LMFDB - Embedded newform 9.24.a.d.1.4

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			1.4
Root			$159.463$ of defining polynomial
Character	$\chi$	$=$	9.1

$q$-expansion

$f(q)$	$=$	$q+4783.90 q^{2} +1.44971e7 q^{4} +1.42520e8 q^{5} +6.49474e9 q^{7} +2.92224e10 q^{8} +O(q^{10})$	\(q+4783.90 q^{2} +1.44971e7 q^{4} +1.42520e8 q^{5} +6.49474e9 q^{7} +2.92224e10 q^{8} +6.81799e11 q^{10} -6.56843e11 q^{11} -1.03084e13 q^{13} +3.10702e13 q^{14} +1.81865e13 q^{16} +2.36156e14 q^{17} +5.53329e14 q^{19} +2.06612e15 q^{20} -3.14227e15 q^{22} +3.36472e15 q^{23} +8.39088e15 q^{25} -4.93141e16 q^{26} +9.41548e16 q^{28} -8.75953e15 q^{29} -1.61750e17 q^{31} -1.58133e17 q^{32} +1.12975e18 q^{34} +9.25627e17 q^{35} +1.42652e18 q^{37} +2.64707e18 q^{38} +4.16476e18 q^{40} -4.86336e18 q^{41} -3.63780e18 q^{43} -9.52231e18 q^{44} +1.60965e19 q^{46} -1.28817e19 q^{47} +1.48129e19 q^{49} +4.01411e19 q^{50} -1.49441e20 q^{52} +1.00229e19 q^{53} -9.36129e19 q^{55} +1.89792e20 q^{56} -4.19047e19 q^{58} +1.29320e19 q^{59} +1.60482e20 q^{61} -7.73794e20 q^{62} -9.09050e20 q^{64} -1.46914e21 q^{65} +3.50743e20 q^{67} +3.42358e21 q^{68} +4.42811e21 q^{70} -2.05647e21 q^{71} -9.56238e20 q^{73} +6.82431e21 q^{74} +8.02166e21 q^{76} -4.26602e21 q^{77} -4.09106e21 q^{79} +2.59193e21 q^{80} -2.32658e22 q^{82} -1.62083e22 q^{83} +3.36569e22 q^{85} -1.74029e22 q^{86} -1.91945e22 q^{88} -2.59976e21 q^{89} -6.69501e22 q^{91} +4.87787e22 q^{92} -6.16249e22 q^{94} +7.88602e22 q^{95} +2.98925e22 q^{97} +7.08634e22 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	4783.90	1.65172	0.825861	−	0.563873i	$-0.190689\pi$
$2$	4783.90	1.65172	0.825861	+	0.563873i	$0.190689\pi$
$3$	0	0
$3$	0	0
$4$	1.44971e7	1.72819
$5$	1.42520e8	1.30533	0.652663	−	0.757648i	$-0.273652\pi$
$5$	1.42520e8	1.30533	0.652663	+	0.757648i	$0.273652\pi$
$6$	0	0
$7$	6.49474e9	1.24146	0.620732	−	0.784023i	$-0.286835\pi$
$7$	6.49474e9	1.24146	0.620732	+	0.784023i	$0.286835\pi$
$8$	2.92224e10	1.20276
$9$	0	0
$10$	6.81799e11	2.15604
$11$	−6.56843e11	−0.694138	−0.347069	−	0.937840i	$-0.612823\pi$
$11$	−6.56843e11	−0.694138	−0.347069	+	0.937840i	$0.612823\pi$
$12$	0	0
$13$	−1.03084e13	−1.59529	−0.797647	−	0.603125i	$-0.793922\pi$
$13$	−1.03084e13	−1.59529	−0.797647	+	0.603125i	$0.793922\pi$
$14$	3.10702e13	2.05056
$15$	0	0
$16$	1.81865e13	0.258445
$17$	2.36156e14	1.67123	0.835617	−	0.549313i	$-0.185110\pi$
$17$	2.36156e14	1.67123	0.835617	+	0.549313i	$0.185110\pi$
$18$	0	0
$19$	5.53329e14	1.08973	0.544863	−	0.838525i	$-0.316582\pi$
$19$	5.53329e14	1.08973	0.544863	+	0.838525i	$0.316582\pi$
$20$	2.06612e15	2.25585
$21$	0	0
$22$	−3.14227e15	−1.14652
$23$	3.36472e15	0.736341	0.368171	−	0.929758i	$-0.379984\pi$
$23$	3.36472e15	0.736341	0.368171	+	0.929758i	$0.379984\pi$
$24$	0	0
$25$	8.39088e15	0.703878
$26$	−4.93141e16	−2.63498
$27$	0	0
$28$	9.41548e16	2.14548
$29$	−8.75953e15	−0.133323	−0.0666613	−	0.997776i	$-0.521235\pi$
$29$	−8.75953e15	−0.133323	−0.0666613	+	0.997776i	$0.521235\pi$
$30$	0	0
$31$	−1.61750e17	−1.14336	−0.571681	−	0.820476i	$-0.693709\pi$
$31$	−1.61750e17	−1.14336	−0.571681	+	0.820476i	$0.693709\pi$
$32$	−1.58133e17	−0.775884
$33$	0	0
$34$	1.12975e18	2.76041
$35$	9.25627e17	1.62052
$36$	0	0
$37$	1.42652e18	1.31813	0.659063	−	0.752088i	$-0.270953\pi$
$37$	1.42652e18	1.31813	0.659063	+	0.752088i	$0.270953\pi$
$38$	2.64707e18	1.79992
$39$	0	0
$40$	4.16476e18	1.57000
$41$	−4.86336e18	−1.38014	−0.690069	−	0.723744i	$-0.742420\pi$
$41$	−4.86336e18	−1.38014	−0.690069	+	0.723744i	$0.742420\pi$
$42$	0	0
$43$	−3.63780e18	−0.596969	−0.298485	−	0.954414i	$-0.596481\pi$
$43$	−3.63780e18	−0.596969	−0.298485	+	0.954414i	$0.596481\pi$
$44$	−9.52231e18	−1.19960
$45$	0	0
$46$	1.60965e19	1.21623
$47$	−1.28817e19	−0.760062	−0.380031	−	0.924974i	$-0.624087\pi$
$47$	−1.28817e19	−0.760062	−0.380031	+	0.924974i	$0.624087\pi$
$48$	0	0
$49$	1.48129e19	0.541234
$50$	4.01411e19	1.16261
$51$	0	0
$52$	−1.49441e20	−2.75697
$53$	1.00229e19	0.148533	0.0742664	−	0.997238i	$-0.476338\pi$
$53$	1.00229e19	0.148533	0.0742664	+	0.997238i	$0.476338\pi$
$54$	0	0
$55$	−9.36129e19	−0.906077
$56$	1.89792e20	1.49319
$57$	0	0
$58$	−4.19047e19	−0.220212
$59$	1.29320e19	0.0558301	0.0279150	−	0.999610i	$-0.491113\pi$
$59$	1.29320e19	0.0558301	0.0279150	+	0.999610i	$0.491113\pi$
$60$	0	0
$61$	1.60482e20	0.472207	0.236104	−	0.971728i	$-0.424130\pi$
$61$	1.60482e20	0.472207	0.236104	+	0.971728i	$0.424130\pi$
$62$	−7.73794e20	−1.88852
$63$	0	0
$64$	−9.09050e20	−1.53999
$65$	−1.46914e21	−2.08238
$66$	0	0
$67$	3.50743e20	0.350856	0.175428	−	0.984492i	$-0.443869\pi$
$67$	3.50743e20	0.350856	0.175428	+	0.984492i	$0.443869\pi$
$68$	3.42358e21	2.88821
$69$	0	0
$70$	4.42811e21	2.67664
$71$	−2.05647e21	−1.05597	−0.527985	−	0.849254i	$-0.677052\pi$
$71$	−2.05647e21	−1.05597	−0.527985	+	0.849254i	$0.677052\pi$
$72$	0	0
$73$	−9.56238e20	−0.356741	−0.178370	−	0.983963i	$-0.557083\pi$
$73$	−9.56238e20	−0.356741	−0.178370	+	0.983963i	$0.557083\pi$
$74$	6.82431e21	2.17718
$75$	0	0
$76$	8.02166e21	1.88325
$77$	−4.26602e21	−0.861747
$78$	0	0
$79$	−4.09106e21	−0.615353	−0.307677	−	0.951491i	$-0.599551\pi$
$79$	−4.09106e21	−0.615353	−0.307677	+	0.951491i	$0.599551\pi$
$80$	2.59193e21	0.337356
$81$	0	0
$82$	−2.32658e22	−2.27960
$83$	−1.62083e22	−1.38146	−0.690730	−	0.723113i	$-0.742711\pi$
$83$	−1.62083e22	−1.38146	−0.690730	+	0.723113i	$0.742711\pi$
$84$	0	0
$85$	3.36569e22	2.18151
$86$	−1.74029e22	−0.986028
$87$	0	0
$88$	−1.91945e22	−0.834884
$89$	−2.59976e21	−0.0992997	−0.0496499	−	0.998767i	$-0.515811\pi$
$89$	−2.59976e21	−0.0992997	−0.0496499	+	0.998767i	$0.515811\pi$
$90$	0	0
$91$	−6.69501e22	−1.98050
$92$	4.87787e22	1.27254
$93$	0	0
$94$	−6.16249e22	−1.25541
$95$	7.88602e22	1.42245
$96$	0	0
$97$	2.98925e22	0.424314	0.212157	−	0.977236i	$-0.431951\pi$
$97$	2.98925e22	0.424314	0.212157	+	0.977236i	$0.431951\pi$
$98$	7.08634e22	0.893968
$99$	0	0
$100$	1.21643e23	1.21643
$101$	−1.72257e23	−1.53632	−0.768159	−	0.640259i	$-0.778827\pi$
$101$	−1.72257e23	−1.53632	−0.768159	+	0.640259i	$0.778827\pi$
$102$	0	0
$103$	1.51134e23	1.07580	0.537902	−	0.843007i	$-0.319217\pi$
$103$	1.51134e23	1.07580	0.537902	+	0.843007i	$0.319217\pi$
$104$	−3.01234e23	−1.91876
$105$	0	0
$106$	4.79487e22	0.245335
$107$	3.38525e23	1.55481	0.777405	−	0.629000i	$-0.216536\pi$
$107$	3.38525e23	1.55481	0.777405	+	0.629000i	$0.216536\pi$
$108$	0	0
$109$	−1.51292e22	−0.0561579	−0.0280789	−	0.999606i	$-0.508939\pi$
$109$	−1.51292e22	−0.0561579	−0.0280789	+	0.999606i	$0.508939\pi$
$110$	−4.47835e23	−1.49659
$111$	0	0
$112$	1.18116e23	0.320851
$113$	2.35991e23	0.578753	0.289377	−	0.957215i	$-0.406552\pi$
$113$	2.35991e23	0.578753	0.289377	+	0.957215i	$0.406552\pi$
$114$	0	0
$115$	4.79539e23	0.961166
$116$	−1.26988e23	−0.230406
$117$	0	0
$118$	6.18655e22	0.0922158
$119$	1.53377e24	2.07478
$120$	0	0
$121$	−4.63987e23	−0.518173
$122$	7.67728e23	0.779955
$123$	0	0
$124$	−2.34490e24	−1.97595
$125$	−5.03100e23	−0.386536
$126$	0	0
$127$	1.15026e24	0.736295	0.368147	−	0.929767i	$-0.379992\pi$
$127$	1.15026e24	0.736295	0.368147	+	0.929767i	$0.379992\pi$
$128$	−3.02229e24	−1.76775
$129$	0	0
$130$	−7.02822e24	−3.43951
$131$	−2.28724e24	−1.02493	−0.512463	−	0.858709i	$-0.671267\pi$
$131$	−2.28724e24	−1.02493	−0.512463	+	0.858709i	$0.671267\pi$
$132$	0	0
$133$	3.59373e24	1.35285
$134$	1.67792e24	0.579517
$135$	0	0
$136$	6.90105e24	2.01010
$137$	−6.77439e24	−1.81378	−0.906888	−	0.421372i	$-0.861549\pi$
$137$	−6.77439e24	−1.81378	−0.906888	+	0.421372i	$0.861549\pi$
$138$	0	0
$139$	2.54141e24	0.575976	0.287988	−	0.957634i	$-0.407014\pi$
$139$	2.54141e24	0.575976	0.287988	+	0.957634i	$0.407014\pi$
$140$	1.34189e25	2.80056
$141$	0	0
$142$	−9.83795e24	−1.74417
$143$	6.77097e24	1.10735
$144$	0	0
$145$	−1.24840e24	−0.174030
$146$	−4.57455e24	−0.589237
$147$	0	0
$148$	2.06803e25	2.27797
$149$	−5.05631e24	−0.515456	−0.257728	−	0.966217i	$-0.582974\pi$
$149$	−5.05631e24	−0.515456	−0.257728	+	0.966217i	$0.582974\pi$
$150$	0	0
$151$	1.95057e25	1.70579	0.852897	−	0.522079i	$-0.174843\pi$
$151$	1.95057e25	1.70579	0.852897	+	0.522079i	$0.174843\pi$
$152$	1.61696e25	1.31068
$153$	0	0
$154$	−2.04082e25	−1.42337
$155$	−2.30525e25	−1.49246
$156$	0	0
$157$	1.88837e25	1.05497	0.527487	−	0.849563i	$-0.323134\pi$
$157$	1.88837e25	1.05497	0.527487	+	0.849563i	$0.323134\pi$
$158$	−1.95712e25	−1.01639
$159$	0	0
$160$	−2.25370e25	−1.01278
$161$	2.18530e25	0.914141
$162$	0	0
$163$	−3.04668e25	−1.10578	−0.552891	−	0.833253i	$-0.686475\pi$
$163$	−3.04668e25	−1.10578	−0.552891	+	0.833253i	$0.686475\pi$
$164$	−7.05046e25	−2.38514
$165$	0	0
$166$	−7.75387e25	−2.28179
$167$	−1.11101e24	−0.0305127	−0.0152563	−	0.999884i	$-0.504856\pi$
$167$	−1.11101e24	−0.0305127	−0.0152563	+	0.999884i	$0.504856\pi$
$168$	0	0
$169$	6.45082e25	1.54496
$170$	1.61011e26	3.60324
$171$	0	0
$172$	−5.27376e25	−1.03168
$173$	2.97282e25	0.544049	0.272025	−	0.962290i	$-0.412307\pi$
$173$	2.97282e25	0.544049	0.272025	+	0.962290i	$0.412307\pi$
$174$	0	0
$175$	5.44966e25	0.873840
$176$	−1.19457e25	−0.179397
$177$	0	0
$178$	−1.24370e25	−0.164016
$179$	3.56213e25	0.440453	0.220227	−	0.975449i	$-0.429320\pi$
$179$	3.56213e25	0.440453	0.220227	+	0.975449i	$0.429320\pi$
$180$	0	0
$181$	−2.89955e25	−0.315520	−0.157760	−	0.987478i	$-0.550427\pi$
$181$	−2.89955e25	−0.315520	−0.157760	+	0.987478i	$0.550427\pi$
$182$	−3.20282e26	−3.27124
$183$	0	0
$184$	9.83252e25	0.885645
$185$	2.03306e26	1.72059
$186$	0	0
$187$	−1.55118e26	−1.16007
$188$	−1.86748e26	−1.31353
$189$	0	0
$190$	3.77259e26	2.34949
$191$	−1.36977e26	−0.803090	−0.401545	−	0.915839i	$-0.631527\pi$
$191$	−1.36977e26	−0.803090	−0.401545	+	0.915839i	$0.631527\pi$
$192$	0	0
$193$	9.06538e25	0.471495	0.235747	−	0.971814i	$-0.424246\pi$
$193$	9.06538e25	0.471495	0.235747	+	0.971814i	$0.424246\pi$
$194$	1.43003e26	0.700848
$195$	0	0
$196$	2.14744e26	0.935354
$197$	8.23482e25	0.338292	0.169146	−	0.985591i	$-0.445899\pi$
$197$	8.23482e25	0.338292	0.169146	+	0.985591i	$0.445899\pi$
$198$	0	0
$199$	4.07308e26	1.48975	0.744874	−	0.667205i	$-0.232510\pi$
$199$	4.07308e26	1.48975	0.744874	+	0.667205i	$0.232510\pi$
$200$	2.45201e26	0.846600
$201$	0	0
$202$	−8.24061e26	−2.53757
$203$	−5.68908e25	−0.165515
$204$	0	0
$205$	−6.93124e26	−1.80153
$206$	7.23008e26	1.77693
$207$	0	0
$208$	−1.87473e26	−0.412296
$209$	−3.63450e26	−0.756419
$210$	0	0
$211$	−2.71522e26	−0.506474	−0.253237	−	0.967404i	$-0.581495\pi$
$211$	−2.71522e26	−0.506474	−0.253237	+	0.967404i	$0.581495\pi$
$212$	1.45303e26	0.256693
$213$	0	0
$214$	1.61947e27	2.56812
$215$	−5.18458e26	−0.779240
$216$	0	0
$217$	−1.05052e27	−1.41944
$218$	−7.23765e25	−0.0927572
$219$	0	0
$220$	−1.35712e27	−1.56587
$221$	−2.43438e27	−2.66611
$222$	0	0
$223$	−3.26296e26	−0.322185	−0.161093	−	0.986939i	$-0.551502\pi$
$223$	−3.26296e26	−0.322185	−0.161093	+	0.986939i	$0.551502\pi$
$224$	−1.02703e27	−0.963233
$225$	0	0
$226$	1.12896e27	0.955940
$227$	2.34616e27	1.88826	0.944128	−	0.329580i	$-0.106907\pi$
$227$	2.34616e27	1.88826	0.944128	+	0.329580i	$0.106907\pi$
$228$	0	0
$229$	1.12251e27	0.816739	0.408370	−	0.912817i	$-0.366097\pi$
$229$	1.12251e27	0.816739	0.408370	+	0.912817i	$0.366097\pi$
$230$	2.29406e27	1.58758
$231$	0	0
$232$	−2.55974e26	−0.160356
$233$	1.38324e27	0.824719	0.412359	−	0.911021i	$-0.364705\pi$
$233$	1.38324e27	0.824719	0.412359	+	0.911021i	$0.364705\pi$
$234$	0	0
$235$	−1.83590e27	−0.992129
$236$	1.87477e26	0.0964849
$237$	0	0
$238$	7.33742e27	3.42696
$239$	3.66637e26	0.163178	0.0815888	−	0.996666i	$-0.474001\pi$
$239$	3.66637e26	0.163178	0.0815888	+	0.996666i	$0.474001\pi$
$240$	0	0
$241$	1.56991e27	0.634862	0.317431	−	0.948281i	$-0.397180\pi$
$241$	1.56991e27	0.634862	0.317431	+	0.948281i	$0.397180\pi$
$242$	−2.21967e27	−0.855878
$243$	0	0
$244$	2.32652e27	0.816062
$245$	2.11113e27	0.706487
$246$	0	0
$247$	−5.70391e27	−1.73843
$248$	−4.72671e27	−1.37520
$249$	0	0
$250$	−2.40678e27	−0.638450
$251$	−3.08385e27	−0.781349	−0.390674	−	0.920529i	$-0.627758\pi$
$251$	−3.08385e27	−0.781349	−0.390674	+	0.920529i	$0.627758\pi$
$252$	0	0
$253$	−2.21009e27	−0.511122
$254$	5.50271e27	1.21615
$255$	0	0
$256$	−6.83268e27	−1.37985
$257$	−1.78781e27	−0.345216	−0.172608	−	0.984991i	$-0.555219\pi$
$257$	−1.78781e27	−0.345216	−0.172608	+	0.984991i	$0.555219\pi$
$258$	0	0
$259$	9.26485e27	1.63641
$260$	−2.12983e28	−3.59874
$261$	0	0
$262$	−1.09419e28	−1.69289
$263$	1.23162e28	1.82384	0.911921	−	0.410367i	$-0.134599\pi$
$263$	1.23162e28	1.82384	0.911921	+	0.410367i	$0.134599\pi$
$264$	0	0
$265$	1.42846e27	0.193884
$266$	1.71920e28	2.23454
$267$	0	0
$268$	5.08476e27	0.606345
$269$	5.19149e27	0.593118	0.296559	−	0.955015i	$-0.404161\pi$
$269$	5.19149e27	0.593118	0.296559	+	0.955015i	$0.404161\pi$
$270$	0	0
$271$	4.52253e27	0.474498	0.237249	−	0.971449i	$-0.423754\pi$
$271$	4.52253e27	0.474498	0.237249	+	0.971449i	$0.423754\pi$
$272$	4.29485e27	0.431923
$273$	0	0
$274$	−3.24080e28	−2.99585
$275$	−5.51149e27	−0.488589
$276$	0	0
$277$	−7.44055e27	−0.606859	−0.303429	−	0.952854i	$-0.598132\pi$
$277$	−7.44055e27	−0.606859	−0.303429	+	0.952854i	$0.598132\pi$
$278$	1.21578e28	0.951353
$279$	0	0
$280$	2.70490e28	1.94910
$281$	2.32245e27	0.160629	0.0803145	−	0.996770i	$-0.474408\pi$
$281$	2.32245e27	0.160629	0.0803145	+	0.996770i	$0.474408\pi$
$282$	0	0
$283$	7.10788e27	0.453102	0.226551	−	0.973999i	$-0.427255\pi$
$283$	7.10788e27	0.453102	0.226551	+	0.973999i	$0.427255\pi$
$284$	−2.98128e28	−1.82491
$285$	0	0
$286$	3.23916e28	1.82904
$287$	−3.15863e28	−1.71339
$288$	0	0
$289$	3.58023e28	1.79302
$290$	−5.97224e27	−0.287449
$291$	0	0
$292$	−1.38627e28	−0.616515
$293$	−1.07564e28	−0.459928	−0.229964	−	0.973199i	$-0.573861\pi$
$293$	−1.07564e28	−0.459928	−0.229964	+	0.973199i	$0.573861\pi$
$294$	0	0
$295$	1.84307e27	0.0728765
$296$	4.16862e28	1.58540
$297$	0	0
$298$	−2.41889e28	−0.851391
$299$	−3.46847e28	−1.17468
$300$	0	0
$301$	−2.36266e28	−0.741116
$302$	9.33133e28	2.81750
$303$	0	0
$304$	1.00631e28	0.281634
$305$	2.28718e28	0.616384
$306$	0	0
$307$	4.66353e28	1.16580	0.582899	−	0.812545i	$-0.301918\pi$
$307$	4.66353e28	1.16580	0.582899	+	0.812545i	$0.301918\pi$
$308$	−6.18449e28	−1.48926
$309$	0	0
$310$	−1.10281e29	−2.46513
$311$	−1.35856e28	−0.292639	−0.146320	−	0.989237i	$-0.546743\pi$
$311$	−1.35856e28	−0.292639	−0.146320	+	0.989237i	$0.546743\pi$
$312$	0	0
$313$	2.25524e28	0.451267	0.225633	−	0.974212i	$-0.427555\pi$
$313$	2.25524e28	0.451267	0.225633	+	0.974212i	$0.427555\pi$
$314$	9.03379e28	1.74252
$315$	0	0
$316$	−5.93085e28	−1.06345
$317$	2.03254e28	0.351445	0.175723	−	0.984440i	$-0.443774\pi$
$317$	2.03254e28	0.351445	0.175723	+	0.984440i	$0.443774\pi$
$318$	0	0
$319$	5.75363e27	0.0925442
$320$	−1.29557e29	−2.01019
$321$	0	0
$322$	1.04543e29	1.50991
$323$	1.30672e29	1.82119
$324$	0	0
$325$	−8.64962e28	−1.12289
$326$	−1.45750e29	−1.82645
$327$	0	0
$328$	−1.42119e29	−1.65998
$329$	−8.36635e28	−0.943590
$330$	0	0
$331$	−7.51986e28	−0.791020	−0.395510	−	0.918462i	$-0.629432\pi$
$331$	−7.51986e28	−0.791020	−0.395510	+	0.918462i	$0.629432\pi$
$332$	−2.34973e29	−2.38742
$333$	0	0
$334$	−5.31498e27	−0.0503984
$335$	4.99878e28	0.457982
$336$	0	0
$337$	1.54497e29	1.32183	0.660916	−	0.750460i	$-0.270168\pi$
$337$	1.54497e29	1.32183	0.660916	+	0.750460i	$0.270168\pi$
$338$	3.08601e29	2.55185
$339$	0	0
$340$	4.87927e29	3.77005
$341$	1.06244e29	0.793651
$342$	0	0
$343$	−8.15470e28	−0.569542
$344$	−1.06305e29	−0.718013
$345$	0	0
$346$	1.42217e29	0.898618
$347$	2.64051e29	1.61398	0.806991	−	0.590564i	$-0.201095\pi$
$347$	2.64051e29	1.61398	0.806991	+	0.590564i	$0.201095\pi$
$348$	0	0
$349$	−1.13409e29	−0.648864	−0.324432	−	0.945909i	$-0.605173\pi$
$349$	−1.13409e29	−0.648864	−0.324432	+	0.945909i	$0.605173\pi$
$350$	2.60706e29	1.44334
$351$	0	0
$352$	1.03868e29	0.538571
$353$	−2.02443e29	−1.01600	−0.508000	−	0.861357i	$-0.669615\pi$
$353$	−2.02443e29	−1.01600	−0.508000	+	0.861357i	$0.669615\pi$
$354$	0	0
$355$	−2.93087e29	−1.37839
$356$	−3.76889e28	−0.171609
$357$	0	0
$358$	1.70409e29	0.727506
$359$	−3.35901e29	−1.38875	−0.694377	−	0.719612i	$-0.744320\pi$
$359$	−3.35901e29	−1.38875	−0.694377	+	0.719612i	$0.744320\pi$
$360$	0	0
$361$	4.83433e28	0.187501
$362$	−1.38711e29	−0.521151
$363$	0	0
$364$	−9.70581e29	−3.42268
$365$	−1.36283e29	−0.465664
$366$	0	0
$367$	4.32640e29	1.38825	0.694125	−	0.719855i	$-0.255792\pi$
$367$	4.32640e29	1.38825	0.694125	+	0.719855i	$0.255792\pi$
$368$	6.11924e28	0.190304
$369$	0	0
$370$	9.72598e29	2.84193
$371$	6.50964e28	0.184398
$372$	0	0
$373$	−6.52581e29	−1.73773	−0.868866	−	0.495048i	$-0.835151\pi$
$373$	−6.52581e29	−1.73773	−0.868866	+	0.495048i	$0.835151\pi$
$374$	−7.42068e29	−1.91611
$375$	0	0
$376$	−3.76435e29	−0.914175
$377$	9.02963e28	0.212689
$378$	0	0
$379$	−3.54426e29	−0.785551	−0.392775	−	0.919634i	$-0.628485\pi$
$379$	−3.54426e29	−0.785551	−0.392775	+	0.919634i	$0.628485\pi$
$380$	1.14324e30	2.45826
$381$	0	0
$382$	−6.55285e29	−1.32648
$383$	−2.02823e29	−0.398411	−0.199206	−	0.979958i	$-0.563836\pi$
$383$	−2.02823e29	−0.398411	−0.199206	+	0.979958i	$0.563836\pi$
$384$	0	0
$385$	−6.07992e29	−1.12486
$386$	4.33679e29	0.778779
$387$	0	0
$388$	4.33354e29	0.733293
$389$	−2.72682e29	−0.447957	−0.223979	−	0.974594i	$-0.571905\pi$
$389$	−2.72682e29	−0.447957	−0.223979	+	0.974594i	$0.571905\pi$
$390$	0	0
$391$	7.94601e29	1.23060
$392$	4.32868e29	0.650977
$393$	0	0
$394$	3.93945e29	0.558765
$395$	−5.83056e29	−0.803237
$396$	0	0
$397$	1.06349e29	0.138243	0.0691213	−	0.997608i	$-0.477980\pi$
$397$	1.06349e29	0.138243	0.0691213	+	0.997608i	$0.477980\pi$
$398$	1.94852e30	2.46065
$399$	0	0
$400$	1.52601e29	0.181914
$401$	9.48539e29	1.09874	0.549370	−	0.835579i	$-0.314868\pi$
$401$	9.48539e29	1.09874	0.549370	+	0.835579i	$0.314868\pi$
$402$	0	0
$403$	1.66737e30	1.82400
$404$	−2.49723e30	−2.65505
$405$	0	0
$406$	−2.72160e29	−0.273385
$407$	−9.36997e29	−0.914961
$408$	0	0
$409$	8.99433e29	0.830139	0.415069	−	0.909790i	$-0.363757\pi$
$409$	8.99433e29	0.830139	0.415069	+	0.909790i	$0.363757\pi$
$410$	−3.31584e30	−2.97563
$411$	0	0
$412$	2.19100e30	1.85919
$413$	8.39901e28	0.0693111
$414$	0	0
$415$	−2.30999e30	−1.80326
$416$	1.63009e30	1.23776
$417$	0	0
$418$	−1.73871e30	−1.24940
$419$	5.82139e29	0.406973	0.203487	−	0.979078i	$-0.434773\pi$
$419$	5.82139e29	0.406973	0.203487	+	0.979078i	$0.434773\pi$
$420$	0	0
$421$	−8.99599e29	−0.595394	−0.297697	−	0.954660i	$-0.596219\pi$
$421$	−8.99599e29	−0.595394	−0.297697	+	0.954660i	$0.596219\pi$
$422$	−1.29894e30	−0.836555
$423$	0	0
$424$	2.92894e29	0.178650
$425$	1.98156e30	1.17635
$426$	0	0
$427$	1.04229e30	0.586228
$428$	4.90763e30	2.68700
$429$	0	0
$430$	−2.48025e30	−1.28709
$431$	1.72002e30	0.869051	0.434525	−	0.900660i	$-0.356916\pi$
$431$	1.72002e30	0.869051	0.434525	+	0.900660i	$0.356916\pi$
$432$	0	0
$433$	4.04694e30	1.93872	0.969362	−	0.245635i	$-0.0789965\pi$
$433$	4.04694e30	1.93872	0.969362	+	0.245635i	$0.0789965\pi$
$434$	−5.02559e30	−2.34453
$435$	0	0
$436$	−2.19329e29	−0.0970513
$437$	1.86180e30	0.802410
$438$	0	0
$439$	1.61226e30	0.659317	0.329659	−	0.944100i	$-0.393066\pi$
$439$	1.61226e30	0.659317	0.329659	+	0.944100i	$0.393066\pi$
$440$	−2.73559e30	−1.08980
$441$	0	0
$442$	−1.16458e31	−4.40367
$443$	4.83888e30	1.78280	0.891398	−	0.453221i	$-0.149725\pi$
$443$	4.83888e30	1.78280	0.891398	+	0.453221i	$0.149725\pi$
$444$	0	0
$445$	−3.70516e29	−0.129619
$446$	−1.56097e30	−0.532161
$447$	0	0
$448$	−5.90404e30	−1.91184
$449$	−4.75344e30	−1.50029	−0.750145	−	0.661273i	$-0.770016\pi$
$449$	−4.75344e30	−1.50029	−0.750145	+	0.661273i	$0.770016\pi$
$450$	0	0
$451$	3.19447e30	0.958005
$452$	3.42118e30	1.00019
$453$	0	0
$454$	1.12238e31	3.11887
$455$	−9.54169e30	−2.58520
$456$	0	0
$457$	−6.88472e30	−1.77358	−0.886788	−	0.462176i	$-0.847069\pi$
$457$	−6.88472e30	−1.77358	−0.886788	+	0.462176i	$0.847069\pi$
$458$	5.36999e30	1.34903
$459$	0	0
$460$	6.95191e30	1.66108
$461$	−7.02180e30	−1.63639	−0.818197	−	0.574938i	$-0.805026\pi$
$461$	−7.02180e30	−1.63639	−0.818197	+	0.574938i	$0.805026\pi$
$462$	0	0
$463$	4.31929e29	0.0957703	0.0478852	−	0.998853i	$-0.484752\pi$
$463$	4.31929e29	0.0957703	0.0478852	+	0.998853i	$0.484752\pi$
$464$	−1.59305e29	−0.0344566
$465$	0	0
$466$	6.61730e30	1.36221
$467$	8.75520e30	1.75842	0.879208	−	0.476437i	$-0.158072\pi$
$467$	8.75520e30	1.75842	0.879208	+	0.476437i	$0.158072\pi$
$468$	0	0
$469$	2.27799e30	0.435576
$470$	−8.78275e30	−1.63872
$471$	0	0
$472$	3.77904e29	0.0671504
$473$	2.38947e30	0.414379
$474$	0	0
$475$	4.64292e30	0.767034
$476$	2.22353e31	3.58560
$477$	0	0
$478$	1.75395e30	0.269524
$479$	−1.07807e31	−1.61729	−0.808647	−	0.588294i	$-0.799800\pi$
$479$	−1.07807e31	−1.61729	−0.808647	+	0.588294i	$0.799800\pi$
$480$	0	0
$481$	−1.47050e31	−2.10280
$482$	7.51030e30	1.04862
$483$	0	0
$484$	−6.72647e30	−0.895500
$485$	4.26026e30	0.553868
$486$	0	0
$487$	9.49272e30	1.17708	0.588542	−	0.808466i	$-0.299702\pi$
$487$	9.49272e30	1.17708	0.588542	+	0.808466i	$0.299702\pi$
$488$	4.68966e30	0.567954
$489$	0	0
$490$	1.00994e31	1.16692
$491$	5.62204e30	0.634537	0.317269	−	0.948336i	$-0.397234\pi$
$491$	5.62204e30	0.634537	0.317269	+	0.948336i	$0.397234\pi$
$492$	0	0
$493$	−2.06862e30	−0.222813
$494$	−2.72869e31	−2.87141
$495$	0	0
$496$	−2.94166e30	−0.295497
$497$	−1.33562e31	−1.31095
$498$	0	0
$499$	−1.40826e29	−0.0131986	−0.00659928	−	0.999978i	$-0.502101\pi$
$499$	−1.40826e29	−0.0131986	−0.00659928	+	0.999978i	$0.502101\pi$
$500$	−7.29349e30	−0.668006
$501$	0	0
$502$	−1.47528e31	−1.29057
$503$	7.85202e30	0.671351	0.335675	−	0.941978i	$-0.391035\pi$
$503$	7.85202e30	0.671351	0.335675	+	0.941978i	$0.391035\pi$
$504$	0	0
$505$	−2.45500e31	−2.00540
$506$	−1.05729e31	−0.844232
$507$	0	0
$508$	1.66754e31	1.27246
$509$	−6.86794e30	−0.512356	−0.256178	−	0.966630i	$-0.582463\pi$
$509$	−6.86794e30	−0.512356	−0.256178	+	0.966630i	$0.582463\pi$
$510$	0	0
$511$	−6.21052e30	−0.442881
$512$	−7.33402e30	−0.511373
$513$	0	0
$514$	−8.55272e30	−0.570202
$515$	2.15395e31	1.40428
$516$	0	0
$517$	8.46127e30	0.527588
$518$	4.43221e31	2.70289
$519$	0	0
$520$	−4.29318e31	−2.50461
$521$	2.37730e31	1.35660	0.678299	−	0.734786i	$-0.262718\pi$
$521$	2.37730e31	1.35660	0.678299	+	0.734786i	$0.262718\pi$
$522$	0	0
$523$	−1.46642e31	−0.800733	−0.400366	−	0.916355i	$-0.631117\pi$
$523$	−1.46642e31	−0.800733	−0.400366	+	0.916355i	$0.631117\pi$
$524$	−3.31584e31	−1.77127
$525$	0	0
$526$	5.89197e31	3.01248
$527$	−3.81982e31	−1.91083
$528$	0	0
$529$	−9.55911e30	−0.457802
$530$	6.83363e30	0.320243
$531$	0	0
$532$	5.20986e31	2.33799
$533$	5.01332e31	2.20172
$534$	0	0
$535$	4.82464e31	2.02954
$536$	1.02496e31	0.421997
$537$	0	0
$538$	2.48356e31	0.979666
$539$	−9.72975e30	−0.375691
$540$	0	0
$541$	−4.35565e31	−1.61170	−0.805851	−	0.592119i	$-0.798292\pi$
$541$	−4.35565e31	−1.61170	−0.805851	+	0.592119i	$0.798292\pi$
$542$	2.16353e31	0.783740
$543$	0	0
$544$	−3.73441e31	−1.29668
$545$	−2.15620e30	−0.0733044
$546$	0	0
$547$	2.29429e31	0.747816	0.373908	−	0.927466i	$-0.378018\pi$
$547$	2.29429e31	0.747816	0.373908	+	0.927466i	$0.378018\pi$
$548$	−9.82089e31	−3.13455
$549$	0	0
$550$	−2.63664e31	−0.807013
$551$	−4.84690e30	−0.145285
$552$	0	0
$553$	−2.65704e31	−0.763939
$554$	−3.55948e31	−1.00236
$555$	0	0
$556$	3.68430e31	0.995395
$557$	3.17560e29	0.00840410	0.00420205	−	0.999991i	$-0.498662\pi$
$557$	3.17560e29	0.00840410	0.00420205	+	0.999991i	$0.498662\pi$
$558$	0	0
$559$	3.74998e31	0.952341
$560$	1.68339e31	0.418815
$561$	0	0
$562$	1.11104e31	0.265315
$563$	−2.42453e31	−0.567260	−0.283630	−	0.958934i	$-0.591539\pi$
$563$	−2.42453e31	−0.567260	−0.283630	+	0.958934i	$0.591539\pi$
$564$	0	0
$565$	3.36333e31	0.755462
$566$	3.40034e31	0.748399
$567$	0	0
$568$	−6.00949e31	−1.27008
$569$	−1.73034e31	−0.358376	−0.179188	−	0.983815i	$-0.557347\pi$
$569$	−1.73034e31	−0.358376	−0.179188	+	0.983815i	$0.557347\pi$
$570$	0	0
$571$	1.56501e31	0.311317	0.155658	−	0.987811i	$-0.450250\pi$
$571$	1.56501e31	0.311317	0.155658	+	0.987811i	$0.450250\pi$
$572$	9.81593e31	1.91372
$573$	0	0
$574$	−1.51106e32	−2.83005
$575$	2.82330e31	0.518295
$576$	0	0
$577$	2.29233e30	0.0404348	0.0202174	−	0.999796i	$-0.493564\pi$
$577$	2.29233e30	0.0404348	0.0202174	+	0.999796i	$0.493564\pi$
$578$	1.71275e32	2.96158
$579$	0	0
$580$	−1.80982e31	−0.300756
$581$	−1.05268e32	−1.71503
$582$	0	0
$583$	−6.58350e30	−0.103102
$584$	−2.79435e31	−0.429075
$585$	0	0
$586$	−5.14576e31	−0.759674
$587$	7.82714e30	0.113309	0.0566545	−	0.998394i	$-0.481957\pi$
$587$	7.82714e30	0.113309	0.0566545	+	0.998394i	$0.481957\pi$
$588$	0	0
$589$	−8.95008e31	−1.24595
$590$	8.81704e30	0.120372
$591$	0	0
$592$	2.59433e31	0.340664
$593$	−5.20855e31	−0.670792	−0.335396	−	0.942077i	$-0.608870\pi$
$593$	−5.20855e31	−0.670792	−0.335396	+	0.942077i	$0.608870\pi$
$594$	0	0
$595$	2.18593e32	2.70826
$596$	−7.33018e31	−0.890805
$597$	0	0
$598$	−1.65928e32	−1.94025
$599$	−2.46078e31	−0.282270	−0.141135	−	0.989990i	$-0.545075\pi$
$599$	−2.46078e31	−0.282270	−0.141135	+	0.989990i	$0.545075\pi$
$600$	0	0
$601$	−6.01876e31	−0.664432	−0.332216	−	0.943203i	$-0.607796\pi$
$601$	−6.01876e31	−0.664432	−0.332216	+	0.943203i	$0.607796\pi$
$602$	−1.13027e32	−1.22412
$603$	0	0
$604$	2.82776e32	2.94793
$605$	−6.61273e31	−0.676385
$606$	0	0
$607$	1.78920e32	1.76193	0.880964	−	0.473183i	$-0.156895\pi$
$607$	1.78920e32	1.76193	0.880964	+	0.473183i	$0.156895\pi$
$608$	−8.74994e31	−0.845501
$609$	0	0
$610$	1.09416e32	1.01810
$611$	1.32789e32	1.21252
$612$	0	0
$613$	−1.26160e32	−1.10950	−0.554748	−	0.832018i	$-0.687185\pi$
$613$	−1.26160e32	−1.10950	−0.554748	+	0.832018i	$0.687185\pi$
$614$	2.23099e32	1.92558
$615$	0	0
$616$	−1.24663e32	−1.03648
$617$	1.03069e32	0.841101	0.420550	−	0.907269i	$-0.361837\pi$
$617$	1.03069e32	0.841101	0.420550	+	0.907269i	$0.361837\pi$
$618$	0	0
$619$	1.21348e32	0.954096	0.477048	−	0.878877i	$-0.341707\pi$
$619$	1.21348e32	0.954096	0.477048	+	0.878877i	$0.341707\pi$
$620$	−3.34194e32	−2.57925
$621$	0	0
$622$	−6.49919e31	−0.483359
$623$	−1.68848e31	−0.123277
$624$	0	0
$625$	−1.71729e32	−1.20843
$626$	1.07889e32	0.745368
$627$	0	0
$628$	2.73759e32	1.82319
$629$	3.36881e32	2.20290
$630$	0	0
$631$	3.50812e31	0.221175	0.110588	−	0.993866i	$-0.464727\pi$
$631$	3.50812e31	0.221175	0.110588	+	0.993866i	$0.464727\pi$
$632$	−1.19550e32	−0.740125
$633$	0	0
$634$	9.72346e31	0.580490
$635$	1.63934e32	0.961105
$636$	0	0
$637$	−1.52697e32	−0.863427
$638$	2.75248e31	0.152857
$639$	0	0
$640$	−4.30736e32	−2.30750
$641$	−1.36428e32	−0.717855	−0.358927	−	0.933366i	$-0.616857\pi$
$641$	−1.36428e32	−0.717855	−0.358927	+	0.933366i	$0.616857\pi$
$642$	0	0
$643$	8.01130e31	0.406702	0.203351	−	0.979106i	$-0.434817\pi$
$643$	8.01130e31	0.406702	0.203351	+	0.979106i	$0.434817\pi$
$644$	3.16805e32	1.57981
$645$	0	0
$646$	6.25123e32	3.00809
$647$	1.62553e32	0.768416	0.384208	−	0.923246i	$-0.374474\pi$
$647$	1.62553e32	0.768416	0.384208	+	0.923246i	$0.374474\pi$
$648$	0	0
$649$	−8.49431e30	−0.0387538
$650$	−4.13789e32	−1.85471
$651$	0	0
$652$	−4.41680e32	−1.91100
$653$	−4.57884e32	−1.94650	−0.973249	−	0.229751i	$-0.926209\pi$
$653$	−4.57884e32	−1.94650	−0.973249	+	0.229751i	$0.926209\pi$
$654$	0	0
$655$	−3.25977e32	−1.33786
$656$	−8.84474e31	−0.356690
$657$	0	0
$658$	−4.00238e32	−1.55855
$659$	−4.46949e32	−1.71031	−0.855157	−	0.518369i	$-0.826539\pi$
$659$	−4.46949e32	−1.71031	−0.855157	+	0.518369i	$0.826539\pi$
$660$	0	0
$661$	−1.68932e32	−0.624301	−0.312151	−	0.950033i	$-0.601049\pi$
$661$	−1.68932e32	−0.624301	−0.312151	+	0.950033i	$0.601049\pi$
$662$	−3.59742e32	−1.30655
$663$	0	0
$664$	−4.73644e32	−1.66157
$665$	5.12176e32	1.76592
$666$	0	0
$667$	−2.94734e31	−0.0981709
$668$	−1.61065e31	−0.0527316
$669$	0	0
$670$	2.39136e32	0.756459
$671$	−1.05411e32	−0.327777
$672$	0	0
$673$	5.31206e32	1.59621	0.798104	−	0.602519i	$-0.205836\pi$
$673$	5.31206e32	1.59621	0.798104	+	0.602519i	$0.205836\pi$
$674$	7.39098e32	2.18330
$675$	0	0
$676$	9.35181e32	2.66998
$677$	1.10916e32	0.311333	0.155666	−	0.987810i	$-0.450248\pi$
$677$	1.10916e32	0.311333	0.155666	+	0.987810i	$0.450248\pi$
$678$	0	0
$679$	1.94144e32	0.526770
$680$	9.83534e32	2.62384
$681$	0	0
$682$	5.08261e32	1.31089
$683$	−4.30126e32	−1.09083	−0.545415	−	0.838166i	$-0.683628\pi$
$683$	−4.30126e32	−1.09083	−0.545415	+	0.838166i	$0.683628\pi$
$684$	0	0
$685$	−9.65483e32	−2.36757
$686$	−3.90113e32	−0.940725
$687$	0	0
$688$	−6.61588e31	−0.154284
$689$	−1.03320e32	−0.236954
$690$	0	0
$691$	−1.48977e32	−0.330461	−0.165231	−	0.986255i	$-0.552837\pi$
$691$	−1.48977e32	−0.330461	−0.165231	+	0.986255i	$0.552837\pi$
$692$	4.30972e32	0.940219
$693$	0	0
$694$	1.26319e33	2.66585
$695$	3.62200e32	0.751837
$696$	0	0
$697$	−1.14851e33	−2.30653
$698$	−5.42535e32	−1.07174
$699$	0	0
$700$	7.90042e32	1.51016
$701$	1.71421e32	0.322334	0.161167	−	0.986927i	$-0.448474\pi$
$701$	1.71421e32	0.322334	0.161167	+	0.986927i	$0.448474\pi$
$702$	0	0
$703$	7.89333e32	1.43640
$704$	5.97103e32	1.06897
$705$	0	0
$706$	−9.68467e32	−1.67815
$707$	−1.11876e33	−1.90728
$708$	0	0
$709$	1.24223e32	0.205008	0.102504	−	0.994733i	$-0.467315\pi$
$709$	1.24223e32	0.205008	0.102504	+	0.994733i	$0.467315\pi$
$710$	−1.40210e33	−2.27671
$711$	0	0
$712$	−7.59711e31	−0.119434
$713$	−5.44243e32	−0.841905
$714$	0	0
$715$	9.64995e32	1.44546
$716$	5.16405e32	0.761186
$717$	0	0
$718$	−1.60692e33	−2.29384
$719$	1.12436e33	1.57951	0.789753	−	0.613425i	$-0.210209\pi$
$719$	1.12436e33	1.57951	0.789753	+	0.613425i	$0.210209\pi$
$720$	0	0
$721$	9.81574e32	1.33557
$722$	2.31269e32	0.309699
$723$	0	0
$724$	−4.20350e32	−0.545277
$725$	−7.35002e31	−0.0938429
$726$	0	0
$727$	5.20007e32	0.643225	0.321613	−	0.946871i	$-0.395775\pi$
$727$	5.20007e32	0.643225	0.321613	+	0.946871i	$0.395775\pi$
$728$	−1.95644e33	−2.38208
$729$	0	0
$730$	−6.51962e32	−0.769147
$731$	−8.59091e32	−0.997675
$732$	0	0
$733$	−1.16237e33	−1.30812	−0.654060	−	0.756443i	$-0.726935\pi$
$733$	−1.16237e33	−1.30812	−0.654060	+	0.756443i	$0.726935\pi$
$734$	2.06971e33	2.29300
$735$	0	0
$736$	−5.32073e32	−0.571316
$737$	−2.30383e32	−0.243543
$738$	0	0
$739$	−1.47548e33	−1.51190	−0.755950	−	0.654629i	$-0.772825\pi$
$739$	−1.47548e33	−1.51190	−0.755950	+	0.654629i	$0.772825\pi$
$740$	2.94735e33	2.97349
$741$	0	0
$742$	3.11415e32	0.304575
$743$	1.35734e33	1.30712	0.653561	−	0.756874i	$-0.273274\pi$
$743$	1.35734e33	1.30712	0.653561	+	0.756874i	$0.273274\pi$
$744$	0	0
$745$	−7.20623e32	−0.672839
$746$	−3.12188e33	−2.87025
$747$	0	0
$748$	−2.24876e33	−2.00481
$749$	2.19863e33	1.93024
$750$	0	0
$751$	1.72188e33	1.46603	0.733016	−	0.680211i	$-0.238112\pi$
$751$	1.72188e33	1.46603	0.733016	+	0.680211i	$0.238112\pi$
$752$	−2.34273e32	−0.196434
$753$	0	0
$754$	4.31968e32	0.351303
$755$	2.77994e33	2.22662
$756$	0	0
$757$	2.32468e33	1.80618	0.903091	−	0.429450i	$-0.141293\pi$
$757$	2.32468e33	1.80618	0.903091	+	0.429450i	$0.141293\pi$
$758$	−1.69554e33	−1.29751
$759$	0	0
$760$	2.30448e33	1.71087
$761$	−1.60288e33	−1.17214	−0.586068	−	0.810262i	$-0.699325\pi$
$761$	−1.60288e33	−1.17214	−0.586068	+	0.810262i	$0.699325\pi$
$762$	0	0
$763$	−9.82601e31	−0.0697180
$764$	−1.98577e33	−1.38789
$765$	0	0
$766$	−9.70285e32	−0.658065
$767$	−1.33308e32	−0.0890654
$768$	0	0
$769$	1.94080e33	1.25842	0.629212	−	0.777234i	$-0.283378\pi$
$769$	1.94080e33	1.25842	0.629212	+	0.777234i	$0.283378\pi$
$770$	−2.90857e33	−1.85796
$771$	0	0
$772$	1.31422e33	0.814831
$773$	−1.83275e33	−1.11954	−0.559768	−	0.828649i	$-0.689110\pi$
$773$	−1.83275e33	−1.11954	−0.559768	+	0.828649i	$0.689110\pi$
$774$	0	0
$775$	−1.35722e33	−0.804788
$776$	8.73529e32	0.510349
$777$	0	0
$778$	−1.30448e33	−0.739901
$779$	−2.69104e33	−1.50397
$780$	0	0
$781$	1.35078e33	0.732989
$782$	3.80129e33	2.03261
$783$	0	0
$784$	2.69394e32	0.139879
$785$	2.69130e33	1.37709
$786$	0	0
$787$	8.71269e32	0.432955	0.216477	−	0.976288i	$-0.430543\pi$
$787$	8.71269e32	0.432955	0.216477	+	0.976288i	$0.430543\pi$
$788$	1.19381e33	0.584633
$789$	0	0
$790$	−2.78928e33	−1.32672
$791$	1.53270e33	0.718502
$792$	0	0
$793$	−1.65430e33	−0.753309
$794$	5.08762e32	0.228338
$795$	0	0
$796$	5.90478e33	2.57456
$797$	−2.50263e33	−1.07554	−0.537769	−	0.843092i	$-0.680733\pi$
$797$	−2.50263e33	−1.07554	−0.537769	+	0.843092i	$0.680733\pi$
$798$	0	0
$799$	−3.04210e33	−1.27024
$800$	−1.32687e33	−0.546128
$801$	0	0
$802$	4.53772e33	1.81481
$803$	6.28098e32	0.247627
$804$	0	0
$805$	3.11448e33	1.19325
$806$	7.97654e33	3.01274
$807$	0	0
$808$	−5.03376e33	−1.84783
$809$	−1.81340e33	−0.656273	−0.328136	−	0.944630i	$-0.606421\pi$
$809$	−1.81340e33	−0.656273	−0.328136	+	0.944630i	$0.606421\pi$
$810$	0	0
$811$	1.10505e33	0.388726	0.194363	−	0.980930i	$-0.437736\pi$
$811$	1.10505e33	0.388726	0.194363	+	0.980930i	$0.437736\pi$
$812$	−8.24752e32	−0.286041
$813$	0	0
$814$	−4.48250e33	−1.51126
$815$	−4.34212e33	−1.44341
$816$	0	0
$817$	−2.01290e33	−0.650532
$818$	4.30280e33	1.37116
$819$	0	0
$820$	−1.00483e34	−3.11338
$821$	−2.80900e33	−0.858234	−0.429117	−	0.903249i	$-0.641175\pi$
$821$	−2.80900e33	−0.858234	−0.429117	+	0.903249i	$0.641175\pi$
$822$	0	0
$823$	−5.09759e33	−1.51449	−0.757245	−	0.653131i	$-0.773455\pi$
$823$	−5.09759e33	−1.51449	−0.757245	+	0.653131i	$0.773455\pi$
$824$	4.41648e33	1.29394
$825$	0	0
$826$	4.01800e32	0.114483
$827$	−7.25325e32	−0.203807	−0.101904	−	0.994794i	$-0.532493\pi$
$827$	−7.25325e32	−0.203807	−0.101904	+	0.994794i	$0.532493\pi$
$828$	0	0
$829$	2.48272e33	0.678500	0.339250	−	0.940696i	$-0.389827\pi$
$829$	2.48272e33	0.678500	0.339250	+	0.940696i	$0.389827\pi$
$830$	−1.10508e34	−2.97848
$831$	0	0
$832$	9.37081e33	2.45674
$833$	3.49816e33	0.904528
$834$	0	0
$835$	−1.58341e32	−0.0398290
$836$	−5.26897e33	−1.30723
$837$	0	0
$838$	2.78490e33	0.672207
$839$	−6.12814e33	−1.45904	−0.729519	−	0.683961i	$-0.760256\pi$
$839$	−6.12814e33	−1.45904	−0.729519	+	0.683961i	$0.760256\pi$
$840$	0	0
$841$	−4.23999e33	−0.982225
$842$	−4.30359e33	−0.983427
$843$	0	0
$844$	−3.93628e33	−0.875283
$845$	9.19368e33	2.01668
$846$	0	0
$847$	−3.01348e33	−0.643293
$848$	1.82282e32	0.0383876
$849$	0	0
$850$	9.47959e33	1.94300
$851$	4.79983e33	0.970591
$852$	0	0
$853$	5.01234e33	0.986567	0.493283	−	0.869869i	$-0.335797\pi$
$853$	5.01234e33	0.986567	0.493283	+	0.869869i	$0.335797\pi$
$854$	4.98620e33	0.968286
$855$	0	0
$856$	9.89250e33	1.87007
$857$	3.25628e33	0.607355	0.303678	−	0.952775i	$-0.401785\pi$
$857$	3.25628e33	0.607355	0.303678	+	0.952775i	$0.401785\pi$
$858$	0	0
$859$	−5.33067e33	−0.967968	−0.483984	−	0.875077i	$-0.660811\pi$
$859$	−5.33067e33	−0.967968	−0.483984	+	0.875077i	$0.660811\pi$
$860$	−7.51613e33	−1.34667
$861$	0	0
$862$	8.22841e33	1.43543
$863$	−1.52320e32	−0.0262199	−0.0131100	−	0.999914i	$-0.504173\pi$
$863$	−1.52320e32	−0.0262199	−0.0131100	+	0.999914i	$0.504173\pi$
$864$	0	0
$865$	4.23685e33	0.710162
$866$	1.93602e34	3.20224
$867$	0	0
$868$	−1.52295e34	−2.45307
$869$	2.68718e33	0.427140
$870$	0	0
$871$	−3.61559e33	−0.559719
$872$	−4.42110e32	−0.0675447
$873$	0	0
$874$	8.90666e33	1.32536
$875$	−3.26751e33	−0.479870
$876$	0	0
$877$	−5.21094e33	−0.745454	−0.372727	−	0.927941i	$-0.621577\pi$
$877$	−5.21094e33	−0.745454	−0.372727	+	0.927941i	$0.621577\pi$
$878$	7.71291e33	1.08901
$879$	0	0
$880$	−1.70249e33	−0.234171
$881$	−3.11935e33	−0.423488	−0.211744	−	0.977325i	$-0.567914\pi$
$881$	−3.11935e33	−0.423488	−0.211744	+	0.977325i	$0.567914\pi$
$882$	0	0
$883$	−5.95233e33	−0.787298	−0.393649	−	0.919261i	$-0.628787\pi$
$883$	−5.95233e33	−0.787298	−0.393649	+	0.919261i	$0.628787\pi$
$884$	−3.52915e34	−4.60754
$885$	0	0
$886$	2.31487e34	2.94469
$887$	1.22417e34	1.53716	0.768582	−	0.639751i	$-0.220963\pi$
$887$	1.22417e34	1.53716	0.768582	+	0.639751i	$0.220963\pi$
$888$	0	0
$889$	7.47061e33	0.914084
$890$	−1.77251e33	−0.214094
$891$	0	0
$892$	−4.73034e33	−0.556796
$893$	−7.12783e33	−0.828258
$894$	0	0
$895$	5.07673e33	0.574935
$896$	−1.96290e34	−2.19460
$897$	0	0
$898$	−2.27400e34	−2.47806
$899$	1.41685e33	0.152436
$900$	0	0
$901$	2.36698e33	0.248233
$902$	1.52820e34	1.58236
$903$	0	0
$904$	6.89622e33	0.696104
$905$	−4.13242e33	−0.411856
$906$	0	0
$907$	−1.82802e34	−1.77622	−0.888109	−	0.459632i	$-0.847981\pi$
$907$	−1.82802e34	−1.77622	−0.888109	+	0.459632i	$0.847981\pi$
$908$	3.40125e34	3.26326
$909$	0	0
$910$	−4.56465e34	−4.27003
$911$	2.70847e33	0.250185	0.125093	−	0.992145i	$-0.460077\pi$
$911$	2.70847e33	0.250185	0.125093	+	0.992145i	$0.460077\pi$
$912$	0	0
$913$	1.06463e34	0.958924
$914$	−3.29358e34	−2.92946
$915$	0	0
$916$	1.62732e34	1.41148
$917$	−1.48551e34	−1.27241
$918$	0	0
$919$	3.04477e32	0.0254346	0.0127173	−	0.999919i	$-0.495952\pi$
$919$	3.04477e32	0.0254346	0.0127173	+	0.999919i	$0.495952\pi$
$920$	1.40133e34	1.15606
$921$	0	0
$922$	−3.35916e34	−2.70287
$923$	2.11988e34	1.68458
$924$	0	0
$925$	1.19697e34	0.927800
$926$	2.06630e33	0.158186
$927$	0	0
$928$	1.38517e33	0.103443
$929$	8.26259e33	0.609446	0.304723	−	0.952441i	$-0.401436\pi$
$929$	8.26259e33	0.609446	0.304723	+	0.952441i	$0.401436\pi$
$930$	0	0
$931$	8.19640e33	0.589796
$932$	2.00530e34	1.42527
$933$	0	0
$934$	4.18840e34	2.90442
$935$	−2.21073e34	−1.51427
$936$	0	0
$937$	2.63092e34	1.75834	0.879169	−	0.476510i	$-0.158098\pi$
$937$	2.63092e34	1.75834	0.879169	+	0.476510i	$0.158098\pi$
$938$	1.08977e34	0.719450
$939$	0	0
$940$	−2.66152e34	−1.71459
$941$	5.07997e33	0.323281	0.161641	−	0.986850i	$-0.448321\pi$
$941$	5.07997e33	0.323281	0.161641	+	0.986850i	$0.448321\pi$
$942$	0	0
$943$	−1.63639e34	−1.01625
$944$	2.35188e32	0.0144290
$945$	0	0
$946$	1.14310e34	0.684439
$947$	−3.02144e33	−0.178727	−0.0893634	−	0.995999i	$-0.528483\pi$
$947$	−3.02144e33	−0.178727	−0.0893634	+	0.995999i	$0.528483\pi$
$948$	0	0
$949$	9.85723e33	0.569107
$950$	2.22113e34	1.26693
$951$	0	0
$952$	4.48205e34	2.49547
$953$	−2.84405e34	−1.56448	−0.782239	−	0.622979i	$-0.785922\pi$
$953$	−2.84405e34	−1.56448	−0.782239	+	0.622979i	$0.785922\pi$
$954$	0	0
$955$	−1.95219e34	−1.04830
$956$	5.31517e33	0.282001
$957$	0	0
$958$	−5.15738e34	−2.67132
$959$	−4.39979e34	−2.25174
$960$	0	0
$961$	6.14965e33	0.307278
$962$	−7.03474e34	−3.47324
$963$	0	0
$964$	2.27592e34	1.09716
$965$	1.29199e34	0.615455
$966$	0	0
$967$	1.35564e33	0.0630581	0.0315291	−	0.999503i	$-0.489962\pi$
$967$	1.35564e33	0.0630581	0.0315291	+	0.999503i	$0.489962\pi$
$968$	−1.35588e34	−0.623240
$969$	0	0
$970$	2.03807e34	0.914836
$971$	1.58853e34	0.704651	0.352326	−	0.935877i	$-0.385391\pi$
$971$	1.58853e34	0.704651	0.352326	+	0.935877i	$0.385391\pi$
$972$	0	0
$973$	1.65058e34	0.715054
$974$	4.54122e34	1.94422
$975$	0	0
$976$	2.91860e33	0.122040
$977$	1.73918e33	0.0718714	0.0359357	−	0.999354i	$-0.488559\pi$
$977$	1.73918e33	0.0718714	0.0359357	+	0.999354i	$0.488559\pi$
$978$	0	0
$979$	1.70763e33	0.0689277
$980$	3.06052e34	1.22094
$981$	0	0
$982$	2.68953e34	1.04808
$983$	7.60342e33	0.292849	0.146424	−	0.989222i	$-0.453223\pi$
$983$	7.60342e33	0.292849	0.146424	+	0.989222i	$0.453223\pi$
$984$	0	0
$985$	1.17362e34	0.441582
$986$	−9.89606e33	−0.368026
$987$	0	0
$988$	−8.26901e34	−3.00434
$989$	−1.22402e34	−0.439573
$990$	0	0
$991$	3.03285e34	1.06415	0.532076	−	0.846697i	$-0.321412\pi$
$991$	3.03285e34	1.06415	0.532076	+	0.846697i	$0.321412\pi$
$992$	2.55779e34	0.887117
$993$	0	0
$994$	−6.38949e34	−2.16532
$995$	5.80494e34	1.94461
$996$	0	0
$997$	−1.74476e34	−0.571137	−0.285568	−	0.958358i	$-0.592182\pi$
$997$	−1.74476e34	−0.571137	−0.285568	+	0.958358i	$0.592182\pi$
$998$	−6.73697e32	−0.0218004
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9.24.a.d.1.4	yes	4
3.2	odd	2	inner	9.24.a.d.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.24.a.d.1.1	✓	4	3.2	odd	2	inner
9.24.a.d.1.4	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)

\(f(q)\)	\(=\)	\(q+4783.90 q^{2} +1.44971e7 q^{4} +1.42520e8 q^{5} +6.49474e9 q^{7} +2.92224e10 q^{8} +O(q^{10})\)	\(q+4783.90 q^{2} +1.44971e7 q^{4} +1.42520e8 q^{5} +6.49474e9 q^{7} +2.92224e10 q^{8} +6.81799e11 q^{10} -6.56843e11 q^{11} -1.03084e13 q^{13} +3.10702e13 q^{14} +1.81865e13 q^{16} +2.36156e14 q^{17} +5.53329e14 q^{19} +2.06612e15 q^{20} -3.14227e15 q^{22} +3.36472e15 q^{23} +8.39088e15 q^{25} -4.93141e16 q^{26} +9.41548e16 q^{28} -8.75953e15 q^{29} -1.61750e17 q^{31} -1.58133e17 q^{32} +1.12975e18 q^{34} +9.25627e17 q^{35} +1.42652e18 q^{37} +2.64707e18 q^{38} +4.16476e18 q^{40} -4.86336e18 q^{41} -3.63780e18 q^{43} -9.52231e18 q^{44} +1.60965e19 q^{46} -1.28817e19 q^{47} +1.48129e19 q^{49} +4.01411e19 q^{50} -1.49441e20 q^{52} +1.00229e19 q^{53} -9.36129e19 q^{55} +1.89792e20 q^{56} -4.19047e19 q^{58} +1.29320e19 q^{59} +1.60482e20 q^{61} -7.73794e20 q^{62} -9.09050e20 q^{64} -1.46914e21 q^{65} +3.50743e20 q^{67} +3.42358e21 q^{68} +4.42811e21 q^{70} -2.05647e21 q^{71} -9.56238e20 q^{73} +6.82431e21 q^{74} +8.02166e21 q^{76} -4.26602e21 q^{77} -4.09106e21 q^{79} +2.59193e21 q^{80} -2.32658e22 q^{82} -1.62083e22 q^{83} +3.36569e22 q^{85} -1.74029e22 q^{86} -1.91945e22 q^{88} -2.59976e21 q^{89} -6.69501e22 q^{91} +4.87787e22 q^{92} -6.16249e22 q^{94} +7.88602e22 q^{95} +2.98925e22 q^{97} +7.08634e22 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 19109968 q^{4} + 8561438480 q^{7}+O(q^{10}) \) 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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