LMFDB - Embedded newform 1350.3.d.n.701.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1350,3,Mod(701,1350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1350, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1350.701");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1350 = 2 \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1350.d (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$36.7848356886$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			701.1
Root			$0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1350.701
Dual form			1350.3.d.n.701.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{2} -2.00000 q^{4} -0.242641 q^{7} +2.82843i q^{8} +O(q^{10})$	$q-1.41421i q^{2} -2.00000 q^{4} -0.242641 q^{7} +2.82843i q^{8} -3.00000i q^{11} +3.48528 q^{13} +0.343146i q^{14} +4.00000 q^{16} -7.75736i q^{17} -8.97056 q^{19} -4.24264 q^{22} +6.51472i q^{23} -4.92893i q^{26} +0.485281 q^{28} -7.02944i q^{29} +29.2132 q^{31} -5.65685i q^{32} -10.9706 q^{34} -36.4558 q^{37} +12.6863i q^{38} -57.2132i q^{41} +11.0294 q^{43} +6.00000i q^{44} +9.21320 q^{46} -6.51472i q^{47} -48.9411 q^{49} -6.97056 q^{52} +26.9117i q^{53} -0.686292i q^{56} -9.94113 q^{58} -84.9411i q^{59} +57.3675 q^{61} -41.3137i q^{62} -8.00000 q^{64} -97.0955 q^{67} +15.5147i q^{68} -72.5147i q^{71} -106.912 q^{73} +51.5563i q^{74} +17.9411 q^{76} +0.727922i q^{77} -78.1838 q^{79} -80.9117 q^{82} -25.8823i q^{83} -15.5980i q^{86} +8.48528 q^{88} -31.7574i q^{89} -0.845671 q^{91} -13.0294i q^{92} -9.21320 q^{94} -110.765 q^{97} +69.2132i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4} + 16 q^{7}+O(q^{10}) $ 4 * q - 8 * q^4 + 16 * q^7	$ 4 q - 8 q^{4} + 16 q^{7} - 20 q^{13} + 16 q^{16} + 32 q^{19} - 32 q^{28} + 32 q^{31} + 24 q^{34} - 44 q^{37} + 112 q^{43} - 48 q^{46} - 60 q^{49} + 40 q^{52} + 96 q^{58} - 76 q^{61} - 32 q^{64} - 32 q^{67} - 224 q^{73} - 64 q^{76} - 160 q^{79} - 120 q^{82} - 224 q^{91} + 48 q^{94} + 100 q^{97}+O(q^{100}) $ 4 * q - 8 * q^4 + 16 * q^7 - 20 * q^13 + 16 * q^16 + 32 * q^19 - 32 * q^28 + 32 * q^31 + 24 * q^34 - 44 * q^37 + 112 * q^43 - 48 * q^46 - 60 * q^49 + 40 * q^52 + 96 * q^58 - 76 * q^61 - 32 * q^64 - 32 * q^67 - 224 * q^73 - 64 * q^76 - 160 * q^79 - 120 * q^82 - 224 * q^91 + 48 * q^94 + 100 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times$.

$n$	$1001$	$1027$
$\chi(n)$	$-1$	$1$

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 $

$2$	− 1.41421i	− 0.707107i
$2$	− 1.41421i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−2.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.242641	−0.0346630	−0.0173315	−	0.999850i	$-0.505517\pi$
$7$	−0.242641	−0.0346630	−0.0173315	+	0.999850i	$0.505517\pi$
$8$	2.82843i	0.353553i
$9$	0	0
$10$	0	0
$11$	− 3.00000i	− 0.272727i	−0.990659	−	0.136364i	$-0.956458\pi$
$11$	− 3.00000i	− 0.272727i	0.990659	−	0.136364i	$-0.0435416\pi$
$12$	0	0
$13$	3.48528	0.268099	0.134049	−	0.990975i	$-0.457202\pi$
$13$	3.48528	0.268099	0.134049	+	0.990975i	$0.457202\pi$
$14$	0.343146i	0.0245104i
$15$	0	0
$16$	4.00000	0.250000
$17$	− 7.75736i	− 0.456315i	−0.973624	−	0.228158i	$-0.926730\pi$
$17$	− 7.75736i	− 0.456315i	0.973624	−	0.228158i	$-0.0732702\pi$
$18$	0	0
$19$	−8.97056	−0.472135	−0.236067	−	0.971737i	$-0.575859\pi$
$19$	−8.97056	−0.472135	−0.236067	+	0.971737i	$0.575859\pi$
$20$	0	0
$21$	0	0
$22$	−4.24264	−0.192847
$23$	6.51472i	0.283249i	0.989920	+	0.141624i	$0.0452325\pi$
$23$	6.51472i	0.283249i	−0.989920	+	0.141624i	$0.954768\pi$
$24$	0	0
$25$	0	0
$26$	− 4.92893i	− 0.189574i
$27$	0	0
$28$	0.485281	0.0173315
$29$	− 7.02944i	− 0.242394i	−0.992628	−	0.121197i	$-0.961327\pi$
$29$	− 7.02944i	− 0.242394i	0.992628	−	0.121197i	$-0.0386733\pi$
$30$	0	0
$31$	29.2132	0.942361	0.471181	−	0.882037i	$-0.343828\pi$
$31$	29.2132	0.942361	0.471181	+	0.882037i	$0.343828\pi$
$32$	− 5.65685i	− 0.176777i
$33$	0	0
$34$	−10.9706	−0.322664
$35$	0	0
$36$	0	0
$37$	−36.4558	−0.985293	−0.492647	−	0.870230i	$-0.663970\pi$
$37$	−36.4558	−0.985293	−0.492647	+	0.870230i	$0.663970\pi$
$38$	12.6863i	0.333850i
$39$	0	0
$40$	0	0
$41$	− 57.2132i	− 1.39544i	−0.716369	−	0.697722i	$-0.754197\pi$
$41$	− 57.2132i	− 1.39544i	0.716369	−	0.697722i	$-0.245803\pi$
$42$	0	0
$43$	11.0294	0.256499	0.128249	−	0.991742i	$-0.459064\pi$
$43$	11.0294	0.256499	0.128249	+	0.991742i	$0.459064\pi$
$44$	6.00000i	0.136364i
$45$	0	0
$46$	9.21320	0.200287
$47$	− 6.51472i	− 0.138611i	−0.997595	−	0.0693055i	$-0.977922\pi$
$47$	− 6.51472i	− 0.138611i	0.997595	−	0.0693055i	$-0.0220783\pi$
$48$	0	0
$49$	−48.9411	−0.998798
$50$	0	0
$51$	0	0
$52$	−6.97056	−0.134049
$53$	26.9117i	0.507768i	0.967235	+	0.253884i	$0.0817081\pi$
$53$	26.9117i	0.507768i	−0.967235	+	0.253884i	$0.918292\pi$
$54$	0	0
$55$	0	0
$56$	− 0.686292i	− 0.0122552i
$57$	0	0
$58$	−9.94113	−0.171399
$59$	− 84.9411i	− 1.43968i	−0.694140	−	0.719840i	$-0.744215\pi$
$59$	− 84.9411i	− 1.43968i	0.694140	−	0.719840i	$-0.255785\pi$
$60$	0	0
$61$	57.3675	0.940451	0.470226	−	0.882546i	$-0.344173\pi$
$61$	57.3675	0.940451	0.470226	+	0.882546i	$0.344173\pi$
$62$	− 41.3137i	− 0.666350i
$63$	0	0
$64$	−8.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	−97.0955	−1.44919	−0.724593	−	0.689177i	$-0.757972\pi$
$67$	−97.0955	−1.44919	−0.724593	+	0.689177i	$0.757972\pi$
$68$	15.5147i	0.228158i
$69$	0	0
$70$	0	0
$71$	− 72.5147i	− 1.02133i	−0.859779	−	0.510667i	$-0.829398\pi$
$71$	− 72.5147i	− 1.02133i	0.859779	−	0.510667i	$-0.170602\pi$
$72$	0	0
$73$	−106.912	−1.46454	−0.732272	−	0.681012i	$-0.761540\pi$
$73$	−106.912	−1.46454	−0.732272	+	0.681012i	$0.761540\pi$
$74$	51.5563i	0.696707i
$75$	0	0
$76$	17.9411	0.236067
$77$	0.727922i	0.00945353i
$78$	0	0
$79$	−78.1838	−0.989668	−0.494834	−	0.868988i	$-0.664771\pi$
$79$	−78.1838	−0.989668	−0.494834	+	0.868988i	$0.664771\pi$
$80$	0	0
$81$	0	0
$82$	−80.9117	−0.986728
$83$	− 25.8823i	− 0.311834i	−0.987770	−	0.155917i	$-0.950167\pi$
$83$	− 25.8823i	− 0.311834i	0.987770	−	0.155917i	$-0.0498333\pi$
$84$	0	0
$85$	0	0
$86$	− 15.5980i	− 0.181372i
$87$	0	0
$88$	8.48528	0.0964237
$89$	− 31.7574i	− 0.356824i	−0.983956	−	0.178412i	$-0.942904\pi$
$89$	− 31.7574i	− 0.356824i	0.983956	−	0.178412i	$-0.0570960\pi$
$90$	0	0
$91$	−0.845671	−0.00929309
$92$	− 13.0294i	− 0.141624i
$93$	0	0
$94$	−9.21320	−0.0980128
$95$	0	0
$96$	0	0
$97$	−110.765	−1.14190	−0.570951	−	0.820984i	$-0.693425\pi$
$97$	−110.765	−1.14190	−0.570951	+	0.820984i	$0.693425\pi$
$98$	69.2132i	0.706257i
$99$	0	0
$100$	0	0
$101$	75.0366i	0.742936i	0.928446	+	0.371468i	$0.121146\pi$
$101$	75.0366i	0.742936i	−0.928446	+	0.371468i	$0.878854\pi$
$102$	0	0
$103$	−61.0955	−0.593160	−0.296580	−	0.955008i	$-0.595846\pi$
$103$	−61.0955	−0.593160	−0.296580	+	0.955008i	$0.595846\pi$
$104$	9.85786i	0.0947872i
$105$	0	0
$106$	38.0589	0.359046
$107$	− 33.8528i	− 0.316381i	−0.987409	−	0.158191i	$-0.949434\pi$
$107$	− 33.8528i	− 0.316381i	0.987409	−	0.158191i	$-0.0505661\pi$
$108$	0	0
$109$	19.1472	0.175662	0.0878311	−	0.996135i	$-0.472006\pi$
$109$	19.1472	0.175662	0.0878311	+	0.996135i	$0.472006\pi$
$110$	0	0
$111$	0	0
$112$	−0.970563	−0.00866574
$113$	− 26.7868i	− 0.237051i	−0.992951	−	0.118526i	$-0.962183\pi$
$113$	− 26.7868i	− 0.237051i	0.992951	−	0.118526i	$-0.0378168\pi$
$114$	0	0
$115$	0	0
$116$	14.0589i	0.121197i
$117$	0	0
$118$	−120.125	−1.01801
$119$	1.88225i	0.0158172i
$120$	0	0
$121$	112.000	0.925620
$122$	− 81.1299i	− 0.665000i
$123$	0	0
$124$	−58.4264	−0.471181
$125$	0	0
$126$	0	0
$127$	−174.919	−1.37731	−0.688657	−	0.725087i	$-0.741799\pi$
$127$	−174.919	−1.37731	−0.688657	+	0.725087i	$0.741799\pi$
$128$	11.3137i	0.0883883i
$129$	0	0
$130$	0	0
$131$	− 149.735i	− 1.14302i	−0.820597	−	0.571508i	$-0.806359\pi$
$131$	− 149.735i	− 1.14302i	0.820597	−	0.571508i	$-0.193641\pi$
$132$	0	0
$133$	2.17662	0.0163656
$134$	137.314i	1.02473i
$135$	0	0
$136$	21.9411	0.161332
$137$	192.853i	1.40768i	0.710356	+	0.703842i	$0.248534\pi$
$137$	192.853i	1.40768i	−0.710356	+	0.703842i	$0.751466\pi$
$138$	0	0
$139$	−117.095	−0.842413	−0.421207	−	0.906965i	$-0.638393\pi$
$139$	−117.095	−0.842413	−0.421207	+	0.906965i	$0.638393\pi$
$140$	0	0
$141$	0	0
$142$	−102.551	−0.722192
$143$	− 10.4558i	− 0.0731178i
$144$	0	0
$145$	0	0
$146$	151.196i	1.03559i
$147$	0	0
$148$	72.9117	0.492647
$149$	− 140.007i	− 0.939645i	−0.882761	−	0.469823i	$-0.844318\pi$
$149$	− 140.007i	− 0.939645i	0.882761	−	0.469823i	$-0.155682\pi$
$150$	0	0
$151$	130.912	0.866965	0.433482	−	0.901162i	$-0.357285\pi$
$151$	130.912	0.866965	0.433482	+	0.901162i	$0.357285\pi$
$152$	− 25.3726i	− 0.166925i
$153$	0	0
$154$	1.02944	0.00668466
$155$	0	0
$156$	0	0
$157$	−153.706	−0.979017	−0.489508	−	0.871999i	$-0.662824\pi$
$157$	−153.706	−0.979017	−0.489508	+	0.871999i	$0.662824\pi$
$158$	110.569i	0.699801i
$159$	0	0
$160$	0	0
$161$	− 1.58074i	− 0.00981823i
$162$	0	0
$163$	−182.302	−1.11841	−0.559207	−	0.829028i	$-0.688894\pi$
$163$	−182.302	−1.11841	−0.559207	+	0.829028i	$0.688894\pi$
$164$	114.426i	0.697722i
$165$	0	0
$166$	−36.6030	−0.220500
$167$	31.7208i	0.189945i	0.995480	+	0.0949724i	$0.0302763\pi$
$167$	31.7208i	0.189945i	−0.995480	+	0.0949724i	$0.969724\pi$
$168$	0	0
$169$	−156.853	−0.928123
$170$	0	0
$171$	0	0
$172$	−22.0589	−0.128249
$173$	− 331.154i	− 1.91419i	−0.289779	−	0.957093i	$-0.593582\pi$
$173$	− 331.154i	− 1.91419i	0.289779	−	0.957093i	$-0.406418\pi$
$174$	0	0
$175$	0	0
$176$	− 12.0000i	− 0.0681818i
$177$	0	0
$178$	−44.9117	−0.252313
$179$	− 144.088i	− 0.804963i	−0.915428	−	0.402481i	$-0.868148\pi$
$179$	− 144.088i	− 0.804963i	0.915428	−	0.402481i	$-0.131852\pi$
$180$	0	0
$181$	−46.6913	−0.257963	−0.128982	−	0.991647i	$-0.541171\pi$
$181$	−46.6913	−0.257963	−0.128982	+	0.991647i	$0.541171\pi$
$182$	1.19596i	0.00657121i
$183$	0	0
$184$	−18.4264	−0.100144
$185$	0	0
$186$	0	0
$187$	−23.2721	−0.124450
$188$	13.0294i	0.0693055i
$189$	0	0
$190$	0	0
$191$	298.441i	1.56252i	0.624208	+	0.781258i	$0.285422\pi$
$191$	298.441i	1.56252i	−0.624208	+	0.781258i	$0.714578\pi$
$192$	0	0
$193$	66.9117	0.346693	0.173346	−	0.984861i	$-0.444542\pi$
$193$	66.9117	0.346693	0.173346	+	0.984861i	$0.444542\pi$
$194$	156.645i	0.807447i
$195$	0	0
$196$	97.8823	0.499399
$197$	122.059i	0.619588i	0.950804	+	0.309794i	$0.100260\pi$
$197$	122.059i	0.619588i	−0.950804	+	0.309794i	$0.899740\pi$
$198$	0	0
$199$	94.7868	0.476316	0.238158	−	0.971226i	$-0.423456\pi$
$199$	94.7868	0.476316	0.238158	+	0.971226i	$0.423456\pi$
$200$	0	0
$201$	0	0
$202$	106.118	0.525335
$203$	1.70563i	0.00840211i
$204$	0	0
$205$	0	0
$206$	86.4020i	0.419427i
$207$	0	0
$208$	13.9411	0.0670246
$209$	26.9117i	0.128764i
$210$	0	0
$211$	165.706	0.785335	0.392667	−	0.919681i	$-0.371552\pi$
$211$	165.706	0.785335	0.392667	+	0.919681i	$0.371552\pi$
$212$	− 53.8234i	− 0.253884i
$213$	0	0
$214$	−47.8751	−0.223715
$215$	0	0
$216$	0	0
$217$	−7.08831	−0.0326650
$218$	− 27.0782i	− 0.124212i
$219$	0	0
$220$	0	0
$221$	− 27.0366i	− 0.122337i
$222$	0	0
$223$	200.971	0.901213	0.450607	−	0.892723i	$-0.351208\pi$
$223$	200.971	0.901213	0.450607	+	0.892723i	$0.351208\pi$
$224$	1.37258i	0.00612760i
$225$	0	0
$226$	−37.8823	−0.167621
$227$	349.441i	1.53939i	0.638414	+	0.769693i	$0.279591\pi$
$227$	349.441i	1.53939i	−0.638414	+	0.769693i	$0.720409\pi$
$228$	0	0
$229$	−172.515	−0.753339	−0.376670	−	0.926348i	$-0.622931\pi$
$229$	−172.515	−0.753339	−0.376670	+	0.926348i	$0.622931\pi$
$230$	0	0
$231$	0	0
$232$	19.8823	0.0856994
$233$	− 148.243i	− 0.636235i	−0.948051	−	0.318117i	$-0.896949\pi$
$233$	− 148.243i	− 0.636235i	0.948051	−	0.318117i	$-0.103051\pi$
$234$	0	0
$235$	0	0
$236$	169.882i	0.719840i
$237$	0	0
$238$	2.66190	0.0111845
$239$	− 141.603i	− 0.592481i	−0.955113	−	0.296241i	$-0.904267\pi$
$239$	− 141.603i	− 0.592481i	0.955113	−	0.296241i	$-0.0957330\pi$
$240$	0	0
$241$	83.8528	0.347937	0.173968	−	0.984751i	$-0.444341\pi$
$241$	83.8528	0.347937	0.173968	+	0.984751i	$0.444341\pi$
$242$	− 158.392i	− 0.654512i
$243$	0	0
$244$	−114.735	−0.470226
$245$	0	0
$246$	0	0
$247$	−31.2649	−0.126579
$248$	82.6274i	0.333175i
$249$	0	0
$250$	0	0
$251$	− 202.529i	− 0.806888i	−0.915004	−	0.403444i	$-0.867813\pi$
$251$	− 202.529i	− 0.806888i	0.915004	−	0.403444i	$-0.132187\pi$
$252$	0	0
$253$	19.5442	0.0772496
$254$	247.373i	0.973908i
$255$	0	0
$256$	16.0000	0.0625000
$257$	− 39.8894i	− 0.155212i	−0.996984	−	0.0776058i	$-0.975272\pi$
$257$	− 39.8894i	− 0.155212i	0.996984	−	0.0776058i	$-0.0247276\pi$
$258$	0	0
$259$	8.84567	0.0341532
$260$	0	0
$261$	0	0
$262$	−211.757	−0.808234
$263$	− 295.014i	− 1.12173i	−0.827908	−	0.560864i	$-0.810469\pi$
$263$	− 295.014i	− 1.12173i	0.827908	−	0.560864i	$-0.189531\pi$
$264$	0	0
$265$	0	0
$266$	− 3.07821i	− 0.0115722i
$267$	0	0
$268$	194.191	0.724593
$269$	− 181.706i	− 0.675486i	−0.941238	−	0.337743i	$-0.890337\pi$
$269$	− 181.706i	− 0.675486i	0.941238	−	0.337743i	$-0.109663\pi$
$270$	0	0
$271$	−261.823	−0.966138	−0.483069	−	0.875582i	$-0.660478\pi$
$271$	−261.823	−0.966138	−0.483069	+	0.875582i	$0.660478\pi$
$272$	− 31.0294i	− 0.114079i
$273$	0	0
$274$	272.735	0.995383
$275$	0	0
$276$	0	0
$277$	305.882	1.10427	0.552134	−	0.833755i	$-0.313814\pi$
$277$	305.882	1.10427	0.552134	+	0.833755i	$0.313814\pi$
$278$	165.598i	0.595676i
$279$	0	0
$280$	0	0
$281$	558.323i	1.98691i	0.114205	+	0.993457i	$0.463568\pi$
$281$	558.323i	1.98691i	−0.114205	+	0.993457i	$0.536432\pi$
$282$	0	0
$283$	8.97056	0.0316981	0.0158491	−	0.999874i	$-0.494955\pi$
$283$	8.97056	0.0316981	0.0158491	+	0.999874i	$0.494955\pi$
$284$	145.029i	0.510667i
$285$	0	0
$286$	−14.7868	−0.0517021
$287$	13.8823i	0.0483702i
$288$	0	0
$289$	228.823	0.791776
$290$	0	0
$291$	0	0
$292$	213.823	0.732272
$293$	146.683i	0.500626i	0.968165	+	0.250313i	$0.0805335\pi$
$293$	146.683i	0.500626i	−0.968165	+	0.250313i	$0.919467\pi$
$294$	0	0
$295$	0	0
$296$	− 103.113i	− 0.348354i
$297$	0	0
$298$	−198.000	−0.664430
$299$	22.7056i	0.0759386i
$300$	0	0
$301$	−2.67619	−0.00889100
$302$	− 185.137i	− 0.613037i
$303$	0	0
$304$	−35.8823	−0.118034
$305$	0	0
$306$	0	0
$307$	343.640	1.11935	0.559674	−	0.828713i	$-0.310926\pi$
$307$	343.640	1.11935	0.559674	+	0.828713i	$0.310926\pi$
$308$	− 1.45584i	− 0.00472677i
$309$	0	0
$310$	0	0
$311$	− 512.044i	− 1.64644i	−0.567720	−	0.823221i	$-0.692175\pi$
$311$	− 512.044i	− 1.64644i	0.567720	−	0.823221i	$-0.307825\pi$
$312$	0	0
$313$	68.6173	0.219225	0.109612	−	0.993974i	$-0.465039\pi$
$313$	68.6173	0.219225	0.109612	+	0.993974i	$0.465039\pi$
$314$	217.373i	0.692269i
$315$	0	0
$316$	156.368	0.494834
$317$	516.500i	1.62934i	0.579928	+	0.814668i	$0.303081\pi$
$317$	516.500i	1.62934i	−0.579928	+	0.814668i	$0.696919\pi$
$318$	0	0
$319$	−21.0883	−0.0661076
$320$	0	0
$321$	0	0
$322$	−2.23550	−0.00694254
$323$	69.5879i	0.215442i
$324$	0	0
$325$	0	0
$326$	257.813i	0.790838i
$327$	0	0
$328$	161.823	0.493364
$329$	1.58074i	0.00480467i
$330$	0	0
$331$	−265.110	−0.800936	−0.400468	−	0.916311i	$-0.631152\pi$
$331$	−265.110	−0.800936	−0.400468	+	0.916311i	$0.631152\pi$
$332$	51.7645i	0.155917i
$333$	0	0
$334$	44.8600	0.134311
$335$	0	0
$336$	0	0
$337$	−72.4710	−0.215047	−0.107524	−	0.994203i	$-0.534292\pi$
$337$	−72.4710	−0.215047	−0.107524	+	0.994203i	$0.534292\pi$
$338$	221.823i	0.656282i
$339$	0	0
$340$	0	0
$341$	− 87.6396i	− 0.257008i
$342$	0	0
$343$	23.7645	0.0692843
$344$	31.1960i	0.0906859i
$345$	0	0
$346$	−468.323	−1.35353
$347$	− 529.706i	− 1.52653i	−0.646086	−	0.763265i	$-0.723595\pi$
$347$	− 529.706i	− 1.52653i	0.646086	−	0.763265i	$-0.276405\pi$
$348$	0	0
$349$	337.470	0.966963	0.483482	−	0.875355i	$-0.339372\pi$
$349$	337.470	0.966963	0.483482	+	0.875355i	$0.339372\pi$
$350$	0	0
$351$	0	0
$352$	−16.9706	−0.0482118
$353$	− 490.087i	− 1.38835i	−0.719806	−	0.694175i	$-0.755769\pi$
$353$	− 490.087i	− 1.38835i	0.719806	−	0.694175i	$-0.244231\pi$
$354$	0	0
$355$	0	0
$356$	63.5147i	0.178412i
$357$	0	0
$358$	−203.772	−0.569195
$359$	− 370.955i	− 1.03330i	−0.856196	−	0.516651i	$-0.827178\pi$
$359$	− 370.955i	− 1.03330i	0.856196	−	0.516651i	$-0.172822\pi$
$360$	0	0
$361$	−280.529	−0.777089
$362$	66.0315i	0.182408i
$363$	0	0
$364$	1.69134	0.00464654
$365$	0	0
$366$	0	0
$367$	393.823	1.07309	0.536544	−	0.843872i	$-0.319730\pi$
$367$	393.823	1.07309	0.536544	+	0.843872i	$0.319730\pi$
$368$	26.0589i	0.0708122i
$369$	0	0
$370$	0	0
$371$	− 6.52987i	− 0.0176007i
$372$	0	0
$373$	−90.1177	−0.241603	−0.120801	−	0.992677i	$-0.538546\pi$
$373$	−90.1177	−0.241603	−0.120801	+	0.992677i	$0.538546\pi$
$374$	32.9117i	0.0879992i
$375$	0	0
$376$	18.4264	0.0490064
$377$	− 24.4996i	− 0.0649856i
$378$	0	0
$379$	327.051	0.862931	0.431466	−	0.902129i	$-0.357997\pi$
$379$	327.051	0.862931	0.431466	+	0.902129i	$0.357997\pi$
$380$	0	0
$381$	0	0
$382$	422.059	1.10487
$383$	468.338i	1.22281i	0.791316	+	0.611407i	$0.209396\pi$
$383$	468.338i	1.22281i	−0.791316	+	0.611407i	$0.790604\pi$
$384$	0	0
$385$	0	0
$386$	− 94.6274i	− 0.245149i
$387$	0	0
$388$	221.529	0.570951
$389$	80.3818i	0.206637i	0.994648	+	0.103319i	$0.0329461\pi$
$389$	80.3818i	0.206637i	−0.994648	+	0.103319i	$0.967054\pi$
$390$	0	0
$391$	50.5370	0.129251
$392$	− 138.426i	− 0.353129i
$393$	0	0
$394$	172.617	0.438115
$395$	0	0
$396$	0	0
$397$	489.632	1.23333	0.616664	−	0.787226i	$-0.288484\pi$
$397$	489.632	1.23333	0.616664	+	0.787226i	$0.288484\pi$
$398$	− 134.049i	− 0.336806i
$399$	0	0
$400$	0	0
$401$	− 263.418i	− 0.656904i	−0.944521	−	0.328452i	$-0.893473\pi$
$401$	− 263.418i	− 0.656904i	0.944521	−	0.328452i	$-0.106527\pi$
$402$	0	0
$403$	101.816	0.252646
$404$	− 150.073i	− 0.371468i
$405$	0	0
$406$	2.41212	0.00594119
$407$	109.368i	0.268716i
$408$	0	0
$409$	170.411	0.416653	0.208327	−	0.978059i	$-0.433198\pi$
$409$	170.411	0.416653	0.208327	+	0.978059i	$0.433198\pi$
$410$	0	0
$411$	0	0
$412$	122.191	0.296580
$413$	20.6102i	0.0499036i
$414$	0	0
$415$	0	0
$416$	− 19.7157i	− 0.0473936i
$417$	0	0
$418$	38.0589	0.0910499
$419$	− 353.647i	− 0.844026i	−0.906590	−	0.422013i	$-0.861324\pi$
$419$	− 353.647i	− 0.844026i	0.906590	−	0.422013i	$-0.138676\pi$
$420$	0	0
$421$	−661.749	−1.57185	−0.785926	−	0.618321i	$-0.787813\pi$
$421$	−661.749	−1.57185	−0.785926	+	0.618321i	$0.787813\pi$
$422$	− 234.343i	− 0.555316i
$423$	0	0
$424$	−76.1177	−0.179523
$425$	0	0
$426$	0	0
$427$	−13.9197	−0.0325988
$428$	67.7056i	0.158191i
$429$	0	0
$430$	0	0
$431$	− 757.544i	− 1.75764i	−0.477151	−	0.878822i	$-0.658330\pi$
$431$	− 757.544i	− 1.75764i	0.477151	−	0.878822i	$-0.341670\pi$
$432$	0	0
$433$	293.294	0.677354	0.338677	−	0.940903i	$-0.390021\pi$
$433$	293.294	0.677354	0.338677	+	0.940903i	$0.390021\pi$
$434$	10.0244i	0.0230977i
$435$	0	0
$436$	−38.2944	−0.0878311
$437$	− 58.4407i	− 0.133732i
$438$	0	0
$439$	441.831	1.00645	0.503224	−	0.864156i	$-0.332147\pi$
$439$	441.831	1.00645	0.503224	+	0.864156i	$0.332147\pi$
$440$	0	0
$441$	0	0
$442$	−38.2355	−0.0865057
$443$	46.3827i	0.104701i	0.998629	+	0.0523507i	$0.0166714\pi$
$443$	46.3827i	0.104701i	−0.998629	+	0.0523507i	$0.983329\pi$
$444$	0	0
$445$	0	0
$446$	− 284.215i	− 0.637254i
$447$	0	0
$448$	1.94113	0.00433287
$449$	− 69.2132i	− 0.154150i	−0.997025	−	0.0770748i	$-0.975442\pi$
$449$	− 69.2132i	− 0.154150i	0.997025	−	0.0770748i	$-0.0245580\pi$
$450$	0	0
$451$	−171.640	−0.380576
$452$	53.5736i	0.118526i
$453$	0	0
$454$	494.184	1.08851
$455$	0	0
$456$	0	0
$457$	189.382	0.414402	0.207201	−	0.978298i	$-0.433565\pi$
$457$	189.382	0.414402	0.207201	+	0.978298i	$0.433565\pi$
$458$	243.973i	0.532691i
$459$	0	0
$460$	0	0
$461$	615.058i	1.33418i	0.744976	+	0.667091i	$0.232461\pi$
$461$	615.058i	1.33418i	−0.744976	+	0.667091i	$0.767539\pi$
$462$	0	0
$463$	−137.088	−0.296087	−0.148044	−	0.988981i	$-0.547298\pi$
$463$	−137.088	−0.296087	−0.148044	+	0.988981i	$0.547298\pi$
$464$	− 28.1177i	− 0.0605986i
$465$	0	0
$466$	−209.647	−0.449886
$467$	− 337.118i	− 0.721880i	−0.932589	−	0.360940i	$-0.882456\pi$
$467$	− 337.118i	− 0.721880i	0.932589	−	0.360940i	$-0.117544\pi$
$468$	0	0
$469$	23.5593	0.0502331
$470$	0	0
$471$	0	0
$472$	240.250	0.509004
$473$	− 33.0883i	− 0.0699541i
$474$	0	0
$475$	0	0
$476$	− 3.76450i	− 0.00790862i
$477$	0	0
$478$	−200.257	−0.418948
$479$	− 227.647i	− 0.475254i	−0.971356	−	0.237627i	$-0.923630\pi$
$479$	− 227.647i	− 0.475254i	0.971356	−	0.237627i	$-0.0763696\pi$
$480$	0	0
$481$	−127.059	−0.264156
$482$	− 118.586i	− 0.246029i
$483$	0	0
$484$	−224.000	−0.462810
$485$	0	0
$486$	0	0
$487$	640.874	1.31596	0.657982	−	0.753034i	$-0.271410\pi$
$487$	640.874	1.31596	0.657982	+	0.753034i	$0.271410\pi$
$488$	162.260i	0.332500i
$489$	0	0
$490$	0	0
$491$	− 610.118i	− 1.24260i	−0.783572	−	0.621301i	$-0.786604\pi$
$491$	− 610.118i	− 1.24260i	0.783572	−	0.621301i	$-0.213396\pi$
$492$	0	0
$493$	−54.5299	−0.110608
$494$	44.2153i	0.0895046i
$495$	0	0
$496$	116.853	0.235590
$497$	17.5950i	0.0354025i
$498$	0	0
$499$	43.2935	0.0867605	0.0433803	−	0.999059i	$-0.486187\pi$
$499$	43.2935	0.0867605	0.0433803	+	0.999059i	$0.486187\pi$
$500$	0	0
$501$	0	0
$502$	−286.419	−0.570556
$503$	303.765i	0.603906i	0.953323	+	0.301953i	$0.0976385\pi$
$503$	303.765i	0.603906i	−0.953323	+	0.301953i	$0.902362\pi$
$504$	0	0
$505$	0	0
$506$	− 27.6396i	− 0.0546237i
$507$	0	0
$508$	349.838	0.688657
$509$	40.2426i	0.0790622i	0.999218	+	0.0395311i	$0.0125864\pi$
$509$	40.2426i	0.0790622i	−0.999218	+	0.0395311i	$0.987414\pi$
$510$	0	0
$511$	25.9411	0.0507654
$512$	− 22.6274i	− 0.0441942i
$513$	0	0
$514$	−56.4121	−0.109751
$515$	0	0
$516$	0	0
$517$	−19.5442	−0.0378030
$518$	− 12.5097i	− 0.0241499i
$519$	0	0
$520$	0	0
$521$	88.5959i	0.170050i	0.996379	+	0.0850248i	$0.0270970\pi$
$521$	88.5959i	0.170050i	−0.996379	+	0.0850248i	$0.972903\pi$
$522$	0	0
$523$	711.647	1.36070	0.680351	−	0.732887i	$-0.261828\pi$
$523$	711.647	1.36070	0.680351	+	0.732887i	$0.261828\pi$
$524$	299.470i	0.571508i
$525$	0	0
$526$	−417.213	−0.793181
$527$	− 226.617i	− 0.430014i
$528$	0	0
$529$	486.558	0.919770
$530$	0	0
$531$	0	0
$532$	−4.35325	−0.00818280
$533$	− 199.404i	− 0.374117i
$534$	0	0
$535$	0	0
$536$	− 274.627i	− 0.512365i
$537$	0	0
$538$	−256.971	−0.477640
$539$	146.823i	0.272400i
$540$	0	0
$541$	197.426	0.364929	0.182464	−	0.983212i	$-0.441593\pi$
$541$	197.426	0.364929	0.182464	+	0.983212i	$0.441593\pi$
$542$	370.274i	0.683163i
$543$	0	0
$544$	−43.8823	−0.0806659
$545$	0	0
$546$	0	0
$547$	453.470	0.829013	0.414507	−	0.910046i	$-0.363954\pi$
$547$	453.470	0.829013	0.414507	+	0.910046i	$0.363954\pi$
$548$	− 385.706i	− 0.703842i
$549$	0	0
$550$	0	0
$551$	63.0580i	0.114443i
$552$	0	0
$553$	18.9706	0.0343048
$554$	− 432.583i	− 0.780835i
$555$	0	0
$556$	234.191	0.421207
$557$	156.978i	0.281827i	0.990022	+	0.140914i	$0.0450040\pi$
$557$	156.978i	0.281827i	−0.990022	+	0.140914i	$0.954996\pi$
$558$	0	0
$559$	38.4407	0.0687669
$560$	0	0
$561$	0	0
$562$	789.588	1.40496
$563$	699.323i	1.24214i	0.783756	+	0.621068i	$0.213301\pi$
$563$	699.323i	1.24214i	−0.783756	+	0.621068i	$0.786699\pi$
$564$	0	0
$565$	0	0
$566$	− 12.6863i	− 0.0224139i
$567$	0	0
$568$	205.103	0.361096
$569$	792.978i	1.39363i	0.717249	+	0.696817i	$0.245401\pi$
$569$	792.978i	1.39363i	−0.717249	+	0.696817i	$0.754599\pi$
$570$	0	0
$571$	−327.418	−0.573412	−0.286706	−	0.958019i	$-0.592560\pi$
$571$	−327.418	−0.573412	−0.286706	+	0.958019i	$0.592560\pi$
$572$	20.9117i	0.0365589i
$573$	0	0
$574$	19.6325	0.0342029
$575$	0	0
$576$	0	0
$577$	844.088	1.46289	0.731446	−	0.681900i	$-0.238846\pi$
$577$	844.088	1.46289	0.731446	+	0.681900i	$0.238846\pi$
$578$	− 323.605i	− 0.559870i
$579$	0	0
$580$	0	0
$581$	6.28009i	0.0108091i
$582$	0	0
$583$	80.7351	0.138482
$584$	− 302.392i	− 0.517794i
$585$	0	0
$586$	207.442	0.353996
$587$	− 304.471i	− 0.518690i	−0.965785	−	0.259345i	$-0.916493\pi$
$587$	− 304.471i	− 0.518690i	0.965785	−	0.259345i	$-0.0835067\pi$
$588$	0	0
$589$	−262.059	−0.444922
$590$	0	0
$591$	0	0
$592$	−145.823	−0.246323
$593$	403.882i	0.681083i	0.940229	+	0.340542i	$0.110610\pi$
$593$	403.882i	0.681083i	−0.940229	+	0.340542i	$0.889390\pi$
$594$	0	0
$595$	0	0
$596$	280.014i	0.469823i
$597$	0	0
$598$	32.1106	0.0536967
$599$	− 133.176i	− 0.222330i	−0.993802	−	0.111165i	$-0.964542\pi$
$599$	− 133.176i	− 0.222330i	0.993802	−	0.111165i	$-0.0354582\pi$
$600$	0	0
$601$	−263.294	−0.438094	−0.219047	−	0.975714i	$-0.570295\pi$
$601$	−263.294	−0.438094	−0.219047	+	0.975714i	$0.570295\pi$
$602$	3.78470i	0.00628688i
$603$	0	0
$604$	−261.823	−0.433482
$605$	0	0
$606$	0	0
$607$	−613.449	−1.01062	−0.505312	−	0.862937i	$-0.668623\pi$
$607$	−613.449	−1.01062	−0.505312	+	0.862937i	$0.668623\pi$
$608$	50.7452i	0.0834624i
$609$	0	0
$610$	0	0
$611$	− 22.7056i	− 0.0371614i
$612$	0	0
$613$	712.838	1.16287	0.581434	−	0.813594i	$-0.302492\pi$
$613$	712.838	1.16287	0.581434	+	0.813594i	$0.302492\pi$
$614$	− 485.980i	− 0.791498i
$615$	0	0
$616$	−2.05887	−0.00334233
$617$	192.728i	0.312363i	0.987728	+	0.156181i	$0.0499185\pi$
$617$	192.728i	0.312363i	−0.987728	+	0.156181i	$0.950082\pi$
$618$	0	0
$619$	−660.403	−1.06689	−0.533444	−	0.845836i	$-0.679102\pi$
$619$	−660.403	−1.06689	−0.533444	+	0.845836i	$0.679102\pi$
$620$	0	0
$621$	0	0
$622$	−724.139	−1.16421
$623$	7.70563i	0.0123686i
$624$	0	0
$625$	0	0
$626$	− 97.0395i	− 0.155015i
$627$	0	0
$628$	307.411	0.489508
$629$	282.801i	0.449604i
$630$	0	0
$631$	−128.971	−0.204391	−0.102195	−	0.994764i	$-0.532587\pi$
$631$	−128.971	−0.204391	−0.102195	+	0.994764i	$0.532587\pi$
$632$	− 221.137i	− 0.349900i
$633$	0	0
$634$	730.441	1.15211
$635$	0	0
$636$	0	0
$637$	−170.574	−0.267776
$638$	29.8234i	0.0467451i
$639$	0	0
$640$	0	0
$641$	526.773i	0.821798i	0.911681	+	0.410899i	$0.134785\pi$
$641$	526.773i	0.821798i	−0.911681	+	0.410899i	$0.865215\pi$
$642$	0	0
$643$	−558.169	−0.868071	−0.434035	−	0.900896i	$-0.642911\pi$
$643$	−558.169	−0.868071	−0.434035	+	0.900896i	$0.642911\pi$
$644$	3.16147i	0.00490912i
$645$	0	0
$646$	98.4121	0.152341
$647$	− 990.131i	− 1.53034i	−0.643827	−	0.765171i	$-0.722654\pi$
$647$	− 990.131i	− 1.53034i	0.643827	−	0.765171i	$-0.277346\pi$
$648$	0	0
$649$	−254.823	−0.392640
$650$	0	0
$651$	0	0
$652$	364.603	0.559207
$653$	− 832.139i	− 1.27433i	−0.770726	−	0.637166i	$-0.780106\pi$
$653$	− 832.139i	− 1.27433i	0.770726	−	0.637166i	$-0.219894\pi$
$654$	0	0
$655$	0	0
$656$	− 228.853i	− 0.348861i
$657$	0	0
$658$	2.23550	0.00339741
$659$	239.793i	0.363874i	0.983310	+	0.181937i	$0.0582367\pi$
$659$	239.793i	0.363874i	−0.983310	+	0.181937i	$0.941763\pi$
$660$	0	0
$661$	1053.01	1.59306	0.796531	−	0.604597i	$-0.206666\pi$
$661$	1053.01	1.59306	0.796531	+	0.604597i	$0.206666\pi$
$662$	374.922i	0.566347i
$663$	0	0
$664$	73.2061	0.110250
$665$	0	0
$666$	0	0
$667$	45.7948	0.0686579
$668$	− 63.4416i	− 0.0949724i
$669$	0	0
$670$	0	0
$671$	− 172.103i	− 0.256487i
$672$	0	0
$673$	−721.912	−1.07268	−0.536339	−	0.844003i	$-0.680193\pi$
$673$	−721.912	−1.07268	−0.536339	+	0.844003i	$0.680193\pi$
$674$	102.489i	0.152062i
$675$	0	0
$676$	313.706	0.464062
$677$	538.419i	0.795302i	0.917537	+	0.397651i	$0.130174\pi$
$677$	538.419i	0.795302i	−0.917537	+	0.397651i	$0.869826\pi$
$678$	0	0
$679$	26.8760	0.0395817
$680$	0	0
$681$	0	0
$682$	−123.941	−0.181732
$683$	335.412i	0.491087i	0.969386	+	0.245543i	$0.0789663\pi$
$683$	335.412i	0.491087i	−0.969386	+	0.245543i	$0.921034\pi$
$684$	0	0
$685$	0	0
$686$	− 33.6081i	− 0.0489914i
$687$	0	0
$688$	44.1177	0.0641246
$689$	93.7948i	0.136132i
$690$	0	0
$691$	225.477	0.326306	0.163153	−	0.986601i	$-0.447834\pi$
$691$	225.477	0.326306	0.163153	+	0.986601i	$0.447834\pi$
$692$	662.309i	0.957093i
$693$	0	0
$694$	−749.117	−1.07942
$695$	0	0
$696$	0	0
$697$	−443.823	−0.636762
$698$	− 477.255i	− 0.683746i
$699$	0	0
$700$	0	0
$701$	709.684i	1.01239i	0.862420	+	0.506194i	$0.168948\pi$
$701$	709.684i	1.01239i	−0.862420	+	0.506194i	$0.831052\pi$
$702$	0	0
$703$	327.029	0.465191
$704$	24.0000i	0.0340909i
$705$	0	0
$706$	−693.088	−0.981711
$707$	− 18.2069i	− 0.0257524i
$708$	0	0
$709$	663.867	0.936343	0.468171	−	0.883638i	$-0.344913\pi$
$709$	663.867	0.936343	0.468171	+	0.883638i	$0.344913\pi$
$710$	0	0
$711$	0	0
$712$	89.8234	0.126156
$713$	190.316i	0.266923i
$714$	0	0
$715$	0	0
$716$	288.177i	0.402481i
$717$	0	0
$718$	−524.610	−0.730655
$719$	402.161i	0.559334i	0.960097	+	0.279667i	$0.0902241\pi$
$719$	402.161i	0.559334i	−0.960097	+	0.279667i	$0.909776\pi$
$720$	0	0
$721$	14.8242	0.0205607
$722$	396.728i	0.549485i
$723$	0	0
$724$	93.3827	0.128982
$725$	0	0
$726$	0	0
$727$	−643.294	−0.884860	−0.442430	−	0.896803i	$-0.645884\pi$
$727$	−643.294	−0.884860	−0.442430	+	0.896803i	$0.645884\pi$
$728$	− 2.39192i	− 0.00328560i
$729$	0	0
$730$	0	0
$731$	− 85.5593i	− 0.117044i
$732$	0	0
$733$	−159.191	−0.217177	−0.108589	−	0.994087i	$-0.534633\pi$
$733$	−159.191	−0.217177	−0.108589	+	0.994087i	$0.534633\pi$
$734$	− 556.950i	− 0.758788i
$735$	0	0
$736$	36.8528	0.0500718
$737$	291.286i	0.395233i
$738$	0	0
$739$	115.272	0.155984	0.0779919	−	0.996954i	$-0.475149\pi$
$739$	115.272	0.155984	0.0779919	+	0.996954i	$0.475149\pi$
$740$	0	0
$741$	0	0
$742$	−9.23463	−0.0124456
$743$	− 93.0732i	− 0.125267i	−0.998037	−	0.0626334i	$-0.980050\pi$
$743$	− 93.0732i	− 0.125267i	0.998037	−	0.0626334i	$-0.0199499\pi$
$744$	0	0
$745$	0	0
$746$	127.446i	0.170839i
$747$	0	0
$748$	46.5442	0.0622248
$749$	8.21407i	0.0109667i
$750$	0	0
$751$	109.823	0.146236	0.0731181	−	0.997323i	$-0.476705\pi$
$751$	109.823	0.146236	0.0731181	+	0.997323i	$0.476705\pi$
$752$	− 26.0589i	− 0.0346528i
$753$	0	0
$754$	−34.6476	−0.0459518
$755$	0	0
$756$	0	0
$757$	−305.573	−0.403663	−0.201831	−	0.979420i	$-0.564689\pi$
$757$	−305.573	−0.403663	−0.201831	+	0.979420i	$0.564689\pi$
$758$	− 462.520i	− 0.610184i
$759$	0	0
$760$	0	0
$761$	1323.79i	1.73953i	0.493462	+	0.869767i	$0.335731\pi$
$761$	1323.79i	1.73953i	−0.493462	+	0.869767i	$0.664269\pi$
$762$	0	0
$763$	−4.64589	−0.00608897
$764$	− 596.881i	− 0.781258i
$765$	0	0
$766$	662.330	0.864661
$767$	− 296.044i	− 0.385976i
$768$	0	0
$769$	−1383.79	−1.79947	−0.899735	−	0.436436i	$-0.856241\pi$
$769$	−1383.79	−1.79947	−0.899735	+	0.436436i	$0.856241\pi$
$770$	0	0
$771$	0	0
$772$	−133.823	−0.173346
$773$	− 1058.56i	− 1.36942i	−0.728818	−	0.684708i	$-0.759930\pi$
$773$	− 1058.56i	− 1.36942i	0.728818	−	0.684708i	$-0.240070\pi$
$774$	0	0
$775$	0	0
$776$	− 313.289i	− 0.403723i
$777$	0	0
$778$	113.677	0.146114
$779$	513.235i	0.658838i
$780$	0	0
$781$	−217.544	−0.278546
$782$	− 71.4701i	− 0.0913940i
$783$	0	0
$784$	−195.765	−0.249700
$785$	0	0
$786$	0	0
$787$	−1092.76	−1.38852	−0.694259	−	0.719725i	$-0.744268\pi$
$787$	−1092.76	−1.38852	−0.694259	+	0.719725i	$0.744268\pi$
$788$	− 244.118i	− 0.309794i
$789$	0	0
$790$	0	0
$791$	6.49957i	0.00821690i
$792$	0	0
$793$	199.942	0.252134
$794$	− 692.444i	− 0.872095i
$795$	0	0
$796$	−189.574	−0.238158
$797$	464.029i	0.582219i	0.956690	+	0.291110i	$0.0940244\pi$
$797$	464.029i	0.582219i	−0.956690	+	0.291110i	$0.905976\pi$
$798$	0	0
$799$	−50.5370	−0.0632503
$800$	0	0
$801$	0	0
$802$	−372.530	−0.464501
$803$	320.735i	0.399421i
$804$	0	0
$805$	0	0
$806$	− 143.990i	− 0.178648i
$807$	0	0
$808$	−212.235	−0.262668
$809$	1324.26i	1.63691i	0.574567	+	0.818457i	$0.305170\pi$
$809$	1324.26i	1.63691i	−0.574567	+	0.818457i	$0.694830\pi$
$810$	0	0
$811$	527.433	0.650349	0.325174	−	0.945654i	$-0.394577\pi$
$811$	527.433	0.650349	0.325174	+	0.945654i	$0.394577\pi$
$812$	− 3.41125i	− 0.00420105i
$813$	0	0
$814$	154.669	0.190011
$815$	0	0
$816$	0	0
$817$	−98.9403	−0.121102
$818$	− 240.998i	− 0.294618i
$819$	0	0
$820$	0	0
$821$	718.169i	0.874750i	0.899279	+	0.437375i	$0.144092\pi$
$821$	718.169i	0.874750i	−0.899279	+	0.437375i	$0.855908\pi$
$822$	0	0
$823$	−864.617	−1.05057	−0.525284	−	0.850927i	$-0.676041\pi$
$823$	−864.617	−1.05057	−0.525284	+	0.850927i	$0.676041\pi$
$824$	− 172.804i	− 0.209714i
$825$	0	0
$826$	29.1472	0.0352872
$827$	555.323i	0.671491i	0.941953	+	0.335745i	$0.108988\pi$
$827$	555.323i	0.671491i	−0.941953	+	0.335745i	$0.891012\pi$
$828$	0	0
$829$	−347.721	−0.419446	−0.209723	−	0.977761i	$-0.567256\pi$
$829$	−347.721	−0.419446	−0.209723	+	0.977761i	$0.567256\pi$
$830$	0	0
$831$	0	0
$832$	−27.8823	−0.0335123
$833$	379.654i	0.455767i
$834$	0	0
$835$	0	0
$836$	− 53.8234i	− 0.0643820i
$837$	0	0
$838$	−500.132	−0.596816
$839$	529.721i	0.631372i	0.948864	+	0.315686i	$0.102235\pi$
$839$	529.721i	0.631372i	−0.948864	+	0.315686i	$0.897765\pi$
$840$	0	0
$841$	791.587	0.941245
$842$	935.855i	1.11147i
$843$	0	0
$844$	−331.411	−0.392667
$845$	0	0
$846$	0	0
$847$	−27.1758	−0.0320847
$848$	107.647i	0.126942i
$849$	0	0
$850$	0	0
$851$	− 237.500i	− 0.279083i
$852$	0	0
$853$	−917.043	−1.07508	−0.537540	−	0.843238i	$-0.680646\pi$
$853$	−917.043	−1.07508	−0.537540	+	0.843238i	$0.680646\pi$
$854$	19.6854i	0.0230508i
$855$	0	0
$856$	95.7502	0.111858
$857$	− 287.294i	− 0.335232i	−0.985852	−	0.167616i	$-0.946393\pi$
$857$	− 287.294i	− 0.335232i	0.985852	−	0.167616i	$-0.0536068\pi$
$858$	0	0
$859$	82.5584	0.0961099	0.0480550	−	0.998845i	$-0.484698\pi$
$859$	82.5584	0.0961099	0.0480550	+	0.998845i	$0.484698\pi$
$860$	0	0
$861$	0	0
$862$	−1071.33	−1.24284
$863$	1214.56i	1.40737i	0.710513	+	0.703684i	$0.248463\pi$
$863$	1214.56i	1.40737i	−0.710513	+	0.703684i	$0.751537\pi$
$864$	0	0
$865$	0	0
$866$	− 414.781i	− 0.478962i
$867$	0	0
$868$	14.1766	0.0163325
$869$	234.551i	0.269909i
$870$	0	0
$871$	−338.405	−0.388525
$872$	54.1564i	0.0621060i
$873$	0	0
$874$	−82.6476	−0.0945625
$875$	0	0
$876$	0	0
$877$	−1489.63	−1.69855	−0.849277	−	0.527948i	$-0.822962\pi$
$877$	−1489.63	−1.69855	−0.849277	+	0.527948i	$0.822962\pi$
$878$	− 624.843i	− 0.711666i
$879$	0	0
$880$	0	0
$881$	1438.94i	1.63330i	0.577131	+	0.816652i	$0.304172\pi$
$881$	1438.94i	1.63330i	−0.577131	+	0.816652i	$0.695828\pi$
$882$	0	0
$883$	−684.139	−0.774790	−0.387395	−	0.921914i	$-0.626625\pi$
$883$	−684.139	−0.774790	−0.387395	+	0.921914i	$0.626625\pi$
$884$	54.0732i	0.0611687i
$885$	0	0
$886$	65.5950	0.0740350
$887$	− 1005.57i	− 1.13368i	−0.823828	−	0.566839i	$-0.808166\pi$
$887$	− 1005.57i	− 1.13368i	0.823828	−	0.566839i	$-0.191834\pi$
$888$	0	0
$889$	42.4424	0.0477418
$890$	0	0
$891$	0	0
$892$	−401.941	−0.450607
$893$	58.4407i	0.0654431i
$894$	0	0
$895$	0	0
$896$	− 2.74517i	− 0.00306380i
$897$	0	0
$898$	−97.8823	−0.109000
$899$	− 205.352i	− 0.228423i
$900$	0	0
$901$	208.764	0.231702
$902$	242.735i	0.269108i
$903$	0	0
$904$	75.7645	0.0838103
$905$	0	0
$906$	0	0
$907$	73.4416	0.0809720	0.0404860	−	0.999180i	$-0.487109\pi$
$907$	73.4416	0.0809720	0.0404860	+	0.999180i	$0.487109\pi$
$908$	− 698.881i	− 0.769693i
$909$	0	0
$910$	0	0
$911$	− 1557.69i	− 1.70987i	−0.518738	−	0.854933i	$-0.673598\pi$
$911$	− 1557.69i	− 1.70987i	0.518738	−	0.854933i	$-0.326402\pi$
$912$	0	0
$913$	−77.6468	−0.0850457
$914$	− 267.826i	− 0.293027i
$915$	0	0
$916$	345.029	0.376670
$917$	36.3318i	0.0396203i
$918$	0	0
$919$	−635.322	−0.691319	−0.345659	−	0.938360i	$-0.612345\pi$
$919$	−635.322	−0.691319	−0.345659	+	0.938360i	$0.612345\pi$
$920$	0	0
$921$	0	0
$922$	869.823	0.943409
$923$	− 252.734i	− 0.273818i
$924$	0	0
$925$	0	0
$926$	193.872i	0.209365i
$927$	0	0
$928$	−39.7645	−0.0428497
$929$	1051.26i	1.13160i	0.824542	+	0.565801i	$0.191433\pi$
$929$	1051.26i	1.13160i	−0.824542	+	0.565801i	$0.808567\pi$
$930$	0	0
$931$	439.029	0.471568
$932$	296.485i	0.318117i
$933$	0	0
$934$	−476.756	−0.510446
$935$	0	0
$936$	0	0
$937$	−782.381	−0.834985	−0.417493	−	0.908680i	$-0.637091\pi$
$937$	−782.381	−0.834985	−0.417493	+	0.908680i	$0.637091\pi$
$938$	− 33.3179i	− 0.0355201i
$939$	0	0
$940$	0	0
$941$	1085.14i	1.15318i	0.817035	+	0.576588i	$0.195616\pi$
$941$	1085.14i	1.15318i	−0.817035	+	0.576588i	$0.804384\pi$
$942$	0	0
$943$	372.728	0.395258
$944$	− 339.765i	− 0.359920i
$945$	0	0
$946$	−46.7939	−0.0494651
$947$	− 1358.56i	− 1.43459i	−0.696769	−	0.717296i	$-0.745380\pi$
$947$	− 1358.56i	− 1.43459i	0.696769	−	0.717296i	$-0.254620\pi$
$948$	0	0
$949$	−372.617	−0.392642
$950$	0	0
$951$	0	0
$952$	−5.32381	−0.00559224
$953$	− 790.648i	− 0.829641i	−0.909903	−	0.414820i	$-0.863844\pi$
$953$	− 790.648i	− 0.829641i	0.909903	−	0.414820i	$-0.136156\pi$
$954$	0	0
$955$	0	0
$956$	283.206i	0.296241i
$957$	0	0
$958$	−321.941	−0.336055
$959$	− 46.7939i	− 0.0487945i
$960$	0	0
$961$	−107.589	−0.111955
$962$	179.688i	0.186786i
$963$	0	0
$964$	−167.706	−0.173968
$965$	0	0
$966$	0	0
$967$	−1062.23	−1.09848	−0.549242	−	0.835663i	$-0.685083\pi$
$967$	−1062.23	−1.09848	−0.549242	+	0.835663i	$0.685083\pi$
$968$	316.784i	0.327256i
$969$	0	0
$970$	0	0
$971$	− 194.500i	− 0.200309i	−0.994972	−	0.100155i	$-0.968066\pi$
$971$	− 194.500i	− 0.200309i	0.994972	−	0.100155i	$-0.0319338\pi$
$972$	0	0
$973$	28.4121	0.0292005
$974$	− 906.333i	− 0.930527i
$975$	0	0
$976$	229.470	0.235113
$977$	715.882i	0.732735i	0.930470	+	0.366368i	$0.119399\pi$
$977$	715.882i	0.732735i	−0.930470	+	0.366368i	$0.880601\pi$
$978$	0	0
$979$	−95.2721	−0.0973157
$980$	0	0
$981$	0	0
$982$	−862.837	−0.878653
$983$	1679.98i	1.70904i	0.519420	+	0.854519i	$0.326148\pi$
$983$	1679.98i	1.70904i	−0.519420	+	0.854519i	$0.673852\pi$
$984$	0	0
$985$	0	0
$986$	77.1169i	0.0782118i
$987$	0	0
$988$	62.5299	0.0632893
$989$	71.8537i	0.0726529i
$990$	0	0
$991$	655.294	0.661245	0.330622	−	0.943763i	$-0.392741\pi$
$991$	655.294	0.661245	0.330622	+	0.943763i	$0.392741\pi$
$992$	− 165.255i	− 0.166588i
$993$	0	0
$994$	24.8831	0.0250333
$995$	0	0
$996$	0	0
$997$	−1169.57	−1.17309	−0.586546	−	0.809916i	$-0.699513\pi$
$997$	−1169.57	−1.17309	−0.586546	+	0.809916i	$0.699513\pi$
$998$	− 61.2263i	− 0.0613490i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1350.3.d.n.701.1	yes	4
3.2	odd	2	inner	1350.3.d.n.701.3	yes	4
5.2	odd	4		1350.3.b.f.1349.3		4
5.3	odd	4		1350.3.b.a.1349.2		4
5.4	even	2		1350.3.d.l.701.4	yes	4
15.2	even	4		1350.3.b.a.1349.1		4
15.8	even	4		1350.3.b.f.1349.4		4
15.14	odd	2		1350.3.d.l.701.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1350.3.b.a.1349.1		4	15.2	even	4
1350.3.b.a.1349.2		4	5.3	odd	4
1350.3.b.f.1349.3		4	5.2	odd	4
1350.3.b.f.1349.4		4	15.8	even	4
1350.3.d.l.701.2	✓	4	15.14	odd	2
1350.3.d.l.701.4	yes	4	5.4	even	2
1350.3.d.n.701.1	yes	4	1.1	even	1	trivial
1350.3.d.n.701.3	yes	4	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1350.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -2.00000 q^{4} -0.242641 q^{7} +2.82843i q^{8} +O(q^{10})\)	\(q-1.41421i q^{2} -2.00000 q^{4} -0.242641 q^{7} +2.82843i q^{8} -3.00000i q^{11} +3.48528 q^{13} +0.343146i q^{14} +4.00000 q^{16} -7.75736i q^{17} -8.97056 q^{19} -4.24264 q^{22} +6.51472i q^{23} -4.92893i q^{26} +0.485281 q^{28} -7.02944i q^{29} +29.2132 q^{31} -5.65685i q^{32} -10.9706 q^{34} -36.4558 q^{37} +12.6863i q^{38} -57.2132i q^{41} +11.0294 q^{43} +6.00000i q^{44} +9.21320 q^{46} -6.51472i q^{47} -48.9411 q^{49} -6.97056 q^{52} +26.9117i q^{53} -0.686292i q^{56} -9.94113 q^{58} -84.9411i q^{59} +57.3675 q^{61} -41.3137i q^{62} -8.00000 q^{64} -97.0955 q^{67} +15.5147i q^{68} -72.5147i q^{71} -106.912 q^{73} +51.5563i q^{74} +17.9411 q^{76} +0.727922i q^{77} -78.1838 q^{79} -80.9117 q^{82} -25.8823i q^{83} -15.5980i q^{86} +8.48528 q^{88} -31.7574i q^{89} -0.845671 q^{91} -13.0294i q^{92} -9.21320 q^{94} -110.765 q^{97} +69.2132i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4} + 16 q^{7}+O(q^{10}) \) 4 * q - 8 * q^4 + 16 * q^7	\( 4 q - 8 q^{4} + 16 q^{7} - 20 q^{13} + 16 q^{16} + 32 q^{19} - 32 q^{28} + 32 q^{31} + 24 q^{34} - 44 q^{37} + 112 q^{43} - 48 q^{46} - 60 q^{49} + 40 q^{52} + 96 q^{58} - 76 q^{61} - 32 q^{64} - 32 q^{67} - 224 q^{73} - 64 q^{76} - 160 q^{79} - 120 q^{82} - 224 q^{91} + 48 q^{94} + 100 q^{97}+O(q^{100}) \) 4 * q - 8 * q^4 + 16 * q^7 - 20 * q^13 + 16 * q^16 + 32 * q^19 - 32 * q^28 + 32 * q^31 + 24 * q^34 - 44 * q^37 + 112 * q^43 - 48 * q^46 - 60 * q^49 + 40 * q^52 + 96 * q^58 - 76 * q^61 - 32 * q^64 - 32 * q^67 - 224 * q^73 - 64 * q^76 - 160 * q^79 - 120 * q^82 - 224 * q^91 + 48 * q^94 + 100 * q^97

Introduction

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