LMFDB - Embedded newform 2394.2.a.w.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2394,2,Mod(1,2394)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2394, 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("2394.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2394.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:	$19.1161862439$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 266)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	2394.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -3.85410 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -3.85410 q^{5} +1.00000 q^{7} +1.00000 q^{8} -3.85410 q^{10} -2.38197 q^{11} +0.763932 q^{13} +1.00000 q^{14} +1.00000 q^{16} +6.47214 q^{17} -1.00000 q^{19} -3.85410 q^{20} -2.38197 q^{22} -7.70820 q^{23} +9.85410 q^{25} +0.763932 q^{26} +1.00000 q^{28} +2.14590 q^{29} +8.47214 q^{31} +1.00000 q^{32} +6.47214 q^{34} -3.85410 q^{35} -1.85410 q^{37} -1.00000 q^{38} -3.85410 q^{40} +10.3262 q^{41} -5.38197 q^{43} -2.38197 q^{44} -7.70820 q^{46} -2.85410 q^{47} +1.00000 q^{49} +9.85410 q^{50} +0.763932 q^{52} +13.6180 q^{53} +9.18034 q^{55} +1.00000 q^{56} +2.14590 q^{58} +2.14590 q^{59} +13.3262 q^{61} +8.47214 q^{62} +1.00000 q^{64} -2.94427 q^{65} -6.94427 q^{67} +6.47214 q^{68} -3.85410 q^{70} +13.0344 q^{71} -2.76393 q^{73} -1.85410 q^{74} -1.00000 q^{76} -2.38197 q^{77} +2.32624 q^{79} -3.85410 q^{80} +10.3262 q^{82} +8.94427 q^{83} -24.9443 q^{85} -5.38197 q^{86} -2.38197 q^{88} +3.61803 q^{89} +0.763932 q^{91} -7.70820 q^{92} -2.85410 q^{94} +3.85410 q^{95} -2.85410 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - q^{5} + 2 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 - q^5 + 2 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} - q^{5} + 2 q^{7} + 2 q^{8} - q^{10} - 7 q^{11} + 6 q^{13} + 2 q^{14} + 2 q^{16} + 4 q^{17} - 2 q^{19} - q^{20} - 7 q^{22} - 2 q^{23} + 13 q^{25} + 6 q^{26} + 2 q^{28} + 11 q^{29} + 8 q^{31} + 2 q^{32} + 4 q^{34} - q^{35} + 3 q^{37} - 2 q^{38} - q^{40} + 5 q^{41} - 13 q^{43} - 7 q^{44} - 2 q^{46} + q^{47} + 2 q^{49} + 13 q^{50} + 6 q^{52} + 25 q^{53} - 4 q^{55} + 2 q^{56} + 11 q^{58} + 11 q^{59} + 11 q^{61} + 8 q^{62} + 2 q^{64} + 12 q^{65} + 4 q^{67} + 4 q^{68} - q^{70} - 3 q^{71} - 10 q^{73} + 3 q^{74} - 2 q^{76} - 7 q^{77} - 11 q^{79} - q^{80} + 5 q^{82} - 32 q^{85} - 13 q^{86} - 7 q^{88} + 5 q^{89} + 6 q^{91} - 2 q^{92} + q^{94} + q^{95} + q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - q^5 + 2 * q^7 + 2 * q^8 - q^10 - 7 * q^11 + 6 * q^13 + 2 * q^14 + 2 * q^16 + 4 * q^17 - 2 * q^19 - q^20 - 7 * q^22 - 2 * q^23 + 13 * q^25 + 6 * q^26 + 2 * q^28 + 11 * q^29 + 8 * q^31 + 2 * q^32 + 4 * q^34 - q^35 + 3 * q^37 - 2 * q^38 - q^40 + 5 * q^41 - 13 * q^43 - 7 * q^44 - 2 * q^46 + q^47 + 2 * q^49 + 13 * q^50 + 6 * q^52 + 25 * q^53 - 4 * q^55 + 2 * q^56 + 11 * q^58 + 11 * q^59 + 11 * q^61 + 8 * q^62 + 2 * q^64 + 12 * q^65 + 4 * q^67 + 4 * q^68 - q^70 - 3 * q^71 - 10 * q^73 + 3 * q^74 - 2 * q^76 - 7 * q^77 - 11 * q^79 - q^80 + 5 * q^82 - 32 * q^85 - 13 * q^86 - 7 * q^88 + 5 * q^89 + 6 * q^91 - 2 * q^92 + q^94 + q^95 + q^97 + 2 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−3.85410	−1.72361	−0.861803	−	0.507242i	$-0.830665\pi$
$5$	−3.85410	−1.72361	−0.861803	+	0.507242i	$0.830665\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	0	0
$10$	−3.85410	−1.21877
$11$	−2.38197	−0.718190	−0.359095	−	0.933301i	$-0.616915\pi$
$11$	−2.38197	−0.718190	−0.359095	+	0.933301i	$0.616915\pi$
$12$	0	0
$13$	0.763932	0.211877	0.105938	−	0.994373i	$-0.466215\pi$
$13$	0.763932	0.211877	0.105938	+	0.994373i	$0.466215\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	6.47214	1.56972	0.784862	−	0.619671i	$-0.212734\pi$
$17$	6.47214	1.56972	0.784862	+	0.619671i	$0.212734\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−3.85410	−0.861803
$21$	0	0
$22$	−2.38197	−0.507837
$23$	−7.70820	−1.60727	−0.803636	−	0.595121i	$-0.797104\pi$
$23$	−7.70820	−1.60727	−0.803636	+	0.595121i	$0.797104\pi$
$24$	0	0
$25$	9.85410	1.97082
$26$	0.763932	0.149819
$27$	0	0
$28$	1.00000	0.188982
$29$	2.14590	0.398483	0.199242	−	0.979950i	$-0.436152\pi$
$29$	2.14590	0.398483	0.199242	+	0.979950i	$0.436152\pi$
$30$	0	0
$31$	8.47214	1.52164	0.760820	−	0.648963i	$-0.224797\pi$
$31$	8.47214	1.52164	0.760820	+	0.648963i	$0.224797\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	6.47214	1.10996
$35$	−3.85410	−0.651462
$36$	0	0
$37$	−1.85410	−0.304812	−0.152406	−	0.988318i	$-0.548702\pi$
$37$	−1.85410	−0.304812	−0.152406	+	0.988318i	$0.548702\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	−3.85410	−0.609387
$41$	10.3262	1.61269	0.806344	−	0.591447i	$-0.201443\pi$
$41$	10.3262	1.61269	0.806344	+	0.591447i	$0.201443\pi$
$42$	0	0
$43$	−5.38197	−0.820742	−0.410371	−	0.911919i	$-0.634601\pi$
$43$	−5.38197	−0.820742	−0.410371	+	0.911919i	$0.634601\pi$
$44$	−2.38197	−0.359095
$45$	0	0
$46$	−7.70820	−1.13651
$47$	−2.85410	−0.416314	−0.208157	−	0.978095i	$-0.566746\pi$
$47$	−2.85410	−0.416314	−0.208157	+	0.978095i	$0.566746\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	9.85410	1.39358
$51$	0	0
$52$	0.763932	0.105938
$53$	13.6180	1.87058	0.935290	−	0.353881i	$-0.115138\pi$
$53$	13.6180	1.87058	0.935290	+	0.353881i	$0.115138\pi$
$54$	0	0
$55$	9.18034	1.23788
$56$	1.00000	0.133631
$57$	0	0
$58$	2.14590	0.281770
$59$	2.14590	0.279372	0.139686	−	0.990196i	$-0.455391\pi$
$59$	2.14590	0.279372	0.139686	+	0.990196i	$0.455391\pi$
$60$	0	0
$61$	13.3262	1.70625	0.853125	−	0.521707i	$-0.174704\pi$
$61$	13.3262	1.70625	0.853125	+	0.521707i	$0.174704\pi$
$62$	8.47214	1.07596
$63$	0	0
$64$	1.00000	0.125000
$65$	−2.94427	−0.365192
$66$	0	0
$67$	−6.94427	−0.848378	−0.424189	−	0.905574i	$-0.639441\pi$
$67$	−6.94427	−0.848378	−0.424189	+	0.905574i	$0.639441\pi$
$68$	6.47214	0.784862
$69$	0	0
$70$	−3.85410	−0.460653
$71$	13.0344	1.54690	0.773452	−	0.633855i	$-0.218528\pi$
$71$	13.0344	1.54690	0.773452	+	0.633855i	$0.218528\pi$
$72$	0	0
$73$	−2.76393	−0.323494	−0.161747	−	0.986832i	$-0.551713\pi$
$73$	−2.76393	−0.323494	−0.161747	+	0.986832i	$0.551713\pi$
$74$	−1.85410	−0.215535
$75$	0	0
$76$	−1.00000	−0.114708
$77$	−2.38197	−0.271450
$78$	0	0
$79$	2.32624	0.261722	0.130861	−	0.991401i	$-0.458226\pi$
$79$	2.32624	0.261722	0.130861	+	0.991401i	$0.458226\pi$
$80$	−3.85410	−0.430902
$81$	0	0
$82$	10.3262	1.14034
$83$	8.94427	0.981761	0.490881	−	0.871227i	$-0.336675\pi$
$83$	8.94427	0.981761	0.490881	+	0.871227i	$0.336675\pi$
$84$	0	0
$85$	−24.9443	−2.70559
$86$	−5.38197	−0.580352
$87$	0	0
$88$	−2.38197	−0.253918
$89$	3.61803	0.383511	0.191755	−	0.981443i	$-0.438582\pi$
$89$	3.61803	0.383511	0.191755	+	0.981443i	$0.438582\pi$
$90$	0	0
$91$	0.763932	0.0800818
$92$	−7.70820	−0.803636
$93$	0	0
$94$	−2.85410	−0.294378
$95$	3.85410	0.395423
$96$	0	0
$97$	−2.85410	−0.289790	−0.144895	−	0.989447i	$-0.546284\pi$
$97$	−2.85410	−0.289790	−0.144895	+	0.989447i	$0.546284\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	9.85410	0.985410
$101$	2.94427	0.292966	0.146483	−	0.989213i	$-0.453205\pi$
$101$	2.94427	0.292966	0.146483	+	0.989213i	$0.453205\pi$
$102$	0	0
$103$	13.2361	1.30419	0.652094	−	0.758138i	$-0.273891\pi$
$103$	13.2361	1.30419	0.652094	+	0.758138i	$0.273891\pi$
$104$	0.763932	0.0749097
$105$	0	0
$106$	13.6180	1.32270
$107$	5.23607	0.506190	0.253095	−	0.967441i	$-0.418552\pi$
$107$	5.23607	0.506190	0.253095	+	0.967441i	$0.418552\pi$
$108$	0	0
$109$	7.32624	0.701726	0.350863	−	0.936427i	$-0.385888\pi$
$109$	7.32624	0.701726	0.350863	+	0.936427i	$0.385888\pi$
$110$	9.18034	0.875311
$111$	0	0
$112$	1.00000	0.0944911
$113$	−8.94427	−0.841406	−0.420703	−	0.907198i	$-0.638217\pi$
$113$	−8.94427	−0.841406	−0.420703	+	0.907198i	$0.638217\pi$
$114$	0	0
$115$	29.7082	2.77030
$116$	2.14590	0.199242
$117$	0	0
$118$	2.14590	0.197546
$119$	6.47214	0.593300
$120$	0	0
$121$	−5.32624	−0.484203
$122$	13.3262	1.20650
$123$	0	0
$124$	8.47214	0.760820
$125$	−18.7082	−1.67331
$126$	0	0
$127$	−13.6180	−1.20841	−0.604203	−	0.796831i	$-0.706508\pi$
$127$	−13.6180	−1.20841	−0.604203	+	0.796831i	$0.706508\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−2.94427	−0.258230
$131$	−16.1803	−1.41368	−0.706841	−	0.707372i	$-0.749881\pi$
$131$	−16.1803	−1.41368	−0.706841	+	0.707372i	$0.749881\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	−6.94427	−0.599894
$135$	0	0
$136$	6.47214	0.554981
$137$	12.6180	1.07803	0.539016	−	0.842296i	$-0.318796\pi$
$137$	12.6180	1.07803	0.539016	+	0.842296i	$0.318796\pi$
$138$	0	0
$139$	1.52786	0.129592	0.0647959	−	0.997899i	$-0.479360\pi$
$139$	1.52786	0.129592	0.0647959	+	0.997899i	$0.479360\pi$
$140$	−3.85410	−0.325731
$141$	0	0
$142$	13.0344	1.09383
$143$	−1.81966	−0.152168
$144$	0	0
$145$	−8.27051	−0.686828
$146$	−2.76393	−0.228745
$147$	0	0
$148$	−1.85410	−0.152406
$149$	−4.00000	−0.327693	−0.163846	−	0.986486i	$-0.552390\pi$
$149$	−4.00000	−0.327693	−0.163846	+	0.986486i	$0.552390\pi$
$150$	0	0
$151$	13.8885	1.13023	0.565117	−	0.825011i	$-0.308831\pi$
$151$	13.8885	1.13023	0.565117	+	0.825011i	$0.308831\pi$
$152$	−1.00000	−0.0811107
$153$	0	0
$154$	−2.38197	−0.191944
$155$	−32.6525	−2.62271
$156$	0	0
$157$	−14.0902	−1.12452	−0.562259	−	0.826961i	$-0.690068\pi$
$157$	−14.0902	−1.12452	−0.562259	+	0.826961i	$0.690068\pi$
$158$	2.32624	0.185066
$159$	0	0
$160$	−3.85410	−0.304694
$161$	−7.70820	−0.607492
$162$	0	0
$163$	−4.09017	−0.320367	−0.160183	−	0.987087i	$-0.551209\pi$
$163$	−4.09017	−0.320367	−0.160183	+	0.987087i	$0.551209\pi$
$164$	10.3262	0.806344
$165$	0	0
$166$	8.94427	0.694210
$167$	14.9443	1.15642	0.578211	−	0.815887i	$-0.303751\pi$
$167$	14.9443	1.15642	0.578211	+	0.815887i	$0.303751\pi$
$168$	0	0
$169$	−12.4164	−0.955108
$170$	−24.9443	−1.91314
$171$	0	0
$172$	−5.38197	−0.410371
$173$	4.29180	0.326299	0.163150	−	0.986601i	$-0.447835\pi$
$173$	4.29180	0.326299	0.163150	+	0.986601i	$0.447835\pi$
$174$	0	0
$175$	9.85410	0.744900
$176$	−2.38197	−0.179547
$177$	0	0
$178$	3.61803	0.271183
$179$	−14.7639	−1.10351	−0.551754	−	0.834007i	$-0.686041\pi$
$179$	−14.7639	−1.10351	−0.551754	+	0.834007i	$0.686041\pi$
$180$	0	0
$181$	18.1803	1.35133	0.675667	−	0.737207i	$-0.263856\pi$
$181$	18.1803	1.35133	0.675667	+	0.737207i	$0.263856\pi$
$182$	0.763932	0.0566264
$183$	0	0
$184$	−7.70820	−0.568256
$185$	7.14590	0.525377
$186$	0	0
$187$	−15.4164	−1.12736
$188$	−2.85410	−0.208157
$189$	0	0
$190$	3.85410	0.279606
$191$	8.65248	0.626071	0.313036	−	0.949741i	$-0.398654\pi$
$191$	8.65248	0.626071	0.313036	+	0.949741i	$0.398654\pi$
$192$	0	0
$193$	−0.291796	−0.0210039	−0.0105020	−	0.999945i	$-0.503343\pi$
$193$	−0.291796	−0.0210039	−0.0105020	+	0.999945i	$0.503343\pi$
$194$	−2.85410	−0.204913
$195$	0	0
$196$	1.00000	0.0714286
$197$	−6.94427	−0.494759	−0.247379	−	0.968919i	$-0.579569\pi$
$197$	−6.94427	−0.494759	−0.247379	+	0.968919i	$0.579569\pi$
$198$	0	0
$199$	−4.03444	−0.285994	−0.142997	−	0.989723i	$-0.545674\pi$
$199$	−4.03444	−0.285994	−0.142997	+	0.989723i	$0.545674\pi$
$200$	9.85410	0.696790
$201$	0	0
$202$	2.94427	0.207158
$203$	2.14590	0.150613
$204$	0	0
$205$	−39.7984	−2.77964
$206$	13.2361	0.922201
$207$	0	0
$208$	0.763932	0.0529692
$209$	2.38197	0.164764
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	13.6180	0.935290
$213$	0	0
$214$	5.23607	0.357930
$215$	20.7426	1.41464
$216$	0	0
$217$	8.47214	0.575126
$218$	7.32624	0.496195
$219$	0	0
$220$	9.18034	0.618938
$221$	4.94427	0.332588
$222$	0	0
$223$	18.6525	1.24906	0.624531	−	0.781000i	$-0.285290\pi$
$223$	18.6525	1.24906	0.624531	+	0.781000i	$0.285290\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	−8.94427	−0.594964
$227$	−20.9443	−1.39012	−0.695060	−	0.718952i	$-0.744622\pi$
$227$	−20.9443	−1.39012	−0.695060	+	0.718952i	$0.744622\pi$
$228$	0	0
$229$	−6.03444	−0.398767	−0.199384	−	0.979922i	$-0.563894\pi$
$229$	−6.03444	−0.398767	−0.199384	+	0.979922i	$0.563894\pi$
$230$	29.7082	1.95890
$231$	0	0
$232$	2.14590	0.140885
$233$	26.7426	1.75197	0.875984	−	0.482339i	$-0.160213\pi$
$233$	26.7426	1.75197	0.875984	+	0.482339i	$0.160213\pi$
$234$	0	0
$235$	11.0000	0.717561
$236$	2.14590	0.139686
$237$	0	0
$238$	6.47214	0.419526
$239$	−25.1246	−1.62518	−0.812588	−	0.582838i	$-0.801942\pi$
$239$	−25.1246	−1.62518	−0.812588	+	0.582838i	$0.801942\pi$
$240$	0	0
$241$	15.8541	1.02125	0.510626	−	0.859803i	$-0.329414\pi$
$241$	15.8541	1.02125	0.510626	+	0.859803i	$0.329414\pi$
$242$	−5.32624	−0.342384
$243$	0	0
$244$	13.3262	0.853125
$245$	−3.85410	−0.246230
$246$	0	0
$247$	−0.763932	−0.0486078
$248$	8.47214	0.537981
$249$	0	0
$250$	−18.7082	−1.18321
$251$	−8.76393	−0.553174	−0.276587	−	0.960989i	$-0.589203\pi$
$251$	−8.76393	−0.553174	−0.276587	+	0.960989i	$0.589203\pi$
$252$	0	0
$253$	18.3607	1.15433
$254$	−13.6180	−0.854471
$255$	0	0
$256$	1.00000	0.0625000
$257$	−22.9787	−1.43337	−0.716686	−	0.697396i	$-0.754342\pi$
$257$	−22.9787	−1.43337	−0.716686	+	0.697396i	$0.754342\pi$
$258$	0	0
$259$	−1.85410	−0.115208
$260$	−2.94427	−0.182596
$261$	0	0
$262$	−16.1803	−0.999625
$263$	4.76393	0.293757	0.146878	−	0.989155i	$-0.453077\pi$
$263$	4.76393	0.293757	0.146878	+	0.989155i	$0.453077\pi$
$264$	0	0
$265$	−52.4853	−3.22415
$266$	−1.00000	−0.0613139
$267$	0	0
$268$	−6.94427	−0.424189
$269$	−19.5279	−1.19063	−0.595317	−	0.803491i	$-0.702974\pi$
$269$	−19.5279	−1.19063	−0.595317	+	0.803491i	$0.702974\pi$
$270$	0	0
$271$	2.14590	0.130354	0.0651770	−	0.997874i	$-0.479239\pi$
$271$	2.14590	0.130354	0.0651770	+	0.997874i	$0.479239\pi$
$272$	6.47214	0.392431
$273$	0	0
$274$	12.6180	0.762283
$275$	−23.4721	−1.41542
$276$	0	0
$277$	−20.6525	−1.24089	−0.620444	−	0.784251i	$-0.713047\pi$
$277$	−20.6525	−1.24089	−0.620444	+	0.784251i	$0.713047\pi$
$278$	1.52786	0.0916352
$279$	0	0
$280$	−3.85410	−0.230327
$281$	−16.0000	−0.954480	−0.477240	−	0.878773i	$-0.658363\pi$
$281$	−16.0000	−0.954480	−0.477240	+	0.878773i	$0.658363\pi$
$282$	0	0
$283$	9.52786	0.566373	0.283186	−	0.959065i	$-0.408609\pi$
$283$	9.52786	0.566373	0.283186	+	0.959065i	$0.408609\pi$
$284$	13.0344	0.773452
$285$	0	0
$286$	−1.81966	−0.107599
$287$	10.3262	0.609539
$288$	0	0
$289$	24.8885	1.46403
$290$	−8.27051	−0.485661
$291$	0	0
$292$	−2.76393	−0.161747
$293$	−3.81966	−0.223147	−0.111573	−	0.993756i	$-0.535589\pi$
$293$	−3.81966	−0.223147	−0.111573	+	0.993756i	$0.535589\pi$
$294$	0	0
$295$	−8.27051	−0.481528
$296$	−1.85410	−0.107767
$297$	0	0
$298$	−4.00000	−0.231714
$299$	−5.88854	−0.340543
$300$	0	0
$301$	−5.38197	−0.310211
$302$	13.8885	0.799196
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	−51.3607	−2.94090
$306$	0	0
$307$	4.03444	0.230258	0.115129	−	0.993351i	$-0.463272\pi$
$307$	4.03444	0.230258	0.115129	+	0.993351i	$0.463272\pi$
$308$	−2.38197	−0.135725
$309$	0	0
$310$	−32.6525	−1.85454
$311$	−0.381966	−0.0216593	−0.0108297	−	0.999941i	$-0.503447\pi$
$311$	−0.381966	−0.0216593	−0.0108297	+	0.999941i	$0.503447\pi$
$312$	0	0
$313$	17.8885	1.01112	0.505560	−	0.862791i	$-0.331286\pi$
$313$	17.8885	1.01112	0.505560	+	0.862791i	$0.331286\pi$
$314$	−14.0902	−0.795154
$315$	0	0
$316$	2.32624	0.130861
$317$	18.3820	1.03243	0.516217	−	0.856458i	$-0.327340\pi$
$317$	18.3820	1.03243	0.516217	+	0.856458i	$0.327340\pi$
$318$	0	0
$319$	−5.11146	−0.286187
$320$	−3.85410	−0.215451
$321$	0	0
$322$	−7.70820	−0.429561
$323$	−6.47214	−0.360119
$324$	0	0
$325$	7.52786	0.417571
$326$	−4.09017	−0.226534
$327$	0	0
$328$	10.3262	0.570171
$329$	−2.85410	−0.157352
$330$	0	0
$331$	−29.4164	−1.61687	−0.808436	−	0.588584i	$-0.799686\pi$
$331$	−29.4164	−1.61687	−0.808436	+	0.588584i	$0.799686\pi$
$332$	8.94427	0.490881
$333$	0	0
$334$	14.9443	0.817714
$335$	26.7639	1.46227
$336$	0	0
$337$	−12.7639	−0.695296	−0.347648	−	0.937625i	$-0.613020\pi$
$337$	−12.7639	−0.695296	−0.347648	+	0.937625i	$0.613020\pi$
$338$	−12.4164	−0.675364
$339$	0	0
$340$	−24.9443	−1.35279
$341$	−20.1803	−1.09283
$342$	0	0
$343$	1.00000	0.0539949
$344$	−5.38197	−0.290176
$345$	0	0
$346$	4.29180	0.230728
$347$	−4.58359	−0.246060	−0.123030	−	0.992403i	$-0.539261\pi$
$347$	−4.58359	−0.246060	−0.123030	+	0.992403i	$0.539261\pi$
$348$	0	0
$349$	−6.00000	−0.321173	−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000	−0.321173	−0.160586	+	0.987022i	$0.551338\pi$
$350$	9.85410	0.526724
$351$	0	0
$352$	−2.38197	−0.126959
$353$	−36.3607	−1.93528	−0.967642	−	0.252328i	$-0.918804\pi$
$353$	−36.3607	−1.93528	−0.967642	+	0.252328i	$0.918804\pi$
$354$	0	0
$355$	−50.2361	−2.66625
$356$	3.61803	0.191755
$357$	0	0
$358$	−14.7639	−0.780298
$359$	−27.7082	−1.46238	−0.731192	−	0.682172i	$-0.761035\pi$
$359$	−27.7082	−1.46238	−0.731192	+	0.682172i	$0.761035\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	18.1803	0.955537
$363$	0	0
$364$	0.763932	0.0400409
$365$	10.6525	0.557576
$366$	0	0
$367$	13.3262	0.695624	0.347812	−	0.937564i	$-0.386925\pi$
$367$	13.3262	0.695624	0.347812	+	0.937564i	$0.386925\pi$
$368$	−7.70820	−0.401818
$369$	0	0
$370$	7.14590	0.371498
$371$	13.6180	0.707013
$372$	0	0
$373$	23.7426	1.22935	0.614674	−	0.788781i	$-0.289288\pi$
$373$	23.7426	1.22935	0.614674	+	0.788781i	$0.289288\pi$
$374$	−15.4164	−0.797163
$375$	0	0
$376$	−2.85410	−0.147189
$377$	1.63932	0.0844293
$378$	0	0
$379$	38.6525	1.98544	0.992722	−	0.120427i	$-0.0384264\pi$
$379$	38.6525	1.98544	0.992722	+	0.120427i	$0.0384264\pi$
$380$	3.85410	0.197711
$381$	0	0
$382$	8.65248	0.442699
$383$	25.2361	1.28950	0.644751	−	0.764392i	$-0.276961\pi$
$383$	25.2361	1.28950	0.644751	+	0.764392i	$0.276961\pi$
$384$	0	0
$385$	9.18034	0.467873
$386$	−0.291796	−0.0148520
$387$	0	0
$388$	−2.85410	−0.144895
$389$	22.3607	1.13373	0.566866	−	0.823810i	$-0.308156\pi$
$389$	22.3607	1.13373	0.566866	+	0.823810i	$0.308156\pi$
$390$	0	0
$391$	−49.8885	−2.52297
$392$	1.00000	0.0505076
$393$	0	0
$394$	−6.94427	−0.349847
$395$	−8.96556	−0.451106
$396$	0	0
$397$	9.90983	0.497360	0.248680	−	0.968586i	$-0.420003\pi$
$397$	9.90983	0.497360	0.248680	+	0.968586i	$0.420003\pi$
$398$	−4.03444	−0.202228
$399$	0	0
$400$	9.85410	0.492705
$401$	−28.9443	−1.44541	−0.722704	−	0.691158i	$-0.757101\pi$
$401$	−28.9443	−1.44541	−0.722704	+	0.691158i	$0.757101\pi$
$402$	0	0
$403$	6.47214	0.322400
$404$	2.94427	0.146483
$405$	0	0
$406$	2.14590	0.106499
$407$	4.41641	0.218913
$408$	0	0
$409$	−7.43769	−0.367770	−0.183885	−	0.982948i	$-0.558867\pi$
$409$	−7.43769	−0.367770	−0.183885	+	0.982948i	$0.558867\pi$
$410$	−39.7984	−1.96550
$411$	0	0
$412$	13.2361	0.652094
$413$	2.14590	0.105593
$414$	0	0
$415$	−34.4721	−1.69217
$416$	0.763932	0.0374548
$417$	0	0
$418$	2.38197	0.116506
$419$	9.41641	0.460022	0.230011	−	0.973188i	$-0.426124\pi$
$419$	9.41641	0.460022	0.230011	+	0.973188i	$0.426124\pi$
$420$	0	0
$421$	30.3607	1.47969	0.739844	−	0.672778i	$-0.234899\pi$
$421$	30.3607	1.47969	0.739844	+	0.672778i	$0.234899\pi$
$422$	−4.00000	−0.194717
$423$	0	0
$424$	13.6180	0.661350
$425$	63.7771	3.09364
$426$	0	0
$427$	13.3262	0.644902
$428$	5.23607	0.253095
$429$	0	0
$430$	20.7426	1.00030
$431$	28.2705	1.36174	0.680871	−	0.732403i	$-0.261601\pi$
$431$	28.2705	1.36174	0.680871	+	0.732403i	$0.261601\pi$
$432$	0	0
$433$	10.9098	0.524293	0.262146	−	0.965028i	$-0.415570\pi$
$433$	10.9098	0.524293	0.262146	+	0.965028i	$0.415570\pi$
$434$	8.47214	0.406676
$435$	0	0
$436$	7.32624	0.350863
$437$	7.70820	0.368733
$438$	0	0
$439$	−4.47214	−0.213443	−0.106722	−	0.994289i	$-0.534035\pi$
$439$	−4.47214	−0.213443	−0.106722	+	0.994289i	$0.534035\pi$
$440$	9.18034	0.437656
$441$	0	0
$442$	4.94427	0.235175
$443$	−16.7984	−0.798115	−0.399057	−	0.916926i	$-0.630663\pi$
$443$	−16.7984	−0.798115	−0.399057	+	0.916926i	$0.630663\pi$
$444$	0	0
$445$	−13.9443	−0.661022
$446$	18.6525	0.883220
$447$	0	0
$448$	1.00000	0.0472456
$449$	30.0689	1.41904	0.709519	−	0.704686i	$-0.248912\pi$
$449$	30.0689	1.41904	0.709519	+	0.704686i	$0.248912\pi$
$450$	0	0
$451$	−24.5967	−1.15822
$452$	−8.94427	−0.420703
$453$	0	0
$454$	−20.9443	−0.982963
$455$	−2.94427	−0.138030
$456$	0	0
$457$	15.5066	0.725367	0.362684	−	0.931912i	$-0.381861\pi$
$457$	15.5066	0.725367	0.362684	+	0.931912i	$0.381861\pi$
$458$	−6.03444	−0.281971
$459$	0	0
$460$	29.7082	1.38515
$461$	7.61803	0.354807	0.177404	−	0.984138i	$-0.443230\pi$
$461$	7.61803	0.354807	0.177404	+	0.984138i	$0.443230\pi$
$462$	0	0
$463$	−7.23607	−0.336289	−0.168144	−	0.985762i	$-0.553777\pi$
$463$	−7.23607	−0.336289	−0.168144	+	0.985762i	$0.553777\pi$
$464$	2.14590	0.0996208
$465$	0	0
$466$	26.7426	1.23883
$467$	−8.29180	−0.383699	−0.191849	−	0.981424i	$-0.561448\pi$
$467$	−8.29180	−0.383699	−0.191849	+	0.981424i	$0.561448\pi$
$468$	0	0
$469$	−6.94427	−0.320657
$470$	11.0000	0.507392
$471$	0	0
$472$	2.14590	0.0987730
$473$	12.8197	0.589449
$474$	0	0
$475$	−9.85410	−0.452137
$476$	6.47214	0.296650
$477$	0	0
$478$	−25.1246	−1.14917
$479$	33.2705	1.52017	0.760084	−	0.649825i	$-0.225158\pi$
$479$	33.2705	1.52017	0.760084	+	0.649825i	$0.225158\pi$
$480$	0	0
$481$	−1.41641	−0.0645826
$482$	15.8541	0.722135
$483$	0	0
$484$	−5.32624	−0.242102
$485$	11.0000	0.499484
$486$	0	0
$487$	16.4377	0.744863	0.372432	−	0.928060i	$-0.378524\pi$
$487$	16.4377	0.744863	0.372432	+	0.928060i	$0.378524\pi$
$488$	13.3262	0.603250
$489$	0	0
$490$	−3.85410	−0.174111
$491$	−37.8885	−1.70989	−0.854943	−	0.518722i	$-0.826408\pi$
$491$	−37.8885	−1.70989	−0.854943	+	0.518722i	$0.826408\pi$
$492$	0	0
$493$	13.8885	0.625509
$494$	−0.763932	−0.0343709
$495$	0	0
$496$	8.47214	0.380410
$497$	13.0344	0.584675
$498$	0	0
$499$	−9.67376	−0.433057	−0.216529	−	0.976276i	$-0.569473\pi$
$499$	−9.67376	−0.433057	−0.216529	+	0.976276i	$0.569473\pi$
$500$	−18.7082	−0.836656
$501$	0	0
$502$	−8.76393	−0.391153
$503$	0.909830	0.0405673	0.0202837	−	0.999794i	$-0.493543\pi$
$503$	0.909830	0.0405673	0.0202837	+	0.999794i	$0.493543\pi$
$504$	0	0
$505$	−11.3475	−0.504958
$506$	18.3607	0.816232
$507$	0	0
$508$	−13.6180	−0.604203
$509$	−28.2918	−1.25401	−0.627006	−	0.779015i	$-0.715720\pi$
$509$	−28.2918	−1.25401	−0.627006	+	0.779015i	$0.715720\pi$
$510$	0	0
$511$	−2.76393	−0.122269
$512$	1.00000	0.0441942
$513$	0	0
$514$	−22.9787	−1.01355
$515$	−51.0132	−2.24791
$516$	0	0
$517$	6.79837	0.298992
$518$	−1.85410	−0.0814646
$519$	0	0
$520$	−2.94427	−0.129115
$521$	3.52786	0.154559	0.0772793	−	0.997009i	$-0.475377\pi$
$521$	3.52786	0.154559	0.0772793	+	0.997009i	$0.475377\pi$
$522$	0	0
$523$	24.0000	1.04945	0.524723	−	0.851273i	$-0.324169\pi$
$523$	24.0000	1.04945	0.524723	+	0.851273i	$0.324169\pi$
$524$	−16.1803	−0.706841
$525$	0	0
$526$	4.76393	0.207717
$527$	54.8328	2.38855
$528$	0	0
$529$	36.4164	1.58332
$530$	−52.4853	−2.27982
$531$	0	0
$532$	−1.00000	−0.0433555
$533$	7.88854	0.341691
$534$	0	0
$535$	−20.1803	−0.872472
$536$	−6.94427	−0.299947
$537$	0	0
$538$	−19.5279	−0.841906
$539$	−2.38197	−0.102599
$540$	0	0
$541$	8.29180	0.356492	0.178246	−	0.983986i	$-0.442958\pi$
$541$	8.29180	0.356492	0.178246	+	0.983986i	$0.442958\pi$
$542$	2.14590	0.0921742
$543$	0	0
$544$	6.47214	0.277491
$545$	−28.2361	−1.20950
$546$	0	0
$547$	−9.23607	−0.394906	−0.197453	−	0.980312i	$-0.563267\pi$
$547$	−9.23607	−0.394906	−0.197453	+	0.980312i	$0.563267\pi$
$548$	12.6180	0.539016
$549$	0	0
$550$	−23.4721	−1.00086
$551$	−2.14590	−0.0914183
$552$	0	0
$553$	2.32624	0.0989217
$554$	−20.6525	−0.877440
$555$	0	0
$556$	1.52786	0.0647959
$557$	20.7639	0.879796	0.439898	−	0.898048i	$-0.355015\pi$
$557$	20.7639	0.879796	0.439898	+	0.898048i	$0.355015\pi$
$558$	0	0
$559$	−4.11146	−0.173896
$560$	−3.85410	−0.162866
$561$	0	0
$562$	−16.0000	−0.674919
$563$	6.85410	0.288866	0.144433	−	0.989515i	$-0.453864\pi$
$563$	6.85410	0.288866	0.144433	+	0.989515i	$0.453864\pi$
$564$	0	0
$565$	34.4721	1.45025
$566$	9.52786	0.400486
$567$	0	0
$568$	13.0344	0.546913
$569$	23.7082	0.993900	0.496950	−	0.867779i	$-0.334453\pi$
$569$	23.7082	0.993900	0.496950	+	0.867779i	$0.334453\pi$
$570$	0	0
$571$	1.03444	0.0432900	0.0216450	−	0.999766i	$-0.493110\pi$
$571$	1.03444	0.0432900	0.0216450	+	0.999766i	$0.493110\pi$
$572$	−1.81966	−0.0760838
$573$	0	0
$574$	10.3262	0.431009
$575$	−75.9574	−3.16764
$576$	0	0
$577$	−17.0557	−0.710039	−0.355020	−	0.934859i	$-0.615526\pi$
$577$	−17.0557	−0.710039	−0.355020	+	0.934859i	$0.615526\pi$
$578$	24.8885	1.03523
$579$	0	0
$580$	−8.27051	−0.343414
$581$	8.94427	0.371071
$582$	0	0
$583$	−32.4377	−1.34343
$584$	−2.76393	−0.114372
$585$	0	0
$586$	−3.81966	−0.157789
$587$	−6.76393	−0.279177	−0.139589	−	0.990210i	$-0.544578\pi$
$587$	−6.76393	−0.279177	−0.139589	+	0.990210i	$0.544578\pi$
$588$	0	0
$589$	−8.47214	−0.349088
$590$	−8.27051	−0.340492
$591$	0	0
$592$	−1.85410	−0.0762031
$593$	35.7082	1.46636	0.733180	−	0.680035i	$-0.238035\pi$
$593$	35.7082	1.46636	0.733180	+	0.680035i	$0.238035\pi$
$594$	0	0
$595$	−24.9443	−1.02262
$596$	−4.00000	−0.163846
$597$	0	0
$598$	−5.88854	−0.240800
$599$	2.56231	0.104693	0.0523465	−	0.998629i	$-0.483330\pi$
$599$	2.56231	0.104693	0.0523465	+	0.998629i	$0.483330\pi$
$600$	0	0
$601$	−18.0000	−0.734235	−0.367118	−	0.930175i	$-0.619655\pi$
$601$	−18.0000	−0.734235	−0.367118	+	0.930175i	$0.619655\pi$
$602$	−5.38197	−0.219353
$603$	0	0
$604$	13.8885	0.565117
$605$	20.5279	0.834576
$606$	0	0
$607$	−32.1803	−1.30616	−0.653080	−	0.757289i	$-0.726523\pi$
$607$	−32.1803	−1.30616	−0.653080	+	0.757289i	$0.726523\pi$
$608$	−1.00000	−0.0405554
$609$	0	0
$610$	−51.3607	−2.07953
$611$	−2.18034	−0.0882071
$612$	0	0
$613$	−31.7771	−1.28346	−0.641732	−	0.766929i	$-0.721784\pi$
$613$	−31.7771	−1.28346	−0.641732	+	0.766929i	$0.721784\pi$
$614$	4.03444	0.162817
$615$	0	0
$616$	−2.38197	−0.0959721
$617$	41.8541	1.68498	0.842491	−	0.538710i	$-0.181088\pi$
$617$	41.8541	1.68498	0.842491	+	0.538710i	$0.181088\pi$
$618$	0	0
$619$	−44.0689	−1.77128	−0.885639	−	0.464374i	$-0.846279\pi$
$619$	−44.0689	−1.77128	−0.885639	+	0.464374i	$0.846279\pi$
$620$	−32.6525	−1.31135
$621$	0	0
$622$	−0.381966	−0.0153154
$623$	3.61803	0.144953
$624$	0	0
$625$	22.8328	0.913313
$626$	17.8885	0.714970
$627$	0	0
$628$	−14.0902	−0.562259
$629$	−12.0000	−0.478471
$630$	0	0
$631$	−12.3607	−0.492071	−0.246035	−	0.969261i	$-0.579128\pi$
$631$	−12.3607	−0.492071	−0.246035	+	0.969261i	$0.579128\pi$
$632$	2.32624	0.0925328
$633$	0	0
$634$	18.3820	0.730041
$635$	52.4853	2.08282
$636$	0	0
$637$	0.763932	0.0302681
$638$	−5.11146	−0.202364
$639$	0	0
$640$	−3.85410	−0.152347
$641$	38.1803	1.50803	0.754016	−	0.656856i	$-0.228114\pi$
$641$	38.1803	1.50803	0.754016	+	0.656856i	$0.228114\pi$
$642$	0	0
$643$	1.52786	0.0602531	0.0301265	−	0.999546i	$-0.490409\pi$
$643$	1.52786	0.0602531	0.0301265	+	0.999546i	$0.490409\pi$
$644$	−7.70820	−0.303746
$645$	0	0
$646$	−6.47214	−0.254643
$647$	−48.3262	−1.89990	−0.949950	−	0.312401i	$-0.898867\pi$
$647$	−48.3262	−1.89990	−0.949950	+	0.312401i	$0.898867\pi$
$648$	0	0
$649$	−5.11146	−0.200642
$650$	7.52786	0.295267
$651$	0	0
$652$	−4.09017	−0.160183
$653$	−25.7082	−1.00604	−0.503020	−	0.864275i	$-0.667778\pi$
$653$	−25.7082	−1.00604	−0.503020	+	0.864275i	$0.667778\pi$
$654$	0	0
$655$	62.3607	2.43663
$656$	10.3262	0.403172
$657$	0	0
$658$	−2.85410	−0.111264
$659$	29.8885	1.16429	0.582146	−	0.813084i	$-0.302213\pi$
$659$	29.8885	1.16429	0.582146	+	0.813084i	$0.302213\pi$
$660$	0	0
$661$	31.8885	1.24032	0.620160	−	0.784475i	$-0.287068\pi$
$661$	31.8885	1.24032	0.620160	+	0.784475i	$0.287068\pi$
$662$	−29.4164	−1.14330
$663$	0	0
$664$	8.94427	0.347105
$665$	3.85410	0.149456
$666$	0	0
$667$	−16.5410	−0.640471
$668$	14.9443	0.578211
$669$	0	0
$670$	26.7639	1.03398
$671$	−31.7426	−1.22541
$672$	0	0
$673$	−6.65248	−0.256434	−0.128217	−	0.991746i	$-0.540925\pi$
$673$	−6.65248	−0.256434	−0.128217	+	0.991746i	$0.540925\pi$
$674$	−12.7639	−0.491648
$675$	0	0
$676$	−12.4164	−0.477554
$677$	−15.8885	−0.610646	−0.305323	−	0.952249i	$-0.598764\pi$
$677$	−15.8885	−0.610646	−0.305323	+	0.952249i	$0.598764\pi$
$678$	0	0
$679$	−2.85410	−0.109530
$680$	−24.9443	−0.956569
$681$	0	0
$682$	−20.1803	−0.772745
$683$	34.0000	1.30097	0.650487	−	0.759517i	$-0.274565\pi$
$683$	34.0000	1.30097	0.650487	+	0.759517i	$0.274565\pi$
$684$	0	0
$685$	−48.6312	−1.85810
$686$	1.00000	0.0381802
$687$	0	0
$688$	−5.38197	−0.205186
$689$	10.4033	0.396332
$690$	0	0
$691$	−3.12461	−0.118866	−0.0594329	−	0.998232i	$-0.518929\pi$
$691$	−3.12461	−0.118866	−0.0594329	+	0.998232i	$0.518929\pi$
$692$	4.29180	0.163150
$693$	0	0
$694$	−4.58359	−0.173991
$695$	−5.88854	−0.223365
$696$	0	0
$697$	66.8328	2.53147
$698$	−6.00000	−0.227103
$699$	0	0
$700$	9.85410	0.372450
$701$	−28.6525	−1.08219	−0.541095	−	0.840962i	$-0.681990\pi$
$701$	−28.6525	−1.08219	−0.541095	+	0.840962i	$0.681990\pi$
$702$	0	0
$703$	1.85410	0.0699288
$704$	−2.38197	−0.0897737
$705$	0	0
$706$	−36.3607	−1.36845
$707$	2.94427	0.110731
$708$	0	0
$709$	5.70820	0.214376	0.107188	−	0.994239i	$-0.465815\pi$
$709$	5.70820	0.214376	0.107188	+	0.994239i	$0.465815\pi$
$710$	−50.2361	−1.88533
$711$	0	0
$712$	3.61803	0.135592
$713$	−65.3050	−2.44569
$714$	0	0
$715$	7.01316	0.262277
$716$	−14.7639	−0.551754
$717$	0	0
$718$	−27.7082	−1.03406
$719$	−6.11146	−0.227919	−0.113959	−	0.993485i	$-0.536353\pi$
$719$	−6.11146	−0.227919	−0.113959	+	0.993485i	$0.536353\pi$
$720$	0	0
$721$	13.2361	0.492937
$722$	1.00000	0.0372161
$723$	0	0
$724$	18.1803	0.675667
$725$	21.1459	0.785339
$726$	0	0
$727$	−25.5066	−0.945987	−0.472993	−	0.881066i	$-0.656827\pi$
$727$	−25.5066	−0.945987	−0.472993	+	0.881066i	$0.656827\pi$
$728$	0.763932	0.0283132
$729$	0	0
$730$	10.6525	0.394266
$731$	−34.8328	−1.28834
$732$	0	0
$733$	7.85410	0.290098	0.145049	−	0.989424i	$-0.453666\pi$
$733$	7.85410	0.290098	0.145049	+	0.989424i	$0.453666\pi$
$734$	13.3262	0.491880
$735$	0	0
$736$	−7.70820	−0.284128
$737$	16.5410	0.609296
$738$	0	0
$739$	−0.798374	−0.0293687	−0.0146843	−	0.999892i	$-0.504674\pi$
$739$	−0.798374	−0.0293687	−0.0146843	+	0.999892i	$0.504674\pi$
$740$	7.14590	0.262688
$741$	0	0
$742$	13.6180	0.499934
$743$	−29.6738	−1.08862	−0.544312	−	0.838883i	$-0.683209\pi$
$743$	−29.6738	−1.08862	−0.544312	+	0.838883i	$0.683209\pi$
$744$	0	0
$745$	15.4164	0.564813
$746$	23.7426	0.869280
$747$	0	0
$748$	−15.4164	−0.563680
$749$	5.23607	0.191322
$750$	0	0
$751$	−9.38197	−0.342353	−0.171176	−	0.985240i	$-0.554757\pi$
$751$	−9.38197	−0.342353	−0.171176	+	0.985240i	$0.554757\pi$
$752$	−2.85410	−0.104078
$753$	0	0
$754$	1.63932	0.0597005
$755$	−53.5279	−1.94808
$756$	0	0
$757$	−35.4164	−1.28723	−0.643616	−	0.765349i	$-0.722566\pi$
$757$	−35.4164	−1.28723	−0.643616	+	0.765349i	$0.722566\pi$
$758$	38.6525	1.40392
$759$	0	0
$760$	3.85410	0.139803
$761$	12.1115	0.439040	0.219520	−	0.975608i	$-0.429551\pi$
$761$	12.1115	0.439040	0.219520	+	0.975608i	$0.429551\pi$
$762$	0	0
$763$	7.32624	0.265228
$764$	8.65248	0.313036
$765$	0	0
$766$	25.2361	0.911816
$767$	1.63932	0.0591924
$768$	0	0
$769$	−9.81966	−0.354106	−0.177053	−	0.984201i	$-0.556656\pi$
$769$	−9.81966	−0.354106	−0.177053	+	0.984201i	$0.556656\pi$
$770$	9.18034	0.330836
$771$	0	0
$772$	−0.291796	−0.0105020
$773$	−31.7082	−1.14046	−0.570232	−	0.821483i	$-0.693147\pi$
$773$	−31.7082	−1.14046	−0.570232	+	0.821483i	$0.693147\pi$
$774$	0	0
$775$	83.4853	2.99888
$776$	−2.85410	−0.102456
$777$	0	0
$778$	22.3607	0.801669
$779$	−10.3262	−0.369976
$780$	0	0
$781$	−31.0476	−1.11097
$782$	−49.8885	−1.78401
$783$	0	0
$784$	1.00000	0.0357143
$785$	54.3050	1.93823
$786$	0	0
$787$	39.4508	1.40627	0.703135	−	0.711056i	$-0.251783\pi$
$787$	39.4508	1.40627	0.703135	+	0.711056i	$0.251783\pi$
$788$	−6.94427	−0.247379
$789$	0	0
$790$	−8.96556	−0.318980
$791$	−8.94427	−0.318022
$792$	0	0
$793$	10.1803	0.361514
$794$	9.90983	0.351687
$795$	0	0
$796$	−4.03444	−0.142997
$797$	−41.9574	−1.48621	−0.743104	−	0.669176i	$-0.766647\pi$
$797$	−41.9574	−1.48621	−0.743104	+	0.669176i	$0.766647\pi$
$798$	0	0
$799$	−18.4721	−0.653497
$800$	9.85410	0.348395
$801$	0	0
$802$	−28.9443	−1.02206
$803$	6.58359	0.232330
$804$	0	0
$805$	29.7082	1.04708
$806$	6.47214	0.227971
$807$	0	0
$808$	2.94427	0.103579
$809$	−15.7426	−0.553482	−0.276741	−	0.960945i	$-0.589254\pi$
$809$	−15.7426	−0.553482	−0.276741	+	0.960945i	$0.589254\pi$
$810$	0	0
$811$	43.6180	1.53164	0.765818	−	0.643057i	$-0.222334\pi$
$811$	43.6180	1.53164	0.765818	+	0.643057i	$0.222334\pi$
$812$	2.14590	0.0753063
$813$	0	0
$814$	4.41641	0.154795
$815$	15.7639	0.552186
$816$	0	0
$817$	5.38197	0.188291
$818$	−7.43769	−0.260053
$819$	0	0
$820$	−39.7984	−1.38982
$821$	9.41641	0.328635	0.164317	−	0.986408i	$-0.447458\pi$
$821$	9.41641	0.328635	0.164317	+	0.986408i	$0.447458\pi$
$822$	0	0
$823$	−14.3607	−0.500582	−0.250291	−	0.968171i	$-0.580526\pi$
$823$	−14.3607	−0.500582	−0.250291	+	0.968171i	$0.580526\pi$
$824$	13.2361	0.461100
$825$	0	0
$826$	2.14590	0.0746653
$827$	−9.81966	−0.341463	−0.170732	−	0.985318i	$-0.554613\pi$
$827$	−9.81966	−0.341463	−0.170732	+	0.985318i	$0.554613\pi$
$828$	0	0
$829$	27.4164	0.952211	0.476106	−	0.879388i	$-0.342048\pi$
$829$	27.4164	0.952211	0.476106	+	0.879388i	$0.342048\pi$
$830$	−34.4721	−1.19655
$831$	0	0
$832$	0.763932	0.0264846
$833$	6.47214	0.224246
$834$	0	0
$835$	−57.5967	−1.99322
$836$	2.38197	0.0823820
$837$	0	0
$838$	9.41641	0.325284
$839$	−39.8885	−1.37711	−0.688553	−	0.725186i	$-0.741754\pi$
$839$	−39.8885	−1.37711	−0.688553	+	0.725186i	$0.741754\pi$
$840$	0	0
$841$	−24.3951	−0.841211
$842$	30.3607	1.04630
$843$	0	0
$844$	−4.00000	−0.137686
$845$	47.8541	1.64623
$846$	0	0
$847$	−5.32624	−0.183012
$848$	13.6180	0.467645
$849$	0	0
$850$	63.7771	2.18754
$851$	14.2918	0.489916
$852$	0	0
$853$	−25.1591	−0.861430	−0.430715	−	0.902488i	$-0.641739\pi$
$853$	−25.1591	−0.861430	−0.430715	+	0.902488i	$0.641739\pi$
$854$	13.3262	0.456014
$855$	0	0
$856$	5.23607	0.178965
$857$	23.8885	0.816017	0.408009	−	0.912978i	$-0.366223\pi$
$857$	23.8885	0.816017	0.408009	+	0.912978i	$0.366223\pi$
$858$	0	0
$859$	−31.1246	−1.06196	−0.530979	−	0.847385i	$-0.678176\pi$
$859$	−31.1246	−1.06196	−0.530979	+	0.847385i	$0.678176\pi$
$860$	20.7426	0.707318
$861$	0	0
$862$	28.2705	0.962897
$863$	−22.0344	−0.750061	−0.375031	−	0.927012i	$-0.622368\pi$
$863$	−22.0344	−0.750061	−0.375031	+	0.927012i	$0.622368\pi$
$864$	0	0
$865$	−16.5410	−0.562412
$866$	10.9098	0.370731
$867$	0	0
$868$	8.47214	0.287563
$869$	−5.54102	−0.187966
$870$	0	0
$871$	−5.30495	−0.179751
$872$	7.32624	0.248098
$873$	0	0
$874$	7.70820	0.260734
$875$	−18.7082	−0.632453
$876$	0	0
$877$	8.27051	0.279275	0.139638	−	0.990203i	$-0.455406\pi$
$877$	8.27051	0.279275	0.139638	+	0.990203i	$0.455406\pi$
$878$	−4.47214	−0.150927
$879$	0	0
$880$	9.18034	0.309469
$881$	−7.23607	−0.243789	−0.121895	−	0.992543i	$-0.538897\pi$
$881$	−7.23607	−0.243789	−0.121895	+	0.992543i	$0.538897\pi$
$882$	0	0
$883$	−2.56231	−0.0862285	−0.0431142	−	0.999070i	$-0.513728\pi$
$883$	−2.56231	−0.0862285	−0.0431142	+	0.999070i	$0.513728\pi$
$884$	4.94427	0.166294
$885$	0	0
$886$	−16.7984	−0.564352
$887$	−32.2492	−1.08282	−0.541411	−	0.840758i	$-0.682110\pi$
$887$	−32.2492	−1.08282	−0.541411	+	0.840758i	$0.682110\pi$
$888$	0	0
$889$	−13.6180	−0.456734
$890$	−13.9443	−0.467413
$891$	0	0
$892$	18.6525	0.624531
$893$	2.85410	0.0955089
$894$	0	0
$895$	56.9017	1.90201
$896$	1.00000	0.0334077
$897$	0	0
$898$	30.0689	1.00341
$899$	18.1803	0.606348
$900$	0	0
$901$	88.1378	2.93629
$902$	−24.5967	−0.818982
$903$	0	0
$904$	−8.94427	−0.297482
$905$	−70.0689	−2.32917
$906$	0	0
$907$	−26.0689	−0.865603	−0.432802	−	0.901489i	$-0.642475\pi$
$907$	−26.0689	−0.865603	−0.432802	+	0.901489i	$0.642475\pi$
$908$	−20.9443	−0.695060
$909$	0	0
$910$	−2.94427	−0.0976017
$911$	40.2148	1.33238	0.666188	−	0.745784i	$-0.267925\pi$
$911$	40.2148	1.33238	0.666188	+	0.745784i	$0.267925\pi$
$912$	0	0
$913$	−21.3050	−0.705091
$914$	15.5066	0.512912
$915$	0	0
$916$	−6.03444	−0.199384
$917$	−16.1803	−0.534322
$918$	0	0
$919$	29.2361	0.964409	0.482204	−	0.876059i	$-0.339836\pi$
$919$	29.2361	0.964409	0.482204	+	0.876059i	$0.339836\pi$
$920$	29.7082	0.979450
$921$	0	0
$922$	7.61803	0.250887
$923$	9.95743	0.327753
$924$	0	0
$925$	−18.2705	−0.600731
$926$	−7.23607	−0.237792
$927$	0	0
$928$	2.14590	0.0704426
$929$	−13.7082	−0.449752	−0.224876	−	0.974387i	$-0.572198\pi$
$929$	−13.7082	−0.449752	−0.224876	+	0.974387i	$0.572198\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	26.7426	0.875984
$933$	0	0
$934$	−8.29180	−0.271316
$935$	59.4164	1.94312
$936$	0	0
$937$	−6.83282	−0.223218	−0.111609	−	0.993752i	$-0.535600\pi$
$937$	−6.83282	−0.223218	−0.111609	+	0.993752i	$0.535600\pi$
$938$	−6.94427	−0.226739
$939$	0	0
$940$	11.0000	0.358780
$941$	50.0689	1.63220	0.816099	−	0.577911i	$-0.196132\pi$
$941$	50.0689	1.63220	0.816099	+	0.577911i	$0.196132\pi$
$942$	0	0
$943$	−79.5967	−2.59203
$944$	2.14590	0.0698430
$945$	0	0
$946$	12.8197	0.416803
$947$	−21.6180	−0.702492	−0.351246	−	0.936283i	$-0.614242\pi$
$947$	−21.6180	−0.702492	−0.351246	+	0.936283i	$0.614242\pi$
$948$	0	0
$949$	−2.11146	−0.0685408
$950$	−9.85410	−0.319709
$951$	0	0
$952$	6.47214	0.209763
$953$	0.472136	0.0152940	0.00764699	−	0.999971i	$-0.497566\pi$
$953$	0.472136	0.0152940	0.00764699	+	0.999971i	$0.497566\pi$
$954$	0	0
$955$	−33.3475	−1.07910
$956$	−25.1246	−0.812588
$957$	0	0
$958$	33.2705	1.07492
$959$	12.6180	0.407458
$960$	0	0
$961$	40.7771	1.31539
$962$	−1.41641	−0.0456668
$963$	0	0
$964$	15.8541	0.510626
$965$	1.12461	0.0362025
$966$	0	0
$967$	53.3050	1.71417	0.857086	−	0.515174i	$-0.172273\pi$
$967$	53.3050	1.71417	0.857086	+	0.515174i	$0.172273\pi$
$968$	−5.32624	−0.171192
$969$	0	0
$970$	11.0000	0.353189
$971$	53.5066	1.71711	0.858554	−	0.512723i	$-0.171363\pi$
$971$	53.5066	1.71711	0.858554	+	0.512723i	$0.171363\pi$
$972$	0	0
$973$	1.52786	0.0489811
$974$	16.4377	0.526698
$975$	0	0
$976$	13.3262	0.426562
$977$	−33.2361	−1.06332	−0.531658	−	0.846959i	$-0.678431\pi$
$977$	−33.2361	−1.06332	−0.531658	+	0.846959i	$0.678431\pi$
$978$	0	0
$979$	−8.61803	−0.275434
$980$	−3.85410	−0.123115
$981$	0	0
$982$	−37.8885	−1.20907
$983$	16.9443	0.540438	0.270219	−	0.962799i	$-0.412904\pi$
$983$	16.9443	0.540438	0.270219	+	0.962799i	$0.412904\pi$
$984$	0	0
$985$	26.7639	0.852770
$986$	13.8885	0.442301
$987$	0	0
$988$	−0.763932	−0.0243039
$989$	41.4853	1.31916
$990$	0	0
$991$	7.68692	0.244183	0.122091	−	0.992519i	$-0.461040\pi$
$991$	7.68692	0.244183	0.122091	+	0.992519i	$0.461040\pi$
$992$	8.47214	0.268991
$993$	0	0
$994$	13.0344	0.413427
$995$	15.5492	0.492941
$996$	0	0
$997$	29.0902	0.921295	0.460647	−	0.887583i	$-0.347617\pi$
$997$	29.0902	0.921295	0.460647	+	0.887583i	$0.347617\pi$
$998$	−9.67376	−0.306218
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2394.2.a.w.1.1		2
3.2	odd	2		266.2.a.b.1.1	✓	2
12.11	even	2		2128.2.a.b.1.2		2
15.14	odd	2		6650.2.a.bq.1.2		2
21.20	even	2		1862.2.a.g.1.2		2
24.5	odd	2		8512.2.a.h.1.2		2
24.11	even	2		8512.2.a.bc.1.1		2
57.56	even	2		5054.2.a.k.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
266.2.a.b.1.1	✓	2	3.2	odd	2
1862.2.a.g.1.2		2	21.20	even	2
2128.2.a.b.1.2		2	12.11	even	2
2394.2.a.w.1.1		2	1.1	even	1	trivial
5054.2.a.k.1.2		2	57.56	even	2
6650.2.a.bq.1.2		2	15.14	odd	2
8512.2.a.h.1.2		2	24.5	odd	2
8512.2.a.bc.1.1		2	24.11	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:	\( N \)	\(=\)	\( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2394.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -3.85410 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -3.85410 q^{5} +1.00000 q^{7} +1.00000 q^{8} -3.85410 q^{10} -2.38197 q^{11} +0.763932 q^{13} +1.00000 q^{14} +1.00000 q^{16} +6.47214 q^{17} -1.00000 q^{19} -3.85410 q^{20} -2.38197 q^{22} -7.70820 q^{23} +9.85410 q^{25} +0.763932 q^{26} +1.00000 q^{28} +2.14590 q^{29} +8.47214 q^{31} +1.00000 q^{32} +6.47214 q^{34} -3.85410 q^{35} -1.85410 q^{37} -1.00000 q^{38} -3.85410 q^{40} +10.3262 q^{41} -5.38197 q^{43} -2.38197 q^{44} -7.70820 q^{46} -2.85410 q^{47} +1.00000 q^{49} +9.85410 q^{50} +0.763932 q^{52} +13.6180 q^{53} +9.18034 q^{55} +1.00000 q^{56} +2.14590 q^{58} +2.14590 q^{59} +13.3262 q^{61} +8.47214 q^{62} +1.00000 q^{64} -2.94427 q^{65} -6.94427 q^{67} +6.47214 q^{68} -3.85410 q^{70} +13.0344 q^{71} -2.76393 q^{73} -1.85410 q^{74} -1.00000 q^{76} -2.38197 q^{77} +2.32624 q^{79} -3.85410 q^{80} +10.3262 q^{82} +8.94427 q^{83} -24.9443 q^{85} -5.38197 q^{86} -2.38197 q^{88} +3.61803 q^{89} +0.763932 q^{91} -7.70820 q^{92} -2.85410 q^{94} +3.85410 q^{95} -2.85410 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - q^{5} + 2 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 - q^5 + 2 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} - q^{5} + 2 q^{7} + 2 q^{8} - q^{10} - 7 q^{11} + 6 q^{13} + 2 q^{14} + 2 q^{16} + 4 q^{17} - 2 q^{19} - q^{20} - 7 q^{22} - 2 q^{23} + 13 q^{25} + 6 q^{26} + 2 q^{28} + 11 q^{29} + 8 q^{31} + 2 q^{32} + 4 q^{34} - q^{35} + 3 q^{37} - 2 q^{38} - q^{40} + 5 q^{41} - 13 q^{43} - 7 q^{44} - 2 q^{46} + q^{47} + 2 q^{49} + 13 q^{50} + 6 q^{52} + 25 q^{53} - 4 q^{55} + 2 q^{56} + 11 q^{58} + 11 q^{59} + 11 q^{61} + 8 q^{62} + 2 q^{64} + 12 q^{65} + 4 q^{67} + 4 q^{68} - q^{70} - 3 q^{71} - 10 q^{73} + 3 q^{74} - 2 q^{76} - 7 q^{77} - 11 q^{79} - q^{80} + 5 q^{82} - 32 q^{85} - 13 q^{86} - 7 q^{88} + 5 q^{89} + 6 q^{91} - 2 q^{92} + q^{94} + q^{95} + q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - q^5 + 2 * q^7 + 2 * q^8 - q^10 - 7 * q^11 + 6 * q^13 + 2 * q^14 + 2 * q^16 + 4 * q^17 - 2 * q^19 - q^20 - 7 * q^22 - 2 * q^23 + 13 * q^25 + 6 * q^26 + 2 * q^28 + 11 * q^29 + 8 * q^31 + 2 * q^32 + 4 * q^34 - q^35 + 3 * q^37 - 2 * q^38 - q^40 + 5 * q^41 - 13 * q^43 - 7 * q^44 - 2 * q^46 + q^47 + 2 * q^49 + 13 * q^50 + 6 * q^52 + 25 * q^53 - 4 * q^55 + 2 * q^56 + 11 * q^58 + 11 * q^59 + 11 * q^61 + 8 * q^62 + 2 * q^64 + 12 * q^65 + 4 * q^67 + 4 * q^68 - q^70 - 3 * q^71 - 10 * q^73 + 3 * q^74 - 2 * q^76 - 7 * q^77 - 11 * q^79 - q^80 + 5 * q^82 - 32 * q^85 - 13 * q^86 - 7 * q^88 + 5 * q^89 + 6 * q^91 - 2 * q^92 + q^94 + q^95 + q^97 + 2 * q^98

Introduction

L-functions

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