LMFDB - Embedded newform 693.2.a.e.1.1

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			1.1
Root			$1.61803$ 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)

$f(q)$	$=$	$q-2.61803 q^{2} +4.85410 q^{4} -1.00000 q^{5} +1.00000 q^{7} -7.47214 q^{8} +O(q^{10})$	\(q-2.61803 q^{2} +4.85410 q^{4} -1.00000 q^{5} +1.00000 q^{7} -7.47214 q^{8} +2.61803 q^{10} +1.00000 q^{11} -4.23607 q^{13} -2.61803 q^{14} +9.85410 q^{16} +2.47214 q^{17} -5.47214 q^{19} -4.85410 q^{20} -2.61803 q^{22} +0.472136 q^{23} -4.00000 q^{25} +11.0902 q^{26} +4.85410 q^{28} -6.23607 q^{29} +8.47214 q^{31} -10.8541 q^{32} -6.47214 q^{34} -1.00000 q^{35} +3.47214 q^{37} +14.3262 q^{38} +7.47214 q^{40} +4.47214 q^{41} -4.00000 q^{43} +4.85410 q^{44} -1.23607 q^{46} -4.23607 q^{47} +1.00000 q^{49} +10.4721 q^{50} -20.5623 q^{52} -14.4721 q^{53} -1.00000 q^{55} -7.47214 q^{56} +16.3262 q^{58} +7.18034 q^{59} -0.472136 q^{61} -22.1803 q^{62} +8.70820 q^{64} +4.23607 q^{65} -13.1803 q^{67} +12.0000 q^{68} +2.61803 q^{70} -4.47214 q^{71} -14.2361 q^{73} -9.09017 q^{74} -26.5623 q^{76} +1.00000 q^{77} -4.47214 q^{79} -9.85410 q^{80} -11.7082 q^{82} -15.4164 q^{83} -2.47214 q^{85} +10.4721 q^{86} -7.47214 q^{88} +10.0000 q^{89} -4.23607 q^{91} +2.29180 q^{92} +11.0902 q^{94} +5.47214 q^{95} -8.00000 q^{97} -2.61803 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.61803	−1.85123	−0.925615	−	0.378467i	$-0.876451\pi$
$2$	−2.61803	−1.85123	−0.925615	+	0.378467i	$0.876451\pi$
$3$	0	0
$3$	0	0
$4$	4.85410	2.42705
$5$	−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
$8$	−7.47214	−2.64180
$9$	0	0
$10$	2.61803	0.827895
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−4.23607	−1.17487	−0.587437	−	0.809270i	$-0.699863\pi$
$13$	−4.23607	−1.17487	−0.587437	+	0.809270i	$0.699863\pi$
$14$	−2.61803	−0.699699
$15$	0	0
$16$	9.85410	2.46353
$17$	2.47214	0.599581	0.299791	−	0.954005i	$-0.403083\pi$
$17$	2.47214	0.599581	0.299791	+	0.954005i	$0.403083\pi$
$18$	0	0
$19$	−5.47214	−1.25539	−0.627697	−	0.778458i	$-0.716002\pi$
$19$	−5.47214	−1.25539	−0.627697	+	0.778458i	$0.716002\pi$
$20$	−4.85410	−1.08541
$21$	0	0
$22$	−2.61803	−0.558167
$23$	0.472136	0.0984472	0.0492236	−	0.998788i	$-0.484325\pi$
$23$	0.472136	0.0984472	0.0492236	+	0.998788i	$0.484325\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	11.0902	2.17496
$27$	0	0
$28$	4.85410	0.917339
$29$	−6.23607	−1.15801	−0.579004	−	0.815324i	$-0.696559\pi$
$29$	−6.23607	−1.15801	−0.579004	+	0.815324i	$0.696559\pi$
$30$	0	0
$31$	8.47214	1.52164	0.760820	−	0.648963i	$-0.224797\pi$
$31$	8.47214	1.52164	0.760820	+	0.648963i	$0.224797\pi$
$32$	−10.8541	−1.91875
$33$	0	0
$34$	−6.47214	−1.10996
$35$	−1.00000	−0.169031
$36$	0	0
$37$	3.47214	0.570816	0.285408	−	0.958406i	$-0.407871\pi$
$37$	3.47214	0.570816	0.285408	+	0.958406i	$0.407871\pi$
$38$	14.3262	2.32402
$39$	0	0
$40$	7.47214	1.18145
$41$	4.47214	0.698430	0.349215	−	0.937043i	$-0.386448\pi$
$41$	4.47214	0.698430	0.349215	+	0.937043i	$0.386448\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	4.85410	0.731783
$45$	0	0
$46$	−1.23607	−0.182248
$47$	−4.23607	−0.617894	−0.308947	−	0.951079i	$-0.599977\pi$
$47$	−4.23607	−0.617894	−0.308947	+	0.951079i	$0.599977\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	10.4721	1.48098
$51$	0	0
$52$	−20.5623	−2.85148
$53$	−14.4721	−1.98790	−0.993950	−	0.109830i	$-0.964969\pi$
$53$	−14.4721	−1.98790	−0.993950	+	0.109830i	$0.964969\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	−7.47214	−0.998506
$57$	0	0
$58$	16.3262	2.14374
$59$	7.18034	0.934801	0.467400	−	0.884046i	$-0.345191\pi$
$59$	7.18034	0.934801	0.467400	+	0.884046i	$0.345191\pi$
$60$	0	0
$61$	−0.472136	−0.0604508	−0.0302254	−	0.999543i	$-0.509623\pi$
$61$	−0.472136	−0.0604508	−0.0302254	+	0.999543i	$0.509623\pi$
$62$	−22.1803	−2.81691
$63$	0	0
$64$	8.70820	1.08853
$65$	4.23607	0.525420
$66$	0	0
$67$	−13.1803	−1.61023	−0.805117	−	0.593115i	$-0.797898\pi$
$67$	−13.1803	−1.61023	−0.805117	+	0.593115i	$0.797898\pi$
$68$	12.0000	1.45521
$69$	0	0
$70$	2.61803	0.312915
$71$	−4.47214	−0.530745	−0.265372	−	0.964146i	$-0.585495\pi$
$71$	−4.47214	−0.530745	−0.265372	+	0.964146i	$0.585495\pi$
$72$	0	0
$73$	−14.2361	−1.66621	−0.833103	−	0.553118i	$-0.813438\pi$
$73$	−14.2361	−1.66621	−0.833103	+	0.553118i	$0.813438\pi$
$74$	−9.09017	−1.05671
$75$	0	0
$76$	−26.5623	−3.04691
$77$	1.00000	0.113961
$78$	0	0
$79$	−4.47214	−0.503155	−0.251577	−	0.967837i	$-0.580949\pi$
$79$	−4.47214	−0.503155	−0.251577	+	0.967837i	$0.580949\pi$
$80$	−9.85410	−1.10172
$81$	0	0
$82$	−11.7082	−1.29295
$83$	−15.4164	−1.69217	−0.846085	−	0.533048i	$-0.821047\pi$
$83$	−15.4164	−1.69217	−0.846085	+	0.533048i	$0.821047\pi$
$84$	0	0
$85$	−2.47214	−0.268141
$86$	10.4721	1.12924
$87$	0	0
$88$	−7.47214	−0.796532
$89$	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
$91$	−4.23607	−0.444061
$92$	2.29180	0.238936
$93$	0	0
$94$	11.0902	1.14386
$95$	5.47214	0.561429
$96$	0	0
$97$	−8.00000	−0.812277	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000	−0.812277	−0.406138	+	0.913812i	$0.633125\pi$
$98$	−2.61803	−0.264461
$99$	0	0
$100$	−19.4164	−1.94164
$101$	−0.472136	−0.0469793	−0.0234896	−	0.999724i	$-0.507478\pi$
$101$	−0.472136	−0.0469793	−0.0234896	+	0.999724i	$0.507478\pi$
$102$	0	0
$103$	11.4164	1.12489	0.562446	−	0.826834i	$-0.309860\pi$
$103$	11.4164	1.12489	0.562446	+	0.826834i	$0.309860\pi$
$104$	31.6525	3.10378
$105$	0	0
$106$	37.8885	3.68006
$107$	−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
$109$	−7.52786	−0.721039	−0.360519	−	0.932752i	$-0.617401\pi$
$109$	−7.52786	−0.721039	−0.360519	+	0.932752i	$0.617401\pi$
$110$	2.61803	0.249620
$111$	0	0
$112$	9.85410	0.931125
$113$	−15.4164	−1.45025	−0.725127	−	0.688615i	$-0.758219\pi$
$113$	−15.4164	−1.45025	−0.725127	+	0.688615i	$0.758219\pi$
$114$	0	0
$115$	−0.472136	−0.0440269
$116$	−30.2705	−2.81055
$117$	0	0
$118$	−18.7984	−1.73053
$119$	2.47214	0.226620
$120$	0	0
$121$	1.00000	0.0909091
$122$	1.23607	0.111908
$123$	0	0
$124$	41.1246	3.69310
$125$	9.00000	0.804984
$126$	0	0
$127$	−4.47214	−0.396838	−0.198419	−	0.980117i	$-0.563581\pi$
$127$	−4.47214	−0.396838	−0.198419	+	0.980117i	$0.563581\pi$
$128$	−1.09017	−0.0963583
$129$	0	0
$130$	−11.0902	−0.972672
$131$	−6.94427	−0.606724	−0.303362	−	0.952875i	$-0.598109\pi$
$131$	−6.94427	−0.606724	−0.303362	+	0.952875i	$0.598109\pi$
$132$	0	0
$133$	−5.47214	−0.474494
$134$	34.5066	2.98091
$135$	0	0
$136$	−18.4721	−1.58397
$137$	10.4721	0.894695	0.447347	−	0.894360i	$-0.352369\pi$
$137$	10.4721	0.894695	0.447347	+	0.894360i	$0.352369\pi$
$138$	0	0
$139$	8.94427	0.758643	0.379322	−	0.925265i	$-0.376157\pi$
$139$	8.94427	0.758643	0.379322	+	0.925265i	$0.376157\pi$
$140$	−4.85410	−0.410246
$141$	0	0
$142$	11.7082	0.982531
$143$	−4.23607	−0.354238
$144$	0	0
$145$	6.23607	0.517877
$146$	37.2705	3.08453
$147$	0	0
$148$	16.8541	1.38540
$149$	−1.29180	−0.105828	−0.0529140	−	0.998599i	$-0.516851\pi$
$149$	−1.29180	−0.105828	−0.0529140	+	0.998599i	$0.516851\pi$
$150$	0	0
$151$	−1.52786	−0.124336	−0.0621679	−	0.998066i	$-0.519801\pi$
$151$	−1.52786	−0.124336	−0.0621679	+	0.998066i	$0.519801\pi$
$152$	40.8885	3.31650
$153$	0	0
$154$	−2.61803	−0.210967
$155$	−8.47214	−0.680498
$156$	0	0
$157$	24.4721	1.95309	0.976545	−	0.215316i	$-0.0690780\pi$
$157$	24.4721	1.95309	0.976545	+	0.215316i	$0.0690780\pi$
$158$	11.7082	0.931455
$159$	0	0
$160$	10.8541	0.858092
$161$	0.472136	0.0372095
$162$	0	0
$163$	16.7082	1.30869	0.654344	−	0.756197i	$-0.272945\pi$
$163$	16.7082	1.30869	0.654344	+	0.756197i	$0.272945\pi$
$164$	21.7082	1.69513
$165$	0	0
$166$	40.3607	3.13260
$167$	18.9443	1.46595	0.732976	−	0.680255i	$-0.238131\pi$
$167$	18.9443	1.46595	0.732976	+	0.680255i	$0.238131\pi$
$168$	0	0
$169$	4.94427	0.380329
$170$	6.47214	0.496390
$171$	0	0
$172$	−19.4164	−1.48049
$173$	−11.8885	−0.903869	−0.451935	−	0.892051i	$-0.649266\pi$
$173$	−11.8885	−0.903869	−0.451935	+	0.892051i	$0.649266\pi$
$174$	0	0
$175$	−4.00000	−0.302372
$176$	9.85410	0.742781
$177$	0	0
$178$	−26.1803	−1.96230
$179$	11.8885	0.888591	0.444296	−	0.895880i	$-0.353454\pi$
$179$	11.8885	0.888591	0.444296	+	0.895880i	$0.353454\pi$
$180$	0	0
$181$	−13.4164	−0.997234	−0.498617	−	0.866822i	$-0.666159\pi$
$181$	−13.4164	−0.997234	−0.498617	+	0.866822i	$0.666159\pi$
$182$	11.0902	0.822058
$183$	0	0
$184$	−3.52786	−0.260078
$185$	−3.47214	−0.255277
$186$	0	0
$187$	2.47214	0.180780
$188$	−20.5623	−1.49966
$189$	0	0
$190$	−14.3262	−1.03933
$191$	−23.8885	−1.72851	−0.864257	−	0.503050i	$-0.832211\pi$
$191$	−23.8885	−1.72851	−0.864257	+	0.503050i	$0.832211\pi$
$192$	0	0
$193$	1.52786	0.109978	0.0549890	−	0.998487i	$-0.482488\pi$
$193$	1.52786	0.109978	0.0549890	+	0.998487i	$0.482488\pi$
$194$	20.9443	1.50371
$195$	0	0
$196$	4.85410	0.346722
$197$	21.4164	1.52586	0.762928	−	0.646484i	$-0.223761\pi$
$197$	21.4164	1.52586	0.762928	+	0.646484i	$0.223761\pi$
$198$	0	0
$199$	3.05573	0.216615	0.108307	−	0.994117i	$-0.465457\pi$
$199$	3.05573	0.216615	0.108307	+	0.994117i	$0.465457\pi$
$200$	29.8885	2.11344
$201$	0	0
$202$	1.23607	0.0869694
$203$	−6.23607	−0.437686
$204$	0	0
$205$	−4.47214	−0.312348
$206$	−29.8885	−2.08243
$207$	0	0
$208$	−41.7426	−2.89433
$209$	−5.47214	−0.378516
$210$	0	0
$211$	−4.94427	−0.340378	−0.170189	−	0.985411i	$-0.554438\pi$
$211$	−4.94427	−0.340378	−0.170189	+	0.985411i	$0.554438\pi$
$212$	−70.2492	−4.82474
$213$	0	0
$214$	44.5066	3.04241
$215$	4.00000	0.272798
$216$	0	0
$217$	8.47214	0.575126
$218$	19.7082	1.33481
$219$	0	0
$220$	−4.85410	−0.327263
$221$	−10.4721	−0.704432
$222$	0	0
$223$	−16.9443	−1.13467	−0.567336	−	0.823486i	$-0.692026\pi$
$223$	−16.9443	−1.13467	−0.567336	+	0.823486i	$0.692026\pi$
$224$	−10.8541	−0.725220
$225$	0	0
$226$	40.3607	2.68475
$227$	−26.9443	−1.78835	−0.894177	−	0.447713i	$-0.852238\pi$
$227$	−26.9443	−1.78835	−0.894177	+	0.447713i	$0.852238\pi$
$228$	0	0
$229$	−2.94427	−0.194563	−0.0972815	−	0.995257i	$-0.531015\pi$
$229$	−2.94427	−0.194563	−0.0972815	+	0.995257i	$0.531015\pi$
$230$	1.23607	0.0815039
$231$	0	0
$232$	46.5967	3.05923
$233$	17.4164	1.14099	0.570493	−	0.821302i	$-0.306752\pi$
$233$	17.4164	1.14099	0.570493	+	0.821302i	$0.306752\pi$
$234$	0	0
$235$	4.23607	0.276331
$236$	34.8541	2.26881
$237$	0	0
$238$	−6.47214	−0.419526
$239$	−9.47214	−0.612702	−0.306351	−	0.951919i	$-0.599108\pi$
$239$	−9.47214	−0.612702	−0.306351	+	0.951919i	$0.599108\pi$
$240$	0	0
$241$	24.1246	1.55400	0.777001	−	0.629499i	$-0.216740\pi$
$241$	24.1246	1.55400	0.777001	+	0.629499i	$0.216740\pi$
$242$	−2.61803	−0.168294
$243$	0	0
$244$	−2.29180	−0.146717
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	23.1803	1.47493
$248$	−63.3050	−4.01987
$249$	0	0
$250$	−23.5623	−1.49021
$251$	−28.1246	−1.77521	−0.887605	−	0.460606i	$-0.847632\pi$
$251$	−28.1246	−1.77521	−0.887605	+	0.460606i	$0.847632\pi$
$252$	0	0
$253$	0.472136	0.0296829
$254$	11.7082	0.734638
$255$	0	0
$256$	−14.5623	−0.910144
$257$	−16.4164	−1.02403	−0.512014	−	0.858977i	$-0.671100\pi$
$257$	−16.4164	−1.02403	−0.512014	+	0.858977i	$0.671100\pi$
$258$	0	0
$259$	3.47214	0.215748
$260$	20.5623	1.27522
$261$	0	0
$262$	18.1803	1.12319
$263$	10.4164	0.642303	0.321152	−	0.947028i	$-0.395930\pi$
$263$	10.4164	0.642303	0.321152	+	0.947028i	$0.395930\pi$
$264$	0	0
$265$	14.4721	0.889016
$266$	14.3262	0.878398
$267$	0	0
$268$	−63.9787	−3.90812
$269$	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
$271$	14.8885	0.904415	0.452207	−	0.891913i	$-0.350637\pi$
$271$	14.8885	0.904415	0.452207	+	0.891913i	$0.350637\pi$
$272$	24.3607	1.47708
$273$	0	0
$274$	−27.4164	−1.65629
$275$	−4.00000	−0.241209
$276$	0	0
$277$	2.47214	0.148536	0.0742681	−	0.997238i	$-0.476338\pi$
$277$	2.47214	0.148536	0.0742681	+	0.997238i	$0.476338\pi$
$278$	−23.4164	−1.40442
$279$	0	0
$280$	7.47214	0.446546
$281$	25.1803	1.50213	0.751067	−	0.660226i	$-0.229540\pi$
$281$	25.1803	1.50213	0.751067	+	0.660226i	$0.229540\pi$
$282$	0	0
$283$	25.3607	1.50754	0.753768	−	0.657141i	$-0.228234\pi$
$283$	25.3607	1.50754	0.753768	+	0.657141i	$0.228234\pi$
$284$	−21.7082	−1.28814
$285$	0	0
$286$	11.0902	0.655775
$287$	4.47214	0.263982
$288$	0	0
$289$	−10.8885	−0.640503
$290$	−16.3262	−0.958710
$291$	0	0
$292$	−69.1033	−4.04397
$293$	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
$295$	−7.18034	−0.418056
$296$	−25.9443	−1.50798
$297$	0	0
$298$	3.38197	0.195912
$299$	−2.00000	−0.115663
$300$	0	0
$301$	−4.00000	−0.230556
$302$	4.00000	0.230174
$303$	0	0
$304$	−53.9230	−3.09270
$305$	0.472136	0.0270344
$306$	0	0
$307$	−3.05573	−0.174400	−0.0871998	−	0.996191i	$-0.527792\pi$
$307$	−3.05573	−0.174400	−0.0871998	+	0.996191i	$0.527792\pi$
$308$	4.85410	0.276588
$309$	0	0
$310$	22.1803	1.25976
$311$	−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
$313$	−16.3607	−0.924760	−0.462380	−	0.886682i	$-0.653004\pi$
$313$	−16.3607	−0.924760	−0.462380	+	0.886682i	$0.653004\pi$
$314$	−64.0689	−3.61562
$315$	0	0
$316$	−21.7082	−1.22118
$317$	17.4164	0.978203	0.489101	−	0.872227i	$-0.337325\pi$
$317$	17.4164	0.978203	0.489101	+	0.872227i	$0.337325\pi$
$318$	0	0
$319$	−6.23607	−0.349153
$320$	−8.70820	−0.486803
$321$	0	0
$322$	−1.23607	−0.0688834
$323$	−13.5279	−0.752710
$324$	0	0
$325$	16.9443	0.939899
$326$	−43.7426	−2.42268
$327$	0	0
$328$	−33.4164	−1.84511
$329$	−4.23607	−0.233542
$330$	0	0
$331$	−4.94427	−0.271762	−0.135881	−	0.990725i	$-0.543386\pi$
$331$	−4.94427	−0.271762	−0.135881	+	0.990725i	$0.543386\pi$
$332$	−74.8328	−4.10698
$333$	0	0
$334$	−49.5967	−2.71381
$335$	13.1803	0.720119
$336$	0	0
$337$	−20.4721	−1.11519	−0.557594	−	0.830114i	$-0.688275\pi$
$337$	−20.4721	−1.11519	−0.557594	+	0.830114i	$0.688275\pi$
$338$	−12.9443	−0.704076
$339$	0	0
$340$	−12.0000	−0.650791
$341$	8.47214	0.458792
$342$	0	0
$343$	1.00000	0.0539949
$344$	29.8885	1.61148
$345$	0	0
$346$	31.1246	1.67327
$347$	8.94427	0.480154	0.240077	−	0.970754i	$-0.422827\pi$
$347$	8.94427	0.480154	0.240077	+	0.970754i	$0.422827\pi$
$348$	0	0
$349$	13.6525	0.730800	0.365400	−	0.930851i	$-0.380932\pi$
$349$	13.6525	0.730800	0.365400	+	0.930851i	$0.380932\pi$
$350$	10.4721	0.559759
$351$	0	0
$352$	−10.8541	−0.578526
$353$	35.4721	1.88799	0.943996	−	0.329958i	$-0.107035\pi$
$353$	35.4721	1.88799	0.943996	+	0.329958i	$0.107035\pi$
$354$	0	0
$355$	4.47214	0.237356
$356$	48.5410	2.57267
$357$	0	0
$358$	−31.1246	−1.64499
$359$	9.88854	0.521897	0.260949	−	0.965353i	$-0.415965\pi$
$359$	9.88854	0.521897	0.260949	+	0.965353i	$0.415965\pi$
$360$	0	0
$361$	10.9443	0.576014
$362$	35.1246	1.84611
$363$	0	0
$364$	−20.5623	−1.07776
$365$	14.2361	0.745150
$366$	0	0
$367$	6.00000	0.313197	0.156599	−	0.987662i	$-0.449947\pi$
$367$	6.00000	0.313197	0.156599	+	0.987662i	$0.449947\pi$
$368$	4.65248	0.242527
$369$	0	0
$370$	9.09017	0.472575
$371$	−14.4721	−0.751356
$372$	0	0
$373$	20.8328	1.07868	0.539341	−	0.842087i	$-0.318673\pi$
$373$	20.8328	1.07868	0.539341	+	0.842087i	$0.318673\pi$
$374$	−6.47214	−0.334666
$375$	0	0
$376$	31.6525	1.63235
$377$	26.4164	1.36051
$378$	0	0
$379$	−30.5967	−1.57165	−0.785825	−	0.618449i	$-0.787761\pi$
$379$	−30.5967	−1.57165	−0.785825	+	0.618449i	$0.787761\pi$
$380$	26.5623	1.36262
$381$	0	0
$382$	62.5410	3.19988
$383$	−28.9443	−1.47898	−0.739492	−	0.673166i	$-0.764934\pi$
$383$	−28.9443	−1.47898	−0.739492	+	0.673166i	$0.764934\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	−4.00000	−0.203595
$387$	0	0
$388$	−38.8328	−1.97144
$389$	31.8885	1.61681	0.808407	−	0.588624i	$-0.200330\pi$
$389$	31.8885	1.61681	0.808407	+	0.588624i	$0.200330\pi$
$390$	0	0
$391$	1.16718	0.0590270
$392$	−7.47214	−0.377400
$393$	0	0
$394$	−56.0689	−2.82471
$395$	4.47214	0.225018
$396$	0	0
$397$	37.3050	1.87228	0.936141	−	0.351625i	$-0.114371\pi$
$397$	37.3050	1.87228	0.936141	+	0.351625i	$0.114371\pi$
$398$	−8.00000	−0.401004
$399$	0	0
$400$	−39.4164	−1.97082
$401$	−25.8885	−1.29281	−0.646406	−	0.762994i	$-0.723729\pi$
$401$	−25.8885	−1.29281	−0.646406	+	0.762994i	$0.723729\pi$
$402$	0	0
$403$	−35.8885	−1.78774
$404$	−2.29180	−0.114021
$405$	0	0
$406$	16.3262	0.810258
$407$	3.47214	0.172107
$408$	0	0
$409$	29.4164	1.45455	0.727274	−	0.686347i	$-0.240787\pi$
$409$	29.4164	1.45455	0.727274	+	0.686347i	$0.240787\pi$
$410$	11.7082	0.578227
$411$	0	0
$412$	55.4164	2.73017
$413$	7.18034	0.353321
$414$	0	0
$415$	15.4164	0.756762
$416$	45.9787	2.25429
$417$	0	0
$418$	14.3262	0.700719
$419$	27.6525	1.35091	0.675456	−	0.737400i	$-0.263947\pi$
$419$	27.6525	1.35091	0.675456	+	0.737400i	$0.263947\pi$
$420$	0	0
$421$	13.4721	0.656592	0.328296	−	0.944575i	$-0.393526\pi$
$421$	13.4721	0.656592	0.328296	+	0.944575i	$0.393526\pi$
$422$	12.9443	0.630117
$423$	0	0
$424$	108.138	5.25163
$425$	−9.88854	−0.479665
$426$	0	0
$427$	−0.472136	−0.0228483
$428$	−82.5197	−3.98874
$429$	0	0
$430$	−10.4721	−0.505011
$431$	7.58359	0.365289	0.182644	−	0.983179i	$-0.441534\pi$
$431$	7.58359	0.365289	0.182644	+	0.983179i	$0.441534\pi$
$432$	0	0
$433$	−14.3607	−0.690130	−0.345065	−	0.938579i	$-0.612143\pi$
$433$	−14.3607	−0.690130	−0.345065	+	0.938579i	$0.612143\pi$
$434$	−22.1803	−1.06469
$435$	0	0
$436$	−36.5410	−1.75000
$437$	−2.58359	−0.123590
$438$	0	0
$439$	−32.8885	−1.56968	−0.784842	−	0.619696i	$-0.787256\pi$
$439$	−32.8885	−1.56968	−0.784842	+	0.619696i	$0.787256\pi$
$440$	7.47214	0.356220
$441$	0	0
$442$	27.4164	1.30407
$443$	−21.5279	−1.02282	−0.511410	−	0.859337i	$-0.670877\pi$
$443$	−21.5279	−1.02282	−0.511410	+	0.859337i	$0.670877\pi$
$444$	0	0
$445$	−10.0000	−0.474045
$446$	44.3607	2.10054
$447$	0	0
$448$	8.70820	0.411424
$449$	−22.4721	−1.06053	−0.530263	−	0.847833i	$-0.677907\pi$
$449$	−22.4721	−1.06053	−0.530263	+	0.847833i	$0.677907\pi$
$450$	0	0
$451$	4.47214	0.210585
$452$	−74.8328	−3.51984
$453$	0	0
$454$	70.5410	3.31065
$455$	4.23607	0.198590
$456$	0	0
$457$	5.41641	0.253369	0.126684	−	0.991943i	$-0.459566\pi$
$457$	5.41641	0.253369	0.126684	+	0.991943i	$0.459566\pi$
$458$	7.70820	0.360181
$459$	0	0
$460$	−2.29180	−0.106856
$461$	29.5279	1.37525	0.687625	−	0.726066i	$-0.258653\pi$
$461$	29.5279	1.37525	0.687625	+	0.726066i	$0.258653\pi$
$462$	0	0
$463$	25.0689	1.16505	0.582525	−	0.812813i	$-0.302065\pi$
$463$	25.0689	1.16505	0.582525	+	0.812813i	$0.302065\pi$
$464$	−61.4508	−2.85278
$465$	0	0
$466$	−45.5967	−2.11223
$467$	−13.7639	−0.636919	−0.318459	−	0.947936i	$-0.603165\pi$
$467$	−13.7639	−0.636919	−0.318459	+	0.947936i	$0.603165\pi$
$468$	0	0
$469$	−13.1803	−0.608612
$470$	−11.0902	−0.511551
$471$	0	0
$472$	−53.6525	−2.46956
$473$	−4.00000	−0.183920
$474$	0	0
$475$	21.8885	1.00432
$476$	12.0000	0.550019
$477$	0	0
$478$	24.7984	1.13425
$479$	7.52786	0.343957	0.171978	−	0.985101i	$-0.444984\pi$
$479$	7.52786	0.343957	0.171978	+	0.985101i	$0.444984\pi$
$480$	0	0
$481$	−14.7082	−0.670636
$482$	−63.1591	−2.87682
$483$	0	0
$484$	4.85410	0.220641
$485$	8.00000	0.363261
$486$	0	0
$487$	−1.88854	−0.0855781	−0.0427890	−	0.999084i	$-0.513624\pi$
$487$	−1.88854	−0.0855781	−0.0427890	+	0.999084i	$0.513624\pi$
$488$	3.52786	0.159699
$489$	0	0
$490$	2.61803	0.118271
$491$	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
$493$	−15.4164	−0.694320
$494$	−60.6869	−2.73043
$495$	0	0
$496$	83.4853	3.74860
$497$	−4.47214	−0.200603
$498$	0	0
$499$	5.18034	0.231904	0.115952	−	0.993255i	$-0.463008\pi$
$499$	5.18034	0.231904	0.115952	+	0.993255i	$0.463008\pi$
$500$	43.6869	1.95374
$501$	0	0
$502$	73.6312	3.28632
$503$	6.58359	0.293548	0.146774	−	0.989170i	$-0.453111\pi$
$503$	6.58359	0.293548	0.146774	+	0.989170i	$0.453111\pi$
$504$	0	0
$505$	0.472136	0.0210098
$506$	−1.23607	−0.0549499
$507$	0	0
$508$	−21.7082	−0.963146
$509$	−14.9443	−0.662393	−0.331197	−	0.943562i	$-0.607452\pi$
$509$	−14.9443	−0.662393	−0.331197	+	0.943562i	$0.607452\pi$
$510$	0	0
$511$	−14.2361	−0.629767
$512$	40.3050	1.78124
$513$	0	0
$514$	42.9787	1.89571
$515$	−11.4164	−0.503067
$516$	0	0
$517$	−4.23607	−0.186302
$518$	−9.09017	−0.399399
$519$	0	0
$520$	−31.6525	−1.38805
$521$	32.5279	1.42507	0.712536	−	0.701636i	$-0.247547\pi$
$521$	32.5279	1.42507	0.712536	+	0.701636i	$0.247547\pi$
$522$	0	0
$523$	−20.4164	−0.892747	−0.446374	−	0.894847i	$-0.647285\pi$
$523$	−20.4164	−0.892747	−0.446374	+	0.894847i	$0.647285\pi$
$524$	−33.7082	−1.47255
$525$	0	0
$526$	−27.2705	−1.18905
$527$	20.9443	0.912347
$528$	0	0
$529$	−22.7771	−0.990308
$530$	−37.8885	−1.64577
$531$	0	0
$532$	−26.5623	−1.15162
$533$	−18.9443	−0.820568
$534$	0	0
$535$	17.0000	0.734974
$536$	98.4853	4.25392
$537$	0	0
$538$	−15.7082	−0.677229
$539$	1.00000	0.0430730
$540$	0	0
$541$	−1.41641	−0.0608961	−0.0304481	−	0.999536i	$-0.509693\pi$
$541$	−1.41641	−0.0608961	−0.0304481	+	0.999536i	$0.509693\pi$
$542$	−38.9787	−1.67428
$543$	0	0
$544$	−26.8328	−1.15045
$545$	7.52786	0.322458
$546$	0	0
$547$	11.4164	0.488130	0.244065	−	0.969759i	$-0.421519\pi$
$547$	11.4164	0.488130	0.244065	+	0.969759i	$0.421519\pi$
$548$	50.8328	2.17147
$549$	0	0
$550$	10.4721	0.446533
$551$	34.1246	1.45376
$552$	0	0
$553$	−4.47214	−0.190175
$554$	−6.47214	−0.274975
$555$	0	0
$556$	43.4164	1.84127
$557$	−17.0689	−0.723232	−0.361616	−	0.932327i	$-0.617775\pi$
$557$	−17.0689	−0.723232	−0.361616	+	0.932327i	$0.617775\pi$
$558$	0	0
$559$	16.9443	0.716666
$560$	−9.85410	−0.416412
$561$	0	0
$562$	−65.9230	−2.78079
$563$	−18.5836	−0.783205	−0.391603	−	0.920134i	$-0.628079\pi$
$563$	−18.5836	−0.783205	−0.391603	+	0.920134i	$0.628079\pi$
$564$	0	0
$565$	15.4164	0.648573
$566$	−66.3951	−2.79080
$567$	0	0
$568$	33.4164	1.40212
$569$	−19.3050	−0.809306	−0.404653	−	0.914470i	$-0.632608\pi$
$569$	−19.3050	−0.809306	−0.404653	+	0.914470i	$0.632608\pi$
$570$	0	0
$571$	37.3050	1.56116	0.780582	−	0.625054i	$-0.214923\pi$
$571$	37.3050	1.56116	0.780582	+	0.625054i	$0.214923\pi$
$572$	−20.5623	−0.859753
$573$	0	0
$574$	−11.7082	−0.488691
$575$	−1.88854	−0.0787577
$576$	0	0
$577$	26.9443	1.12170	0.560852	−	0.827916i	$-0.310474\pi$
$577$	26.9443	1.12170	0.560852	+	0.827916i	$0.310474\pi$
$578$	28.5066	1.18572
$579$	0	0
$580$	30.2705	1.25691
$581$	−15.4164	−0.639580
$582$	0	0
$583$	−14.4721	−0.599375
$584$	106.374	4.40178
$585$	0	0
$586$	−20.9443	−0.865200
$587$	−9.29180	−0.383513	−0.191757	−	0.981442i	$-0.561418\pi$
$587$	−9.29180	−0.383513	−0.191757	+	0.981442i	$0.561418\pi$
$588$	0	0
$589$	−46.3607	−1.91026
$590$	18.7984	0.773917
$591$	0	0
$592$	34.2148	1.40622
$593$	19.4164	0.797336	0.398668	−	0.917095i	$-0.369473\pi$
$593$	19.4164	0.797336	0.398668	+	0.917095i	$0.369473\pi$
$594$	0	0
$595$	−2.47214	−0.101348
$596$	−6.27051	−0.256850
$597$	0	0
$598$	5.23607	0.214119
$599$	−28.8328	−1.17808	−0.589038	−	0.808105i	$-0.700493\pi$
$599$	−28.8328	−1.17808	−0.589038	+	0.808105i	$0.700493\pi$
$600$	0	0
$601$	−25.2918	−1.03167	−0.515837	−	0.856687i	$-0.672519\pi$
$601$	−25.2918	−1.03167	−0.515837	+	0.856687i	$0.672519\pi$
$602$	10.4721	0.426812
$603$	0	0
$604$	−7.41641	−0.301769
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	16.0557	0.651682	0.325841	−	0.945425i	$-0.394353\pi$
$607$	16.0557	0.651682	0.325841	+	0.945425i	$0.394353\pi$
$608$	59.3951	2.40879
$609$	0	0
$610$	−1.23607	−0.0500469
$611$	17.9443	0.725948
$612$	0	0
$613$	−33.8885	−1.36875	−0.684373	−	0.729132i	$-0.739924\pi$
$613$	−33.8885	−1.36875	−0.684373	+	0.729132i	$0.739924\pi$
$614$	8.00000	0.322854
$615$	0	0
$616$	−7.47214	−0.301061
$617$	−21.4164	−0.862192	−0.431096	−	0.902306i	$-0.641873\pi$
$617$	−21.4164	−0.862192	−0.431096	+	0.902306i	$0.641873\pi$
$618$	0	0
$619$	−36.0000	−1.44696	−0.723481	−	0.690344i	$-0.757459\pi$
$619$	−36.0000	−1.44696	−0.723481	+	0.690344i	$0.757459\pi$
$620$	−41.1246	−1.65160
$621$	0	0
$622$	31.4164	1.25968
$623$	10.0000	0.400642
$624$	0	0
$625$	11.0000	0.440000
$626$	42.8328	1.71194
$627$	0	0
$628$	118.790	4.74025
$629$	8.58359	0.342250
$630$	0	0
$631$	−36.0000	−1.43314	−0.716569	−	0.697517i	$-0.754288\pi$
$631$	−36.0000	−1.43314	−0.716569	+	0.697517i	$0.754288\pi$
$632$	33.4164	1.32923
$633$	0	0
$634$	−45.5967	−1.81088
$635$	4.47214	0.177471
$636$	0	0
$637$	−4.23607	−0.167839
$638$	16.3262	0.646362
$639$	0	0
$640$	1.09017	0.0430928
$641$	5.05573	0.199689	0.0998446	−	0.995003i	$-0.468165\pi$
$641$	5.05573	0.199689	0.0998446	+	0.995003i	$0.468165\pi$
$642$	0	0
$643$	−24.0000	−0.946468	−0.473234	−	0.880937i	$-0.656913\pi$
$643$	−24.0000	−0.946468	−0.473234	+	0.880937i	$0.656913\pi$
$644$	2.29180	0.0903094
$645$	0	0
$646$	35.4164	1.39344
$647$	4.23607	0.166537	0.0832685	−	0.996527i	$-0.473464\pi$
$647$	4.23607	0.166537	0.0832685	+	0.996527i	$0.473464\pi$
$648$	0	0
$649$	7.18034	0.281853
$650$	−44.3607	−1.73997
$651$	0	0
$652$	81.1033	3.17625
$653$	13.8885	0.543501	0.271750	−	0.962368i	$-0.412397\pi$
$653$	13.8885	0.543501	0.271750	+	0.962368i	$0.412397\pi$
$654$	0	0
$655$	6.94427	0.271335
$656$	44.0689	1.72060
$657$	0	0
$658$	11.0902	0.432340
$659$	−0.0557281	−0.00217086	−0.00108543	−	0.999999i	$-0.500346\pi$
$659$	−0.0557281	−0.00217086	−0.00108543	+	0.999999i	$0.500346\pi$
$660$	0	0
$661$	−3.41641	−0.132883	−0.0664414	−	0.997790i	$-0.521165\pi$
$661$	−3.41641	−0.132883	−0.0664414	+	0.997790i	$0.521165\pi$
$662$	12.9443	0.503093
$663$	0	0
$664$	115.193	4.47037
$665$	5.47214	0.212200
$666$	0	0
$667$	−2.94427	−0.114003
$668$	91.9574	3.55794
$669$	0	0
$670$	−34.5066	−1.33311
$671$	−0.472136	−0.0182266
$672$	0	0
$673$	−40.8328	−1.57399	−0.786995	−	0.616960i	$-0.788364\pi$
$673$	−40.8328	−1.57399	−0.786995	+	0.616960i	$0.788364\pi$
$674$	53.5967	2.06447
$675$	0	0
$676$	24.0000	0.923077
$677$	19.4164	0.746233	0.373117	−	0.927784i	$-0.378289\pi$
$677$	19.4164	0.746233	0.373117	+	0.927784i	$0.378289\pi$
$678$	0	0
$679$	−8.00000	−0.307012
$680$	18.4721	0.708374
$681$	0	0
$682$	−22.1803	−0.849329
$683$	20.9443	0.801410	0.400705	−	0.916207i	$-0.368765\pi$
$683$	20.9443	0.801410	0.400705	+	0.916207i	$0.368765\pi$
$684$	0	0
$685$	−10.4721	−0.400120
$686$	−2.61803	−0.0999570
$687$	0	0
$688$	−39.4164	−1.50274
$689$	61.3050	2.33553
$690$	0	0
$691$	30.0000	1.14125	0.570627	−	0.821209i	$-0.306700\pi$
$691$	30.0000	1.14125	0.570627	+	0.821209i	$0.306700\pi$
$692$	−57.7082	−2.19374
$693$	0	0
$694$	−23.4164	−0.888875
$695$	−8.94427	−0.339276
$696$	0	0
$697$	11.0557	0.418766
$698$	−35.7426	−1.35288
$699$	0	0
$700$	−19.4164	−0.733871
$701$	12.4721	0.471066	0.235533	−	0.971866i	$-0.424316\pi$
$701$	12.4721	0.471066	0.235533	+	0.971866i	$0.424316\pi$
$702$	0	0
$703$	−19.0000	−0.716599
$704$	8.70820	0.328203
$705$	0	0
$706$	−92.8673	−3.49511
$707$	−0.472136	−0.0177565
$708$	0	0
$709$	−19.4721	−0.731291	−0.365646	−	0.930754i	$-0.619152\pi$
$709$	−19.4721	−0.731291	−0.365646	+	0.930754i	$0.619152\pi$
$710$	−11.7082	−0.439401
$711$	0	0
$712$	−74.7214	−2.80030
$713$	4.00000	0.149801
$714$	0	0
$715$	4.23607	0.158420
$716$	57.7082	2.15666
$717$	0	0
$718$	−25.8885	−0.966152
$719$	17.1803	0.640719	0.320359	−	0.947296i	$-0.396196\pi$
$719$	17.1803	0.640719	0.320359	+	0.947296i	$0.396196\pi$
$720$	0	0
$721$	11.4164	0.425169
$722$	−28.6525	−1.06633
$723$	0	0
$724$	−65.1246	−2.42034
$725$	24.9443	0.926407
$726$	0	0
$727$	27.5279	1.02095	0.510476	−	0.859892i	$-0.329469\pi$
$727$	27.5279	1.02095	0.510476	+	0.859892i	$0.329469\pi$
$728$	31.6525	1.17312
$729$	0	0
$730$	−37.2705	−1.37944
$731$	−9.88854	−0.365741
$732$	0	0
$733$	18.3607	0.678167	0.339084	−	0.940756i	$-0.389883\pi$
$733$	18.3607	0.678167	0.339084	+	0.940756i	$0.389883\pi$
$734$	−15.7082	−0.579800
$735$	0	0
$736$	−5.12461	−0.188896
$737$	−13.1803	−0.485504
$738$	0	0
$739$	3.88854	0.143042	0.0715212	−	0.997439i	$-0.477215\pi$
$739$	3.88854	0.143042	0.0715212	+	0.997439i	$0.477215\pi$
$740$	−16.8541	−0.619569
$741$	0	0
$742$	37.8885	1.39093
$743$	−21.3607	−0.783647	−0.391824	−	0.920040i	$-0.628156\pi$
$743$	−21.3607	−0.783647	−0.391824	+	0.920040i	$0.628156\pi$
$744$	0	0
$745$	1.29180	0.0473277
$746$	−54.5410	−1.99689
$747$	0	0
$748$	12.0000	0.438763
$749$	−17.0000	−0.621166
$750$	0	0
$751$	−35.6525	−1.30098	−0.650489	−	0.759516i	$-0.725436\pi$
$751$	−35.6525	−1.30098	−0.650489	+	0.759516i	$0.725436\pi$
$752$	−41.7426	−1.52220
$753$	0	0
$754$	−69.1591	−2.51862
$755$	1.52786	0.0556047
$756$	0	0
$757$	2.52786	0.0918768	0.0459384	−	0.998944i	$-0.485372\pi$
$757$	2.52786	0.0918768	0.0459384	+	0.998944i	$0.485372\pi$
$758$	80.1033	2.90948
$759$	0	0
$760$	−40.8885	−1.48318
$761$	33.7771	1.22442	0.612209	−	0.790696i	$-0.290281\pi$
$761$	33.7771	1.22442	0.612209	+	0.790696i	$0.290281\pi$
$762$	0	0
$763$	−7.52786	−0.272527
$764$	−115.957	−4.19519
$765$	0	0
$766$	75.7771	2.73794
$767$	−30.4164	−1.09827
$768$	0	0
$769$	12.1246	0.437225	0.218612	−	0.975812i	$-0.429847\pi$
$769$	12.1246	0.437225	0.218612	+	0.975812i	$0.429847\pi$
$770$	2.61803	0.0943474
$771$	0	0
$772$	7.41641	0.266922
$773$	10.0557	0.361679	0.180840	−	0.983513i	$-0.442118\pi$
$773$	10.0557	0.361679	0.180840	+	0.983513i	$0.442118\pi$
$774$	0	0
$775$	−33.8885	−1.21731
$776$	59.7771	2.14587
$777$	0	0
$778$	−83.4853	−2.99309
$779$	−24.4721	−0.876805
$780$	0	0
$781$	−4.47214	−0.160026
$782$	−3.05573	−0.109273
$783$	0	0
$784$	9.85410	0.351932
$785$	−24.4721	−0.873448
$786$	0	0
$787$	−16.4164	−0.585182	−0.292591	−	0.956238i	$-0.594517\pi$
$787$	−16.4164	−0.585182	−0.292591	+	0.956238i	$0.594517\pi$
$788$	103.957	3.70333
$789$	0	0
$790$	−11.7082	−0.416559
$791$	−15.4164	−0.548144
$792$	0	0
$793$	2.00000	0.0710221
$794$	−97.6656	−3.46602
$795$	0	0
$796$	14.8328	0.525735
$797$	14.8885	0.527379	0.263690	−	0.964608i	$-0.415061\pi$
$797$	14.8885	0.527379	0.263690	+	0.964608i	$0.415061\pi$
$798$	0	0
$799$	−10.4721	−0.370478
$800$	43.4164	1.53500
$801$	0	0
$802$	67.7771	2.39329
$803$	−14.2361	−0.502380
$804$	0	0
$805$	−0.472136	−0.0166406
$806$	93.9574	3.30951
$807$	0	0
$808$	3.52786	0.124110
$809$	14.3475	0.504432	0.252216	−	0.967671i	$-0.418841\pi$
$809$	14.3475	0.504432	0.252216	+	0.967671i	$0.418841\pi$
$810$	0	0
$811$	−51.2492	−1.79960	−0.899802	−	0.436299i	$-0.856289\pi$
$811$	−51.2492	−1.79960	−0.899802	+	0.436299i	$0.856289\pi$
$812$	−30.2705	−1.06229
$813$	0	0
$814$	−9.09017	−0.318610
$815$	−16.7082	−0.585263
$816$	0	0
$817$	21.8885	0.765783
$818$	−77.0132	−2.69270
$819$	0	0
$820$	−21.7082	−0.758083
$821$	5.29180	0.184685	0.0923425	−	0.995727i	$-0.470565\pi$
$821$	5.29180	0.184685	0.0923425	+	0.995727i	$0.470565\pi$
$822$	0	0
$823$	−5.76393	−0.200918	−0.100459	−	0.994941i	$-0.532031\pi$
$823$	−5.76393	−0.200918	−0.100459	+	0.994941i	$0.532031\pi$
$824$	−85.3050	−2.97174
$825$	0	0
$826$	−18.7984	−0.654079
$827$	24.8885	0.865459	0.432730	−	0.901524i	$-0.357550\pi$
$827$	24.8885	0.865459	0.432730	+	0.901524i	$0.357550\pi$
$828$	0	0
$829$	28.0000	0.972480	0.486240	−	0.873825i	$-0.338368\pi$
$829$	28.0000	0.972480	0.486240	+	0.873825i	$0.338368\pi$
$830$	−40.3607	−1.40094
$831$	0	0
$832$	−36.8885	−1.27888
$833$	2.47214	0.0856544
$834$	0	0
$835$	−18.9443	−0.655594
$836$	−26.5623	−0.918677
$837$	0	0
$838$	−72.3951	−2.50085
$839$	18.8197	0.649727	0.324863	−	0.945761i	$-0.394682\pi$
$839$	18.8197	0.649727	0.324863	+	0.945761i	$0.394682\pi$
$840$	0	0
$841$	9.88854	0.340984
$842$	−35.2705	−1.21550
$843$	0	0
$844$	−24.0000	−0.826114
$845$	−4.94427	−0.170088
$846$	0	0
$847$	1.00000	0.0343604
$848$	−142.610	−4.89724
$849$	0	0
$850$	25.8885	0.887970
$851$	1.63932	0.0561952
$852$	0	0
$853$	−31.3050	−1.07186	−0.535931	−	0.844262i	$-0.680039\pi$
$853$	−31.3050	−1.07186	−0.535931	+	0.844262i	$0.680039\pi$
$854$	1.23607	0.0422974
$855$	0	0
$856$	127.026	4.34167
$857$	52.9443	1.80854	0.904271	−	0.426959i	$-0.140415\pi$
$857$	52.9443	1.80854	0.904271	+	0.426959i	$0.140415\pi$
$858$	0	0
$859$	−26.0000	−0.887109	−0.443554	−	0.896248i	$-0.646283\pi$
$859$	−26.0000	−0.887109	−0.443554	+	0.896248i	$0.646283\pi$
$860$	19.4164	0.662094
$861$	0	0
$862$	−19.8541	−0.676233
$863$	2.94427	0.100224	0.0501121	−	0.998744i	$-0.484042\pi$
$863$	2.94427	0.100224	0.0501121	+	0.998744i	$0.484042\pi$
$864$	0	0
$865$	11.8885	0.404223
$866$	37.5967	1.27759
$867$	0	0
$868$	41.1246	1.39586
$869$	−4.47214	−0.151707
$870$	0	0
$871$	55.8328	1.89182
$872$	56.2492	1.90484
$873$	0	0
$874$	6.76393	0.228793
$875$	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	$0.543126\pi$
$878$	86.1033	2.90585
$879$	0	0
$880$	−9.85410	−0.332182
$881$	−8.41641	−0.283556	−0.141778	−	0.989898i	$-0.545282\pi$
$881$	−8.41641	−0.283556	−0.141778	+	0.989898i	$0.545282\pi$
$882$	0	0
$883$	20.2361	0.680998	0.340499	−	0.940245i	$-0.389404\pi$
$883$	20.2361	0.680998	0.340499	+	0.940245i	$0.389404\pi$
$884$	−50.8328	−1.70969
$885$	0	0
$886$	56.3607	1.89347
$887$	−44.9443	−1.50908	−0.754540	−	0.656254i	$-0.772140\pi$
$887$	−44.9443	−1.50908	−0.754540	+	0.656254i	$0.772140\pi$
$888$	0	0
$889$	−4.47214	−0.149991
$890$	26.1803	0.877567
$891$	0	0
$892$	−82.2492	−2.75391
$893$	23.1803	0.775700
$894$	0	0
$895$	−11.8885	−0.397390
$896$	−1.09017	−0.0364200
$897$	0	0
$898$	58.8328	1.96328
$899$	−52.8328	−1.76207
$900$	0	0
$901$	−35.7771	−1.19191
$902$	−11.7082	−0.389841
$903$	0	0
$904$	115.193	3.83128
$905$	13.4164	0.445976
$906$	0	0
$907$	−3.05573	−0.101464	−0.0507319	−	0.998712i	$-0.516155\pi$
$907$	−3.05573	−0.101464	−0.0507319	+	0.998712i	$0.516155\pi$
$908$	−130.790	−4.34043
$909$	0	0
$910$	−11.0902	−0.367636
$911$	−33.5279	−1.11083	−0.555414	−	0.831574i	$-0.687440\pi$
$911$	−33.5279	−1.11083	−0.555414	+	0.831574i	$0.687440\pi$
$912$	0	0
$913$	−15.4164	−0.510209
$914$	−14.1803	−0.469044
$915$	0	0
$916$	−14.2918	−0.472214
$917$	−6.94427	−0.229320
$918$	0	0
$919$	55.1935	1.82067	0.910333	−	0.413877i	$-0.135826\pi$
$919$	55.1935	1.82067	0.910333	+	0.413877i	$0.135826\pi$
$920$	3.52786	0.116310
$921$	0	0
$922$	−77.3050	−2.54590
$923$	18.9443	0.623558
$924$	0	0
$925$	−13.8885	−0.456653
$926$	−65.6312	−2.15677
$927$	0	0
$928$	67.6869	2.22193
$929$	−19.5836	−0.642517	−0.321258	−	0.946992i	$-0.604106\pi$
$929$	−19.5836	−0.642517	−0.321258	+	0.946992i	$0.604106\pi$
$930$	0	0
$931$	−5.47214	−0.179342
$932$	84.5410	2.76923
$933$	0	0
$934$	36.0344	1.17908
$935$	−2.47214	−0.0808475
$936$	0	0
$937$	−5.63932	−0.184229	−0.0921143	−	0.995748i	$-0.529363\pi$
$937$	−5.63932	−0.184229	−0.0921143	+	0.995748i	$0.529363\pi$
$938$	34.5066	1.12668
$939$	0	0
$940$	20.5623	0.670668
$941$	−18.9443	−0.617566	−0.308783	−	0.951133i	$-0.599922\pi$
$941$	−18.9443	−0.617566	−0.308783	+	0.951133i	$0.599922\pi$
$942$	0	0
$943$	2.11146	0.0687585
$944$	70.7558	2.30291
$945$	0	0
$946$	10.4721	0.340479
$947$	−48.8328	−1.58685	−0.793427	−	0.608666i	$-0.791705\pi$
$947$	−48.8328	−1.58685	−0.793427	+	0.608666i	$0.791705\pi$
$948$	0	0
$949$	60.3050	1.95758
$950$	−57.3050	−1.85922
$951$	0	0
$952$	−18.4721	−0.598685
$953$	3.06888	0.0994109	0.0497054	−	0.998764i	$-0.484172\pi$
$953$	3.06888	0.0994109	0.0497054	+	0.998764i	$0.484172\pi$
$954$	0	0
$955$	23.8885	0.773015
$956$	−45.9787	−1.48706
$957$	0	0
$958$	−19.7082	−0.636743
$959$	10.4721	0.338163
$960$	0	0
$961$	40.7771	1.31539
$962$	38.5066	1.24150
$963$	0	0
$964$	117.103	3.77164
$965$	−1.52786	−0.0491837
$966$	0	0
$967$	−12.3607	−0.397493	−0.198746	−	0.980051i	$-0.563687\pi$
$967$	−12.3607	−0.397493	−0.198746	+	0.980051i	$0.563687\pi$
$968$	−7.47214	−0.240164
$969$	0	0
$970$	−20.9443	−0.672480
$971$	−17.2918	−0.554920	−0.277460	−	0.960737i	$-0.589493\pi$
$971$	−17.2918	−0.554920	−0.277460	+	0.960737i	$0.589493\pi$
$972$	0	0
$973$	8.94427	0.286740
$974$	4.94427	0.158425
$975$	0	0
$976$	−4.65248	−0.148922
$977$	21.7771	0.696711	0.348355	−	0.937363i	$-0.386740\pi$
$977$	21.7771	0.696711	0.348355	+	0.937363i	$0.386740\pi$
$978$	0	0
$979$	10.0000	0.319601
$980$	−4.85410	−0.155059
$981$	0	0
$982$	−7.85410	−0.250634
$983$	−15.0557	−0.480203	−0.240102	−	0.970748i	$-0.577181\pi$
$983$	−15.0557	−0.480203	−0.240102	+	0.970748i	$0.577181\pi$
$984$	0	0
$985$	−21.4164	−0.682383
$986$	40.3607	1.28535
$987$	0	0
$988$	112.520	3.57973
$989$	−1.88854	−0.0600522
$990$	0	0
$991$	−37.5410	−1.19253	−0.596265	−	0.802788i	$-0.703349\pi$
$991$	−37.5410	−1.19253	−0.596265	+	0.802788i	$0.703349\pi$
$992$	−91.9574	−2.91965
$993$	0	0
$994$	11.7082	0.371362
$995$	−3.05573	−0.0968731
$996$	0	0
$997$	−44.4721	−1.40845	−0.704223	−	0.709979i	$-0.748705\pi$
$997$	−44.4721	−1.40845	−0.704223	+	0.709979i	$0.748705\pi$
$998$	−13.5623	−0.429307
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q-2.61803 q^{2} +4.85410 q^{4} -1.00000 q^{5} +1.00000 q^{7} -7.47214 q^{8} +O(q^{10})\)	\(q-2.61803 q^{2} +4.85410 q^{4} -1.00000 q^{5} +1.00000 q^{7} -7.47214 q^{8} +2.61803 q^{10} +1.00000 q^{11} -4.23607 q^{13} -2.61803 q^{14} +9.85410 q^{16} +2.47214 q^{17} -5.47214 q^{19} -4.85410 q^{20} -2.61803 q^{22} +0.472136 q^{23} -4.00000 q^{25} +11.0902 q^{26} +4.85410 q^{28} -6.23607 q^{29} +8.47214 q^{31} -10.8541 q^{32} -6.47214 q^{34} -1.00000 q^{35} +3.47214 q^{37} +14.3262 q^{38} +7.47214 q^{40} +4.47214 q^{41} -4.00000 q^{43} +4.85410 q^{44} -1.23607 q^{46} -4.23607 q^{47} +1.00000 q^{49} +10.4721 q^{50} -20.5623 q^{52} -14.4721 q^{53} -1.00000 q^{55} -7.47214 q^{56} +16.3262 q^{58} +7.18034 q^{59} -0.472136 q^{61} -22.1803 q^{62} +8.70820 q^{64} +4.23607 q^{65} -13.1803 q^{67} +12.0000 q^{68} +2.61803 q^{70} -4.47214 q^{71} -14.2361 q^{73} -9.09017 q^{74} -26.5623 q^{76} +1.00000 q^{77} -4.47214 q^{79} -9.85410 q^{80} -11.7082 q^{82} -15.4164 q^{83} -2.47214 q^{85} +10.4721 q^{86} -7.47214 q^{88} +10.0000 q^{89} -4.23607 q^{91} +2.29180 q^{92} +11.0902 q^{94} +5.47214 q^{95} -8.00000 q^{97} -2.61803 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{2} + 3 q^{4} - 2 q^{5} + 2 q^{7} - 6 q^{8}+O(q^{10}) \) 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

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Representations

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Database

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