LMFDB - Embedded newform 2394.2.a.w.1.2

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			1.2
Root			$-0.618034$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +2.85410 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +2.85410 q^{5} +1.00000 q^{7} +1.00000 q^{8} +2.85410 q^{10} -4.61803 q^{11} +5.23607 q^{13} +1.00000 q^{14} +1.00000 q^{16} -2.47214 q^{17} -1.00000 q^{19} +2.85410 q^{20} -4.61803 q^{22} +5.70820 q^{23} +3.14590 q^{25} +5.23607 q^{26} +1.00000 q^{28} +8.85410 q^{29} -0.472136 q^{31} +1.00000 q^{32} -2.47214 q^{34} +2.85410 q^{35} +4.85410 q^{37} -1.00000 q^{38} +2.85410 q^{40} -5.32624 q^{41} -7.61803 q^{43} -4.61803 q^{44} +5.70820 q^{46} +3.85410 q^{47} +1.00000 q^{49} +3.14590 q^{50} +5.23607 q^{52} +11.3820 q^{53} -13.1803 q^{55} +1.00000 q^{56} +8.85410 q^{58} +8.85410 q^{59} -2.32624 q^{61} -0.472136 q^{62} +1.00000 q^{64} +14.9443 q^{65} +10.9443 q^{67} -2.47214 q^{68} +2.85410 q^{70} -16.0344 q^{71} -7.23607 q^{73} +4.85410 q^{74} -1.00000 q^{76} -4.61803 q^{77} -13.3262 q^{79} +2.85410 q^{80} -5.32624 q^{82} -8.94427 q^{83} -7.05573 q^{85} -7.61803 q^{86} -4.61803 q^{88} +1.38197 q^{89} +5.23607 q^{91} +5.70820 q^{92} +3.85410 q^{94} -2.85410 q^{95} +3.85410 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	2.85410	1.27639	0.638197	−	0.769873i	$-0.279681\pi$
$5$	2.85410	1.27639	0.638197	+	0.769873i	$0.279681\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	0	0
$10$	2.85410	0.902546
$11$	−4.61803	−1.39239	−0.696195	−	0.717853i	$-0.745125\pi$
$11$	−4.61803	−1.39239	−0.696195	+	0.717853i	$0.745125\pi$
$12$	0	0
$13$	5.23607	1.45222	0.726112	−	0.687576i	$-0.241325\pi$
$13$	5.23607	1.45222	0.726112	+	0.687576i	$0.241325\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.47214	−0.599581	−0.299791	−	0.954005i	$-0.596917\pi$
$17$	−2.47214	−0.599581	−0.299791	+	0.954005i	$0.596917\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	2.85410	0.638197
$21$	0	0
$22$	−4.61803	−0.984568
$23$	5.70820	1.19024	0.595121	−	0.803636i	$-0.297104\pi$
$23$	5.70820	1.19024	0.595121	+	0.803636i	$0.297104\pi$
$24$	0	0
$25$	3.14590	0.629180
$26$	5.23607	1.02688
$27$	0	0
$28$	1.00000	0.188982
$29$	8.85410	1.64417	0.822083	−	0.569368i	$-0.192812\pi$
$29$	8.85410	1.64417	0.822083	+	0.569368i	$0.192812\pi$
$30$	0	0
$31$	−0.472136	−0.0847981	−0.0423991	−	0.999101i	$-0.513500\pi$
$31$	−0.472136	−0.0847981	−0.0423991	+	0.999101i	$0.513500\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−2.47214	−0.423968
$35$	2.85410	0.482431
$36$	0	0
$37$	4.85410	0.798009	0.399005	−	0.916949i	$-0.369356\pi$
$37$	4.85410	0.798009	0.399005	+	0.916949i	$0.369356\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	2.85410	0.451273
$41$	−5.32624	−0.831819	−0.415909	−	0.909406i	$-0.636537\pi$
$41$	−5.32624	−0.831819	−0.415909	+	0.909406i	$0.636537\pi$
$42$	0	0
$43$	−7.61803	−1.16174	−0.580870	−	0.813997i	$-0.697287\pi$
$43$	−7.61803	−1.16174	−0.580870	+	0.813997i	$0.697287\pi$
$44$	−4.61803	−0.696195
$45$	0	0
$46$	5.70820	0.841629
$47$	3.85410	0.562179	0.281089	−	0.959682i	$-0.409304\pi$
$47$	3.85410	0.562179	0.281089	+	0.959682i	$0.409304\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	3.14590	0.444897
$51$	0	0
$52$	5.23607	0.726112
$53$	11.3820	1.56343	0.781717	−	0.623634i	$-0.214344\pi$
$53$	11.3820	1.56343	0.781717	+	0.623634i	$0.214344\pi$
$54$	0	0
$55$	−13.1803	−1.77724
$56$	1.00000	0.133631
$57$	0	0
$58$	8.85410	1.16260
$59$	8.85410	1.15271	0.576353	−	0.817201i	$-0.304475\pi$
$59$	8.85410	1.15271	0.576353	+	0.817201i	$0.304475\pi$
$60$	0	0
$61$	−2.32624	−0.297844	−0.148922	−	0.988849i	$-0.547580\pi$
$61$	−2.32624	−0.297844	−0.148922	+	0.988849i	$0.547580\pi$
$62$	−0.472136	−0.0599613
$63$	0	0
$64$	1.00000	0.125000
$65$	14.9443	1.85361
$66$	0	0
$67$	10.9443	1.33706	0.668528	−	0.743687i	$-0.266925\pi$
$67$	10.9443	1.33706	0.668528	+	0.743687i	$0.266925\pi$
$68$	−2.47214	−0.299791
$69$	0	0
$70$	2.85410	0.341130
$71$	−16.0344	−1.90294	−0.951469	−	0.307744i	$-0.900426\pi$
$71$	−16.0344	−1.90294	−0.951469	+	0.307744i	$0.900426\pi$
$72$	0	0
$73$	−7.23607	−0.846918	−0.423459	−	0.905915i	$-0.639184\pi$
$73$	−7.23607	−0.846918	−0.423459	+	0.905915i	$0.639184\pi$
$74$	4.85410	0.564278
$75$	0	0
$76$	−1.00000	−0.114708
$77$	−4.61803	−0.526274
$78$	0	0
$79$	−13.3262	−1.49932	−0.749659	−	0.661824i	$-0.769783\pi$
$79$	−13.3262	−1.49932	−0.749659	+	0.661824i	$0.769783\pi$
$80$	2.85410	0.319098
$81$	0	0
$82$	−5.32624	−0.588185
$83$	−8.94427	−0.981761	−0.490881	−	0.871227i	$-0.663325\pi$
$83$	−8.94427	−0.981761	−0.490881	+	0.871227i	$0.663325\pi$
$84$	0	0
$85$	−7.05573	−0.765301
$86$	−7.61803	−0.821474
$87$	0	0
$88$	−4.61803	−0.492284
$89$	1.38197	0.146488	0.0732441	−	0.997314i	$-0.476665\pi$
$89$	1.38197	0.146488	0.0732441	+	0.997314i	$0.476665\pi$
$90$	0	0
$91$	5.23607	0.548889
$92$	5.70820	0.595121
$93$	0	0
$94$	3.85410	0.397520
$95$	−2.85410	−0.292825
$96$	0	0
$97$	3.85410	0.391325	0.195662	−	0.980671i	$-0.437314\pi$
$97$	3.85410	0.391325	0.195662	+	0.980671i	$0.437314\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	3.14590	0.314590
$101$	−14.9443	−1.48701	−0.743505	−	0.668730i	$-0.766838\pi$
$101$	−14.9443	−1.48701	−0.743505	+	0.668730i	$0.766838\pi$
$102$	0	0
$103$	8.76393	0.863536	0.431768	−	0.901985i	$-0.357890\pi$
$103$	8.76393	0.863536	0.431768	+	0.901985i	$0.357890\pi$
$104$	5.23607	0.513439
$105$	0	0
$106$	11.3820	1.10551
$107$	0.763932	0.0738521	0.0369260	−	0.999318i	$-0.488243\pi$
$107$	0.763932	0.0738521	0.0369260	+	0.999318i	$0.488243\pi$
$108$	0	0
$109$	−8.32624	−0.797509	−0.398754	−	0.917058i	$-0.630557\pi$
$109$	−8.32624	−0.797509	−0.398754	+	0.917058i	$0.630557\pi$
$110$	−13.1803	−1.25670
$111$	0	0
$112$	1.00000	0.0944911
$113$	8.94427	0.841406	0.420703	−	0.907198i	$-0.361783\pi$
$113$	8.94427	0.841406	0.420703	+	0.907198i	$0.361783\pi$
$114$	0	0
$115$	16.2918	1.51922
$116$	8.85410	0.822083
$117$	0	0
$118$	8.85410	0.815086
$119$	−2.47214	−0.226620
$120$	0	0
$121$	10.3262	0.938749
$122$	−2.32624	−0.210608
$123$	0	0
$124$	−0.472136	−0.0423991
$125$	−5.29180	−0.473313
$126$	0	0
$127$	−11.3820	−1.00999	−0.504993	−	0.863123i	$-0.668505\pi$
$127$	−11.3820	−1.00999	−0.504993	+	0.863123i	$0.668505\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	14.9443	1.31070
$131$	6.18034	0.539979	0.269989	−	0.962863i	$-0.412980\pi$
$131$	6.18034	0.539979	0.269989	+	0.962863i	$0.412980\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	10.9443	0.945441
$135$	0	0
$136$	−2.47214	−0.211984
$137$	10.3820	0.886991	0.443496	−	0.896277i	$-0.353738\pi$
$137$	10.3820	0.886991	0.443496	+	0.896277i	$0.353738\pi$
$138$	0	0
$139$	10.4721	0.888235	0.444117	−	0.895969i	$-0.353517\pi$
$139$	10.4721	0.888235	0.444117	+	0.895969i	$0.353517\pi$
$140$	2.85410	0.241216
$141$	0	0
$142$	−16.0344	−1.34558
$143$	−24.1803	−2.02206
$144$	0	0
$145$	25.2705	2.09860
$146$	−7.23607	−0.598861
$147$	0	0
$148$	4.85410	0.399005
$149$	−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
$151$	−21.8885	−1.78126	−0.890632	−	0.454724i	$-0.849738\pi$
$151$	−21.8885	−1.78126	−0.890632	+	0.454724i	$0.849738\pi$
$152$	−1.00000	−0.0811107
$153$	0	0
$154$	−4.61803	−0.372132
$155$	−1.34752	−0.108236
$156$	0	0
$157$	−2.90983	−0.232230	−0.116115	−	0.993236i	$-0.537044\pi$
$157$	−2.90983	−0.232230	−0.116115	+	0.993236i	$0.537044\pi$
$158$	−13.3262	−1.06018
$159$	0	0
$160$	2.85410	0.225637
$161$	5.70820	0.449869
$162$	0	0
$163$	7.09017	0.555345	0.277672	−	0.960676i	$-0.410437\pi$
$163$	7.09017	0.555345	0.277672	+	0.960676i	$0.410437\pi$
$164$	−5.32624	−0.415909
$165$	0	0
$166$	−8.94427	−0.694210
$167$	−2.94427	−0.227835	−0.113917	−	0.993490i	$-0.536340\pi$
$167$	−2.94427	−0.227835	−0.113917	+	0.993490i	$0.536340\pi$
$168$	0	0
$169$	14.4164	1.10895
$170$	−7.05573	−0.541150
$171$	0	0
$172$	−7.61803	−0.580870
$173$	17.7082	1.34633	0.673165	−	0.739492i	$-0.264934\pi$
$173$	17.7082	1.34633	0.673165	+	0.739492i	$0.264934\pi$
$174$	0	0
$175$	3.14590	0.237808
$176$	−4.61803	−0.348097
$177$	0	0
$178$	1.38197	0.103583
$179$	−19.2361	−1.43777	−0.718886	−	0.695128i	$-0.755348\pi$
$179$	−19.2361	−1.43777	−0.718886	+	0.695128i	$0.755348\pi$
$180$	0	0
$181$	−4.18034	−0.310722	−0.155361	−	0.987858i	$-0.549654\pi$
$181$	−4.18034	−0.310722	−0.155361	+	0.987858i	$0.549654\pi$
$182$	5.23607	0.388123
$183$	0	0
$184$	5.70820	0.420814
$185$	13.8541	1.01857
$186$	0	0
$187$	11.4164	0.834850
$188$	3.85410	0.281089
$189$	0	0
$190$	−2.85410	−0.207058
$191$	−22.6525	−1.63908	−0.819538	−	0.573025i	$-0.805770\pi$
$191$	−22.6525	−1.63908	−0.819538	+	0.573025i	$0.805770\pi$
$192$	0	0
$193$	−13.7082	−0.986738	−0.493369	−	0.869820i	$-0.664235\pi$
$193$	−13.7082	−0.986738	−0.493369	+	0.869820i	$0.664235\pi$
$194$	3.85410	0.276708
$195$	0	0
$196$	1.00000	0.0714286
$197$	10.9443	0.779747	0.389874	−	0.920868i	$-0.372519\pi$
$197$	10.9443	0.779747	0.389874	+	0.920868i	$0.372519\pi$
$198$	0	0
$199$	25.0344	1.77464	0.887322	−	0.461150i	$-0.152563\pi$
$199$	25.0344	1.77464	0.887322	+	0.461150i	$0.152563\pi$
$200$	3.14590	0.222449
$201$	0	0
$202$	−14.9443	−1.05148
$203$	8.85410	0.621436
$204$	0	0
$205$	−15.2016	−1.06173
$206$	8.76393	0.610612
$207$	0	0
$208$	5.23607	0.363056
$209$	4.61803	0.319436
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	11.3820	0.781717
$213$	0	0
$214$	0.763932	0.0522213
$215$	−21.7426	−1.48284
$216$	0	0
$217$	−0.472136	−0.0320507
$218$	−8.32624	−0.563924
$219$	0	0
$220$	−13.1803	−0.888618
$221$	−12.9443	−0.870726
$222$	0	0
$223$	−12.6525	−0.847272	−0.423636	−	0.905832i	$-0.639247\pi$
$223$	−12.6525	−0.847272	−0.423636	+	0.905832i	$0.639247\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	8.94427	0.594964
$227$	−3.05573	−0.202816	−0.101408	−	0.994845i	$-0.532335\pi$
$227$	−3.05573	−0.202816	−0.101408	+	0.994845i	$0.532335\pi$
$228$	0	0
$229$	23.0344	1.52216	0.761079	−	0.648659i	$-0.224670\pi$
$229$	23.0344	1.52216	0.761079	+	0.648659i	$0.224670\pi$
$230$	16.2918	1.07425
$231$	0	0
$232$	8.85410	0.581300
$233$	−15.7426	−1.03134	−0.515668	−	0.856789i	$-0.672456\pi$
$233$	−15.7426	−1.03134	−0.515668	+	0.856789i	$0.672456\pi$
$234$	0	0
$235$	11.0000	0.717561
$236$	8.85410	0.576353
$237$	0	0
$238$	−2.47214	−0.160245
$239$	15.1246	0.978330	0.489165	−	0.872191i	$-0.337302\pi$
$239$	15.1246	0.978330	0.489165	+	0.872191i	$0.337302\pi$
$240$	0	0
$241$	9.14590	0.589139	0.294570	−	0.955630i	$-0.404824\pi$
$241$	9.14590	0.589139	0.294570	+	0.955630i	$0.404824\pi$
$242$	10.3262	0.663796
$243$	0	0
$244$	−2.32624	−0.148922
$245$	2.85410	0.182342
$246$	0	0
$247$	−5.23607	−0.333163
$248$	−0.472136	−0.0299807
$249$	0	0
$250$	−5.29180	−0.334683
$251$	−13.2361	−0.835453	−0.417727	−	0.908573i	$-0.637173\pi$
$251$	−13.2361	−0.835453	−0.417727	+	0.908573i	$0.637173\pi$
$252$	0	0
$253$	−26.3607	−1.65728
$254$	−11.3820	−0.714168
$255$	0	0
$256$	1.00000	0.0625000
$257$	23.9787	1.49575	0.747876	−	0.663839i	$-0.231074\pi$
$257$	23.9787	1.49575	0.747876	+	0.663839i	$0.231074\pi$
$258$	0	0
$259$	4.85410	0.301619
$260$	14.9443	0.926804
$261$	0	0
$262$	6.18034	0.381823
$263$	9.23607	0.569520	0.284760	−	0.958599i	$-0.408086\pi$
$263$	9.23607	0.569520	0.284760	+	0.958599i	$0.408086\pi$
$264$	0	0
$265$	32.4853	1.99556
$266$	−1.00000	−0.0613139
$267$	0	0
$268$	10.9443	0.668528
$269$	−28.4721	−1.73598	−0.867988	−	0.496584i	$-0.834587\pi$
$269$	−28.4721	−1.73598	−0.867988	+	0.496584i	$0.834587\pi$
$270$	0	0
$271$	8.85410	0.537848	0.268924	−	0.963161i	$-0.413332\pi$
$271$	8.85410	0.537848	0.268924	+	0.963161i	$0.413332\pi$
$272$	−2.47214	−0.149895
$273$	0	0
$274$	10.3820	0.627198
$275$	−14.5279	−0.876063
$276$	0	0
$277$	10.6525	0.640045	0.320023	−	0.947410i	$-0.396309\pi$
$277$	10.6525	0.640045	0.320023	+	0.947410i	$0.396309\pi$
$278$	10.4721	0.628077
$279$	0	0
$280$	2.85410	0.170565
$281$	−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
$283$	18.4721	1.09805	0.549027	−	0.835804i	$-0.314998\pi$
$283$	18.4721	1.09805	0.549027	+	0.835804i	$0.314998\pi$
$284$	−16.0344	−0.951469
$285$	0	0
$286$	−24.1803	−1.42981
$287$	−5.32624	−0.314398
$288$	0	0
$289$	−10.8885	−0.640503
$290$	25.2705	1.48394
$291$	0	0
$292$	−7.23607	−0.423459
$293$	−26.1803	−1.52947	−0.764736	−	0.644344i	$-0.777131\pi$
$293$	−26.1803	−1.52947	−0.764736	+	0.644344i	$0.777131\pi$
$294$	0	0
$295$	25.2705	1.47131
$296$	4.85410	0.282139
$297$	0	0
$298$	−4.00000	−0.231714
$299$	29.8885	1.72850
$300$	0	0
$301$	−7.61803	−0.439096
$302$	−21.8885	−1.25954
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	−6.63932	−0.380166
$306$	0	0
$307$	−25.0344	−1.42879	−0.714396	−	0.699742i	$-0.753298\pi$
$307$	−25.0344	−1.42879	−0.714396	+	0.699742i	$0.753298\pi$
$308$	−4.61803	−0.263137
$309$	0	0
$310$	−1.34752	−0.0765342
$311$	−2.61803	−0.148455	−0.0742275	−	0.997241i	$-0.523649\pi$
$311$	−2.61803	−0.148455	−0.0742275	+	0.997241i	$0.523649\pi$
$312$	0	0
$313$	−17.8885	−1.01112	−0.505560	−	0.862791i	$-0.668714\pi$
$313$	−17.8885	−1.01112	−0.505560	+	0.862791i	$0.668714\pi$
$314$	−2.90983	−0.164211
$315$	0	0
$316$	−13.3262	−0.749659
$317$	20.6180	1.15802	0.579012	−	0.815319i	$-0.303438\pi$
$317$	20.6180	1.15802	0.579012	+	0.815319i	$0.303438\pi$
$318$	0	0
$319$	−40.8885	−2.28932
$320$	2.85410	0.159549
$321$	0	0
$322$	5.70820	0.318106
$323$	2.47214	0.137553
$324$	0	0
$325$	16.4721	0.913710
$326$	7.09017	0.392688
$327$	0	0
$328$	−5.32624	−0.294092
$329$	3.85410	0.212484
$330$	0	0
$331$	−2.58359	−0.142007	−0.0710035	−	0.997476i	$-0.522620\pi$
$331$	−2.58359	−0.142007	−0.0710035	+	0.997476i	$0.522620\pi$
$332$	−8.94427	−0.490881
$333$	0	0
$334$	−2.94427	−0.161103
$335$	31.2361	1.70661
$336$	0	0
$337$	−17.2361	−0.938908	−0.469454	−	0.882957i	$-0.655549\pi$
$337$	−17.2361	−0.938908	−0.469454	+	0.882957i	$0.655549\pi$
$338$	14.4164	0.784149
$339$	0	0
$340$	−7.05573	−0.382651
$341$	2.18034	0.118072
$342$	0	0
$343$	1.00000	0.0539949
$344$	−7.61803	−0.410737
$345$	0	0
$346$	17.7082	0.951999
$347$	−31.4164	−1.68652	−0.843261	−	0.537505i	$-0.819367\pi$
$347$	−31.4164	−1.68652	−0.843261	+	0.537505i	$0.819367\pi$
$348$	0	0
$349$	−6.00000	−0.321173	−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000	−0.321173	−0.160586	+	0.987022i	$0.551338\pi$
$350$	3.14590	0.168155
$351$	0	0
$352$	−4.61803	−0.246142
$353$	8.36068	0.444994	0.222497	−	0.974933i	$-0.428579\pi$
$353$	8.36068	0.444994	0.222497	+	0.974933i	$0.428579\pi$
$354$	0	0
$355$	−45.7639	−2.42890
$356$	1.38197	0.0732441
$357$	0	0
$358$	−19.2361	−1.01666
$359$	−14.2918	−0.754292	−0.377146	−	0.926154i	$-0.623095\pi$
$359$	−14.2918	−0.754292	−0.377146	+	0.926154i	$0.623095\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−4.18034	−0.219714
$363$	0	0
$364$	5.23607	0.274445
$365$	−20.6525	−1.08100
$366$	0	0
$367$	−2.32624	−0.121429	−0.0607143	−	0.998155i	$-0.519338\pi$
$367$	−2.32624	−0.121429	−0.0607143	+	0.998155i	$0.519338\pi$
$368$	5.70820	0.297561
$369$	0	0
$370$	13.8541	0.720240
$371$	11.3820	0.590922
$372$	0	0
$373$	−18.7426	−0.970457	−0.485229	−	0.874387i	$-0.661264\pi$
$373$	−18.7426	−0.970457	−0.485229	+	0.874387i	$0.661264\pi$
$374$	11.4164	0.590328
$375$	0	0
$376$	3.85410	0.198760
$377$	46.3607	2.38770
$378$	0	0
$379$	7.34752	0.377417	0.188708	−	0.982033i	$-0.439570\pi$
$379$	7.34752	0.377417	0.188708	+	0.982033i	$0.439570\pi$
$380$	−2.85410	−0.146412
$381$	0	0
$382$	−22.6525	−1.15900
$383$	20.7639	1.06099	0.530494	−	0.847689i	$-0.322007\pi$
$383$	20.7639	1.06099	0.530494	+	0.847689i	$0.322007\pi$
$384$	0	0
$385$	−13.1803	−0.671732
$386$	−13.7082	−0.697729
$387$	0	0
$388$	3.85410	0.195662
$389$	−22.3607	−1.13373	−0.566866	−	0.823810i	$-0.691844\pi$
$389$	−22.3607	−1.13373	−0.566866	+	0.823810i	$0.691844\pi$
$390$	0	0
$391$	−14.1115	−0.713647
$392$	1.00000	0.0505076
$393$	0	0
$394$	10.9443	0.551364
$395$	−38.0344	−1.91372
$396$	0	0
$397$	21.0902	1.05849	0.529243	−	0.848471i	$-0.322476\pi$
$397$	21.0902	1.05849	0.529243	+	0.848471i	$0.322476\pi$
$398$	25.0344	1.25486
$399$	0	0
$400$	3.14590	0.157295
$401$	−11.0557	−0.552097	−0.276048	−	0.961144i	$-0.589025\pi$
$401$	−11.0557	−0.552097	−0.276048	+	0.961144i	$0.589025\pi$
$402$	0	0
$403$	−2.47214	−0.123146
$404$	−14.9443	−0.743505
$405$	0	0
$406$	8.85410	0.439422
$407$	−22.4164	−1.11114
$408$	0	0
$409$	−27.5623	−1.36287	−0.681434	−	0.731879i	$-0.738643\pi$
$409$	−27.5623	−1.36287	−0.681434	+	0.731879i	$0.738643\pi$
$410$	−15.2016	−0.750755
$411$	0	0
$412$	8.76393	0.431768
$413$	8.85410	0.435682
$414$	0	0
$415$	−25.5279	−1.25311
$416$	5.23607	0.256719
$417$	0	0
$418$	4.61803	0.225875
$419$	−17.4164	−0.850847	−0.425424	−	0.904994i	$-0.639875\pi$
$419$	−17.4164	−0.850847	−0.425424	+	0.904994i	$0.639875\pi$
$420$	0	0
$421$	−14.3607	−0.699897	−0.349948	−	0.936769i	$-0.613801\pi$
$421$	−14.3607	−0.699897	−0.349948	+	0.936769i	$0.613801\pi$
$422$	−4.00000	−0.194717
$423$	0	0
$424$	11.3820	0.552757
$425$	−7.77709	−0.377244
$426$	0	0
$427$	−2.32624	−0.112575
$428$	0.763932	0.0369260
$429$	0	0
$430$	−21.7426	−1.04852
$431$	−5.27051	−0.253872	−0.126936	−	0.991911i	$-0.540514\pi$
$431$	−5.27051	−0.253872	−0.126936	+	0.991911i	$0.540514\pi$
$432$	0	0
$433$	22.0902	1.06159	0.530793	−	0.847502i	$-0.321894\pi$
$433$	22.0902	1.06159	0.530793	+	0.847502i	$0.321894\pi$
$434$	−0.472136	−0.0226633
$435$	0	0
$436$	−8.32624	−0.398754
$437$	−5.70820	−0.273060
$438$	0	0
$439$	4.47214	0.213443	0.106722	−	0.994289i	$-0.465965\pi$
$439$	4.47214	0.213443	0.106722	+	0.994289i	$0.465965\pi$
$440$	−13.1803	−0.628348
$441$	0	0
$442$	−12.9443	−0.615696
$443$	7.79837	0.370512	0.185256	−	0.982690i	$-0.440689\pi$
$443$	7.79837	0.370512	0.185256	+	0.982690i	$0.440689\pi$
$444$	0	0
$445$	3.94427	0.186976
$446$	−12.6525	−0.599112
$447$	0	0
$448$	1.00000	0.0472456
$449$	−28.0689	−1.32465	−0.662326	−	0.749216i	$-0.730431\pi$
$449$	−28.0689	−1.32465	−0.662326	+	0.749216i	$0.730431\pi$
$450$	0	0
$451$	24.5967	1.15822
$452$	8.94427	0.420703
$453$	0	0
$454$	−3.05573	−0.143412
$455$	14.9443	0.700598
$456$	0	0
$457$	−22.5066	−1.05281	−0.526407	−	0.850233i	$-0.676461\pi$
$457$	−22.5066	−1.05281	−0.526407	+	0.850233i	$0.676461\pi$
$458$	23.0344	1.07633
$459$	0	0
$460$	16.2918	0.759609
$461$	5.38197	0.250663	0.125332	−	0.992115i	$-0.460001\pi$
$461$	5.38197	0.250663	0.125332	+	0.992115i	$0.460001\pi$
$462$	0	0
$463$	−2.76393	−0.128451	−0.0642254	−	0.997935i	$-0.520458\pi$
$463$	−2.76393	−0.128451	−0.0642254	+	0.997935i	$0.520458\pi$
$464$	8.85410	0.411041
$465$	0	0
$466$	−15.7426	−0.729264
$467$	−21.7082	−1.00454	−0.502268	−	0.864712i	$-0.667501\pi$
$467$	−21.7082	−1.00454	−0.502268	+	0.864712i	$0.667501\pi$
$468$	0	0
$469$	10.9443	0.505360
$470$	11.0000	0.507392
$471$	0	0
$472$	8.85410	0.407543
$473$	35.1803	1.61759
$474$	0	0
$475$	−3.14590	−0.144344
$476$	−2.47214	−0.113310
$477$	0	0
$478$	15.1246	0.691784
$479$	−0.270510	−0.0123599	−0.00617995	−	0.999981i	$-0.501967\pi$
$479$	−0.270510	−0.0123599	−0.00617995	+	0.999981i	$0.501967\pi$
$480$	0	0
$481$	25.4164	1.15889
$482$	9.14590	0.416584
$483$	0	0
$484$	10.3262	0.469374
$485$	11.0000	0.499484
$486$	0	0
$487$	36.5623	1.65680	0.828398	−	0.560140i	$-0.189253\pi$
$487$	36.5623	1.65680	0.828398	+	0.560140i	$0.189253\pi$
$488$	−2.32624	−0.105304
$489$	0	0
$490$	2.85410	0.128935
$491$	−2.11146	−0.0952887	−0.0476443	−	0.998864i	$-0.515171\pi$
$491$	−2.11146	−0.0952887	−0.0476443	+	0.998864i	$0.515171\pi$
$492$	0	0
$493$	−21.8885	−0.985810
$494$	−5.23607	−0.235582
$495$	0	0
$496$	−0.472136	−0.0211995
$497$	−16.0344	−0.719243
$498$	0	0
$499$	−25.3262	−1.13376	−0.566879	−	0.823801i	$-0.691849\pi$
$499$	−25.3262	−1.13376	−0.566879	+	0.823801i	$0.691849\pi$
$500$	−5.29180	−0.236656
$501$	0	0
$502$	−13.2361	−0.590755
$503$	12.0902	0.539074	0.269537	−	0.962990i	$-0.413129\pi$
$503$	12.0902	0.539074	0.269537	+	0.962990i	$0.413129\pi$
$504$	0	0
$505$	−42.6525	−1.89801
$506$	−26.3607	−1.17188
$507$	0	0
$508$	−11.3820	−0.504993
$509$	−41.7082	−1.84868	−0.924342	−	0.381565i	$-0.875385\pi$
$509$	−41.7082	−1.84868	−0.924342	+	0.381565i	$0.875385\pi$
$510$	0	0
$511$	−7.23607	−0.320105
$512$	1.00000	0.0441942
$513$	0	0
$514$	23.9787	1.05766
$515$	25.0132	1.10221
$516$	0	0
$517$	−17.7984	−0.782772
$518$	4.85410	0.213277
$519$	0	0
$520$	14.9443	0.655350
$521$	12.4721	0.546414	0.273207	−	0.961955i	$-0.411916\pi$
$521$	12.4721	0.546414	0.273207	+	0.961955i	$0.411916\pi$
$522$	0	0
$523$	24.0000	1.04945	0.524723	−	0.851273i	$-0.324169\pi$
$523$	24.0000	1.04945	0.524723	+	0.851273i	$0.324169\pi$
$524$	6.18034	0.269989
$525$	0	0
$526$	9.23607	0.402712
$527$	1.16718	0.0508433
$528$	0	0
$529$	9.58359	0.416678
$530$	32.4853	1.41107
$531$	0	0
$532$	−1.00000	−0.0433555
$533$	−27.8885	−1.20799
$534$	0	0
$535$	2.18034	0.0942643
$536$	10.9443	0.472721
$537$	0	0
$538$	−28.4721	−1.22752
$539$	−4.61803	−0.198913
$540$	0	0
$541$	21.7082	0.933309	0.466654	−	0.884440i	$-0.345459\pi$
$541$	21.7082	0.933309	0.466654	+	0.884440i	$0.345459\pi$
$542$	8.85410	0.380316
$543$	0	0
$544$	−2.47214	−0.105992
$545$	−23.7639	−1.01794
$546$	0	0
$547$	−4.76393	−0.203691	−0.101846	−	0.994800i	$-0.532475\pi$
$547$	−4.76393	−0.203691	−0.101846	+	0.994800i	$0.532475\pi$
$548$	10.3820	0.443496
$549$	0	0
$550$	−14.5279	−0.619470
$551$	−8.85410	−0.377197
$552$	0	0
$553$	−13.3262	−0.566689
$554$	10.6525	0.452580
$555$	0	0
$556$	10.4721	0.444117
$557$	25.2361	1.06929	0.534643	−	0.845078i	$-0.320446\pi$
$557$	25.2361	1.06929	0.534643	+	0.845078i	$0.320446\pi$
$558$	0	0
$559$	−39.8885	−1.68711
$560$	2.85410	0.120608
$561$	0	0
$562$	−16.0000	−0.674919
$563$	0.145898	0.00614887	0.00307443	−	0.999995i	$-0.499021\pi$
$563$	0.145898	0.00614887	0.00307443	+	0.999995i	$0.499021\pi$
$564$	0	0
$565$	25.5279	1.07397
$566$	18.4721	0.776442
$567$	0	0
$568$	−16.0344	−0.672790
$569$	10.2918	0.431455	0.215727	−	0.976454i	$-0.430788\pi$
$569$	10.2918	0.431455	0.215727	+	0.976454i	$0.430788\pi$
$570$	0	0
$571$	−28.0344	−1.17320	−0.586602	−	0.809875i	$-0.699535\pi$
$571$	−28.0344	−1.17320	−0.586602	+	0.809875i	$0.699535\pi$
$572$	−24.1803	−1.01103
$573$	0	0
$574$	−5.32624	−0.222313
$575$	17.9574	0.748876
$576$	0	0
$577$	−34.9443	−1.45475	−0.727375	−	0.686241i	$-0.759260\pi$
$577$	−34.9443	−1.45475	−0.727375	+	0.686241i	$0.759260\pi$
$578$	−10.8885	−0.452904
$579$	0	0
$580$	25.2705	1.04930
$581$	−8.94427	−0.371071
$582$	0	0
$583$	−52.5623	−2.17691
$584$	−7.23607	−0.299431
$585$	0	0
$586$	−26.1803	−1.08150
$587$	−11.2361	−0.463762	−0.231881	−	0.972744i	$-0.574488\pi$
$587$	−11.2361	−0.463762	−0.231881	+	0.972744i	$0.574488\pi$
$588$	0	0
$589$	0.472136	0.0194540
$590$	25.2705	1.04037
$591$	0	0
$592$	4.85410	0.199502
$593$	22.2918	0.915414	0.457707	−	0.889103i	$-0.348671\pi$
$593$	22.2918	0.915414	0.457707	+	0.889103i	$0.348671\pi$
$594$	0	0
$595$	−7.05573	−0.289257
$596$	−4.00000	−0.163846
$597$	0	0
$598$	29.8885	1.22223
$599$	−17.5623	−0.717576	−0.358788	−	0.933419i	$-0.616810\pi$
$599$	−17.5623	−0.717576	−0.358788	+	0.933419i	$0.616810\pi$
$600$	0	0
$601$	−18.0000	−0.734235	−0.367118	−	0.930175i	$-0.619655\pi$
$601$	−18.0000	−0.734235	−0.367118	+	0.930175i	$0.619655\pi$
$602$	−7.61803	−0.310488
$603$	0	0
$604$	−21.8885	−0.890632
$605$	29.4721	1.19821
$606$	0	0
$607$	−9.81966	−0.398568	−0.199284	−	0.979942i	$-0.563862\pi$
$607$	−9.81966	−0.398568	−0.199284	+	0.979942i	$0.563862\pi$
$608$	−1.00000	−0.0405554
$609$	0	0
$610$	−6.63932	−0.268818
$611$	20.1803	0.816409
$612$	0	0
$613$	39.7771	1.60658	0.803291	−	0.595587i	$-0.203081\pi$
$613$	39.7771	1.60658	0.803291	+	0.595587i	$0.203081\pi$
$614$	−25.0344	−1.01031
$615$	0	0
$616$	−4.61803	−0.186066
$617$	35.1459	1.41492	0.707460	−	0.706753i	$-0.249841\pi$
$617$	35.1459	1.41492	0.707460	+	0.706753i	$0.249841\pi$
$618$	0	0
$619$	14.0689	0.565476	0.282738	−	0.959197i	$-0.408757\pi$
$619$	14.0689	0.565476	0.282738	+	0.959197i	$0.408757\pi$
$620$	−1.34752	−0.0541179
$621$	0	0
$622$	−2.61803	−0.104974
$623$	1.38197	0.0553673
$624$	0	0
$625$	−30.8328	−1.23331
$626$	−17.8885	−0.714970
$627$	0	0
$628$	−2.90983	−0.116115
$629$	−12.0000	−0.478471
$630$	0	0
$631$	32.3607	1.28826	0.644129	−	0.764917i	$-0.277220\pi$
$631$	32.3607	1.28826	0.644129	+	0.764917i	$0.277220\pi$
$632$	−13.3262	−0.530089
$633$	0	0
$634$	20.6180	0.818847
$635$	−32.4853	−1.28914
$636$	0	0
$637$	5.23607	0.207461
$638$	−40.8885	−1.61879
$639$	0	0
$640$	2.85410	0.112818
$641$	15.8197	0.624839	0.312420	−	0.949944i	$-0.398861\pi$
$641$	15.8197	0.624839	0.312420	+	0.949944i	$0.398861\pi$
$642$	0	0
$643$	10.4721	0.412981	0.206490	−	0.978449i	$-0.433796\pi$
$643$	10.4721	0.412981	0.206490	+	0.978449i	$0.433796\pi$
$644$	5.70820	0.224935
$645$	0	0
$646$	2.47214	0.0972649
$647$	−32.6738	−1.28454	−0.642269	−	0.766479i	$-0.722007\pi$
$647$	−32.6738	−1.28454	−0.642269	+	0.766479i	$0.722007\pi$
$648$	0	0
$649$	−40.8885	−1.60502
$650$	16.4721	0.646090
$651$	0	0
$652$	7.09017	0.277672
$653$	−12.2918	−0.481015	−0.240508	−	0.970647i	$-0.577314\pi$
$653$	−12.2918	−0.481015	−0.240508	+	0.970647i	$0.577314\pi$
$654$	0	0
$655$	17.6393	0.689225
$656$	−5.32624	−0.207955
$657$	0	0
$658$	3.85410	0.150249
$659$	−5.88854	−0.229385	−0.114693	−	0.993401i	$-0.536588\pi$
$659$	−5.88854	−0.229385	−0.114693	+	0.993401i	$0.536588\pi$
$660$	0	0
$661$	−3.88854	−0.151247	−0.0756234	−	0.997136i	$-0.524095\pi$
$661$	−3.88854	−0.151247	−0.0756234	+	0.997136i	$0.524095\pi$
$662$	−2.58359	−0.100414
$663$	0	0
$664$	−8.94427	−0.347105
$665$	−2.85410	−0.110677
$666$	0	0
$667$	50.5410	1.95696
$668$	−2.94427	−0.113917
$669$	0	0
$670$	31.2361	1.20675
$671$	10.7426	0.414715
$672$	0	0
$673$	24.6525	0.950283	0.475142	−	0.879909i	$-0.342397\pi$
$673$	24.6525	0.950283	0.475142	+	0.879909i	$0.342397\pi$
$674$	−17.2361	−0.663909
$675$	0	0
$676$	14.4164	0.554477
$677$	19.8885	0.764379	0.382189	−	0.924084i	$-0.375170\pi$
$677$	19.8885	0.764379	0.382189	+	0.924084i	$0.375170\pi$
$678$	0	0
$679$	3.85410	0.147907
$680$	−7.05573	−0.270575
$681$	0	0
$682$	2.18034	0.0834895
$683$	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
$685$	29.6312	1.13215
$686$	1.00000	0.0381802
$687$	0	0
$688$	−7.61803	−0.290435
$689$	59.5967	2.27046
$690$	0	0
$691$	37.1246	1.41229	0.706143	−	0.708069i	$-0.250433\pi$
$691$	37.1246	1.41229	0.706143	+	0.708069i	$0.250433\pi$
$692$	17.7082	0.673165
$693$	0	0
$694$	−31.4164	−1.19255
$695$	29.8885	1.13374
$696$	0	0
$697$	13.1672	0.498743
$698$	−6.00000	−0.227103
$699$	0	0
$700$	3.14590	0.118904
$701$	2.65248	0.100183	0.0500913	−	0.998745i	$-0.484049\pi$
$701$	2.65248	0.100183	0.0500913	+	0.998745i	$0.484049\pi$
$702$	0	0
$703$	−4.85410	−0.183076
$704$	−4.61803	−0.174049
$705$	0	0
$706$	8.36068	0.314658
$707$	−14.9443	−0.562037
$708$	0	0
$709$	−7.70820	−0.289488	−0.144744	−	0.989469i	$-0.546236\pi$
$709$	−7.70820	−0.289488	−0.144744	+	0.989469i	$0.546236\pi$
$710$	−45.7639	−1.71749
$711$	0	0
$712$	1.38197	0.0517914
$713$	−2.69505	−0.100930
$714$	0	0
$715$	−69.0132	−2.58095
$716$	−19.2361	−0.718886
$717$	0	0
$718$	−14.2918	−0.533365
$719$	−41.8885	−1.56218	−0.781090	−	0.624419i	$-0.785336\pi$
$719$	−41.8885	−1.56218	−0.781090	+	0.624419i	$0.785336\pi$
$720$	0	0
$721$	8.76393	0.326386
$722$	1.00000	0.0372161
$723$	0	0
$724$	−4.18034	−0.155361
$725$	27.8541	1.03448
$726$	0	0
$727$	12.5066	0.463843	0.231922	−	0.972734i	$-0.425499\pi$
$727$	12.5066	0.463843	0.231922	+	0.972734i	$0.425499\pi$
$728$	5.23607	0.194062
$729$	0	0
$730$	−20.6525	−0.764382
$731$	18.8328	0.696557
$732$	0	0
$733$	1.14590	0.0423247	0.0211624	−	0.999776i	$-0.493263\pi$
$733$	1.14590	0.0423247	0.0211624	+	0.999776i	$0.493263\pi$
$734$	−2.32624	−0.0858630
$735$	0	0
$736$	5.70820	0.210407
$737$	−50.5410	−1.86170
$738$	0	0
$739$	23.7984	0.875437	0.437719	−	0.899112i	$-0.355787\pi$
$739$	23.7984	0.875437	0.437719	+	0.899112i	$0.355787\pi$
$740$	13.8541	0.509287
$741$	0	0
$742$	11.3820	0.417845
$743$	−45.3262	−1.66286	−0.831429	−	0.555631i	$-0.812477\pi$
$743$	−45.3262	−1.66286	−0.831429	+	0.555631i	$0.812477\pi$
$744$	0	0
$745$	−11.4164	−0.418265
$746$	−18.7426	−0.686217
$747$	0	0
$748$	11.4164	0.417425
$749$	0.763932	0.0279135
$750$	0	0
$751$	−11.6180	−0.423948	−0.211974	−	0.977275i	$-0.567989\pi$
$751$	−11.6180	−0.423948	−0.211974	+	0.977275i	$0.567989\pi$
$752$	3.85410	0.140545
$753$	0	0
$754$	46.3607	1.68836
$755$	−62.4721	−2.27359
$756$	0	0
$757$	−8.58359	−0.311976	−0.155988	−	0.987759i	$-0.549856\pi$
$757$	−8.58359	−0.311976	−0.155988	+	0.987759i	$0.549856\pi$
$758$	7.34752	0.266874
$759$	0	0
$760$	−2.85410	−0.103529
$761$	47.8885	1.73596	0.867979	−	0.496601i	$-0.165419\pi$
$761$	47.8885	1.73596	0.867979	+	0.496601i	$0.165419\pi$
$762$	0	0
$763$	−8.32624	−0.301430
$764$	−22.6525	−0.819538
$765$	0	0
$766$	20.7639	0.750231
$767$	46.3607	1.67399
$768$	0	0
$769$	−32.1803	−1.16045	−0.580226	−	0.814455i	$-0.697036\pi$
$769$	−32.1803	−1.16045	−0.580226	+	0.814455i	$0.697036\pi$
$770$	−13.1803	−0.474986
$771$	0	0
$772$	−13.7082	−0.493369
$773$	−18.2918	−0.657910	−0.328955	−	0.944346i	$-0.606696\pi$
$773$	−18.2918	−0.657910	−0.328955	+	0.944346i	$0.606696\pi$
$774$	0	0
$775$	−1.48529	−0.0533532
$776$	3.85410	0.138354
$777$	0	0
$778$	−22.3607	−0.801669
$779$	5.32624	0.190832
$780$	0	0
$781$	74.0476	2.64963
$782$	−14.1115	−0.504625
$783$	0	0
$784$	1.00000	0.0357143
$785$	−8.30495	−0.296416
$786$	0	0
$787$	−16.4508	−0.586409	−0.293205	−	0.956050i	$-0.594722\pi$
$787$	−16.4508	−0.586409	−0.293205	+	0.956050i	$0.594722\pi$
$788$	10.9443	0.389874
$789$	0	0
$790$	−38.0344	−1.35320
$791$	8.94427	0.318022
$792$	0	0
$793$	−12.1803	−0.432537
$794$	21.0902	0.748462
$795$	0	0
$796$	25.0344	0.887322
$797$	51.9574	1.84043	0.920213	−	0.391417i	$-0.128015\pi$
$797$	51.9574	1.84043	0.920213	+	0.391417i	$0.128015\pi$
$798$	0	0
$799$	−9.52786	−0.337072
$800$	3.14590	0.111224
$801$	0	0
$802$	−11.0557	−0.390391
$803$	33.4164	1.17924
$804$	0	0
$805$	16.2918	0.574210
$806$	−2.47214	−0.0870773
$807$	0	0
$808$	−14.9443	−0.525738
$809$	26.7426	0.940221	0.470111	−	0.882607i	$-0.344214\pi$
$809$	26.7426	0.940221	0.470111	+	0.882607i	$0.344214\pi$
$810$	0	0
$811$	41.3820	1.45312	0.726559	−	0.687104i	$-0.241119\pi$
$811$	41.3820	1.45312	0.726559	+	0.687104i	$0.241119\pi$
$812$	8.85410	0.310718
$813$	0	0
$814$	−22.4164	−0.785695
$815$	20.2361	0.708839
$816$	0	0
$817$	7.61803	0.266521
$818$	−27.5623	−0.963693
$819$	0	0
$820$	−15.2016	−0.530864
$821$	−17.4164	−0.607837	−0.303918	−	0.952698i	$-0.598295\pi$
$821$	−17.4164	−0.607837	−0.303918	+	0.952698i	$0.598295\pi$
$822$	0	0
$823$	30.3607	1.05831	0.529153	−	0.848526i	$-0.322510\pi$
$823$	30.3607	1.05831	0.529153	+	0.848526i	$0.322510\pi$
$824$	8.76393	0.305306
$825$	0	0
$826$	8.85410	0.308074
$827$	−32.1803	−1.11902	−0.559510	−	0.828824i	$-0.689011\pi$
$827$	−32.1803	−1.11902	−0.559510	+	0.828824i	$0.689011\pi$
$828$	0	0
$829$	0.583592	0.0202690	0.0101345	−	0.999949i	$-0.496774\pi$
$829$	0.583592	0.0202690	0.0101345	+	0.999949i	$0.496774\pi$
$830$	−25.5279	−0.886085
$831$	0	0
$832$	5.23607	0.181528
$833$	−2.47214	−0.0856544
$834$	0	0
$835$	−8.40325	−0.290807
$836$	4.61803	0.159718
$837$	0	0
$838$	−17.4164	−0.601640
$839$	−4.11146	−0.141943	−0.0709716	−	0.997478i	$-0.522610\pi$
$839$	−4.11146	−0.141943	−0.0709716	+	0.997478i	$0.522610\pi$
$840$	0	0
$841$	49.3951	1.70328
$842$	−14.3607	−0.494902
$843$	0	0
$844$	−4.00000	−0.137686
$845$	41.1459	1.41546
$846$	0	0
$847$	10.3262	0.354814
$848$	11.3820	0.390858
$849$	0	0
$850$	−7.77709	−0.266752
$851$	27.7082	0.949825
$852$	0	0
$853$	44.1591	1.51198	0.755989	−	0.654585i	$-0.227157\pi$
$853$	44.1591	1.51198	0.755989	+	0.654585i	$0.227157\pi$
$854$	−2.32624	−0.0796022
$855$	0	0
$856$	0.763932	0.0261107
$857$	−11.8885	−0.406105	−0.203052	−	0.979168i	$-0.565086\pi$
$857$	−11.8885	−0.406105	−0.203052	+	0.979168i	$0.565086\pi$
$858$	0	0
$859$	9.12461	0.311328	0.155664	−	0.987810i	$-0.450248\pi$
$859$	9.12461	0.311328	0.155664	+	0.987810i	$0.450248\pi$
$860$	−21.7426	−0.741418
$861$	0	0
$862$	−5.27051	−0.179514
$863$	7.03444	0.239455	0.119728	−	0.992807i	$-0.461798\pi$
$863$	7.03444	0.239455	0.119728	+	0.992807i	$0.461798\pi$
$864$	0	0
$865$	50.5410	1.71845
$866$	22.0902	0.750655
$867$	0	0
$868$	−0.472136	−0.0160253
$869$	61.5410	2.08764
$870$	0	0
$871$	57.3050	1.94170
$872$	−8.32624	−0.281962
$873$	0	0
$874$	−5.70820	−0.193083
$875$	−5.29180	−0.178895
$876$	0	0
$877$	−25.2705	−0.853324	−0.426662	−	0.904411i	$-0.640311\pi$
$877$	−25.2705	−0.853324	−0.426662	+	0.904411i	$0.640311\pi$
$878$	4.47214	0.150927
$879$	0	0
$880$	−13.1803	−0.444309
$881$	−2.76393	−0.0931192	−0.0465596	−	0.998916i	$-0.514826\pi$
$881$	−2.76393	−0.0931192	−0.0465596	+	0.998916i	$0.514826\pi$
$882$	0	0
$883$	17.5623	0.591019	0.295509	−	0.955340i	$-0.404511\pi$
$883$	17.5623	0.591019	0.295509	+	0.955340i	$0.404511\pi$
$884$	−12.9443	−0.435363
$885$	0	0
$886$	7.79837	0.261991
$887$	48.2492	1.62005	0.810025	−	0.586395i	$-0.199453\pi$
$887$	48.2492	1.62005	0.810025	+	0.586395i	$0.199453\pi$
$888$	0	0
$889$	−11.3820	−0.381739
$890$	3.94427	0.132212
$891$	0	0
$892$	−12.6525	−0.423636
$893$	−3.85410	−0.128973
$894$	0	0
$895$	−54.9017	−1.83516
$896$	1.00000	0.0334077
$897$	0	0
$898$	−28.0689	−0.936671
$899$	−4.18034	−0.139422
$900$	0	0
$901$	−28.1378	−0.937405
$902$	24.5967	0.818982
$903$	0	0
$904$	8.94427	0.297482
$905$	−11.9311	−0.396604
$906$	0	0
$907$	32.0689	1.06483	0.532415	−	0.846484i	$-0.321285\pi$
$907$	32.0689	1.06483	0.532415	+	0.846484i	$0.321285\pi$
$908$	−3.05573	−0.101408
$909$	0	0
$910$	14.9443	0.495398
$911$	−11.2148	−0.371562	−0.185781	−	0.982591i	$-0.559482\pi$
$911$	−11.2148	−0.371562	−0.185781	+	0.982591i	$0.559482\pi$
$912$	0	0
$913$	41.3050	1.36699
$914$	−22.5066	−0.744451
$915$	0	0
$916$	23.0344	0.761079
$917$	6.18034	0.204093
$918$	0	0
$919$	24.7639	0.816887	0.408443	−	0.912784i	$-0.366072\pi$
$919$	24.7639	0.816887	0.408443	+	0.912784i	$0.366072\pi$
$920$	16.2918	0.537125
$921$	0	0
$922$	5.38197	0.177246
$923$	−83.9574	−2.76349
$924$	0	0
$925$	15.2705	0.502091
$926$	−2.76393	−0.0908284
$927$	0	0
$928$	8.85410	0.290650
$929$	−0.291796	−0.00957352	−0.00478676	−	0.999989i	$-0.501524\pi$
$929$	−0.291796	−0.00957352	−0.00478676	+	0.999989i	$0.501524\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	−15.7426	−0.515668
$933$	0	0
$934$	−21.7082	−0.710314
$935$	32.5836	1.06560
$936$	0	0
$937$	46.8328	1.52996	0.764981	−	0.644053i	$-0.222748\pi$
$937$	46.8328	1.52996	0.764981	+	0.644053i	$0.222748\pi$
$938$	10.9443	0.357343
$939$	0	0
$940$	11.0000	0.358780
$941$	−8.06888	−0.263038	−0.131519	−	0.991314i	$-0.541985\pi$
$941$	−8.06888	−0.263038	−0.131519	+	0.991314i	$0.541985\pi$
$942$	0	0
$943$	−30.4033	−0.990066
$944$	8.85410	0.288176
$945$	0	0
$946$	35.1803	1.14381
$947$	−19.3820	−0.629829	−0.314915	−	0.949120i	$-0.601976\pi$
$947$	−19.3820	−0.629829	−0.314915	+	0.949120i	$0.601976\pi$
$948$	0	0
$949$	−37.8885	−1.22991
$950$	−3.14590	−0.102066
$951$	0	0
$952$	−2.47214	−0.0801224
$953$	−8.47214	−0.274439	−0.137220	−	0.990541i	$-0.543817\pi$
$953$	−8.47214	−0.274439	−0.137220	+	0.990541i	$0.543817\pi$
$954$	0	0
$955$	−64.6525	−2.09210
$956$	15.1246	0.489165
$957$	0	0
$958$	−0.270510	−0.00873978
$959$	10.3820	0.335251
$960$	0	0
$961$	−30.7771	−0.992809
$962$	25.4164	0.819458
$963$	0	0
$964$	9.14590	0.294570
$965$	−39.1246	−1.25947
$966$	0	0
$967$	−9.30495	−0.299227	−0.149614	−	0.988745i	$-0.547803\pi$
$967$	−9.30495	−0.299227	−0.149614	+	0.988745i	$0.547803\pi$
$968$	10.3262	0.331898
$969$	0	0
$970$	11.0000	0.353189
$971$	15.4934	0.497208	0.248604	−	0.968605i	$-0.420028\pi$
$971$	15.4934	0.497208	0.248604	+	0.968605i	$0.420028\pi$
$972$	0	0
$973$	10.4721	0.335721
$974$	36.5623	1.17153
$975$	0	0
$976$	−2.32624	−0.0744611
$977$	−28.7639	−0.920240	−0.460120	−	0.887857i	$-0.652194\pi$
$977$	−28.7639	−0.920240	−0.460120	+	0.887857i	$0.652194\pi$
$978$	0	0
$979$	−6.38197	−0.203969
$980$	2.85410	0.0911709
$981$	0	0
$982$	−2.11146	−0.0673793
$983$	−0.944272	−0.0301176	−0.0150588	−	0.999887i	$-0.504794\pi$
$983$	−0.944272	−0.0301176	−0.0150588	+	0.999887i	$0.504794\pi$
$984$	0	0
$985$	31.2361	0.995264
$986$	−21.8885	−0.697073
$987$	0	0
$988$	−5.23607	−0.166582
$989$	−43.4853	−1.38275
$990$	0	0
$991$	−52.6869	−1.67366	−0.836828	−	0.547467i	$-0.815592\pi$
$991$	−52.6869	−1.67366	−0.836828	+	0.547467i	$0.815592\pi$
$992$	−0.472136	−0.0149903
$993$	0	0
$994$	−16.0344	−0.508582
$995$	71.4508	2.26514
$996$	0	0
$997$	17.9098	0.567210	0.283605	−	0.958941i	$-0.408470\pi$
$997$	17.9098	0.567210	0.283605	+	0.958941i	$0.408470\pi$
$998$	−25.3262	−0.801688
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +2.85410 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +2.85410 q^{5} +1.00000 q^{7} +1.00000 q^{8} +2.85410 q^{10} -4.61803 q^{11} +5.23607 q^{13} +1.00000 q^{14} +1.00000 q^{16} -2.47214 q^{17} -1.00000 q^{19} +2.85410 q^{20} -4.61803 q^{22} +5.70820 q^{23} +3.14590 q^{25} +5.23607 q^{26} +1.00000 q^{28} +8.85410 q^{29} -0.472136 q^{31} +1.00000 q^{32} -2.47214 q^{34} +2.85410 q^{35} +4.85410 q^{37} -1.00000 q^{38} +2.85410 q^{40} -5.32624 q^{41} -7.61803 q^{43} -4.61803 q^{44} +5.70820 q^{46} +3.85410 q^{47} +1.00000 q^{49} +3.14590 q^{50} +5.23607 q^{52} +11.3820 q^{53} -13.1803 q^{55} +1.00000 q^{56} +8.85410 q^{58} +8.85410 q^{59} -2.32624 q^{61} -0.472136 q^{62} +1.00000 q^{64} +14.9443 q^{65} +10.9443 q^{67} -2.47214 q^{68} +2.85410 q^{70} -16.0344 q^{71} -7.23607 q^{73} +4.85410 q^{74} -1.00000 q^{76} -4.61803 q^{77} -13.3262 q^{79} +2.85410 q^{80} -5.32624 q^{82} -8.94427 q^{83} -7.05573 q^{85} -7.61803 q^{86} -4.61803 q^{88} +1.38197 q^{89} +5.23607 q^{91} +5.70820 q^{92} +3.85410 q^{94} -2.85410 q^{95} +3.85410 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - q^{5} + 2 q^{7} + 2 q^{8}+O(q^{10}) \) 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

Modular forms

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Database

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