LMFDB - Embedded newform 2394.2.a.bc.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:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.48119$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -2.96239 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -2.96239 q^{5} -1.00000 q^{7} +1.00000 q^{8} -2.96239 q^{10} +5.35026 q^{11} -0.962389 q^{13} -1.00000 q^{14} +1.00000 q^{16} -3.35026 q^{17} -1.00000 q^{19} -2.96239 q^{20} +5.35026 q^{22} +4.96239 q^{23} +3.77575 q^{25} -0.962389 q^{26} -1.00000 q^{28} +1.22425 q^{29} -0.387873 q^{31} +1.00000 q^{32} -3.35026 q^{34} +2.96239 q^{35} +1.61213 q^{37} -1.00000 q^{38} -2.96239 q^{40} +7.92478 q^{41} -0.775746 q^{43} +5.35026 q^{44} +4.96239 q^{46} +12.3127 q^{47} +1.00000 q^{49} +3.77575 q^{50} -0.962389 q^{52} +2.00000 q^{53} -15.8496 q^{55} -1.00000 q^{56} +1.22425 q^{58} +2.77575 q^{61} -0.387873 q^{62} +1.00000 q^{64} +2.85097 q^{65} +3.35026 q^{67} -3.35026 q^{68} +2.96239 q^{70} +14.7005 q^{71} -8.70052 q^{73} +1.61213 q^{74} -1.00000 q^{76} -5.35026 q^{77} +11.5877 q^{79} -2.96239 q^{80} +7.92478 q^{82} +11.9248 q^{83} +9.92478 q^{85} -0.775746 q^{86} +5.35026 q^{88} -0.700523 q^{89} +0.962389 q^{91} +4.96239 q^{92} +12.3127 q^{94} +2.96239 q^{95} -11.2750 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} + 2 q^{5} - 3 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q + 3 * q^2 + 3 * q^4 + 2 * q^5 - 3 * q^7 + 3 * q^8	$ 3 q + 3 q^{2} + 3 q^{4} + 2 q^{5} - 3 q^{7} + 3 q^{8} + 2 q^{10} + 6 q^{11} + 8 q^{13} - 3 q^{14} + 3 q^{16} - 3 q^{19} + 2 q^{20} + 6 q^{22} + 4 q^{23} + 13 q^{25} + 8 q^{26} - 3 q^{28} + 2 q^{29} - 2 q^{31} + 3 q^{32} - 2 q^{35} + 4 q^{37} - 3 q^{38} + 2 q^{40} + 2 q^{41} - 4 q^{43} + 6 q^{44} + 4 q^{46} + 16 q^{47} + 3 q^{49} + 13 q^{50} + 8 q^{52} + 6 q^{53} - 4 q^{55} - 3 q^{56} + 2 q^{58} + 10 q^{61} - 2 q^{62} + 3 q^{64} + 32 q^{65} - 2 q^{70} + 24 q^{71} - 6 q^{73} + 4 q^{74} - 3 q^{76} - 6 q^{77} - 18 q^{79} + 2 q^{80} + 2 q^{82} + 14 q^{83} + 8 q^{85} - 4 q^{86} + 6 q^{88} + 18 q^{89} - 8 q^{91} + 4 q^{92} + 16 q^{94} - 2 q^{95} - 2 q^{97} + 3 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 + 2 * q^5 - 3 * q^7 + 3 * q^8 + 2 * q^10 + 6 * q^11 + 8 * q^13 - 3 * q^14 + 3 * q^16 - 3 * q^19 + 2 * q^20 + 6 * q^22 + 4 * q^23 + 13 * q^25 + 8 * q^26 - 3 * q^28 + 2 * q^29 - 2 * q^31 + 3 * q^32 - 2 * q^35 + 4 * q^37 - 3 * q^38 + 2 * q^40 + 2 * q^41 - 4 * q^43 + 6 * q^44 + 4 * q^46 + 16 * q^47 + 3 * q^49 + 13 * q^50 + 8 * q^52 + 6 * q^53 - 4 * q^55 - 3 * q^56 + 2 * q^58 + 10 * q^61 - 2 * q^62 + 3 * q^64 + 32 * q^65 - 2 * q^70 + 24 * q^71 - 6 * q^73 + 4 * q^74 - 3 * q^76 - 6 * q^77 - 18 * q^79 + 2 * q^80 + 2 * q^82 + 14 * q^83 + 8 * q^85 - 4 * q^86 + 6 * q^88 + 18 * q^89 - 8 * q^91 + 4 * q^92 + 16 * q^94 - 2 * q^95 - 2 * q^97 + 3 * 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
$5$	−2.96239	−1.32482	−0.662410	−	0.749141i	$-0.730466\pi$
$5$	−2.96239	−1.32482	−0.662410	+	0.749141i	$0.730466\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	0	0
$10$	−2.96239	−0.936790
$11$	5.35026	1.61316	0.806582	−	0.591122i	$-0.201315\pi$
$11$	5.35026	1.61316	0.806582	+	0.591122i	$0.201315\pi$
$12$	0	0
$13$	−0.962389	−0.266919	−0.133459	−	0.991054i	$-0.542609\pi$
$13$	−0.962389	−0.266919	−0.133459	+	0.991054i	$0.542609\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.35026	−0.812558	−0.406279	−	0.913749i	$-0.633174\pi$
$17$	−3.35026	−0.812558	−0.406279	+	0.913749i	$0.633174\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−2.96239	−0.662410
$21$	0	0
$22$	5.35026	1.14068
$23$	4.96239	1.03473	0.517365	−	0.855765i	$-0.326913\pi$
$23$	4.96239	1.03473	0.517365	+	0.855765i	$0.326913\pi$
$24$	0	0
$25$	3.77575	0.755149
$26$	−0.962389	−0.188740
$27$	0	0
$28$	−1.00000	−0.188982
$29$	1.22425	0.227338	0.113669	−	0.993519i	$-0.463740\pi$
$29$	1.22425	0.227338	0.113669	+	0.993519i	$0.463740\pi$
$30$	0	0
$31$	−0.387873	−0.0696641	−0.0348320	−	0.999393i	$-0.511090\pi$
$31$	−0.387873	−0.0696641	−0.0348320	+	0.999393i	$0.511090\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−3.35026	−0.574565
$35$	2.96239	0.500735
$36$	0	0
$37$	1.61213	0.265032	0.132516	−	0.991181i	$-0.457694\pi$
$37$	1.61213	0.265032	0.132516	+	0.991181i	$0.457694\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	−2.96239	−0.468395
$41$	7.92478	1.23764	0.618821	−	0.785532i	$-0.287611\pi$
$41$	7.92478	1.23764	0.618821	+	0.785532i	$0.287611\pi$
$42$	0	0
$43$	−0.775746	−0.118300	−0.0591501	−	0.998249i	$-0.518839\pi$
$43$	−0.775746	−0.118300	−0.0591501	+	0.998249i	$0.518839\pi$
$44$	5.35026	0.806582
$45$	0	0
$46$	4.96239	0.731664
$47$	12.3127	1.79598	0.897992	−	0.440011i	$-0.145026\pi$
$47$	12.3127	1.79598	0.897992	+	0.440011i	$0.145026\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	3.77575	0.533971
$51$	0	0
$52$	−0.962389	−0.133459
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	−15.8496	−2.13715
$56$	−1.00000	−0.133631
$57$	0	0
$58$	1.22425	0.160752
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	2.77575	0.355398	0.177699	−	0.984085i	$-0.443135\pi$
$61$	2.77575	0.355398	0.177699	+	0.984085i	$0.443135\pi$
$62$	−0.387873	−0.0492599
$63$	0	0
$64$	1.00000	0.125000
$65$	2.85097	0.353619
$66$	0	0
$67$	3.35026	0.409300	0.204650	−	0.978835i	$-0.434394\pi$
$67$	3.35026	0.409300	0.204650	+	0.978835i	$0.434394\pi$
$68$	−3.35026	−0.406279
$69$	0	0
$70$	2.96239	0.354073
$71$	14.7005	1.74463	0.872316	−	0.488943i	$-0.162617\pi$
$71$	14.7005	1.74463	0.872316	+	0.488943i	$0.162617\pi$
$72$	0	0
$73$	−8.70052	−1.01832	−0.509160	−	0.860672i	$-0.670044\pi$
$73$	−8.70052	−1.01832	−0.509160	+	0.860672i	$0.670044\pi$
$74$	1.61213	0.187406
$75$	0	0
$76$	−1.00000	−0.114708
$77$	−5.35026	−0.609719
$78$	0	0
$79$	11.5877	1.30372	0.651858	−	0.758341i	$-0.273990\pi$
$79$	11.5877	1.30372	0.651858	+	0.758341i	$0.273990\pi$
$80$	−2.96239	−0.331205
$81$	0	0
$82$	7.92478	0.875145
$83$	11.9248	1.30891	0.654457	−	0.756099i	$-0.272897\pi$
$83$	11.9248	1.30891	0.654457	+	0.756099i	$0.272897\pi$
$84$	0	0
$85$	9.92478	1.07649
$86$	−0.775746	−0.0836509
$87$	0	0
$88$	5.35026	0.570340
$89$	−0.700523	−0.0742553	−0.0371277	−	0.999311i	$-0.511821\pi$
$89$	−0.700523	−0.0742553	−0.0371277	+	0.999311i	$0.511821\pi$
$90$	0	0
$91$	0.962389	0.100886
$92$	4.96239	0.517365
$93$	0	0
$94$	12.3127	1.26995
$95$	2.96239	0.303935
$96$	0	0
$97$	−11.2750	−1.14481	−0.572403	−	0.819972i	$-0.693989\pi$
$97$	−11.2750	−1.14481	−0.572403	+	0.819972i	$0.693989\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	3.77575	0.377575
$101$	−6.96239	−0.692784	−0.346392	−	0.938090i	$-0.612593\pi$
$101$	−6.96239	−0.692784	−0.346392	+	0.938090i	$0.612593\pi$
$102$	0	0
$103$	8.23743	0.811658	0.405829	−	0.913949i	$-0.366983\pi$
$103$	8.23743	0.811658	0.405829	+	0.913949i	$0.366983\pi$
$104$	−0.962389	−0.0943700
$105$	0	0
$106$	2.00000	0.194257
$107$	1.29948	0.125625	0.0628126	−	0.998025i	$-0.479993\pi$
$107$	1.29948	0.125625	0.0628126	+	0.998025i	$0.479993\pi$
$108$	0	0
$109$	−3.16362	−0.303020	−0.151510	−	0.988456i	$-0.548414\pi$
$109$	−3.16362	−0.303020	−0.151510	+	0.988456i	$0.548414\pi$
$110$	−15.8496	−1.51120
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	−18.6253	−1.75212	−0.876060	−	0.482201i	$-0.839837\pi$
$113$	−18.6253	−1.75212	−0.876060	+	0.482201i	$0.839837\pi$
$114$	0	0
$115$	−14.7005	−1.37083
$116$	1.22425	0.113669
$117$	0	0
$118$	0	0
$119$	3.35026	0.307118
$120$	0	0
$121$	17.6253	1.60230
$122$	2.77575	0.251304
$123$	0	0
$124$	−0.387873	−0.0348320
$125$	3.62672	0.324383
$126$	0	0
$127$	17.6629	1.56733	0.783665	−	0.621184i	$-0.213348\pi$
$127$	17.6629	1.56733	0.783665	+	0.621184i	$0.213348\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	2.85097	0.250047
$131$	14.6253	1.27782	0.638909	−	0.769282i	$-0.279386\pi$
$131$	14.6253	1.27782	0.638909	+	0.769282i	$0.279386\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	3.35026	0.289419
$135$	0	0
$136$	−3.35026	−0.287283
$137$	9.92478	0.847931	0.423965	−	0.905678i	$-0.360638\pi$
$137$	9.92478	0.847931	0.423965	+	0.905678i	$0.360638\pi$
$138$	0	0
$139$	12.6253	1.07086	0.535432	−	0.844578i	$-0.320149\pi$
$139$	12.6253	1.07086	0.535432	+	0.844578i	$0.320149\pi$
$140$	2.96239	0.250368
$141$	0	0
$142$	14.7005	1.23364
$143$	−5.14903	−0.430584
$144$	0	0
$145$	−3.62672	−0.301182
$146$	−8.70052	−0.720060
$147$	0	0
$148$	1.61213	0.132516
$149$	−1.53690	−0.125908	−0.0629540	−	0.998016i	$-0.520052\pi$
$149$	−1.53690	−0.125908	−0.0629540	+	0.998016i	$0.520052\pi$
$150$	0	0
$151$	−12.1114	−0.985613	−0.492807	−	0.870139i	$-0.664029\pi$
$151$	−12.1114	−0.985613	−0.492807	+	0.870139i	$0.664029\pi$
$152$	−1.00000	−0.0811107
$153$	0	0
$154$	−5.35026	−0.431136
$155$	1.14903	0.0922924
$156$	0	0
$157$	23.4010	1.86761	0.933803	−	0.357786i	$-0.116468\pi$
$157$	23.4010	1.86761	0.933803	+	0.357786i	$0.116468\pi$
$158$	11.5877	0.921867
$159$	0	0
$160$	−2.96239	−0.234197
$161$	−4.96239	−0.391091
$162$	0	0
$163$	−23.8496	−1.86804	−0.934021	−	0.357219i	$-0.883725\pi$
$163$	−23.8496	−1.86804	−0.934021	+	0.357219i	$0.883725\pi$
$164$	7.92478	0.618821
$165$	0	0
$166$	11.9248	0.925542
$167$	1.92478	0.148944	0.0744719	−	0.997223i	$-0.476273\pi$
$167$	1.92478	0.148944	0.0744719	+	0.997223i	$0.476273\pi$
$168$	0	0
$169$	−12.0738	−0.928754
$170$	9.92478	0.761196
$171$	0	0
$172$	−0.775746	−0.0591501
$173$	7.40105	0.562691	0.281346	−	0.959607i	$-0.409219\pi$
$173$	7.40105	0.562691	0.281346	+	0.959607i	$0.409219\pi$
$174$	0	0
$175$	−3.77575	−0.285420
$176$	5.35026	0.403291
$177$	0	0
$178$	−0.700523	−0.0525065
$179$	−21.2506	−1.58834	−0.794172	−	0.607693i	$-0.792095\pi$
$179$	−21.2506	−1.58834	−0.794172	+	0.607693i	$0.792095\pi$
$180$	0	0
$181$	−14.8872	−1.10655	−0.553277	−	0.832997i	$-0.686623\pi$
$181$	−14.8872	−1.10655	−0.553277	+	0.832997i	$0.686623\pi$
$182$	0.962389	0.0713370
$183$	0	0
$184$	4.96239	0.365832
$185$	−4.77575	−0.351120
$186$	0	0
$187$	−17.9248	−1.31079
$188$	12.3127	0.897992
$189$	0	0
$190$	2.96239	0.214914
$191$	7.41090	0.536234	0.268117	−	0.963386i	$-0.413599\pi$
$191$	7.41090	0.536234	0.268117	+	0.963386i	$0.413599\pi$
$192$	0	0
$193$	−6.77575	−0.487729	−0.243864	−	0.969809i	$-0.578415\pi$
$193$	−6.77575	−0.487729	−0.243864	+	0.969809i	$0.578415\pi$
$194$	−11.2750	−0.809501
$195$	0	0
$196$	1.00000	0.0714286
$197$	2.46310	0.175488	0.0877442	−	0.996143i	$-0.472034\pi$
$197$	2.46310	0.175488	0.0877442	+	0.996143i	$0.472034\pi$
$198$	0	0
$199$	−10.4485	−0.740675	−0.370338	−	0.928897i	$-0.620758\pi$
$199$	−10.4485	−0.740675	−0.370338	+	0.928897i	$0.620758\pi$
$200$	3.77575	0.266986
$201$	0	0
$202$	−6.96239	−0.489872
$203$	−1.22425	−0.0859258
$204$	0	0
$205$	−23.4763	−1.63965
$206$	8.23743	0.573929
$207$	0	0
$208$	−0.962389	−0.0667296
$209$	−5.35026	−0.370085
$210$	0	0
$211$	−10.8265	−0.745329	−0.372665	−	0.927966i	$-0.621556\pi$
$211$	−10.8265	−0.745329	−0.372665	+	0.927966i	$0.621556\pi$
$212$	2.00000	0.137361
$213$	0	0
$214$	1.29948	0.0888304
$215$	2.29806	0.156727
$216$	0	0
$217$	0.387873	0.0263305
$218$	−3.16362	−0.214267
$219$	0	0
$220$	−15.8496	−1.06858
$221$	3.22425	0.216887
$222$	0	0
$223$	−22.9380	−1.53604	−0.768019	−	0.640427i	$-0.778758\pi$
$223$	−22.9380	−1.53604	−0.768019	+	0.640427i	$0.778758\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−18.6253	−1.23894
$227$	−23.0738	−1.53146	−0.765731	−	0.643161i	$-0.777623\pi$
$227$	−23.0738	−1.53146	−0.765731	+	0.643161i	$0.777623\pi$
$228$	0	0
$229$	−8.07522	−0.533626	−0.266813	−	0.963748i	$-0.585971\pi$
$229$	−8.07522	−0.533626	−0.266813	+	0.963748i	$0.585971\pi$
$230$	−14.7005	−0.969324
$231$	0	0
$232$	1.22425	0.0803762
$233$	1.29948	0.0851315	0.0425658	−	0.999094i	$-0.486447\pi$
$233$	1.29948	0.0851315	0.0425658	+	0.999094i	$0.486447\pi$
$234$	0	0
$235$	−36.4749	−2.37936
$236$	0	0
$237$	0	0
$238$	3.35026	0.217165
$239$	0.186642	0.0120729	0.00603644	−	0.999982i	$-0.498079\pi$
$239$	0.186642	0.0120729	0.00603644	+	0.999982i	$0.498079\pi$
$240$	0	0
$241$	15.2750	0.983952	0.491976	−	0.870609i	$-0.336275\pi$
$241$	15.2750	0.983952	0.491976	+	0.870609i	$0.336275\pi$
$242$	17.6253	1.13300
$243$	0	0
$244$	2.77575	0.177699
$245$	−2.96239	−0.189260
$246$	0	0
$247$	0.962389	0.0612353
$248$	−0.387873	−0.0246300
$249$	0	0
$250$	3.62672	0.229374
$251$	14.7757	0.932637	0.466318	−	0.884617i	$-0.345580\pi$
$251$	14.7757	0.932637	0.466318	+	0.884617i	$0.345580\pi$
$252$	0	0
$253$	26.5501	1.66919
$254$	17.6629	1.10827
$255$	0	0
$256$	1.00000	0.0625000
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	0	0
$259$	−1.61213	−0.100173
$260$	2.85097	0.176810
$261$	0	0
$262$	14.6253	0.903554
$263$	−24.8119	−1.52997	−0.764985	−	0.644048i	$-0.777254\pi$
$263$	−24.8119	−1.52997	−0.764985	+	0.644048i	$0.777254\pi$
$264$	0	0
$265$	−5.92478	−0.363956
$266$	1.00000	0.0613139
$267$	0	0
$268$	3.35026	0.204650
$269$	15.4010	0.939018	0.469509	−	0.882928i	$-0.344431\pi$
$269$	15.4010	0.939018	0.469509	+	0.882928i	$0.344431\pi$
$270$	0	0
$271$	4.62530	0.280967	0.140484	−	0.990083i	$-0.455134\pi$
$271$	4.62530	0.280967	0.140484	+	0.990083i	$0.455134\pi$
$272$	−3.35026	−0.203139
$273$	0	0
$274$	9.92478	0.599578
$275$	20.2012	1.21818
$276$	0	0
$277$	−30.4749	−1.83106	−0.915528	−	0.402254i	$-0.868227\pi$
$277$	−30.4749	−1.83106	−0.915528	+	0.402254i	$0.868227\pi$
$278$	12.6253	0.757215
$279$	0	0
$280$	2.96239	0.177037
$281$	−9.22425	−0.550273	−0.275136	−	0.961405i	$-0.588723\pi$
$281$	−9.22425	−0.550273	−0.275136	+	0.961405i	$0.588723\pi$
$282$	0	0
$283$	−0.775746	−0.0461133	−0.0230567	−	0.999734i	$-0.507340\pi$
$283$	−0.775746	−0.0461133	−0.0230567	+	0.999734i	$0.507340\pi$
$284$	14.7005	0.872316
$285$	0	0
$286$	−5.14903	−0.304469
$287$	−7.92478	−0.467785
$288$	0	0
$289$	−5.77575	−0.339750
$290$	−3.62672	−0.212968
$291$	0	0
$292$	−8.70052	−0.509160
$293$	0.448507	0.0262021	0.0131010	−	0.999914i	$-0.495830\pi$
$293$	0.448507	0.0262021	0.0131010	+	0.999914i	$0.495830\pi$
$294$	0	0
$295$	0	0
$296$	1.61213	0.0937030
$297$	0	0
$298$	−1.53690	−0.0890305
$299$	−4.77575	−0.276189
$300$	0	0
$301$	0.775746	0.0447133
$302$	−12.1114	−0.696934
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	−8.22284	−0.470838
$306$	0	0
$307$	−2.07522	−0.118439	−0.0592196	−	0.998245i	$-0.518861\pi$
$307$	−2.07522	−0.118439	−0.0592196	+	0.998245i	$0.518861\pi$
$308$	−5.35026	−0.304859
$309$	0	0
$310$	1.14903	0.0652606
$311$	−3.93937	−0.223381	−0.111690	−	0.993743i	$-0.535627\pi$
$311$	−3.93937	−0.223381	−0.111690	+	0.993743i	$0.535627\pi$
$312$	0	0
$313$	−14.0000	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	23.4010	1.32060
$315$	0	0
$316$	11.5877	0.651858
$317$	11.5515	0.648796	0.324398	−	0.945921i	$-0.394838\pi$
$317$	11.5515	0.648796	0.324398	+	0.945921i	$0.394838\pi$
$318$	0	0
$319$	6.55008	0.366734
$320$	−2.96239	−0.165603
$321$	0	0
$322$	−4.96239	−0.276543
$323$	3.35026	0.186414
$324$	0	0
$325$	−3.63374	−0.201563
$326$	−23.8496	−1.32090
$327$	0	0
$328$	7.92478	0.437573
$329$	−12.3127	−0.678818
$330$	0	0
$331$	17.9003	0.983892	0.491946	−	0.870626i	$-0.336286\pi$
$331$	17.9003	0.983892	0.491946	+	0.870626i	$0.336286\pi$
$332$	11.9248	0.654457
$333$	0	0
$334$	1.92478	0.105319
$335$	−9.92478	−0.542249
$336$	0	0
$337$	28.7005	1.56342	0.781709	−	0.623644i	$-0.214348\pi$
$337$	28.7005	1.56342	0.781709	+	0.623644i	$0.214348\pi$
$338$	−12.0738	−0.656729
$339$	0	0
$340$	9.92478	0.538247
$341$	−2.07522	−0.112380
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−0.775746	−0.0418254
$345$	0	0
$346$	7.40105	0.397883
$347$	−27.6483	−1.48424	−0.742120	−	0.670267i	$-0.766180\pi$
$347$	−27.6483	−1.48424	−0.742120	+	0.670267i	$0.766180\pi$
$348$	0	0
$349$	3.92478	0.210089	0.105044	−	0.994468i	$-0.466502\pi$
$349$	3.92478	0.210089	0.105044	+	0.994468i	$0.466502\pi$
$350$	−3.77575	−0.201822
$351$	0	0
$352$	5.35026	0.285170
$353$	−28.1260	−1.49700	−0.748498	−	0.663137i	$-0.769225\pi$
$353$	−28.1260	−1.49700	−0.748498	+	0.663137i	$0.769225\pi$
$354$	0	0
$355$	−43.5487	−2.31132
$356$	−0.700523	−0.0371277
$357$	0	0
$358$	−21.2506	−1.12313
$359$	30.1114	1.58922	0.794610	−	0.607120i	$-0.207675\pi$
$359$	30.1114	1.58922	0.794610	+	0.607120i	$0.207675\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−14.8872	−0.782452
$363$	0	0
$364$	0.962389	0.0504429
$365$	25.7743	1.34909
$366$	0	0
$367$	27.0738	1.41324	0.706621	−	0.707593i	$-0.250219\pi$
$367$	27.0738	1.41324	0.706621	+	0.707593i	$0.250219\pi$
$368$	4.96239	0.258682
$369$	0	0
$370$	−4.77575	−0.248279
$371$	−2.00000	−0.103835
$372$	0	0
$373$	6.76116	0.350079	0.175040	−	0.984561i	$-0.443995\pi$
$373$	6.76116	0.350079	0.175040	+	0.984561i	$0.443995\pi$
$374$	−17.9248	−0.926868
$375$	0	0
$376$	12.3127	0.634976
$377$	−1.17821	−0.0606808
$378$	0	0
$379$	15.6023	0.801435	0.400718	−	0.916202i	$-0.368761\pi$
$379$	15.6023	0.801435	0.400718	+	0.916202i	$0.368761\pi$
$380$	2.96239	0.151967
$381$	0	0
$382$	7.41090	0.379174
$383$	−7.37470	−0.376830	−0.188415	−	0.982090i	$-0.560335\pi$
$383$	−7.37470	−0.376830	−0.188415	+	0.982090i	$0.560335\pi$
$384$	0	0
$385$	15.8496	0.807768
$386$	−6.77575	−0.344876
$387$	0	0
$388$	−11.2750	−0.572403
$389$	28.2374	1.43169	0.715847	−	0.698257i	$-0.246041\pi$
$389$	28.2374	1.43169	0.715847	+	0.698257i	$0.246041\pi$
$390$	0	0
$391$	−16.6253	−0.840778
$392$	1.00000	0.0505076
$393$	0	0
$394$	2.46310	0.124089
$395$	−34.3272	−1.72719
$396$	0	0
$397$	−2.77575	−0.139311	−0.0696554	−	0.997571i	$-0.522190\pi$
$397$	−2.77575	−0.139311	−0.0696554	+	0.997571i	$0.522190\pi$
$398$	−10.4485	−0.523736
$399$	0	0
$400$	3.77575	0.188787
$401$	14.2520	0.711712	0.355856	−	0.934541i	$-0.384189\pi$
$401$	14.2520	0.711712	0.355856	+	0.934541i	$0.384189\pi$
$402$	0	0
$403$	0.373285	0.0185946
$404$	−6.96239	−0.346392
$405$	0	0
$406$	−1.22425	−0.0607587
$407$	8.62530	0.427540
$408$	0	0
$409$	15.0230	0.742841	0.371420	−	0.928465i	$-0.378871\pi$
$409$	15.0230	0.742841	0.371420	+	0.928465i	$0.378871\pi$
$410$	−23.4763	−1.15941
$411$	0	0
$412$	8.23743	0.405829
$413$	0	0
$414$	0	0
$415$	−35.3258	−1.73408
$416$	−0.962389	−0.0471850
$417$	0	0
$418$	−5.35026	−0.261690
$419$	24.9525	1.21901	0.609506	−	0.792782i	$-0.291368\pi$
$419$	24.9525	1.21901	0.609506	+	0.792782i	$0.291368\pi$
$420$	0	0
$421$	−1.08840	−0.0530452	−0.0265226	−	0.999648i	$-0.508443\pi$
$421$	−1.08840	−0.0530452	−0.0265226	+	0.999648i	$0.508443\pi$
$422$	−10.8265	−0.527027
$423$	0	0
$424$	2.00000	0.0971286
$425$	−12.6497	−0.613602
$426$	0	0
$427$	−2.77575	−0.134328
$428$	1.29948	0.0628126
$429$	0	0
$430$	2.29806	0.110822
$431$	3.84955	0.185427	0.0927133	−	0.995693i	$-0.470446\pi$
$431$	3.84955	0.185427	0.0927133	+	0.995693i	$0.470446\pi$
$432$	0	0
$433$	23.3766	1.12341	0.561704	−	0.827338i	$-0.310146\pi$
$433$	23.3766	1.12341	0.561704	+	0.827338i	$0.310146\pi$
$434$	0.387873	0.0186185
$435$	0	0
$436$	−3.16362	−0.151510
$437$	−4.96239	−0.237383
$438$	0	0
$439$	−17.7889	−0.849019	−0.424509	−	0.905424i	$-0.639553\pi$
$439$	−17.7889	−0.849019	−0.424509	+	0.905424i	$0.639553\pi$
$440$	−15.8496	−0.755598
$441$	0	0
$442$	3.22425	0.153362
$443$	9.19982	0.437096	0.218548	−	0.975826i	$-0.429868\pi$
$443$	9.19982	0.437096	0.218548	+	0.975826i	$0.429868\pi$
$444$	0	0
$445$	2.07522	0.0983750
$446$	−22.9380	−1.08614
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	−21.6991	−1.02404	−0.512022	−	0.858972i	$-0.671103\pi$
$449$	−21.6991	−1.02404	−0.512022	+	0.858972i	$0.671103\pi$
$450$	0	0
$451$	42.3996	1.99652
$452$	−18.6253	−0.876060
$453$	0	0
$454$	−23.0738	−1.08291
$455$	−2.85097	−0.133655
$456$	0	0
$457$	−7.67276	−0.358917	−0.179458	−	0.983766i	$-0.557434\pi$
$457$	−7.67276	−0.358917	−0.179458	+	0.983766i	$0.557434\pi$
$458$	−8.07522	−0.377330
$459$	0	0
$460$	−14.7005	−0.685415
$461$	−12.6351	−0.588478	−0.294239	−	0.955732i	$-0.595066\pi$
$461$	−12.6351	−0.588478	−0.294239	+	0.955732i	$0.595066\pi$
$462$	0	0
$463$	10.0752	0.468235	0.234118	−	0.972208i	$-0.424780\pi$
$463$	10.0752	0.468235	0.234118	+	0.972208i	$0.424780\pi$
$464$	1.22425	0.0568346
$465$	0	0
$466$	1.29948	0.0601971
$467$	−25.4763	−1.17890	−0.589451	−	0.807804i	$-0.700656\pi$
$467$	−25.4763	−1.17890	−0.589451	+	0.807804i	$0.700656\pi$
$468$	0	0
$469$	−3.35026	−0.154701
$470$	−36.4749	−1.68246
$471$	0	0
$472$	0	0
$473$	−4.15045	−0.190838
$474$	0	0
$475$	−3.77575	−0.173243
$476$	3.35026	0.153559
$477$	0	0
$478$	0.186642	0.00853682
$479$	15.7889	0.721414	0.360707	−	0.932679i	$-0.382535\pi$
$479$	15.7889	0.721414	0.360707	+	0.932679i	$0.382535\pi$
$480$	0	0
$481$	−1.55149	−0.0707420
$482$	15.2750	0.695759
$483$	0	0
$484$	17.6253	0.801150
$485$	33.4010	1.51666
$486$	0	0
$487$	26.1378	1.18442	0.592208	−	0.805785i	$-0.298257\pi$
$487$	26.1378	1.18442	0.592208	+	0.805785i	$0.298257\pi$
$488$	2.77575	0.125652
$489$	0	0
$490$	−2.96239	−0.133827
$491$	8.97698	0.405125	0.202563	−	0.979269i	$-0.435073\pi$
$491$	8.97698	0.405125	0.202563	+	0.979269i	$0.435073\pi$
$492$	0	0
$493$	−4.10157	−0.184725
$494$	0.962389	0.0432999
$495$	0	0
$496$	−0.387873	−0.0174160
$497$	−14.7005	−0.659409
$498$	0	0
$499$	−25.7743	−1.15382	−0.576909	−	0.816809i	$-0.695741\pi$
$499$	−25.7743	−1.15382	−0.576909	+	0.816809i	$0.695741\pi$
$500$	3.62672	0.162192
$501$	0	0
$502$	14.7757	0.659474
$503$	−15.6385	−0.697285	−0.348643	−	0.937256i	$-0.613357\pi$
$503$	−15.6385	−0.697285	−0.348643	+	0.937256i	$0.613357\pi$
$504$	0	0
$505$	20.6253	0.917814
$506$	26.5501	1.18029
$507$	0	0
$508$	17.6629	0.783665
$509$	5.32582	0.236063	0.118032	−	0.993010i	$-0.462342\pi$
$509$	5.32582	0.236063	0.118032	+	0.993010i	$0.462342\pi$
$510$	0	0
$511$	8.70052	0.384888
$512$	1.00000	0.0441942
$513$	0	0
$514$	2.00000	0.0882162
$515$	−24.4025	−1.07530
$516$	0	0
$517$	65.8759	2.89722
$518$	−1.61213	−0.0708328
$519$	0	0
$520$	2.85097	0.125023
$521$	36.6516	1.60574	0.802869	−	0.596156i	$-0.203306\pi$
$521$	36.6516	1.60574	0.802869	+	0.596156i	$0.203306\pi$
$522$	0	0
$523$	−24.3733	−1.06577	−0.532885	−	0.846188i	$-0.678892\pi$
$523$	−24.3733	−1.06577	−0.532885	+	0.846188i	$0.678892\pi$
$524$	14.6253	0.638909
$525$	0	0
$526$	−24.8119	−1.08185
$527$	1.29948	0.0566061
$528$	0	0
$529$	1.62530	0.0706652
$530$	−5.92478	−0.257356
$531$	0	0
$532$	1.00000	0.0433555
$533$	−7.62672	−0.330350
$534$	0	0
$535$	−3.84955	−0.166431
$536$	3.35026	0.144709
$537$	0	0
$538$	15.4010	0.663986
$539$	5.35026	0.230452
$540$	0	0
$541$	−11.1490	−0.479334	−0.239667	−	0.970855i	$-0.577038\pi$
$541$	−11.1490	−0.479334	−0.239667	+	0.970855i	$0.577038\pi$
$542$	4.62530	0.198674
$543$	0	0
$544$	−3.35026	−0.143641
$545$	9.37187	0.401447
$546$	0	0
$547$	−30.0508	−1.28488	−0.642439	−	0.766336i	$-0.722078\pi$
$547$	−30.0508	−1.28488	−0.642439	+	0.766336i	$0.722078\pi$
$548$	9.92478	0.423965
$549$	0	0
$550$	20.2012	0.861383
$551$	−1.22425	−0.0521550
$552$	0	0
$553$	−11.5877	−0.492759
$554$	−30.4749	−1.29475
$555$	0	0
$556$	12.6253	0.535432
$557$	−33.5369	−1.42100	−0.710502	−	0.703695i	$-0.751532\pi$
$557$	−33.5369	−1.42100	−0.710502	+	0.703695i	$0.751532\pi$
$558$	0	0
$559$	0.746569	0.0315765
$560$	2.96239	0.125184
$561$	0	0
$562$	−9.22425	−0.389102
$563$	−35.6991	−1.50454	−0.752269	−	0.658856i	$-0.771041\pi$
$563$	−35.6991	−1.50454	−0.752269	+	0.658856i	$0.771041\pi$
$564$	0	0
$565$	55.1754	2.32125
$566$	−0.775746	−0.0326070
$567$	0	0
$568$	14.7005	0.616820
$569$	−30.8773	−1.29444	−0.647222	−	0.762301i	$-0.724069\pi$
$569$	−30.8773	−1.29444	−0.647222	+	0.762301i	$0.724069\pi$
$570$	0	0
$571$	29.6239	1.23972	0.619861	−	0.784712i	$-0.287189\pi$
$571$	29.6239	1.23972	0.619861	+	0.784712i	$0.287189\pi$
$572$	−5.14903	−0.215292
$573$	0	0
$574$	−7.92478	−0.330774
$575$	18.7367	0.781375
$576$	0	0
$577$	−23.6531	−0.984690	−0.492345	−	0.870400i	$-0.663860\pi$
$577$	−23.6531	−0.984690	−0.492345	+	0.870400i	$0.663860\pi$
$578$	−5.77575	−0.240239
$579$	0	0
$580$	−3.62672	−0.150591
$581$	−11.9248	−0.494723
$582$	0	0
$583$	10.7005	0.443170
$584$	−8.70052	−0.360030
$585$	0	0
$586$	0.448507	0.0185277
$587$	−22.1016	−0.912229	−0.456115	−	0.889921i	$-0.650759\pi$
$587$	−22.1016	−0.912229	−0.456115	+	0.889921i	$0.650759\pi$
$588$	0	0
$589$	0.387873	0.0159820
$590$	0	0
$591$	0	0
$592$	1.61213	0.0662580
$593$	−9.42548	−0.387058	−0.193529	−	0.981095i	$-0.561993\pi$
$593$	−9.42548	−0.387058	−0.193529	+	0.981095i	$0.561993\pi$
$594$	0	0
$595$	−9.92478	−0.406876
$596$	−1.53690	−0.0629540
$597$	0	0
$598$	−4.77575	−0.195295
$599$	−19.4763	−0.795779	−0.397889	−	0.917433i	$-0.630257\pi$
$599$	−19.4763	−0.795779	−0.397889	+	0.917433i	$0.630257\pi$
$600$	0	0
$601$	26.7513	1.09121	0.545604	−	0.838043i	$-0.316300\pi$
$601$	26.7513	1.09121	0.545604	+	0.838043i	$0.316300\pi$
$602$	0.775746	0.0316171
$603$	0	0
$604$	−12.1114	−0.492807
$605$	−52.2130	−2.12276
$606$	0	0
$607$	−35.4128	−1.43736	−0.718681	−	0.695340i	$-0.755254\pi$
$607$	−35.4128	−1.43736	−0.718681	+	0.695340i	$0.755254\pi$
$608$	−1.00000	−0.0405554
$609$	0	0
$610$	−8.22284	−0.332933
$611$	−11.8496	−0.479382
$612$	0	0
$613$	29.8496	1.20561	0.602806	−	0.797888i	$-0.294049\pi$
$613$	29.8496	1.20561	0.602806	+	0.797888i	$0.294049\pi$
$614$	−2.07522	−0.0837492
$615$	0	0
$616$	−5.35026	−0.215568
$617$	46.5501	1.87404	0.937018	−	0.349282i	$-0.113574\pi$
$617$	46.5501	1.87404	0.937018	+	0.349282i	$0.113574\pi$
$618$	0	0
$619$	−20.3733	−0.818871	−0.409436	−	0.912339i	$-0.634274\pi$
$619$	−20.3733	−0.818871	−0.409436	+	0.912339i	$0.634274\pi$
$620$	1.14903	0.0461462
$621$	0	0
$622$	−3.93937	−0.157954
$623$	0.700523	0.0280659
$624$	0	0
$625$	−29.6225	−1.18490
$626$	−14.0000	−0.559553
$627$	0	0
$628$	23.4010	0.933803
$629$	−5.40105	−0.215354
$630$	0	0
$631$	8.52373	0.339324	0.169662	−	0.985502i	$-0.445732\pi$
$631$	8.52373	0.339324	0.169662	+	0.985502i	$0.445732\pi$
$632$	11.5877	0.460934
$633$	0	0
$634$	11.5515	0.458768
$635$	−52.3244	−2.07643
$636$	0	0
$637$	−0.962389	−0.0381312
$638$	6.55008	0.259320
$639$	0	0
$640$	−2.96239	−0.117099
$641$	−0.448507	−0.0177150	−0.00885749	−	0.999961i	$-0.502819\pi$
$641$	−0.448507	−0.0177150	−0.00885749	+	0.999961i	$0.502819\pi$
$642$	0	0
$643$	33.6531	1.32715	0.663574	−	0.748111i	$-0.269039\pi$
$643$	33.6531	1.32715	0.663574	+	0.748111i	$0.269039\pi$
$644$	−4.96239	−0.195546
$645$	0	0
$646$	3.35026	0.131814
$647$	7.63847	0.300299	0.150150	−	0.988663i	$-0.452024\pi$
$647$	7.63847	0.300299	0.150150	+	0.988663i	$0.452024\pi$
$648$	0	0
$649$	0	0
$650$	−3.63374	−0.142527
$651$	0	0
$652$	−23.8496	−0.934021
$653$	−9.78892	−0.383070	−0.191535	−	0.981486i	$-0.561347\pi$
$653$	−9.78892	−0.383070	−0.191535	+	0.981486i	$0.561347\pi$
$654$	0	0
$655$	−43.3258	−1.69288
$656$	7.92478	0.309411
$657$	0	0
$658$	−12.3127	−0.479997
$659$	31.9511	1.24464	0.622320	−	0.782763i	$-0.286190\pi$
$659$	31.9511	1.24464	0.622320	+	0.782763i	$0.286190\pi$
$660$	0	0
$661$	24.1866	0.940751	0.470376	−	0.882466i	$-0.344118\pi$
$661$	24.1866	0.940751	0.470376	+	0.882466i	$0.344118\pi$
$662$	17.9003	0.695716
$663$	0	0
$664$	11.9248	0.462771
$665$	−2.96239	−0.114877
$666$	0	0
$667$	6.07522	0.235234
$668$	1.92478	0.0744719
$669$	0	0
$670$	−9.92478	−0.383428
$671$	14.8510	0.573315
$672$	0	0
$673$	22.3733	0.862427	0.431213	−	0.902250i	$-0.358086\pi$
$673$	22.3733	0.862427	0.431213	+	0.902250i	$0.358086\pi$
$674$	28.7005	1.10550
$675$	0	0
$676$	−12.0738	−0.464377
$677$	−13.6267	−0.523717	−0.261859	−	0.965106i	$-0.584335\pi$
$677$	−13.6267	−0.523717	−0.261859	+	0.965106i	$0.584335\pi$
$678$	0	0
$679$	11.2750	0.432696
$680$	9.92478	0.380598
$681$	0	0
$682$	−2.07522	−0.0794644
$683$	−47.5487	−1.81940	−0.909700	−	0.415267i	$-0.863688\pi$
$683$	−47.5487	−1.81940	−0.909700	+	0.415267i	$0.863688\pi$
$684$	0	0
$685$	−29.4010	−1.12336
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	−0.775746	−0.0295750
$689$	−1.92478	−0.0733282
$690$	0	0
$691$	40.3536	1.53512	0.767561	−	0.640975i	$-0.221470\pi$
$691$	40.3536	1.53512	0.767561	+	0.640975i	$0.221470\pi$
$692$	7.40105	0.281346
$693$	0	0
$694$	−27.6483	−1.04952
$695$	−37.4010	−1.41870
$696$	0	0
$697$	−26.5501	−1.00566
$698$	3.92478	0.148555
$699$	0	0
$700$	−3.77575	−0.142710
$701$	16.5383	0.624644	0.312322	−	0.949976i	$-0.398893\pi$
$701$	16.5383	0.624644	0.312322	+	0.949976i	$0.398893\pi$
$702$	0	0
$703$	−1.61213	−0.0608025
$704$	5.35026	0.201646
$705$	0	0
$706$	−28.1260	−1.05854
$707$	6.96239	0.261848
$708$	0	0
$709$	−38.7269	−1.45442	−0.727209	−	0.686416i	$-0.759183\pi$
$709$	−38.7269	−1.45442	−0.727209	+	0.686416i	$0.759183\pi$
$710$	−43.5487	−1.63435
$711$	0	0
$712$	−0.700523	−0.0262532
$713$	−1.92478	−0.0720835
$714$	0	0
$715$	15.2534	0.570446
$716$	−21.2506	−0.794172
$717$	0	0
$718$	30.1114	1.12375
$719$	15.1636	0.565508	0.282754	−	0.959193i	$-0.408752\pi$
$719$	15.1636	0.565508	0.282754	+	0.959193i	$0.408752\pi$
$720$	0	0
$721$	−8.23743	−0.306778
$722$	1.00000	0.0372161
$723$	0	0
$724$	−14.8872	−0.553277
$725$	4.62247	0.171674
$726$	0	0
$727$	41.2506	1.52990	0.764950	−	0.644090i	$-0.222764\pi$
$727$	41.2506	1.52990	0.764950	+	0.644090i	$0.222764\pi$
$728$	0.962389	0.0356685
$729$	0	0
$730$	25.7743	0.953951
$731$	2.59895	0.0961258
$732$	0	0
$733$	9.62672	0.355571	0.177785	−	0.984069i	$-0.443107\pi$
$733$	9.62672	0.355571	0.177785	+	0.984069i	$0.443107\pi$
$734$	27.0738	0.999312
$735$	0	0
$736$	4.96239	0.182916
$737$	17.9248	0.660268
$738$	0	0
$739$	−3.74798	−0.137872	−0.0689359	−	0.997621i	$-0.521960\pi$
$739$	−3.74798	−0.137872	−0.0689359	+	0.997621i	$0.521960\pi$
$740$	−4.77575	−0.175560
$741$	0	0
$742$	−2.00000	−0.0734223
$743$	1.29948	0.0476732	0.0238366	−	0.999716i	$-0.492412\pi$
$743$	1.29948	0.0476732	0.0238366	+	0.999716i	$0.492412\pi$
$744$	0	0
$745$	4.55291	0.166806
$746$	6.76116	0.247544
$747$	0	0
$748$	−17.9248	−0.655395
$749$	−1.29948	−0.0474818
$750$	0	0
$751$	39.1852	1.42989	0.714945	−	0.699181i	$-0.246452\pi$
$751$	39.1852	1.42989	0.714945	+	0.699181i	$0.246452\pi$
$752$	12.3127	0.448996
$753$	0	0
$754$	−1.17821	−0.0429078
$755$	35.8787	1.30576
$756$	0	0
$757$	−44.6516	−1.62289	−0.811446	−	0.584428i	$-0.801319\pi$
$757$	−44.6516	−1.62289	−0.811446	+	0.584428i	$0.801319\pi$
$758$	15.6023	0.566700
$759$	0	0
$760$	2.96239	0.107457
$761$	−11.6023	−0.420582	−0.210291	−	0.977639i	$-0.567441\pi$
$761$	−11.6023	−0.420582	−0.210291	+	0.977639i	$0.567441\pi$
$762$	0	0
$763$	3.16362	0.114531
$764$	7.41090	0.268117
$765$	0	0
$766$	−7.37470	−0.266459
$767$	0	0
$768$	0	0
$769$	−9.10299	−0.328262	−0.164131	−	0.986439i	$-0.552482\pi$
$769$	−9.10299	−0.328262	−0.164131	+	0.986439i	$0.552482\pi$
$770$	15.8496	0.571178
$771$	0	0
$772$	−6.77575	−0.243864
$773$	−9.62672	−0.346249	−0.173124	−	0.984900i	$-0.555386\pi$
$773$	−9.62672	−0.346249	−0.173124	+	0.984900i	$0.555386\pi$
$774$	0	0
$775$	−1.46451	−0.0526068
$776$	−11.2750	−0.404750
$777$	0	0
$778$	28.2374	1.01236
$779$	−7.92478	−0.283935
$780$	0	0
$781$	78.6516	2.81438
$782$	−16.6253	−0.594520
$783$	0	0
$784$	1.00000	0.0357143
$785$	−69.3230	−2.47424
$786$	0	0
$787$	−34.3996	−1.22621	−0.613107	−	0.790000i	$-0.710081\pi$
$787$	−34.3996	−1.22621	−0.613107	+	0.790000i	$0.710081\pi$
$788$	2.46310	0.0877442
$789$	0	0
$790$	−34.3272	−1.22131
$791$	18.6253	0.662239
$792$	0	0
$793$	−2.67135	−0.0948623
$794$	−2.77575	−0.0985075
$795$	0	0
$796$	−10.4485	−0.370338
$797$	45.0249	1.59486	0.797432	−	0.603408i	$-0.206191\pi$
$797$	45.0249	1.59486	0.797432	+	0.603408i	$0.206191\pi$
$798$	0	0
$799$	−41.2506	−1.45934
$800$	3.77575	0.133493
$801$	0	0
$802$	14.2520	0.503256
$803$	−46.5501	−1.64272
$804$	0	0
$805$	14.7005	0.518125
$806$	0.373285	0.0131484
$807$	0	0
$808$	−6.96239	−0.244936
$809$	27.5778	0.969585	0.484793	−	0.874629i	$-0.338895\pi$
$809$	27.5778	0.969585	0.484793	+	0.874629i	$0.338895\pi$
$810$	0	0
$811$	−30.4288	−1.06850	−0.534250	−	0.845327i	$-0.679406\pi$
$811$	−30.4288	−1.06850	−0.534250	+	0.845327i	$0.679406\pi$
$812$	−1.22425	−0.0429629
$813$	0	0
$814$	8.62530	0.302317
$815$	70.6516	2.47482
$816$	0	0
$817$	0.775746	0.0271399
$818$	15.0230	0.525268
$819$	0	0
$820$	−23.4763	−0.819827
$821$	8.28630	0.289194	0.144597	−	0.989491i	$-0.453811\pi$
$821$	8.28630	0.289194	0.144597	+	0.989491i	$0.453811\pi$
$822$	0	0
$823$	−4.37328	−0.152443	−0.0762216	−	0.997091i	$-0.524286\pi$
$823$	−4.37328	−0.152443	−0.0762216	+	0.997091i	$0.524286\pi$
$824$	8.23743	0.286964
$825$	0	0
$826$	0	0
$827$	−25.1002	−0.872818	−0.436409	−	0.899748i	$-0.643750\pi$
$827$	−25.1002	−0.872818	−0.436409	+	0.899748i	$0.643750\pi$
$828$	0	0
$829$	31.5125	1.09447	0.547237	−	0.836978i	$-0.315680\pi$
$829$	31.5125	1.09447	0.547237	+	0.836978i	$0.315680\pi$
$830$	−35.3258	−1.22618
$831$	0	0
$832$	−0.962389	−0.0333648
$833$	−3.35026	−0.116080
$834$	0	0
$835$	−5.70194	−0.197324
$836$	−5.35026	−0.185043
$837$	0	0
$838$	24.9525	0.861971
$839$	−46.3996	−1.60189	−0.800947	−	0.598736i	$-0.795670\pi$
$839$	−46.3996	−1.60189	−0.800947	+	0.598736i	$0.795670\pi$
$840$	0	0
$841$	−27.5012	−0.948317
$842$	−1.08840	−0.0375086
$843$	0	0
$844$	−10.8265	−0.372665
$845$	35.7673	1.23043
$846$	0	0
$847$	−17.6253	−0.605612
$848$	2.00000	0.0686803
$849$	0	0
$850$	−12.6497	−0.433882
$851$	8.00000	0.274236
$852$	0	0
$853$	−7.65306	−0.262036	−0.131018	−	0.991380i	$-0.541825\pi$
$853$	−7.65306	−0.262036	−0.131018	+	0.991380i	$0.541825\pi$
$854$	−2.77575	−0.0949841
$855$	0	0
$856$	1.29948	0.0444152
$857$	−5.89843	−0.201487	−0.100743	−	0.994912i	$-0.532122\pi$
$857$	−5.89843	−0.201487	−0.100743	+	0.994912i	$0.532122\pi$
$858$	0	0
$859$	−35.3258	−1.20530	−0.602651	−	0.798005i	$-0.705889\pi$
$859$	−35.3258	−1.20530	−0.602651	+	0.798005i	$0.705889\pi$
$860$	2.29806	0.0783633
$861$	0	0
$862$	3.84955	0.131116
$863$	47.9511	1.63228	0.816138	−	0.577858i	$-0.196111\pi$
$863$	47.9511	1.63228	0.816138	+	0.577858i	$0.196111\pi$
$864$	0	0
$865$	−21.9248	−0.745465
$866$	23.3766	0.794370
$867$	0	0
$868$	0.387873	0.0131653
$869$	61.9972	2.10311
$870$	0	0
$871$	−3.22425	−0.109250
$872$	−3.16362	−0.107134
$873$	0	0
$874$	−4.96239	−0.167855
$875$	−3.62672	−0.122605
$876$	0	0
$877$	48.2638	1.62975	0.814876	−	0.579635i	$-0.196805\pi$
$877$	48.2638	1.62975	0.814876	+	0.579635i	$0.196805\pi$
$878$	−17.7889	−0.600347
$879$	0	0
$880$	−15.8496	−0.534288
$881$	27.8740	0.939099	0.469549	−	0.882906i	$-0.344416\pi$
$881$	27.8740	0.939099	0.469549	+	0.882906i	$0.344416\pi$
$882$	0	0
$883$	−2.07522	−0.0698368	−0.0349184	−	0.999390i	$-0.511117\pi$
$883$	−2.07522	−0.0698368	−0.0349184	+	0.999390i	$0.511117\pi$
$884$	3.22425	0.108443
$885$	0	0
$886$	9.19982	0.309074
$887$	−25.5515	−0.857935	−0.428968	−	0.903320i	$-0.641123\pi$
$887$	−25.5515	−0.857935	−0.428968	+	0.903320i	$0.641123\pi$
$888$	0	0
$889$	−17.6629	−0.592395
$890$	2.07522	0.0695616
$891$	0	0
$892$	−22.9380	−0.768019
$893$	−12.3127	−0.412027
$894$	0	0
$895$	62.9525	2.10427
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−21.6991	−0.724109
$899$	−0.474855	−0.0158373
$900$	0	0
$901$	−6.70052	−0.223227
$902$	42.3996	1.41175
$903$	0	0
$904$	−18.6253	−0.619468
$905$	44.1016	1.46599
$906$	0	0
$907$	−7.50071	−0.249057	−0.124528	−	0.992216i	$-0.539742\pi$
$907$	−7.50071	−0.249057	−0.124528	+	0.992216i	$0.539742\pi$
$908$	−23.0738	−0.765731
$909$	0	0
$910$	−2.85097	−0.0945087
$911$	7.32582	0.242715	0.121358	−	0.992609i	$-0.461275\pi$
$911$	7.32582	0.242715	0.121358	+	0.992609i	$0.461275\pi$
$912$	0	0
$913$	63.8007	2.11149
$914$	−7.67276	−0.253792
$915$	0	0
$916$	−8.07522	−0.266813
$917$	−14.6253	−0.482970
$918$	0	0
$919$	20.3733	0.672053	0.336026	−	0.941853i	$-0.390917\pi$
$919$	20.3733	0.672053	0.336026	+	0.941853i	$0.390917\pi$
$920$	−14.7005	−0.484662
$921$	0	0
$922$	−12.6351	−0.416116
$923$	−14.1476	−0.465674
$924$	0	0
$925$	6.08698	0.200139
$926$	10.0752	0.331092
$927$	0	0
$928$	1.22425	0.0401881
$929$	−12.7513	−0.418357	−0.209178	−	0.977877i	$-0.567079\pi$
$929$	−12.7513	−0.418357	−0.209178	+	0.977877i	$0.567079\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	1.29948	0.0425658
$933$	0	0
$934$	−25.4763	−0.833609
$935$	53.1002	1.73656
$936$	0	0
$937$	37.6991	1.23158	0.615788	−	0.787912i	$-0.288838\pi$
$937$	37.6991	1.23158	0.615788	+	0.787912i	$0.288838\pi$
$938$	−3.35026	−0.109390
$939$	0	0
$940$	−36.4749	−1.18968
$941$	−34.5764	−1.12716	−0.563580	−	0.826062i	$-0.690576\pi$
$941$	−34.5764	−1.12716	−0.563580	+	0.826062i	$0.690576\pi$
$942$	0	0
$943$	39.3258	1.28063
$944$	0	0
$945$	0	0
$946$	−4.15045	−0.134943
$947$	24.9770	0.811643	0.405821	−	0.913952i	$-0.366986\pi$
$947$	24.9770	0.811643	0.405821	+	0.913952i	$0.366986\pi$
$948$	0	0
$949$	8.37328	0.271808
$950$	−3.77575	−0.122501
$951$	0	0
$952$	3.35026	0.108583
$953$	−18.7757	−0.608206	−0.304103	−	0.952639i	$-0.598357\pi$
$953$	−18.7757	−0.608206	−0.304103	+	0.952639i	$0.598357\pi$
$954$	0	0
$955$	−21.9540	−0.710413
$956$	0.186642	0.00603644
$957$	0	0
$958$	15.7889	0.510117
$959$	−9.92478	−0.320488
$960$	0	0
$961$	−30.8496	−0.995147
$962$	−1.55149	−0.0500221
$963$	0	0
$964$	15.2750	0.491976
$965$	20.0724	0.646153
$966$	0	0
$967$	−16.0000	−0.514525	−0.257263	−	0.966342i	$-0.582821\pi$
$967$	−16.0000	−0.514525	−0.257263	+	0.966342i	$0.582821\pi$
$968$	17.6253	0.566499
$969$	0	0
$970$	33.4010	1.07244
$971$	−53.7255	−1.72413	−0.862066	−	0.506796i	$-0.830830\pi$
$971$	−53.7255	−1.72413	−0.862066	+	0.506796i	$0.830830\pi$
$972$	0	0
$973$	−12.6253	−0.404749
$974$	26.1378	0.837508
$975$	0	0
$976$	2.77575	0.0888495
$977$	20.7005	0.662268	0.331134	−	0.943584i	$-0.392569\pi$
$977$	20.7005	0.662268	0.331134	+	0.943584i	$0.392569\pi$
$978$	0	0
$979$	−3.74798	−0.119786
$980$	−2.96239	−0.0946300
$981$	0	0
$982$	8.97698	0.286467
$983$	−17.9248	−0.571712	−0.285856	−	0.958273i	$-0.592278\pi$
$983$	−17.9248	−0.571712	−0.285856	+	0.958273i	$0.592278\pi$
$984$	0	0
$985$	−7.29665	−0.232491
$986$	−4.10157	−0.130621
$987$	0	0
$988$	0.962389	0.0306177
$989$	−3.84955	−0.122409
$990$	0	0
$991$	−40.5863	−1.28927	−0.644633	−	0.764492i	$-0.722990\pi$
$991$	−40.5863	−1.28927	−0.644633	+	0.764492i	$0.722990\pi$
$992$	−0.387873	−0.0123150
$993$	0	0
$994$	−14.7005	−0.466272
$995$	30.9525	0.981261
$996$	0	0
$997$	19.2506	0.609673	0.304836	−	0.952405i	$-0.401398\pi$
$997$	19.2506	0.609673	0.304836	+	0.952405i	$0.401398\pi$
$998$	−25.7743	−0.815872
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2394.2.a.bc.1.1	yes	3
3.2	odd	2		2394.2.a.bb.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2394.2.a.bb.1.3	✓	3	3.2	odd	2
2394.2.a.bc.1.1	yes	3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2394.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -2.96239 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -2.96239 q^{5} -1.00000 q^{7} +1.00000 q^{8} -2.96239 q^{10} +5.35026 q^{11} -0.962389 q^{13} -1.00000 q^{14} +1.00000 q^{16} -3.35026 q^{17} -1.00000 q^{19} -2.96239 q^{20} +5.35026 q^{22} +4.96239 q^{23} +3.77575 q^{25} -0.962389 q^{26} -1.00000 q^{28} +1.22425 q^{29} -0.387873 q^{31} +1.00000 q^{32} -3.35026 q^{34} +2.96239 q^{35} +1.61213 q^{37} -1.00000 q^{38} -2.96239 q^{40} +7.92478 q^{41} -0.775746 q^{43} +5.35026 q^{44} +4.96239 q^{46} +12.3127 q^{47} +1.00000 q^{49} +3.77575 q^{50} -0.962389 q^{52} +2.00000 q^{53} -15.8496 q^{55} -1.00000 q^{56} +1.22425 q^{58} +2.77575 q^{61} -0.387873 q^{62} +1.00000 q^{64} +2.85097 q^{65} +3.35026 q^{67} -3.35026 q^{68} +2.96239 q^{70} +14.7005 q^{71} -8.70052 q^{73} +1.61213 q^{74} -1.00000 q^{76} -5.35026 q^{77} +11.5877 q^{79} -2.96239 q^{80} +7.92478 q^{82} +11.9248 q^{83} +9.92478 q^{85} -0.775746 q^{86} +5.35026 q^{88} -0.700523 q^{89} +0.962389 q^{91} +4.96239 q^{92} +12.3127 q^{94} +2.96239 q^{95} -11.2750 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} + 2 q^{5} - 3 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^4 + 2 * q^5 - 3 * q^7 + 3 * q^8	\( 3 q + 3 q^{2} + 3 q^{4} + 2 q^{5} - 3 q^{7} + 3 q^{8} + 2 q^{10} + 6 q^{11} + 8 q^{13} - 3 q^{14} + 3 q^{16} - 3 q^{19} + 2 q^{20} + 6 q^{22} + 4 q^{23} + 13 q^{25} + 8 q^{26} - 3 q^{28} + 2 q^{29} - 2 q^{31} + 3 q^{32} - 2 q^{35} + 4 q^{37} - 3 q^{38} + 2 q^{40} + 2 q^{41} - 4 q^{43} + 6 q^{44} + 4 q^{46} + 16 q^{47} + 3 q^{49} + 13 q^{50} + 8 q^{52} + 6 q^{53} - 4 q^{55} - 3 q^{56} + 2 q^{58} + 10 q^{61} - 2 q^{62} + 3 q^{64} + 32 q^{65} - 2 q^{70} + 24 q^{71} - 6 q^{73} + 4 q^{74} - 3 q^{76} - 6 q^{77} - 18 q^{79} + 2 q^{80} + 2 q^{82} + 14 q^{83} + 8 q^{85} - 4 q^{86} + 6 q^{88} + 18 q^{89} - 8 q^{91} + 4 q^{92} + 16 q^{94} - 2 q^{95} - 2 q^{97} + 3 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 + 2 * q^5 - 3 * q^7 + 3 * q^8 + 2 * q^10 + 6 * q^11 + 8 * q^13 - 3 * q^14 + 3 * q^16 - 3 * q^19 + 2 * q^20 + 6 * q^22 + 4 * q^23 + 13 * q^25 + 8 * q^26 - 3 * q^28 + 2 * q^29 - 2 * q^31 + 3 * q^32 - 2 * q^35 + 4 * q^37 - 3 * q^38 + 2 * q^40 + 2 * q^41 - 4 * q^43 + 6 * q^44 + 4 * q^46 + 16 * q^47 + 3 * q^49 + 13 * q^50 + 8 * q^52 + 6 * q^53 - 4 * q^55 - 3 * q^56 + 2 * q^58 + 10 * q^61 - 2 * q^62 + 3 * q^64 + 32 * q^65 - 2 * q^70 + 24 * q^71 - 6 * q^73 + 4 * q^74 - 3 * q^76 - 6 * q^77 - 18 * q^79 + 2 * q^80 + 2 * q^82 + 14 * q^83 + 8 * q^85 - 4 * q^86 + 6 * q^88 + 18 * q^89 - 8 * q^91 + 4 * q^92 + 16 * q^94 - 2 * q^95 - 2 * q^97 + 3 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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