LMFDB - Embedded newform 693.2.a.l.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [693,2,Mod(1,693)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(693, 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("693.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 693 = 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	693.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:	$5.53363286007$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.86081$ of defining polynomial
Character	$\chi$	$=$	693.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.46260 q^{2} +0.139194 q^{4} -2.39821 q^{5} -1.00000 q^{7} +2.72161 q^{8} +O(q^{10})$	\(q-1.46260 q^{2} +0.139194 q^{4} -2.39821 q^{5} -1.00000 q^{7} +2.72161 q^{8} +3.50761 q^{10} +1.00000 q^{11} +5.04502 q^{13} +1.46260 q^{14} -4.25901 q^{16} +6.36842 q^{17} -5.32340 q^{19} -0.333816 q^{20} -1.46260 q^{22} -4.92520 q^{23} +0.751399 q^{25} -7.37883 q^{26} -0.139194 q^{28} -5.04502 q^{29} -7.57201 q^{31} +0.786003 q^{32} -9.31444 q^{34} +2.39821 q^{35} +4.24860 q^{37} +7.78600 q^{38} -6.52699 q^{40} +0.646809 q^{41} -10.5180 q^{43} +0.139194 q^{44} +7.20359 q^{46} -0.526989 q^{47} +1.00000 q^{49} -1.09899 q^{50} +0.702237 q^{52} -3.72161 q^{53} -2.39821 q^{55} -2.72161 q^{56} +7.37883 q^{58} -7.97021 q^{59} -2.00000 q^{61} +11.0748 q^{62} +7.36842 q^{64} -12.0990 q^{65} +8.76663 q^{67} +0.886447 q^{68} -3.50761 q^{70} +11.4432 q^{71} -13.0450 q^{73} -6.21400 q^{74} -0.740987 q^{76} -1.00000 q^{77} +11.4432 q^{79} +10.2140 q^{80} -0.946021 q^{82} -13.1648 q^{83} -15.2728 q^{85} +15.3836 q^{86} +2.72161 q^{88} -11.8504 q^{89} -5.04502 q^{91} -0.685559 q^{92} +0.770774 q^{94} +12.7666 q^{95} -1.87122 q^{97} -1.46260 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{2} + 6 q^{4} - 4 q^{5} - 3 q^{7} - 3 q^{8}+O(q^{10}) $ 3 * q - 2 * q^2 + 6 * q^4 - 4 * q^5 - 3 * q^7 - 3 * q^8	$ 3 q - 2 q^{2} + 6 q^{4} - 4 q^{5} - 3 q^{7} - 3 q^{8} - 11 q^{10} + 3 q^{11} - 4 q^{13} + 2 q^{14} - 4 q^{16} - 8 q^{17} - 8 q^{19} + 3 q^{20} - 2 q^{22} - 10 q^{23} + 15 q^{25} + q^{26} - 6 q^{28} + 4 q^{29} - 2 q^{31} - 8 q^{32} - 4 q^{34} + 4 q^{35} + 13 q^{38} - 18 q^{40} - 14 q^{41} - 14 q^{43} + 6 q^{44} + 28 q^{46} + 3 q^{49} + 19 q^{50} - 29 q^{52} - 4 q^{55} + 3 q^{56} - q^{58} - 6 q^{61} + 38 q^{62} - 5 q^{64} - 14 q^{65} - 4 q^{67} - 42 q^{68} + 11 q^{70} + 12 q^{71} - 20 q^{73} - 29 q^{74} - 11 q^{76} - 3 q^{77} + 12 q^{79} + 41 q^{80} - 6 q^{82} - 6 q^{83} - 6 q^{85} - 24 q^{86} - 3 q^{88} - 26 q^{89} + 4 q^{91} - 26 q^{92} + 35 q^{94} + 8 q^{95} - 4 q^{97} - 2 q^{98}+O(q^{100}) $ 3 * q - 2 * q^2 + 6 * q^4 - 4 * q^5 - 3 * q^7 - 3 * q^8 - 11 * q^10 + 3 * q^11 - 4 * q^13 + 2 * q^14 - 4 * q^16 - 8 * q^17 - 8 * q^19 + 3 * q^20 - 2 * q^22 - 10 * q^23 + 15 * q^25 + q^26 - 6 * q^28 + 4 * q^29 - 2 * q^31 - 8 * q^32 - 4 * q^34 + 4 * q^35 + 13 * q^38 - 18 * q^40 - 14 * q^41 - 14 * q^43 + 6 * q^44 + 28 * q^46 + 3 * q^49 + 19 * q^50 - 29 * q^52 - 4 * q^55 + 3 * q^56 - q^58 - 6 * q^61 + 38 * q^62 - 5 * q^64 - 14 * q^65 - 4 * q^67 - 42 * q^68 + 11 * q^70 + 12 * q^71 - 20 * q^73 - 29 * q^74 - 11 * q^76 - 3 * q^77 + 12 * q^79 + 41 * q^80 - 6 * q^82 - 6 * q^83 - 6 * q^85 - 24 * q^86 - 3 * q^88 - 26 * q^89 + 4 * q^91 - 26 * q^92 + 35 * q^94 + 8 * q^95 - 4 * 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 $

$2$	−1.46260	−1.03421	−0.517107	−	0.855921i	$-0.672991\pi$
$2$	−1.46260	−1.03421	−0.517107	+	0.855921i	$0.672991\pi$
$3$	0	0
$3$	0	0
$4$	0.139194	0.0695971
$5$	−2.39821	−1.07251	−0.536255	−	0.844056i	$-0.680162\pi$
$5$	−2.39821	−1.07251	−0.536255	+	0.844056i	$0.680162\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	2.72161	0.962235
$9$	0	0
$10$	3.50761	1.10921
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	5.04502	1.39924	0.699618	−	0.714517i	$-0.253354\pi$
$13$	5.04502	1.39924	0.699618	+	0.714517i	$0.253354\pi$
$14$	1.46260	0.390896
$15$	0	0
$16$	−4.25901	−1.06475
$17$	6.36842	1.54457	0.772284	−	0.635277i	$-0.219114\pi$
$17$	6.36842	1.54457	0.772284	+	0.635277i	$0.219114\pi$
$18$	0	0
$19$	−5.32340	−1.22127	−0.610636	−	0.791911i	$-0.709086\pi$
$19$	−5.32340	−1.22127	−0.610636	+	0.791911i	$0.709086\pi$
$20$	−0.333816	−0.0746436
$21$	0	0
$22$	−1.46260	−0.311827
$23$	−4.92520	−1.02697	−0.513487	−	0.858097i	$-0.671647\pi$
$23$	−4.92520	−1.02697	−0.513487	+	0.858097i	$0.671647\pi$
$24$	0	0
$25$	0.751399	0.150280
$26$	−7.37883	−1.44711
$27$	0	0
$28$	−0.139194	−0.0263052
$29$	−5.04502	−0.936836	−0.468418	−	0.883507i	$-0.655176\pi$
$29$	−5.04502	−0.936836	−0.468418	+	0.883507i	$0.655176\pi$
$30$	0	0
$31$	−7.57201	−1.35997	−0.679986	−	0.733225i	$-0.738014\pi$
$31$	−7.57201	−1.35997	−0.679986	+	0.733225i	$0.738014\pi$
$32$	0.786003	0.138947
$33$	0	0
$34$	−9.31444	−1.59741
$35$	2.39821	0.405371
$36$	0	0
$37$	4.24860	0.698466	0.349233	−	0.937036i	$-0.386442\pi$
$37$	4.24860	0.698466	0.349233	+	0.937036i	$0.386442\pi$
$38$	7.78600	1.26306
$39$	0	0
$40$	−6.52699	−1.03201
$41$	0.646809	0.101015	0.0505073	−	0.998724i	$-0.483916\pi$
$41$	0.646809	0.101015	0.0505073	+	0.998724i	$0.483916\pi$
$42$	0	0
$43$	−10.5180	−1.60398	−0.801992	−	0.597335i	$-0.796226\pi$
$43$	−10.5180	−1.60398	−0.801992	+	0.597335i	$0.796226\pi$
$44$	0.139194	0.0209843
$45$	0	0
$46$	7.20359	1.06211
$47$	−0.526989	−0.0768693	−0.0384347	−	0.999261i	$-0.512237\pi$
$47$	−0.526989	−0.0768693	−0.0384347	+	0.999261i	$0.512237\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−1.09899	−0.155421
$51$	0	0
$52$	0.702237	0.0973827
$53$	−3.72161	−0.511203	−0.255601	−	0.966782i	$-0.582273\pi$
$53$	−3.72161	−0.511203	−0.255601	+	0.966782i	$0.582273\pi$
$54$	0	0
$55$	−2.39821	−0.323374
$56$	−2.72161	−0.363691
$57$	0	0
$58$	7.37883	0.968888
$59$	−7.97021	−1.03763	−0.518817	−	0.854886i	$-0.673627\pi$
$59$	−7.97021	−1.03763	−0.518817	+	0.854886i	$0.673627\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	11.0748	1.40650
$63$	0	0
$64$	7.36842	0.921053
$65$	−12.0990	−1.50070
$66$	0	0
$67$	8.76663	1.07101	0.535507	−	0.844531i	$-0.320121\pi$
$67$	8.76663	1.07101	0.535507	+	0.844531i	$0.320121\pi$
$68$	0.886447	0.107497
$69$	0	0
$70$	−3.50761	−0.419240
$71$	11.4432	1.35806	0.679030	−	0.734110i	$-0.262400\pi$
$71$	11.4432	1.35806	0.679030	+	0.734110i	$0.262400\pi$
$72$	0	0
$73$	−13.0450	−1.52680	−0.763402	−	0.645924i	$-0.776472\pi$
$73$	−13.0450	−1.52680	−0.763402	+	0.645924i	$0.776472\pi$
$74$	−6.21400	−0.722363
$75$	0	0
$76$	−0.740987	−0.0849970
$77$	−1.00000	−0.113961
$78$	0	0
$79$	11.4432	1.28746	0.643732	−	0.765251i	$-0.277385\pi$
$79$	11.4432	1.28746	0.643732	+	0.765251i	$0.277385\pi$
$80$	10.2140	1.14196
$81$	0	0
$82$	−0.946021	−0.104471
$83$	−13.1648	−1.44503	−0.722514	−	0.691356i	$-0.757014\pi$
$83$	−13.1648	−1.44503	−0.722514	+	0.691356i	$0.757014\pi$
$84$	0	0
$85$	−15.2728	−1.65657
$86$	15.3836	1.65886
$87$	0	0
$88$	2.72161	0.290125
$89$	−11.8504	−1.25614	−0.628070	−	0.778157i	$-0.716155\pi$
$89$	−11.8504	−1.25614	−0.628070	+	0.778157i	$0.716155\pi$
$90$	0	0
$91$	−5.04502	−0.528861
$92$	−0.685559	−0.0714744
$93$	0	0
$94$	0.770774	0.0794993
$95$	12.7666	1.30983
$96$	0	0
$97$	−1.87122	−0.189993	−0.0949967	−	0.995478i	$-0.530284\pi$
$97$	−1.87122	−0.189993	−0.0949967	+	0.995478i	$0.530284\pi$
$98$	−1.46260	−0.147745
$99$	0	0
$100$	0.104590	0.0104590
$101$	−4.51803	−0.449560	−0.224780	−	0.974409i	$-0.572166\pi$
$101$	−4.51803	−0.449560	−0.224780	+	0.974409i	$0.572166\pi$
$102$	0	0
$103$	−10.6468	−1.04906	−0.524531	−	0.851392i	$-0.675759\pi$
$103$	−10.6468	−1.04906	−0.524531	+	0.851392i	$0.675759\pi$
$104$	13.7306	1.34639
$105$	0	0
$106$	5.44322	0.528693
$107$	−15.9702	−1.54390	−0.771949	−	0.635684i	$-0.780718\pi$
$107$	−15.9702	−1.54390	−0.771949	+	0.635684i	$0.780718\pi$
$108$	0	0
$109$	12.7756	1.22368	0.611840	−	0.790982i	$-0.290430\pi$
$109$	12.7756	1.22368	0.611840	+	0.790982i	$0.290430\pi$
$110$	3.50761	0.334438
$111$	0	0
$112$	4.25901	0.402439
$113$	−18.7368	−1.76261	−0.881307	−	0.472544i	$-0.843336\pi$
$113$	−18.7368	−1.76261	−0.881307	+	0.472544i	$0.843336\pi$
$114$	0	0
$115$	11.8116	1.10144
$116$	−0.702237	−0.0652010
$117$	0	0
$118$	11.6572	1.07313
$119$	−6.36842	−0.583792
$120$	0	0
$121$	1.00000	0.0909091
$122$	2.92520	0.264835
$123$	0	0
$124$	−1.05398	−0.0946501
$125$	10.1890	0.911334
$126$	0	0
$127$	−2.27839	−0.202174	−0.101087	−	0.994878i	$-0.532232\pi$
$127$	−2.27839	−0.202174	−0.101087	+	0.994878i	$0.532232\pi$
$128$	−12.3490	−1.09151
$129$	0	0
$130$	17.6960	1.55204
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	5.32340	0.461598
$134$	−12.8221	−1.10766
$135$	0	0
$136$	17.3324	1.48624
$137$	4.77559	0.408006	0.204003	−	0.978970i	$-0.434605\pi$
$137$	4.77559	0.408006	0.204003	+	0.978970i	$0.434605\pi$
$138$	0	0
$139$	−15.4432	−1.30988	−0.654939	−	0.755682i	$-0.727306\pi$
$139$	−15.4432	−1.30988	−0.654939	+	0.755682i	$0.727306\pi$
$140$	0.333816	0.0282126
$141$	0	0
$142$	−16.7368	−1.40452
$143$	5.04502	0.421885
$144$	0	0
$145$	12.0990	1.00477
$146$	19.0796	1.57904
$147$	0	0
$148$	0.591380	0.0486112
$149$	9.84143	0.806241	0.403121	−	0.915147i	$-0.367925\pi$
$149$	9.84143	0.806241	0.403121	+	0.915147i	$0.367925\pi$
$150$	0	0
$151$	−4.12878	−0.335996	−0.167998	−	0.985787i	$-0.553730\pi$
$151$	−4.12878	−0.335996	−0.167998	+	0.985787i	$0.553730\pi$
$152$	−14.4882	−1.17515
$153$	0	0
$154$	1.46260	0.117860
$155$	18.1592	1.45859
$156$	0	0
$157$	−0.946021	−0.0755007	−0.0377504	−	0.999287i	$-0.512019\pi$
$157$	−0.946021	−0.0755007	−0.0377504	+	0.999287i	$0.512019\pi$
$158$	−16.7368	−1.33151
$159$	0	0
$160$	−1.88500	−0.149022
$161$	4.92520	0.388160
$162$	0	0
$163$	8.76663	0.686655	0.343328	−	0.939216i	$-0.388446\pi$
$163$	8.76663	0.686655	0.343328	+	0.939216i	$0.388446\pi$
$164$	0.0900320	0.00703032
$165$	0	0
$166$	19.2549	1.49447
$167$	24.3684	1.88568	0.942842	−	0.333239i	$-0.108142\pi$
$167$	24.3684	1.88568	0.942842	+	0.333239i	$0.108142\pi$
$168$	0	0
$169$	12.4522	0.957860
$170$	22.3380	1.71324
$171$	0	0
$172$	−1.46405	−0.111633
$173$	−12.3476	−0.938770	−0.469385	−	0.882994i	$-0.655524\pi$
$173$	−12.3476	−0.938770	−0.469385	+	0.882994i	$0.655524\pi$
$174$	0	0
$175$	−0.751399	−0.0568004
$176$	−4.25901	−0.321035
$177$	0	0
$178$	17.3324	1.29912
$179$	5.59283	0.418028	0.209014	−	0.977913i	$-0.432975\pi$
$179$	5.59283	0.418028	0.209014	+	0.977913i	$0.432975\pi$
$180$	0	0
$181$	13.5720	1.00880	0.504400	−	0.863470i	$-0.331714\pi$
$181$	13.5720	1.00880	0.504400	+	0.863470i	$0.331714\pi$
$182$	7.37883	0.546955
$183$	0	0
$184$	−13.4045	−0.988191
$185$	−10.1890	−0.749112
$186$	0	0
$187$	6.36842	0.465705
$188$	−0.0733538	−0.00534988
$189$	0	0
$190$	−18.6724	−1.35464
$191$	9.42240	0.681781	0.340890	−	0.940103i	$-0.389271\pi$
$191$	9.42240	0.681781	0.340890	+	0.940103i	$0.389271\pi$
$192$	0	0
$193$	−10.1288	−0.729086	−0.364543	−	0.931187i	$-0.618775\pi$
$193$	−10.1288	−0.729086	−0.364543	+	0.931187i	$0.618775\pi$
$194$	2.73684	0.196494
$195$	0	0
$196$	0.139194	0.00994244
$197$	2.25756	0.160845	0.0804224	−	0.996761i	$-0.474373\pi$
$197$	2.25756	0.160845	0.0804224	+	0.996761i	$0.474373\pi$
$198$	0	0
$199$	3.07480	0.217967	0.108984	−	0.994044i	$-0.465240\pi$
$199$	3.07480	0.217967	0.108984	+	0.994044i	$0.465240\pi$
$200$	2.04502	0.144604
$201$	0	0
$202$	6.60806	0.464941
$203$	5.04502	0.354091
$204$	0	0
$205$	−1.55118	−0.108339
$206$	15.5720	1.08495
$207$	0	0
$208$	−21.4868	−1.48984
$209$	−5.32340	−0.368228
$210$	0	0
$211$	−14.6468	−1.00833	−0.504164	−	0.863608i	$-0.668199\pi$
$211$	−14.6468	−1.00833	−0.504164	+	0.863608i	$0.668199\pi$
$212$	−0.518027	−0.0355782
$213$	0	0
$214$	23.3580	1.59672
$215$	25.2244	1.72029
$216$	0	0
$217$	7.57201	0.514021
$218$	−18.6856	−1.26555
$219$	0	0
$220$	−0.333816	−0.0225059
$221$	32.1288	2.16122
$222$	0	0
$223$	−1.90997	−0.127901	−0.0639505	−	0.997953i	$-0.520370\pi$
$223$	−1.90997	−0.127901	−0.0639505	+	0.997953i	$0.520370\pi$
$224$	−0.786003	−0.0525170
$225$	0	0
$226$	27.4045	1.82292
$227$	3.20359	0.212629	0.106315	−	0.994333i	$-0.466095\pi$
$227$	3.20359	0.212629	0.106315	+	0.994333i	$0.466095\pi$
$228$	0	0
$229$	−18.3088	−1.20988	−0.604941	−	0.796270i	$-0.706803\pi$
$229$	−18.3088	−1.20988	−0.604941	+	0.796270i	$0.706803\pi$
$230$	−17.2757	−1.13913
$231$	0	0
$232$	−13.7306	−0.901456
$233$	16.5872	1.08667	0.543333	−	0.839517i	$-0.317162\pi$
$233$	16.5872	1.08667	0.543333	+	0.839517i	$0.317162\pi$
$234$	0	0
$235$	1.26383	0.0824432
$236$	−1.10941	−0.0722162
$237$	0	0
$238$	9.31444	0.603766
$239$	−2.91623	−0.188635	−0.0943177	−	0.995542i	$-0.530067\pi$
$239$	−2.91623	−0.188635	−0.0943177	+	0.995542i	$0.530067\pi$
$240$	0	0
$241$	−6.09899	−0.392871	−0.196435	−	0.980517i	$-0.562937\pi$
$241$	−6.09899	−0.392871	−0.196435	+	0.980517i	$0.562937\pi$
$242$	−1.46260	−0.0940194
$243$	0	0
$244$	−0.278388	−0.0178220
$245$	−2.39821	−0.153216
$246$	0	0
$247$	−26.8567	−1.70885
$248$	−20.6081	−1.30861
$249$	0	0
$250$	−14.9025	−0.942514
$251$	−1.62262	−0.102419	−0.0512093	−	0.998688i	$-0.516308\pi$
$251$	−1.62262	−0.102419	−0.0512093	+	0.998688i	$0.516308\pi$
$252$	0	0
$253$	−4.92520	−0.309644
$254$	3.33237	0.209091
$255$	0	0
$256$	3.32485	0.207803
$257$	6.89541	0.430124	0.215062	−	0.976600i	$-0.431005\pi$
$257$	6.89541	0.430124	0.215062	+	0.976600i	$0.431005\pi$
$258$	0	0
$259$	−4.24860	−0.263995
$260$	−1.68411	−0.104444
$261$	0	0
$262$	5.85039	0.361439
$263$	−5.08377	−0.313478	−0.156739	−	0.987640i	$-0.550098\pi$
$263$	−5.08377	−0.313478	−0.156739	+	0.987640i	$0.550098\pi$
$264$	0	0
$265$	8.92520	0.548270
$266$	−7.78600	−0.477390
$267$	0	0
$268$	1.22026	0.0745394
$269$	0.886447	0.0540476	0.0270238	−	0.999635i	$-0.491397\pi$
$269$	0.886447	0.0540476	0.0270238	+	0.999635i	$0.491397\pi$
$270$	0	0
$271$	−25.3234	−1.53829	−0.769144	−	0.639076i	$-0.779317\pi$
$271$	−25.3234	−1.53829	−0.769144	+	0.639076i	$0.779317\pi$
$272$	−27.1232	−1.64458
$273$	0	0
$274$	−6.98477	−0.421965
$275$	0.751399	0.0453111
$276$	0	0
$277$	−24.8269	−1.49170	−0.745851	−	0.666113i	$-0.767957\pi$
$277$	−24.8269	−1.49170	−0.745851	+	0.666113i	$0.767957\pi$
$278$	22.5872	1.35469
$279$	0	0
$280$	6.52699	0.390062
$281$	−1.90101	−0.113404	−0.0567022	−	0.998391i	$-0.518059\pi$
$281$	−1.90101	−0.113404	−0.0567022	+	0.998391i	$0.518059\pi$
$282$	0	0
$283$	−22.3178	−1.32666	−0.663328	−	0.748329i	$-0.730857\pi$
$283$	−22.3178	−1.32666	−0.663328	+	0.748329i	$0.730857\pi$
$284$	1.59283	0.0945171
$285$	0	0
$286$	−7.37883	−0.436320
$287$	−0.646809	−0.0381799
$288$	0	0
$289$	23.5568	1.38569
$290$	−17.6960	−1.03914
$291$	0	0
$292$	−1.81579	−0.106261
$293$	−12.0900	−0.706307	−0.353154	−	0.935565i	$-0.614891\pi$
$293$	−12.0900	−0.706307	−0.353154	+	0.935565i	$0.614891\pi$
$294$	0	0
$295$	19.1142	1.11287
$296$	11.5630	0.672088
$297$	0	0
$298$	−14.3941	−0.833826
$299$	−24.8477	−1.43698
$300$	0	0
$301$	10.5180	0.606249
$302$	6.03875	0.347491
$303$	0	0
$304$	22.6724	1.30035
$305$	4.79641	0.274642
$306$	0	0
$307$	−13.5928	−0.775784	−0.387892	−	0.921705i	$-0.626797\pi$
$307$	−13.5928	−0.775784	−0.387892	+	0.921705i	$0.626797\pi$
$308$	−0.139194	−0.00793132
$309$	0	0
$310$	−26.5597	−1.50849
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	14.9252	0.843622	0.421811	−	0.906684i	$-0.361395\pi$
$313$	14.9252	0.843622	0.421811	+	0.906684i	$0.361395\pi$
$314$	1.38365	0.0780838
$315$	0	0
$316$	1.59283	0.0896037
$317$	−3.97918	−0.223493	−0.111746	−	0.993737i	$-0.535644\pi$
$317$	−3.97918	−0.223493	−0.111746	+	0.993737i	$0.535644\pi$
$318$	0	0
$319$	−5.04502	−0.282467
$320$	−17.6710	−0.987839
$321$	0	0
$322$	−7.20359	−0.401440
$323$	−33.9017	−1.88634
$324$	0	0
$325$	3.79082	0.210277
$326$	−12.8221	−0.710148
$327$	0	0
$328$	1.76036	0.0971997
$329$	0.526989	0.0290539
$330$	0	0
$331$	23.4432	1.28856	0.644278	−	0.764791i	$-0.277158\pi$
$331$	23.4432	1.28856	0.644278	+	0.764791i	$0.277158\pi$
$332$	−1.83247	−0.100570
$333$	0	0
$334$	−35.6412	−1.95020
$335$	−21.0242	−1.14867
$336$	0	0
$337$	11.1648	0.608187	0.304094	−	0.952642i	$-0.401646\pi$
$337$	11.1648	0.608187	0.304094	+	0.952642i	$0.401646\pi$
$338$	−18.2125	−0.990632
$339$	0	0
$340$	−2.12588	−0.115292
$341$	−7.57201	−0.410047
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−28.6260	−1.54341
$345$	0	0
$346$	18.0596	0.970889
$347$	22.5872	1.21255	0.606273	−	0.795256i	$-0.292664\pi$
$347$	22.5872	1.21255	0.606273	+	0.795256i	$0.292664\pi$
$348$	0	0
$349$	27.9315	1.49514	0.747568	−	0.664185i	$-0.231221\pi$
$349$	27.9315	1.49514	0.747568	+	0.664185i	$0.231221\pi$
$350$	1.09899	0.0587437
$351$	0	0
$352$	0.786003	0.0418941
$353$	16.5478	0.880751	0.440376	−	0.897814i	$-0.354845\pi$
$353$	16.5478	0.880751	0.440376	+	0.897814i	$0.354845\pi$
$354$	0	0
$355$	−27.4432	−1.45654
$356$	−1.64951	−0.0874236
$357$	0	0
$358$	−8.18006	−0.432330
$359$	−22.0305	−1.16272	−0.581362	−	0.813645i	$-0.697480\pi$
$359$	−22.0305	−1.16272	−0.581362	+	0.813645i	$0.697480\pi$
$360$	0	0
$361$	9.33863	0.491507
$362$	−19.8504	−1.04331
$363$	0	0
$364$	−0.702237	−0.0368072
$365$	31.2847	1.63751
$366$	0	0
$367$	−19.3836	−1.01182	−0.505909	−	0.862587i	$-0.668843\pi$
$367$	−19.3836	−1.01182	−0.505909	+	0.862587i	$0.668843\pi$
$368$	20.9765	1.09347
$369$	0	0
$370$	14.9025	0.774742
$371$	3.72161	0.193216
$372$	0	0
$373$	29.2549	1.51476	0.757380	−	0.652975i	$-0.226479\pi$
$373$	29.2549	1.51476	0.757380	+	0.652975i	$0.226479\pi$
$374$	−9.31444	−0.481638
$375$	0	0
$376$	−1.43426	−0.0739663
$377$	−25.4522	−1.31085
$378$	0	0
$379$	12.5270	0.643468	0.321734	−	0.946830i	$-0.395734\pi$
$379$	12.5270	0.643468	0.321734	+	0.946830i	$0.395734\pi$
$380$	1.77704	0.0911602
$381$	0	0
$382$	−13.7812	−0.705107
$383$	17.5928	0.898952	0.449476	−	0.893293i	$-0.351611\pi$
$383$	17.5928	0.898952	0.449476	+	0.893293i	$0.351611\pi$
$384$	0	0
$385$	2.39821	0.122224
$386$	14.8143	0.754030
$387$	0	0
$388$	−0.260463	−0.0132230
$389$	−20.0900	−1.01861	−0.509303	−	0.860588i	$-0.670097\pi$
$389$	−20.0900	−1.01861	−0.509303	+	0.860588i	$0.670097\pi$
$390$	0	0
$391$	−31.3657	−1.58623
$392$	2.72161	0.137462
$393$	0	0
$394$	−3.30191	−0.166348
$395$	−27.4432	−1.38082
$396$	0	0
$397$	−35.1053	−1.76188	−0.880941	−	0.473226i	$-0.843090\pi$
$397$	−35.1053	−1.76188	−0.880941	+	0.473226i	$0.843090\pi$
$398$	−4.49720	−0.225424
$399$	0	0
$400$	−3.20022	−0.160011
$401$	−9.57201	−0.478003	−0.239002	−	0.971019i	$-0.576820\pi$
$401$	−9.57201	−0.478003	−0.239002	+	0.971019i	$0.576820\pi$
$402$	0	0
$403$	−38.2009	−1.90292
$404$	−0.628883	−0.0312881
$405$	0	0
$406$	−7.37883	−0.366205
$407$	4.24860	0.210595
$408$	0	0
$409$	38.1801	1.88788	0.943941	−	0.330113i	$-0.107087\pi$
$409$	38.1801	1.88788	0.943941	+	0.330113i	$0.107087\pi$
$410$	2.26875	0.112046
$411$	0	0
$412$	−1.48197	−0.0730116
$413$	7.97021	0.392189
$414$	0	0
$415$	31.5720	1.54981
$416$	3.96540	0.194420
$417$	0	0
$418$	7.78600	0.380826
$419$	−7.17380	−0.350463	−0.175231	−	0.984527i	$-0.556067\pi$
$419$	−7.17380	−0.350463	−0.175231	+	0.984527i	$0.556067\pi$
$420$	0	0
$421$	15.1530	0.738511	0.369255	−	0.929328i	$-0.379613\pi$
$421$	15.1530	0.738511	0.369255	+	0.929328i	$0.379613\pi$
$422$	21.4224	1.04283
$423$	0	0
$424$	−10.1288	−0.491897
$425$	4.78522	0.232117
$426$	0	0
$427$	2.00000	0.0967868
$428$	−2.22296	−0.107451
$429$	0	0
$430$	−36.8932	−1.77915
$431$	−5.56304	−0.267962	−0.133981	−	0.990984i	$-0.542776\pi$
$431$	−5.56304	−0.267962	−0.133981	+	0.990984i	$0.542776\pi$
$432$	0	0
$433$	−25.6412	−1.23224	−0.616119	−	0.787653i	$-0.711296\pi$
$433$	−25.6412	−1.23224	−0.616119	+	0.787653i	$0.711296\pi$
$434$	−11.0748	−0.531608
$435$	0	0
$436$	1.77829	0.0851645
$437$	26.2188	1.25422
$438$	0	0
$439$	23.6710	1.12976	0.564878	−	0.825175i	$-0.308923\pi$
$439$	23.6710	1.12976	0.564878	+	0.825175i	$0.308923\pi$
$440$	−6.52699	−0.311162
$441$	0	0
$442$	−46.9915	−2.23516
$443$	18.0305	0.856653	0.428326	−	0.903624i	$-0.359103\pi$
$443$	18.0305	0.856653	0.428326	+	0.903624i	$0.359103\pi$
$444$	0	0
$445$	28.4197	1.34722
$446$	2.79352	0.132277
$447$	0	0
$448$	−7.36842	−0.348125
$449$	34.9765	1.65064	0.825321	−	0.564664i	$-0.190994\pi$
$449$	34.9765	1.65064	0.825321	+	0.564664i	$0.190994\pi$
$450$	0	0
$451$	0.646809	0.0304570
$452$	−2.60806	−0.122673
$453$	0	0
$454$	−4.68556	−0.219904
$455$	12.0990	0.567210
$456$	0	0
$457$	4.53595	0.212183	0.106091	−	0.994356i	$-0.466166\pi$
$457$	4.53595	0.212183	0.106091	+	0.994356i	$0.466166\pi$
$458$	26.7785	1.25128
$459$	0	0
$460$	1.64411	0.0766571
$461$	2.79641	0.130242	0.0651210	−	0.997877i	$-0.479257\pi$
$461$	2.79641	0.130242	0.0651210	+	0.997877i	$0.479257\pi$
$462$	0	0
$463$	38.3595	1.78272	0.891358	−	0.453301i	$-0.149754\pi$
$463$	38.3595	1.78272	0.891358	+	0.453301i	$0.149754\pi$
$464$	21.4868	0.997499
$465$	0	0
$466$	−24.2605	−1.12384
$467$	20.4674	0.947119	0.473560	−	0.880762i	$-0.342969\pi$
$467$	20.4674	0.947119	0.473560	+	0.880762i	$0.342969\pi$
$468$	0	0
$469$	−8.76663	−0.404805
$470$	−1.84848	−0.0852638
$471$	0	0
$472$	−21.6918	−0.998447
$473$	−10.5180	−0.483619
$474$	0	0
$475$	−4.00000	−0.183533
$476$	−0.886447	−0.0406302
$477$	0	0
$478$	4.26528	0.195089
$479$	11.6137	0.530641	0.265321	−	0.964160i	$-0.414522\pi$
$479$	11.6137	0.530641	0.265321	+	0.964160i	$0.414522\pi$
$480$	0	0
$481$	21.4343	0.977318
$482$	8.92038	0.406312
$483$	0	0
$484$	0.139194	0.00632701
$485$	4.48757	0.203770
$486$	0	0
$487$	32.4793	1.47178	0.735888	−	0.677103i	$-0.236765\pi$
$487$	32.4793	1.47178	0.735888	+	0.677103i	$0.236765\pi$
$488$	−5.44322	−0.246403
$489$	0	0
$490$	3.50761	0.158458
$491$	−26.6766	−1.20390	−0.601949	−	0.798535i	$-0.705609\pi$
$491$	−26.6766	−1.20390	−0.601949	+	0.798535i	$0.705609\pi$
$492$	0	0
$493$	−32.1288	−1.44701
$494$	39.2805	1.76731
$495$	0	0
$496$	32.2493	1.44804
$497$	−11.4432	−0.513299
$498$	0	0
$499$	−41.2459	−1.84642	−0.923210	−	0.384296i	$-0.874444\pi$
$499$	−41.2459	−1.84642	−0.923210	+	0.384296i	$0.874444\pi$
$500$	1.41825	0.0634262
$501$	0	0
$502$	2.37324	0.105923
$503$	−30.5180	−1.36073	−0.680366	−	0.732873i	$-0.738179\pi$
$503$	−30.5180	−1.36073	−0.680366	+	0.732873i	$0.738179\pi$
$504$	0	0
$505$	10.8352	0.482159
$506$	7.20359	0.320238
$507$	0	0
$508$	−0.317138	−0.0140707
$509$	18.9944	0.841912	0.420956	−	0.907081i	$-0.361695\pi$
$509$	18.9944	0.841912	0.420956	+	0.907081i	$0.361695\pi$
$510$	0	0
$511$	13.0450	0.577078
$512$	19.8352	0.876599
$513$	0	0
$514$	−10.0852	−0.444840
$515$	25.5333	1.12513
$516$	0	0
$517$	−0.526989	−0.0231770
$518$	6.21400	0.273027
$519$	0	0
$520$	−32.9288	−1.44402
$521$	−25.2430	−1.10592	−0.552958	−	0.833209i	$-0.686501\pi$
$521$	−25.2430	−1.10592	−0.552958	+	0.833209i	$0.686501\pi$
$522$	0	0
$523$	−2.93416	−0.128302	−0.0641509	−	0.997940i	$-0.520434\pi$
$523$	−2.93416	−0.128302	−0.0641509	+	0.997940i	$0.520434\pi$
$524$	−0.556777	−0.0243229
$525$	0	0
$526$	7.43551	0.324204
$527$	−48.2217	−2.10057
$528$	0	0
$529$	1.25756	0.0546767
$530$	−13.0540	−0.567029
$531$	0	0
$532$	0.740987	0.0321258
$533$	3.26316	0.141343
$534$	0	0
$535$	38.2999	1.65585
$536$	23.8594	1.03057
$537$	0	0
$538$	−1.29652	−0.0558968
$539$	1.00000	0.0430730
$540$	0	0
$541$	8.90437	0.382829	0.191414	−	0.981509i	$-0.438693\pi$
$541$	8.90437	0.382829	0.191414	+	0.981509i	$0.438693\pi$
$542$	37.0380	1.59092
$543$	0	0
$544$	5.00560	0.214613
$545$	−30.6385	−1.31241
$546$	0	0
$547$	−29.4737	−1.26020	−0.630102	−	0.776513i	$-0.716987\pi$
$547$	−29.4737	−1.26020	−0.630102	+	0.776513i	$0.716987\pi$
$548$	0.664734	0.0283960
$549$	0	0
$550$	−1.09899	−0.0468613
$551$	26.8567	1.14413
$552$	0	0
$553$	−11.4432	−0.486615
$554$	36.3117	1.54274
$555$	0	0
$556$	−2.14961	−0.0911636
$557$	−14.8954	−0.631139	−0.315569	−	0.948903i	$-0.602196\pi$
$557$	−14.8954	−0.631139	−0.315569	+	0.948903i	$0.602196\pi$
$558$	0	0
$559$	−53.0636	−2.24435
$560$	−10.2140	−0.431620
$561$	0	0
$562$	2.78041	0.117284
$563$	7.81164	0.329222	0.164611	−	0.986359i	$-0.447363\pi$
$563$	7.81164	0.329222	0.164611	+	0.986359i	$0.447363\pi$
$564$	0	0
$565$	44.9348	1.89042
$566$	32.6420	1.37205
$567$	0	0
$568$	31.1440	1.30677
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	0	0
$571$	25.5512	1.06928	0.534642	−	0.845079i	$-0.320447\pi$
$571$	25.5512	1.06928	0.534642	+	0.845079i	$0.320447\pi$
$572$	0.702237	0.0293620
$573$	0	0
$574$	0.946021	0.0394862
$575$	−3.70079	−0.154334
$576$	0	0
$577$	−29.5124	−1.22862	−0.614309	−	0.789065i	$-0.710565\pi$
$577$	−29.5124	−1.22862	−0.614309	+	0.789065i	$0.710565\pi$
$578$	−34.4541	−1.43310
$579$	0	0
$580$	1.68411	0.0699288
$581$	13.1648	0.546169
$582$	0	0
$583$	−3.72161	−0.154133
$584$	−35.5035	−1.46914
$585$	0	0
$586$	17.6829	0.730472
$587$	−31.3955	−1.29583	−0.647916	−	0.761712i	$-0.724359\pi$
$587$	−31.3955	−1.29583	−0.647916	+	0.761712i	$0.724359\pi$
$588$	0	0
$589$	40.3088	1.66090
$590$	−27.9564	−1.15095
$591$	0	0
$592$	−18.0948	−0.743694
$593$	7.90997	0.324823	0.162412	−	0.986723i	$-0.448073\pi$
$593$	7.90997	0.324823	0.162412	+	0.986723i	$0.448073\pi$
$594$	0	0
$595$	15.2728	0.626123
$596$	1.36987	0.0561120
$597$	0	0
$598$	36.3422	1.48614
$599$	−27.4432	−1.12130	−0.560650	−	0.828053i	$-0.689449\pi$
$599$	−27.4432	−1.12130	−0.560650	+	0.828053i	$0.689449\pi$
$600$	0	0
$601$	31.9910	1.30494	0.652471	−	0.757814i	$-0.273732\pi$
$601$	31.9910	1.30494	0.652471	+	0.757814i	$0.273732\pi$
$602$	−15.3836	−0.626991
$603$	0	0
$604$	−0.574702	−0.0233843
$605$	−2.39821	−0.0975010
$606$	0	0
$607$	7.41344	0.300902	0.150451	−	0.988617i	$-0.451927\pi$
$607$	7.41344	0.300902	0.150451	+	0.988617i	$0.451927\pi$
$608$	−4.18421	−0.169692
$609$	0	0
$610$	−7.01523	−0.284038
$611$	−2.65867	−0.107558
$612$	0	0
$613$	−33.9917	−1.37291	−0.686456	−	0.727171i	$-0.740835\pi$
$613$	−33.9917	−1.37291	−0.686456	+	0.727171i	$0.740835\pi$
$614$	19.8809	0.802326
$615$	0	0
$616$	−2.72161	−0.109657
$617$	44.0305	1.77260	0.886300	−	0.463112i	$-0.153267\pi$
$617$	44.0305	1.77260	0.886300	+	0.463112i	$0.153267\pi$
$618$	0	0
$619$	40.0096	1.60812	0.804061	−	0.594546i	$-0.202668\pi$
$619$	40.0096	1.60812	0.804061	+	0.594546i	$0.202668\pi$
$620$	2.52766	0.101513
$621$	0	0
$622$	11.7008	0.469159
$623$	11.8504	0.474776
$624$	0	0
$625$	−28.1924	−1.12770
$626$	−21.8296	−0.872485
$627$	0	0
$628$	−0.131681	−0.00525463
$629$	27.0569	1.07883
$630$	0	0
$631$	28.5568	1.13683	0.568414	−	0.822743i	$-0.307557\pi$
$631$	28.5568	1.13683	0.568414	+	0.822743i	$0.307557\pi$
$632$	31.1440	1.23884
$633$	0	0
$634$	5.81994	0.231139
$635$	5.46405	0.216834
$636$	0	0
$637$	5.04502	0.199891
$638$	7.37883	0.292131
$639$	0	0
$640$	29.6156	1.17066
$641$	31.1053	1.22858	0.614292	−	0.789079i	$-0.289442\pi$
$641$	31.1053	1.22858	0.614292	+	0.789079i	$0.289442\pi$
$642$	0	0
$643$	−5.48197	−0.216188	−0.108094	−	0.994141i	$-0.534475\pi$
$643$	−5.48197	−0.216188	−0.108094	+	0.994141i	$0.534475\pi$
$644$	0.685559	0.0270148
$645$	0	0
$646$	49.5845	1.95088
$647$	9.26383	0.364199	0.182099	−	0.983280i	$-0.441711\pi$
$647$	9.26383	0.364199	0.182099	+	0.983280i	$0.441711\pi$
$648$	0	0
$649$	−7.97021	−0.312858
$650$	−5.54445	−0.217471
$651$	0	0
$652$	1.22026	0.0477892
$653$	29.9821	1.17329	0.586645	−	0.809844i	$-0.300449\pi$
$653$	29.9821	1.17329	0.586645	+	0.809844i	$0.300449\pi$
$654$	0	0
$655$	9.59283	0.374823
$656$	−2.75477	−0.107556
$657$	0	0
$658$	−0.770774	−0.0300479
$659$	−23.9702	−0.933747	−0.466873	−	0.884324i	$-0.654620\pi$
$659$	−23.9702	−0.933747	−0.466873	+	0.884324i	$0.654620\pi$
$660$	0	0
$661$	−40.4585	−1.57365	−0.786826	−	0.617175i	$-0.788277\pi$
$661$	−40.4585	−1.57365	−0.786826	+	0.617175i	$0.788277\pi$
$662$	−34.2880	−1.33264
$663$	0	0
$664$	−35.8296	−1.39046
$665$	−12.7666	−0.495069
$666$	0	0
$667$	24.8477	0.962107
$668$	3.39194	0.131238
$669$	0	0
$670$	30.7499	1.18797
$671$	−2.00000	−0.0772091
$672$	0	0
$673$	21.8712	0.843073	0.421537	−	0.906811i	$-0.361491\pi$
$673$	21.8712	0.843073	0.421537	+	0.906811i	$0.361491\pi$
$674$	−16.3297	−0.628995
$675$	0	0
$676$	1.73327	0.0666643
$677$	1.26316	0.0485472	0.0242736	−	0.999705i	$-0.492273\pi$
$677$	1.26316	0.0485472	0.0242736	+	0.999705i	$0.492273\pi$
$678$	0	0
$679$	1.87122	0.0718108
$680$	−41.5666	−1.59401
$681$	0	0
$682$	11.0748	0.424076
$683$	−37.6441	−1.44041	−0.720206	−	0.693760i	$-0.755953\pi$
$683$	−37.6441	−1.44041	−0.720206	+	0.693760i	$0.755953\pi$
$684$	0	0
$685$	−11.4529	−0.437591
$686$	1.46260	0.0558423
$687$	0	0
$688$	44.7964	1.70785
$689$	−18.7756	−0.715293
$690$	0	0
$691$	14.3892	0.547393	0.273696	−	0.961816i	$-0.411754\pi$
$691$	14.3892	0.547393	0.273696	+	0.961816i	$0.411754\pi$
$692$	−1.71871	−0.0653357
$693$	0	0
$694$	−33.0361	−1.25403
$695$	37.0361	1.40486
$696$	0	0
$697$	4.11915	0.156024
$698$	−40.8525	−1.54629
$699$	0	0
$700$	−0.104590	−0.00395314
$701$	39.2936	1.48410	0.742050	−	0.670345i	$-0.233854\pi$
$701$	39.2936	1.48410	0.742050	+	0.670345i	$0.233854\pi$
$702$	0	0
$703$	−22.6170	−0.853017
$704$	7.36842	0.277708
$705$	0	0
$706$	−24.2028	−0.910885
$707$	4.51803	0.169918
$708$	0	0
$709$	−49.2430	−1.84936	−0.924680	−	0.380745i	$-0.875667\pi$
$709$	−49.2430	−1.84936	−0.924680	+	0.380745i	$0.875667\pi$
$710$	40.1384	1.50637
$711$	0	0
$712$	−32.2522	−1.20870
$713$	37.2936	1.39666
$714$	0	0
$715$	−12.0990	−0.452477
$716$	0.778489	0.0290935
$717$	0	0
$718$	32.2217	1.20250
$719$	7.41344	0.276475	0.138237	−	0.990399i	$-0.455856\pi$
$719$	7.41344	0.276475	0.138237	+	0.990399i	$0.455856\pi$
$720$	0	0
$721$	10.6468	0.396508
$722$	−13.6587	−0.508323
$723$	0	0
$724$	1.88914	0.0702095
$725$	−3.79082	−0.140787
$726$	0	0
$727$	18.9557	0.703026	0.351513	−	0.936183i	$-0.385667\pi$
$727$	18.9557	0.703026	0.351513	+	0.936183i	$0.385667\pi$
$728$	−13.7306	−0.508889
$729$	0	0
$730$	−45.7569	−1.69354
$731$	−66.9832	−2.47746
$732$	0	0
$733$	−3.59283	−0.132704	−0.0663521	−	0.997796i	$-0.521136\pi$
$733$	−3.59283	−0.132704	−0.0663521	+	0.997796i	$0.521136\pi$
$734$	28.3505	1.04644
$735$	0	0
$736$	−3.87122	−0.142695
$737$	8.76663	0.322923
$738$	0	0
$739$	−26.7756	−0.984956	−0.492478	−	0.870325i	$-0.663909\pi$
$739$	−26.7756	−0.984956	−0.492478	+	0.870325i	$0.663909\pi$
$740$	−1.41825	−0.0521360
$741$	0	0
$742$	−5.44322	−0.199827
$743$	33.8027	1.24010	0.620050	−	0.784562i	$-0.287112\pi$
$743$	33.8027	1.24010	0.620050	+	0.784562i	$0.287112\pi$
$744$	0	0
$745$	−23.6018	−0.864703
$746$	−42.7881	−1.56658
$747$	0	0
$748$	0.886447	0.0324117
$749$	15.9702	0.583539
$750$	0	0
$751$	35.3955	1.29160	0.645800	−	0.763506i	$-0.276524\pi$
$751$	35.3955	1.29160	0.645800	+	0.763506i	$0.276524\pi$
$752$	2.24445	0.0818468
$753$	0	0
$754$	37.2263	1.35570
$755$	9.90168	0.360359
$756$	0	0
$757$	−29.3442	−1.06653	−0.533267	−	0.845947i	$-0.679036\pi$
$757$	−29.3442	−1.06653	−0.533267	+	0.845947i	$0.679036\pi$
$758$	−18.3220	−0.665483
$759$	0	0
$760$	34.7458	1.26036
$761$	−12.9044	−0.467783	−0.233892	−	0.972263i	$-0.575146\pi$
$761$	−12.9044	−0.467783	−0.233892	+	0.972263i	$0.575146\pi$
$762$	0	0
$763$	−12.7756	−0.462507
$764$	1.31154	0.0474500
$765$	0	0
$766$	−25.7312	−0.929708
$767$	−40.2099	−1.45189
$768$	0	0
$769$	9.78186	0.352743	0.176371	−	0.984324i	$-0.443564\pi$
$769$	9.78186	0.352743	0.176371	+	0.984324i	$0.443564\pi$
$770$	−3.50761	−0.126406
$771$	0	0
$772$	−1.40987	−0.0507422
$773$	−29.7223	−1.06904	−0.534518	−	0.845157i	$-0.679507\pi$
$773$	−29.7223	−1.06904	−0.534518	+	0.845157i	$0.679507\pi$
$774$	0	0
$775$	−5.68960	−0.204376
$776$	−5.09273	−0.182818
$777$	0	0
$778$	29.3836	1.05345
$779$	−3.44322	−0.123366
$780$	0	0
$781$	11.4432	0.409471
$782$	45.8755	1.64050
$783$	0	0
$784$	−4.25901	−0.152108
$785$	2.26875	0.0809753
$786$	0	0
$787$	17.0242	0.606847	0.303423	−	0.952856i	$-0.401870\pi$
$787$	17.0242	0.606847	0.303423	+	0.952856i	$0.401870\pi$
$788$	0.314240	0.0111943
$789$	0	0
$790$	40.1384	1.42806
$791$	18.7368	0.666205
$792$	0	0
$793$	−10.0900	−0.358308
$794$	51.3449	1.82216
$795$	0	0
$796$	0.427995	0.0151699
$797$	36.4287	1.29037	0.645185	−	0.764027i	$-0.276780\pi$
$797$	36.4287	1.29037	0.645185	+	0.764027i	$0.276780\pi$
$798$	0	0
$799$	−3.35609	−0.118730
$800$	0.590602	0.0208809
$801$	0	0
$802$	14.0000	0.494357
$803$	−13.0450	−0.460349
$804$	0	0
$805$	−11.8116	−0.416306
$806$	55.8726	1.96803
$807$	0	0
$808$	−12.2963	−0.432583
$809$	−44.4882	−1.56412	−0.782062	−	0.623201i	$-0.785832\pi$
$809$	−44.4882	−1.56412	−0.782062	+	0.623201i	$0.785832\pi$
$810$	0	0
$811$	7.65307	0.268736	0.134368	−	0.990932i	$-0.457100\pi$
$811$	7.65307	0.268736	0.134368	+	0.990932i	$0.457100\pi$
$812$	0.702237	0.0246437
$813$	0	0
$814$	−6.21400	−0.217800
$815$	−21.0242	−0.736445
$816$	0	0
$817$	55.9917	1.95890
$818$	−55.8421	−1.95247
$819$	0	0
$820$	−0.215915	−0.00754009
$821$	44.3691	1.54849	0.774246	−	0.632885i	$-0.218129\pi$
$821$	44.3691	1.54849	0.774246	+	0.632885i	$0.218129\pi$
$822$	0	0
$823$	6.61702	0.230655	0.115327	−	0.993328i	$-0.463208\pi$
$823$	6.61702	0.230655	0.115327	+	0.993328i	$0.463208\pi$
$824$	−28.9765	−1.00944
$825$	0	0
$826$	−11.6572	−0.405607
$827$	−39.7126	−1.38094	−0.690472	−	0.723359i	$-0.742597\pi$
$827$	−39.7126	−1.38094	−0.690472	+	0.723359i	$0.742597\pi$
$828$	0	0
$829$	−3.90997	−0.135799	−0.0678994	−	0.997692i	$-0.521630\pi$
$829$	−3.90997	−0.135799	−0.0678994	+	0.997692i	$0.521630\pi$
$830$	−46.1772	−1.60283
$831$	0	0
$832$	37.1738	1.28877
$833$	6.36842	0.220653
$834$	0	0
$835$	−58.4405	−2.02242
$836$	−0.740987	−0.0256276
$837$	0	0
$838$	10.4924	0.362453
$839$	9.58097	0.330772	0.165386	−	0.986229i	$-0.447113\pi$
$839$	9.58097	0.330772	0.165386	+	0.986229i	$0.447113\pi$
$840$	0	0
$841$	−3.54781	−0.122338
$842$	−22.1627	−0.763778
$843$	0	0
$844$	−2.03875	−0.0701767
$845$	−29.8629	−1.02732
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	15.8504	0.544305
$849$	0	0
$850$	−6.99886	−0.240059
$851$	−20.9252	−0.717307
$852$	0	0
$853$	−14.5568	−0.498415	−0.249207	−	0.968450i	$-0.580170\pi$
$853$	−14.5568	−0.498415	−0.249207	+	0.968450i	$0.580170\pi$
$854$	−2.92520	−0.100098
$855$	0	0
$856$	−43.4647	−1.48559
$857$	−10.4793	−0.357965	−0.178983	−	0.983852i	$-0.557281\pi$
$857$	−10.4793	−0.357965	−0.178983	+	0.983852i	$0.557281\pi$
$858$	0	0
$859$	2.88645	0.0984843	0.0492421	−	0.998787i	$-0.484319\pi$
$859$	2.88645	0.0984843	0.0492421	+	0.998787i	$0.484319\pi$
$860$	3.51109	0.119727
$861$	0	0
$862$	8.13650	0.277130
$863$	20.5485	0.699479	0.349739	−	0.936847i	$-0.386270\pi$
$863$	20.5485	0.699479	0.349739	+	0.936847i	$0.386270\pi$
$864$	0	0
$865$	29.6121	1.00684
$866$	37.5028	1.27440
$867$	0	0
$868$	1.05398	0.0357744
$869$	11.4432	0.388185
$870$	0	0
$871$	44.2278	1.49860
$872$	34.7702	1.17747
$873$	0	0
$874$	−38.3476	−1.29713
$875$	−10.1890	−0.344452
$876$	0	0
$877$	59.1149	1.99617	0.998084	−	0.0618724i	$-0.0197072\pi$
$877$	59.1149	1.99617	0.998084	+	0.0618724i	$0.0197072\pi$
$878$	−34.6212	−1.16841
$879$	0	0
$880$	10.2140	0.344314
$881$	−30.3982	−1.02414	−0.512071	−	0.858943i	$-0.671121\pi$
$881$	−30.3982	−1.02414	−0.512071	+	0.858943i	$0.671121\pi$
$882$	0	0
$883$	−35.6114	−1.19842	−0.599210	−	0.800592i	$-0.704519\pi$
$883$	−35.6114	−1.19842	−0.599210	+	0.800592i	$0.704519\pi$
$884$	4.47214	0.150414
$885$	0	0
$886$	−26.3713	−0.885962
$887$	−22.9736	−0.771377	−0.385689	−	0.922629i	$-0.626036\pi$
$887$	−22.9736	−0.771377	−0.385689	+	0.922629i	$0.626036\pi$
$888$	0	0
$889$	2.27839	0.0764147
$890$	−41.5666	−1.39332
$891$	0	0
$892$	−0.265856	−0.00890153
$893$	2.80538	0.0938784
$894$	0	0
$895$	−13.4128	−0.448339
$896$	12.3490	0.412553
$897$	0	0
$898$	−51.1565	−1.70712
$899$	38.2009	1.27407
$900$	0	0
$901$	−23.7008	−0.789588
$902$	−0.946021	−0.0314991
$903$	0	0
$904$	−50.9944	−1.69605
$905$	−32.5485	−1.08195
$906$	0	0
$907$	−57.1745	−1.89845	−0.949224	−	0.314602i	$-0.898129\pi$
$907$	−57.1745	−1.89845	−0.949224	+	0.314602i	$0.898129\pi$
$908$	0.445920	0.0147984
$909$	0	0
$910$	−17.6960	−0.586616
$911$	−6.82687	−0.226184	−0.113092	−	0.993584i	$-0.536076\pi$
$911$	−6.82687	−0.226184	−0.113092	+	0.993584i	$0.536076\pi$
$912$	0	0
$913$	−13.1648	−0.435692
$914$	−6.63428	−0.219442
$915$	0	0
$916$	−2.54848	−0.0842043
$917$	4.00000	0.132092
$918$	0	0
$919$	12.0692	0.398126	0.199063	−	0.979987i	$-0.436210\pi$
$919$	12.0692	0.398126	0.199063	+	0.979987i	$0.436210\pi$
$920$	32.1467	1.05985
$921$	0	0
$922$	−4.09003	−0.134698
$923$	57.7312	1.90025
$924$	0	0
$925$	3.19239	0.104965
$926$	−56.1045	−1.84371
$927$	0	0
$928$	−3.96540	−0.130171
$929$	−26.8954	−0.882410	−0.441205	−	0.897406i	$-0.645449\pi$
$929$	−26.8954	−0.882410	−0.441205	+	0.897406i	$0.645449\pi$
$930$	0	0
$931$	−5.32340	−0.174468
$932$	2.30885	0.0756288
$933$	0	0
$934$	−29.9356	−0.979523
$935$	−15.2728	−0.499474
$936$	0	0
$937$	−14.9944	−0.489846	−0.244923	−	0.969543i	$-0.578763\pi$
$937$	−14.9944	−0.489846	−0.244923	+	0.969543i	$0.578763\pi$
$938$	12.8221	0.418655
$939$	0	0
$940$	0.175918	0.00573780
$941$	30.1205	0.981900	0.490950	−	0.871188i	$-0.336650\pi$
$941$	30.1205	0.981900	0.490950	+	0.871188i	$0.336650\pi$
$942$	0	0
$943$	−3.18566	−0.103739
$944$	33.9452	1.10482
$945$	0	0
$946$	15.3836	0.500166
$947$	17.3532	0.563903	0.281951	−	0.959429i	$-0.409018\pi$
$947$	17.3532	0.563903	0.281951	+	0.959429i	$0.409018\pi$
$948$	0	0
$949$	−65.8123	−2.13636
$950$	5.85039	0.189812
$951$	0	0
$952$	−17.3324	−0.561745
$953$	2.14064	0.0693422	0.0346711	−	0.999399i	$-0.488962\pi$
$953$	2.14064	0.0693422	0.0346711	+	0.999399i	$0.488962\pi$
$954$	0	0
$955$	−22.5969	−0.731217
$956$	−0.405923	−0.0131285
$957$	0	0
$958$	−16.9861	−0.548796
$959$	−4.77559	−0.154212
$960$	0	0
$961$	26.3353	0.849525
$962$	−31.3497	−1.01076
$963$	0	0
$964$	−0.848944	−0.0273427
$965$	24.2909	0.781952
$966$	0	0
$967$	−1.53326	−0.0493062	−0.0246531	−	0.999696i	$-0.507848\pi$
$967$	−1.53326	−0.0493062	−0.0246531	+	0.999696i	$0.507848\pi$
$968$	2.72161	0.0874759
$969$	0	0
$970$	−6.56351	−0.210742
$971$	26.5574	0.852269	0.426135	−	0.904660i	$-0.359875\pi$
$971$	26.5574	0.852269	0.426135	+	0.904660i	$0.359875\pi$
$972$	0	0
$973$	15.4432	0.495087
$974$	−47.5041	−1.52213
$975$	0	0
$976$	8.51803	0.272655
$977$	55.9017	1.78845	0.894227	−	0.447615i	$-0.147726\pi$
$977$	55.9017	1.78845	0.894227	+	0.447615i	$0.147726\pi$
$978$	0	0
$979$	−11.8504	−0.378740
$980$	−0.333816	−0.0106634
$981$	0	0
$982$	39.0171	1.24509
$983$	53.0361	1.69159	0.845794	−	0.533510i	$-0.179127\pi$
$983$	53.0361	1.69159	0.845794	+	0.533510i	$0.179127\pi$
$984$	0	0
$985$	−5.41411	−0.172508
$986$	46.9915	1.49651
$987$	0	0
$988$	−3.73829	−0.118931
$989$	51.8034	1.64725
$990$	0	0
$991$	14.7362	0.468110	0.234055	−	0.972223i	$-0.424800\pi$
$991$	14.7362	0.468110	0.234055	+	0.972223i	$0.424800\pi$
$992$	−5.95162	−0.188964
$993$	0	0
$994$	16.7368	0.530860
$995$	−7.37402	−0.233772
$996$	0	0
$997$	−45.0665	−1.42727	−0.713635	−	0.700517i	$-0.752953\pi$
$997$	−45.0665	−1.42727	−0.713635	+	0.700517i	$0.752953\pi$
$998$	60.3262	1.90959
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.2.a.l.1.2		3
3.2	odd	2		231.2.a.e.1.2	✓	3
7.6	odd	2		4851.2.a.bi.1.2		3
11.10	odd	2		7623.2.a.cd.1.2		3
12.11	even	2		3696.2.a.bo.1.2		3
15.14	odd	2		5775.2.a.bp.1.2		3
21.20	even	2		1617.2.a.t.1.2		3
33.32	even	2		2541.2.a.bg.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.a.e.1.2	✓	3	3.2	odd	2
693.2.a.l.1.2		3	1.1	even	1	trivial
1617.2.a.t.1.2		3	21.20	even	2
2541.2.a.bg.1.2		3	33.32	even	2
3696.2.a.bo.1.2		3	12.11	even	2
4851.2.a.bi.1.2		3	7.6	odd	2
5775.2.a.bp.1.2		3	15.14	odd	2
7623.2.a.cd.1.2		3	11.10	odd	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:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	693.a (trivial)

\(f(q)\)	\(=\)	\(q-1.46260 q^{2} +0.139194 q^{4} -2.39821 q^{5} -1.00000 q^{7} +2.72161 q^{8} +O(q^{10})\)	\(q-1.46260 q^{2} +0.139194 q^{4} -2.39821 q^{5} -1.00000 q^{7} +2.72161 q^{8} +3.50761 q^{10} +1.00000 q^{11} +5.04502 q^{13} +1.46260 q^{14} -4.25901 q^{16} +6.36842 q^{17} -5.32340 q^{19} -0.333816 q^{20} -1.46260 q^{22} -4.92520 q^{23} +0.751399 q^{25} -7.37883 q^{26} -0.139194 q^{28} -5.04502 q^{29} -7.57201 q^{31} +0.786003 q^{32} -9.31444 q^{34} +2.39821 q^{35} +4.24860 q^{37} +7.78600 q^{38} -6.52699 q^{40} +0.646809 q^{41} -10.5180 q^{43} +0.139194 q^{44} +7.20359 q^{46} -0.526989 q^{47} +1.00000 q^{49} -1.09899 q^{50} +0.702237 q^{52} -3.72161 q^{53} -2.39821 q^{55} -2.72161 q^{56} +7.37883 q^{58} -7.97021 q^{59} -2.00000 q^{61} +11.0748 q^{62} +7.36842 q^{64} -12.0990 q^{65} +8.76663 q^{67} +0.886447 q^{68} -3.50761 q^{70} +11.4432 q^{71} -13.0450 q^{73} -6.21400 q^{74} -0.740987 q^{76} -1.00000 q^{77} +11.4432 q^{79} +10.2140 q^{80} -0.946021 q^{82} -13.1648 q^{83} -15.2728 q^{85} +15.3836 q^{86} +2.72161 q^{88} -11.8504 q^{89} -5.04502 q^{91} -0.685559 q^{92} +0.770774 q^{94} +12.7666 q^{95} -1.87122 q^{97} -1.46260 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{2} + 6 q^{4} - 4 q^{5} - 3 q^{7} - 3 q^{8}+O(q^{10}) \) 3 * q - 2 * q^2 + 6 * q^4 - 4 * q^5 - 3 * q^7 - 3 * q^8	\( 3 q - 2 q^{2} + 6 q^{4} - 4 q^{5} - 3 q^{7} - 3 q^{8} - 11 q^{10} + 3 q^{11} - 4 q^{13} + 2 q^{14} - 4 q^{16} - 8 q^{17} - 8 q^{19} + 3 q^{20} - 2 q^{22} - 10 q^{23} + 15 q^{25} + q^{26} - 6 q^{28} + 4 q^{29} - 2 q^{31} - 8 q^{32} - 4 q^{34} + 4 q^{35} + 13 q^{38} - 18 q^{40} - 14 q^{41} - 14 q^{43} + 6 q^{44} + 28 q^{46} + 3 q^{49} + 19 q^{50} - 29 q^{52} - 4 q^{55} + 3 q^{56} - q^{58} - 6 q^{61} + 38 q^{62} - 5 q^{64} - 14 q^{65} - 4 q^{67} - 42 q^{68} + 11 q^{70} + 12 q^{71} - 20 q^{73} - 29 q^{74} - 11 q^{76} - 3 q^{77} + 12 q^{79} + 41 q^{80} - 6 q^{82} - 6 q^{83} - 6 q^{85} - 24 q^{86} - 3 q^{88} - 26 q^{89} + 4 q^{91} - 26 q^{92} + 35 q^{94} + 8 q^{95} - 4 q^{97} - 2 q^{98}+O(q^{100}) \) 3 * q - 2 * q^2 + 6 * q^4 - 4 * q^5 - 3 * q^7 - 3 * q^8 - 11 * q^10 + 3 * q^11 - 4 * q^13 + 2 * q^14 - 4 * q^16 - 8 * q^17 - 8 * q^19 + 3 * q^20 - 2 * q^22 - 10 * q^23 + 15 * q^25 + q^26 - 6 * q^28 + 4 * q^29 - 2 * q^31 - 8 * q^32 - 4 * q^34 + 4 * q^35 + 13 * q^38 - 18 * q^40 - 14 * q^41 - 14 * q^43 + 6 * q^44 + 28 * q^46 + 3 * q^49 + 19 * q^50 - 29 * q^52 - 4 * q^55 + 3 * q^56 - q^58 - 6 * q^61 + 38 * q^62 - 5 * q^64 - 14 * q^65 - 4 * q^67 - 42 * q^68 + 11 * q^70 + 12 * q^71 - 20 * q^73 - 29 * q^74 - 11 * q^76 - 3 * q^77 + 12 * q^79 + 41 * q^80 - 6 * q^82 - 6 * q^83 - 6 * q^85 - 24 * q^86 - 3 * q^88 - 26 * q^89 + 4 * q^91 - 26 * q^92 + 35 * q^94 + 8 * q^95 - 4 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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