LMFDB - Embedded newform 1350.3.d.l.701.3

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			701.3
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1350.701
Dual form			1350.3.d.l.701.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} -2.00000 q^{4} -8.24264 q^{7} -2.82843i q^{8} +O(q^{10})$	$q+1.41421i q^{2} -2.00000 q^{4} -8.24264 q^{7} -2.82843i q^{8} +3.00000i q^{11} +13.4853 q^{13} -11.6569i q^{14} +4.00000 q^{16} -16.2426i q^{17} +24.9706 q^{19} -4.24264 q^{22} +23.4853i q^{23} +19.0711i q^{26} +16.4853 q^{28} +40.9706i q^{29} -13.2132 q^{31} +5.65685i q^{32} +22.9706 q^{34} -14.4558 q^{37} +35.3137i q^{38} +14.7868i q^{41} -44.9706 q^{43} -6.00000i q^{44} -33.2132 q^{46} -23.4853i q^{47} +18.9411 q^{49} -26.9706 q^{52} -74.9117i q^{53} +23.3137i q^{56} -57.9411 q^{58} +17.0589i q^{59} -95.3675 q^{61} -18.6863i q^{62} -8.00000 q^{64} -81.0955 q^{67} +32.4853i q^{68} +89.4853i q^{71} +5.08831 q^{73} -20.4437i q^{74} -49.9411 q^{76} -24.7279i q^{77} -1.81623 q^{79} -20.9117 q^{82} +109.882i q^{83} -63.5980i q^{86} +8.48528 q^{88} +40.2426i q^{89} -111.154 q^{91} -46.9706i q^{92} +33.2132 q^{94} -160.765 q^{97} +26.7868i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	−8.24264	−1.17752	−0.588760	−	0.808308i	$-0.700384\pi$
$7$	−8.24264	−1.17752	−0.588760	+	0.808308i	$0.700384\pi$
$8$	− 2.82843i	− 0.353553i
$9$	0	0
$10$	0	0
$11$	3.00000i	0.272727i	0.990659	+	0.136364i	$0.0435416\pi$
$11$	3.00000i	0.272727i	−0.990659	+	0.136364i	$0.956458\pi$
$12$	0	0
$13$	13.4853	1.03733	0.518665	−	0.854978i	$-0.326429\pi$
$13$	13.4853	1.03733	0.518665	+	0.854978i	$0.326429\pi$
$14$	− 11.6569i	− 0.832632i
$15$	0	0
$16$	4.00000	0.250000
$17$	− 16.2426i	− 0.955449i	−0.878510	−	0.477725i	$-0.841462\pi$
$17$	− 16.2426i	− 0.955449i	0.878510	−	0.477725i	$-0.158538\pi$
$18$	0	0
$19$	24.9706	1.31424	0.657120	−	0.753786i	$-0.271774\pi$
$19$	24.9706	1.31424	0.657120	+	0.753786i	$0.271774\pi$
$20$	0	0
$21$	0	0
$22$	−4.24264	−0.192847
$23$	23.4853i	1.02110i	0.859848	+	0.510550i	$0.170558\pi$
$23$	23.4853i	1.02110i	−0.859848	+	0.510550i	$0.829442\pi$
$24$	0	0
$25$	0	0
$26$	19.0711i	0.733503i
$27$	0	0
$28$	16.4853	0.588760
$29$	40.9706i	1.41278i	0.707824	+	0.706389i	$0.249677\pi$
$29$	40.9706i	1.41278i	−0.707824	+	0.706389i	$0.750323\pi$
$30$	0	0
$31$	−13.2132	−0.426232	−0.213116	−	0.977027i	$-0.568361\pi$
$31$	−13.2132	−0.426232	−0.213116	+	0.977027i	$0.568361\pi$
$32$	5.65685i	0.176777i
$33$	0	0
$34$	22.9706	0.675605
$35$	0	0
$36$	0	0
$37$	−14.4558	−0.390698	−0.195349	−	0.980734i	$-0.562584\pi$
$37$	−14.4558	−0.390698	−0.195349	+	0.980734i	$0.562584\pi$
$38$	35.3137i	0.929308i
$39$	0	0
$40$	0	0
$41$	14.7868i	0.360654i	0.983607	+	0.180327i	$0.0577155\pi$
$41$	14.7868i	0.360654i	−0.983607	+	0.180327i	$0.942284\pi$
$42$	0	0
$43$	−44.9706	−1.04583	−0.522914	−	0.852386i	$-0.675155\pi$
$43$	−44.9706	−1.04583	−0.522914	+	0.852386i	$0.675155\pi$
$44$	− 6.00000i	− 0.136364i
$45$	0	0
$46$	−33.2132	−0.722026
$47$	− 23.4853i	− 0.499687i	−0.968286	−	0.249843i	$-0.919621\pi$
$47$	− 23.4853i	− 0.499687i	0.968286	−	0.249843i	$-0.0803791\pi$
$48$	0	0
$49$	18.9411	0.386554
$50$	0	0
$51$	0	0
$52$	−26.9706	−0.518665
$53$	− 74.9117i	− 1.41343i	−0.707499	−	0.706714i	$-0.750177\pi$
$53$	− 74.9117i	− 1.41343i	0.707499	−	0.706714i	$-0.249823\pi$
$54$	0	0
$55$	0	0
$56$	23.3137i	0.416316i
$57$	0	0
$58$	−57.9411	−0.998985
$59$	17.0589i	0.289133i	0.989495	+	0.144567i	$0.0461788\pi$
$59$	17.0589i	0.289133i	−0.989495	+	0.144567i	$0.953821\pi$
$60$	0	0
$61$	−95.3675	−1.56340	−0.781701	−	0.623653i	$-0.785648\pi$
$61$	−95.3675	−1.56340	−0.781701	+	0.623653i	$0.785648\pi$
$62$	− 18.6863i	− 0.301392i
$63$	0	0
$64$	−8.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	−81.0955	−1.21038	−0.605190	−	0.796081i	$-0.706903\pi$
$67$	−81.0955	−1.21038	−0.605190	+	0.796081i	$0.706903\pi$
$68$	32.4853i	0.477725i
$69$	0	0
$70$	0	0
$71$	89.4853i	1.26036i	0.776451	+	0.630178i	$0.217018\pi$
$71$	89.4853i	1.26036i	−0.776451	+	0.630178i	$0.782982\pi$
$72$	0	0
$73$	5.08831	0.0697029	0.0348515	−	0.999393i	$-0.488904\pi$
$73$	5.08831	0.0697029	0.0348515	+	0.999393i	$0.488904\pi$
$74$	− 20.4437i	− 0.276266i
$75$	0	0
$76$	−49.9411	−0.657120
$77$	− 24.7279i	− 0.321142i
$78$	0	0
$79$	−1.81623	−0.0229903	−0.0114952	−	0.999934i	$-0.503659\pi$
$79$	−1.81623	−0.0229903	−0.0114952	+	0.999934i	$0.503659\pi$
$80$	0	0
$81$	0	0
$82$	−20.9117	−0.255021
$83$	109.882i	1.32388i	0.749556	+	0.661941i	$0.230267\pi$
$83$	109.882i	1.32388i	−0.749556	+	0.661941i	$0.769733\pi$
$84$	0	0
$85$	0	0
$86$	− 63.5980i	− 0.739511i
$87$	0	0
$88$	8.48528	0.0964237
$89$	40.2426i	0.452165i	0.974108	+	0.226082i	$0.0725918\pi$
$89$	40.2426i	0.452165i	−0.974108	+	0.226082i	$0.927408\pi$
$90$	0	0
$91$	−111.154	−1.22148
$92$	− 46.9706i	− 0.510550i
$93$	0	0
$94$	33.2132	0.353332
$95$	0	0
$96$	0	0
$97$	−160.765	−1.65737	−0.828683	−	0.559718i	$-0.810909\pi$
$97$	−160.765	−1.65737	−0.828683	+	0.559718i	$0.810909\pi$
$98$	26.7868i	0.273335i
$99$	0	0
$100$	0	0
$101$	171.037i	1.69343i	0.532046	+	0.846716i	$0.321424\pi$
$101$	171.037i	1.69343i	−0.532046	+	0.846716i	$0.678576\pi$
$102$	0	0
$103$	−117.095	−1.13685	−0.568425	−	0.822735i	$-0.692447\pi$
$103$	−117.095	−1.13685	−0.568425	+	0.822735i	$0.692447\pi$
$104$	− 38.1421i	− 0.366751i
$105$	0	0
$106$	105.941	0.999445
$107$	135.853i	1.26965i	0.772655	+	0.634826i	$0.218928\pi$
$107$	135.853i	1.26965i	−0.772655	+	0.634826i	$0.781072\pi$
$108$	0	0
$109$	188.853	1.73259	0.866297	−	0.499529i	$-0.166493\pi$
$109$	188.853	1.73259	0.866297	+	0.499529i	$0.166493\pi$
$110$	0	0
$111$	0	0
$112$	−32.9706	−0.294380
$113$	− 69.2132i	− 0.612506i	−0.951950	−	0.306253i	$-0.900925\pi$
$113$	− 69.2132i	− 0.612506i	0.951950	−	0.306253i	$-0.0990754\pi$
$114$	0	0
$115$	0	0
$116$	− 81.9411i	− 0.706389i
$117$	0	0
$118$	−24.1249	−0.204448
$119$	133.882i	1.12506i
$120$	0	0
$121$	112.000	0.925620
$122$	− 134.870i	− 1.10549i
$123$	0	0
$124$	26.4264	0.213116
$125$	0	0
$126$	0	0
$127$	−206.919	−1.62928	−0.814641	−	0.579965i	$-0.803066\pi$
$127$	−206.919	−1.62928	−0.814641	+	0.579965i	$0.803066\pi$
$128$	− 11.3137i	− 0.0883883i
$129$	0	0
$130$	0	0
$131$	− 155.735i	− 1.18882i	−0.804163	−	0.594409i	$-0.797386\pi$
$131$	− 155.735i	− 1.18882i	0.804163	−	0.594409i	$-0.202614\pi$
$132$	0	0
$133$	−205.823	−1.54754
$134$	− 114.686i	− 0.855868i
$135$	0	0
$136$	−45.9411	−0.337802
$137$	23.1472i	0.168958i	0.996425	+	0.0844788i	$0.0269225\pi$
$137$	23.1472i	0.168958i	−0.996425	+	0.0844788i	$0.973077\pi$
$138$	0	0
$139$	61.0955	0.439536	0.219768	−	0.975552i	$-0.429470\pi$
$139$	61.0955	0.439536	0.219768	+	0.975552i	$0.429470\pi$
$140$	0	0
$141$	0	0
$142$	−126.551	−0.891206
$143$	40.4558i	0.282908i
$144$	0	0
$145$	0	0
$146$	7.19596i	0.0492874i
$147$	0	0
$148$	28.9117	0.195349
$149$	− 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
$151$	29.0883	0.192638	0.0963189	−	0.995351i	$-0.469293\pi$
$151$	29.0883	0.192638	0.0963189	+	0.995351i	$0.469293\pi$
$152$	− 70.6274i	− 0.464654i
$153$	0	0
$154$	34.9706	0.227082
$155$	0	0
$156$	0	0
$157$	−185.706	−1.18284	−0.591419	−	0.806364i	$-0.701432\pi$
$157$	−185.706	−1.18284	−0.591419	+	0.806364i	$0.701432\pi$
$158$	− 2.56854i	− 0.0162566i
$159$	0	0
$160$	0	0
$161$	− 193.581i	− 1.20236i
$162$	0	0
$163$	241.698	1.48281	0.741406	−	0.671056i	$-0.234159\pi$
$163$	241.698	1.48281	0.741406	+	0.671056i	$0.234159\pi$
$164$	− 29.5736i	− 0.180327i
$165$	0	0
$166$	−155.397	−0.936126
$167$	286.279i	1.71425i	0.515111	+	0.857123i	$0.327751\pi$
$167$	286.279i	1.71425i	−0.515111	+	0.857123i	$0.672249\pi$
$168$	0	0
$169$	12.8528	0.0760522
$170$	0	0
$171$	0	0
$172$	89.9411	0.522914
$173$	− 220.846i	− 1.27656i	−0.769802	−	0.638282i	$-0.779645\pi$
$173$	− 220.846i	− 1.27656i	0.769802	−	0.638282i	$-0.220355\pi$
$174$	0	0
$175$	0	0
$176$	12.0000i	0.0681818i
$177$	0	0
$178$	−56.9117	−0.319729
$179$	245.912i	1.37381i	0.726748	+	0.686904i	$0.241031\pi$
$179$	245.912i	1.37381i	−0.726748	+	0.686904i	$0.758969\pi$
$180$	0	0
$181$	−267.309	−1.47684	−0.738422	−	0.674339i	$-0.764429\pi$
$181$	−267.309	−1.47684	−0.738422	+	0.674339i	$0.764429\pi$
$182$	− 157.196i	− 0.863714i
$183$	0	0
$184$	66.4264	0.361013
$185$	0	0
$186$	0	0
$187$	48.7279	0.260577
$188$	46.9706i	0.249843i
$189$	0	0
$190$	0	0
$191$	346.441i	1.81383i	0.421318	+	0.906913i	$0.361568\pi$
$191$	346.441i	1.81383i	−0.421318	+	0.906913i	$0.638432\pi$
$192$	0	0
$193$	34.9117	0.180890	0.0904448	−	0.995901i	$-0.471171\pi$
$193$	34.9117	0.180890	0.0904448	+	0.995901i	$0.471171\pi$
$194$	− 227.355i	− 1.17193i
$195$	0	0
$196$	−37.8823	−0.193277
$197$	189.941i	0.964168i	0.876125	+	0.482084i	$0.160120\pi$
$197$	189.941i	0.964168i	−0.876125	+	0.482084i	$0.839880\pi$
$198$	0	0
$199$	137.213	0.689514	0.344757	−	0.938692i	$-0.387961\pi$
$199$	137.213	0.689514	0.344757	+	0.938692i	$0.387961\pi$
$200$	0	0
$201$	0	0
$202$	−241.882	−1.19744
$203$	− 337.706i	− 1.66357i
$204$	0	0
$205$	0	0
$206$	− 165.598i	− 0.803874i
$207$	0	0
$208$	53.9411	0.259332
$209$	74.9117i	0.358429i
$210$	0	0
$211$	−173.706	−0.823249	−0.411625	−	0.911353i	$-0.635039\pi$
$211$	−173.706	−0.823249	−0.411625	+	0.911353i	$0.635039\pi$
$212$	149.823i	0.706714i
$213$	0	0
$214$	−192.125	−0.897780
$215$	0	0
$216$	0	0
$217$	108.912	0.501897
$218$	267.078i	1.22513i
$219$	0	0
$220$	0	0
$221$	− 219.037i	− 0.991116i
$222$	0	0
$223$	−167.029	−0.749011	−0.374505	−	0.927225i	$-0.622188\pi$
$223$	−167.029	−0.749011	−0.374505	+	0.927225i	$0.622188\pi$
$224$	− 46.6274i	− 0.208158i
$225$	0	0
$226$	97.8823	0.433107
$227$	− 295.441i	− 1.30150i	−0.759292	−	0.650750i	$-0.774454\pi$
$227$	− 295.441i	− 1.30150i	0.759292	−	0.650750i	$-0.225546\pi$
$228$	0	0
$229$	−189.485	−0.827447	−0.413723	−	0.910403i	$-0.635772\pi$
$229$	−189.485	−0.827447	−0.413723	+	0.910403i	$0.635772\pi$
$230$	0	0
$231$	0	0
$232$	115.882	0.499492
$233$	− 139.757i	− 0.599817i	−0.953968	−	0.299908i	$-0.903044\pi$
$233$	− 139.757i	− 0.599817i	0.953968	−	0.299908i	$-0.0969562\pi$
$234$	0	0
$235$	0	0
$236$	− 34.1177i	− 0.144567i
$237$	0	0
$238$	−189.338	−0.795538
$239$	260.397i	1.08953i	0.838590	+	0.544764i	$0.183381\pi$
$239$	260.397i	1.08953i	−0.838590	+	0.544764i	$0.816619\pi$
$240$	0	0
$241$	−85.8528	−0.356236	−0.178118	−	0.984009i	$-0.557001\pi$
$241$	−85.8528	−0.356236	−0.178118	+	0.984009i	$0.557001\pi$
$242$	158.392i	0.654512i
$243$	0	0
$244$	190.735	0.781701
$245$	0	0
$246$	0	0
$247$	336.735	1.36330
$248$	37.3726i	0.150696i
$249$	0	0
$250$	0	0
$251$	− 340.529i	− 1.35669i	−0.734744	−	0.678345i	$-0.762698\pi$
$251$	− 340.529i	− 1.35669i	0.734744	−	0.678345i	$-0.237302\pi$
$252$	0	0
$253$	−70.4558	−0.278482
$254$	− 292.627i	− 1.15208i
$255$	0	0
$256$	16.0000	0.0625000
$257$	375.889i	1.46260i	0.682053	+	0.731302i	$0.261087\pi$
$257$	375.889i	1.46260i	−0.682053	+	0.731302i	$0.738913\pi$
$258$	0	0
$259$	119.154	0.460055
$260$	0	0
$261$	0	0
$262$	220.243	0.840621
$263$	265.014i	1.00766i	0.863803	+	0.503829i	$0.168076\pi$
$263$	265.014i	1.00766i	−0.863803	+	0.503829i	$0.831924\pi$
$264$	0	0
$265$	0	0
$266$	− 291.078i	− 1.09428i
$267$	0	0
$268$	162.191	0.605190
$269$	− 157.706i	− 0.586266i	−0.956072	−	0.293133i	$-0.905302\pi$
$269$	− 157.706i	− 0.586266i	0.956072	−	0.293133i	$-0.0946979\pi$
$270$	0	0
$271$	−58.1766	−0.214674	−0.107337	−	0.994223i	$-0.534232\pi$
$271$	−58.1766	−0.214674	−0.107337	+	0.994223i	$0.534232\pi$
$272$	− 64.9706i	− 0.238862i
$273$	0	0
$274$	−32.7351	−0.119471
$275$	0	0
$276$	0	0
$277$	−170.118	−0.614143	−0.307072	−	0.951686i	$-0.599349\pi$
$277$	−170.118	−0.614143	−0.307072	+	0.951686i	$0.599349\pi$
$278$	86.4020i	0.310799i
$279$	0	0
$280$	0	0
$281$	222.323i	0.791185i	0.918426	+	0.395592i	$0.129461\pi$
$281$	222.323i	0.791185i	−0.918426	+	0.395592i	$0.870539\pi$
$282$	0	0
$283$	24.9706	0.0882352	0.0441176	−	0.999026i	$-0.485952\pi$
$283$	24.9706	0.0882352	0.0441176	+	0.999026i	$0.485952\pi$
$284$	− 178.971i	− 0.630178i
$285$	0	0
$286$	−57.2132	−0.200046
$287$	− 121.882i	− 0.424677i
$288$	0	0
$289$	25.1766	0.0871163
$290$	0	0
$291$	0	0
$292$	−10.1766	−0.0348515
$293$	− 506.683i	− 1.72929i	−0.502379	−	0.864647i	$-0.667542\pi$
$293$	− 506.683i	− 1.72929i	0.502379	−	0.864647i	$-0.332458\pi$
$294$	0	0
$295$	0	0
$296$	40.8873i	0.138133i
$297$	0	0
$298$	198.000	0.664430
$299$	316.706i	1.05922i
$300$	0	0
$301$	370.676	1.23148
$302$	41.1371i	0.136216i
$303$	0	0
$304$	99.8823	0.328560
$305$	0	0
$306$	0	0
$307$	−216.360	−0.704757	−0.352378	−	0.935858i	$-0.614627\pi$
$307$	−216.360	−0.704757	−0.352378	+	0.935858i	$0.614627\pi$
$308$	49.4558i	0.160571i
$309$	0	0
$310$	0	0
$311$	− 14.0437i	− 0.0451567i	−0.999745	−	0.0225783i	$-0.992812\pi$
$311$	− 14.0437i	− 0.0451567i	0.999745	−	0.0225783i	$-0.00718752\pi$
$312$	0	0
$313$	372.617	1.19047	0.595235	−	0.803551i	$-0.297059\pi$
$313$	372.617	1.19047	0.595235	+	0.803551i	$0.297059\pi$
$314$	− 262.627i	− 0.836393i
$315$	0	0
$316$	3.63247	0.0114952
$317$	− 60.4996i	− 0.190850i	−0.995437	−	0.0954252i	$-0.969579\pi$
$317$	− 60.4996i	− 0.190850i	0.995437	−	0.0954252i	$-0.0304211\pi$
$318$	0	0
$319$	−122.912	−0.385303
$320$	0	0
$321$	0	0
$322$	273.765	0.850200
$323$	− 405.588i	− 1.25569i
$324$	0	0
$325$	0	0
$326$	341.813i	1.04851i
$327$	0	0
$328$	41.8234	0.127510
$329$	193.581i	0.588391i
$330$	0	0
$331$	473.110	1.42933	0.714667	−	0.699465i	$-0.246578\pi$
$331$	473.110	1.42933	0.714667	+	0.699465i	$0.246578\pi$
$332$	− 219.765i	− 0.661941i
$333$	0	0
$334$	−404.860	−1.21216
$335$	0	0
$336$	0	0
$337$	615.529	1.82650	0.913248	−	0.407405i	$-0.133566\pi$
$337$	615.529	1.82650	0.913248	+	0.407405i	$0.133566\pi$
$338$	18.1766i	0.0537770i
$339$	0	0
$340$	0	0
$341$	− 39.6396i	− 0.116245i
$342$	0	0
$343$	247.765	0.722345
$344$	127.196i	0.369756i
$345$	0	0
$346$	312.323	0.902667
$347$	− 190.294i	− 0.548399i	−0.961673	−	0.274199i	$-0.911587\pi$
$347$	− 190.294i	− 0.548399i	0.961673	−	0.274199i	$-0.0884128\pi$
$348$	0	0
$349$	−273.470	−0.783582	−0.391791	−	0.920054i	$-0.628144\pi$
$349$	−273.470	−0.783582	−0.391791	+	0.920054i	$0.628144\pi$
$350$	0	0
$351$	0	0
$352$	−16.9706	−0.0482118
$353$	562.087i	1.59232i	0.605089	+	0.796158i	$0.293138\pi$
$353$	562.087i	1.59232i	−0.605089	+	0.796158i	$0.706862\pi$
$354$	0	0
$355$	0	0
$356$	− 80.4853i	− 0.226082i
$357$	0	0
$358$	−347.772	−0.971429
$359$	− 256.955i	− 0.715753i	−0.933769	−	0.357877i	$-0.883501\pi$
$359$	− 256.955i	− 0.715753i	0.933769	−	0.357877i	$-0.116499\pi$
$360$	0	0
$361$	262.529	0.727227
$362$	− 378.032i	− 1.04429i
$363$	0	0
$364$	222.309	0.610738
$365$	0	0
$366$	0	0
$367$	−190.177	−0.518192	−0.259096	−	0.965852i	$-0.583425\pi$
$367$	−190.177	−0.518192	−0.259096	+	0.965852i	$0.583425\pi$
$368$	93.9411i	0.255275i
$369$	0	0
$370$	0	0
$371$	617.470i	1.66434i
$372$	0	0
$373$	225.882	0.605582	0.302791	−	0.953057i	$-0.402082\pi$
$373$	225.882	0.605582	0.302791	+	0.953057i	$0.402082\pi$
$374$	68.9117i	0.184256i
$375$	0	0
$376$	−66.4264	−0.176666
$377$	552.500i	1.46552i
$378$	0	0
$379$	−479.051	−1.26399	−0.631993	−	0.774974i	$-0.717763\pi$
$379$	−479.051	−1.26399	−0.631993	+	0.774974i	$0.717763\pi$
$380$	0	0
$381$	0	0
$382$	−489.941	−1.28257
$383$	281.662i	0.735410i	0.929943	+	0.367705i	$0.119856\pi$
$383$	281.662i	0.735410i	−0.929943	+	0.367705i	$0.880144\pi$
$384$	0	0
$385$	0	0
$386$	49.3726i	0.127908i
$387$	0	0
$388$	321.529	0.828683
$389$	632.382i	1.62566i	0.582501	+	0.812830i	$0.302074\pi$
$389$	632.382i	1.62566i	−0.582501	+	0.812830i	$0.697926\pi$
$390$	0	0
$391$	381.463	0.975609
$392$	− 53.5736i	− 0.136667i
$393$	0	0
$394$	−268.617	−0.681770
$395$	0	0
$396$	0	0
$397$	511.632	1.28874	0.644372	−	0.764712i	$-0.277119\pi$
$397$	511.632	1.28874	0.644372	+	0.764712i	$0.277119\pi$
$398$	194.049i	0.487560i
$399$	0	0
$400$	0	0
$401$	− 695.418i	− 1.73421i	−0.498125	−	0.867105i	$-0.665978\pi$
$401$	− 695.418i	− 1.73421i	0.498125	−	0.867105i	$-0.334022\pi$
$402$	0	0
$403$	−178.184	−0.442143
$404$	− 342.073i	− 0.846716i
$405$	0	0
$406$	477.588	1.17632
$407$	− 43.3675i	− 0.106554i
$408$	0	0
$409$	−508.411	−1.24306	−0.621530	−	0.783391i	$-0.713488\pi$
$409$	−508.411	−1.24306	−0.621530	+	0.783391i	$0.713488\pi$
$410$	0	0
$411$	0	0
$412$	234.191	0.568425
$413$	− 140.610i	− 0.340460i
$414$	0	0
$415$	0	0
$416$	76.2843i	0.183376i
$417$	0	0
$418$	−105.941	−0.253448
$419$	− 53.6468i	− 0.128035i	−0.997949	−	0.0640176i	$-0.979609\pi$
$419$	− 53.6468i	− 0.128035i	0.997949	−	0.0640176i	$-0.0203914\pi$
$420$	0	0
$421$	203.749	0.483965	0.241983	−	0.970281i	$-0.422202\pi$
$421$	203.749	0.483965	0.241983	+	0.970281i	$0.422202\pi$
$422$	− 245.657i	− 0.582125i
$423$	0	0
$424$	−211.882	−0.499722
$425$	0	0
$426$	0	0
$427$	786.080	1.84094
$428$	− 271.706i	− 0.634826i
$429$	0	0
$430$	0	0
$431$	808.456i	1.87577i	0.346950	+	0.937884i	$0.387218\pi$
$431$	808.456i	1.87577i	−0.346950	+	0.937884i	$0.612782\pi$
$432$	0	0
$433$	−632.706	−1.46121	−0.730607	−	0.682798i	$-0.760763\pi$
$433$	−632.706	−1.46121	−0.730607	+	0.682798i	$0.760763\pi$
$434$	154.024i	0.354895i
$435$	0	0
$436$	−377.706	−0.866297
$437$	586.441i	1.34197i
$438$	0	0
$439$	−41.8305	−0.0952859	−0.0476430	−	0.998864i	$-0.515171\pi$
$439$	−41.8305	−0.0952859	−0.0476430	+	0.998864i	$0.515171\pi$
$440$	0	0
$441$	0	0
$442$	309.765	0.700825
$443$	487.617i	1.10072i	0.834929	+	0.550358i	$0.185509\pi$
$443$	487.617i	1.10072i	−0.834929	+	0.550358i	$0.814491\pi$
$444$	0	0
$445$	0	0
$446$	− 236.215i	− 0.529631i
$447$	0	0
$448$	65.9411	0.147190
$449$	26.7868i	0.0596588i	0.999555	+	0.0298294i	$0.00949640\pi$
$449$	26.7868i	0.0596588i	−0.999555	+	0.0298294i	$0.990504\pi$
$450$	0	0
$451$	−44.3604	−0.0983601
$452$	138.426i	0.306253i
$453$	0	0
$454$	417.816	0.920300
$455$	0	0
$456$	0	0
$457$	523.382	1.14526	0.572628	−	0.819815i	$-0.305924\pi$
$457$	523.382	1.14526	0.572628	+	0.819815i	$0.305924\pi$
$458$	− 267.973i	− 0.585093i
$459$	0	0
$460$	0	0
$461$	471.058i	1.02182i	0.859635	+	0.510909i	$0.170691\pi$
$461$	471.058i	1.02182i	−0.859635	+	0.510909i	$0.829309\pi$
$462$	0	0
$463$	238.912	0.516008	0.258004	−	0.966144i	$-0.416935\pi$
$463$	238.912	0.516008	0.258004	+	0.966144i	$0.416935\pi$
$464$	163.882i	0.353195i
$465$	0	0
$466$	197.647	0.424135
$467$	− 472.882i	− 1.01260i	−0.862359	−	0.506298i	$-0.831014\pi$
$467$	− 472.882i	− 1.01260i	0.862359	−	0.506298i	$-0.168986\pi$
$468$	0	0
$469$	668.441	1.42525
$470$	0	0
$471$	0	0
$472$	48.2498	0.102224
$473$	− 134.912i	− 0.285226i
$474$	0	0
$475$	0	0
$476$	− 267.765i	− 0.562530i
$477$	0	0
$478$	−368.257	−0.770412
$479$	− 179.647i	− 0.375045i	−0.982260	−	0.187523i	$-0.939954\pi$
$479$	− 179.647i	− 0.375045i	0.982260	−	0.187523i	$-0.0600458\pi$
$480$	0	0
$481$	−194.941	−0.405283
$482$	− 121.414i	− 0.251897i
$483$	0	0
$484$	−224.000	−0.462810
$485$	0	0
$486$	0	0
$487$	368.874	0.757442	0.378721	−	0.925511i	$-0.376364\pi$
$487$	368.874	0.757442	0.378721	+	0.925511i	$0.376364\pi$
$488$	269.740i	0.552746i
$489$	0	0
$490$	0	0
$491$	745.882i	1.51911i	0.650444	+	0.759554i	$0.274583\pi$
$491$	745.882i	1.51911i	−0.650444	+	0.759554i	$0.725417\pi$
$492$	0	0
$493$	665.470	1.34984
$494$	476.215i	0.963999i
$495$	0	0
$496$	−52.8528	−0.106558
$497$	− 737.595i	− 1.48409i
$498$	0	0
$499$	−771.294	−1.54568	−0.772839	−	0.634602i	$-0.781164\pi$
$499$	−771.294	−1.54568	−0.772839	+	0.634602i	$0.781164\pi$
$500$	0	0
$501$	0	0
$502$	481.581	0.959324
$503$	32.2355i	0.0640865i	0.999486	+	0.0320432i	$0.0102014\pi$
$503$	32.2355i	0.0640865i	−0.999486	+	0.0320432i	$0.989799\pi$
$504$	0	0
$505$	0	0
$506$	− 99.6396i	− 0.196916i
$507$	0	0
$508$	413.838	0.814641
$509$	− 31.7574i	− 0.0623917i	−0.999513	−	0.0311958i	$-0.990068\pi$
$509$	− 31.7574i	− 0.0623917i	0.999513	−	0.0311958i	$-0.00993155\pi$
$510$	0	0
$511$	−41.9411	−0.0820766
$512$	22.6274i	0.0441942i
$513$	0	0
$514$	−531.588	−1.03422
$515$	0	0
$516$	0	0
$517$	70.4558	0.136278
$518$	168.510i	0.325308i
$519$	0	0
$520$	0	0
$521$	− 487.404i	− 0.935517i	−0.883856	−	0.467758i	$-0.845062\pi$
$521$	− 487.404i	− 0.935517i	0.883856	−	0.467758i	$-0.154938\pi$
$522$	0	0
$523$	−304.353	−0.581937	−0.290969	−	0.956733i	$-0.593978\pi$
$523$	−304.353	−0.581937	−0.290969	+	0.956733i	$0.593978\pi$
$524$	311.470i	0.594409i
$525$	0	0
$526$	−374.787	−0.712522
$527$	214.617i	0.407243i
$528$	0	0
$529$	−22.5584	−0.0426436
$530$	0	0
$531$	0	0
$532$	411.647	0.773772
$533$	199.404i	0.374117i
$534$	0	0
$535$	0	0
$536$	229.373i	0.427934i
$537$	0	0
$538$	223.029	0.414553
$539$	56.8234i	0.105424i
$540$	0	0
$541$	112.574	0.208084	0.104042	−	0.994573i	$-0.466822\pi$
$541$	112.574	0.208084	0.104042	+	0.994573i	$0.466822\pi$
$542$	− 82.2742i	− 0.151797i
$543$	0	0
$544$	91.8823	0.168901
$545$	0	0
$546$	0	0
$547$	157.470	0.287880	0.143940	−	0.989586i	$-0.454023\pi$
$547$	157.470	0.287880	0.143940	+	0.989586i	$0.454023\pi$
$548$	− 46.2944i	− 0.0844788i
$549$	0	0
$550$	0	0
$551$	1023.06i	1.85673i
$552$	0	0
$553$	14.9706	0.0270715
$554$	− 240.583i	− 0.434265i
$555$	0	0
$556$	−122.191	−0.219768
$557$	− 156.978i	− 0.281827i	−0.990022	−	0.140914i	$-0.954996\pi$
$557$	− 156.978i	− 0.281827i	0.990022	−	0.140914i	$-0.0450040\pi$
$558$	0	0
$559$	−606.441	−1.08487
$560$	0	0
$561$	0	0
$562$	−314.412	−0.559452
$563$	− 81.3229i	− 0.144446i	−0.997389	−	0.0722229i	$-0.976991\pi$
$563$	− 81.3229i	− 0.144446i	0.997389	−	0.0722229i	$-0.0230093\pi$
$564$	0	0
$565$	0	0
$566$	35.3137i	0.0623917i
$567$	0	0
$568$	253.103	0.445603
$569$	− 479.022i	− 0.841867i	−0.907092	−	0.420933i	$-0.861703\pi$
$569$	− 479.022i	− 0.841867i	0.907092	−	0.420933i	$-0.138297\pi$
$570$	0	0
$571$	631.418	1.10581	0.552906	−	0.833244i	$-0.313519\pi$
$571$	631.418	1.10581	0.552906	+	0.833244i	$0.313519\pi$
$572$	− 80.9117i	− 0.141454i
$573$	0	0
$574$	172.368	0.300292
$575$	0	0
$576$	0	0
$577$	−945.912	−1.63936	−0.819681	−	0.572820i	$-0.805849\pi$
$577$	−945.912	−1.63936	−0.819681	+	0.572820i	$0.805849\pi$
$578$	35.6051i	0.0616006i
$579$	0	0
$580$	0	0
$581$	− 905.720i	− 1.55890i
$582$	0	0
$583$	224.735	0.385480
$584$	− 14.3919i	− 0.0246437i
$585$	0	0
$586$	716.558	1.22280
$587$	− 847.529i	− 1.44383i	−0.691981	−	0.721916i	$-0.743262\pi$
$587$	− 847.529i	− 1.44383i	0.691981	−	0.721916i	$-0.256738\pi$
$588$	0	0
$589$	−329.941	−0.560172
$590$	0	0
$591$	0	0
$592$	−57.8234	−0.0976746
$593$	268.118i	0.452138i	0.974111	+	0.226069i	$0.0725875\pi$
$593$	268.118i	0.452138i	−0.974111	+	0.226069i	$0.927413\pi$
$594$	0	0
$595$	0	0
$596$	280.014i	0.469823i
$597$	0	0
$598$	−447.889	−0.748979
$599$	− 817.176i	− 1.36423i	−0.731243	−	0.682117i	$-0.761060\pi$
$599$	− 817.176i	− 1.36423i	0.731243	−	0.682117i	$-0.238940\pi$
$600$	0	0
$601$	−602.706	−1.00284	−0.501419	−	0.865205i	$-0.667188\pi$
$601$	−602.706	−1.00284	−0.501419	+	0.865205i	$0.667188\pi$
$602$	524.215i	0.870790i
$603$	0	0
$604$	−58.1766	−0.0963189
$605$	0	0
$606$	0	0
$607$	842.551	1.38806	0.694029	−	0.719947i	$-0.255834\pi$
$607$	842.551	1.38806	0.694029	+	0.719947i	$0.255834\pi$
$608$	141.255i	0.232327i
$609$	0	0
$610$	0	0
$611$	− 316.706i	− 0.518340i
$612$	0	0
$613$	50.8377	0.0829326	0.0414663	−	0.999140i	$-0.486797\pi$
$613$	50.8377	0.0829326	0.0414663	+	0.999140i	$0.486797\pi$
$614$	− 305.980i	− 0.498338i
$615$	0	0
$616$	−69.9411	−0.113541
$617$	167.272i	0.271105i	0.990770	+	0.135553i	$0.0432810\pi$
$617$	167.272i	0.271105i	−0.990770	+	0.135553i	$0.956719\pi$
$618$	0	0
$619$	892.403	1.44169	0.720843	−	0.693099i	$-0.243755\pi$
$619$	892.403	1.44169	0.720843	+	0.693099i	$0.243755\pi$
$620$	0	0
$621$	0	0
$622$	19.8608	0.0319306
$623$	− 331.706i	− 0.532433i
$624$	0	0
$625$	0	0
$626$	526.960i	0.841790i
$627$	0	0
$628$	371.411	0.591419
$629$	234.801i	0.373293i
$630$	0	0
$631$	−95.0294	−0.150601	−0.0753007	−	0.997161i	$-0.523992\pi$
$631$	−95.0294	−0.150601	−0.0753007	+	0.997161i	$0.523992\pi$
$632$	5.13708i	0.00812830i
$633$	0	0
$634$	85.5593	0.134952
$635$	0	0
$636$	0	0
$637$	255.426	0.400983
$638$	− 173.823i	− 0.272450i
$639$	0	0
$640$	0	0
$641$	− 1129.23i	− 1.76167i	−0.473428	−	0.880833i	$-0.656983\pi$
$641$	− 1129.23i	− 1.76167i	0.473428	−	0.880833i	$-0.343017\pi$
$642$	0	0
$643$	1041.83	1.62027	0.810133	−	0.586247i	$-0.199395\pi$
$643$	1041.83	1.62027	0.810133	+	0.586247i	$0.199395\pi$
$644$	387.161i	0.601182i
$645$	0	0
$646$	573.588	0.887907
$647$	588.131i	0.909013i	0.890744	+	0.454506i	$0.150184\pi$
$647$	588.131i	0.909013i	−0.890744	+	0.454506i	$0.849816\pi$
$648$	0	0
$649$	−51.1766	−0.0788546
$650$	0	0
$651$	0	0
$652$	−483.397	−0.741406
$653$	− 127.861i	− 0.195805i	−0.995196	−	0.0979026i	$-0.968787\pi$
$653$	− 127.861i	− 0.195805i	0.995196	−	0.0979026i	$-0.0312134\pi$
$654$	0	0
$655$	0	0
$656$	59.1472i	0.0901634i
$657$	0	0
$658$	−273.765	−0.416055
$659$	1151.79i	1.74779i	0.486115	+	0.873895i	$0.338413\pi$
$659$	1151.79i	1.74779i	−0.486115	+	0.873895i	$0.661587\pi$
$660$	0	0
$661$	492.986	0.745818	0.372909	−	0.927868i	$-0.378360\pi$
$661$	492.986	0.745818	0.372909	+	0.927868i	$0.378360\pi$
$662$	669.078i	1.01069i
$663$	0	0
$664$	310.794	0.468063
$665$	0	0
$666$	0	0
$667$	−962.205	−1.44259
$668$	− 572.558i	− 0.857123i
$669$	0	0
$670$	0	0
$671$	− 286.103i	− 0.426382i
$672$	0	0
$673$	620.088	0.921379	0.460690	−	0.887561i	$-0.347602\pi$
$673$	620.088	0.921379	0.460690	+	0.887561i	$0.347602\pi$
$674$	870.489i	1.29153i
$675$	0	0
$676$	−25.7056	−0.0380261
$677$	733.581i	1.08358i	0.840515	+	0.541788i	$0.182252\pi$
$677$	733.581i	1.08358i	−0.840515	+	0.541788i	$0.817748\pi$
$678$	0	0
$679$	1325.12	1.95158
$680$	0	0
$681$	0	0
$682$	56.0589	0.0821978
$683$	810.588i	1.18681i	0.804906	+	0.593403i	$0.202216\pi$
$683$	810.588i	1.18681i	−0.804906	+	0.593403i	$0.797784\pi$
$684$	0	0
$685$	0	0
$686$	350.392i	0.510775i
$687$	0	0
$688$	−179.882	−0.261457
$689$	− 1010.21i	− 1.46619i
$690$	0	0
$691$	−665.477	−0.963064	−0.481532	−	0.876428i	$-0.659919\pi$
$691$	−665.477	−0.963064	−0.481532	+	0.876428i	$0.659919\pi$
$692$	441.691i	0.638282i
$693$	0	0
$694$	269.117	0.387776
$695$	0	0
$696$	0	0
$697$	240.177	0.344586
$698$	− 386.745i	− 0.554076i
$699$	0	0
$700$	0	0
$701$	− 1210.32i	− 1.72656i	−0.504729	−	0.863278i	$-0.668407\pi$
$701$	− 1210.32i	− 1.72656i	0.504729	−	0.863278i	$-0.331593\pi$
$702$	0	0
$703$	−360.971	−0.513472
$704$	− 24.0000i	− 0.0340909i
$705$	0	0
$706$	−794.912	−1.12594
$707$	− 1409.79i	− 1.99405i
$708$	0	0
$709$	−65.8671	−0.0929014	−0.0464507	−	0.998921i	$-0.514791\pi$
$709$	−65.8671	−0.0929014	−0.0464507	+	0.998921i	$0.514791\pi$
$710$	0	0
$711$	0	0
$712$	113.823	0.159864
$713$	− 310.316i	− 0.435226i
$714$	0	0
$715$	0	0
$716$	− 491.823i	− 0.686904i
$717$	0	0
$718$	363.390	0.506114
$719$	− 11.8385i	− 0.0164653i	−0.999966	−	0.00823263i	$-0.997379\pi$
$719$	− 11.8385i	− 0.0164653i	0.999966	−	0.00823263i	$-0.00262056\pi$
$720$	0	0
$721$	965.176	1.33866
$722$	371.272i	0.514227i
$723$	0	0
$724$	534.617	0.738422
$725$	0	0
$726$	0	0
$727$	−171.294	−0.235617	−0.117808	−	0.993036i	$-0.537587\pi$
$727$	−171.294	−0.235617	−0.117808	+	0.993036i	$0.537587\pi$
$728$	314.392i	0.431857i
$729$	0	0
$730$	0	0
$731$	730.441i	0.999235i
$732$	0	0
$733$	−197.191	−0.269019	−0.134509	−	0.990912i	$-0.542946\pi$
$733$	−197.191	−0.269019	−0.134509	+	0.990912i	$0.542946\pi$
$734$	− 268.950i	− 0.366417i
$735$	0	0
$736$	−132.853	−0.180507
$737$	− 243.286i	− 0.330104i
$738$	0	0
$739$	140.728	0.190430	0.0952151	−	0.995457i	$-0.469646\pi$
$739$	140.728	0.190430	0.0952151	+	0.995457i	$0.469646\pi$
$740$	0	0
$741$	0	0
$742$	−873.235	−1.17687
$743$	399.073i	0.537111i	0.963264	+	0.268555i	$0.0865462\pi$
$743$	399.073i	0.537111i	−0.963264	+	0.268555i	$0.913454\pi$
$744$	0	0
$745$	0	0
$746$	319.446i	0.428211i
$747$	0	0
$748$	−97.4558	−0.130289
$749$	− 1119.79i	− 1.49504i
$750$	0	0
$751$	−93.8234	−0.124931	−0.0624656	−	0.998047i	$-0.519896\pi$
$751$	−93.8234	−0.124931	−0.0624656	+	0.998047i	$0.519896\pi$
$752$	− 93.9411i	− 0.124922i
$753$	0	0
$754$	−781.352	−1.03628
$755$	0	0
$756$	0	0
$757$	−763.573	−1.00868	−0.504341	−	0.863504i	$-0.668265\pi$
$757$	−763.573	−1.00868	−0.504341	+	0.863504i	$0.668265\pi$
$758$	− 677.480i	− 0.893773i
$759$	0	0
$760$	0	0
$761$	− 212.214i	− 0.278862i	−0.990232	−	0.139431i	$-0.955473\pi$
$761$	− 212.214i	− 0.278862i	0.990232	−	0.139431i	$-0.0445274\pi$
$762$	0	0
$763$	−1556.65	−2.04016
$764$	− 692.881i	− 0.906913i
$765$	0	0
$766$	−398.330	−0.520013
$767$	230.044i	0.299927i
$768$	0	0
$769$	7.79307	0.0101340	0.00506702	−	0.999987i	$-0.498387\pi$
$769$	7.79307	0.0101340	0.00506702	+	0.999987i	$0.498387\pi$
$770$	0	0
$771$	0	0
$772$	−69.8234	−0.0904448
$773$	− 549.442i	− 0.710791i	−0.934716	−	0.355396i	$-0.884346\pi$
$773$	− 549.442i	− 0.710791i	0.934716	−	0.355396i	$-0.115654\pi$
$774$	0	0
$775$	0	0
$776$	454.711i	0.585967i
$777$	0	0
$778$	−894.323	−1.14952
$779$	369.235i	0.473985i
$780$	0	0
$781$	−268.456	−0.343733
$782$	539.470i	0.689860i
$783$	0	0
$784$	75.7645	0.0966384
$785$	0	0
$786$	0	0
$787$	−332.764	−0.422825	−0.211413	−	0.977397i	$-0.567806\pi$
$787$	−332.764	−0.422825	−0.211413	+	0.977397i	$0.567806\pi$
$788$	− 379.882i	− 0.482084i
$789$	0	0
$790$	0	0
$791$	570.500i	0.721238i
$792$	0	0
$793$	−1286.06	−1.62176
$794$	723.556i	0.911280i
$795$	0	0
$796$	−274.426	−0.344757
$797$	− 656.029i	− 0.823122i	−0.911382	−	0.411561i	$-0.864984\pi$
$797$	− 656.029i	− 0.823122i	0.911382	−	0.411561i	$-0.135016\pi$
$798$	0	0
$799$	−381.463	−0.477426
$800$	0	0
$801$	0	0
$802$	983.470	1.22627
$803$	15.2649i	0.0190099i
$804$	0	0
$805$	0	0
$806$	− 251.990i	− 0.312643i
$807$	0	0
$808$	483.765	0.598718
$809$	− 475.736i	− 0.588054i	−0.955797	−	0.294027i	$-0.905004\pi$
$809$	− 475.736i	− 0.588054i	0.955797	−	0.294027i	$-0.0949956\pi$
$810$	0	0
$811$	−991.433	−1.22248	−0.611241	−	0.791445i	$-0.709329\pi$
$811$	−991.433	−1.22248	−0.611241	+	0.791445i	$0.709329\pi$
$812$	675.411i	0.831787i
$813$	0	0
$814$	61.3310	0.0753452
$815$	0	0
$816$	0	0
$817$	−1122.94	−1.37447
$818$	− 719.002i	− 0.878976i
$819$	0	0
$820$	0	0
$821$	− 1201.83i	− 1.46386i	−0.681379	−	0.731931i	$-0.738619\pi$
$821$	− 1201.83i	− 1.46386i	0.681379	−	0.731931i	$-0.261381\pi$
$822$	0	0
$823$	423.383	0.514438	0.257219	−	0.966353i	$-0.417194\pi$
$823$	423.383	0.514438	0.257219	+	0.966353i	$0.417194\pi$
$824$	331.196i	0.401937i
$825$	0	0
$826$	198.853	0.240742
$827$	− 225.323i	− 0.272458i	−0.990677	−	0.136229i	$-0.956502\pi$
$827$	− 225.323i	− 0.272458i	0.990677	−	0.136229i	$-0.0434983\pi$
$828$	0	0
$829$	−602.279	−0.726513	−0.363256	−	0.931689i	$-0.618335\pi$
$829$	−602.279	−0.726513	−0.363256	+	0.931689i	$0.618335\pi$
$830$	0	0
$831$	0	0
$832$	−107.882	−0.129666
$833$	− 307.654i	− 0.369332i
$834$	0	0
$835$	0	0
$836$	− 149.823i	− 0.179215i
$837$	0	0
$838$	75.8680	0.0905346
$839$	− 784.279i	− 0.934779i	−0.884052	−	0.467389i	$-0.845195\pi$
$839$	− 784.279i	− 0.934779i	0.884052	−	0.467389i	$-0.154805\pi$
$840$	0	0
$841$	−837.587	−0.995942
$842$	288.145i	0.342215i
$843$	0	0
$844$	347.411	0.411625
$845$	0	0
$846$	0	0
$847$	−923.176	−1.08994
$848$	− 299.647i	− 0.353357i
$849$	0	0
$850$	0	0
$851$	− 339.500i	− 0.398942i
$852$	0	0
$853$	−763.043	−0.894540	−0.447270	−	0.894399i	$-0.647604\pi$
$853$	−763.043	−0.894540	−0.447270	+	0.894399i	$0.647604\pi$
$854$	1111.69i	1.30174i
$855$	0	0
$856$	384.250	0.448890
$857$	527.294i	0.615278i	0.951503	+	0.307639i	$0.0995390\pi$
$857$	527.294i	0.615278i	−0.951503	+	0.307639i	$0.900461\pi$
$858$	0	0
$859$	−426.558	−0.496576	−0.248288	−	0.968686i	$-0.579868\pi$
$859$	−426.558	−0.496576	−0.248288	+	0.968686i	$0.579868\pi$
$860$	0	0
$861$	0	0
$862$	−1143.33	−1.32637
$863$	705.442i	0.817429i	0.912662	+	0.408715i	$0.134023\pi$
$863$	705.442i	0.817429i	−0.912662	+	0.408715i	$0.865977\pi$
$864$	0	0
$865$	0	0
$866$	− 894.781i	− 1.03323i
$867$	0	0
$868$	−217.823	−0.250949
$869$	− 5.44870i	− 0.00627008i
$870$	0	0
$871$	−1093.60	−1.25556
$872$	− 534.156i	− 0.612565i
$873$	0	0
$874$	−829.352	−0.948916
$875$	0	0
$876$	0	0
$877$	488.368	0.556862	0.278431	−	0.960456i	$-0.410186\pi$
$877$	488.368	0.556862	0.278431	+	0.960456i	$0.410186\pi$
$878$	− 59.1573i	− 0.0673773i
$879$	0	0
$880$	0	0
$881$	− 217.060i	− 0.246379i	−0.992383	−	0.123189i	$-0.960688\pi$
$881$	− 217.060i	− 0.246379i	0.992383	−	0.123189i	$-0.0393123\pi$
$882$	0	0
$883$	−20.1392	−0.0228077	−0.0114038	−	0.999935i	$-0.503630\pi$
$883$	−20.1392	−0.0228077	−0.0114038	+	0.999935i	$0.503630\pi$
$884$	438.073i	0.495558i
$885$	0	0
$886$	−689.595	−0.778324
$887$	63.5727i	0.0716716i	0.999358	+	0.0358358i	$0.0114093\pi$
$887$	63.5727i	0.0716716i	−0.999358	+	0.0358358i	$0.988591\pi$
$888$	0	0
$889$	1705.56	1.91851
$890$	0	0
$891$	0	0
$892$	334.059	0.374505
$893$	− 586.441i	− 0.656709i
$894$	0	0
$895$	0	0
$896$	93.2548i	0.104079i
$897$	0	0
$898$	−37.8823	−0.0421851
$899$	− 541.352i	− 0.602172i
$900$	0	0
$901$	−1216.76	−1.35046
$902$	− 62.7351i	− 0.0695511i
$903$	0	0
$904$	−195.765	−0.216554
$905$	0	0
$906$	0	0
$907$	−582.558	−0.642292	−0.321146	−	0.947030i	$-0.604068\pi$
$907$	−582.558	−0.642292	−0.321146	+	0.947030i	$0.604068\pi$
$908$	590.881i	0.650750i
$909$	0	0
$910$	0	0
$911$	− 1683.69i	− 1.84818i	−0.382179	−	0.924088i	$-0.624826\pi$
$911$	− 1683.69i	− 1.84818i	0.382179	−	0.924088i	$-0.375174\pi$
$912$	0	0
$913$	−329.647	−0.361059
$914$	740.174i	0.809818i
$915$	0	0
$916$	378.971	0.413723
$917$	1283.67i	1.39986i
$918$	0	0
$919$	1299.32	1.41384	0.706922	−	0.707292i	$-0.250083\pi$
$919$	1299.32	1.41384	0.706922	+	0.707292i	$0.250083\pi$
$920$	0	0
$921$	0	0
$922$	−666.177	−0.722534
$923$	1206.73i	1.30740i
$924$	0	0
$925$	0	0
$926$	337.872i	0.364873i
$927$	0	0
$928$	−231.765	−0.249746
$929$	− 1636.74i	− 1.76183i	−0.473272	−	0.880916i	$-0.656927\pi$
$929$	− 1636.74i	− 1.76183i	0.473272	−	0.880916i	$-0.343073\pi$
$930$	0	0
$931$	472.971	0.508024
$932$	279.515i	0.299908i
$933$	0	0
$934$	668.756	0.716013
$935$	0	0
$936$	0	0
$937$	−1084.38	−1.15729	−0.578645	−	0.815579i	$-0.696418\pi$
$937$	−1084.38	−1.15729	−0.578645	+	0.815579i	$0.696418\pi$
$938$	945.318i	1.00780i
$939$	0	0
$940$	0	0
$941$	773.138i	0.821614i	0.911722	+	0.410807i	$0.134753\pi$
$941$	773.138i	0.821614i	−0.911722	+	0.410807i	$0.865247\pi$
$942$	0	0
$943$	−347.272	−0.368263
$944$	68.2355i	0.0722834i
$945$	0	0
$946$	190.794	0.201685
$947$	− 849.442i	− 0.896982i	−0.893787	−	0.448491i	$-0.851962\pi$
$947$	− 849.442i	− 0.896982i	0.893787	−	0.448491i	$-0.148038\pi$
$948$	0	0
$949$	68.6173	0.0723049
$950$	0	0
$951$	0	0
$952$	378.676	0.397769
$953$	− 1537.35i	− 1.61317i	−0.591117	−	0.806586i	$-0.701313\pi$
$953$	− 1537.35i	− 1.61317i	0.591117	−	0.806586i	$-0.298687\pi$
$954$	0	0
$955$	0	0
$956$	− 520.794i	− 0.544764i
$957$	0	0
$958$	254.059	0.265197
$959$	− 190.794i	− 0.198951i
$960$	0	0
$961$	−786.411	−0.818326
$962$	− 275.688i	− 0.286578i
$963$	0	0
$964$	171.706	0.178118
$965$	0	0
$966$	0	0
$967$	−974.234	−1.00748	−0.503740	−	0.863855i	$-0.668043\pi$
$967$	−974.234	−1.00748	−0.503740	+	0.863855i	$0.668043\pi$
$968$	− 316.784i	− 0.327256i
$969$	0	0
$970$	0	0
$971$	771.500i	0.794541i	0.917702	+	0.397271i	$0.130043\pi$
$971$	771.500i	0.794541i	−0.917702	+	0.397271i	$0.869957\pi$
$972$	0	0
$973$	−503.588	−0.517562
$974$	521.667i	0.535592i
$975$	0	0
$976$	−381.470	−0.390851
$977$	580.118i	0.593775i	0.954913	+	0.296887i	$0.0959486\pi$
$977$	580.118i	0.593775i	−0.954913	+	0.296887i	$0.904051\pi$
$978$	0	0
$979$	−120.728	−0.123318
$980$	0	0
$981$	0	0
$982$	−1054.84	−1.07417
$983$	1086.02i	1.10480i	0.833580	+	0.552398i	$0.186287\pi$
$983$	1086.02i	1.10480i	−0.833580	+	0.552398i	$0.813713\pi$
$984$	0	0
$985$	0	0
$986$	941.117i	0.954480i
$987$	0	0
$988$	−673.470	−0.681650
$989$	− 1056.15i	− 1.06789i
$990$	0	0
$991$	−159.294	−0.160740	−0.0803701	−	0.996765i	$-0.525610\pi$
$991$	−159.294	−0.160740	−0.0803701	+	0.996765i	$0.525610\pi$
$992$	− 74.7452i	− 0.0753479i
$993$	0	0
$994$	1043.12	1.04941
$995$	0	0
$996$	0	0
$997$	100.427	0.100729	0.0503647	−	0.998731i	$-0.483962\pi$
$997$	100.427	0.100729	0.0503647	+	0.998731i	$0.483962\pi$
$998$	− 1090.77i	− 1.09296i
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1350.3.b.a.1349.3		4	15.2	even	4
1350.3.b.a.1349.4		4	5.3	odd	4
1350.3.b.f.1349.1		4	5.2	odd	4
1350.3.b.f.1349.2		4	15.8	even	4
1350.3.d.l.701.1	✓	4	3.2	odd	2	inner
1350.3.d.l.701.3	yes	4	1.1	even	1	trivial
1350.3.d.n.701.2	yes	4	5.4	even	2
1350.3.d.n.701.4	yes	4	15.14	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:	\( N \)	\(=\)	\( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1350.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -2.00000 q^{4} -8.24264 q^{7} -2.82843i q^{8} +O(q^{10})\)	\(q+1.41421i q^{2} -2.00000 q^{4} -8.24264 q^{7} -2.82843i q^{8} +3.00000i q^{11} +13.4853 q^{13} -11.6569i q^{14} +4.00000 q^{16} -16.2426i q^{17} +24.9706 q^{19} -4.24264 q^{22} +23.4853i q^{23} +19.0711i q^{26} +16.4853 q^{28} +40.9706i q^{29} -13.2132 q^{31} +5.65685i q^{32} +22.9706 q^{34} -14.4558 q^{37} +35.3137i q^{38} +14.7868i q^{41} -44.9706 q^{43} -6.00000i q^{44} -33.2132 q^{46} -23.4853i q^{47} +18.9411 q^{49} -26.9706 q^{52} -74.9117i q^{53} +23.3137i q^{56} -57.9411 q^{58} +17.0589i q^{59} -95.3675 q^{61} -18.6863i q^{62} -8.00000 q^{64} -81.0955 q^{67} +32.4853i q^{68} +89.4853i q^{71} +5.08831 q^{73} -20.4437i q^{74} -49.9411 q^{76} -24.7279i q^{77} -1.81623 q^{79} -20.9117 q^{82} +109.882i q^{83} -63.5980i q^{86} +8.48528 q^{88} +40.2426i q^{89} -111.154 q^{91} -46.9706i q^{92} +33.2132 q^{94} -160.765 q^{97} +26.7868i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4} - 16 q^{7}+O(q^{10}) \) 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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