017	0.735084	−	0.677976i	$-0.237143\pi$
$757$	40.4497	1.47017	0.735084	+	0.677976i	$0.237143\pi$
$758$	−52.1730	−1.89501
$759$	0	0
$760$	2.48825	0.0902581
$761$	−20.6529	−0.748666	−0.374333	−	0.927294i	$-0.622128\pi$
$761$	−20.6529	−0.748666	−0.374333	+	0.927294i	$0.622128\pi$
$762$	0	0
$763$	−42.6406	−1.54369
$764$	−5.34481	−0.193368
$765$	0	0
$766$	45.5479	1.64571
$767$	19.8502	0.716749
$768$	0	0
$769$	−2.53650	−0.0914686	−0.0457343	−	0.998954i	$-0.514563\pi$
$769$	−2.53650	−0.0914686	−0.0457343	+	0.998954i	$0.514563\pi$
$770$	6.43436	0.231878
$771$	0	0
$772$	3.70744	0.133434
$773$	−27.4490	−0.987273	−0.493637	−	0.869668i	$-0.664333\pi$
$773$	−27.4490	−0.987273	−0.493637	+	0.869668i	$0.664333\pi$
$774$	0	0
$775$	−8.51574	−0.305895
$776$	9.27223	0.332854
$777$	0	0
$778$	38.0100	1.36272
$779$	3.85750	0.138209
$780$	0	0
$781$	4.55334	0.162931
$782$	−41.1486	−1.47147
$783$	0	0
$784$	−47.7564	−1.70559
$785$	17.3831	0.620429
$786$	0	0
$787$	39.4380	1.40581	0.702907	−	0.711281i	$-0.251885\pi$
$787$	39.4380	1.40581	0.702907	+	0.711281i	$0.251885\pi$
$788$	6.72989	0.239742
$789$	0	0
$790$	−14.9733	−0.532725
$791$	−29.9220	−1.06390
$792$	0	0
$793$	13.4536	0.477751
$794$	34.9367	1.23986
$795$	0	0
$796$	5.65034	0.200271
$797$	−38.1840	−1.35255	−0.676273	−	0.736651i	$-0.736406\pi$
$797$	−38.1840	−1.35255	−0.676273	+	0.736651i	$0.736406\pi$
$798$	0	0
$799$	27.6744	0.979049
$800$	−2.18056	−0.0770943
$801$	0	0
$802$	16.9230	0.597572
$803$	−6.03445	−0.212951
$804$	0	0
$805$	23.0476	0.812322
$806$	−24.0775	−0.848094
$807$	0	0
$808$	−35.0677	−1.23368
$809$	15.2742	0.537011	0.268506	−	0.963278i	$-0.413470\pi$
$809$	15.2742	0.537011	0.268506	+	0.963278i	$0.413470\pi$
$810$	0	0
$811$	9.16787	0.321928	0.160964	−	0.986960i	$-0.448540\pi$
$811$	9.16787	0.321928	0.160964	+	0.986960i	$0.448540\pi$
$812$	6.48668	0.227638
$813$	0	0
$814$	−10.9944	−0.385353
$815$	−17.0128	−0.595934
$816$	0	0
$817$	−12.0581	−0.421859
$818$	47.5206	1.66152
$819$	0	0
$820$	−1.50726	−0.0526359
$821$	8.97266	0.313148	0.156574	−	0.987666i	$-0.449955\pi$
$821$	8.97266	0.313148	0.156574	+	0.987666i	$0.449955\pi$
$822$	0	0
$823$	−11.8425	−0.412805	−0.206402	−	0.978467i	$-0.566176\pi$
$823$	−11.8425	−0.412805	−0.206402	+	0.978467i	$0.566176\pi$
$824$	−26.9910	−0.940277
$825$	0	0
$826$	−69.8468	−2.43028
$827$	35.1524	1.22237	0.611184	−	0.791488i	$-0.290693\pi$
$827$	35.1524	1.22237	0.611184	+	0.791488i	$0.290693\pi$
$828$	0	0
$829$	−37.8166	−1.31343	−0.656713	−	0.754141i	$-0.728054\pi$
$829$	−37.8166	−1.31343	−0.656713	+	0.754141i	$0.728054\pi$
$830$	0.786185	0.0272889
$831$	0	0
$832$	10.7633	0.373149
$833$	−49.5754	−1.71769
$834$	0	0
$835$	−10.3330	−0.357588
$836$	0.390736	0.0135139
$837$	0	0
$838$	54.6733	1.88866
$839$	−25.0571	−0.865067	−0.432533	−	0.901618i	$-0.642380\pi$
$839$	−25.0571	−0.865067	−0.432533	+	0.901618i	$0.642380\pi$
$840$	0	0
$841$	−13.0852	−0.451215
$842$	−59.9445	−2.06582
$843$	0	0
$844$	0.446342	0.0153637
$845$	9.65616	0.332182
$846$	0	0
$847$	−4.16140	−0.142987
$848$	−13.0955	−0.449702
$849$	0	0
$850$	−7.42966	−0.254835
$851$	−39.3815	−1.34998
$852$	0	0
$853$	23.8418	0.816329	0.408164	−	0.912908i	$-0.366169\pi$
$853$	23.8418	0.816329	0.408164	+	0.912908i	$0.366169\pi$
$854$	−47.3391	−1.61991
$855$	0	0
$856$	−23.6311	−0.807696
$857$	34.6121	1.18233	0.591164	−	0.806551i	$-0.298669\pi$
$857$	34.6121	1.18233	0.591164	+	0.806551i	$0.298669\pi$
$858$	0	0
$859$	32.6520	1.11407	0.557035	−	0.830489i	$-0.311939\pi$
$859$	32.6520	1.11407	0.557035	+	0.830489i	$0.311939\pi$
$860$	4.71152	0.160662
$861$	0	0
$862$	−45.9794	−1.56606
$863$	10.1698	0.346182	0.173091	−	0.984906i	$-0.444624\pi$
$863$	10.1698	0.346182	0.173091	+	0.984906i	$0.444624\pi$
$864$	0	0
$865$	15.1982	0.516753
$866$	−13.2095	−0.448876
$867$	0	0
$868$	13.8467	0.469986
$869$	9.68391	0.328504
$870$	0	0
$871$	−21.5192	−0.729150
$872$	−25.4963	−0.863414
$873$	0	0
$874$	8.56352	0.289665
$875$	4.16140	0.140681
$876$	0	0
$877$	8.91797	0.301138	0.150569	−	0.988599i	$-0.451889\pi$
$877$	8.91797	0.301138	0.150569	+	0.988599i	$0.451889\pi$
$878$	−3.20449	−0.108146
$879$	0	0
$880$	4.62880	0.156037
$881$	9.08968	0.306239	0.153120	−	0.988208i	$-0.451068\pi$
$881$	9.08968	0.306239	0.153120	+	0.988208i	$0.451068\pi$
$882$	0	0
$883$	−45.0015	−1.51442	−0.757210	−	0.653171i	$-0.773438\pi$
$883$	−45.0015	−1.51442	−0.757210	+	0.653171i	$0.773438\pi$
$884$	−3.43328	−0.115474
$885$	0	0
$886$	−36.8703	−1.23868
$887$	21.4687	0.720850	0.360425	−	0.932788i	$-0.382632\pi$
$887$	21.4687	0.720850	0.360425	+	0.932788i	$0.382632\pi$
$888$	0	0
$889$	13.9550	0.468037
$890$	10.6094	0.355629
$891$	0	0
$892$	2.68362	0.0898542
$893$	−5.75936	−0.192730
$894$	0	0
$895$	−10.2858	−0.343815
$896$	−56.0211	−1.87153
$897$	0	0
$898$	−8.41772	−0.280903
$899$	33.9721	1.13303
$900$	0	0
$901$	−13.5943	−0.452893
$902$	5.96447	0.198595
$903$	0	0
$904$	−17.8914	−0.595059
$905$	−23.0997	−0.767861
$906$	0	0
$907$	12.5080	0.415322	0.207661	−	0.978201i	$-0.433415\pi$
$907$	12.5080	0.415322	0.207661	+	0.978201i	$0.433415\pi$
$908$	0.0410156	0.00136115
$909$	0	0
$910$	11.7660	0.390038
$911$	4.62780	0.153326	0.0766629	−	0.997057i	$-0.475573\pi$
$911$	4.62780	0.153326	0.0766629	+	0.997057i	$0.475573\pi$
$912$	0	0
$913$	−0.508463	−0.0168277
$914$	−36.2559	−1.19924
$915$	0	0
$916$	5.84849	0.193239
$917$	−21.4474	−0.708254
$918$	0	0
$919$	−11.1376	−0.367395	−0.183698	−	0.982983i	$-0.558807\pi$
$919$	−11.1376	−0.367395	−0.183698	+	0.982983i	$0.558807\pi$
$920$	13.7810	0.454345
$921$	0	0
$922$	58.1343	1.91455
$923$	8.32632	0.274064
$924$	0	0
$925$	−7.11059	−0.233795
$926$	42.1678	1.38572
$927$	0	0
$928$	8.69896	0.285557
$929$	27.3911	0.898672	0.449336	−	0.893363i	$-0.351661\pi$
$929$	27.3911	0.898672	0.449336	+	0.893363i	$0.351661\pi$
$930$	0	0
$931$	10.3172	0.338134
$932$	6.56452	0.215028
$933$	0	0
$934$	−21.9889	−0.719500
$935$	4.80511	0.157144
$936$	0	0
$937$	41.4439	1.35391	0.676957	−	0.736022i	$-0.263298\pi$
$937$	41.4439	1.35391	0.676957	+	0.736022i	$0.263298\pi$
$938$	75.7195	2.47233
$939$	0	0
$940$	2.25039	0.0733996
$941$	43.5568	1.41991	0.709956	−	0.704246i	$-0.248715\pi$
$941$	43.5568	1.41991	0.709956	+	0.704246i	$0.248715\pi$
$942$	0	0
$943$	21.3645	0.695724
$944$	−50.2469	−1.63540
$945$	0	0
$946$	−18.6442	−0.606176
$947$	33.0491	1.07395	0.536976	−	0.843598i	$-0.319567\pi$
$947$	33.0491	1.07395	0.536976	+	0.843598i	$0.319567\pi$
$948$	0	0
$949$	−11.0347	−0.358201
$950$	1.54620	0.0501654
$951$	0	0
$952$	−49.7549	−1.61257
$953$	−0.505736	−0.0163824	−0.00819120	−	0.999966i	$-0.502607\pi$
$953$	−0.505736	−0.0163824	−0.00819120	+	0.999966i	$0.502607\pi$
$954$	0	0
$955$	13.6788	0.442637
$956$	1.34547	0.0435157
$957$	0	0
$958$	−2.41293	−0.0779584
$959$	−47.0672	−1.51988
$960$	0	0
$961$	41.5179	1.33929
$962$	−20.1045	−0.648196
$963$	0	0
$964$	−5.78817	−0.186424
$965$	−9.48835	−0.305441
$966$	0	0
$967$	−46.1493	−1.48406	−0.742031	−	0.670366i	$-0.766137\pi$
$967$	−46.1493	−1.48406	−0.742031	+	0.670366i	$0.766137\pi$
$968$	−2.48825	−0.0799752
$969$	0	0
$970$	5.76178	0.185000
$971$	27.0101	0.866795	0.433398	−	0.901203i	$-0.357315\pi$
$971$	27.0101	0.866795	0.433398	+	0.901203i	$0.357315\pi$
$972$	0	0
$973$	−21.4152	−0.686538
$974$	−27.5295	−0.882101
$975$	0	0
$976$	−34.0552	−1.09008
$977$	30.5323	0.976814	0.488407	−	0.872616i	$-0.337578\pi$
$977$	30.5323	0.976814	0.488407	+	0.872616i	$0.337578\pi$
$978$	0	0
$979$	−6.86161	−0.219298
$980$	−4.03131	−0.128776
$981$	0	0
$982$	−7.69940	−0.245698
$983$	−27.5882	−0.879926	−0.439963	−	0.898016i	$-0.645008\pi$
$983$	−27.5882	−0.879926	−0.439963	+	0.898016i	$0.645008\pi$
$984$	0	0
$985$	−17.2237	−0.548791
$986$	29.6394	0.943910
$987$	0	0
$988$	0.714506	0.0227315
$989$	−66.7828	−2.12357
$990$	0	0
$991$	−57.2710	−1.81927	−0.909636	−	0.415405i	$-0.863640\pi$
$991$	−57.2710	−1.81927	−0.909636	+	0.415405i	$0.863640\pi$
$992$	18.5691	0.589568
$993$	0	0
$994$	−29.2978	−0.929271
$995$	−14.4608	−0.458437
$996$	0	0
$997$	−50.4928	−1.59912	−0.799561	−	0.600584i	$-0.794935\pi$
$997$	−50.4928	−1.59912	−0.799561	+	0.600584i	$0.794935\pi$
$998$	11.0407	0.349486
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9405.2.a.v.1.4		5
3.2	odd	2		1045.2.a.d.1.2	✓	5
15.14	odd	2		5225.2.a.j.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.a.d.1.2	✓	5	3.2	odd	2
5225.2.a.j.1.4		5	15.14	odd	2
9405.2.a.v.1.4		5	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)

\(f(q)\)	\(=\)	\(q+1.54620 q^{2} +0.390736 q^{4} -1.00000 q^{5} -4.16140 q^{7} -2.48825 q^{8} -1.54620 q^{10} +1.00000 q^{11} +1.82862 q^{13} -6.43436 q^{14} -4.62880 q^{16} -4.80511 q^{17} +1.00000 q^{19} -0.390736 q^{20} +1.54620 q^{22} +5.53843 q^{23} +1.00000 q^{25} +2.82741 q^{26} -1.62601 q^{28} -3.98933 q^{29} -8.51574 q^{31} -2.18056 q^{32} -7.42966 q^{34} +4.16140 q^{35} -7.11059 q^{37} +1.54620 q^{38} +2.48825 q^{40} +3.85750 q^{41} -12.0581 q^{43} +0.390736 q^{44} +8.56352 q^{46} -5.75936 q^{47} +10.3172 q^{49} +1.54620 q^{50} +0.714506 q^{52} +2.82914 q^{53} -1.00000 q^{55} +10.3546 q^{56} -6.16831 q^{58} +10.8553 q^{59} +7.35724 q^{61} -13.1670 q^{62} +5.88602 q^{64} -1.82862 q^{65} -11.7680 q^{67} -1.87753 q^{68} +6.43436 q^{70} +4.55334 q^{71} -6.03445 q^{73} -10.9944 q^{74} +0.390736 q^{76} -4.16140 q^{77} +9.68391 q^{79} +4.62880 q^{80} +5.96447 q^{82} -0.508463 q^{83} +4.80511 q^{85} -18.6442 q^{86} -2.48825 q^{88} -6.86161 q^{89} -7.60961 q^{91} +2.16406 q^{92} -8.90513 q^{94} -1.00000 q^{95} -3.72641 q^{97} +15.9525 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 3 q^{2} + 5 q^{4} - 5 q^{5} - 11 q^{7} - 3 q^{8} - 3 q^{10} + 5 q^{11} + q^{13} - 3 q^{16} + 3 q^{17} + 5 q^{19} - 5 q^{20} + 3 q^{22} + 8 q^{23} + 5 q^{25} + 16 q^{26} - 22 q^{28} - 11 q^{29} - 5 q^{31}+ \cdots - 10 q^{98}+O(q^{100}) \) 5 * q + 3 * q^2 + 5 * q^4 - 5 * q^5 - 11 * q^7 - 3 * q^8 - 3 * q^10 + 5 * q^11 + q^13 - 3 * q^16 + 3 * q^17 + 5 * q^19 - 5 * q^20 + 3 * q^22 + 8 * q^23 + 5 * q^25 + 16 * q^26 - 22 * q^28 - 11 * q^29 - 5 * q^31 + 2 * q^32 + 4 * q^34 + 11 * q^35 - 9 * q^37 + 3 * q^38 + 3 * q^40 - 15 * q^41 - 13 * q^43 + 5 * q^44 + 18 * q^46 + 20 * q^47 + 20 * q^49 + 3 * q^50 + q^52 + 5 * q^53 - 5 * q^55 - 33 * q^58 + 17 * q^59 + 3 * q^61 - 14 * q^62 - 17 * q^64 - q^65 - 28 * q^67 + 25 * q^68 + 6 * q^71 - 16 * q^73 + 21 * q^74 + 5 * q^76 - 11 * q^77 + 3 * q^79 + 3 * q^80 + 2 * q^82 + 33 * q^83 - 3 * q^85 - 10 * q^86 - 3 * q^88 + 16 * q^89 - 22 * q^91 + 19 * q^92 - 10 * q^94 - 5 * q^95 - 14 * q^97 - 10 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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