LMFDB - Embedded newform 693.2.a.e.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:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ 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)

$f(q)$	$=$	$q-0.381966 q^{2} -1.85410 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.47214 q^{8} +O(q^{10})$	\(q-0.381966 q^{2} -1.85410 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.47214 q^{8} +0.381966 q^{10} +1.00000 q^{11} +0.236068 q^{13} -0.381966 q^{14} +3.14590 q^{16} -6.47214 q^{17} +3.47214 q^{19} +1.85410 q^{20} -0.381966 q^{22} -8.47214 q^{23} -4.00000 q^{25} -0.0901699 q^{26} -1.85410 q^{28} -1.76393 q^{29} -0.472136 q^{31} -4.14590 q^{32} +2.47214 q^{34} -1.00000 q^{35} -5.47214 q^{37} -1.32624 q^{38} -1.47214 q^{40} -4.47214 q^{41} -4.00000 q^{43} -1.85410 q^{44} +3.23607 q^{46} +0.236068 q^{47} +1.00000 q^{49} +1.52786 q^{50} -0.437694 q^{52} -5.52786 q^{53} -1.00000 q^{55} +1.47214 q^{56} +0.673762 q^{58} -15.1803 q^{59} +8.47214 q^{61} +0.180340 q^{62} -4.70820 q^{64} -0.236068 q^{65} +9.18034 q^{67} +12.0000 q^{68} +0.381966 q^{70} +4.47214 q^{71} -9.76393 q^{73} +2.09017 q^{74} -6.43769 q^{76} +1.00000 q^{77} +4.47214 q^{79} -3.14590 q^{80} +1.70820 q^{82} +11.4164 q^{83} +6.47214 q^{85} +1.52786 q^{86} +1.47214 q^{88} +10.0000 q^{89} +0.236068 q^{91} +15.7082 q^{92} -0.0901699 q^{94} -3.47214 q^{95} -8.00000 q^{97} -0.381966 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{2} + 3 q^{4} - 2 q^{5} + 2 q^{7} - 6 q^{8}+O(q^{10}) $ 2 * q - 3 * q^2 + 3 * q^4 - 2 * q^5 + 2 * q^7 - 6 * q^8	$ 2 q - 3 q^{2} + 3 q^{4} - 2 q^{5} + 2 q^{7} - 6 q^{8} + 3 q^{10} + 2 q^{11} - 4 q^{13} - 3 q^{14} + 13 q^{16} - 4 q^{17} - 2 q^{19} - 3 q^{20} - 3 q^{22} - 8 q^{23} - 8 q^{25} + 11 q^{26} + 3 q^{28} - 8 q^{29} + 8 q^{31} - 15 q^{32} - 4 q^{34} - 2 q^{35} - 2 q^{37} + 13 q^{38} + 6 q^{40} - 8 q^{43} + 3 q^{44} + 2 q^{46} - 4 q^{47} + 2 q^{49} + 12 q^{50} - 21 q^{52} - 20 q^{53} - 2 q^{55} - 6 q^{56} + 17 q^{58} - 8 q^{59} + 8 q^{61} - 22 q^{62} + 4 q^{64} + 4 q^{65} - 4 q^{67} + 24 q^{68} + 3 q^{70} - 24 q^{73} - 7 q^{74} - 33 q^{76} + 2 q^{77} - 13 q^{80} - 10 q^{82} - 4 q^{83} + 4 q^{85} + 12 q^{86} - 6 q^{88} + 20 q^{89} - 4 q^{91} + 18 q^{92} + 11 q^{94} + 2 q^{95} - 16 q^{97} - 3 q^{98}+O(q^{100}) $ 2 * q - 3 * q^2 + 3 * q^4 - 2 * q^5 + 2 * q^7 - 6 * q^8 + 3 * q^10 + 2 * q^11 - 4 * q^13 - 3 * q^14 + 13 * q^16 - 4 * q^17 - 2 * q^19 - 3 * q^20 - 3 * q^22 - 8 * q^23 - 8 * q^25 + 11 * q^26 + 3 * q^28 - 8 * q^29 + 8 * q^31 - 15 * q^32 - 4 * q^34 - 2 * q^35 - 2 * q^37 + 13 * q^38 + 6 * q^40 - 8 * q^43 + 3 * q^44 + 2 * q^46 - 4 * q^47 + 2 * q^49 + 12 * q^50 - 21 * q^52 - 20 * q^53 - 2 * q^55 - 6 * q^56 + 17 * q^58 - 8 * q^59 + 8 * q^61 - 22 * q^62 + 4 * q^64 + 4 * q^65 - 4 * q^67 + 24 * q^68 + 3 * q^70 - 24 * q^73 - 7 * q^74 - 33 * q^76 + 2 * q^77 - 13 * q^80 - 10 * q^82 - 4 * q^83 + 4 * q^85 + 12 * q^86 - 6 * q^88 + 20 * q^89 - 4 * q^91 + 18 * q^92 + 11 * q^94 + 2 * q^95 - 16 * 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 $

$2$	−0.381966	−0.270091	−0.135045	−	0.990839i	$-0.543118\pi$
$2$	−0.381966	−0.270091	−0.135045	+	0.990839i	$0.543118\pi$
$3$	0	0
$3$	0	0
$4$	−1.85410	−0.927051
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.47214	0.520479
$9$	0	0
$10$	0.381966	0.120788
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	0.236068	0.0654735	0.0327367	−	0.999464i	$-0.489578\pi$
$13$	0.236068	0.0654735	0.0327367	+	0.999464i	$0.489578\pi$
$14$	−0.381966	−0.102085
$15$	0	0
$16$	3.14590	0.786475
$17$	−6.47214	−1.56972	−0.784862	−	0.619671i	$-0.787266\pi$
$17$	−6.47214	−1.56972	−0.784862	+	0.619671i	$0.787266\pi$
$18$	0	0
$19$	3.47214	0.796563	0.398281	−	0.917263i	$-0.369607\pi$
$19$	3.47214	0.796563	0.398281	+	0.917263i	$0.369607\pi$
$20$	1.85410	0.414590
$21$	0	0
$22$	−0.381966	−0.0814354
$23$	−8.47214	−1.76656	−0.883281	−	0.468844i	$-0.844671\pi$
$23$	−8.47214	−1.76656	−0.883281	+	0.468844i	$0.844671\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	−0.0901699	−0.0176838
$27$	0	0
$28$	−1.85410	−0.350392
$29$	−1.76393	−0.327554	−0.163777	−	0.986497i	$-0.552368\pi$
$29$	−1.76393	−0.327554	−0.163777	+	0.986497i	$0.552368\pi$
$30$	0	0
$31$	−0.472136	−0.0847981	−0.0423991	−	0.999101i	$-0.513500\pi$
$31$	−0.472136	−0.0847981	−0.0423991	+	0.999101i	$0.513500\pi$
$32$	−4.14590	−0.732898
$33$	0	0
$34$	2.47214	0.423968
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−5.47214	−0.899614	−0.449807	−	0.893126i	$-0.648507\pi$
$37$	−5.47214	−0.899614	−0.449807	+	0.893126i	$0.648507\pi$
$38$	−1.32624	−0.215144
$39$	0	0
$40$	−1.47214	−0.232765
$41$	−4.47214	−0.698430	−0.349215	−	0.937043i	$-0.613552\pi$
$41$	−4.47214	−0.698430	−0.349215	+	0.937043i	$0.613552\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	−1.85410	−0.279516
$45$	0	0
$46$	3.23607	0.477132
$47$	0.236068	0.0344341	0.0172170	−	0.999852i	$-0.494519\pi$
$47$	0.236068	0.0344341	0.0172170	+	0.999852i	$0.494519\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	1.52786	0.216073
$51$	0	0
$52$	−0.437694	−0.0606973
$53$	−5.52786	−0.759311	−0.379655	−	0.925128i	$-0.623957\pi$
$53$	−5.52786	−0.759311	−0.379655	+	0.925128i	$0.623957\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	1.47214	0.196722
$57$	0	0
$58$	0.673762	0.0884693
$59$	−15.1803	−1.97631	−0.988156	−	0.153453i	$-0.950961\pi$
$59$	−15.1803	−1.97631	−0.988156	+	0.153453i	$0.950961\pi$
$60$	0	0
$61$	8.47214	1.08475	0.542373	−	0.840138i	$-0.317526\pi$
$61$	8.47214	1.08475	0.542373	+	0.840138i	$0.317526\pi$
$62$	0.180340	0.0229032
$63$	0	0
$64$	−4.70820	−0.588525
$65$	−0.236068	−0.0292806
$66$	0	0
$67$	9.18034	1.12156	0.560779	−	0.827966i	$-0.310502\pi$
$67$	9.18034	1.12156	0.560779	+	0.827966i	$0.310502\pi$
$68$	12.0000	1.45521
$69$	0	0
$70$	0.381966	0.0456537
$71$	4.47214	0.530745	0.265372	−	0.964146i	$-0.414505\pi$
$71$	4.47214	0.530745	0.265372	+	0.964146i	$0.414505\pi$
$72$	0	0
$73$	−9.76393	−1.14278	−0.571391	−	0.820678i	$-0.693596\pi$
$73$	−9.76393	−1.14278	−0.571391	+	0.820678i	$0.693596\pi$
$74$	2.09017	0.242977
$75$	0	0
$76$	−6.43769	−0.738454
$77$	1.00000	0.113961
$78$	0	0
$79$	4.47214	0.503155	0.251577	−	0.967837i	$-0.419051\pi$
$79$	4.47214	0.503155	0.251577	+	0.967837i	$0.419051\pi$
$80$	−3.14590	−0.351722
$81$	0	0
$82$	1.70820	0.188640
$83$	11.4164	1.25311	0.626557	−	0.779376i	$-0.284464\pi$
$83$	11.4164	1.25311	0.626557	+	0.779376i	$0.284464\pi$
$84$	0	0
$85$	6.47214	0.702002
$86$	1.52786	0.164754
$87$	0	0
$88$	1.47214	0.156930
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	0.236068	0.0247466
$92$	15.7082	1.63769
$93$	0	0
$94$	−0.0901699	−0.00930032
$95$	−3.47214	−0.356234
$96$	0	0
$97$	−8.00000	−0.812277	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000	−0.812277	−0.406138	+	0.913812i	$0.633125\pi$
$98$	−0.381966	−0.0385844
$99$	0	0
$100$	7.41641	0.741641
$101$	8.47214	0.843009	0.421505	−	0.906826i	$-0.361502\pi$
$101$	8.47214	0.843009	0.421505	+	0.906826i	$0.361502\pi$
$102$	0	0
$103$	−15.4164	−1.51902	−0.759512	−	0.650493i	$-0.774562\pi$
$103$	−15.4164	−1.51902	−0.759512	+	0.650493i	$0.774562\pi$
$104$	0.347524	0.0340775
$105$	0	0
$106$	2.11146	0.205083
$107$	−17.0000	−1.64345	−0.821726	−	0.569883i	$-0.806989\pi$
$107$	−17.0000	−1.64345	−0.821726	+	0.569883i	$0.806989\pi$
$108$	0	0
$109$	−16.4721	−1.57774	−0.788872	−	0.614557i	$-0.789335\pi$
$109$	−16.4721	−1.57774	−0.788872	+	0.614557i	$0.789335\pi$
$110$	0.381966	0.0364190
$111$	0	0
$112$	3.14590	0.297259
$113$	11.4164	1.07397	0.536983	−	0.843593i	$-0.319564\pi$
$113$	11.4164	1.07397	0.536983	+	0.843593i	$0.319564\pi$
$114$	0	0
$115$	8.47214	0.790031
$116$	3.27051	0.303659
$117$	0	0
$118$	5.79837	0.533784
$119$	−6.47214	−0.593300
$120$	0	0
$121$	1.00000	0.0909091
$122$	−3.23607	−0.292980
$123$	0	0
$124$	0.875388	0.0786122
$125$	9.00000	0.804984
$126$	0	0
$127$	4.47214	0.396838	0.198419	−	0.980117i	$-0.436419\pi$
$127$	4.47214	0.396838	0.198419	+	0.980117i	$0.436419\pi$
$128$	10.0902	0.891853
$129$	0	0
$130$	0.0901699	0.00790843
$131$	10.9443	0.956205	0.478103	−	0.878304i	$-0.341325\pi$
$131$	10.9443	0.956205	0.478103	+	0.878304i	$0.341325\pi$
$132$	0	0
$133$	3.47214	0.301072
$134$	−3.50658	−0.302922
$135$	0	0
$136$	−9.52786	−0.817008
$137$	1.52786	0.130534	0.0652671	−	0.997868i	$-0.479210\pi$
$137$	1.52786	0.130534	0.0652671	+	0.997868i	$0.479210\pi$
$138$	0	0
$139$	−8.94427	−0.758643	−0.379322	−	0.925265i	$-0.623843\pi$
$139$	−8.94427	−0.758643	−0.379322	+	0.925265i	$0.623843\pi$
$140$	1.85410	0.156700
$141$	0	0
$142$	−1.70820	−0.143349
$143$	0.236068	0.0197410
$144$	0	0
$145$	1.76393	0.146487
$146$	3.72949	0.308655
$147$	0	0
$148$	10.1459	0.833988
$149$	−14.7082	−1.20494	−0.602472	−	0.798140i	$-0.705817\pi$
$149$	−14.7082	−1.20494	−0.602472	+	0.798140i	$0.705817\pi$
$150$	0	0
$151$	−10.4721	−0.852210	−0.426105	−	0.904674i	$-0.640115\pi$
$151$	−10.4721	−0.852210	−0.426105	+	0.904674i	$0.640115\pi$
$152$	5.11146	0.414594
$153$	0	0
$154$	−0.381966	−0.0307797
$155$	0.472136	0.0379229
$156$	0	0
$157$	15.5279	1.23926	0.619629	−	0.784895i	$-0.287283\pi$
$157$	15.5279	1.23926	0.619629	+	0.784895i	$0.287283\pi$
$158$	−1.70820	−0.135897
$159$	0	0
$160$	4.14590	0.327762
$161$	−8.47214	−0.667698
$162$	0	0
$163$	3.29180	0.257833	0.128917	−	0.991655i	$-0.458850\pi$
$163$	3.29180	0.257833	0.128917	+	0.991655i	$0.458850\pi$
$164$	8.29180	0.647480
$165$	0	0
$166$	−4.36068	−0.338454
$167$	1.05573	0.0816947	0.0408473	−	0.999165i	$-0.486994\pi$
$167$	1.05573	0.0816947	0.0408473	+	0.999165i	$0.486994\pi$
$168$	0	0
$169$	−12.9443	−0.995713
$170$	−2.47214	−0.189604
$171$	0	0
$172$	7.41641	0.565496
$173$	23.8885	1.81621	0.908106	−	0.418740i	$-0.137528\pi$
$173$	23.8885	1.81621	0.908106	+	0.418740i	$0.137528\pi$
$174$	0	0
$175$	−4.00000	−0.302372
$176$	3.14590	0.237131
$177$	0	0
$178$	−3.81966	−0.286296
$179$	−23.8885	−1.78551	−0.892757	−	0.450539i	$-0.851232\pi$
$179$	−23.8885	−1.78551	−0.892757	+	0.450539i	$0.851232\pi$
$180$	0	0
$181$	13.4164	0.997234	0.498617	−	0.866822i	$-0.333841\pi$
$181$	13.4164	0.997234	0.498617	+	0.866822i	$0.333841\pi$
$182$	−0.0901699	−0.00668384
$183$	0	0
$184$	−12.4721	−0.919458
$185$	5.47214	0.402319
$186$	0	0
$187$	−6.47214	−0.473289
$188$	−0.437694	−0.0319221
$189$	0	0
$190$	1.32624	0.0962154
$191$	11.8885	0.860225	0.430112	−	0.902775i	$-0.358474\pi$
$191$	11.8885	0.860225	0.430112	+	0.902775i	$0.358474\pi$
$192$	0	0
$193$	10.4721	0.753801	0.376900	−	0.926254i	$-0.376990\pi$
$193$	10.4721	0.753801	0.376900	+	0.926254i	$0.376990\pi$
$194$	3.05573	0.219388
$195$	0	0
$196$	−1.85410	−0.132436
$197$	−5.41641	−0.385903	−0.192952	−	0.981208i	$-0.561806\pi$
$197$	−5.41641	−0.385903	−0.192952	+	0.981208i	$0.561806\pi$
$198$	0	0
$199$	20.9443	1.48470	0.742350	−	0.670012i	$-0.233711\pi$
$199$	20.9443	1.48470	0.742350	+	0.670012i	$0.233711\pi$
$200$	−5.88854	−0.416383
$201$	0	0
$202$	−3.23607	−0.227689
$203$	−1.76393	−0.123804
$204$	0	0
$205$	4.47214	0.312348
$206$	5.88854	0.410274
$207$	0	0
$208$	0.742646	0.0514932
$209$	3.47214	0.240173
$210$	0	0
$211$	12.9443	0.891120	0.445560	−	0.895252i	$-0.353005\pi$
$211$	12.9443	0.891120	0.445560	+	0.895252i	$0.353005\pi$
$212$	10.2492	0.703920
$213$	0	0
$214$	6.49342	0.443881
$215$	4.00000	0.272798
$216$	0	0
$217$	−0.472136	−0.0320507
$218$	6.29180	0.426134
$219$	0	0
$220$	1.85410	0.125004
$221$	−1.52786	−0.102775
$222$	0	0
$223$	0.944272	0.0632331	0.0316166	−	0.999500i	$-0.489934\pi$
$223$	0.944272	0.0632331	0.0316166	+	0.999500i	$0.489934\pi$
$224$	−4.14590	−0.277009
$225$	0	0
$226$	−4.36068	−0.290068
$227$	−9.05573	−0.601050	−0.300525	−	0.953774i	$-0.597162\pi$
$227$	−9.05573	−0.601050	−0.300525	+	0.953774i	$0.597162\pi$
$228$	0	0
$229$	14.9443	0.987545	0.493773	−	0.869591i	$-0.335618\pi$
$229$	14.9443	0.987545	0.493773	+	0.869591i	$0.335618\pi$
$230$	−3.23607	−0.213380
$231$	0	0
$232$	−2.59675	−0.170485
$233$	−9.41641	−0.616889	−0.308445	−	0.951242i	$-0.599808\pi$
$233$	−9.41641	−0.616889	−0.308445	+	0.951242i	$0.599808\pi$
$234$	0	0
$235$	−0.236068	−0.0153994
$236$	28.1459	1.83214
$237$	0	0
$238$	2.47214	0.160245
$239$	−0.527864	−0.0341447	−0.0170723	−	0.999854i	$-0.505435\pi$
$239$	−0.527864	−0.0341447	−0.0170723	+	0.999854i	$0.505435\pi$
$240$	0	0
$241$	−16.1246	−1.03868	−0.519339	−	0.854568i	$-0.673822\pi$
$241$	−16.1246	−1.03868	−0.519339	+	0.854568i	$0.673822\pi$
$242$	−0.381966	−0.0245537
$243$	0	0
$244$	−15.7082	−1.00561
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	0.819660	0.0521537
$248$	−0.695048	−0.0441356
$249$	0	0
$250$	−3.43769	−0.217419
$251$	12.1246	0.765299	0.382649	−	0.923894i	$-0.375012\pi$
$251$	12.1246	0.765299	0.382649	+	0.923894i	$0.375012\pi$
$252$	0	0
$253$	−8.47214	−0.532639
$254$	−1.70820	−0.107182
$255$	0	0
$256$	5.56231	0.347644
$257$	10.4164	0.649758	0.324879	−	0.945756i	$-0.394676\pi$
$257$	10.4164	0.649758	0.324879	+	0.945756i	$0.394676\pi$
$258$	0	0
$259$	−5.47214	−0.340022
$260$	0.437694	0.0271446
$261$	0	0
$262$	−4.18034	−0.258262
$263$	−16.4164	−1.01228	−0.506140	−	0.862452i	$-0.668928\pi$
$263$	−16.4164	−1.01228	−0.506140	+	0.862452i	$0.668928\pi$
$264$	0	0
$265$	5.52786	0.339574
$266$	−1.32624	−0.0813169
$267$	0	0
$268$	−17.0213	−1.03974
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	−20.8885	−1.26889	−0.634444	−	0.772969i	$-0.718771\pi$
$271$	−20.8885	−1.26889	−0.634444	+	0.772969i	$0.718771\pi$
$272$	−20.3607	−1.23455
$273$	0	0
$274$	−0.583592	−0.0352561
$275$	−4.00000	−0.241209
$276$	0	0
$277$	−6.47214	−0.388873	−0.194436	−	0.980915i	$-0.562288\pi$
$277$	−6.47214	−0.388873	−0.194436	+	0.980915i	$0.562288\pi$
$278$	3.41641	0.204903
$279$	0	0
$280$	−1.47214	−0.0879770
$281$	2.81966	0.168207	0.0841034	−	0.996457i	$-0.473197\pi$
$281$	2.81966	0.168207	0.0841034	+	0.996457i	$0.473197\pi$
$282$	0	0
$283$	−19.3607	−1.15087	−0.575436	−	0.817846i	$-0.695168\pi$
$283$	−19.3607	−1.15087	−0.575436	+	0.817846i	$0.695168\pi$
$284$	−8.29180	−0.492028
$285$	0	0
$286$	−0.0901699	−0.00533186
$287$	−4.47214	−0.263982
$288$	0	0
$289$	24.8885	1.46403
$290$	−0.673762	−0.0395647
$291$	0	0
$292$	18.1033	1.05942
$293$	8.00000	0.467365	0.233682	−	0.972313i	$-0.424922\pi$
$293$	8.00000	0.467365	0.233682	+	0.972313i	$0.424922\pi$
$294$	0	0
$295$	15.1803	0.883834
$296$	−8.05573	−0.468230
$297$	0	0
$298$	5.61803	0.325444
$299$	−2.00000	−0.115663
$300$	0	0
$301$	−4.00000	−0.230556
$302$	4.00000	0.230174
$303$	0	0
$304$	10.9230	0.626476
$305$	−8.47214	−0.485113
$306$	0	0
$307$	−20.9443	−1.19535	−0.597676	−	0.801737i	$-0.703909\pi$
$307$	−20.9443	−1.19535	−0.597676	+	0.801737i	$0.703909\pi$
$308$	−1.85410	−0.105647
$309$	0	0
$310$	−0.180340	−0.0102426
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	0	0
$313$	28.3607	1.60304	0.801520	−	0.597968i	$-0.204025\pi$
$313$	28.3607	1.60304	0.801520	+	0.597968i	$0.204025\pi$
$314$	−5.93112	−0.334712
$315$	0	0
$316$	−8.29180	−0.466450
$317$	−9.41641	−0.528878	−0.264439	−	0.964402i	$-0.585187\pi$
$317$	−9.41641	−0.528878	−0.264439	+	0.964402i	$0.585187\pi$
$318$	0	0
$319$	−1.76393	−0.0987612
$320$	4.70820	0.263197
$321$	0	0
$322$	3.23607	0.180339
$323$	−22.4721	−1.25038
$324$	0	0
$325$	−0.944272	−0.0523788
$326$	−1.25735	−0.0696384
$327$	0	0
$328$	−6.58359	−0.363518
$329$	0.236068	0.0130148
$330$	0	0
$331$	12.9443	0.711482	0.355741	−	0.934585i	$-0.384229\pi$
$331$	12.9443	0.711482	0.355741	+	0.934585i	$0.384229\pi$
$332$	−21.1672	−1.16170
$333$	0	0
$334$	−0.403252	−0.0220650
$335$	−9.18034	−0.501576
$336$	0	0
$337$	−11.5279	−0.627963	−0.313981	−	0.949429i	$-0.601663\pi$
$337$	−11.5279	−0.627963	−0.313981	+	0.949429i	$0.601663\pi$
$338$	4.94427	0.268933
$339$	0	0
$340$	−12.0000	−0.650791
$341$	−0.472136	−0.0255676
$342$	0	0
$343$	1.00000	0.0539949
$344$	−5.88854	−0.317489
$345$	0	0
$346$	−9.12461	−0.490542
$347$	−8.94427	−0.480154	−0.240077	−	0.970754i	$-0.577173\pi$
$347$	−8.94427	−0.480154	−0.240077	+	0.970754i	$0.577173\pi$
$348$	0	0
$349$	−17.6525	−0.944915	−0.472458	−	0.881353i	$-0.656633\pi$
$349$	−17.6525	−0.944915	−0.472458	+	0.881353i	$0.656633\pi$
$350$	1.52786	0.0816678
$351$	0	0
$352$	−4.14590	−0.220977
$353$	26.5279	1.41194	0.705968	−	0.708244i	$-0.250512\pi$
$353$	26.5279	1.41194	0.705968	+	0.708244i	$0.250512\pi$
$354$	0	0
$355$	−4.47214	−0.237356
$356$	−18.5410	−0.982672
$357$	0	0
$358$	9.12461	0.482251
$359$	−25.8885	−1.36635	−0.683173	−	0.730257i	$-0.739400\pi$
$359$	−25.8885	−1.36635	−0.683173	+	0.730257i	$0.739400\pi$
$360$	0	0
$361$	−6.94427	−0.365488
$362$	−5.12461	−0.269344
$363$	0	0
$364$	−0.437694	−0.0229414
$365$	9.76393	0.511068
$366$	0	0
$367$	6.00000	0.313197	0.156599	−	0.987662i	$-0.449947\pi$
$367$	6.00000	0.313197	0.156599	+	0.987662i	$0.449947\pi$
$368$	−26.6525	−1.38936
$369$	0	0
$370$	−2.09017	−0.108663
$371$	−5.52786	−0.286992
$372$	0	0
$373$	−32.8328	−1.70002	−0.850009	−	0.526768i	$-0.823404\pi$
$373$	−32.8328	−1.70002	−0.850009	+	0.526768i	$0.823404\pi$
$374$	2.47214	0.127831
$375$	0	0
$376$	0.347524	0.0179222
$377$	−0.416408	−0.0214461
$378$	0	0
$379$	18.5967	0.955251	0.477625	−	0.878564i	$-0.341498\pi$
$379$	18.5967	0.955251	0.477625	+	0.878564i	$0.341498\pi$
$380$	6.43769	0.330247
$381$	0	0
$382$	−4.54102	−0.232339
$383$	−11.0557	−0.564921	−0.282461	−	0.959279i	$-0.591151\pi$
$383$	−11.0557	−0.564921	−0.282461	+	0.959279i	$0.591151\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	−4.00000	−0.203595
$387$	0	0
$388$	14.8328	0.753022
$389$	−3.88854	−0.197157	−0.0985785	−	0.995129i	$-0.531430\pi$
$389$	−3.88854	−0.197157	−0.0985785	+	0.995129i	$0.531430\pi$
$390$	0	0
$391$	54.8328	2.77301
$392$	1.47214	0.0743541
$393$	0	0
$394$	2.06888	0.104229
$395$	−4.47214	−0.225018
$396$	0	0
$397$	−25.3050	−1.27002	−0.635010	−	0.772504i	$-0.719004\pi$
$397$	−25.3050	−1.27002	−0.635010	+	0.772504i	$0.719004\pi$
$398$	−8.00000	−0.401004
$399$	0	0
$400$	−12.5836	−0.629180
$401$	9.88854	0.493810	0.246905	−	0.969040i	$-0.420586\pi$
$401$	9.88854	0.493810	0.246905	+	0.969040i	$0.420586\pi$
$402$	0	0
$403$	−0.111456	−0.00555203
$404$	−15.7082	−0.781512
$405$	0	0
$406$	0.673762	0.0334383
$407$	−5.47214	−0.271244
$408$	0	0
$409$	2.58359	0.127750	0.0638752	−	0.997958i	$-0.479654\pi$
$409$	2.58359	0.127750	0.0638752	+	0.997958i	$0.479654\pi$
$410$	−1.70820	−0.0843622
$411$	0	0
$412$	28.5836	1.40821
$413$	−15.1803	−0.746976
$414$	0	0
$415$	−11.4164	−0.560409
$416$	−0.978714	−0.0479854
$417$	0	0
$418$	−1.32624	−0.0648684
$419$	−3.65248	−0.178435	−0.0892176	−	0.996012i	$-0.528437\pi$
$419$	−3.65248	−0.178435	−0.0892176	+	0.996012i	$0.528437\pi$
$420$	0	0
$421$	4.52786	0.220675	0.110337	−	0.993894i	$-0.464807\pi$
$421$	4.52786	0.220675	0.110337	+	0.993894i	$0.464807\pi$
$422$	−4.94427	−0.240683
$423$	0	0
$424$	−8.13777	−0.395205
$425$	25.8885	1.25578
$426$	0	0
$427$	8.47214	0.409995
$428$	31.5197	1.52356
$429$	0	0
$430$	−1.52786	−0.0736801
$431$	34.4164	1.65778	0.828890	−	0.559412i	$-0.188973\pi$
$431$	34.4164	1.65778	0.828890	+	0.559412i	$0.188973\pi$
$432$	0	0
$433$	30.3607	1.45904	0.729521	−	0.683959i	$-0.239743\pi$
$433$	30.3607	1.45904	0.729521	+	0.683959i	$0.239743\pi$
$434$	0.180340	0.00865659
$435$	0	0
$436$	30.5410	1.46265
$437$	−29.4164	−1.40718
$438$	0	0
$439$	2.88854	0.137863	0.0689313	−	0.997621i	$-0.478041\pi$
$439$	2.88854	0.137863	0.0689313	+	0.997621i	$0.478041\pi$
$440$	−1.47214	−0.0701813
$441$	0	0
$442$	0.583592	0.0277586
$443$	−30.4721	−1.44777	−0.723887	−	0.689918i	$-0.757647\pi$
$443$	−30.4721	−1.44777	−0.723887	+	0.689918i	$0.757647\pi$
$444$	0	0
$445$	−10.0000	−0.474045
$446$	−0.360680	−0.0170787
$447$	0	0
$448$	−4.70820	−0.222442
$449$	−13.5279	−0.638419	−0.319210	−	0.947684i	$-0.603417\pi$
$449$	−13.5279	−0.638419	−0.319210	+	0.947684i	$0.603417\pi$
$450$	0	0
$451$	−4.47214	−0.210585
$452$	−21.1672	−0.995621
$453$	0	0
$454$	3.45898	0.162338
$455$	−0.236068	−0.0110670
$456$	0	0
$457$	−21.4164	−1.00182	−0.500909	−	0.865500i	$-0.667001\pi$
$457$	−21.4164	−1.00182	−0.500909	+	0.865500i	$0.667001\pi$
$458$	−5.70820	−0.266727
$459$	0	0
$460$	−15.7082	−0.732399
$461$	38.4721	1.79183	0.895913	−	0.444230i	$-0.146523\pi$
$461$	38.4721	1.79183	0.895913	+	0.444230i	$0.146523\pi$
$462$	0	0
$463$	−33.0689	−1.53684	−0.768421	−	0.639945i	$-0.778957\pi$
$463$	−33.0689	−1.53684	−0.768421	+	0.639945i	$0.778957\pi$
$464$	−5.54915	−0.257613
$465$	0	0
$466$	3.59675	0.166616
$467$	−18.2361	−0.843865	−0.421932	−	0.906627i	$-0.638648\pi$
$467$	−18.2361	−0.843865	−0.421932	+	0.906627i	$0.638648\pi$
$468$	0	0
$469$	9.18034	0.423909
$470$	0.0901699	0.00415923
$471$	0	0
$472$	−22.3475	−1.02863
$473$	−4.00000	−0.183920
$474$	0	0
$475$	−13.8885	−0.637250
$476$	12.0000	0.550019
$477$	0	0
$478$	0.201626	0.00922216
$479$	16.4721	0.752631	0.376316	−	0.926492i	$-0.377191\pi$
$479$	16.4721	0.752631	0.376316	+	0.926492i	$0.377191\pi$
$480$	0	0
$481$	−1.29180	−0.0589008
$482$	6.15905	0.280537
$483$	0	0
$484$	−1.85410	−0.0842774
$485$	8.00000	0.363261
$486$	0	0
$487$	33.8885	1.53564	0.767818	−	0.640668i	$-0.221342\pi$
$487$	33.8885	1.53564	0.767818	+	0.640668i	$0.221342\pi$
$488$	12.4721	0.564587
$489$	0	0
$490$	0.381966	0.0172555
$491$	3.00000	0.135388	0.0676941	−	0.997706i	$-0.478436\pi$
$491$	3.00000	0.135388	0.0676941	+	0.997706i	$0.478436\pi$
$492$	0	0
$493$	11.4164	0.514169
$494$	−0.313082	−0.0140862
$495$	0	0
$496$	−1.48529	−0.0666916
$497$	4.47214	0.200603
$498$	0	0
$499$	−17.1803	−0.769098	−0.384549	−	0.923105i	$-0.625643\pi$
$499$	−17.1803	−0.769098	−0.384549	+	0.923105i	$0.625643\pi$
$500$	−16.6869	−0.746262
$501$	0	0
$502$	−4.63119	−0.206700
$503$	33.4164	1.48996	0.744982	−	0.667085i	$-0.232458\pi$
$503$	33.4164	1.48996	0.744982	+	0.667085i	$0.232458\pi$
$504$	0	0
$505$	−8.47214	−0.377005
$506$	3.23607	0.143861
$507$	0	0
$508$	−8.29180	−0.367889
$509$	2.94427	0.130503	0.0652513	−	0.997869i	$-0.479215\pi$
$509$	2.94427	0.130503	0.0652513	+	0.997869i	$0.479215\pi$
$510$	0	0
$511$	−9.76393	−0.431931
$512$	−22.3050	−0.985749
$513$	0	0
$514$	−3.97871	−0.175494
$515$	15.4164	0.679328
$516$	0	0
$517$	0.236068	0.0103823
$518$	2.09017	0.0918368
$519$	0	0
$520$	−0.347524	−0.0152399
$521$	41.4721	1.81693	0.908464	−	0.417964i	$-0.137256\pi$
$521$	41.4721	1.81693	0.908464	+	0.417964i	$0.137256\pi$
$522$	0	0
$523$	6.41641	0.280570	0.140285	−	0.990111i	$-0.455198\pi$
$523$	6.41641	0.280570	0.140285	+	0.990111i	$0.455198\pi$
$524$	−20.2918	−0.886451
$525$	0	0
$526$	6.27051	0.273407
$527$	3.05573	0.133110
$528$	0	0
$529$	48.7771	2.12074
$530$	−2.11146	−0.0917158
$531$	0	0
$532$	−6.43769	−0.279109
$533$	−1.05573	−0.0457287
$534$	0	0
$535$	17.0000	0.734974
$536$	13.5147	0.583746
$537$	0	0
$538$	−2.29180	−0.0988063
$539$	1.00000	0.0430730
$540$	0	0
$541$	25.4164	1.09274	0.546368	−	0.837545i	$-0.316010\pi$
$541$	25.4164	1.09274	0.546368	+	0.837545i	$0.316010\pi$
$542$	7.97871	0.342715
$543$	0	0
$544$	26.8328	1.15045
$545$	16.4721	0.705589
$546$	0	0
$547$	−15.4164	−0.659158	−0.329579	−	0.944128i	$-0.606907\pi$
$547$	−15.4164	−0.659158	−0.329579	+	0.944128i	$0.606907\pi$
$548$	−2.83282	−0.121012
$549$	0	0
$550$	1.52786	0.0651483
$551$	−6.12461	−0.260917
$552$	0	0
$553$	4.47214	0.190175
$554$	2.47214	0.105031
$555$	0	0
$556$	16.5836	0.703301
$557$	41.0689	1.74014	0.870072	−	0.492924i	$-0.164072\pi$
$557$	41.0689	1.74014	0.870072	+	0.492924i	$0.164072\pi$
$558$	0	0
$559$	−0.944272	−0.0399384
$560$	−3.14590	−0.132938
$561$	0	0
$562$	−1.07701	−0.0454311
$563$	−45.4164	−1.91407	−0.957037	−	0.289967i	$-0.906356\pi$
$563$	−45.4164	−1.91407	−0.957037	+	0.289967i	$0.906356\pi$
$564$	0	0
$565$	−11.4164	−0.480292
$566$	7.39512	0.310840
$567$	0	0
$568$	6.58359	0.276241
$569$	43.3050	1.81544	0.907719	−	0.419579i	$-0.137822\pi$
$569$	43.3050	1.81544	0.907719	+	0.419579i	$0.137822\pi$
$570$	0	0
$571$	−25.3050	−1.05898	−0.529490	−	0.848316i	$-0.677617\pi$
$571$	−25.3050	−1.05898	−0.529490	+	0.848316i	$0.677617\pi$
$572$	−0.437694	−0.0183009
$573$	0	0
$574$	1.70820	0.0712991
$575$	33.8885	1.41325
$576$	0	0
$577$	9.05573	0.376995	0.188497	−	0.982074i	$-0.439638\pi$
$577$	9.05573	0.376995	0.188497	+	0.982074i	$0.439638\pi$
$578$	−9.50658	−0.395422
$579$	0	0
$580$	−3.27051	−0.135801
$581$	11.4164	0.473632
$582$	0	0
$583$	−5.52786	−0.228941
$584$	−14.3738	−0.594794
$585$	0	0
$586$	−3.05573	−0.126231
$587$	−22.7082	−0.937268	−0.468634	−	0.883392i	$-0.655254\pi$
$587$	−22.7082	−0.937268	−0.468634	+	0.883392i	$0.655254\pi$
$588$	0	0
$589$	−1.63932	−0.0675470
$590$	−5.79837	−0.238715
$591$	0	0
$592$	−17.2148	−0.707523
$593$	−7.41641	−0.304555	−0.152278	−	0.988338i	$-0.548661\pi$
$593$	−7.41641	−0.304555	−0.152278	+	0.988338i	$0.548661\pi$
$594$	0	0
$595$	6.47214	0.265332
$596$	27.2705	1.11704
$597$	0	0
$598$	0.763932	0.0312395
$599$	24.8328	1.01464	0.507321	−	0.861757i	$-0.330636\pi$
$599$	24.8328	1.01464	0.507321	+	0.861757i	$0.330636\pi$
$600$	0	0
$601$	−38.7082	−1.57894	−0.789470	−	0.613789i	$-0.789645\pi$
$601$	−38.7082	−1.57894	−0.789470	+	0.613789i	$0.789645\pi$
$602$	1.52786	0.0622711
$603$	0	0
$604$	19.4164	0.790042
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	33.9443	1.37776	0.688878	−	0.724878i	$-0.258104\pi$
$607$	33.9443	1.37776	0.688878	+	0.724878i	$0.258104\pi$
$608$	−14.3951	−0.583799
$609$	0	0
$610$	3.23607	0.131025
$611$	0.0557281	0.00225452
$612$	0	0
$613$	1.88854	0.0762776	0.0381388	−	0.999272i	$-0.487857\pi$
$613$	1.88854	0.0762776	0.0381388	+	0.999272i	$0.487857\pi$
$614$	8.00000	0.322854
$615$	0	0
$616$	1.47214	0.0593140
$617$	5.41641	0.218056	0.109028	−	0.994039i	$-0.465226\pi$
$617$	5.41641	0.218056	0.109028	+	0.994039i	$0.465226\pi$
$618$	0	0
$619$	−36.0000	−1.44696	−0.723481	−	0.690344i	$-0.757459\pi$
$619$	−36.0000	−1.44696	−0.723481	+	0.690344i	$0.757459\pi$
$620$	−0.875388	−0.0351564
$621$	0	0
$622$	4.58359	0.183785
$623$	10.0000	0.400642
$624$	0	0
$625$	11.0000	0.440000
$626$	−10.8328	−0.432966
$627$	0	0
$628$	−28.7902	−1.14886
$629$	35.4164	1.41214
$630$	0	0
$631$	−36.0000	−1.43314	−0.716569	−	0.697517i	$-0.754288\pi$
$631$	−36.0000	−1.43314	−0.716569	+	0.697517i	$0.754288\pi$
$632$	6.58359	0.261881
$633$	0	0
$634$	3.59675	0.142845
$635$	−4.47214	−0.177471
$636$	0	0
$637$	0.236068	0.00935335
$638$	0.673762	0.0266745
$639$	0	0
$640$	−10.0902	−0.398849
$641$	22.9443	0.906244	0.453122	−	0.891448i	$-0.350310\pi$
$641$	22.9443	0.906244	0.453122	+	0.891448i	$0.350310\pi$
$642$	0	0
$643$	−24.0000	−0.946468	−0.473234	−	0.880937i	$-0.656913\pi$
$643$	−24.0000	−0.946468	−0.473234	+	0.880937i	$0.656913\pi$
$644$	15.7082	0.618990
$645$	0	0
$646$	8.58359	0.337717
$647$	−0.236068	−0.00928079	−0.00464039	−	0.999989i	$-0.501477\pi$
$647$	−0.236068	−0.00928079	−0.00464039	+	0.999989i	$0.501477\pi$
$648$	0	0
$649$	−15.1803	−0.595880
$650$	0.360680	0.0141470
$651$	0	0
$652$	−6.10333	−0.239025
$653$	−21.8885	−0.856565	−0.428282	−	0.903645i	$-0.640881\pi$
$653$	−21.8885	−0.856565	−0.428282	+	0.903645i	$0.640881\pi$
$654$	0	0
$655$	−10.9443	−0.427628
$656$	−14.0689	−0.549298
$657$	0	0
$658$	−0.0901699	−0.00351519
$659$	−17.9443	−0.699010	−0.349505	−	0.936935i	$-0.613650\pi$
$659$	−17.9443	−0.699010	−0.349505	+	0.936935i	$0.613650\pi$
$660$	0	0
$661$	23.4164	0.910793	0.455396	−	0.890289i	$-0.349498\pi$
$661$	23.4164	0.910793	0.455396	+	0.890289i	$0.349498\pi$
$662$	−4.94427	−0.192165
$663$	0	0
$664$	16.8065	0.652219
$665$	−3.47214	−0.134644
$666$	0	0
$667$	14.9443	0.578645
$668$	−1.95743	−0.0757351
$669$	0	0
$670$	3.50658	0.135471
$671$	8.47214	0.327063
$672$	0	0
$673$	12.8328	0.494669	0.247334	−	0.968930i	$-0.420445\pi$
$673$	12.8328	0.494669	0.247334	+	0.968930i	$0.420445\pi$
$674$	4.40325	0.169607
$675$	0	0
$676$	24.0000	0.923077
$677$	−7.41641	−0.285036	−0.142518	−	0.989792i	$-0.545520\pi$
$677$	−7.41641	−0.285036	−0.142518	+	0.989792i	$0.545520\pi$
$678$	0	0
$679$	−8.00000	−0.307012
$680$	9.52786	0.365377
$681$	0	0
$682$	0.180340	0.00690557
$683$	3.05573	0.116924	0.0584621	−	0.998290i	$-0.481380\pi$
$683$	3.05573	0.116924	0.0584621	+	0.998290i	$0.481380\pi$
$684$	0	0
$685$	−1.52786	−0.0583767
$686$	−0.381966	−0.0145835
$687$	0	0
$688$	−12.5836	−0.479745
$689$	−1.30495	−0.0497147
$690$	0	0
$691$	30.0000	1.14125	0.570627	−	0.821209i	$-0.306700\pi$
$691$	30.0000	1.14125	0.570627	+	0.821209i	$0.306700\pi$
$692$	−44.2918	−1.68372
$693$	0	0
$694$	3.41641	0.129685
$695$	8.94427	0.339276
$696$	0	0
$697$	28.9443	1.09634
$698$	6.74265	0.255213
$699$	0	0
$700$	7.41641	0.280314
$701$	3.52786	0.133246	0.0666228	−	0.997778i	$-0.478778\pi$
$701$	3.52786	0.133246	0.0666228	+	0.997778i	$0.478778\pi$
$702$	0	0
$703$	−19.0000	−0.716599
$704$	−4.70820	−0.177447
$705$	0	0
$706$	−10.1327	−0.381351
$707$	8.47214	0.318627
$708$	0	0
$709$	−10.5279	−0.395382	−0.197691	−	0.980264i	$-0.563344\pi$
$709$	−10.5279	−0.395382	−0.197691	+	0.980264i	$0.563344\pi$
$710$	1.70820	0.0641078
$711$	0	0
$712$	14.7214	0.551706
$713$	4.00000	0.149801
$714$	0	0
$715$	−0.236068	−0.00882844
$716$	44.2918	1.65526
$717$	0	0
$718$	9.88854	0.369037
$719$	−5.18034	−0.193194	−0.0965970	−	0.995324i	$-0.530796\pi$
$719$	−5.18034	−0.193194	−0.0965970	+	0.995324i	$0.530796\pi$
$720$	0	0
$721$	−15.4164	−0.574137
$722$	2.65248	0.0987149
$723$	0	0
$724$	−24.8754	−0.924487
$725$	7.05573	0.262043
$726$	0	0
$727$	36.4721	1.35268	0.676338	−	0.736591i	$-0.263566\pi$
$727$	36.4721	1.35268	0.676338	+	0.736591i	$0.263566\pi$
$728$	0.347524	0.0128801
$729$	0	0
$730$	−3.72949	−0.138035
$731$	25.8885	0.957522
$732$	0	0
$733$	−26.3607	−0.973654	−0.486827	−	0.873498i	$-0.661846\pi$
$733$	−26.3607	−0.973654	−0.486827	+	0.873498i	$0.661846\pi$
$734$	−2.29180	−0.0845917
$735$	0	0
$736$	35.1246	1.29471
$737$	9.18034	0.338162
$738$	0	0
$739$	−31.8885	−1.17304	−0.586520	−	0.809935i	$-0.699502\pi$
$739$	−31.8885	−1.17304	−0.586520	+	0.809935i	$0.699502\pi$
$740$	−10.1459	−0.372971
$741$	0	0
$742$	2.11146	0.0775140
$743$	23.3607	0.857020	0.428510	−	0.903537i	$-0.359039\pi$
$743$	23.3607	0.857020	0.428510	+	0.903537i	$0.359039\pi$
$744$	0	0
$745$	14.7082	0.538867
$746$	12.5410	0.459159
$747$	0	0
$748$	12.0000	0.438763
$749$	−17.0000	−0.621166
$750$	0	0
$751$	−4.34752	−0.158643	−0.0793217	−	0.996849i	$-0.525275\pi$
$751$	−4.34752	−0.158643	−0.0793217	+	0.996849i	$0.525275\pi$
$752$	0.742646	0.0270815
$753$	0	0
$754$	0.159054	0.00579239
$755$	10.4721	0.381120
$756$	0	0
$757$	11.4721	0.416962	0.208481	−	0.978026i	$-0.433148\pi$
$757$	11.4721	0.416962	0.208481	+	0.978026i	$0.433148\pi$
$758$	−7.10333	−0.258004
$759$	0	0
$760$	−5.11146	−0.185412
$761$	−37.7771	−1.36942	−0.684709	−	0.728816i	$-0.740071\pi$
$761$	−37.7771	−1.36942	−0.684709	+	0.728816i	$0.740071\pi$
$762$	0	0
$763$	−16.4721	−0.596331
$764$	−22.0426	−0.797472
$765$	0	0
$766$	4.22291	0.152580
$767$	−3.58359	−0.129396
$768$	0	0
$769$	−28.1246	−1.01420	−0.507100	−	0.861887i	$-0.669282\pi$
$769$	−28.1246	−1.01420	−0.507100	+	0.861887i	$0.669282\pi$
$770$	0.381966	0.0137651
$771$	0	0
$772$	−19.4164	−0.698812
$773$	27.9443	1.00509	0.502543	−	0.864552i	$-0.332398\pi$
$773$	27.9443	1.00509	0.502543	+	0.864552i	$0.332398\pi$
$774$	0	0
$775$	1.88854	0.0678385
$776$	−11.7771	−0.422773
$777$	0	0
$778$	1.48529	0.0532503
$779$	−15.5279	−0.556343
$780$	0	0
$781$	4.47214	0.160026
$782$	−20.9443	−0.748966
$783$	0	0
$784$	3.14590	0.112354
$785$	−15.5279	−0.554213
$786$	0	0
$787$	10.4164	0.371305	0.185652	−	0.982615i	$-0.440560\pi$
$787$	10.4164	0.371305	0.185652	+	0.982615i	$0.440560\pi$
$788$	10.0426	0.357752
$789$	0	0
$790$	1.70820	0.0607752
$791$	11.4164	0.405921
$792$	0	0
$793$	2.00000	0.0710221
$794$	9.66563	0.343020
$795$	0	0
$796$	−38.8328	−1.37639
$797$	−20.8885	−0.739910	−0.369955	−	0.929050i	$-0.620627\pi$
$797$	−20.8885	−0.739910	−0.369955	+	0.929050i	$0.620627\pi$
$798$	0	0
$799$	−1.52786	−0.0540519
$800$	16.5836	0.586319
$801$	0	0
$802$	−3.77709	−0.133374
$803$	−9.76393	−0.344562
$804$	0	0
$805$	8.47214	0.298604
$806$	0.0425725	0.00149955
$807$	0	0
$808$	12.4721	0.438768
$809$	45.6525	1.60506	0.802528	−	0.596615i	$-0.203488\pi$
$809$	45.6525	1.60506	0.802528	+	0.596615i	$0.203488\pi$
$810$	0	0
$811$	29.2492	1.02708	0.513540	−	0.858066i	$-0.328334\pi$
$811$	29.2492	1.02708	0.513540	+	0.858066i	$0.328334\pi$
$812$	3.27051	0.114772
$813$	0	0
$814$	2.09017	0.0732604
$815$	−3.29180	−0.115307
$816$	0	0
$817$	−13.8885	−0.485899
$818$	−0.986844	−0.0345042
$819$	0	0
$820$	−8.29180	−0.289562
$821$	18.7082	0.652921	0.326460	−	0.945211i	$-0.394144\pi$
$821$	18.7082	0.652921	0.326460	+	0.945211i	$0.394144\pi$
$822$	0	0
$823$	−10.2361	−0.356807	−0.178403	−	0.983957i	$-0.557093\pi$
$823$	−10.2361	−0.356807	−0.178403	+	0.983957i	$0.557093\pi$
$824$	−22.6950	−0.790619
$825$	0	0
$826$	5.79837	0.201751
$827$	−10.8885	−0.378632	−0.189316	−	0.981916i	$-0.560627\pi$
$827$	−10.8885	−0.378632	−0.189316	+	0.981916i	$0.560627\pi$
$828$	0	0
$829$	28.0000	0.972480	0.486240	−	0.873825i	$-0.338368\pi$
$829$	28.0000	0.972480	0.486240	+	0.873825i	$0.338368\pi$
$830$	4.36068	0.151361
$831$	0	0
$832$	−1.11146	−0.0385328
$833$	−6.47214	−0.224246
$834$	0	0
$835$	−1.05573	−0.0365350
$836$	−6.43769	−0.222652
$837$	0	0
$838$	1.39512	0.0481937
$839$	41.1803	1.42170	0.710852	−	0.703342i	$-0.248310\pi$
$839$	41.1803	1.42170	0.710852	+	0.703342i	$0.248310\pi$
$840$	0	0
$841$	−25.8885	−0.892708
$842$	−1.72949	−0.0596022
$843$	0	0
$844$	−24.0000	−0.826114
$845$	12.9443	0.445296
$846$	0	0
$847$	1.00000	0.0343604
$848$	−17.3901	−0.597178
$849$	0	0
$850$	−9.88854	−0.339174
$851$	46.3607	1.58922
$852$	0	0
$853$	31.3050	1.07186	0.535931	−	0.844262i	$-0.319961\pi$
$853$	31.3050	1.07186	0.535931	+	0.844262i	$0.319961\pi$
$854$	−3.23607	−0.110736
$855$	0	0
$856$	−25.0263	−0.855382
$857$	35.0557	1.19748	0.598740	−	0.800943i	$-0.295668\pi$
$857$	35.0557	1.19748	0.598740	+	0.800943i	$0.295668\pi$
$858$	0	0
$859$	−26.0000	−0.887109	−0.443554	−	0.896248i	$-0.646283\pi$
$859$	−26.0000	−0.887109	−0.443554	+	0.896248i	$0.646283\pi$
$860$	−7.41641	−0.252897
$861$	0	0
$862$	−13.1459	−0.447751
$863$	−14.9443	−0.508709	−0.254354	−	0.967111i	$-0.581863\pi$
$863$	−14.9443	−0.508709	−0.254354	+	0.967111i	$0.581863\pi$
$864$	0	0
$865$	−23.8885	−0.812235
$866$	−11.5967	−0.394074
$867$	0	0
$868$	0.875388	0.0297126
$869$	4.47214	0.151707
$870$	0	0
$871$	2.16718	0.0734322
$872$	−24.2492	−0.821182
$873$	0	0
$874$	11.2361	0.380066
$875$	9.00000	0.304256
$876$	0	0
$877$	−8.00000	−0.270141	−0.135070	−	0.990836i	$-0.543126\pi$
$877$	−8.00000	−0.270141	−0.135070	+	0.990836i	$0.543126\pi$
$878$	−1.10333	−0.0372354
$879$	0	0
$880$	−3.14590	−0.106048
$881$	18.4164	0.620465	0.310232	−	0.950661i	$-0.399593\pi$
$881$	18.4164	0.620465	0.310232	+	0.950661i	$0.399593\pi$
$882$	0	0
$883$	15.7639	0.530499	0.265249	−	0.964180i	$-0.414546\pi$
$883$	15.7639	0.530499	0.265249	+	0.964180i	$0.414546\pi$
$884$	2.83282	0.0952779
$885$	0	0
$886$	11.6393	0.391031
$887$	−27.0557	−0.908442	−0.454221	−	0.890889i	$-0.650082\pi$
$887$	−27.0557	−0.908442	−0.454221	+	0.890889i	$0.650082\pi$
$888$	0	0
$889$	4.47214	0.149991
$890$	3.81966	0.128035
$891$	0	0
$892$	−1.75078	−0.0586203
$893$	0.819660	0.0274289
$894$	0	0
$895$	23.8885	0.798506
$896$	10.0902	0.337089
$897$	0	0
$898$	5.16718	0.172431
$899$	0.832816	0.0277760
$900$	0	0
$901$	35.7771	1.19191
$902$	1.70820	0.0568770
$903$	0	0
$904$	16.8065	0.558976
$905$	−13.4164	−0.445976
$906$	0	0
$907$	−20.9443	−0.695443	−0.347722	−	0.937598i	$-0.613045\pi$
$907$	−20.9443	−0.695443	−0.347722	+	0.937598i	$0.613045\pi$
$908$	16.7902	0.557204
$909$	0	0
$910$	0.0901699	0.00298910
$911$	−42.4721	−1.40716	−0.703582	−	0.710614i	$-0.748417\pi$
$911$	−42.4721	−1.40716	−0.703582	+	0.710614i	$0.748417\pi$
$912$	0	0
$913$	11.4164	0.377828
$914$	8.18034	0.270582
$915$	0	0
$916$	−27.7082	−0.915505
$917$	10.9443	0.361412
$918$	0	0
$919$	−43.1935	−1.42482	−0.712411	−	0.701762i	$-0.752397\pi$
$919$	−43.1935	−1.42482	−0.712411	+	0.701762i	$0.752397\pi$
$920$	12.4721	0.411194
$921$	0	0
$922$	−14.6950	−0.483956
$923$	1.05573	0.0347497
$924$	0	0
$925$	21.8885	0.719691
$926$	12.6312	0.415087
$927$	0	0
$928$	7.31308	0.240064
$929$	−46.4164	−1.52287	−0.761436	−	0.648240i	$-0.775506\pi$
$929$	−46.4164	−1.52287	−0.761436	+	0.648240i	$0.775506\pi$
$930$	0	0
$931$	3.47214	0.113795
$932$	17.4590	0.571888
$933$	0	0
$934$	6.96556	0.227920
$935$	6.47214	0.211661
$936$	0	0
$937$	−50.3607	−1.64521	−0.822606	−	0.568612i	$-0.807481\pi$
$937$	−50.3607	−1.64521	−0.822606	+	0.568612i	$0.807481\pi$
$938$	−3.50658	−0.114494
$939$	0	0
$940$	0.437694	0.0142760
$941$	−1.05573	−0.0344158	−0.0172079	−	0.999852i	$-0.505478\pi$
$941$	−1.05573	−0.0344158	−0.0172079	+	0.999852i	$0.505478\pi$
$942$	0	0
$943$	37.8885	1.23382
$944$	−47.7558	−1.55432
$945$	0	0
$946$	1.52786	0.0496751
$947$	4.83282	0.157045	0.0785227	−	0.996912i	$-0.474980\pi$
$947$	4.83282	0.157045	0.0785227	+	0.996912i	$0.474980\pi$
$948$	0	0
$949$	−2.30495	−0.0748219
$950$	5.30495	0.172115
$951$	0	0
$952$	−9.52786	−0.308800
$953$	−55.0689	−1.78386	−0.891928	−	0.452177i	$-0.850647\pi$
$953$	−55.0689	−1.78386	−0.891928	+	0.452177i	$0.850647\pi$
$954$	0	0
$955$	−11.8885	−0.384704
$956$	0.978714	0.0316539
$957$	0	0
$958$	−6.29180	−0.203279
$959$	1.52786	0.0493373
$960$	0	0
$961$	−30.7771	−0.992809
$962$	0.493422	0.0159086
$963$	0	0
$964$	29.8967	0.962907
$965$	−10.4721	−0.337110
$966$	0	0
$967$	32.3607	1.04065	0.520325	−	0.853969i	$-0.325811\pi$
$967$	32.3607	1.04065	0.520325	+	0.853969i	$0.325811\pi$
$968$	1.47214	0.0473162
$969$	0	0
$970$	−3.05573	−0.0981135
$971$	−30.7082	−0.985473	−0.492737	−	0.870179i	$-0.664003\pi$
$971$	−30.7082	−0.985473	−0.492737	+	0.870179i	$0.664003\pi$
$972$	0	0
$973$	−8.94427	−0.286740
$974$	−12.9443	−0.414761
$975$	0	0
$976$	26.6525	0.853125
$977$	−49.7771	−1.59251	−0.796255	−	0.604961i	$-0.793189\pi$
$977$	−49.7771	−1.59251	−0.796255	+	0.604961i	$0.793189\pi$
$978$	0	0
$979$	10.0000	0.319601
$980$	1.85410	0.0592271
$981$	0	0
$982$	−1.14590	−0.0365671
$983$	−32.9443	−1.05076	−0.525380	−	0.850868i	$-0.676077\pi$
$983$	−32.9443	−1.05076	−0.525380	+	0.850868i	$0.676077\pi$
$984$	0	0
$985$	5.41641	0.172581
$986$	−4.36068	−0.138872
$987$	0	0
$988$	−1.51973	−0.0483492
$989$	33.8885	1.07759
$990$	0	0
$991$	29.5410	0.938401	0.469201	−	0.883092i	$-0.344542\pi$
$991$	29.5410	0.938401	0.469201	+	0.883092i	$0.344542\pi$
$992$	1.95743	0.0621484
$993$	0	0
$994$	−1.70820	−0.0541809
$995$	−20.9443	−0.663978
$996$	0	0
$997$	−35.5279	−1.12518	−0.562589	−	0.826736i	$-0.690195\pi$
$997$	−35.5279	−1.12518	−0.562589	+	0.826736i	$0.690195\pi$
$998$	6.56231	0.207726
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.2.a.e.1.2	✓	2
3.2	odd	2		693.2.a.k.1.1	yes	2
7.6	odd	2		4851.2.a.u.1.2		2
11.10	odd	2		7623.2.a.bw.1.1		2
21.20	even	2		4851.2.a.bf.1.1		2
33.32	even	2		7623.2.a.v.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
693.2.a.e.1.2	✓	2	1.1	even	1	trivial
693.2.a.k.1.1	yes	2	3.2	odd	2
4851.2.a.u.1.2		2	7.6	odd	2
4851.2.a.bf.1.1		2	21.20	even	2
7623.2.a.v.1.2		2	33.32	even	2
7623.2.a.bw.1.1		2	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)

\(f(q)\)	\(=\)	\(q-0.381966 q^{2} -1.85410 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.47214 q^{8} +O(q^{10})\)	\(q-0.381966 q^{2} -1.85410 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.47214 q^{8} +0.381966 q^{10} +1.00000 q^{11} +0.236068 q^{13} -0.381966 q^{14} +3.14590 q^{16} -6.47214 q^{17} +3.47214 q^{19} +1.85410 q^{20} -0.381966 q^{22} -8.47214 q^{23} -4.00000 q^{25} -0.0901699 q^{26} -1.85410 q^{28} -1.76393 q^{29} -0.472136 q^{31} -4.14590 q^{32} +2.47214 q^{34} -1.00000 q^{35} -5.47214 q^{37} -1.32624 q^{38} -1.47214 q^{40} -4.47214 q^{41} -4.00000 q^{43} -1.85410 q^{44} +3.23607 q^{46} +0.236068 q^{47} +1.00000 q^{49} +1.52786 q^{50} -0.437694 q^{52} -5.52786 q^{53} -1.00000 q^{55} +1.47214 q^{56} +0.673762 q^{58} -15.1803 q^{59} +8.47214 q^{61} +0.180340 q^{62} -4.70820 q^{64} -0.236068 q^{65} +9.18034 q^{67} +12.0000 q^{68} +0.381966 q^{70} +4.47214 q^{71} -9.76393 q^{73} +2.09017 q^{74} -6.43769 q^{76} +1.00000 q^{77} +4.47214 q^{79} -3.14590 q^{80} +1.70820 q^{82} +11.4164 q^{83} +6.47214 q^{85} +1.52786 q^{86} +1.47214 q^{88} +10.0000 q^{89} +0.236068 q^{91} +15.7082 q^{92} -0.0901699 q^{94} -3.47214 q^{95} -8.00000 q^{97} -0.381966 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{2} + 3 q^{4} - 2 q^{5} + 2 q^{7} - 6 q^{8}+O(q^{10}) \) 2 * q - 3 * q^2 + 3 * q^4 - 2 * q^5 + 2 * q^7 - 6 * q^8	\( 2 q - 3 q^{2} + 3 q^{4} - 2 q^{5} + 2 q^{7} - 6 q^{8} + 3 q^{10} + 2 q^{11} - 4 q^{13} - 3 q^{14} + 13 q^{16} - 4 q^{17} - 2 q^{19} - 3 q^{20} - 3 q^{22} - 8 q^{23} - 8 q^{25} + 11 q^{26} + 3 q^{28} - 8 q^{29} + 8 q^{31} - 15 q^{32} - 4 q^{34} - 2 q^{35} - 2 q^{37} + 13 q^{38} + 6 q^{40} - 8 q^{43} + 3 q^{44} + 2 q^{46} - 4 q^{47} + 2 q^{49} + 12 q^{50} - 21 q^{52} - 20 q^{53} - 2 q^{55} - 6 q^{56} + 17 q^{58} - 8 q^{59} + 8 q^{61} - 22 q^{62} + 4 q^{64} + 4 q^{65} - 4 q^{67} + 24 q^{68} + 3 q^{70} - 24 q^{73} - 7 q^{74} - 33 q^{76} + 2 q^{77} - 13 q^{80} - 10 q^{82} - 4 q^{83} + 4 q^{85} + 12 q^{86} - 6 q^{88} + 20 q^{89} - 4 q^{91} + 18 q^{92} + 11 q^{94} + 2 q^{95} - 16 q^{97} - 3 q^{98}+O(q^{100}) \) 2 * q - 3 * q^2 + 3 * q^4 - 2 * q^5 + 2 * q^7 - 6 * q^8 + 3 * q^10 + 2 * q^11 - 4 * q^13 - 3 * q^14 + 13 * q^16 - 4 * q^17 - 2 * q^19 - 3 * q^20 - 3 * q^22 - 8 * q^23 - 8 * q^25 + 11 * q^26 + 3 * q^28 - 8 * q^29 + 8 * q^31 - 15 * q^32 - 4 * q^34 - 2 * q^35 - 2 * q^37 + 13 * q^38 + 6 * q^40 - 8 * q^43 + 3 * q^44 + 2 * q^46 - 4 * q^47 + 2 * q^49 + 12 * q^50 - 21 * q^52 - 20 * q^53 - 2 * q^55 - 6 * q^56 + 17 * q^58 - 8 * q^59 + 8 * q^61 - 22 * q^62 + 4 * q^64 + 4 * q^65 - 4 * q^67 + 24 * q^68 + 3 * q^70 - 24 * q^73 - 7 * q^74 - 33 * q^76 + 2 * q^77 - 13 * q^80 - 10 * q^82 - 4 * q^83 + 4 * q^85 + 12 * q^86 - 6 * q^88 + 20 * q^89 - 4 * q^91 + 18 * q^92 + 11 * q^94 + 2 * q^95 - 16 * q^97 - 3 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more