LMFDB - Embedded newform 9.24.a.d.1.3

Newspace parameters

Level:	$ N $	$=$	$ 9 = 3^{2} $
Weight:	$ k $	$=$	$ 24 $
Character orbit:	$[\chi]$	$=$	9.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$30.1683633611$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
Defining polynomial:	$ x^{4} - 29258x^{2} + 97377280 $ x^4 - 29258*x^2 + 97377280
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{10}\cdot 5^{4} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$61.8825$ of defining polynomial
Character	$\chi$	$=$	9.1

$q$-expansion

$f(q)$	$=$	$q+1856.48 q^{2} -4.94211e6 q^{4} -1.25135e8 q^{5} -2.21402e9 q^{7} -2.47481e10 q^{8} +O(q^{10})$	\(q+1856.48 q^{2} -4.94211e6 q^{4} -1.25135e8 q^{5} -2.21402e9 q^{7} -2.47481e10 q^{8} -2.32310e11 q^{10} +9.54062e11 q^{11} +3.54107e12 q^{13} -4.11027e12 q^{14} -4.48694e12 q^{16} +1.97591e14 q^{17} -1.41772e14 q^{19} +6.18429e14 q^{20} +1.77119e15 q^{22} -1.35827e15 q^{23} +3.73779e15 q^{25} +6.57390e15 q^{26} +1.09419e16 q^{28} -1.17210e17 q^{29} +2.61332e17 q^{31} +1.99273e17 q^{32} +3.66823e17 q^{34} +2.77051e17 q^{35} +1.63064e18 q^{37} -2.63196e17 q^{38} +3.09685e18 q^{40} +1.39986e18 q^{41} +7.83002e18 q^{43} -4.71508e18 q^{44} -2.52160e18 q^{46} -1.10807e19 q^{47} -2.24669e19 q^{49} +6.93911e18 q^{50} -1.75003e19 q^{52} +6.84789e19 q^{53} -1.19386e20 q^{55} +5.47929e19 q^{56} -2.17598e20 q^{58} -3.80399e20 q^{59} -2.54300e20 q^{61} +4.85157e20 q^{62} +4.07584e20 q^{64} -4.43111e20 q^{65} +7.58191e20 q^{67} -9.76516e20 q^{68} +5.14338e20 q^{70} +1.65470e21 q^{71} +3.73535e21 q^{73} +3.02724e21 q^{74} +7.00652e20 q^{76} -2.11231e21 q^{77} +3.30977e21 q^{79} +5.61472e20 q^{80} +2.59880e21 q^{82} -1.73160e22 q^{83} -2.47255e22 q^{85} +1.45362e22 q^{86} -2.36113e22 q^{88} -7.98037e20 q^{89} -7.83999e21 q^{91} +6.71272e21 q^{92} -2.05711e22 q^{94} +1.77406e22 q^{95} +9.51761e20 q^{97} -4.17092e22 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 19109968 q^{4} + 8561438480 q^{7}+O(q^{10}) $ 4 * q + 19109968 * q^4 + 8561438480 * q^7	$ 4 q + 19109968 q^{4} + 8561438480 q^{7} + 898978809600 q^{10} - 13534569998680 q^{13} + 27399081725056 q^{16} + 823113907401824 q^{19} - 27\!\cdots\!00 q^{22}+ \cdots + 61\!\cdots\!80 q^{97}+O(q^{100}) $ 4 * q + 19109968 * q^4 + 8561438480 * q^7 + 898978809600 * q^10 - 13534569998680 * q^13 + 27399081725056 * q^16 + 823113907401824 * q^19 - 2742157478016000 * q^22 + 24257341946701900 * q^25 + 210193505445412160 * q^28 + 199165518593535632 * q^31 + 2993143687606771200 * q^34 + 6114312513425419640 * q^37 + 14523223786943846400 * q^40 + 8384436518889991520 * q^43 + 27149798727824025600 * q^46 - 15307929032390927772 * q^49 - 333882838377412430560 * q^52 - 425998563364306944000 * q^55 - 519004760872044384000 * q^58 - 187635897281945053672 * q^61 - 1002932785066778037248 * q^64 + 2217868901107775746880 * q^67 + 9884892109681717632000 * q^70 + 5558219032297686657560 * q^73 + 17444623005047704385408 * q^76 - 1562583260526294231664 * q^79 - 41334076267904219328000 * q^82 + 17862754059765861580800 * q^85 - 85611540851539743744000 * q^88 - 149580096340642991861600 * q^91 - 164391971318917439846400 * q^94 + 61688465492693971809080 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1856.48	0.640980	0.320490	−	0.947252i	$-0.396152\pi$
$2$	1856.48	0.640980	0.320490	+	0.947252i	$0.396152\pi$
$3$	0	0
$3$	0	0
$4$	−4.94211e6	−0.589145
$5$	−1.25135e8	−1.14610	−0.573051	−	0.819520i	$-0.694240\pi$
$5$	−1.25135e8	−1.14610	−0.573051	+	0.819520i	$0.694240\pi$
$6$	0	0
$7$	−2.21402e9	−0.423208	−0.211604	−	0.977355i	$-0.567869\pi$
$7$	−2.21402e9	−0.423208	−0.211604	+	0.977355i	$0.567869\pi$
$8$	−2.47481e10	−1.01861
$9$	0	0
$10$	−2.32310e11	−0.734628
$11$	9.54062e11	1.00823	0.504116	−	0.863636i	$-0.331818\pi$
$11$	9.54062e11	1.00823	0.504116	+	0.863636i	$0.331818\pi$
$12$	0	0
$13$	3.54107e12	0.548006	0.274003	−	0.961729i	$-0.411652\pi$
$13$	3.54107e12	0.548006	0.274003	+	0.961729i	$0.411652\pi$
$14$	−4.11027e12	−0.271268
$15$	0	0
$16$	−4.48694e12	−0.0637632
$17$	1.97591e14	1.39831	0.699157	−	0.714968i	$-0.253559\pi$
$17$	1.97591e14	1.39831	0.699157	+	0.714968i	$0.253559\pi$
$18$	0	0
$19$	−1.41772e14	−0.279205	−0.139603	−	0.990208i	$-0.544583\pi$
$19$	−1.41772e14	−0.279205	−0.139603	+	0.990208i	$0.544583\pi$
$20$	6.18429e14	0.675220
$21$	0	0
$22$	1.77119e15	0.646257
$23$	−1.35827e15	−0.297246	−0.148623	−	0.988894i	$-0.547484\pi$
$23$	−1.35827e15	−0.297246	−0.148623	+	0.988894i	$0.547484\pi$
$24$	0	0
$25$	3.73779e15	0.313548
$26$	6.57390e15	0.351261
$27$	0	0
$28$	1.09419e16	0.249331
$29$	−1.17210e17	−1.78397	−0.891986	−	0.452063i	$-0.850688\pi$
$29$	−1.17210e17	−1.78397	−0.891986	+	0.452063i	$0.850688\pi$
$30$	0	0
$31$	2.61332e17	1.84728	0.923642	−	0.383255i	$-0.125197\pi$
$31$	2.61332e17	1.84728	0.923642	+	0.383255i	$0.125197\pi$
$32$	1.99273e17	0.977739
$33$	0	0
$34$	3.66823e17	0.896291
$35$	2.77051e17	0.485040
$36$	0	0
$37$	1.63064e18	1.50674	0.753370	−	0.657597i	$-0.228427\pi$
$37$	1.63064e18	1.50674	0.753370	+	0.657597i	$0.228427\pi$
$38$	−2.63196e17	−0.178965
$39$	0	0
$40$	3.09685e18	1.16743
$41$	1.39986e18	0.397255	0.198628	−	0.980075i	$-0.436352\pi$
$41$	1.39986e18	0.397255	0.198628	+	0.980075i	$0.436352\pi$
$42$	0	0
$43$	7.83002e18	1.28492	0.642459	−	0.766320i	$-0.277914\pi$
$43$	7.83002e18	1.28492	0.642459	+	0.766320i	$0.277914\pi$
$44$	−4.71508e18	−0.593995
$45$	0	0
$46$	−2.52160e18	−0.190529
$47$	−1.10807e19	−0.653796	−0.326898	−	0.945060i	$-0.606003\pi$
$47$	−1.10807e19	−0.653796	−0.326898	+	0.945060i	$0.606003\pi$
$48$	0	0
$49$	−2.24669e19	−0.820895
$50$	6.93911e18	0.200978
$51$	0	0
$52$	−1.75003e19	−0.322855
$53$	6.84789e19	1.01481	0.507404	−	0.861708i	$-0.330605\pi$
$53$	6.84789e19	1.01481	0.507404	+	0.861708i	$0.330605\pi$
$54$	0	0
$55$	−1.19386e20	−1.15554
$56$	5.47929e19	0.431084
$57$	0	0
$58$	−2.17598e20	−1.14349
$59$	−3.80399e20	−1.64226	−0.821128	−	0.570744i	$-0.806655\pi$
$59$	−3.80399e20	−1.64226	−0.821128	+	0.570744i	$0.806655\pi$
$60$	0	0
$61$	−2.54300e20	−0.748260	−0.374130	−	0.927376i	$-0.622059\pi$
$61$	−2.54300e20	−0.748260	−0.374130	+	0.927376i	$0.622059\pi$
$62$	4.85157e20	1.18407
$63$	0	0
$64$	4.07584e20	0.690474
$65$	−4.43111e20	−0.628071
$66$	0	0
$67$	7.58191e20	0.758435	0.379218	−	0.925308i	$-0.376193\pi$
$67$	7.58191e20	0.758435	0.379218	+	0.925308i	$0.376193\pi$
$68$	−9.76516e20	−0.823810
$69$	0	0
$70$	5.14338e20	0.310901
$71$	1.65470e21	0.849668	0.424834	−	0.905271i	$-0.360333\pi$
$71$	1.65470e21	0.849668	0.424834	+	0.905271i	$0.360333\pi$
$72$	0	0
$73$	3.73535e21	1.39354	0.696768	−	0.717297i	$-0.254621\pi$
$73$	3.73535e21	1.39354	0.696768	+	0.717297i	$0.254621\pi$
$74$	3.02724e21	0.965790
$75$	0	0
$76$	7.00652e20	0.164493
$77$	−2.11231e21	−0.426692
$78$	0	0
$79$	3.30977e21	0.497836	0.248918	−	0.968525i	$-0.419925\pi$
$79$	3.30977e21	0.497836	0.248918	+	0.968525i	$0.419925\pi$
$80$	5.61472e20	0.0730791
$81$	0	0
$82$	2.59880e21	0.254632
$83$	−1.73160e22	−1.47588	−0.737939	−	0.674868i	$-0.764201\pi$
$83$	−1.73160e22	−1.47588	−0.737939	+	0.674868i	$0.764201\pi$
$84$	0	0
$85$	−2.47255e22	−1.60261
$86$	1.45362e22	0.823607
$87$	0	0
$88$	−2.36113e22	−1.02700
$89$	−7.98037e20	−0.0304816	−0.0152408	−	0.999884i	$-0.504851\pi$
$89$	−7.98037e20	−0.0304816	−0.0152408	+	0.999884i	$0.504851\pi$
$90$	0	0
$91$	−7.83999e21	−0.231921
$92$	6.71272e21	0.175121
$93$	0	0
$94$	−2.05711e22	−0.419070
$95$	1.77406e22	0.319998
$96$	0	0
$97$	9.51761e20	0.0135099	0.00675496	−	0.999977i	$-0.497850\pi$
$97$	9.51761e20	0.0135099	0.00675496	+	0.999977i	$0.497850\pi$
$98$	−4.17092e22	−0.526177
$99$	0	0
$100$	−1.84725e22	−0.184725
$101$	1.42407e23	1.27009	0.635045	−	0.772476i	$-0.280982\pi$
$101$	1.42407e23	1.27009	0.635045	+	0.772476i	$0.280982\pi$
$102$	0	0
$103$	1.25632e23	0.894275	0.447137	−	0.894465i	$-0.352444\pi$
$103$	1.25632e23	0.894275	0.447137	+	0.894465i	$0.352444\pi$
$104$	−8.76348e22	−0.558204
$105$	0	0
$106$	1.27129e23	0.650472
$107$	−2.55948e23	−1.17554	−0.587772	−	0.809027i	$-0.699995\pi$
$107$	−2.55948e23	−1.17554	−0.587772	+	0.809027i	$0.699995\pi$
$108$	0	0
$109$	3.89375e23	1.44532	0.722658	−	0.691205i	$-0.242920\pi$
$109$	3.89375e23	1.44532	0.722658	+	0.691205i	$0.242920\pi$
$110$	−2.21638e23	−0.740675
$111$	0	0
$112$	9.93417e21	0.0269851
$113$	5.25664e23	1.28916	0.644579	−	0.764538i	$-0.277033\pi$
$113$	5.25664e23	1.28916	0.644579	+	0.764538i	$0.277033\pi$
$114$	0	0
$115$	1.69967e23	0.340674
$116$	5.79265e23	1.05102
$117$	0	0
$118$	−7.06201e23	−1.05265
$119$	−4.37471e23	−0.591778
$120$	0	0
$121$	1.48038e22	0.0165326
$122$	−4.72101e23	−0.479620
$123$	0	0
$124$	−1.29153e24	−1.08832
$125$	1.02400e24	0.786743
$126$	0	0
$127$	−7.27123e23	−0.465441	−0.232721	−	0.972544i	$-0.574763\pi$
$127$	−7.27123e23	−0.465441	−0.232721	+	0.972544i	$0.574763\pi$
$128$	−9.14950e23	−0.535159
$129$	0	0
$130$	−8.22624e23	−0.402581
$131$	−1.29816e24	−0.581711	−0.290856	−	0.956767i	$-0.593940\pi$
$131$	−1.29816e24	−0.581711	−0.290856	+	0.956767i	$0.593940\pi$
$132$	0	0
$133$	3.13886e23	0.118162
$134$	1.40756e24	0.486142
$135$	0	0
$136$	−4.89001e24	−1.42434
$137$	−1.56938e24	−0.420187	−0.210093	−	0.977681i	$-0.567377\pi$
$137$	−1.56938e24	−0.420187	−0.210093	+	0.977681i	$0.567377\pi$
$138$	0	0
$139$	5.97766e24	1.35476	0.677378	−	0.735635i	$-0.263116\pi$
$139$	5.97766e24	1.35476	0.677378	+	0.735635i	$0.263116\pi$
$140$	−1.36922e24	−0.285759
$141$	0	0
$142$	3.07192e24	0.544620
$143$	3.37840e24	0.552518
$144$	0	0
$145$	1.46671e25	2.04461
$146$	6.93458e24	0.893228
$147$	0	0
$148$	−8.05880e24	−0.887688
$149$	1.50771e24	0.153700	0.0768502	−	0.997043i	$-0.475514\pi$
$149$	1.50771e24	0.153700	0.0768502	+	0.997043i	$0.475514\pi$
$150$	0	0
$151$	1.13154e25	0.989548	0.494774	−	0.869022i	$-0.335251\pi$
$151$	1.13154e25	0.989548	0.494774	+	0.869022i	$0.335251\pi$
$152$	3.50859e24	0.284401
$153$	0	0
$154$	−3.92146e24	−0.273501
$155$	−3.27018e25	−2.11718
$156$	0	0
$157$	4.43402e24	0.247715	0.123857	−	0.992300i	$-0.460473\pi$
$157$	4.43402e24	0.247715	0.123857	+	0.992300i	$0.460473\pi$
$158$	6.14450e24	0.319103
$159$	0	0
$160$	−2.49359e25	−1.12059
$161$	3.00724e24	0.125797
$162$	0	0
$163$	7.82788e24	0.284110	0.142055	−	0.989859i	$-0.454629\pi$
$163$	7.82788e24	0.284110	0.142055	+	0.989859i	$0.454629\pi$
$164$	−6.91824e24	−0.234041
$165$	0	0
$166$	−3.21468e25	−0.946008
$167$	6.36666e25	1.74853	0.874264	−	0.485452i	$-0.161345\pi$
$167$	6.36666e25	1.74853	0.874264	+	0.485452i	$0.161345\pi$
$168$	0	0
$169$	−2.92148e25	−0.699689
$170$	−4.59023e25	−1.02724
$171$	0	0
$172$	−3.86968e25	−0.757004
$173$	−1.87367e25	−0.342896	−0.171448	−	0.985193i	$-0.554844\pi$
$173$	−1.87367e25	−0.342896	−0.171448	+	0.985193i	$0.554844\pi$
$174$	0	0
$175$	−8.27554e24	−0.132696
$176$	−4.28081e24	−0.0642881
$177$	0	0
$178$	−1.48154e24	−0.0195381
$179$	1.38944e26	1.71802	0.859012	−	0.511955i	$-0.171079\pi$
$179$	1.38944e26	1.71802	0.859012	+	0.511955i	$0.171079\pi$
$180$	0	0
$181$	8.32272e25	0.905652	0.452826	−	0.891599i	$-0.350416\pi$
$181$	8.32272e25	0.905652	0.452826	+	0.891599i	$0.350416\pi$
$182$	−1.45548e25	−0.148657
$183$	0	0
$184$	3.36147e25	0.302778
$185$	−2.04050e26	−1.72688
$186$	0	0
$187$	1.88514e26	1.40983
$188$	5.47621e25	0.385181
$189$	0	0
$190$	3.29350e25	0.205112
$191$	−1.26673e26	−0.742679	−0.371340	−	0.928497i	$-0.621101\pi$
$191$	−1.26673e26	−0.742679	−0.371340	+	0.928497i	$0.621101\pi$
$192$	0	0
$193$	3.03237e25	0.157715	0.0788577	−	0.996886i	$-0.474873\pi$
$193$	3.03237e25	0.157715	0.0788577	+	0.996886i	$0.474873\pi$
$194$	1.76692e24	0.00865958
$195$	0	0
$196$	1.11034e26	0.483626
$197$	2.46843e26	1.01405	0.507025	−	0.861932i	$-0.330745\pi$
$197$	2.46843e26	1.01405	0.507025	+	0.861932i	$0.330745\pi$
$198$	0	0
$199$	−2.44174e26	−0.893078	−0.446539	−	0.894764i	$-0.647344\pi$
$199$	−2.44174e26	−0.893078	−0.446539	+	0.894764i	$0.647344\pi$
$200$	−9.25033e25	−0.319383
$201$	0	0
$202$	2.64374e26	0.814101
$203$	2.59506e26	0.754992
$204$	0	0
$205$	−1.75171e26	−0.455295
$206$	2.33232e26	0.573212
$207$	0	0
$208$	−1.58885e25	−0.0349426
$209$	−1.35259e26	−0.281504
$210$	0	0
$211$	−8.76884e26	−1.63566	−0.817832	−	0.575457i	$-0.804824\pi$
$211$	−8.76884e26	−1.63566	−0.817832	+	0.575457i	$0.804824\pi$
$212$	−3.38430e26	−0.597869
$213$	0	0
$214$	−4.75162e26	−0.753500
$215$	−9.79808e26	−1.47265
$216$	0	0
$217$	−5.78595e26	−0.781786
$218$	7.22865e26	0.926419
$219$	0	0
$220$	5.90020e26	0.680779
$221$	6.99683e26	0.766285
$222$	0	0
$223$	−2.83665e26	−0.280091	−0.140046	−	0.990145i	$-0.544725\pi$
$223$	−2.83665e26	−0.280091	−0.140046	+	0.990145i	$0.544725\pi$
$224$	−4.41194e26	−0.413787
$225$	0	0
$226$	9.75882e26	0.826324
$227$	−7.65440e26	−0.616047	−0.308024	−	0.951379i	$-0.599668\pi$
$227$	−7.65440e26	−0.616047	−0.308024	+	0.951379i	$0.599668\pi$
$228$	0	0
$229$	−9.18669e26	−0.668422	−0.334211	−	0.942498i	$-0.608470\pi$
$229$	−9.18669e26	−0.668422	−0.334211	+	0.942498i	$0.608470\pi$
$230$	3.15539e26	0.218365
$231$	0	0
$232$	2.90073e27	1.81717
$233$	−6.69203e26	−0.398993	−0.199496	−	0.979899i	$-0.563931\pi$
$233$	−6.69203e26	−0.398993	−0.199496	+	0.979899i	$0.563931\pi$
$234$	0	0
$235$	1.38658e27	0.749317
$236$	1.87997e27	0.967527
$237$	0	0
$238$	−8.12154e26	−0.379318
$239$	1.32134e27	0.588081	0.294041	−	0.955793i	$-0.405000\pi$
$239$	1.32134e27	0.588081	0.294041	+	0.955793i	$0.405000\pi$
$240$	0	0
$241$	9.67571e26	0.391279	0.195640	−	0.980676i	$-0.437322\pi$
$241$	9.67571e26	0.391279	0.195640	+	0.980676i	$0.437322\pi$
$242$	2.74828e25	0.0105970
$243$	0	0
$244$	1.25678e27	0.440834
$245$	2.81139e27	0.940829
$246$	0	0
$247$	−5.02024e26	−0.153006
$248$	−6.46749e27	−1.88166
$249$	0	0
$250$	1.90102e27	0.504286
$251$	−2.70487e27	−0.685328	−0.342664	−	0.939458i	$-0.611329\pi$
$251$	−2.70487e27	−0.685328	−0.342664	+	0.939458i	$0.611329\pi$
$252$	0	0
$253$	−1.29587e27	−0.299693
$254$	−1.34989e27	−0.298339
$255$	0	0
$256$	−5.11764e27	−1.03350
$257$	1.75032e27	0.337977	0.168988	−	0.985618i	$-0.445950\pi$
$257$	1.75032e27	0.337977	0.168988	+	0.985618i	$0.445950\pi$
$258$	0	0
$259$	−3.61027e27	−0.637665
$260$	2.18990e27	0.370025
$261$	0	0
$262$	−2.41000e27	−0.372865
$263$	5.05191e27	0.748108	0.374054	−	0.927407i	$-0.377967\pi$
$263$	5.05191e27	0.748108	0.374054	+	0.927407i	$0.377967\pi$
$264$	0	0
$265$	−8.56909e27	−1.16307
$266$	5.82722e26	0.0757395
$267$	0	0
$268$	−3.74706e27	−0.446828
$269$	−2.79663e27	−0.319509	−0.159754	−	0.987157i	$-0.551070\pi$
$269$	−2.79663e27	−0.319509	−0.159754	+	0.987157i	$0.551070\pi$
$270$	0	0
$271$	3.12408e27	0.327775	0.163887	−	0.986479i	$-0.447597\pi$
$271$	3.12408e27	0.327775	0.163887	+	0.986479i	$0.447597\pi$
$272$	−8.86579e26	−0.0891610
$273$	0	0
$274$	−2.91352e27	−0.269331
$275$	3.56608e27	0.316130
$276$	0	0
$277$	2.33237e28	1.90230	0.951152	−	0.308723i	$-0.0999016\pi$
$277$	2.33237e28	1.90230	0.951152	+	0.308723i	$0.0999016\pi$
$278$	1.10974e28	0.868371
$279$	0	0
$280$	−6.85650e27	−0.494066
$281$	−1.54956e28	−1.07173	−0.535866	−	0.844303i	$-0.680015\pi$
$281$	−1.54956e28	−1.07173	−0.535866	+	0.844303i	$0.680015\pi$
$282$	0	0
$283$	5.17195e27	0.329694	0.164847	−	0.986319i	$-0.447287\pi$
$283$	5.17195e27	0.329694	0.164847	+	0.986319i	$0.447287\pi$
$284$	−8.17772e27	−0.500578
$285$	0	0
$286$	6.27191e27	0.354153
$287$	−3.09931e27	−0.168122
$288$	0	0
$289$	1.90747e28	0.955283
$290$	2.72290e28	1.31056
$291$	0	0
$292$	−1.84605e28	−0.820995
$293$	−9.81588e27	−0.419712	−0.209856	−	0.977732i	$-0.567300\pi$
$293$	−9.81588e27	−0.419712	−0.209856	+	0.977732i	$0.567300\pi$
$294$	0	0
$295$	4.76011e28	1.88219
$296$	−4.03553e28	−1.53478
$297$	0	0
$298$	2.79902e27	0.0985189
$299$	−4.80972e27	−0.162893
$300$	0	0
$301$	−1.73358e28	−0.543788
$302$	2.10069e28	0.634280
$303$	0	0
$304$	6.36122e26	0.0178030
$305$	3.18217e28	0.857582
$306$	0	0
$307$	−1.51274e28	−0.378158	−0.189079	−	0.981962i	$-0.560550\pi$
$307$	−1.51274e28	−0.378158	−0.189079	+	0.981962i	$0.560550\pi$
$308$	1.04393e28	0.251384
$309$	0	0
$310$	−6.07101e28	−1.35707
$311$	5.90519e28	1.27201	0.636003	−	0.771686i	$-0.280586\pi$
$311$	5.90519e28	1.27201	0.636003	+	0.771686i	$0.280586\pi$
$312$	0	0
$313$	7.74763e27	0.155028	0.0775138	−	0.996991i	$-0.475302\pi$
$313$	7.74763e27	0.155028	0.0775138	+	0.996991i	$0.475302\pi$
$314$	8.23165e27	0.158780
$315$	0	0
$316$	−1.63572e28	−0.293297
$317$	−8.76539e28	−1.51562	−0.757809	−	0.652476i	$-0.773730\pi$
$317$	−8.76539e28	−1.51562	−0.757809	+	0.652476i	$0.773730\pi$
$318$	0	0
$319$	−1.11826e29	−1.79866
$320$	−5.10029e28	−0.791353
$321$	0	0
$322$	5.58286e27	0.0806333
$323$	−2.80129e28	−0.390417
$324$	0	0
$325$	1.32358e28	0.171826
$326$	1.45323e28	0.182109
$327$	0	0
$328$	−3.46439e28	−0.404648
$329$	2.45329e28	0.276692
$330$	0	0
$331$	1.28558e29	1.35231	0.676155	−	0.736759i	$-0.263645\pi$
$331$	1.28558e29	1.35231	0.676155	+	0.736759i	$0.263645\pi$
$332$	8.55777e28	0.869506
$333$	0	0
$334$	1.18196e29	1.12077
$335$	−9.48761e28	−0.869244
$336$	0	0
$337$	−1.77046e29	−1.51475	−0.757377	−	0.652978i	$-0.773520\pi$
$337$	−1.77046e29	−1.51475	−0.757377	+	0.652978i	$0.773520\pi$
$338$	−5.42365e28	−0.448487
$339$	0	0
$340$	1.22196e29	0.944170
$341$	2.49327e29	1.86249
$342$	0	0
$343$	1.10337e29	0.770618
$344$	−1.93779e29	−1.30883
$345$	0	0
$346$	−3.47841e28	−0.219789
$347$	−1.60648e27	−0.00981942	−0.00490971	−	0.999988i	$-0.501563\pi$
$347$	−1.60648e27	−0.00981942	−0.00490971	+	0.999988i	$0.501563\pi$
$348$	0	0
$349$	1.80178e29	1.03088	0.515442	−	0.856925i	$-0.327628\pi$
$349$	1.80178e29	1.03088	0.515442	+	0.856925i	$0.327628\pi$
$350$	−1.53633e28	−0.0850556
$351$	0	0
$352$	1.90118e29	0.985788
$353$	3.12751e29	1.56960	0.784801	−	0.619748i	$-0.212765\pi$
$353$	3.12751e29	1.56960	0.784801	+	0.619748i	$0.212765\pi$
$354$	0	0
$355$	−2.07061e29	−0.973805
$356$	3.94398e27	0.0179581
$357$	0	0
$358$	2.57946e29	1.10122
$359$	−1.81842e29	−0.751811	−0.375906	−	0.926658i	$-0.622668\pi$
$359$	−1.81842e29	−0.751811	−0.375906	+	0.926658i	$0.622668\pi$
$360$	0	0
$361$	−2.37730e29	−0.922044
$362$	1.54509e29	0.580505
$363$	0	0
$364$	3.87461e28	0.136635
$365$	−4.67422e29	−1.59713
$366$	0	0
$367$	−2.10392e29	−0.675103	−0.337552	−	0.941307i	$-0.609599\pi$
$367$	−2.10392e29	−0.675103	−0.337552	+	0.941307i	$0.609599\pi$
$368$	6.09447e27	0.0189534
$369$	0	0
$370$	−3.78813e29	−1.10689
$371$	−1.51614e29	−0.429475
$372$	0	0
$373$	−6.19125e29	−1.64864	−0.824322	−	0.566122i	$-0.808443\pi$
$373$	−6.19125e29	−1.64864	−0.824322	+	0.566122i	$0.808443\pi$
$374$	3.49972e29	0.903670
$375$	0	0
$376$	2.74227e29	0.665963
$377$	−4.15049e29	−0.977628
$378$	0	0
$379$	−8.16748e28	−0.181024	−0.0905122	−	0.995895i	$-0.528850\pi$
$379$	−8.16748e28	−0.181024	−0.0905122	+	0.995895i	$0.528850\pi$
$380$	−8.76760e28	−0.188525
$381$	0	0
$382$	−2.35166e29	−0.476042
$383$	7.92901e29	1.55752	0.778759	−	0.627323i	$-0.215849\pi$
$383$	7.92901e29	1.55752	0.778759	+	0.627323i	$0.215849\pi$
$384$	0	0
$385$	2.64324e29	0.489033
$386$	5.62953e28	0.101092
$387$	0	0
$388$	−4.70370e27	−0.00795930
$389$	2.41701e29	0.397062	0.198531	−	0.980095i	$-0.436383\pi$
$389$	2.41701e29	0.397062	0.198531	+	0.980095i	$0.436383\pi$
$390$	0	0
$391$	−2.68382e29	−0.415643
$392$	5.56013e29	0.836171
$393$	0	0
$394$	4.58258e29	0.649985
$395$	−4.14167e29	−0.570570
$396$	0	0
$397$	4.74027e29	0.616187	0.308094	−	0.951356i	$-0.400309\pi$
$397$	4.74027e29	0.616187	0.308094	+	0.951356i	$0.400309\pi$
$398$	−4.53303e29	−0.572445
$399$	0	0
$400$	−1.67712e28	−0.0199928
$401$	−9.97411e29	−1.15535	−0.577676	−	0.816266i	$-0.696040\pi$
$401$	−9.97411e29	−1.15535	−0.577676	+	0.816266i	$0.696040\pi$
$402$	0	0
$403$	9.25395e29	1.01232
$404$	−7.03788e29	−0.748267
$405$	0	0
$406$	4.81766e29	0.483934
$407$	1.55573e30	1.51914
$408$	0	0
$409$	1.82542e29	0.168478	0.0842391	−	0.996446i	$-0.473154\pi$
$409$	1.82542e29	0.168478	0.0842391	+	0.996446i	$0.473154\pi$
$410$	−3.25200e29	−0.291835
$411$	0	0
$412$	−6.20885e29	−0.526857
$413$	8.42211e29	0.695016
$414$	0	0
$415$	2.16684e30	1.69151
$416$	7.05637e29	0.535807
$417$	0	0
$418$	−2.51105e29	−0.180438
$419$	−2.31769e30	−1.62029	−0.810146	−	0.586228i	$-0.800612\pi$
$419$	−2.31769e30	−1.62029	−0.810146	+	0.586228i	$0.800612\pi$
$420$	0	0
$421$	2.84805e29	0.188497	0.0942483	−	0.995549i	$-0.469955\pi$
$421$	2.84805e29	0.188497	0.0942483	+	0.995549i	$0.469955\pi$
$422$	−1.62791e30	−1.04843
$423$	0	0
$424$	−1.69473e30	−1.03369
$425$	7.38554e29	0.438439
$426$	0	0
$427$	5.63025e29	0.316670
$428$	1.26492e30	0.692566
$429$	0	0
$430$	−1.81899e30	−0.943937
$431$	2.68991e30	1.35910	0.679548	−	0.733632i	$-0.262176\pi$
$431$	2.68991e30	1.35910	0.679548	+	0.733632i	$0.262176\pi$
$432$	0	0
$433$	−2.83636e30	−1.35879	−0.679393	−	0.733774i	$-0.737757\pi$
$433$	−2.83636e30	−1.35879	−0.679393	+	0.733774i	$0.737757\pi$
$434$	−1.07415e30	−0.501109
$435$	0	0
$436$	−1.92433e30	−0.851501
$437$	1.92565e29	0.0829927
$438$	0	0
$439$	−6.34373e29	−0.259419	−0.129710	−	0.991552i	$-0.541405\pi$
$439$	−6.34373e29	−0.259419	−0.129710	+	0.991552i	$0.541405\pi$
$440$	2.95459e30	1.17704
$441$	0	0
$442$	1.29894e30	0.491173
$443$	−1.41158e30	−0.520072	−0.260036	−	0.965599i	$-0.583734\pi$
$443$	−1.41158e30	−0.520072	−0.260036	+	0.965599i	$0.583734\pi$
$444$	0	0
$445$	9.98622e28	0.0349350
$446$	−5.26617e29	−0.179533
$447$	0	0
$448$	−9.02399e29	−0.292214
$449$	−3.28307e30	−1.03621	−0.518104	−	0.855318i	$-0.673362\pi$
$449$	−3.28307e30	−1.03621	−0.518104	+	0.855318i	$0.673362\pi$
$450$	0	0
$451$	1.33555e30	0.400525
$452$	−2.59789e30	−0.759501
$453$	0	0
$454$	−1.42102e30	−0.394874
$455$	9.81056e29	0.265805
$456$	0	0
$457$	3.77120e29	0.0971501	0.0485750	−	0.998820i	$-0.484532\pi$
$457$	3.77120e29	0.0971501	0.0485750	+	0.998820i	$0.484532\pi$
$458$	−1.70549e30	−0.428445
$459$	0	0
$460$	−8.39994e29	−0.200706
$461$	1.82043e30	0.424243	0.212121	−	0.977243i	$-0.431963\pi$
$461$	1.82043e30	0.424243	0.212121	+	0.977243i	$0.431963\pi$
$462$	0	0
$463$	8.46419e30	1.87674	0.938370	−	0.345633i	$-0.112336\pi$
$463$	8.46419e30	1.87674	0.938370	+	0.345633i	$0.112336\pi$
$464$	5.25914e29	0.113752
$465$	0	0
$466$	−1.24236e30	−0.255746
$467$	−1.55072e30	−0.311451	−0.155725	−	0.987800i	$-0.549771\pi$
$467$	−1.55072e30	−0.311451	−0.155725	+	0.987800i	$0.549771\pi$
$468$	0	0
$469$	−1.67865e30	−0.320976
$470$	2.57416e30	0.480297
$471$	0	0
$472$	9.41416e30	1.67282
$473$	7.47033e30	1.29550
$474$	0	0
$475$	−5.29914e29	−0.0875444
$476$	2.16203e30	0.348643
$477$	0	0
$478$	2.45303e30	0.376948
$479$	−9.58203e30	−1.43747	−0.718736	−	0.695284i	$-0.755279\pi$
$479$	−9.58203e30	−1.43747	−0.718736	+	0.695284i	$0.755279\pi$
$480$	0	0
$481$	5.77420e30	0.825703
$482$	1.79627e30	0.250802
$483$	0	0
$484$	−7.31618e28	−0.00974008
$485$	−1.19098e29	−0.0154837
$486$	0	0
$487$	−4.05601e30	−0.502940	−0.251470	−	0.967865i	$-0.580914\pi$
$487$	−4.05601e30	−0.502940	−0.251470	+	0.967865i	$0.580914\pi$
$488$	6.29344e30	0.762185
$489$	0	0
$490$	5.21927e30	0.603052
$491$	−5.18224e30	−0.584898	−0.292449	−	0.956281i	$-0.594470\pi$
$491$	−5.18224e30	−0.584898	−0.292449	+	0.956281i	$0.594470\pi$
$492$	0	0
$493$	−2.31597e31	−2.49455
$494$	−9.31995e29	−0.0980740
$495$	0	0
$496$	−1.17258e30	−0.117789
$497$	−3.66355e30	−0.359586
$498$	0	0
$499$	4.83971e30	0.453590	0.226795	−	0.973943i	$-0.427175\pi$
$499$	4.83971e30	0.453590	0.226795	+	0.973943i	$0.427175\pi$
$500$	−5.06070e30	−0.463506
$501$	0	0
$502$	−5.02152e30	−0.439281
$503$	1.61218e31	1.37842	0.689211	−	0.724561i	$-0.257957\pi$
$503$	1.61218e31	1.37842	0.689211	+	0.724561i	$0.257957\pi$
$504$	0	0
$505$	−1.78200e31	−1.45565
$506$	−2.40576e30	−0.192097
$507$	0	0
$508$	3.59352e30	0.274212
$509$	−1.33185e31	−0.993577	−0.496789	−	0.867872i	$-0.665488\pi$
$509$	−1.33185e31	−0.993577	−0.496789	+	0.867872i	$0.665488\pi$
$510$	0	0
$511$	−8.27014e30	−0.589756
$512$	−1.82562e30	−0.127294
$513$	0	0
$514$	3.24943e30	0.216636
$515$	−1.57209e31	−1.02493
$516$	0	0
$517$	−1.05717e31	−0.659179
$518$	−6.70238e30	−0.408730
$519$	0	0
$520$	1.09662e31	0.639759
$521$	2.34032e31	1.33549	0.667745	−	0.744390i	$-0.267260\pi$
$521$	2.34032e31	1.33549	0.667745	+	0.744390i	$0.267260\pi$
$522$	0	0
$523$	1.49020e31	0.813722	0.406861	−	0.913490i	$-0.366623\pi$
$523$	1.49020e31	0.813722	0.406861	+	0.913490i	$0.366623\pi$
$524$	6.41563e30	0.342712
$525$	0	0
$526$	9.37874e30	0.479522
$527$	5.16370e31	2.58308
$528$	0	0
$529$	−1.90356e31	−0.911645
$530$	−1.59083e31	−0.745507
$531$	0	0
$532$	−1.55126e30	−0.0696146
$533$	4.95699e30	0.217698
$534$	0	0
$535$	3.20281e31	1.34729
$536$	−1.87638e31	−0.772549
$537$	0	0
$538$	−5.19187e30	−0.204799
$539$	−2.14348e31	−0.827653
$540$	0	0
$541$	8.98916e30	0.332622	0.166311	−	0.986073i	$-0.446814\pi$
$541$	8.98916e30	0.332622	0.166311	+	0.986073i	$0.446814\pi$
$542$	5.79978e30	0.210097
$543$	0	0
$544$	3.93745e31	1.36719
$545$	−4.87243e31	−1.65648
$546$	0	0
$547$	2.92608e31	0.953743	0.476871	−	0.878973i	$-0.341771\pi$
$547$	2.92608e31	0.953743	0.476871	+	0.878973i	$0.341771\pi$
$548$	7.75605e30	0.247551
$549$	0	0
$550$	6.62034e30	0.202633
$551$	1.66171e31	0.498095
$552$	0	0
$553$	−7.32790e30	−0.210688
$554$	4.32999e31	1.21934
$555$	0	0
$556$	−2.95422e31	−0.798148
$557$	−3.88195e31	−1.02734	−0.513671	−	0.857987i	$-0.671715\pi$
$557$	−3.88195e31	−1.02734	−0.513671	+	0.857987i	$0.671715\pi$
$558$	0	0
$559$	2.77266e31	0.704144
$560$	−1.24311e30	−0.0309277
$561$	0	0
$562$	−2.87673e31	−0.686959
$563$	−1.23043e31	−0.287878	−0.143939	−	0.989587i	$-0.545977\pi$
$563$	−1.23043e31	−0.287878	−0.143939	+	0.989587i	$0.545977\pi$
$564$	0	0
$565$	−6.57788e31	−1.47751
$566$	9.60160e30	0.211327
$567$	0	0
$568$	−4.09508e31	−0.865480
$569$	3.60714e31	0.747088	0.373544	−	0.927612i	$-0.378142\pi$
$569$	3.60714e31	0.747088	0.373544	+	0.927612i	$0.378142\pi$
$570$	0	0
$571$	4.10890e31	0.817354	0.408677	−	0.912679i	$-0.365990\pi$
$571$	4.10890e31	0.817354	0.408677	+	0.912679i	$0.365990\pi$
$572$	−1.66964e31	−0.325513
$573$	0	0
$574$	−5.75380e30	−0.107763
$575$	−5.07693e30	−0.0932010
$576$	0	0
$577$	−5.71556e31	−1.00818	−0.504089	−	0.863652i	$-0.668172\pi$
$577$	−5.71556e31	−1.00818	−0.504089	+	0.863652i	$0.668172\pi$
$578$	3.54117e31	0.612317
$579$	0	0
$580$	−7.24862e31	−1.20457
$581$	3.83381e31	0.624604
$582$	0	0
$583$	6.53331e31	1.02316
$584$	−9.24429e31	−1.41947
$585$	0	0
$586$	−1.82229e31	−0.269027
$587$	2.06703e30	0.0299233	0.0149616	−	0.999888i	$-0.495237\pi$
$587$	2.06703e30	0.0299233	0.0149616	+	0.999888i	$0.495237\pi$
$588$	0	0
$589$	−3.70496e31	−0.515772
$590$	8.83703e31	1.20645
$591$	0	0
$592$	−7.31658e30	−0.0960745
$593$	3.79598e31	0.488872	0.244436	−	0.969665i	$-0.421397\pi$
$593$	3.79598e31	0.488872	0.244436	+	0.969665i	$0.421397\pi$
$594$	0	0
$595$	5.47428e31	0.678238
$596$	−7.45125e30	−0.0905519
$597$	0	0
$598$	−8.92914e30	−0.104411
$599$	−1.17051e32	−1.34266	−0.671331	−	0.741158i	$-0.734277\pi$
$599$	−1.17051e32	−1.34266	−0.671331	+	0.741158i	$0.734277\pi$
$600$	0	0
$601$	6.55159e31	0.723254	0.361627	−	0.932323i	$-0.382221\pi$
$601$	6.55159e31	0.723254	0.361627	+	0.932323i	$0.382221\pi$
$602$	−3.21835e31	−0.348557
$603$	0	0
$604$	−5.59222e31	−0.582987
$605$	−1.85247e30	−0.0189480
$606$	0	0
$607$	7.95239e31	0.783119	0.391560	−	0.920153i	$-0.371936\pi$
$607$	7.95239e31	0.783119	0.391560	+	0.920153i	$0.371936\pi$
$608$	−2.82513e31	−0.272990
$609$	0	0
$610$	5.90763e31	0.549693
$611$	−3.92375e31	−0.358284
$612$	0	0
$613$	3.67217e30	0.0322944	0.0161472	−	0.999870i	$-0.494860\pi$
$613$	3.67217e30	0.0322944	0.0161472	+	0.999870i	$0.494860\pi$
$614$	−2.80837e31	−0.242391
$615$	0	0
$616$	5.22758e31	0.434633
$617$	−3.41957e31	−0.279057	−0.139528	−	0.990218i	$-0.544559\pi$
$617$	−3.41957e31	−0.279057	−0.139528	+	0.990218i	$0.544559\pi$
$618$	0	0
$619$	2.21499e32	1.74152	0.870762	−	0.491704i	$-0.163626\pi$
$619$	2.21499e32	1.74152	0.870762	+	0.491704i	$0.163626\pi$
$620$	1.61616e32	1.24732
$621$	0	0
$622$	1.09628e32	0.815330
$623$	1.76687e30	0.0129001
$624$	0	0
$625$	−1.72695e32	−1.21524
$626$	1.43833e31	0.0993695
$627$	0	0
$628$	−2.19134e31	−0.145940
$629$	3.22200e32	2.10690
$630$	0	0
$631$	1.58296e32	0.998002	0.499001	−	0.866601i	$-0.333700\pi$
$631$	1.58296e32	0.998002	0.499001	+	0.866601i	$0.333700\pi$
$632$	−8.19106e31	−0.507100
$633$	0	0
$634$	−1.62727e32	−0.971481
$635$	9.09884e31	0.533443
$636$	0	0
$637$	−7.95566e31	−0.449855
$638$	−2.07602e32	−1.15290
$639$	0	0
$640$	1.14492e32	0.613346
$641$	−1.97318e32	−1.03824	−0.519121	−	0.854700i	$-0.673741\pi$
$641$	−1.97318e32	−1.03824	−0.519121	+	0.854700i	$0.673741\pi$
$642$	0	0
$643$	−2.91777e32	−1.48124	−0.740618	−	0.671926i	$-0.765467\pi$
$643$	−2.91777e32	−1.48124	−0.740618	+	0.671926i	$0.765467\pi$
$644$	−1.48621e31	−0.0741127
$645$	0	0
$646$	−5.20052e31	−0.250249
$647$	6.05387e31	0.286176	0.143088	−	0.989710i	$-0.454297\pi$
$647$	6.05387e31	0.286176	0.143088	+	0.989710i	$0.454297\pi$
$648$	0	0
$649$	−3.62924e32	−1.65578
$650$	2.45719e31	0.110137
$651$	0	0
$652$	−3.86862e31	−0.167382
$653$	−2.70051e32	−1.14801	−0.574003	−	0.818853i	$-0.694610\pi$
$653$	−2.70051e32	−1.14801	−0.574003	+	0.818853i	$0.694610\pi$
$654$	0	0
$655$	1.62445e32	0.666700
$656$	−6.28107e30	−0.0253302
$657$	0	0
$658$	4.55448e31	0.177354
$659$	3.41256e32	1.30587	0.652933	−	0.757416i	$-0.273538\pi$
$659$	3.41256e32	1.30587	0.652933	+	0.757416i	$0.273538\pi$
$660$	0	0
$661$	−4.65199e32	−1.71918	−0.859592	−	0.510982i	$-0.829282\pi$
$661$	−4.65199e32	−1.71918	−0.859592	+	0.510982i	$0.829282\pi$
$662$	2.38664e32	0.866804
$663$	0	0
$664$	4.28540e32	1.50334
$665$	−3.92781e31	−0.135426
$666$	0	0
$667$	1.59203e32	0.530279
$668$	−3.14647e32	−1.03014
$669$	0	0
$670$	−1.76135e32	−0.557168
$671$	−2.42618e32	−0.754420
$672$	0	0
$673$	−4.76445e32	−1.43166	−0.715829	−	0.698276i	$-0.753951\pi$
$673$	−4.76445e32	−1.43166	−0.715829	+	0.698276i	$0.753951\pi$
$674$	−3.28682e32	−0.970927
$675$	0	0
$676$	1.44382e32	0.412218
$677$	−4.41103e31	−0.123814	−0.0619071	−	0.998082i	$-0.519718\pi$
$677$	−4.41103e31	−0.123814	−0.0619071	+	0.998082i	$0.519718\pi$
$678$	0	0
$679$	−2.10722e30	−0.00571751
$680$	6.11911e32	1.63243
$681$	0	0
$682$	4.62870e32	1.19382
$683$	2.77520e32	0.703812	0.351906	−	0.936035i	$-0.385534\pi$
$683$	2.77520e32	0.703812	0.351906	+	0.936035i	$0.385534\pi$
$684$	0	0
$685$	1.96384e32	0.481576
$686$	2.04838e32	0.493950
$687$	0	0
$688$	−3.51328e31	−0.0819305
$689$	2.42488e32	0.556121
$690$	0	0
$691$	8.25247e32	1.83057	0.915286	−	0.402806i	$-0.131965\pi$
$691$	8.25247e32	1.83057	0.915286	+	0.402806i	$0.131965\pi$
$692$	9.25986e31	0.202015
$693$	0	0
$694$	−2.98239e30	−0.00629405
$695$	−7.48014e32	−1.55269
$696$	0	0
$697$	2.76599e32	0.555487
$698$	3.34496e32	0.660775
$699$	0	0
$700$	4.08986e31	0.0781773
$701$	−1.99680e32	−0.375472	−0.187736	−	0.982220i	$-0.560115\pi$
$701$	−1.99680e32	−0.375472	−0.187736	+	0.982220i	$0.560115\pi$
$702$	0	0
$703$	−2.31179e32	−0.420690
$704$	3.88860e32	0.696158
$705$	0	0
$706$	5.80614e32	1.00608
$707$	−3.15291e32	−0.537512
$708$	0	0
$709$	−1.31374e32	−0.216809	−0.108404	−	0.994107i	$-0.534574\pi$
$709$	−1.31374e32	−0.216809	−0.108404	+	0.994107i	$0.534574\pi$
$710$	−3.84404e32	−0.624190
$711$	0	0
$712$	1.97499e31	0.0310489
$713$	−3.54960e32	−0.549098
$714$	0	0
$715$	−4.22755e32	−0.633241
$716$	−6.86675e32	−1.01217
$717$	0	0
$718$	−3.37586e32	−0.481896
$719$	8.81786e32	1.23874	0.619371	−	0.785098i	$-0.287388\pi$
$719$	8.81786e32	1.23874	0.619371	+	0.785098i	$0.287388\pi$
$720$	0	0
$721$	−2.78151e32	−0.378464
$722$	−4.41341e32	−0.591012
$723$	0	0
$724$	−4.11317e32	−0.533560
$725$	−4.38106e32	−0.559362
$726$	0	0
$727$	−1.53127e32	−0.189412	−0.0947059	−	0.995505i	$-0.530191\pi$
$727$	−1.53127e32	−0.189412	−0.0947059	+	0.995505i	$0.530191\pi$
$728$	1.94025e32	0.236237
$729$	0	0
$730$	−8.67757e32	−1.02373
$731$	1.54714e33	1.79672
$732$	0	0
$733$	−4.26963e32	−0.480502	−0.240251	−	0.970711i	$-0.577230\pi$
$733$	−4.26963e32	−0.480502	−0.240251	+	0.970711i	$0.577230\pi$
$734$	−3.90588e32	−0.432728
$735$	0	0
$736$	−2.70666e32	−0.290629
$737$	7.23361e32	0.764679
$738$	0	0
$739$	6.20770e32	0.636091	0.318046	−	0.948075i	$-0.396973\pi$
$739$	6.20770e32	0.636091	0.318046	+	0.948075i	$0.396973\pi$
$740$	1.00844e33	1.01738
$741$	0	0
$742$	−2.81467e32	−0.275285
$743$	−8.09984e32	−0.780019	−0.390010	−	0.920811i	$-0.627528\pi$
$743$	−8.09984e32	−0.780019	−0.390010	+	0.920811i	$0.627528\pi$
$744$	0	0
$745$	−1.88667e32	−0.176156
$746$	−1.14939e33	−1.05675
$747$	0	0
$748$	−9.31657e32	−0.830592
$749$	5.66675e32	0.497500
$750$	0	0
$751$	1.33627e33	1.13772	0.568859	−	0.822435i	$-0.307385\pi$
$751$	1.33627e33	1.13772	0.568859	+	0.822435i	$0.307385\pi$
$752$	4.97185e31	0.0416881
$753$	0	0
$754$	−7.70528e32	−0.626640
$755$	−1.41596e33	−1.13412
$756$	0	0
$757$	6.60404e32	0.513106	0.256553	−	0.966530i	$-0.417413\pi$
$757$	6.60404e32	0.513106	0.256553	+	0.966530i	$0.417413\pi$
$758$	−1.51627e32	−0.116033
$759$	0	0
$760$	−4.39047e32	−0.325953
$761$	1.77866e33	1.30068	0.650338	−	0.759645i	$-0.274627\pi$
$761$	1.77866e33	1.30068	0.650338	+	0.759645i	$0.274627\pi$
$762$	0	0
$763$	−8.62084e32	−0.611670
$764$	6.26033e32	0.437546
$765$	0	0
$766$	1.47200e33	0.998338
$767$	−1.34702e33	−0.899967
$768$	0	0
$769$	2.31911e33	1.50372	0.751861	−	0.659322i	$-0.229156\pi$
$769$	2.31911e33	1.50372	0.751861	+	0.659322i	$0.229156\pi$
$770$	4.90711e32	0.313460
$771$	0	0
$772$	−1.49863e32	−0.0929172
$773$	−7.88012e32	−0.481359	−0.240679	−	0.970605i	$-0.577370\pi$
$773$	−7.88012e32	−0.481359	−0.240679	+	0.970605i	$0.577370\pi$
$774$	0	0
$775$	9.76805e32	0.579213
$776$	−2.35543e31	−0.0137613
$777$	0	0
$778$	4.48712e32	0.254508
$779$	−1.98461e32	−0.110916
$780$	0	0
$781$	1.57869e33	0.856663
$782$	−4.98245e32	−0.266419
$783$	0	0
$784$	1.00807e32	0.0523429
$785$	−5.54850e32	−0.283906
$786$	0	0
$787$	2.06876e33	1.02802	0.514010	−	0.857784i	$-0.328160\pi$
$787$	2.06876e33	1.02802	0.514010	+	0.857784i	$0.328160\pi$
$788$	−1.21992e33	−0.597422
$789$	0	0
$790$	−7.68891e32	−0.365724
$791$	−1.16383e33	−0.545582
$792$	0	0
$793$	−9.00492e32	−0.410051
$794$	8.80020e32	0.394964
$795$	0	0
$796$	1.20673e33	0.526152
$797$	3.07381e33	1.32101	0.660506	−	0.750821i	$-0.270342\pi$
$797$	3.07381e33	1.32101	0.660506	+	0.750821i	$0.270342\pi$
$798$	0	0
$799$	−2.18945e33	−0.914213
$800$	7.44839e32	0.306568
$801$	0	0
$802$	−1.85167e33	−0.740557
$803$	3.56375e33	1.40501
$804$	0	0
$805$	−3.76310e32	−0.144176
$806$	1.71797e33	0.648879
$807$	0	0
$808$	−3.52430e33	−1.29373
$809$	−1.24720e33	−0.451367	−0.225683	−	0.974201i	$-0.572461\pi$
$809$	−1.24720e33	−0.451367	−0.225683	+	0.974201i	$0.572461\pi$
$810$	0	0
$811$	−1.90918e33	−0.671593	−0.335797	−	0.941934i	$-0.609006\pi$
$811$	−1.90918e33	−0.671593	−0.335797	+	0.941934i	$0.609006\pi$
$812$	−1.28250e33	−0.444800
$813$	0	0
$814$	2.88818e33	0.973740
$815$	−9.79540e32	−0.325619
$816$	0	0
$817$	−1.11008e33	−0.358756
$818$	3.38884e32	0.107991
$819$	0	0
$820$	8.65713e32	0.268234
$821$	−5.44054e33	−1.66225	−0.831123	−	0.556088i	$-0.812302\pi$
$821$	−5.44054e33	−1.66225	−0.831123	+	0.556088i	$0.812302\pi$
$822$	0	0
$823$	−4.68984e33	−1.39335	−0.696675	−	0.717387i	$-0.745338\pi$
$823$	−4.68984e33	−1.39335	−0.696675	+	0.717387i	$0.745338\pi$
$824$	−3.10915e33	−0.910917
$825$	0	0
$826$	1.56354e33	0.445491
$827$	2.85324e33	0.801723	0.400861	−	0.916139i	$-0.368711\pi$
$827$	2.85324e33	0.801723	0.400861	+	0.916139i	$0.368711\pi$
$828$	0	0
$829$	−6.68450e32	−0.182680	−0.0913402	−	0.995820i	$-0.529115\pi$
$829$	−6.68450e32	−0.182680	−0.0913402	+	0.995820i	$0.529115\pi$
$830$	4.02268e33	1.08422
$831$	0	0
$832$	1.44328e33	0.378384
$833$	−4.43925e33	−1.14787
$834$	0	0
$835$	−7.96691e33	−2.00399
$836$	6.68465e32	0.165847
$837$	0	0
$838$	−4.30273e33	−1.03857
$839$	−9.91584e32	−0.236084	−0.118042	−	0.993009i	$-0.537662\pi$
$839$	−9.91584e32	−0.236084	−0.118042	+	0.993009i	$0.537662\pi$
$840$	0	0
$841$	9.42149e33	2.18256
$842$	5.28734e32	0.120823
$843$	0	0
$844$	4.33365e33	0.963643
$845$	3.65578e33	0.801915
$846$	0	0
$847$	−3.27758e31	−0.00699672
$848$	−3.07260e32	−0.0647074
$849$	0	0
$850$	1.37111e33	0.281031
$851$	−2.21485e33	−0.447872
$852$	0	0
$853$	−5.46591e33	−1.07584	−0.537921	−	0.842995i	$-0.680790\pi$
$853$	−5.46591e33	−1.07584	−0.537921	+	0.842995i	$0.680790\pi$
$854$	1.04524e33	0.202979
$855$	0	0
$856$	6.33425e33	1.19742
$857$	−4.01521e33	−0.748909	−0.374454	−	0.927245i	$-0.622170\pi$
$857$	−4.01521e33	−0.748909	−0.374454	+	0.927245i	$0.622170\pi$
$858$	0	0
$859$	1.42976e33	0.259622	0.129811	−	0.991539i	$-0.458563\pi$
$859$	1.42976e33	0.259622	0.129811	+	0.991539i	$0.458563\pi$
$860$	4.84232e33	0.867603
$861$	0	0
$862$	4.99376e33	0.871152
$863$	1.03231e34	1.77699	0.888493	−	0.458890i	$-0.151753\pi$
$863$	1.03231e34	1.77699	0.888493	+	0.458890i	$0.151753\pi$
$864$	0	0
$865$	2.34461e33	0.392993
$866$	−5.26564e33	−0.870955
$867$	0	0
$868$	2.85948e33	0.460585
$869$	3.15772e33	0.501934
$870$	0	0
$871$	2.68480e33	0.415627
$872$	−9.63630e33	−1.47221
$873$	0	0
$874$	3.57492e32	0.0531967
$875$	−2.26715e33	−0.332956
$876$	0	0
$877$	1.03125e34	1.47526	0.737629	−	0.675207i	$-0.235946\pi$
$877$	1.03125e34	1.47526	0.737629	+	0.675207i	$0.235946\pi$
$878$	−1.17770e33	−0.166283
$879$	0	0
$880$	5.35679e32	0.0736807
$881$	1.04287e34	1.41582	0.707908	−	0.706305i	$-0.249639\pi$
$881$	1.04287e34	1.41582	0.707908	+	0.706305i	$0.249639\pi$
$882$	0	0
$883$	−1.35131e34	−1.78734	−0.893669	−	0.448727i	$-0.851878\pi$
$883$	−1.35131e34	−1.78734	−0.893669	+	0.448727i	$0.851878\pi$
$884$	−3.45791e33	−0.451453
$885$	0	0
$886$	−2.62057e33	−0.333356
$887$	6.46528e33	0.811831	0.405916	−	0.913911i	$-0.366953\pi$
$887$	6.46528e33	0.811831	0.405916	+	0.913911i	$0.366953\pi$
$888$	0	0
$889$	1.60986e33	0.196979
$890$	1.85392e32	0.0223926
$891$	0	0
$892$	1.40190e33	0.165014
$893$	1.57093e33	0.182543
$894$	0	0
$895$	−1.73867e34	−1.96903
$896$	2.02572e33	0.226484
$897$	0	0
$898$	−6.09493e33	−0.664188
$899$	−3.06308e34	−3.29550
$900$	0	0
$901$	1.35308e34	1.41902
$902$	2.47942e33	0.256729
$903$	0	0
$904$	−1.30092e34	−1.31315
$905$	−1.04146e34	−1.03797
$906$	0	0
$907$	1.63933e33	0.159288	0.0796439	−	0.996823i	$-0.474622\pi$
$907$	1.63933e33	0.159288	0.0796439	+	0.996823i	$0.474622\pi$
$908$	3.78289e33	0.362941
$909$	0	0
$910$	1.82131e33	0.170375
$911$	1.72363e34	1.59214	0.796072	−	0.605202i	$-0.206908\pi$
$911$	1.72363e34	1.59214	0.796072	+	0.605202i	$0.206908\pi$
$912$	0	0
$913$	−1.65206e34	−1.48803
$914$	7.00114e32	0.0622712
$915$	0	0
$916$	4.54016e33	0.393798
$917$	2.87415e33	0.246185
$918$	0	0
$919$	−1.99436e33	−0.166600	−0.0833000	−	0.996525i	$-0.526546\pi$
$919$	−1.99436e33	−0.166600	−0.0833000	+	0.996525i	$0.526546\pi$
$920$	−4.20636e33	−0.347014
$921$	0	0
$922$	3.37959e33	0.271931
$923$	5.85941e33	0.465623
$924$	0	0
$925$	6.09499e33	0.472436
$926$	1.57136e34	1.20295
$927$	0	0
$928$	−2.33568e34	−1.74426
$929$	1.66371e34	1.22715	0.613573	−	0.789638i	$-0.289732\pi$
$929$	1.66371e34	1.22715	0.613573	+	0.789638i	$0.289732\pi$
$930$	0	0
$931$	3.18517e33	0.229198
$932$	3.30727e33	0.235064
$933$	0	0
$934$	−2.87887e33	−0.199634
$935$	−2.35897e34	−1.61580
$936$	0	0
$937$	2.29325e34	1.53266	0.766331	−	0.642446i	$-0.222080\pi$
$937$	2.29325e34	1.53266	0.766331	+	0.642446i	$0.222080\pi$
$938$	−3.11637e33	−0.205739
$939$	0	0
$940$	−6.85264e33	−0.441456
$941$	−9.97814e33	−0.634993	−0.317496	−	0.948259i	$-0.602842\pi$
$941$	−9.97814e33	−0.634993	−0.317496	+	0.948259i	$0.602842\pi$
$942$	0	0
$943$	−1.90138e33	−0.118082
$944$	1.70682e33	0.104715
$945$	0	0
$946$	1.38685e34	0.830387
$947$	−9.45460e33	−0.559266	−0.279633	−	0.960107i	$-0.590213\pi$
$947$	−9.45460e33	−0.559266	−0.279633	+	0.960107i	$0.590213\pi$
$948$	0	0
$949$	1.32271e34	0.763666
$950$	−9.83772e32	−0.0561142
$951$	0	0
$952$	1.08266e34	0.602791
$953$	2.62732e33	0.144525	0.0722627	−	0.997386i	$-0.476978\pi$
$953$	2.62732e33	0.144525	0.0722627	+	0.997386i	$0.476978\pi$
$954$	0	0
$955$	1.58512e34	0.851186
$956$	−6.53018e33	−0.346465
$957$	0	0
$958$	−1.77888e34	−0.921390
$959$	3.47464e33	0.177826
$960$	0	0
$961$	4.82813e34	2.41246
$962$	1.07197e34	0.529259
$963$	0	0
$964$	−4.78184e33	−0.230520
$965$	−3.79456e33	−0.180758
$966$	0	0
$967$	−1.74056e33	−0.0809625	−0.0404813	−	0.999180i	$-0.512889\pi$
$967$	−1.74056e33	−0.0809625	−0.0404813	+	0.999180i	$0.512889\pi$
$968$	−3.66366e32	−0.0168402
$969$	0	0
$970$	−2.21103e32	−0.00992476
$971$	7.22515e33	0.320498	0.160249	−	0.987077i	$-0.448770\pi$
$971$	7.22515e33	0.320498	0.160249	+	0.987077i	$0.448770\pi$
$972$	0	0
$973$	−1.32347e34	−0.573344
$974$	−7.52989e33	−0.322374
$975$	0	0
$976$	1.14103e33	0.0477115
$977$	−4.44312e33	−0.183612	−0.0918059	−	0.995777i	$-0.529264\pi$
$977$	−4.44312e33	−0.183612	−0.0918059	+	0.995777i	$0.529264\pi$
$978$	0	0
$979$	−7.61377e32	−0.0307326
$980$	−1.38942e34	−0.554284
$981$	0	0
$982$	−9.62070e33	−0.374908
$983$	2.62175e34	1.00978	0.504890	−	0.863184i	$-0.331533\pi$
$983$	2.62175e34	1.00978	0.504890	+	0.863184i	$0.331533\pi$
$984$	0	0
$985$	−3.08886e34	−1.16220
$986$	−4.29954e34	−1.59896
$987$	0	0
$988$	2.48106e33	0.0901429
$989$	−1.06353e34	−0.381937
$990$	0	0
$991$	2.84277e33	0.0997458	0.0498729	−	0.998756i	$-0.484118\pi$
$991$	2.84277e33	0.0997458	0.0498729	+	0.998756i	$0.484118\pi$
$992$	5.20764e34	1.80616
$993$	0	0
$994$	−6.80129e33	−0.230488
$995$	3.05547e34	1.02356
$996$	0	0
$997$	8.04071e33	0.263209	0.131604	−	0.991302i	$-0.457987\pi$
$997$	8.04071e33	0.263209	0.131604	+	0.991302i	$0.457987\pi$
$998$	8.98481e33	0.290742
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9.24.a.d.1.3	yes	4
3.2	odd	2	inner	9.24.a.d.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.24.a.d.1.2	✓	4	3.2	odd	2	inner
9.24.a.d.1.3	yes	4	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:	\( N \)	\(=\)	\( 9 = 3^{2} \)
Weight:	\( k \)	\(=\)	\( 24 \)
Character orbit:	\([\chi]\)	\(=\)	9.a (trivial)

\(f(q)\)	\(=\)	\(q+1856.48 q^{2} -4.94211e6 q^{4} -1.25135e8 q^{5} -2.21402e9 q^{7} -2.47481e10 q^{8} +O(q^{10})\)	\(q+1856.48 q^{2} -4.94211e6 q^{4} -1.25135e8 q^{5} -2.21402e9 q^{7} -2.47481e10 q^{8} -2.32310e11 q^{10} +9.54062e11 q^{11} +3.54107e12 q^{13} -4.11027e12 q^{14} -4.48694e12 q^{16} +1.97591e14 q^{17} -1.41772e14 q^{19} +6.18429e14 q^{20} +1.77119e15 q^{22} -1.35827e15 q^{23} +3.73779e15 q^{25} +6.57390e15 q^{26} +1.09419e16 q^{28} -1.17210e17 q^{29} +2.61332e17 q^{31} +1.99273e17 q^{32} +3.66823e17 q^{34} +2.77051e17 q^{35} +1.63064e18 q^{37} -2.63196e17 q^{38} +3.09685e18 q^{40} +1.39986e18 q^{41} +7.83002e18 q^{43} -4.71508e18 q^{44} -2.52160e18 q^{46} -1.10807e19 q^{47} -2.24669e19 q^{49} +6.93911e18 q^{50} -1.75003e19 q^{52} +6.84789e19 q^{53} -1.19386e20 q^{55} +5.47929e19 q^{56} -2.17598e20 q^{58} -3.80399e20 q^{59} -2.54300e20 q^{61} +4.85157e20 q^{62} +4.07584e20 q^{64} -4.43111e20 q^{65} +7.58191e20 q^{67} -9.76516e20 q^{68} +5.14338e20 q^{70} +1.65470e21 q^{71} +3.73535e21 q^{73} +3.02724e21 q^{74} +7.00652e20 q^{76} -2.11231e21 q^{77} +3.30977e21 q^{79} +5.61472e20 q^{80} +2.59880e21 q^{82} -1.73160e22 q^{83} -2.47255e22 q^{85} +1.45362e22 q^{86} -2.36113e22 q^{88} -7.98037e20 q^{89} -7.83999e21 q^{91} +6.71272e21 q^{92} -2.05711e22 q^{94} +1.77406e22 q^{95} +9.51761e20 q^{97} -4.17092e22 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 19109968 q^{4} + 8561438480 q^{7}+O(q^{10}) \) 4 * q + 19109968 * q^4 + 8561438480 * q^7	\( 4 q + 19109968 q^{4} + 8561438480 q^{7} + 898978809600 q^{10} - 13534569998680 q^{13} + 27399081725056 q^{16} + 823113907401824 q^{19} - 27\!\cdots\!00 q^{22}+ \cdots + 61\!\cdots\!80 q^{97}+O(q^{100}) \) 4 * q + 19109968 * q^4 + 8561438480 * q^7 + 898978809600 * q^10 - 13534569998680 * q^13 + 27399081725056 * q^16 + 823113907401824 * q^19 - 2742157478016000 * q^22 + 24257341946701900 * q^25 + 210193505445412160 * q^28 + 199165518593535632 * q^31 + 2993143687606771200 * q^34 + 6114312513425419640 * q^37 + 14523223786943846400 * q^40 + 8384436518889991520 * q^43 + 27149798727824025600 * q^46 - 15307929032390927772 * q^49 - 333882838377412430560 * q^52 - 425998563364306944000 * q^55 - 519004760872044384000 * q^58 - 187635897281945053672 * q^61 - 1002932785066778037248 * q^64 + 2217868901107775746880 * q^67 + 9884892109681717632000 * q^70 + 5558219032297686657560 * q^73 + 17444623005047704385408 * q^76 - 1562583260526294231664 * q^79 - 41334076267904219328000 * q^82 + 17862754059765861580800 * q^85 - 85611540851539743744000 * q^88 - 149580096340642991861600 * q^91 - 164391971318917439846400 * q^94 + 61688465492693971809080 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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