LMFDB - Embedded newform 1755.2.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1755,2,Mod(1,1755)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1755, 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("1755.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1755 = 3^{3} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1755.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:	$14.0137455547$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.3981.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} + 2x + 1 $ x^4 - x^3 - 4x^2 + 2x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.75080$ of defining polynomial
Character	$\chi$	$=$	1755.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.75080 q^{2} +5.56690 q^{4} +1.00000 q^{5} +0.636469 q^{7} -9.81184 q^{8} +O(q^{10})$	\(q-2.75080 q^{2} +5.56690 q^{4} +1.00000 q^{5} +0.636469 q^{7} -9.81184 q^{8} -2.75080 q^{10} -0.314502 q^{11} -1.00000 q^{13} -1.75080 q^{14} +15.8566 q^{16} -3.52960 q^{17} +2.11433 q^{19} +5.56690 q^{20} +0.865131 q^{22} -2.09331 q^{23} +1.00000 q^{25} +2.75080 q^{26} +3.54316 q^{28} +5.56264 q^{29} -9.92617 q^{31} -23.9947 q^{32} +9.70924 q^{34} +0.636469 q^{35} -4.89496 q^{37} -5.81610 q^{38} -9.81184 q^{40} -7.90243 q^{41} -1.11433 q^{43} -1.75080 q^{44} +5.75827 q^{46} -3.59219 q^{47} -6.59491 q^{49} -2.75080 q^{50} -5.56690 q^{52} -6.15163 q^{53} -0.314502 q^{55} -6.24494 q^{56} -15.3017 q^{58} +3.99574 q^{59} +11.7260 q^{61} +27.3049 q^{62} +34.2914 q^{64} -1.00000 q^{65} +10.3222 q^{67} -19.6490 q^{68} -1.75080 q^{70} -8.15434 q^{71} -11.6442 q^{73} +13.4651 q^{74} +11.7703 q^{76} -0.200171 q^{77} -15.4142 q^{79} +15.8566 q^{80} +21.7380 q^{82} +9.55837 q^{83} -3.52960 q^{85} +3.06530 q^{86} +3.08584 q^{88} -0.0695665 q^{89} -0.636469 q^{91} -11.6532 q^{92} +9.88140 q^{94} +2.11433 q^{95} -17.3550 q^{97} +18.1413 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3 q^{2} + 3 q^{4} + 4 q^{5} - q^{7} - 9 q^{8}+O(q^{10}) $ 4 * q - 3 * q^2 + 3 * q^4 + 4 * q^5 - q^7 - 9 * q^8	$ 4 q - 3 q^{2} + 3 q^{4} + 4 q^{5} - q^{7} - 9 q^{8} - 3 q^{10} - 6 q^{11} - 4 q^{13} + q^{14} + 13 q^{16} - 2 q^{17} + 4 q^{19} + 3 q^{20} - 9 q^{22} - 9 q^{23} + 4 q^{25} + 3 q^{26} + 10 q^{28} - 16 q^{29} - 5 q^{31} - 26 q^{32} + 19 q^{34} - q^{35} - 13 q^{37} - 12 q^{38} - 9 q^{40} - 12 q^{41} + q^{44} + 2 q^{46} - 9 q^{47} - 11 q^{49} - 3 q^{50} - 3 q^{52} - 13 q^{53} - 6 q^{55} - 14 q^{56} - 9 q^{58} - 3 q^{59} + 3 q^{61} + 25 q^{62} + 45 q^{64} - 4 q^{65} + 6 q^{67} - 32 q^{68} + q^{70} - 11 q^{71} - 3 q^{73} - 4 q^{74} + 5 q^{76} - 10 q^{77} - 3 q^{79} + 13 q^{80} + 22 q^{82} - 19 q^{83} - 2 q^{85} + 9 q^{86} + 26 q^{88} - 16 q^{89} + q^{91} - 19 q^{92} + 25 q^{94} + 4 q^{95} - 35 q^{97} + 21 q^{98}+O(q^{100}) $ 4 * q - 3 * q^2 + 3 * q^4 + 4 * q^5 - q^7 - 9 * q^8 - 3 * q^10 - 6 * q^11 - 4 * q^13 + q^14 + 13 * q^16 - 2 * q^17 + 4 * q^19 + 3 * q^20 - 9 * q^22 - 9 * q^23 + 4 * q^25 + 3 * q^26 + 10 * q^28 - 16 * q^29 - 5 * q^31 - 26 * q^32 + 19 * q^34 - q^35 - 13 * q^37 - 12 * q^38 - 9 * q^40 - 12 * q^41 + q^44 + 2 * q^46 - 9 * q^47 - 11 * q^49 - 3 * q^50 - 3 * q^52 - 13 * q^53 - 6 * q^55 - 14 * q^56 - 9 * q^58 - 3 * q^59 + 3 * q^61 + 25 * q^62 + 45 * q^64 - 4 * q^65 + 6 * q^67 - 32 * q^68 + q^70 - 11 * q^71 - 3 * q^73 - 4 * q^74 + 5 * q^76 - 10 * q^77 - 3 * q^79 + 13 * q^80 + 22 * q^82 - 19 * q^83 - 2 * q^85 + 9 * q^86 + 26 * q^88 - 16 * q^89 + q^91 - 19 * q^92 + 25 * q^94 + 4 * q^95 - 35 * q^97 + 21 * 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 $

$2$	−2.75080	−1.94511	−0.972555	−	0.232674i	$-0.925253\pi$
$2$	−2.75080	−1.94511	−0.972555	+	0.232674i	$0.925253\pi$
$3$	0	0
$3$	0	0
$4$	5.56690	2.78345
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0.636469	0.240563	0.120281	−	0.992740i	$-0.461620\pi$
$7$	0.636469	0.240563	0.120281	+	0.992740i	$0.461620\pi$
$8$	−9.81184	−3.46901
$9$	0	0
$10$	−2.75080	−0.869879
$11$	−0.314502	−0.0948258	−0.0474129	−	0.998875i	$-0.515098\pi$
$11$	−0.314502	−0.0948258	−0.0474129	+	0.998875i	$0.515098\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−1.75080	−0.467921
$15$	0	0
$16$	15.8566	3.96415
$17$	−3.52960	−0.856055	−0.428027	−	0.903766i	$-0.640791\pi$
$17$	−3.52960	−0.856055	−0.428027	+	0.903766i	$0.640791\pi$
$18$	0	0
$19$	2.11433	0.485061	0.242530	−	0.970144i	$-0.422023\pi$
$19$	2.11433	0.485061	0.242530	+	0.970144i	$0.422023\pi$
$20$	5.56690	1.24480
$21$	0	0
$22$	0.865131	0.184447
$23$	−2.09331	−0.436484	−0.218242	−	0.975895i	$-0.570032\pi$
$23$	−2.09331	−0.436484	−0.218242	+	0.975895i	$0.570032\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	2.75080	0.539476
$27$	0	0
$28$	3.54316	0.669595
$29$	5.56264	1.03296	0.516478	−	0.856300i	$-0.327243\pi$
$29$	5.56264	1.03296	0.516478	+	0.856300i	$0.327243\pi$
$30$	0	0
$31$	−9.92617	−1.78279	−0.891396	−	0.453225i	$-0.850274\pi$
$31$	−9.92617	−1.78279	−0.891396	+	0.453225i	$0.850274\pi$
$32$	−23.9947	−4.24170
$33$	0	0
$34$	9.70924	1.66512
$35$	0.636469	0.107583
$36$	0	0
$37$	−4.89496	−0.804727	−0.402364	−	0.915480i	$-0.631811\pi$
$37$	−4.89496	−0.804727	−0.402364	+	0.915480i	$0.631811\pi$
$38$	−5.81610	−0.943496
$39$	0	0
$40$	−9.81184	−1.55139
$41$	−7.90243	−1.23415	−0.617076	−	0.786903i	$-0.711683\pi$
$41$	−7.90243	−1.23415	−0.617076	+	0.786903i	$0.711683\pi$
$42$	0	0
$43$	−1.11433	−0.169934	−0.0849669	−	0.996384i	$-0.527078\pi$
$43$	−1.11433	−0.169934	−0.0849669	+	0.996384i	$0.527078\pi$
$44$	−1.75080	−0.263943
$45$	0	0
$46$	5.75827	0.849010
$47$	−3.59219	−0.523975	−0.261988	−	0.965071i	$-0.584378\pi$
$47$	−3.59219	−0.523975	−0.261988	+	0.965071i	$0.584378\pi$
$48$	0	0
$49$	−6.59491	−0.942130
$50$	−2.75080	−0.389022
$51$	0	0
$52$	−5.56690	−0.771991
$53$	−6.15163	−0.844991	−0.422496	−	0.906365i	$-0.638846\pi$
$53$	−6.15163	−0.844991	−0.422496	+	0.906365i	$0.638846\pi$
$54$	0	0
$55$	−0.314502	−0.0424074
$56$	−6.24494	−0.834515
$57$	0	0
$58$	−15.3017	−2.00921
$59$	3.99574	0.520200	0.260100	−	0.965582i	$-0.416244\pi$
$59$	3.99574	0.520200	0.260100	+	0.965582i	$0.416244\pi$
$60$	0	0
$61$	11.7260	1.50136	0.750680	−	0.660666i	$-0.229726\pi$
$61$	11.7260	1.50136	0.750680	+	0.660666i	$0.229726\pi$
$62$	27.3049	3.46773
$63$	0	0
$64$	34.2914	4.28642
$65$	−1.00000	−0.124035
$66$	0	0
$67$	10.3222	1.26106	0.630532	−	0.776164i	$-0.282837\pi$
$67$	10.3222	1.26106	0.630532	+	0.776164i	$0.282837\pi$
$68$	−19.6490	−2.38279
$69$	0	0
$70$	−1.75080	−0.209261
$71$	−8.15434	−0.967743	−0.483871	−	0.875139i	$-0.660770\pi$
$71$	−8.15434	−0.967743	−0.483871	+	0.875139i	$0.660770\pi$
$72$	0	0
$73$	−11.6442	−1.36285	−0.681426	−	0.731887i	$-0.738640\pi$
$73$	−11.6442	−1.36285	−0.681426	+	0.731887i	$0.738640\pi$
$74$	13.4651	1.56528
$75$	0	0
$76$	11.7703	1.35014
$77$	−0.200171	−0.0228116
$78$	0	0
$79$	−15.4142	−1.73423	−0.867117	−	0.498104i	$-0.834030\pi$
$79$	−15.4142	−1.73423	−0.867117	+	0.498104i	$0.834030\pi$
$80$	15.8566	1.77282
$81$	0	0
$82$	21.7380	2.40056
$83$	9.55837	1.04917	0.524584	−	0.851359i	$-0.324221\pi$
$83$	9.55837	1.04917	0.524584	+	0.851359i	$0.324221\pi$
$84$	0	0
$85$	−3.52960	−0.382839
$86$	3.06530	0.330540
$87$	0	0
$88$	3.08584	0.328952
$89$	−0.0695665	−0.00737403	−0.00368702	−	0.999993i	$-0.501174\pi$
$89$	−0.0695665	−0.00737403	−0.00368702	+	0.999993i	$0.501174\pi$
$90$	0	0
$91$	−0.636469	−0.0667201
$92$	−11.6532	−1.21493
$93$	0	0
$94$	9.88140	1.01919
$95$	2.11433	0.216926
$96$	0	0
$97$	−17.3550	−1.76213	−0.881067	−	0.472992i	$-0.843174\pi$
$97$	−17.3550	−1.76213	−0.881067	+	0.472992i	$0.843174\pi$
$98$	18.1413	1.83255
$99$	0	0
$100$	5.56690	0.556690
$101$	19.7809	1.96828	0.984139	−	0.177401i	$-0.0567689\pi$
$101$	19.7809	1.96828	0.984139	+	0.177401i	$0.0567689\pi$
$102$	0	0
$103$	−5.36490	−0.528620	−0.264310	−	0.964438i	$-0.585144\pi$
$103$	−5.36490	−0.528620	−0.264310	+	0.964438i	$0.585144\pi$
$104$	9.81184	0.962130
$105$	0	0
$106$	16.9219	1.64360
$107$	6.21965	0.601276	0.300638	−	0.953738i	$-0.402800\pi$
$107$	6.21965	0.601276	0.300638	+	0.953738i	$0.402800\pi$
$108$	0	0
$109$	10.1661	0.973733	0.486867	−	0.873476i	$-0.338140\pi$
$109$	10.1661	0.973733	0.486867	+	0.873476i	$0.338140\pi$
$110$	0.865131	0.0824870
$111$	0	0
$112$	10.0922	0.953627
$113$	11.9677	1.12583	0.562915	−	0.826515i	$-0.309680\pi$
$113$	11.9677	1.12583	0.562915	+	0.826515i	$0.309680\pi$
$114$	0	0
$115$	−2.09331	−0.195202
$116$	30.9667	2.87518
$117$	0	0
$118$	−10.9915	−1.01185
$119$	−2.24649	−0.205935
$120$	0	0
$121$	−10.9011	−0.991008
$122$	−32.2559	−2.92031
$123$	0	0
$124$	−55.2580	−4.96232
$125$	1.00000	0.0894427
$126$	0	0
$127$	13.8939	1.23288	0.616442	−	0.787400i	$-0.288573\pi$
$127$	13.8939	1.23288	0.616442	+	0.787400i	$0.288573\pi$
$128$	−46.3393	−4.09586
$129$	0	0
$130$	2.75080	0.241261
$131$	17.5557	1.53385	0.766923	−	0.641740i	$-0.221787\pi$
$131$	17.5557	1.53385	0.766923	+	0.641740i	$0.221787\pi$
$132$	0	0
$133$	1.34571	0.116688
$134$	−28.3944	−2.45291
$135$	0	0
$136$	34.6319	2.96966
$137$	−7.94168	−0.678503	−0.339252	−	0.940696i	$-0.610174\pi$
$137$	−7.94168	−0.678503	−0.339252	+	0.940696i	$0.610174\pi$
$138$	0	0
$139$	−8.97703	−0.761421	−0.380711	−	0.924694i	$-0.624321\pi$
$139$	−8.97703	−0.761421	−0.380711	+	0.924694i	$0.624321\pi$
$140$	3.54316	0.299452
$141$	0	0
$142$	22.4310	1.88237
$143$	0.314502	0.0263000
$144$	0	0
$145$	5.56264	0.461952
$146$	32.0309	2.65090
$147$	0	0
$148$	−27.2498	−2.23992
$149$	−20.8683	−1.70960	−0.854800	−	0.518957i	$-0.826320\pi$
$149$	−20.8683	−1.70960	−0.854800	+	0.518957i	$0.826320\pi$
$150$	0	0
$151$	3.78307	0.307862	0.153931	−	0.988082i	$-0.450807\pi$
$151$	3.78307	0.307862	0.153931	+	0.988082i	$0.450807\pi$
$152$	−20.7455	−1.68268
$153$	0	0
$154$	0.550630	0.0443710
$155$	−9.92617	−0.797289
$156$	0	0
$157$	−11.0043	−0.878236	−0.439118	−	0.898429i	$-0.644709\pi$
$157$	−11.0043	−0.878236	−0.439118	+	0.898429i	$0.644709\pi$
$158$	42.4014	3.37328
$159$	0	0
$160$	−23.9947	−1.89695
$161$	−1.33233	−0.105002
$162$	0	0
$163$	−12.1133	−0.948784	−0.474392	−	0.880314i	$-0.657332\pi$
$163$	−12.1133	−0.948784	−0.474392	+	0.880314i	$0.657332\pi$
$164$	−43.9921	−3.43520
$165$	0	0
$166$	−26.2932	−2.04075
$167$	−19.3507	−1.49741	−0.748703	−	0.662906i	$-0.769323\pi$
$167$	−19.3507	−1.49741	−0.748703	+	0.662906i	$0.769323\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	9.70924	0.744665
$171$	0	0
$172$	−6.20337	−0.473003
$173$	−11.0285	−0.838481	−0.419240	−	0.907875i	$-0.637704\pi$
$173$	−11.0285	−0.838481	−0.419240	+	0.907875i	$0.637704\pi$
$174$	0	0
$175$	0.636469	0.0481126
$176$	−4.98693	−0.375904
$177$	0	0
$178$	0.191364	0.0143433
$179$	−25.0720	−1.87397	−0.936984	−	0.349372i	$-0.886395\pi$
$179$	−25.0720	−1.87397	−0.936984	+	0.349372i	$0.886395\pi$
$180$	0	0
$181$	2.03121	0.150978	0.0754892	−	0.997147i	$-0.475948\pi$
$181$	2.03121	0.150978	0.0754892	+	0.997147i	$0.475948\pi$
$182$	1.75080	0.129778
$183$	0	0
$184$	20.5392	1.51417
$185$	−4.89496	−0.359885
$186$	0	0
$187$	1.11007	0.0811761
$188$	−19.9974	−1.45846
$189$	0	0
$190$	−5.81610	−0.421944
$191$	−9.36381	−0.677541	−0.338771	−	0.940869i	$-0.610011\pi$
$191$	−9.36381	−0.677541	−0.338771	+	0.940869i	$0.610011\pi$
$192$	0	0
$193$	−5.89585	−0.424393	−0.212196	−	0.977227i	$-0.568062\pi$
$193$	−5.89585	−0.424393	−0.212196	+	0.977227i	$0.568062\pi$
$194$	47.7401	3.42754
$195$	0	0
$196$	−36.7132	−2.62237
$197$	−0.0948560	−0.00675821	−0.00337911	−	0.999994i	$-0.501076\pi$
$197$	−0.0948560	−0.00675821	−0.00337911	+	0.999994i	$0.501076\pi$
$198$	0	0
$199$	−3.54454	−0.251266	−0.125633	−	0.992077i	$-0.540096\pi$
$199$	−3.54454	−0.251266	−0.125633	+	0.992077i	$0.540096\pi$
$200$	−9.81184	−0.693802
$201$	0	0
$202$	−54.4134	−3.82852
$203$	3.54045	0.248491
$204$	0	0
$205$	−7.90243	−0.551930
$206$	14.7578	1.02822
$207$	0	0
$208$	−15.8566	−1.09946
$209$	−0.664961	−0.0459963
$210$	0	0
$211$	11.1543	0.767897	0.383948	−	0.923355i	$-0.374564\pi$
$211$	11.1543	0.767897	0.383948	+	0.923355i	$0.374564\pi$
$212$	−34.2455	−2.35199
$213$	0	0
$214$	−17.1090	−1.16955
$215$	−1.11433	−0.0759967
$216$	0	0
$217$	−6.31770	−0.428874
$218$	−27.9648	−1.89402
$219$	0	0
$220$	−1.75080	−0.118039
$221$	3.52960	0.237427
$222$	0	0
$223$	−9.15163	−0.612838	−0.306419	−	0.951897i	$-0.599131\pi$
$223$	−9.15163	−0.612838	−0.306419	+	0.951897i	$0.599131\pi$
$224$	−15.2719	−1.02040
$225$	0	0
$226$	−32.9208	−2.18986
$227$	20.1761	1.33914	0.669569	−	0.742750i	$-0.266479\pi$
$227$	20.1761	1.33914	0.669569	+	0.742750i	$0.266479\pi$
$228$	0	0
$229$	−5.32468	−0.351865	−0.175932	−	0.984402i	$-0.556294\pi$
$229$	−5.32468	−0.351865	−0.175932	+	0.984402i	$0.556294\pi$
$230$	5.75827	0.379689
$231$	0	0
$232$	−54.5797	−3.58333
$233$	5.03681	0.329972	0.164986	−	0.986296i	$-0.447242\pi$
$233$	5.03681	0.329972	0.164986	+	0.986296i	$0.447242\pi$
$234$	0	0
$235$	−3.59219	−0.234329
$236$	22.2439	1.44795
$237$	0	0
$238$	6.17963	0.400566
$239$	0.992534	0.0642017	0.0321008	−	0.999485i	$-0.489780\pi$
$239$	0.992534	0.0642017	0.0321008	+	0.999485i	$0.489780\pi$
$240$	0	0
$241$	0.0741093	0.00477380	0.00238690	−	0.999997i	$-0.499240\pi$
$241$	0.0741093	0.00477380	0.00238690	+	0.999997i	$0.499240\pi$
$242$	29.9867	1.92762
$243$	0	0
$244$	65.2775	4.17896
$245$	−6.59491	−0.421333
$246$	0	0
$247$	−2.11433	−0.134532
$248$	97.3940	6.18452
$249$	0	0
$250$	−2.75080	−0.173976
$251$	−6.37399	−0.402323	−0.201161	−	0.979558i	$-0.564472\pi$
$251$	−6.37399	−0.402323	−0.201161	+	0.979558i	$0.564472\pi$
$252$	0	0
$253$	0.658348	0.0413900
$254$	−38.2193	−2.39810
$255$	0	0
$256$	58.8875	3.68047
$257$	29.9086	1.86565	0.932824	−	0.360331i	$-0.117336\pi$
$257$	29.9086	1.86565	0.932824	+	0.360331i	$0.117336\pi$
$258$	0	0
$259$	−3.11549	−0.193587
$260$	−5.56690	−0.345245
$261$	0	0
$262$	−48.2921	−2.98350
$263$	11.3609	0.700544	0.350272	−	0.936648i	$-0.386089\pi$
$263$	11.3609	0.700544	0.350272	+	0.936648i	$0.386089\pi$
$264$	0	0
$265$	−6.15163	−0.377892
$266$	−3.70177	−0.226970
$267$	0	0
$268$	57.4629	3.51011
$269$	−32.2378	−1.96557	−0.982786	−	0.184747i	$-0.940853\pi$
$269$	−32.2378	−1.96557	−0.982786	+	0.184747i	$0.940853\pi$
$270$	0	0
$271$	−1.26093	−0.0765960	−0.0382980	−	0.999266i	$-0.512194\pi$
$271$	−1.26093	−0.0765960	−0.0382980	+	0.999266i	$0.512194\pi$
$272$	−55.9675	−3.39353
$273$	0	0
$274$	21.8460	1.31976
$275$	−0.314502	−0.0189652
$276$	0	0
$277$	20.4938	1.23136	0.615678	−	0.787998i	$-0.288882\pi$
$277$	20.4938	1.23136	0.615678	+	0.787998i	$0.288882\pi$
$278$	24.6940	1.48105
$279$	0	0
$280$	−6.24494	−0.373206
$281$	−12.9027	−0.769711	−0.384856	−	0.922977i	$-0.625749\pi$
$281$	−12.9027	−0.769711	−0.384856	+	0.922977i	$0.625749\pi$
$282$	0	0
$283$	−24.9032	−1.48034	−0.740171	−	0.672419i	$-0.765255\pi$
$283$	−24.9032	−1.48034	−0.740171	+	0.672419i	$0.765255\pi$
$284$	−45.3944	−2.69366
$285$	0	0
$286$	−0.865131	−0.0511563
$287$	−5.02966	−0.296891
$288$	0	0
$289$	−4.54189	−0.267170
$290$	−15.3017	−0.898547
$291$	0	0
$292$	−64.8222	−3.79343
$293$	−17.0516	−0.996166	−0.498083	−	0.867129i	$-0.665963\pi$
$293$	−17.0516	−0.996166	−0.498083	+	0.867129i	$0.665963\pi$
$294$	0	0
$295$	3.99574	0.232641
$296$	48.0286	2.79161
$297$	0	0
$298$	57.4046	3.32536
$299$	2.09331	0.121059
$300$	0	0
$301$	−0.709238	−0.0408798
$302$	−10.4065	−0.598825
$303$	0	0
$304$	33.5261	1.92285
$305$	11.7260	0.671429
$306$	0	0
$307$	−1.82065	−0.103910	−0.0519549	−	0.998649i	$-0.516545\pi$
$307$	−1.82065	−0.103910	−0.0519549	+	0.998649i	$0.516545\pi$
$308$	−1.11433	−0.0634949
$309$	0	0
$310$	27.3049	1.55081
$311$	−0.464790	−0.0263558	−0.0131779	−	0.999913i	$-0.504195\pi$
$311$	−0.464790	−0.0263558	−0.0131779	+	0.999913i	$0.504195\pi$
$312$	0	0
$313$	23.8391	1.34746	0.673732	−	0.738976i	$-0.264690\pi$
$313$	23.8391	1.34746	0.673732	+	0.738976i	$0.264690\pi$
$314$	30.2705	1.70827
$315$	0	0
$316$	−85.8094	−4.82716
$317$	−29.8982	−1.67925	−0.839624	−	0.543168i	$-0.817225\pi$
$317$	−29.8982	−1.67925	−0.839624	+	0.543168i	$0.817225\pi$
$318$	0	0
$319$	−1.74946	−0.0979509
$320$	34.2914	1.91694
$321$	0	0
$322$	3.66496	0.204240
$323$	−7.46275	−0.415239
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	33.3212	1.84549
$327$	0	0
$328$	77.5374	4.28128
$329$	−2.28632	−0.126049
$330$	0	0
$331$	22.0200	1.21033	0.605163	−	0.796101i	$-0.293108\pi$
$331$	22.0200	1.21033	0.605163	+	0.796101i	$0.293108\pi$
$332$	53.2105	2.92031
$333$	0	0
$334$	53.2300	2.91262
$335$	10.3222	0.563965
$336$	0	0
$337$	−20.0200	−1.09056	−0.545278	−	0.838255i	$-0.683576\pi$
$337$	−20.0200	−1.09056	−0.545278	+	0.838255i	$0.683576\pi$
$338$	−2.75080	−0.149624
$339$	0	0
$340$	−19.6490	−1.06561
$341$	3.12180	0.169055
$342$	0	0
$343$	−8.65274	−0.467204
$344$	10.9336	0.589502
$345$	0	0
$346$	30.3372	1.63094
$347$	−20.2774	−1.08855	−0.544274	−	0.838908i	$-0.683195\pi$
$347$	−20.2774	−1.08855	−0.544274	+	0.838908i	$0.683195\pi$
$348$	0	0
$349$	26.4273	1.41462	0.707310	−	0.706904i	$-0.249909\pi$
$349$	26.4273	1.41462	0.707310	+	0.706904i	$0.249909\pi$
$350$	−1.75080	−0.0935842
$351$	0	0
$352$	7.54637	0.402223
$353$	4.09562	0.217988	0.108994	−	0.994042i	$-0.465237\pi$
$353$	4.09562	0.217988	0.108994	+	0.994042i	$0.465237\pi$
$354$	0	0
$355$	−8.15434	−0.432788
$356$	−0.387270	−0.0205253
$357$	0	0
$358$	68.9680	3.64507
$359$	−7.69324	−0.406034	−0.203017	−	0.979175i	$-0.565075\pi$
$359$	−7.69324	−0.406034	−0.203017	+	0.979175i	$0.565075\pi$
$360$	0	0
$361$	−14.5296	−0.764716
$362$	−5.58744	−0.293669
$363$	0	0
$364$	−3.54316	−0.185712
$365$	−11.6442	−0.609486
$366$	0	0
$367$	33.6208	1.75499	0.877495	−	0.479585i	$-0.159213\pi$
$367$	33.6208	1.75499	0.877495	+	0.479585i	$0.159213\pi$
$368$	−33.1927	−1.73029
$369$	0	0
$370$	13.4651	0.700016
$371$	−3.91532	−0.203274
$372$	0	0
$373$	0.501322	0.0259575	0.0129787	−	0.999916i	$-0.495869\pi$
$373$	0.501322	0.0259575	0.0129787	+	0.999916i	$0.495869\pi$
$374$	−3.05357	−0.157896
$375$	0	0
$376$	35.2460	1.81767
$377$	−5.56264	−0.286490
$378$	0	0
$379$	−15.4328	−0.792730	−0.396365	−	0.918093i	$-0.629728\pi$
$379$	−15.4328	−0.792730	−0.396365	+	0.918093i	$0.629728\pi$
$380$	11.7703	0.603802
$381$	0	0
$382$	25.7580	1.31789
$383$	24.5785	1.25590	0.627950	−	0.778254i	$-0.283894\pi$
$383$	24.5785	1.25590	0.627950	+	0.778254i	$0.283894\pi$
$384$	0	0
$385$	−0.200171	−0.0102016
$386$	16.2183	0.825490
$387$	0	0
$388$	−96.6136	−4.90481
$389$	−20.8380	−1.05653	−0.528265	−	0.849080i	$-0.677157\pi$
$389$	−20.8380	−1.05653	−0.528265	+	0.849080i	$0.677157\pi$
$390$	0	0
$391$	7.38854	0.373655
$392$	64.7082	3.26826
$393$	0	0
$394$	0.260930	0.0131455
$395$	−15.4142	−0.775573
$396$	0	0
$397$	10.0936	0.506582	0.253291	−	0.967390i	$-0.418487\pi$
$397$	10.0936	0.506582	0.253291	+	0.967390i	$0.418487\pi$
$398$	9.75031	0.488739
$399$	0	0
$400$	15.8566	0.792830
$401$	16.0645	0.802224	0.401112	−	0.916029i	$-0.368624\pi$
$401$	16.0645	0.802224	0.401112	+	0.916029i	$0.368624\pi$
$402$	0	0
$403$	9.92617	0.494458
$404$	110.119	5.47860
$405$	0	0
$406$	−9.73907	−0.483342
$407$	1.53947	0.0763089
$408$	0	0
$409$	−32.0291	−1.58374	−0.791868	−	0.610692i	$-0.790891\pi$
$409$	−32.0291	−1.58374	−0.791868	+	0.610692i	$0.790891\pi$
$410$	21.7380	1.07356
$411$	0	0
$412$	−29.8659	−1.47139
$413$	2.54316	0.125141
$414$	0	0
$415$	9.55837	0.469202
$416$	23.9947	1.17644
$417$	0	0
$418$	1.82917	0.0894678
$419$	7.20231	0.351856	0.175928	−	0.984403i	$-0.443707\pi$
$419$	7.20231	0.351856	0.175928	+	0.984403i	$0.443707\pi$
$420$	0	0
$421$	−34.6308	−1.68780	−0.843902	−	0.536497i	$-0.819747\pi$
$421$	−34.6308	−1.68780	−0.843902	+	0.536497i	$0.819747\pi$
$422$	−30.6834	−1.49364
$423$	0	0
$424$	60.3588	2.93128
$425$	−3.52960	−0.171211
$426$	0	0
$427$	7.46324	0.361171
$428$	34.6242	1.67362
$429$	0	0
$430$	3.06530	0.147822
$431$	23.7127	1.14220	0.571101	−	0.820880i	$-0.306517\pi$
$431$	23.7127	1.14220	0.571101	+	0.820880i	$0.306517\pi$
$432$	0	0
$433$	4.57397	0.219811	0.109906	−	0.993942i	$-0.464945\pi$
$433$	4.57397	0.219811	0.109906	+	0.993942i	$0.464945\pi$
$434$	17.3787	0.834206
$435$	0	0
$436$	56.5935	2.71034
$437$	−4.42594	−0.211721
$438$	0	0
$439$	0.904866	0.0431869	0.0215934	−	0.999767i	$-0.493126\pi$
$439$	0.904866	0.0431869	0.0215934	+	0.999767i	$0.493126\pi$
$440$	3.08584	0.147112
$441$	0	0
$442$	−9.70924	−0.461821
$443$	18.7196	0.889396	0.444698	−	0.895680i	$-0.353311\pi$
$443$	18.7196	0.889396	0.444698	+	0.895680i	$0.353311\pi$
$444$	0	0
$445$	−0.0695665	−0.00329777
$446$	25.1743	1.19204
$447$	0	0
$448$	21.8254	1.03115
$449$	−31.3718	−1.48053	−0.740263	−	0.672318i	$-0.765299\pi$
$449$	−31.3718	−1.48053	−0.740263	+	0.672318i	$0.765299\pi$
$450$	0	0
$451$	2.48533	0.117030
$452$	66.6232	3.13369
$453$	0	0
$454$	−55.5005	−2.60477
$455$	−0.636469	−0.0298381
$456$	0	0
$457$	16.0985	0.753058	0.376529	−	0.926405i	$-0.377117\pi$
$457$	16.0985	0.753058	0.376529	+	0.926405i	$0.377117\pi$
$458$	14.6471	0.684416
$459$	0	0
$460$	−11.6532	−0.543335
$461$	6.07917	0.283135	0.141568	−	0.989929i	$-0.454786\pi$
$461$	6.07917	0.283135	0.141568	+	0.989929i	$0.454786\pi$
$462$	0	0
$463$	17.0597	0.792831	0.396416	−	0.918071i	$-0.370254\pi$
$463$	17.0597	0.792831	0.396416	+	0.918071i	$0.370254\pi$
$464$	88.2045	4.09479
$465$	0	0
$466$	−13.8553	−0.641833
$467$	10.3825	0.480447	0.240224	−	0.970718i	$-0.422779\pi$
$467$	10.3825	0.480447	0.240224	+	0.970718i	$0.422779\pi$
$468$	0	0
$469$	6.56979	0.303365
$470$	9.88140	0.455795
$471$	0	0
$472$	−39.2055	−1.80458
$473$	0.350459	0.0161141
$474$	0	0
$475$	2.11433	0.0970122
$476$	−12.5060	−0.573210
$477$	0	0
$478$	−2.73026	−0.124879
$479$	1.43358	0.0655022	0.0327511	−	0.999464i	$-0.489573\pi$
$479$	1.43358	0.0655022	0.0327511	+	0.999464i	$0.489573\pi$
$480$	0	0
$481$	4.89496	0.223191
$482$	−0.203860	−0.00928557
$483$	0	0
$484$	−60.6853	−2.75842
$485$	−17.3550	−0.788050
$486$	0	0
$487$	−4.79102	−0.217102	−0.108551	−	0.994091i	$-0.534621\pi$
$487$	−4.79102	−0.217102	−0.108551	+	0.994091i	$0.534621\pi$
$488$	−115.054	−5.20823
$489$	0	0
$490$	18.1413	0.819539
$491$	−30.2511	−1.36521	−0.682607	−	0.730785i	$-0.739154\pi$
$491$	−30.2511	−1.36521	−0.682607	+	0.730785i	$0.739154\pi$
$492$	0	0
$493$	−19.6339	−0.884267
$494$	5.81610	0.261679
$495$	0	0
$496$	−157.395	−7.06726
$497$	−5.18999	−0.232803
$498$	0	0
$499$	23.9552	1.07238	0.536192	−	0.844096i	$-0.319862\pi$
$499$	23.9552	1.07238	0.536192	+	0.844096i	$0.319862\pi$
$500$	5.56690	0.248959
$501$	0	0
$502$	17.5336	0.782562
$503$	3.65457	0.162949	0.0814746	−	0.996675i	$-0.474037\pi$
$503$	3.65457	0.162949	0.0814746	+	0.996675i	$0.474037\pi$
$504$	0	0
$505$	19.7809	0.880240
$506$	−1.81098	−0.0805081
$507$	0	0
$508$	77.3460	3.43167
$509$	−26.4561	−1.17265	−0.586323	−	0.810078i	$-0.699425\pi$
$509$	−26.4561	−1.17265	−0.586323	+	0.810078i	$0.699425\pi$
$510$	0	0
$511$	−7.41119	−0.327852
$512$	−69.3092	−3.06306
$513$	0	0
$514$	−82.2727	−3.62889
$515$	−5.36490	−0.236406
$516$	0	0
$517$	1.12975	0.0496864
$518$	8.57010	0.376549
$519$	0	0
$520$	9.81184	0.430278
$521$	−36.2508	−1.58817	−0.794087	−	0.607804i	$-0.792051\pi$
$521$	−36.2508	−1.58817	−0.794087	+	0.607804i	$0.792051\pi$
$522$	0	0
$523$	34.7421	1.51917	0.759583	−	0.650411i	$-0.225403\pi$
$523$	34.7421	1.51917	0.759583	+	0.650411i	$0.225403\pi$
$524$	97.7307	4.26938
$525$	0	0
$526$	−31.2516	−1.36264
$527$	35.0355	1.52617
$528$	0	0
$529$	−18.6181	−0.809481
$530$	16.9219	0.735041
$531$	0	0
$532$	7.49142	0.324794
$533$	7.90243	0.342292
$534$	0	0
$535$	6.21965	0.268899
$536$	−101.280	−4.37464
$537$	0	0
$538$	88.6797	3.82325
$539$	2.07411	0.0893382
$540$	0	0
$541$	33.8121	1.45370	0.726849	−	0.686797i	$-0.240984\pi$
$541$	33.8121	1.45370	0.726849	+	0.686797i	$0.240984\pi$
$542$	3.46857	0.148988
$543$	0	0
$544$	84.6917	3.63113
$545$	10.1661	0.435467
$546$	0	0
$547$	1.85660	0.0793826	0.0396913	−	0.999212i	$-0.487363\pi$
$547$	1.85660	0.0793826	0.0396913	+	0.999212i	$0.487363\pi$
$548$	−44.2105	−1.88858
$549$	0	0
$550$	0.865131	0.0368893
$551$	11.7613	0.501046
$552$	0	0
$553$	−9.81067	−0.417192
$554$	−56.3745	−2.39512
$555$	0	0
$556$	−49.9742	−2.11938
$557$	−14.8514	−0.629272	−0.314636	−	0.949212i	$-0.601883\pi$
$557$	−14.8514	−0.629272	−0.314636	+	0.949212i	$0.601883\pi$
$558$	0	0
$559$	1.11433	0.0471312
$560$	10.0922	0.426475
$561$	0	0
$562$	35.4928	1.49717
$563$	−9.69672	−0.408668	−0.204334	−	0.978901i	$-0.565503\pi$
$563$	−9.69672	−0.408668	−0.204334	+	0.978901i	$0.565503\pi$
$564$	0	0
$565$	11.9677	0.503486
$566$	68.5037	2.87943
$567$	0	0
$568$	80.0091	3.35711
$569$	−33.9922	−1.42503	−0.712514	−	0.701658i	$-0.752444\pi$
$569$	−33.9922	−1.42503	−0.712514	+	0.701658i	$0.752444\pi$
$570$	0	0
$571$	−38.4031	−1.60712	−0.803559	−	0.595225i	$-0.797063\pi$
$571$	−38.4031	−1.60712	−0.803559	+	0.595225i	$0.797063\pi$
$572$	1.75080	0.0732046
$573$	0	0
$574$	13.8356	0.577486
$575$	−2.09331	−0.0872969
$576$	0	0
$577$	28.6156	1.19128	0.595642	−	0.803250i	$-0.296898\pi$
$577$	28.6156	1.19128	0.595642	+	0.803250i	$0.296898\pi$
$578$	12.4938	0.519675
$579$	0	0
$580$	30.9667	1.28582
$581$	6.08361	0.252391
$582$	0	0
$583$	1.93470	0.0801270
$584$	114.251	4.72775
$585$	0	0
$586$	46.9056	1.93765
$587$	−6.28145	−0.259263	−0.129632	−	0.991562i	$-0.541379\pi$
$587$	−6.28145	−0.259263	−0.129632	+	0.991562i	$0.541379\pi$
$588$	0	0
$589$	−20.9872	−0.864763
$590$	−10.9915	−0.452512
$591$	0	0
$592$	−77.6175	−3.19006
$593$	−27.3155	−1.12172	−0.560858	−	0.827912i	$-0.689528\pi$
$593$	−27.3155	−1.12172	−0.560858	+	0.827912i	$0.689528\pi$
$594$	0	0
$595$	−2.24649	−0.0920969
$596$	−116.172	−4.75859
$597$	0	0
$598$	−5.75827	−0.235473
$599$	−34.9568	−1.42830	−0.714148	−	0.699995i	$-0.753186\pi$
$599$	−34.9568	−1.42830	−0.714148	+	0.699995i	$0.753186\pi$
$600$	0	0
$601$	18.4485	0.752531	0.376265	−	0.926512i	$-0.377208\pi$
$601$	18.4485	0.752531	0.376265	+	0.926512i	$0.377208\pi$
$602$	1.95097	0.0795156
$603$	0	0
$604$	21.0600	0.856918
$605$	−10.9011	−0.443192
$606$	0	0
$607$	14.4662	0.587163	0.293582	−	0.955934i	$-0.405153\pi$
$607$	14.4662	0.587163	0.293582	+	0.955934i	$0.405153\pi$
$608$	−50.7327	−2.05748
$609$	0	0
$610$	−32.2559	−1.30600
$611$	3.59219	0.145325
$612$	0	0
$613$	0.131092	0.00529475	0.00264737	−	0.999996i	$-0.499157\pi$
$613$	0.131092	0.00529475	0.00264737	+	0.999996i	$0.499157\pi$
$614$	5.00823	0.202116
$615$	0	0
$616$	1.96404	0.0791335
$617$	−2.31742	−0.0932960	−0.0466480	−	0.998911i	$-0.514854\pi$
$617$	−2.31742	−0.0932960	−0.0466480	+	0.998911i	$0.514854\pi$
$618$	0	0
$619$	32.8281	1.31947	0.659737	−	0.751496i	$-0.270668\pi$
$619$	32.8281	1.31947	0.659737	+	0.751496i	$0.270668\pi$
$620$	−55.2580	−2.21922
$621$	0	0
$622$	1.27854	0.0512650
$623$	−0.0442769	−0.00177392
$624$	0	0
$625$	1.00000	0.0400000
$626$	−65.5765	−2.62096
$627$	0	0
$628$	−61.2597	−2.44453
$629$	17.2773	0.688891
$630$	0	0
$631$	4.91916	0.195829	0.0979143	−	0.995195i	$-0.468783\pi$
$631$	4.91916	0.195829	0.0979143	+	0.995195i	$0.468783\pi$
$632$	151.242	6.01607
$633$	0	0
$634$	82.2439	3.26632
$635$	13.8939	0.551363
$636$	0	0
$637$	6.59491	0.261300
$638$	4.81241	0.190525
$639$	0	0
$640$	−46.3393	−1.83172
$641$	27.0299	1.06762	0.533808	−	0.845606i	$-0.320761\pi$
$641$	27.0299	1.06762	0.533808	+	0.845606i	$0.320761\pi$
$642$	0	0
$643$	−33.1805	−1.30851	−0.654256	−	0.756273i	$-0.727018\pi$
$643$	−33.1805	−1.30851	−0.654256	+	0.756273i	$0.727018\pi$
$644$	−7.41693	−0.292268
$645$	0	0
$646$	20.5285	0.807685
$647$	32.3819	1.27307	0.636533	−	0.771250i	$-0.280368\pi$
$647$	32.3819	1.27307	0.636533	+	0.771250i	$0.280368\pi$
$648$	0	0
$649$	−1.25667	−0.0493284
$650$	2.75080	0.107895
$651$	0	0
$652$	−67.4334	−2.64090
$653$	41.6519	1.62996	0.814982	−	0.579487i	$-0.196747\pi$
$653$	41.6519	1.62996	0.814982	+	0.579487i	$0.196747\pi$
$654$	0	0
$655$	17.5557	0.685956
$656$	−125.306	−4.89237
$657$	0	0
$658$	6.28921	0.245179
$659$	43.0706	1.67779	0.838896	−	0.544291i	$-0.183201\pi$
$659$	43.0706	1.67779	0.838896	+	0.544291i	$0.183201\pi$
$660$	0	0
$661$	24.8428	0.966272	0.483136	−	0.875545i	$-0.339498\pi$
$661$	24.8428	0.966272	0.483136	+	0.875545i	$0.339498\pi$
$662$	−60.5725	−2.35422
$663$	0	0
$664$	−93.7852	−3.63957
$665$	1.34571	0.0521843
$666$	0	0
$667$	−11.6443	−0.450869
$668$	−107.724	−4.16795
$669$	0	0
$670$	−28.3944	−1.09697
$671$	−3.68785	−0.142368
$672$	0	0
$673$	−20.8557	−0.803927	−0.401963	−	0.915656i	$-0.631672\pi$
$673$	−20.8557	−0.803927	−0.401963	+	0.915656i	$0.631672\pi$
$674$	55.0709	2.12125
$675$	0	0
$676$	5.56690	0.214112
$677$	−17.5443	−0.674283	−0.337142	−	0.941454i	$-0.609460\pi$
$677$	−17.5443	−0.674283	−0.337142	+	0.941454i	$0.609460\pi$
$678$	0	0
$679$	−11.0459	−0.423904
$680$	34.6319	1.32807
$681$	0	0
$682$	−8.58744	−0.328830
$683$	33.3876	1.27754	0.638770	−	0.769398i	$-0.279444\pi$
$683$	33.3876	1.27754	0.638770	+	0.769398i	$0.279444\pi$
$684$	0	0
$685$	−7.94168	−0.303436
$686$	23.8020	0.908763
$687$	0	0
$688$	−17.6695	−0.673644
$689$	6.15163	0.234358
$690$	0	0
$691$	37.0095	1.40791	0.703954	−	0.710246i	$-0.251416\pi$
$691$	37.0095	1.40791	0.703954	+	0.710246i	$0.251416\pi$
$692$	−61.3945	−2.33387
$693$	0	0
$694$	55.7791	2.11734
$695$	−8.97703	−0.340518
$696$	0	0
$697$	27.8925	1.05650
$698$	−72.6962	−2.75159
$699$	0	0
$700$	3.54316	0.133919
$701$	−18.1157	−0.684221	−0.342110	−	0.939660i	$-0.611142\pi$
$701$	−18.1157	−0.684221	−0.342110	+	0.939660i	$0.611142\pi$
$702$	0	0
$703$	−10.3496	−0.390342
$704$	−10.7847	−0.406463
$705$	0	0
$706$	−11.2662	−0.424010
$707$	12.5900	0.473494
$708$	0	0
$709$	20.2917	0.762072	0.381036	−	0.924560i	$-0.375567\pi$
$709$	20.2917	0.762072	0.381036	+	0.924560i	$0.375567\pi$
$710$	22.4310	0.841819
$711$	0	0
$712$	0.682575	0.0255806
$713$	20.7785	0.778161
$714$	0	0
$715$	0.314502	0.0117617
$716$	−139.573	−5.21610
$717$	0	0
$718$	21.1626	0.789780
$719$	5.69479	0.212380	0.106190	−	0.994346i	$-0.466135\pi$
$719$	5.69479	0.212380	0.106190	+	0.994346i	$0.466135\pi$
$720$	0	0
$721$	−3.41460	−0.127166
$722$	39.9680	1.48746
$723$	0	0
$724$	11.3075	0.420241
$725$	5.56264	0.206591
$726$	0	0
$727$	−10.4118	−0.386151	−0.193076	−	0.981184i	$-0.561846\pi$
$727$	−10.4118	−0.386151	−0.193076	+	0.981184i	$0.561846\pi$
$728$	6.24494	0.231453
$729$	0	0
$730$	32.0309	1.18552
$731$	3.93315	0.145473
$732$	0	0
$733$	−2.11028	−0.0779448	−0.0389724	−	0.999240i	$-0.512408\pi$
$733$	−2.11028	−0.0779448	−0.0389724	+	0.999240i	$0.512408\pi$
$734$	−92.4841	−3.41365
$735$	0	0
$736$	50.2282	1.85144
$737$	−3.24636	−0.119581
$738$	0	0
$739$	−1.95544	−0.0719322	−0.0359661	−	0.999353i	$-0.511451\pi$
$739$	−1.95544	−0.0719322	−0.0359661	+	0.999353i	$0.511451\pi$
$740$	−27.2498	−1.00172
$741$	0	0
$742$	10.7703	0.395389
$743$	−14.5142	−0.532473	−0.266237	−	0.963908i	$-0.585780\pi$
$743$	−14.5142	−0.532473	−0.266237	+	0.963908i	$0.585780\pi$
$744$	0	0
$745$	−20.8683	−0.764557
$746$	−1.37904	−0.0504902
$747$	0	0
$748$	6.17963	0.225950
$749$	3.95861	0.144645
$750$	0	0
$751$	−20.7503	−0.757190	−0.378595	−	0.925562i	$-0.623593\pi$
$751$	−20.7503	−0.757190	−0.378595	+	0.925562i	$0.623593\pi$
$752$	−56.9600	−2.07712
$753$	0	0
$754$	15.3017	0.557255
$755$	3.78307	0.137680
$756$	0	0
$757$	−36.0917	−1.31178	−0.655888	−	0.754858i	$-0.727706\pi$
$757$	−36.0917	−1.31178	−0.655888	+	0.754858i	$0.727706\pi$
$758$	42.4526	1.54195
$759$	0	0
$760$	−20.7455	−0.752517
$761$	−0.887498	−0.0321718	−0.0160859	−	0.999871i	$-0.505121\pi$
$761$	−0.887498	−0.0321718	−0.0160859	+	0.999871i	$0.505121\pi$
$762$	0	0
$763$	6.47040	0.234244
$764$	−52.1274	−1.88590
$765$	0	0
$766$	−67.6104	−2.44286
$767$	−3.99574	−0.144278
$768$	0	0
$769$	42.4934	1.53235	0.766175	−	0.642632i	$-0.222157\pi$
$769$	42.4934	1.53235	0.766175	+	0.642632i	$0.222157\pi$
$770$	0.550630	0.0198433
$771$	0	0
$772$	−32.8216	−1.18128
$773$	−6.29686	−0.226482	−0.113241	−	0.993568i	$-0.536123\pi$
$773$	−6.29686	−0.226482	−0.113241	+	0.993568i	$0.536123\pi$
$774$	0	0
$775$	−9.92617	−0.356559
$776$	170.284	6.11286
$777$	0	0
$778$	57.3212	2.05507
$779$	−16.7084	−0.598639
$780$	0	0
$781$	2.56455	0.0917670
$782$	−20.3244	−0.726799
$783$	0	0
$784$	−104.573	−3.73474
$785$	−11.0043	−0.392759
$786$	0	0
$787$	35.1703	1.25369	0.626843	−	0.779145i	$-0.284347\pi$
$787$	35.1703	1.25369	0.626843	+	0.779145i	$0.284347\pi$
$788$	−0.528054	−0.0188112
$789$	0	0
$790$	42.4014	1.50857
$791$	7.61710	0.270833
$792$	0	0
$793$	−11.7260	−0.416402
$794$	−27.7654	−0.985358
$795$	0	0
$796$	−19.7321	−0.699385
$797$	−20.3813	−0.721944	−0.360972	−	0.932577i	$-0.617555\pi$
$797$	−20.3813	−0.721944	−0.360972	+	0.932577i	$0.617555\pi$
$798$	0	0
$799$	12.6790	0.448551
$800$	−23.9947	−0.848340
$801$	0	0
$802$	−44.1903	−1.56041
$803$	3.66213	0.129234
$804$	0	0
$805$	−1.33233	−0.0469583
$806$	−27.3049	−0.961774
$807$	0	0
$808$	−194.087	−6.82797
$809$	−31.5123	−1.10791	−0.553956	−	0.832546i	$-0.686882\pi$
$809$	−31.5123	−1.10791	−0.553956	+	0.832546i	$0.686882\pi$
$810$	0	0
$811$	43.3744	1.52308	0.761540	−	0.648117i	$-0.224443\pi$
$811$	43.3744	1.52308	0.761540	+	0.648117i	$0.224443\pi$
$812$	19.7093	0.691662
$813$	0	0
$814$	−4.23479	−0.148429
$815$	−12.1133	−0.424309
$816$	0	0
$817$	−2.35606	−0.0824283
$818$	88.1056	3.08054
$819$	0	0
$820$	−43.9921	−1.53627
$821$	13.0501	0.455451	0.227725	−	0.973725i	$-0.426871\pi$
$821$	13.0501	0.455451	0.227725	+	0.973725i	$0.426871\pi$
$822$	0	0
$823$	42.7756	1.49106	0.745532	−	0.666470i	$-0.232196\pi$
$823$	42.7756	1.49106	0.745532	+	0.666470i	$0.232196\pi$
$824$	52.6396	1.83379
$825$	0	0
$826$	−6.99574	−0.243413
$827$	17.1977	0.598024	0.299012	−	0.954249i	$-0.403343\pi$
$827$	17.1977	0.598024	0.299012	+	0.954249i	$0.403343\pi$
$828$	0	0
$829$	14.9920	0.520695	0.260348	−	0.965515i	$-0.416163\pi$
$829$	14.9920	0.520695	0.260348	+	0.965515i	$0.416163\pi$
$830$	−26.2932	−0.912650
$831$	0	0
$832$	−34.2914	−1.18884
$833$	23.2774	0.806515
$834$	0	0
$835$	−19.3507	−0.669660
$836$	−3.70177	−0.128028
$837$	0	0
$838$	−19.8121	−0.684398
$839$	−11.3878	−0.393151	−0.196576	−	0.980489i	$-0.562982\pi$
$839$	−11.3878	−0.393151	−0.196576	+	0.980489i	$0.562982\pi$
$840$	0	0
$841$	1.94295	0.0669982
$842$	95.2625	3.28296
$843$	0	0
$844$	62.0951	2.13740
$845$	1.00000	0.0344010
$846$	0	0
$847$	−6.93821	−0.238400
$848$	−97.5439	−3.34967
$849$	0	0
$850$	9.70924	0.333024
$851$	10.2467	0.351251
$852$	0	0
$853$	21.1974	0.725784	0.362892	−	0.931831i	$-0.381789\pi$
$853$	21.1974	0.725784	0.362892	+	0.931831i	$0.381789\pi$
$854$	−20.5299	−0.702518
$855$	0	0
$856$	−61.0262	−2.08583
$857$	−37.7119	−1.28822	−0.644108	−	0.764935i	$-0.722771\pi$
$857$	−37.7119	−1.28822	−0.644108	+	0.764935i	$0.722771\pi$
$858$	0	0
$859$	−37.8695	−1.29209	−0.646045	−	0.763299i	$-0.723578\pi$
$859$	−37.8695	−1.29209	−0.646045	+	0.763299i	$0.723578\pi$
$860$	−6.20337	−0.211533
$861$	0	0
$862$	−65.2290	−2.22171
$863$	−3.96714	−0.135043	−0.0675216	−	0.997718i	$-0.521509\pi$
$863$	−3.96714	−0.135043	−0.0675216	+	0.997718i	$0.521509\pi$
$864$	0	0
$865$	−11.0285	−0.374980
$866$	−12.5821	−0.427557
$867$	0	0
$868$	−35.1700	−1.19375
$869$	4.84780	0.164450
$870$	0	0
$871$	−10.3222	−0.349756
$872$	−99.7479	−3.37789
$873$	0	0
$874$	12.1749	0.411821
$875$	0.636469	0.0215166
$876$	0	0
$877$	−18.2445	−0.616072	−0.308036	−	0.951375i	$-0.599672\pi$
$877$	−18.2445	−0.616072	−0.308036	+	0.951375i	$0.599672\pi$
$878$	−2.48910	−0.0840032
$879$	0	0
$880$	−4.98693	−0.168109
$881$	−11.5104	−0.387797	−0.193898	−	0.981022i	$-0.562113\pi$
$881$	−11.5104	−0.387797	−0.193898	+	0.981022i	$0.562113\pi$
$882$	0	0
$883$	−18.1785	−0.611755	−0.305877	−	0.952071i	$-0.598950\pi$
$883$	−18.1785	−0.611755	−0.305877	+	0.952071i	$0.598950\pi$
$884$	19.6490	0.660866
$885$	0	0
$886$	−51.4940	−1.72997
$887$	55.4075	1.86040	0.930201	−	0.367050i	$-0.119632\pi$
$887$	55.4075	1.86040	0.930201	+	0.367050i	$0.119632\pi$
$888$	0	0
$889$	8.84304	0.296586
$890$	0.191364	0.00641452
$891$	0	0
$892$	−50.9462	−1.70581
$893$	−7.59508	−0.254160
$894$	0	0
$895$	−25.0720	−0.838064
$896$	−29.4936	−0.985311
$897$	0	0
$898$	86.2975	2.87978
$899$	−55.2157	−1.84155
$900$	0	0
$901$	21.7128	0.723359
$902$	−6.83664	−0.227635
$903$	0	0
$904$	−117.425	−3.90551
$905$	2.03121	0.0675196
$906$	0	0
$907$	−31.3736	−1.04174	−0.520872	−	0.853635i	$-0.674393\pi$
$907$	−31.3736	−1.04174	−0.520872	+	0.853635i	$0.674393\pi$
$908$	112.319	3.72742
$909$	0	0
$910$	1.75080	0.0580385
$911$	22.8030	0.755497	0.377749	−	0.925908i	$-0.376698\pi$
$911$	22.8030	0.755497	0.377749	+	0.925908i	$0.376698\pi$
$912$	0	0
$913$	−3.00612	−0.0994882
$914$	−44.2839	−1.46478
$915$	0	0
$916$	−29.6420	−0.979399
$917$	11.1736	0.368986
$918$	0	0
$919$	−9.33986	−0.308094	−0.154047	−	0.988064i	$-0.549231\pi$
$919$	−9.33986	−0.308094	−0.154047	+	0.988064i	$0.549231\pi$
$920$	20.5392	0.677157
$921$	0	0
$922$	−16.7226	−0.550729
$923$	8.15434	0.268403
$924$	0	0
$925$	−4.89496	−0.160945
$926$	−46.9278	−1.54214
$927$	0	0
$928$	−133.474	−4.38149
$929$	37.6011	1.23365	0.616827	−	0.787099i	$-0.288418\pi$
$929$	37.6011	1.23365	0.616827	+	0.787099i	$0.288418\pi$
$930$	0	0
$931$	−13.9438	−0.456990
$932$	28.0394	0.918462
$933$	0	0
$934$	−28.5603	−0.934522
$935$	1.11007	0.0363031
$936$	0	0
$937$	33.5559	1.09622	0.548112	−	0.836405i	$-0.315347\pi$
$937$	33.5559	1.09622	0.548112	+	0.836405i	$0.315347\pi$
$938$	−18.0722	−0.590078
$939$	0	0
$940$	−19.9974	−0.652243
$941$	33.5203	1.09273	0.546365	−	0.837547i	$-0.316011\pi$
$941$	33.5203	1.09273	0.546365	+	0.837547i	$0.316011\pi$
$942$	0	0
$943$	16.5422	0.538688
$944$	63.3588	2.06215
$945$	0	0
$946$	−0.964043	−0.0313437
$947$	7.63946	0.248249	0.124125	−	0.992267i	$-0.460388\pi$
$947$	7.63946	0.248249	0.124125	+	0.992267i	$0.460388\pi$
$948$	0	0
$949$	11.6442	0.377987
$950$	−5.81610	−0.188699
$951$	0	0
$952$	22.0422	0.714390
$953$	28.0056	0.907190	0.453595	−	0.891208i	$-0.350141\pi$
$953$	28.0056	0.907190	0.453595	+	0.891208i	$0.350141\pi$
$954$	0	0
$955$	−9.36381	−0.303006
$956$	5.52534	0.178702
$957$	0	0
$958$	−3.94351	−0.127409
$959$	−5.05463	−0.163223
$960$	0	0
$961$	67.5288	2.17835
$962$	−13.4651	−0.434131
$963$	0	0
$964$	0.412559	0.0132876
$965$	−5.89585	−0.189794
$966$	0	0
$967$	−32.6663	−1.05048	−0.525239	−	0.850955i	$-0.676024\pi$
$967$	−32.6663	−1.05048	−0.525239	+	0.850955i	$0.676024\pi$
$968$	106.960	3.43782
$969$	0	0
$970$	47.7401	1.53284
$971$	33.1125	1.06263	0.531316	−	0.847174i	$-0.321698\pi$
$971$	33.1125	1.06263	0.531316	+	0.847174i	$0.321698\pi$
$972$	0	0
$973$	−5.71360	−0.183170
$974$	13.1791	0.422287
$975$	0	0
$976$	185.934	5.95162
$977$	42.2245	1.35088	0.675441	−	0.737414i	$-0.263953\pi$
$977$	42.2245	1.35088	0.675441	+	0.737414i	$0.263953\pi$
$978$	0	0
$979$	0.0218788	0.000699249 0
$980$	−36.7132	−1.17276
$981$	0	0
$982$	83.2148	2.65549
$983$	25.9889	0.828916	0.414458	−	0.910068i	$-0.363971\pi$
$983$	25.9889	0.828916	0.414458	+	0.910068i	$0.363971\pi$
$984$	0	0
$985$	−0.0948560	−0.00302236
$986$	54.0090	1.72000
$987$	0	0
$988$	−11.7703	−0.374462
$989$	2.33264	0.0741735
$990$	0	0
$991$	18.3926	0.584261	0.292130	−	0.956379i	$-0.405636\pi$
$991$	18.3926	0.584261	0.292130	+	0.956379i	$0.405636\pi$
$992$	238.175	7.56207
$993$	0	0
$994$	14.2766	0.452827
$995$	−3.54454	−0.112369
$996$	0	0
$997$	48.9187	1.54927	0.774635	−	0.632408i	$-0.217934\pi$
$997$	48.9187	1.54927	0.774635	+	0.632408i	$0.217934\pi$
$998$	−65.8961	−2.08590
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1755.2.a.n.1.1	✓	4
3.2	odd	2		1755.2.a.t.1.4	yes	4
5.4	even	2		8775.2.a.bs.1.4		4
15.14	odd	2		8775.2.a.bg.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1755.2.a.n.1.1	✓	4	1.1	even	1	trivial
1755.2.a.t.1.4	yes	4	3.2	odd	2
8775.2.a.bg.1.1		4	15.14	odd	2
8775.2.a.bs.1.4		4	5.4	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:	\( N \)	\(=\)	\( 1755 = 3^{3} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1755.a (trivial)

\(f(q)\)	\(=\)	\(q-2.75080 q^{2} +5.56690 q^{4} +1.00000 q^{5} +0.636469 q^{7} -9.81184 q^{8} +O(q^{10})\)	\(q-2.75080 q^{2} +5.56690 q^{4} +1.00000 q^{5} +0.636469 q^{7} -9.81184 q^{8} -2.75080 q^{10} -0.314502 q^{11} -1.00000 q^{13} -1.75080 q^{14} +15.8566 q^{16} -3.52960 q^{17} +2.11433 q^{19} +5.56690 q^{20} +0.865131 q^{22} -2.09331 q^{23} +1.00000 q^{25} +2.75080 q^{26} +3.54316 q^{28} +5.56264 q^{29} -9.92617 q^{31} -23.9947 q^{32} +9.70924 q^{34} +0.636469 q^{35} -4.89496 q^{37} -5.81610 q^{38} -9.81184 q^{40} -7.90243 q^{41} -1.11433 q^{43} -1.75080 q^{44} +5.75827 q^{46} -3.59219 q^{47} -6.59491 q^{49} -2.75080 q^{50} -5.56690 q^{52} -6.15163 q^{53} -0.314502 q^{55} -6.24494 q^{56} -15.3017 q^{58} +3.99574 q^{59} +11.7260 q^{61} +27.3049 q^{62} +34.2914 q^{64} -1.00000 q^{65} +10.3222 q^{67} -19.6490 q^{68} -1.75080 q^{70} -8.15434 q^{71} -11.6442 q^{73} +13.4651 q^{74} +11.7703 q^{76} -0.200171 q^{77} -15.4142 q^{79} +15.8566 q^{80} +21.7380 q^{82} +9.55837 q^{83} -3.52960 q^{85} +3.06530 q^{86} +3.08584 q^{88} -0.0695665 q^{89} -0.636469 q^{91} -11.6532 q^{92} +9.88140 q^{94} +2.11433 q^{95} -17.3550 q^{97} +18.1413 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3 q^{2} + 3 q^{4} + 4 q^{5} - q^{7} - 9 q^{8}+O(q^{10}) \) 4 * q - 3 * q^2 + 3 * q^4 + 4 * q^5 - q^7 - 9 * q^8	\( 4 q - 3 q^{2} + 3 q^{4} + 4 q^{5} - q^{7} - 9 q^{8} - 3 q^{10} - 6 q^{11} - 4 q^{13} + q^{14} + 13 q^{16} - 2 q^{17} + 4 q^{19} + 3 q^{20} - 9 q^{22} - 9 q^{23} + 4 q^{25} + 3 q^{26} + 10 q^{28} - 16 q^{29} - 5 q^{31} - 26 q^{32} + 19 q^{34} - q^{35} - 13 q^{37} - 12 q^{38} - 9 q^{40} - 12 q^{41} + q^{44} + 2 q^{46} - 9 q^{47} - 11 q^{49} - 3 q^{50} - 3 q^{52} - 13 q^{53} - 6 q^{55} - 14 q^{56} - 9 q^{58} - 3 q^{59} + 3 q^{61} + 25 q^{62} + 45 q^{64} - 4 q^{65} + 6 q^{67} - 32 q^{68} + q^{70} - 11 q^{71} - 3 q^{73} - 4 q^{74} + 5 q^{76} - 10 q^{77} - 3 q^{79} + 13 q^{80} + 22 q^{82} - 19 q^{83} - 2 q^{85} + 9 q^{86} + 26 q^{88} - 16 q^{89} + q^{91} - 19 q^{92} + 25 q^{94} + 4 q^{95} - 35 q^{97} + 21 q^{98}+O(q^{100}) \) 4 * q - 3 * q^2 + 3 * q^4 + 4 * q^5 - q^7 - 9 * q^8 - 3 * q^10 - 6 * q^11 - 4 * q^13 + q^14 + 13 * q^16 - 2 * q^17 + 4 * q^19 + 3 * q^20 - 9 * q^22 - 9 * q^23 + 4 * q^25 + 3 * q^26 + 10 * q^28 - 16 * q^29 - 5 * q^31 - 26 * q^32 + 19 * q^34 - q^35 - 13 * q^37 - 12 * q^38 - 9 * q^40 - 12 * q^41 + q^44 + 2 * q^46 - 9 * q^47 - 11 * q^49 - 3 * q^50 - 3 * q^52 - 13 * q^53 - 6 * q^55 - 14 * q^56 - 9 * q^58 - 3 * q^59 + 3 * q^61 + 25 * q^62 + 45 * q^64 - 4 * q^65 + 6 * q^67 - 32 * q^68 + q^70 - 11 * q^71 - 3 * q^73 - 4 * q^74 + 5 * q^76 - 10 * q^77 - 3 * q^79 + 13 * q^80 + 22 * q^82 - 19 * q^83 - 2 * q^85 + 9 * q^86 + 26 * q^88 - 16 * q^89 + q^91 - 19 * q^92 + 25 * q^94 + 4 * q^95 - 35 * q^97 + 21 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more