LMFDB - Embedded newform 1035.2.a.o.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1035,2,Mod(1,1035)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1035, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1035.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1035 = 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1035.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$8.26451660920$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.15317.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 5x + 2 $ x^4 - 2x^3 - 4x^2 + 5*x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 115)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.329727$ of defining polynomial
Character	$\chi$	$=$	1035.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.329727 q^{2} -1.89128 q^{4} -1.00000 q^{5} +4.06562 q^{7} -1.28306 q^{8} +O(q^{10})$	\(q+0.329727 q^{2} -1.89128 q^{4} -1.00000 q^{5} +4.06562 q^{7} -1.28306 q^{8} -0.329727 q^{10} -2.65945 q^{11} -5.91023 q^{13} +1.34055 q^{14} +3.35950 q^{16} +3.40617 q^{17} -2.65945 q^{19} +1.89128 q^{20} -0.876894 q^{22} +1.00000 q^{23} +1.00000 q^{25} -1.94876 q^{26} -7.68923 q^{28} -5.84461 q^{29} -9.31640 q^{31} +3.67384 q^{32} +1.12311 q^{34} -4.06562 q^{35} -4.18059 q^{37} -0.876894 q^{38} +1.28306 q^{40} -2.15539 q^{41} +2.34868 q^{43} +5.02977 q^{44} +0.329727 q^{46} -0.242644 q^{47} +9.52927 q^{49} +0.329727 q^{50} +11.1779 q^{52} -9.03585 q^{53} +2.65945 q^{55} -5.21644 q^{56} -1.92713 q^{58} -14.4143 q^{59} +2.46365 q^{61} -3.07187 q^{62} -5.50764 q^{64} +5.91023 q^{65} -7.75485 q^{67} -6.44201 q^{68} -1.34055 q^{70} -12.4395 q^{71} -2.08977 q^{73} -1.37845 q^{74} +5.02977 q^{76} -10.8123 q^{77} -4.15288 q^{79} -3.35950 q^{80} -0.710689 q^{82} +12.5293 q^{83} -3.40617 q^{85} +0.774424 q^{86} +3.41224 q^{88} +9.35682 q^{89} -24.0288 q^{91} -1.89128 q^{92} -0.0800064 q^{94} +2.65945 q^{95} +3.47179 q^{97} +3.14206 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} + 4 q^{4} - 4 q^{5} - 3 q^{7} - 9 q^{8}+O(q^{10}) $ 4 * q - 2 * q^2 + 4 * q^4 - 4 * q^5 - 3 * q^7 - 9 * q^8	$ 4 q - 2 q^{2} + 4 q^{4} - 4 q^{5} - 3 q^{7} - 9 q^{8} + 2 q^{10} - 4 q^{11} + 12 q^{14} + 8 q^{16} + q^{17} - 4 q^{19} - 4 q^{20} - 20 q^{22} + 4 q^{23} + 4 q^{25} + q^{26} - 22 q^{28} - 19 q^{29} - q^{31} - 20 q^{32} - 12 q^{34} + 3 q^{35} - 3 q^{37} - 20 q^{38} + 9 q^{40} - 13 q^{41} - 6 q^{43} + 18 q^{44} - 2 q^{46} - 6 q^{47} + 9 q^{49} - 2 q^{50} - q^{52} - 19 q^{53} + 4 q^{55} + 10 q^{56} + 21 q^{58} - 23 q^{59} + 13 q^{62} + 27 q^{64} - 3 q^{67} + 4 q^{68} - 12 q^{70} + 3 q^{71} - 32 q^{73} + 12 q^{74} + 18 q^{76} - 18 q^{77} + 2 q^{79} - 8 q^{80} - 5 q^{82} + 21 q^{83} - q^{85} + 2 q^{86} - 14 q^{88} - 40 q^{91} + 4 q^{92} + 47 q^{94} + 4 q^{95} - 18 q^{97} - 16 q^{98}+O(q^{100}) $ 4 * q - 2 * q^2 + 4 * q^4 - 4 * q^5 - 3 * q^7 - 9 * q^8 + 2 * q^10 - 4 * q^11 + 12 * q^14 + 8 * q^16 + q^17 - 4 * q^19 - 4 * q^20 - 20 * q^22 + 4 * q^23 + 4 * q^25 + q^26 - 22 * q^28 - 19 * q^29 - q^31 - 20 * q^32 - 12 * q^34 + 3 * q^35 - 3 * q^37 - 20 * q^38 + 9 * q^40 - 13 * q^41 - 6 * q^43 + 18 * q^44 - 2 * q^46 - 6 * q^47 + 9 * q^49 - 2 * q^50 - q^52 - 19 * q^53 + 4 * q^55 + 10 * q^56 + 21 * q^58 - 23 * q^59 + 13 * q^62 + 27 * q^64 - 3 * q^67 + 4 * q^68 - 12 * q^70 + 3 * q^71 - 32 * q^73 + 12 * q^74 + 18 * q^76 - 18 * q^77 + 2 * q^79 - 8 * q^80 - 5 * q^82 + 21 * q^83 - q^85 + 2 * q^86 - 14 * q^88 - 40 * q^91 + 4 * q^92 + 47 * q^94 + 4 * q^95 - 18 * q^97 - 16 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0.329727	0.233152	0.116576	−	0.993182i	$-0.462808\pi$
$2$	0.329727	0.233152	0.116576	+	0.993182i	$0.462808\pi$
$3$	0	0
$3$	0	0
$4$	−1.89128	−0.945640
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	4.06562	1.53666	0.768330	−	0.640054i	$-0.221088\pi$
$7$	4.06562	1.53666	0.768330	+	0.640054i	$0.221088\pi$
$8$	−1.28306	−0.453630
$9$	0	0
$10$	−0.329727	−0.104269
$11$	−2.65945	−0.801856	−0.400928	−	0.916110i	$-0.631312\pi$
$11$	−2.65945	−0.801856	−0.400928	+	0.916110i	$0.631312\pi$
$12$	0	0
$13$	−5.91023	−1.63920	−0.819602	−	0.572933i	$-0.805805\pi$
$13$	−5.91023	−1.63920	−0.819602	+	0.572933i	$0.805805\pi$
$14$	1.34055	0.358276
$15$	0	0
$16$	3.35950	0.839875
$17$	3.40617	0.826117	0.413058	−	0.910705i	$-0.364461\pi$
$17$	3.40617	0.826117	0.413058	+	0.910705i	$0.364461\pi$
$18$	0	0
$19$	−2.65945	−0.610121	−0.305060	−	0.952333i	$-0.598677\pi$
$19$	−2.65945	−0.610121	−0.305060	+	0.952333i	$0.598677\pi$
$20$	1.89128	0.422903
$21$	0	0
$22$	−0.876894	−0.186955
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	−1.94876	−0.382184
$27$	0	0
$28$	−7.68923	−1.45313
$29$	−5.84461	−1.08532	−0.542659	−	0.839953i	$-0.682582\pi$
$29$	−5.84461	−1.08532	−0.542659	+	0.839953i	$0.682582\pi$
$30$	0	0
$31$	−9.31640	−1.67327	−0.836637	−	0.547757i	$-0.815482\pi$
$31$	−9.31640	−1.67327	−0.836637	+	0.547757i	$0.815482\pi$
$32$	3.67384	0.649449
$33$	0	0
$34$	1.12311	0.192611
$35$	−4.06562	−0.687215
$36$	0	0
$37$	−4.18059	−0.687285	−0.343642	−	0.939101i	$-0.611661\pi$
$37$	−4.18059	−0.687285	−0.343642	+	0.939101i	$0.611661\pi$
$38$	−0.876894	−0.142251
$39$	0	0
$40$	1.28306	0.202870
$41$	−2.15539	−0.336615	−0.168307	−	0.985735i	$-0.553830\pi$
$41$	−2.15539	−0.336615	−0.168307	+	0.985735i	$0.553830\pi$
$42$	0	0
$43$	2.34868	0.358171	0.179085	−	0.983834i	$-0.442686\pi$
$43$	2.34868	0.358171	0.179085	+	0.983834i	$0.442686\pi$
$44$	5.02977	0.758267
$45$	0	0
$46$	0.329727	0.0486156
$47$	−0.242644	−0.0353933	−0.0176966	−	0.999843i	$-0.505633\pi$
$47$	−0.242644	−0.0353933	−0.0176966	+	0.999843i	$0.505633\pi$
$48$	0	0
$49$	9.52927	1.36132
$50$	0.329727	0.0466305
$51$	0	0
$52$	11.1779	1.55010
$53$	−9.03585	−1.24117	−0.620585	−	0.784140i	$-0.713105\pi$
$53$	−9.03585	−1.24117	−0.620585	+	0.784140i	$0.713105\pi$
$54$	0	0
$55$	2.65945	0.358601
$56$	−5.21644	−0.697076
$57$	0	0
$58$	−1.92713	−0.253044
$59$	−14.4143	−1.87658	−0.938291	−	0.345846i	$-0.887592\pi$
$59$	−14.4143	−1.87658	−0.938291	+	0.345846i	$0.887592\pi$
$60$	0	0
$61$	2.46365	0.315438	0.157719	−	0.987484i	$-0.449586\pi$
$61$	2.46365	0.315438	0.157719	+	0.987484i	$0.449586\pi$
$62$	−3.07187	−0.390128
$63$	0	0
$64$	−5.50764	−0.688454
$65$	5.91023	0.733074
$66$	0	0
$67$	−7.75485	−0.947405	−0.473703	−	0.880685i	$-0.657083\pi$
$67$	−7.75485	−0.947405	−0.473703	+	0.880685i	$0.657083\pi$
$68$	−6.44201	−0.781209
$69$	0	0
$70$	−1.34055	−0.160226
$71$	−12.4395	−1.47630	−0.738149	−	0.674638i	$-0.764300\pi$
$71$	−12.4395	−1.47630	−0.738149	+	0.674638i	$0.764300\pi$
$72$	0	0
$73$	−2.08977	−0.244589	−0.122294	−	0.992494i	$-0.539025\pi$
$73$	−2.08977	−0.244589	−0.122294	+	0.992494i	$0.539025\pi$
$74$	−1.37845	−0.160242
$75$	0	0
$76$	5.02977	0.576955
$77$	−10.8123	−1.23218
$78$	0	0
$79$	−4.15288	−0.467235	−0.233618	−	0.972329i	$-0.575056\pi$
$79$	−4.15288	−0.467235	−0.233618	+	0.972329i	$0.575056\pi$
$80$	−3.35950	−0.375604
$81$	0	0
$82$	−0.710689	−0.0784825
$83$	12.5293	1.37527	0.687633	−	0.726058i	$-0.258650\pi$
$83$	12.5293	1.37527	0.687633	+	0.726058i	$0.258650\pi$
$84$	0	0
$85$	−3.40617	−0.369451
$86$	0.774424	0.0835083
$87$	0	0
$88$	3.41224	0.363746
$89$	9.35682	0.991821	0.495910	−	0.868374i	$-0.334834\pi$
$89$	9.35682	0.991821	0.495910	+	0.868374i	$0.334834\pi$
$90$	0	0
$91$	−24.0288	−2.51890
$92$	−1.89128	−0.197180
$93$	0	0
$94$	−0.0800064	−0.00825203
$95$	2.65945	0.272854
$96$	0	0
$97$	3.47179	0.352507	0.176253	−	0.984345i	$-0.443602\pi$
$97$	3.47179	0.352507	0.176253	+	0.984345i	$0.443602\pi$
$98$	3.14206	0.317396
$99$	0	0
$100$	−1.89128	−0.189128
$101$	−9.21036	−0.916465	−0.458233	−	0.888832i	$-0.651517\pi$
$101$	−9.21036	−0.916465	−0.458233	+	0.888832i	$0.651517\pi$
$102$	0	0
$103$	4.69736	0.462845	0.231422	−	0.972853i	$-0.425662\pi$
$103$	4.69736	0.462845	0.231422	+	0.972853i	$0.425662\pi$
$104$	7.58319	0.743593
$105$	0	0
$106$	−2.97936	−0.289381
$107$	0.376394	0.0363873	0.0181937	−	0.999834i	$-0.494208\pi$
$107$	0.376394	0.0363873	0.0181937	+	0.999834i	$0.494208\pi$
$108$	0	0
$109$	19.0369	1.82340	0.911702	−	0.410851i	$-0.134768\pi$
$109$	19.0369	1.82340	0.911702	+	0.410851i	$0.134768\pi$
$110$	0.876894	0.0836086
$111$	0	0
$112$	13.6585	1.29060
$113$	−2.94252	−0.276809	−0.138404	−	0.990376i	$-0.544197\pi$
$113$	−2.94252	−0.276809	−0.138404	+	0.990376i	$0.544197\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	11.0538	1.02632
$117$	0	0
$118$	−4.75279	−0.437530
$119$	13.8482	1.26946
$120$	0	0
$121$	−3.92730	−0.357028
$122$	0.812333	0.0735452
$123$	0	0
$124$	17.6199	1.58232
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	6.00357	0.532730	0.266365	−	0.963872i	$-0.414177\pi$
$127$	6.00357	0.532730	0.266365	+	0.963872i	$0.414177\pi$
$128$	−9.16370	−0.809964
$129$	0	0
$130$	1.94876	0.170918
$131$	19.9606	1.74397	0.871985	−	0.489533i	$-0.162833\pi$
$131$	19.9606	1.74397	0.871985	+	0.489533i	$0.162833\pi$
$132$	0	0
$133$	−10.8123	−0.937548
$134$	−2.55698	−0.220890
$135$	0	0
$136$	−4.37032	−0.374752
$137$	11.7609	1.00480	0.502402	−	0.864634i	$-0.332450\pi$
$137$	11.7609	1.00480	0.502402	+	0.864634i	$0.332450\pi$
$138$	0	0
$139$	12.3891	1.05083	0.525415	−	0.850846i	$-0.323910\pi$
$139$	12.3891	1.05083	0.525415	+	0.850846i	$0.323910\pi$
$140$	7.68923	0.649858
$141$	0	0
$142$	−4.10164	−0.344202
$143$	15.7180	1.31441
$144$	0	0
$145$	5.84461	0.485369
$146$	−0.689053	−0.0570264
$147$	0	0
$148$	7.90667	0.649924
$149$	2.13124	0.174598	0.0872991	−	0.996182i	$-0.472176\pi$
$149$	2.13124	0.174598	0.0872991	+	0.996182i	$0.472176\pi$
$150$	0	0
$151$	−0.787129	−0.0640556	−0.0320278	−	0.999487i	$-0.510197\pi$
$151$	−0.787129	−0.0640556	−0.0320278	+	0.999487i	$0.510197\pi$
$152$	3.41224	0.276769
$153$	0	0
$154$	−3.56512	−0.287286
$155$	9.31640	0.748311
$156$	0	0
$157$	−9.18873	−0.733340	−0.366670	−	0.930351i	$-0.619502\pi$
$157$	−9.18873	−0.733340	−0.366670	+	0.930351i	$0.619502\pi$
$158$	−1.36932	−0.108937
$159$	0	0
$160$	−3.67384	−0.290443
$161$	4.06562	0.320416
$162$	0	0
$163$	−17.6928	−1.38581	−0.692903	−	0.721031i	$-0.743669\pi$
$163$	−17.6928	−1.38581	−0.692903	+	0.721031i	$0.743669\pi$
$164$	4.07644	0.318316
$165$	0	0
$166$	4.13124	0.320647
$167$	−14.2462	−1.10240	−0.551202	−	0.834372i	$-0.685831\pi$
$167$	−14.2462	−1.10240	−0.551202	+	0.834372i	$0.685831\pi$
$168$	0	0
$169$	21.9309	1.68699
$170$	−1.12311	−0.0861383
$171$	0	0
$172$	−4.44201	−0.338700
$173$	1.20394	0.0915338	0.0457669	−	0.998952i	$-0.485427\pi$
$173$	1.20394	0.0915338	0.0457669	+	0.998952i	$0.485427\pi$
$174$	0	0
$175$	4.06562	0.307332
$176$	−8.93444	−0.673459
$177$	0	0
$178$	3.08520	0.231245
$179$	−7.42495	−0.554967	−0.277483	−	0.960730i	$-0.589500\pi$
$179$	−7.42495	−0.554967	−0.277483	+	0.960730i	$0.589500\pi$
$180$	0	0
$181$	24.0450	1.78725	0.893627	−	0.448810i	$-0.148152\pi$
$181$	24.0450	1.78725	0.893627	+	0.448810i	$0.148152\pi$
$182$	−7.92294	−0.587287
$183$	0	0
$184$	−1.28306	−0.0945885
$185$	4.18059	0.307363
$186$	0	0
$187$	−9.05854	−0.662426
$188$	0.458908	0.0334693
$189$	0	0
$190$	0.876894	0.0636166
$191$	4.87689	0.352880	0.176440	−	0.984311i	$-0.443542\pi$
$191$	4.87689	0.352880	0.176440	+	0.984311i	$0.443542\pi$
$192$	0	0
$193$	10.2714	0.739353	0.369676	−	0.929161i	$-0.379469\pi$
$193$	10.2714	0.739353	0.369676	+	0.929161i	$0.379469\pi$
$194$	1.14474	0.0821877
$195$	0	0
$196$	−18.0225	−1.28732
$197$	−4.28557	−0.305334	−0.152667	−	0.988278i	$-0.548786\pi$
$197$	−4.28557	−0.305334	−0.152667	+	0.988278i	$0.548786\pi$
$198$	0	0
$199$	4.02164	0.285086	0.142543	−	0.989789i	$-0.454472\pi$
$199$	4.02164	0.285086	0.142543	+	0.989789i	$0.454472\pi$
$200$	−1.28306	−0.0907261
$201$	0	0
$202$	−3.03691	−0.213676
$203$	−23.7620	−1.66776
$204$	0	0
$205$	2.15539	0.150539
$206$	1.54885	0.107913
$207$	0	0
$208$	−19.8554	−1.37673
$209$	7.07270	0.489229
$210$	0	0
$211$	15.3416	1.05616	0.528080	−	0.849195i	$-0.322912\pi$
$211$	15.3416	1.05616	0.528080	+	0.849195i	$0.322912\pi$
$212$	17.0893	1.17370
$213$	0	0
$214$	0.124107	0.00848379
$215$	−2.34868	−0.160179
$216$	0	0
$217$	−37.8770	−2.57126
$218$	6.27699	0.425131
$219$	0	0
$220$	−5.02977	−0.339107
$221$	−20.1312	−1.35417
$222$	0	0
$223$	−17.6801	−1.18395	−0.591973	−	0.805958i	$-0.701651\pi$
$223$	−17.6801	−1.18395	−0.591973	+	0.805958i	$0.701651\pi$
$224$	14.9364	0.997983
$225$	0	0
$226$	−0.970227	−0.0645386
$227$	−8.63817	−0.573335	−0.286668	−	0.958030i	$-0.592548\pi$
$227$	−8.63817	−0.573335	−0.286668	+	0.958030i	$0.592548\pi$
$228$	0	0
$229$	−3.34055	−0.220749	−0.110375	−	0.993890i	$-0.535205\pi$
$229$	−3.34055	−0.220749	−0.110375	+	0.993890i	$0.535205\pi$
$230$	−0.329727	−0.0217416
$231$	0	0
$232$	7.49899	0.492333
$233$	−8.23728	−0.539642	−0.269821	−	0.962910i	$-0.586965\pi$
$233$	−8.23728	−0.539642	−0.269821	+	0.962910i	$0.586965\pi$
$234$	0	0
$235$	0.242644	0.0158284
$236$	27.2615	1.77457
$237$	0	0
$238$	4.56612	0.295978
$239$	2.86525	0.185338	0.0926688	−	0.995697i	$-0.470460\pi$
$239$	2.86525	0.185338	0.0926688	+	0.995697i	$0.470460\pi$
$240$	0	0
$241$	4.68109	0.301536	0.150768	−	0.988569i	$-0.451825\pi$
$241$	4.68109	0.301536	0.150768	+	0.988569i	$0.451825\pi$
$242$	−1.29494	−0.0832418
$243$	0	0
$244$	−4.65945	−0.298291
$245$	−9.52927	−0.608803
$246$	0	0
$247$	15.7180	1.00011
$248$	11.9535	0.759049
$249$	0	0
$250$	−0.329727	−0.0208538
$251$	18.5824	1.17291	0.586455	−	0.809982i	$-0.300523\pi$
$251$	18.5824	1.17291	0.586455	+	0.809982i	$0.300523\pi$
$252$	0	0
$253$	−2.65945	−0.167198
$254$	1.97954	0.124207
$255$	0	0
$256$	7.99375	0.499609
$257$	28.7226	1.79166	0.895832	−	0.444392i	$-0.146580\pi$
$257$	28.7226	1.79166	0.895832	+	0.444392i	$0.146580\pi$
$258$	0	0
$259$	−16.9967	−1.05612
$260$	−11.1779	−0.693224
$261$	0	0
$262$	6.58157	0.406611
$263$	9.20500	0.567604	0.283802	−	0.958883i	$-0.408404\pi$
$263$	9.20500	0.567604	0.283802	+	0.958883i	$0.408404\pi$
$264$	0	0
$265$	9.03585	0.555068
$266$	−3.56512	−0.218592
$267$	0	0
$268$	14.6666	0.895904
$269$	−16.9194	−1.03160	−0.515798	−	0.856710i	$-0.672504\pi$
$269$	−16.9194	−1.03160	−0.515798	+	0.856710i	$0.672504\pi$
$270$	0	0
$271$	1.15082	0.0699072	0.0349536	−	0.999389i	$-0.488872\pi$
$271$	1.15082	0.0699072	0.0349536	+	0.999389i	$0.488872\pi$
$272$	11.4430	0.693835
$273$	0	0
$274$	3.87790	0.234272
$275$	−2.65945	−0.160371
$276$	0	0
$277$	2.57505	0.154720	0.0773600	−	0.997003i	$-0.475351\pi$
$277$	2.57505	0.154720	0.0773600	+	0.997003i	$0.475351\pi$
$278$	4.08502	0.245003
$279$	0	0
$280$	5.21644	0.311742
$281$	−27.4718	−1.63883	−0.819415	−	0.573201i	$-0.805701\pi$
$281$	−27.4718	−1.63883	−0.819415	+	0.573201i	$0.805701\pi$
$282$	0	0
$283$	−2.36389	−0.140519	−0.0702595	−	0.997529i	$-0.522383\pi$
$283$	−2.36389	−0.140519	−0.0702595	+	0.997529i	$0.522383\pi$
$284$	23.5266	1.39605
$285$	0	0
$286$	5.18265	0.306457
$287$	−8.76298	−0.517263
$288$	0	0
$289$	−5.39803	−0.317531
$290$	1.92713	0.113165
$291$	0	0
$292$	3.95233	0.231293
$293$	−34.1198	−1.99330	−0.996650	−	0.0817846i	$-0.973938\pi$
$293$	−34.1198	−1.99330	−0.996650	+	0.0817846i	$0.973938\pi$
$294$	0	0
$295$	14.4143	0.839233
$296$	5.36395	0.311773
$297$	0	0
$298$	0.702728	0.0407080
$299$	−5.91023	−0.341798
$300$	0	0
$301$	9.54885	0.550386
$302$	−0.259538	−0.0149347
$303$	0	0
$304$	−8.93444	−0.512425
$305$	−2.46365	−0.141068
$306$	0	0
$307$	1.49342	0.0852342	0.0426171	−	0.999091i	$-0.486430\pi$
$307$	1.49342	0.0852342	0.0426171	+	0.999091i	$0.486430\pi$
$308$	20.4491	1.16520
$309$	0	0
$310$	3.07187	0.174471
$311$	14.1060	0.799880	0.399940	−	0.916541i	$-0.369031\pi$
$311$	14.1060	0.799880	0.399940	+	0.916541i	$0.369031\pi$
$312$	0	0
$313$	−32.8563	−1.85715	−0.928574	−	0.371146i	$-0.878965\pi$
$313$	−32.8563	−1.85715	−0.928574	+	0.371146i	$0.878965\pi$
$314$	−3.02977	−0.170980
$315$	0	0
$316$	7.85426	0.441836
$317$	−12.5824	−0.706698	−0.353349	−	0.935492i	$-0.614957\pi$
$317$	−12.5824	−0.706698	−0.353349	+	0.935492i	$0.614957\pi$
$318$	0	0
$319$	15.5435	0.870268
$320$	5.50764	0.307886
$321$	0	0
$322$	1.34055	0.0747057
$323$	−9.05854	−0.504031
$324$	0	0
$325$	−5.91023	−0.327841
$326$	−5.83380	−0.323104
$327$	0	0
$328$	2.76549	0.152699
$329$	−0.986499	−0.0543874
$330$	0	0
$331$	1.57677	0.0866669	0.0433334	−	0.999061i	$-0.486202\pi$
$331$	1.57677	0.0866669	0.0433334	+	0.999061i	$0.486202\pi$
$332$	−23.6964	−1.30051
$333$	0	0
$334$	−4.69736	−0.257028
$335$	7.75485	0.423693
$336$	0	0
$337$	−15.4718	−0.842802	−0.421401	−	0.906874i	$-0.638461\pi$
$337$	−15.4718	−0.842802	−0.421401	+	0.906874i	$0.638461\pi$
$338$	7.23120	0.393326
$339$	0	0
$340$	6.44201	0.349367
$341$	24.7765	1.34172
$342$	0	0
$343$	10.2831	0.555233
$344$	−3.01350	−0.162477
$345$	0	0
$346$	0.396971	0.0213413
$347$	9.00312	0.483313	0.241656	−	0.970362i	$-0.422309\pi$
$347$	9.00312	0.483313	0.241656	+	0.970362i	$0.422309\pi$
$348$	0	0
$349$	1.10398	0.0590945	0.0295473	−	0.999563i	$-0.490593\pi$
$349$	1.10398	0.0590945	0.0295473	+	0.999563i	$0.490593\pi$
$350$	1.34055	0.0716552
$351$	0	0
$352$	−9.77041	−0.520765
$353$	−1.96566	−0.104621	−0.0523107	−	0.998631i	$-0.516659\pi$
$353$	−1.96566	−0.104621	−0.0523107	+	0.998631i	$0.516659\pi$
$354$	0	0
$355$	12.4395	0.660220
$356$	−17.6964	−0.937905
$357$	0	0
$358$	−2.44821	−0.129392
$359$	2.60403	0.137435	0.0687177	−	0.997636i	$-0.478109\pi$
$359$	2.60403	0.137435	0.0687177	+	0.997636i	$0.478109\pi$
$360$	0	0
$361$	−11.9273	−0.627753
$362$	7.92830	0.416702
$363$	0	0
$364$	45.4451	2.38197
$365$	2.08977	0.109383
$366$	0	0
$367$	6.98042	0.364375	0.182188	−	0.983264i	$-0.441682\pi$
$367$	6.98042	0.364375	0.182188	+	0.983264i	$0.441682\pi$
$368$	3.35950	0.175126
$369$	0	0
$370$	1.37845	0.0716624
$371$	−36.7363	−1.90726
$372$	0	0
$373$	24.3220	1.25935	0.629673	−	0.776860i	$-0.283189\pi$
$373$	24.3220	1.25935	0.629673	+	0.776860i	$0.283189\pi$
$374$	−2.98685	−0.154446
$375$	0	0
$376$	0.311327	0.0160555
$377$	34.5430	1.77906
$378$	0	0
$379$	−24.6541	−1.26640	−0.633198	−	0.773990i	$-0.718258\pi$
$379$	−24.6541	−1.26640	−0.633198	+	0.773990i	$0.718258\pi$
$380$	−5.02977	−0.258022
$381$	0	0
$382$	1.60804	0.0822747
$383$	4.99292	0.255126	0.127563	−	0.991830i	$-0.459284\pi$
$383$	4.99292	0.255126	0.127563	+	0.991830i	$0.459284\pi$
$384$	0	0
$385$	10.8123	0.551048
$386$	3.38676	0.172382
$387$	0	0
$388$	−6.56612	−0.333344
$389$	32.0234	1.62365	0.811826	−	0.583900i	$-0.198474\pi$
$389$	32.0234	1.62365	0.811826	+	0.583900i	$0.198474\pi$
$390$	0	0
$391$	3.40617	0.172257
$392$	−12.2266	−0.617538
$393$	0	0
$394$	−1.41307	−0.0711894
$395$	4.15288	0.208954
$396$	0	0
$397$	−17.3441	−0.870476	−0.435238	−	0.900315i	$-0.643336\pi$
$397$	−17.3441	−0.870476	−0.435238	+	0.900315i	$0.643336\pi$
$398$	1.32604	0.0664685
$399$	0	0
$400$	3.35950	0.167975
$401$	21.3352	1.06543	0.532714	−	0.846295i	$-0.321172\pi$
$401$	21.3352	1.06543	0.532714	+	0.846295i	$0.321172\pi$
$402$	0	0
$403$	55.0621	2.74284
$404$	17.4194	0.866646
$405$	0	0
$406$	−7.83497	−0.388843
$407$	11.1181	0.551103
$408$	0	0
$409$	−12.5328	−0.619709	−0.309855	−	0.950784i	$-0.600280\pi$
$409$	−12.5328	−0.619709	−0.309855	+	0.950784i	$0.600280\pi$
$410$	0.710689	0.0350985
$411$	0	0
$412$	−8.88403	−0.437685
$413$	−58.6031	−2.88367
$414$	0	0
$415$	−12.5293	−0.615038
$416$	−21.7133	−1.06458
$417$	0	0
$418$	2.33206	0.114065
$419$	−12.0288	−0.587644	−0.293822	−	0.955860i	$-0.594927\pi$
$419$	−12.0288	−0.587644	−0.293822	+	0.955860i	$0.594927\pi$
$420$	0	0
$421$	−38.3721	−1.87014	−0.935071	−	0.354462i	$-0.884664\pi$
$421$	−38.3721	−1.87014	−0.935071	+	0.354462i	$0.884664\pi$
$422$	5.05854	0.246246
$423$	0	0
$424$	11.5935	0.563032
$425$	3.40617	0.165223
$426$	0	0
$427$	10.0163	0.484721
$428$	−0.711866	−0.0344093
$429$	0	0
$430$	−0.774424	−0.0373460
$431$	29.4789	1.41995	0.709975	−	0.704227i	$-0.248706\pi$
$431$	29.4789	1.41995	0.709975	+	0.704227i	$0.248706\pi$
$432$	0	0
$433$	−12.0531	−0.579236	−0.289618	−	0.957142i	$-0.593528\pi$
$433$	−12.0531	−0.579236	−0.289618	+	0.957142i	$0.593528\pi$
$434$	−12.4891	−0.599494
$435$	0	0
$436$	−36.0041	−1.72428
$437$	−2.65945	−0.127219
$438$	0	0
$439$	16.1656	0.771541	0.385771	−	0.922595i	$-0.373936\pi$
$439$	16.1656	0.771541	0.385771	+	0.922595i	$0.373936\pi$
$440$	−3.41224	−0.162672
$441$	0	0
$442$	−6.63782	−0.315729
$443$	−19.3729	−0.920433	−0.460217	−	0.887807i	$-0.652228\pi$
$443$	−19.3729	−0.920433	−0.460217	+	0.887807i	$0.652228\pi$
$444$	0	0
$445$	−9.35682	−0.443556
$446$	−5.82961	−0.276040
$447$	0	0
$448$	−22.3920	−1.05792
$449$	−21.4728	−1.01337	−0.506683	−	0.862132i	$-0.669129\pi$
$449$	−21.4728	−1.01337	−0.506683	+	0.862132i	$0.669129\pi$
$450$	0	0
$451$	5.73215	0.269916
$452$	5.56512	0.261761
$453$	0	0
$454$	−2.84824	−0.133674
$455$	24.0288	1.12649
$456$	0	0
$457$	−5.95602	−0.278611	−0.139305	−	0.990249i	$-0.544487\pi$
$457$	−5.95602	−0.278611	−0.139305	+	0.990249i	$0.544487\pi$
$458$	−1.10147	−0.0514683
$459$	0	0
$460$	1.89128	0.0881814
$461$	−34.6918	−1.61576	−0.807879	−	0.589348i	$-0.799385\pi$
$461$	−34.6918	−1.61576	−0.807879	+	0.589348i	$0.799385\pi$
$462$	0	0
$463$	−30.6399	−1.42396	−0.711979	−	0.702200i	$-0.752201\pi$
$463$	−30.6399	−1.42396	−0.711979	+	0.702200i	$0.752201\pi$
$464$	−19.6350	−0.911531
$465$	0	0
$466$	−2.71605	−0.125819
$467$	8.89547	0.411633	0.205817	−	0.978591i	$-0.434015\pi$
$467$	8.89547	0.411633	0.205817	+	0.978591i	$0.434015\pi$
$468$	0	0
$469$	−31.5283	−1.45584
$470$	0.0800064	0.00369042
$471$	0	0
$472$	18.4944	0.851275
$473$	−6.24621	−0.287201
$474$	0	0
$475$	−2.65945	−0.122024
$476$	−26.1908	−1.20045
$477$	0	0
$478$	0.944750	0.0432119
$479$	9.13224	0.417263	0.208631	−	0.977994i	$-0.433099\pi$
$479$	9.13224	0.417263	0.208631	+	0.977994i	$0.433099\pi$
$480$	0	0
$481$	24.7083	1.12660
$482$	1.54348	0.0703037
$483$	0	0
$484$	7.42763	0.337619
$485$	−3.47179	−0.157646
$486$	0	0
$487$	14.2085	0.643849	0.321924	−	0.946765i	$-0.395670\pi$
$487$	14.2085	0.643849	0.321924	+	0.946765i	$0.395670\pi$
$488$	−3.16101	−0.143092
$489$	0	0
$490$	−3.14206	−0.141944
$491$	23.1760	1.04592	0.522960	−	0.852357i	$-0.324828\pi$
$491$	23.1760	1.04592	0.522960	+	0.852357i	$0.324828\pi$
$492$	0	0
$493$	−19.9077	−0.896599
$494$	5.18265	0.233179
$495$	0	0
$496$	−31.2984	−1.40534
$497$	−50.5743	−2.26857
$498$	0	0
$499$	−2.19831	−0.0984099	−0.0492050	−	0.998789i	$-0.515669\pi$
$499$	−2.19831	−0.0984099	−0.0492050	+	0.998789i	$0.515669\pi$
$500$	1.89128	0.0845806
$501$	0	0
$502$	6.12712	0.273467
$503$	44.0461	1.96392	0.981959	−	0.189092i	$-0.0605545\pi$
$503$	44.0461	1.96392	0.981959	+	0.189092i	$0.0605545\pi$
$504$	0	0
$505$	9.21036	0.409856
$506$	−0.876894	−0.0389827
$507$	0	0
$508$	−11.3544	−0.503771
$509$	9.09254	0.403020	0.201510	−	0.979486i	$-0.435415\pi$
$509$	9.09254	0.403020	0.201510	+	0.979486i	$0.435415\pi$
$510$	0	0
$511$	−8.49619	−0.375850
$512$	20.9632	0.926449
$513$	0	0
$514$	9.47061	0.417731
$515$	−4.69736	−0.206991
$516$	0	0
$517$	0.645301	0.0283803
$518$	−5.60427	−0.246238
$519$	0	0
$520$	−7.58319	−0.332545
$521$	17.1231	0.750177	0.375088	−	0.926989i	$-0.377612\pi$
$521$	17.1231	0.750177	0.375088	+	0.926989i	$0.377612\pi$
$522$	0	0
$523$	−26.6112	−1.16362	−0.581812	−	0.813323i	$-0.697656\pi$
$523$	−26.6112	−1.16362	−0.581812	+	0.813323i	$0.697656\pi$
$524$	−37.7512	−1.64917
$525$	0	0
$526$	3.03514	0.132338
$527$	−31.7332	−1.38232
$528$	0	0
$529$	1.00000	0.0434783
$530$	2.97936	0.129415
$531$	0	0
$532$	20.4491	0.886583
$533$	12.7388	0.551780
$534$	0	0
$535$	−0.376394	−0.0162729
$536$	9.94994	0.429772
$537$	0	0
$538$	−5.57880	−0.240519
$539$	−25.3427	−1.09159
$540$	0	0
$541$	31.5941	1.35834	0.679168	−	0.733983i	$-0.262341\pi$
$541$	31.5941	1.35834	0.679168	+	0.733983i	$0.262341\pi$
$542$	0.379456	0.0162990
$543$	0	0
$544$	12.5137	0.536521
$545$	−19.0369	−0.815452
$546$	0	0
$547$	22.7009	0.970622	0.485311	−	0.874342i	$-0.338706\pi$
$547$	22.7009	0.970622	0.485311	+	0.874342i	$0.338706\pi$
$548$	−22.2432	−0.950182
$549$	0	0
$550$	−0.876894	−0.0373909
$551$	15.5435	0.662175
$552$	0	0
$553$	−16.8840	−0.717982
$554$	0.849065	0.0360733
$555$	0	0
$556$	−23.4313	−0.993706
$557$	−35.1292	−1.48847	−0.744236	−	0.667917i	$-0.767186\pi$
$557$	−35.1292	−1.48847	−0.744236	+	0.667917i	$0.767186\pi$
$558$	0	0
$559$	−13.8813	−0.587115
$560$	−13.6585	−0.577175
$561$	0	0
$562$	−9.05819	−0.382097
$563$	−32.0511	−1.35079	−0.675397	−	0.737455i	$-0.736028\pi$
$563$	−32.0511	−1.35079	−0.675397	+	0.737455i	$0.736028\pi$
$564$	0	0
$565$	2.94252	0.123793
$566$	−0.779440	−0.0327623
$567$	0	0
$568$	15.9606	0.669694
$569$	22.0575	0.924700	0.462350	−	0.886697i	$-0.347006\pi$
$569$	22.0575	0.924700	0.462350	+	0.886697i	$0.347006\pi$
$570$	0	0
$571$	8.49242	0.355397	0.177698	−	0.984085i	$-0.443135\pi$
$571$	8.49242	0.355397	0.177698	+	0.984085i	$0.443135\pi$
$572$	−29.7271	−1.24295
$573$	0	0
$574$	−2.88939	−0.120601
$575$	1.00000	0.0417029
$576$	0	0
$577$	27.1554	1.13050	0.565248	−	0.824921i	$-0.308780\pi$
$577$	27.1554	1.13050	0.565248	+	0.824921i	$0.308780\pi$
$578$	−1.77988	−0.0740331
$579$	0	0
$580$	−11.0538	−0.458984
$581$	50.9393	2.11332
$582$	0	0
$583$	24.0304	0.995239
$584$	2.68130	0.110953
$585$	0	0
$586$	−11.2502	−0.464743
$587$	30.8322	1.27258	0.636290	−	0.771450i	$-0.280468\pi$
$587$	30.8322	1.27258	0.636290	+	0.771450i	$0.280468\pi$
$588$	0	0
$589$	24.7765	1.02090
$590$	4.75279	0.195669
$591$	0	0
$592$	−14.0447	−0.577233
$593$	27.7559	1.13980	0.569899	−	0.821715i	$-0.306982\pi$
$593$	27.7559	1.13980	0.569899	+	0.821715i	$0.306982\pi$
$594$	0	0
$595$	−13.8482	−0.567720
$596$	−4.03077	−0.165107
$597$	0	0
$598$	−1.94876	−0.0796909
$599$	30.7712	1.25728	0.628638	−	0.777698i	$-0.283613\pi$
$599$	30.7712	1.25728	0.628638	+	0.777698i	$0.283613\pi$
$600$	0	0
$601$	−17.9830	−0.733541	−0.366771	−	0.930311i	$-0.619537\pi$
$601$	−17.9830	−0.733541	−0.366771	+	0.930311i	$0.619537\pi$
$602$	3.14851	0.128324
$603$	0	0
$604$	1.48868	0.0605736
$605$	3.92730	0.159668
$606$	0	0
$607$	−18.3220	−0.743668	−0.371834	−	0.928299i	$-0.621271\pi$
$607$	−18.3220	−0.743668	−0.371834	+	0.928299i	$0.621271\pi$
$608$	−9.77041	−0.396242
$609$	0	0
$610$	−0.812333	−0.0328904
$611$	1.43408	0.0580168
$612$	0	0
$613$	35.4451	1.43162	0.715808	−	0.698297i	$-0.246059\pi$
$613$	35.4451	1.43162	0.715808	+	0.698297i	$0.246059\pi$
$614$	0.492423	0.0198726
$615$	0	0
$616$	13.8729	0.558954
$617$	−43.1567	−1.73742	−0.868712	−	0.495318i	$-0.835052\pi$
$617$	−43.1567	−1.73742	−0.868712	+	0.495318i	$0.835052\pi$
$618$	0	0
$619$	5.81133	0.233577	0.116789	−	0.993157i	$-0.462740\pi$
$619$	5.81133	0.233577	0.116789	+	0.993157i	$0.462740\pi$
$620$	−17.6199	−0.707633
$621$	0	0
$622$	4.65114	0.186494
$623$	38.0413	1.52409
$624$	0	0
$625$	1.00000	0.0400000
$626$	−10.8336	−0.432999
$627$	0	0
$628$	17.3785	0.693476
$629$	−14.2398	−0.567777
$630$	0	0
$631$	−27.4626	−1.09327	−0.546635	−	0.837371i	$-0.684092\pi$
$631$	−27.4626	−1.09327	−0.546635	+	0.837371i	$0.684092\pi$
$632$	5.32840	0.211952
$633$	0	0
$634$	−4.14876	−0.164768
$635$	−6.00357	−0.238244
$636$	0	0
$637$	−56.3202	−2.23149
$638$	5.12511	0.202905
$639$	0	0
$640$	9.16370	0.362227
$641$	−37.7938	−1.49277	−0.746383	−	0.665517i	$-0.768211\pi$
$641$	−37.7938	−1.49277	−0.746383	+	0.665517i	$0.768211\pi$
$642$	0	0
$643$	12.5343	0.494304	0.247152	−	0.968977i	$-0.420505\pi$
$643$	12.5343	0.494304	0.247152	+	0.968977i	$0.420505\pi$
$644$	−7.68923	−0.302998
$645$	0	0
$646$	−2.98685	−0.117516
$647$	21.5133	0.845774	0.422887	−	0.906183i	$-0.361017\pi$
$647$	21.5133	0.845774	0.422887	+	0.906183i	$0.361017\pi$
$648$	0	0
$649$	38.3342	1.50475
$650$	−1.94876	−0.0764368
$651$	0	0
$652$	33.4620	1.31047
$653$	−19.9011	−0.778790	−0.389395	−	0.921071i	$-0.627316\pi$
$653$	−19.9011	−0.778790	−0.389395	+	0.921071i	$0.627316\pi$
$654$	0	0
$655$	−19.9606	−0.779927
$656$	−7.24102	−0.282714
$657$	0	0
$658$	−0.325276	−0.0126806
$659$	4.95773	0.193126	0.0965628	−	0.995327i	$-0.469215\pi$
$659$	4.95773	0.193126	0.0965628	+	0.995327i	$0.469215\pi$
$660$	0	0
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	0.519902	0.0202066
$663$	0	0
$664$	−16.0758	−0.623863
$665$	10.8123	0.419284
$666$	0	0
$667$	−5.84461	−0.226304
$668$	26.9436	1.04248
$669$	0	0
$670$	2.55698	0.0987849
$671$	−6.55197	−0.252936
$672$	0	0
$673$	−28.5176	−1.09927	−0.549637	−	0.835404i	$-0.685234\pi$
$673$	−28.5176	−1.09927	−0.549637	+	0.835404i	$0.685234\pi$
$674$	−5.10147	−0.196501
$675$	0	0
$676$	−41.4774	−1.59529
$677$	−28.6214	−1.10001	−0.550004	−	0.835162i	$-0.685374\pi$
$677$	−28.6214	−1.10001	−0.550004	+	0.835162i	$0.685374\pi$
$678$	0	0
$679$	14.1150	0.541683
$680$	4.37032	0.167594
$681$	0	0
$682$	8.16950	0.312826
$683$	−11.3983	−0.436144	−0.218072	−	0.975933i	$-0.569977\pi$
$683$	−11.3983	−0.436144	−0.218072	+	0.975933i	$0.569977\pi$
$684$	0	0
$685$	−11.7609	−0.449362
$686$	3.39060	0.129454
$687$	0	0
$688$	7.89040	0.300819
$689$	53.4040	2.03453
$690$	0	0
$691$	45.1898	1.71910	0.859550	−	0.511051i	$-0.170744\pi$
$691$	45.1898	1.71910	0.859550	+	0.511051i	$0.170744\pi$
$692$	−2.27699	−0.0865580
$693$	0	0
$694$	2.96857	0.112686
$695$	−12.3891	−0.469945
$696$	0	0
$697$	−7.34160	−0.278083
$698$	0.364011	0.0137780
$699$	0	0
$700$	−7.68923	−0.290625
$701$	−40.4871	−1.52918	−0.764588	−	0.644520i	$-0.777057\pi$
$701$	−40.4871	−1.52918	−0.764588	+	0.644520i	$0.777057\pi$
$702$	0	0
$703$	11.1181	0.419327
$704$	14.6473	0.552041
$705$	0	0
$706$	−0.648131	−0.0243927
$707$	−37.4458	−1.40830
$708$	0	0
$709$	−41.6443	−1.56398	−0.781992	−	0.623288i	$-0.785796\pi$
$709$	−41.6443	−1.56398	−0.781992	+	0.623288i	$0.785796\pi$
$710$	4.10164	0.153932
$711$	0	0
$712$	−12.0054	−0.449920
$713$	−9.31640	−0.348902
$714$	0	0
$715$	−15.7180	−0.587820
$716$	14.0427	0.524799
$717$	0	0
$718$	0.858620	0.0320434
$719$	−49.2288	−1.83592	−0.917961	−	0.396670i	$-0.870166\pi$
$719$	−49.2288	−1.83592	−0.917961	+	0.396670i	$0.870166\pi$
$720$	0	0
$721$	19.0977	0.711235
$722$	−3.93276	−0.146362
$723$	0	0
$724$	−45.4759	−1.69010
$725$	−5.84461	−0.217063
$726$	0	0
$727$	27.9581	1.03691	0.518455	−	0.855105i	$-0.326507\pi$
$727$	27.9581	1.03691	0.518455	+	0.855105i	$0.326507\pi$
$728$	30.8304	1.14265
$729$	0	0
$730$	0.689053	0.0255030
$731$	8.00000	0.295891
$732$	0	0
$733$	−18.6655	−0.689427	−0.344714	−	0.938708i	$-0.612024\pi$
$733$	−18.6655	−0.689427	−0.344714	+	0.938708i	$0.612024\pi$
$734$	2.30164	0.0849549
$735$	0	0
$736$	3.67384	0.135420
$737$	20.6237	0.759682
$738$	0	0
$739$	2.50407	0.0921136	0.0460568	−	0.998939i	$-0.485334\pi$
$739$	2.50407	0.0921136	0.0460568	+	0.998939i	$0.485334\pi$
$740$	−7.90667	−0.290655
$741$	0	0
$742$	−12.1130	−0.444681
$743$	−4.69736	−0.172330	−0.0861648	−	0.996281i	$-0.527461\pi$
$743$	−4.69736	−0.172330	−0.0861648	+	0.996281i	$0.527461\pi$
$744$	0	0
$745$	−2.13124	−0.0780826
$746$	8.01963	0.293620
$747$	0	0
$748$	17.1322	0.626417
$749$	1.53027	0.0559150
$750$	0	0
$751$	−16.0720	−0.586477	−0.293239	−	0.956039i	$-0.594733\pi$
$751$	−16.0720	−0.586477	−0.293239	+	0.956039i	$0.594733\pi$
$752$	−0.815163	−0.0297259
$753$	0	0
$754$	11.3898	0.414791
$755$	0.787129	0.0286465
$756$	0	0
$757$	−25.8357	−0.939014	−0.469507	−	0.882929i	$-0.655568\pi$
$757$	−25.8357	−0.939014	−0.469507	+	0.882929i	$0.655568\pi$
$758$	−8.12912	−0.295263
$759$	0	0
$760$	−3.41224	−0.123775
$761$	−15.6005	−0.565518	−0.282759	−	0.959191i	$-0.591250\pi$
$761$	−15.6005	−0.565518	−0.282759	+	0.959191i	$0.591250\pi$
$762$	0	0
$763$	77.3968	2.80195
$764$	−9.22357	−0.333697
$765$	0	0
$766$	1.64630	0.0594833
$767$	85.1919	3.07610
$768$	0	0
$769$	−37.7722	−1.36210	−0.681050	−	0.732237i	$-0.738476\pi$
$769$	−37.7722	−1.36210	−0.681050	+	0.732237i	$0.738476\pi$
$770$	3.56512	0.128478
$771$	0	0
$772$	−19.4261	−0.699161
$773$	42.9078	1.54329	0.771643	−	0.636056i	$-0.219435\pi$
$773$	42.9078	1.54329	0.771643	+	0.636056i	$0.219435\pi$
$774$	0	0
$775$	−9.31640	−0.334655
$776$	−4.45451	−0.159908
$777$	0	0
$778$	10.5590	0.378558
$779$	5.73215	0.205376
$780$	0	0
$781$	33.0823	1.18378
$782$	1.12311	0.0401622
$783$	0	0
$784$	32.0136	1.14334
$785$	9.18873	0.327960
$786$	0	0
$787$	−33.5324	−1.19530	−0.597650	−	0.801757i	$-0.703899\pi$
$787$	−33.5324	−1.19530	−0.597650	+	0.801757i	$0.703899\pi$
$788$	8.10521	0.288736
$789$	0	0
$790$	1.36932	0.0487181
$791$	−11.9632	−0.425361
$792$	0	0
$793$	−14.5608	−0.517068
$794$	−5.71883	−0.202954
$795$	0	0
$796$	−7.60604	−0.269589
$797$	−32.7488	−1.16002	−0.580012	−	0.814608i	$-0.696952\pi$
$797$	−32.7488	−1.16002	−0.580012	+	0.814608i	$0.696952\pi$
$798$	0	0
$799$	−0.826486	−0.0292390
$800$	3.67384	0.129890
$801$	0	0
$802$	7.03479	0.248407
$803$	5.55764	0.196125
$804$	0	0
$805$	−4.06562	−0.143294
$806$	18.1555	0.639499
$807$	0	0
$808$	11.8175	0.415737
$809$	−28.8513	−1.01436	−0.507179	−	0.861841i	$-0.669312\pi$
$809$	−28.8513	−1.01436	−0.507179	+	0.861841i	$0.669312\pi$
$810$	0	0
$811$	36.1791	1.27042	0.635211	−	0.772339i	$-0.280913\pi$
$811$	36.1791	1.27042	0.635211	+	0.772339i	$0.280913\pi$
$812$	44.9406	1.57710
$813$	0	0
$814$	3.66594	0.128491
$815$	17.6928	0.619752
$816$	0	0
$817$	−6.24621	−0.218527
$818$	−4.13242	−0.144487
$819$	0	0
$820$	−4.07644	−0.142355
$821$	−5.73752	−0.200241	−0.100120	−	0.994975i	$-0.531923\pi$
$821$	−5.73752	−0.200241	−0.100120	+	0.994975i	$0.531923\pi$
$822$	0	0
$823$	−6.87333	−0.239589	−0.119795	−	0.992799i	$-0.538224\pi$
$823$	−6.87333	−0.239589	−0.119795	+	0.992799i	$0.538224\pi$
$824$	−6.02700	−0.209961
$825$	0	0
$826$	−19.3230	−0.672334
$827$	19.7332	0.686191	0.343095	−	0.939301i	$-0.388525\pi$
$827$	19.7332	0.686191	0.343095	+	0.939301i	$0.388525\pi$
$828$	0	0
$829$	−2.88833	−0.100316	−0.0501580	−	0.998741i	$-0.515972\pi$
$829$	−2.88833	−0.100316	−0.0501580	+	0.998741i	$0.515972\pi$
$830$	−4.13124	−0.143397
$831$	0	0
$832$	32.5514	1.12852
$833$	32.4583	1.12461
$834$	0	0
$835$	14.2462	0.493010
$836$	−13.3765	−0.462634
$837$	0	0
$838$	−3.96621	−0.137011
$839$	−24.1904	−0.835147	−0.417573	−	0.908643i	$-0.637119\pi$
$839$	−24.1904	−0.835147	−0.417573	+	0.908643i	$0.637119\pi$
$840$	0	0
$841$	5.15951	0.177914
$842$	−12.6523	−0.436028
$843$	0	0
$844$	−29.0153	−0.998747
$845$	−21.9309	−0.754445
$846$	0	0
$847$	−15.9669	−0.548630
$848$	−30.3559	−1.04243
$849$	0	0
$850$	1.12311	0.0385222
$851$	−4.18059	−0.143309
$852$	0	0
$853$	27.5381	0.942887	0.471444	−	0.881896i	$-0.343733\pi$
$853$	27.5381	0.942887	0.471444	+	0.881896i	$0.343733\pi$
$854$	3.30264	0.113014
$855$	0	0
$856$	−0.482936	−0.0165064
$857$	17.3995	0.594357	0.297178	−	0.954822i	$-0.403954\pi$
$857$	17.3995	0.594357	0.297178	+	0.954822i	$0.403954\pi$
$858$	0	0
$859$	1.24560	0.0424993	0.0212497	−	0.999774i	$-0.493236\pi$
$859$	1.24560	0.0424993	0.0212497	+	0.999774i	$0.493236\pi$
$860$	4.44201	0.151471
$861$	0	0
$862$	9.72000	0.331065
$863$	4.60383	0.156716	0.0783580	−	0.996925i	$-0.475032\pi$
$863$	4.60383	0.156716	0.0783580	+	0.996925i	$0.475032\pi$
$864$	0	0
$865$	−1.20394	−0.0409352
$866$	−3.97424	−0.135050
$867$	0	0
$868$	71.6359	2.43148
$869$	11.0444	0.374655
$870$	0	0
$871$	45.8330	1.55299
$872$	−24.4255	−0.827152
$873$	0	0
$874$	−0.876894	−0.0296614
$875$	−4.06562	−0.137443
$876$	0	0
$877$	33.8276	1.14228	0.571138	−	0.820854i	$-0.306502\pi$
$877$	33.8276	1.14228	0.571138	+	0.820854i	$0.306502\pi$
$878$	5.33023	0.179887
$879$	0	0
$880$	8.93444	0.301180
$881$	−26.1134	−0.879782	−0.439891	−	0.898051i	$-0.644983\pi$
$881$	−26.1134	−0.879782	−0.439891	+	0.898051i	$0.644983\pi$
$882$	0	0
$883$	−54.5412	−1.83546	−0.917729	−	0.397206i	$-0.869980\pi$
$883$	−54.5412	−1.83546	−0.917729	+	0.397206i	$0.869980\pi$
$884$	38.0738	1.28056
$885$	0	0
$886$	−6.38777	−0.214601
$887$	−22.6201	−0.759509	−0.379754	−	0.925087i	$-0.623991\pi$
$887$	−22.6201	−0.759509	−0.379754	+	0.925087i	$0.623991\pi$
$888$	0	0
$889$	24.4082	0.818626
$890$	−3.08520	−0.103416
$891$	0	0
$892$	33.4380	1.11959
$893$	0.645301	0.0215942
$894$	0	0
$895$	7.42495	0.248189
$896$	−37.2561	−1.24464
$897$	0	0
$898$	−7.08018	−0.236269
$899$	54.4508	1.81603
$900$	0	0
$901$	−30.7776	−1.02535
$902$	1.89005	0.0629317
$903$	0	0
$904$	3.77543	0.125569
$905$	−24.0450	−0.799284
$906$	0	0
$907$	−3.62149	−0.120250	−0.0601248	−	0.998191i	$-0.519150\pi$
$907$	−3.62149	−0.120250	−0.0601248	+	0.998191i	$0.519150\pi$
$908$	16.3372	0.542169
$909$	0	0
$910$	7.92294	0.262643
$911$	−26.9241	−0.892034	−0.446017	−	0.895025i	$-0.647158\pi$
$911$	−26.9241	−0.892034	−0.446017	+	0.895025i	$0.647158\pi$
$912$	0	0
$913$	−33.3210	−1.10277
$914$	−1.96386	−0.0649587
$915$	0	0
$916$	6.31791	0.208750
$917$	81.1524	2.67989
$918$	0	0
$919$	−19.5059	−0.643441	−0.321721	−	0.946835i	$-0.604261\pi$
$919$	−19.5059	−0.643441	−0.321721	+	0.946835i	$0.604261\pi$
$920$	1.28306	0.0423013
$921$	0	0
$922$	−11.4388	−0.376718
$923$	73.5204	2.41995
$924$	0	0
$925$	−4.18059	−0.137457
$926$	−10.1028	−0.331999
$927$	0	0
$928$	−21.4722	−0.704859
$929$	−27.2139	−0.892860	−0.446430	−	0.894819i	$-0.647305\pi$
$929$	−27.2139	−0.892860	−0.446430	+	0.894819i	$0.647305\pi$
$930$	0	0
$931$	−25.3427	−0.830572
$932$	15.5790	0.510307
$933$	0	0
$934$	2.93308	0.0959732
$935$	9.05854	0.296246
$936$	0	0
$937$	3.85626	0.125978	0.0629892	−	0.998014i	$-0.479937\pi$
$937$	3.85626	0.125978	0.0629892	+	0.998014i	$0.479937\pi$
$938$	−10.3957	−0.339433
$939$	0	0
$940$	−0.458908	−0.0149679
$941$	60.6471	1.97704	0.988519	−	0.151097i	$-0.0482805\pi$
$941$	60.6471	1.97704	0.988519	+	0.151097i	$0.0482805\pi$
$942$	0	0
$943$	−2.15539	−0.0701890
$944$	−48.4248	−1.57609
$945$	0	0
$946$	−2.05955	−0.0669616
$947$	1.11954	0.0363801	0.0181901	−	0.999835i	$-0.494210\pi$
$947$	1.11954	0.0363801	0.0181901	+	0.999835i	$0.494210\pi$
$948$	0	0
$949$	12.3510	0.400931
$950$	−0.876894	−0.0284502
$951$	0	0
$952$	−17.7681	−0.575866
$953$	−14.8590	−0.481331	−0.240666	−	0.970608i	$-0.577366\pi$
$953$	−14.8590	−0.481331	−0.240666	+	0.970608i	$0.577366\pi$
$954$	0	0
$955$	−4.87689	−0.157813
$956$	−5.41899	−0.175263
$957$	0	0
$958$	3.01115	0.0972858
$959$	47.8155	1.54404
$960$	0	0
$961$	55.7953	1.79985
$962$	8.14699	0.262669
$963$	0	0
$964$	−8.85325	−0.285144
$965$	−10.2714	−0.330649
$966$	0	0
$967$	38.5718	1.24039	0.620193	−	0.784449i	$-0.287054\pi$
$967$	38.5718	1.24039	0.620193	+	0.784449i	$0.287054\pi$
$968$	5.03897	0.161959
$969$	0	0
$970$	−1.14474	−0.0367555
$971$	25.4984	0.818284	0.409142	−	0.912471i	$-0.365828\pi$
$971$	25.4984	0.818284	0.409142	+	0.912471i	$0.365828\pi$
$972$	0	0
$973$	50.3694	1.61477
$974$	4.68493	0.150115
$975$	0	0
$976$	8.27664	0.264929
$977$	15.8952	0.508533	0.254267	−	0.967134i	$-0.418166\pi$
$977$	15.8952	0.508533	0.254267	+	0.967134i	$0.418166\pi$
$978$	0	0
$979$	−24.8840	−0.795297
$980$	18.0225	0.575708
$981$	0	0
$982$	7.64176	0.243858
$983$	2.50587	0.0799247	0.0399624	−	0.999201i	$-0.487276\pi$
$983$	2.50587	0.0799247	0.0399624	+	0.999201i	$0.487276\pi$
$984$	0	0
$985$	4.28557	0.136550
$986$	−6.56412	−0.209044
$987$	0	0
$988$	−29.7271	−0.945746
$989$	2.34868	0.0746837
$990$	0	0
$991$	12.5272	0.397938	0.198969	−	0.980006i	$-0.436241\pi$
$991$	12.5272	0.397938	0.198969	+	0.980006i	$0.436241\pi$
$992$	−34.2270	−1.08671
$993$	0	0
$994$	−16.6757	−0.528922
$995$	−4.02164	−0.127494
$996$	0	0
$997$	−9.34803	−0.296055	−0.148028	−	0.988983i	$-0.547292\pi$
$997$	−9.34803	−0.296055	−0.148028	+	0.988983i	$0.547292\pi$
$998$	−0.724843	−0.0229445
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1035.2.a.o.1.3		4
3.2	odd	2		115.2.a.c.1.2	✓	4
5.4	even	2		5175.2.a.bx.1.2		4
12.11	even	2		1840.2.a.u.1.1		4
15.2	even	4		575.2.b.e.24.4		8
15.8	even	4		575.2.b.e.24.5		8
15.14	odd	2		575.2.a.h.1.3		4
21.20	even	2		5635.2.a.v.1.2		4
24.5	odd	2		7360.2.a.cj.1.2		4
24.11	even	2		7360.2.a.cg.1.3		4
60.59	even	2		9200.2.a.cl.1.4		4
69.68	even	2		2645.2.a.m.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.2.a.c.1.2	✓	4	3.2	odd	2
575.2.a.h.1.3		4	15.14	odd	2
575.2.b.e.24.4		8	15.2	even	4
575.2.b.e.24.5		8	15.8	even	4
1035.2.a.o.1.3		4	1.1	even	1	trivial
1840.2.a.u.1.1		4	12.11	even	2
2645.2.a.m.1.2		4	69.68	even	2
5175.2.a.bx.1.2		4	5.4	even	2
5635.2.a.v.1.2		4	21.20	even	2
7360.2.a.cg.1.3		4	24.11	even	2
7360.2.a.cj.1.2		4	24.5	odd	2
9200.2.a.cl.1.4		4	60.59	even	2

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:	\( N \)	\(=\)	\( 1035 = 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1035.a (trivial)

\(f(q)\)	\(=\)	\(q+0.329727 q^{2} -1.89128 q^{4} -1.00000 q^{5} +4.06562 q^{7} -1.28306 q^{8} +O(q^{10})\)	\(q+0.329727 q^{2} -1.89128 q^{4} -1.00000 q^{5} +4.06562 q^{7} -1.28306 q^{8} -0.329727 q^{10} -2.65945 q^{11} -5.91023 q^{13} +1.34055 q^{14} +3.35950 q^{16} +3.40617 q^{17} -2.65945 q^{19} +1.89128 q^{20} -0.876894 q^{22} +1.00000 q^{23} +1.00000 q^{25} -1.94876 q^{26} -7.68923 q^{28} -5.84461 q^{29} -9.31640 q^{31} +3.67384 q^{32} +1.12311 q^{34} -4.06562 q^{35} -4.18059 q^{37} -0.876894 q^{38} +1.28306 q^{40} -2.15539 q^{41} +2.34868 q^{43} +5.02977 q^{44} +0.329727 q^{46} -0.242644 q^{47} +9.52927 q^{49} +0.329727 q^{50} +11.1779 q^{52} -9.03585 q^{53} +2.65945 q^{55} -5.21644 q^{56} -1.92713 q^{58} -14.4143 q^{59} +2.46365 q^{61} -3.07187 q^{62} -5.50764 q^{64} +5.91023 q^{65} -7.75485 q^{67} -6.44201 q^{68} -1.34055 q^{70} -12.4395 q^{71} -2.08977 q^{73} -1.37845 q^{74} +5.02977 q^{76} -10.8123 q^{77} -4.15288 q^{79} -3.35950 q^{80} -0.710689 q^{82} +12.5293 q^{83} -3.40617 q^{85} +0.774424 q^{86} +3.41224 q^{88} +9.35682 q^{89} -24.0288 q^{91} -1.89128 q^{92} -0.0800064 q^{94} +2.65945 q^{95} +3.47179 q^{97} +3.14206 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} + 4 q^{4} - 4 q^{5} - 3 q^{7} - 9 q^{8}+O(q^{10}) \) 4 * q - 2 * q^2 + 4 * q^4 - 4 * q^5 - 3 * q^7 - 9 * q^8	\( 4 q - 2 q^{2} + 4 q^{4} - 4 q^{5} - 3 q^{7} - 9 q^{8} + 2 q^{10} - 4 q^{11} + 12 q^{14} + 8 q^{16} + q^{17} - 4 q^{19} - 4 q^{20} - 20 q^{22} + 4 q^{23} + 4 q^{25} + q^{26} - 22 q^{28} - 19 q^{29} - q^{31} - 20 q^{32} - 12 q^{34} + 3 q^{35} - 3 q^{37} - 20 q^{38} + 9 q^{40} - 13 q^{41} - 6 q^{43} + 18 q^{44} - 2 q^{46} - 6 q^{47} + 9 q^{49} - 2 q^{50} - q^{52} - 19 q^{53} + 4 q^{55} + 10 q^{56} + 21 q^{58} - 23 q^{59} + 13 q^{62} + 27 q^{64} - 3 q^{67} + 4 q^{68} - 12 q^{70} + 3 q^{71} - 32 q^{73} + 12 q^{74} + 18 q^{76} - 18 q^{77} + 2 q^{79} - 8 q^{80} - 5 q^{82} + 21 q^{83} - q^{85} + 2 q^{86} - 14 q^{88} - 40 q^{91} + 4 q^{92} + 47 q^{94} + 4 q^{95} - 18 q^{97} - 16 q^{98}+O(q^{100}) \) 4 * q - 2 * q^2 + 4 * q^4 - 4 * q^5 - 3 * q^7 - 9 * q^8 + 2 * q^10 - 4 * q^11 + 12 * q^14 + 8 * q^16 + q^17 - 4 * q^19 - 4 * q^20 - 20 * q^22 + 4 * q^23 + 4 * q^25 + q^26 - 22 * q^28 - 19 * q^29 - q^31 - 20 * q^32 - 12 * q^34 + 3 * q^35 - 3 * q^37 - 20 * q^38 + 9 * q^40 - 13 * q^41 - 6 * q^43 + 18 * q^44 - 2 * q^46 - 6 * q^47 + 9 * q^49 - 2 * q^50 - q^52 - 19 * q^53 + 4 * q^55 + 10 * q^56 + 21 * q^58 - 23 * q^59 + 13 * q^62 + 27 * q^64 - 3 * q^67 + 4 * q^68 - 12 * q^70 + 3 * q^71 - 32 * q^73 + 12 * q^74 + 18 * q^76 - 18 * q^77 + 2 * q^79 - 8 * q^80 - 5 * q^82 + 21 * q^83 - q^85 + 2 * q^86 - 14 * q^88 - 40 * q^91 + 4 * q^92 + 47 * q^94 + 4 * q^95 - 18 * q^97 - 16 * q^98

Introduction

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