LMFDB - Embedded newform 1350.3.d.n.701.4

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			701.4
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1350.701
Dual form			1350.3.d.n.701.2

$q$-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	$-0.299616\pi$
$7$	8.24264	1.17752	0.588760	+	0.808308i	$0.299616\pi$
$8$	− 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
$13$	−13.4853	−1.03733	−0.518665	−	0.854978i	$-0.673571\pi$
$13$	−13.4853	−1.03733	−0.518665	+	0.854978i	$0.673571\pi$
$14$	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	$-0.750323\pi$
$29$	− 40.9706i	− 1.41278i	0.707824	−	0.706389i	$-0.249677\pi$
$30$	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	$-0.437416\pi$
$37$	14.4558	0.390698	0.195349	+	0.980734i	$0.437416\pi$
$38$	35.3137i	0.929308i
$39$	0	0
$40$	0	0
$41$	− 14.7868i	− 0.360654i	−0.983607	−	0.180327i	$-0.942284\pi$
$41$	− 14.7868i	− 0.360654i	0.983607	−	0.180327i	$-0.0577155\pi$
$42$	0	0
$43$	44.9706	1.04583	0.522914	−	0.852386i	$-0.324845\pi$
$43$	44.9706	1.04583	0.522914	+	0.852386i	$0.324845\pi$
$44$	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	$-0.953821\pi$
$59$	− 17.0589i	− 0.289133i	0.989495	−	0.144567i	$-0.0461788\pi$
$60$	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	$-0.293097\pi$
$67$	81.0955	1.21038	0.605190	+	0.796081i	$0.293097\pi$
$68$	32.4853i	0.477725i
$69$	0	0
$70$	0	0
$71$	− 89.4853i	− 1.26036i	−0.776451	−	0.630178i	$-0.782982\pi$
$71$	− 89.4853i	− 1.26036i	0.776451	−	0.630178i	$-0.217018\pi$
$72$	0	0
$73$	−5.08831	−0.0697029	−0.0348515	−	0.999393i	$-0.511096\pi$
$73$	−5.08831	−0.0697029	−0.0348515	+	0.999393i	$0.511096\pi$
$74$	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	$-0.927408\pi$
$89$	− 40.2426i	− 0.452165i	0.974108	−	0.226082i	$-0.0725918\pi$
$90$	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	$-0.189091\pi$
$97$	160.765	1.65737	0.828683	+	0.559718i	$0.189091\pi$
$98$	26.7868i	0.273335i
$99$	0	0
$100$	0	0
$101$	− 171.037i	− 1.69343i	−0.532046	−	0.846716i	$-0.678576\pi$
$101$	− 171.037i	− 1.69343i	0.532046	−	0.846716i	$-0.321424\pi$
$102$	0	0
$103$	117.095	1.13685	0.568425	−	0.822735i	$-0.307553\pi$
$103$	117.095	1.13685	0.568425	+	0.822735i	$0.307553\pi$
$104$	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	$-0.196934\pi$
$127$	206.919	1.62928	0.814641	+	0.579965i	$0.196934\pi$
$128$	− 11.3137i	− 0.0883883i
$129$	0	0
$130$	0	0
$131$	155.735i	1.18882i	0.804163	+	0.594409i	$0.202614\pi$
$131$	155.735i	1.18882i	−0.804163	+	0.594409i	$0.797386\pi$
$132$	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	$0.155682\pi$
$149$	140.007i	0.939645i	−0.882761	+	0.469823i	$0.844318\pi$
$150$	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	$-0.298568\pi$
$157$	185.706	1.18284	0.591419	+	0.806364i	$0.298568\pi$
$158$	− 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	$-0.765841\pi$
$163$	−241.698	−1.48281	−0.741406	+	0.671056i	$0.765841\pi$
$164$	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	$-0.758969\pi$
$179$	− 245.912i	− 1.37381i	0.726748	−	0.686904i	$-0.241031\pi$
$180$	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	$-0.638432\pi$
$191$	− 346.441i	− 1.81383i	0.421318	−	0.906913i	$-0.361568\pi$
$192$	0	0
$193$	−34.9117	−0.180890	−0.0904448	−	0.995901i	$-0.528829\pi$
$193$	−34.9117	−0.180890	−0.0904448	+	0.995901i	$0.528829\pi$
$194$	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	$-0.377812\pi$
$223$	167.029	0.749011	0.374505	+	0.927225i	$0.377812\pi$
$224$	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	$-0.816619\pi$
$239$	− 260.397i	− 1.08953i	0.838590	−	0.544764i	$-0.183381\pi$
$240$	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	$0.237302\pi$
$251$	340.529i	1.35669i	−0.734744	+	0.678345i	$0.762698\pi$
$252$	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	$0.0946979\pi$
$269$	157.706i	0.586266i	−0.956072	+	0.293133i	$0.905302\pi$
$270$	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	$-0.400651\pi$
$277$	170.118	0.614143	0.307072	+	0.951686i	$0.400651\pi$
$278$	86.4020i	0.310799i
$279$	0	0
$280$	0	0
$281$	− 222.323i	− 0.791185i	−0.918426	−	0.395592i	$-0.870539\pi$
$281$	− 222.323i	− 0.791185i	0.918426	−	0.395592i	$-0.129461\pi$
$282$	0	0
$283$	−24.9706	−0.0882352	−0.0441176	−	0.999026i	$-0.514048\pi$
$283$	−24.9706	−0.0882352	−0.0441176	+	0.999026i	$0.514048\pi$
$284$	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	$-0.385373\pi$
$307$	216.360	0.704757	0.352378	+	0.935858i	$0.385373\pi$
$308$	49.4558i	0.160571i
$309$	0	0
$310$	0	0
$311$	14.0437i	0.0451567i	0.999745	+	0.0225783i	$0.00718752\pi$
$311$	14.0437i	0.0451567i	−0.999745	+	0.0225783i	$0.992812\pi$
$312$	0	0
$313$	−372.617	−1.19047	−0.595235	−	0.803551i	$-0.702941\pi$
$313$	−372.617	−1.19047	−0.595235	+	0.803551i	$0.702941\pi$
$314$	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	$-0.866434\pi$
$337$	−615.529	−1.82650	−0.913248	+	0.407405i	$0.866434\pi$
$338$	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	$0.116499\pi$
$359$	256.955i	0.715753i	−0.933769	+	0.357877i	$0.883501\pi$
$360$	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	$-0.416575\pi$
$367$	190.177	0.518192	0.259096	+	0.965852i	$0.416575\pi$
$368$	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	$-0.597918\pi$
$373$	−225.882	−0.605582	−0.302791	+	0.953057i	$0.597918\pi$
$374$	− 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	$-0.697926\pi$
$389$	− 632.382i	− 1.62566i	0.582501	−	0.812830i	$-0.302074\pi$
$390$	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	$-0.722881\pi$
$397$	−511.632	−1.28874	−0.644372	+	0.764712i	$0.722881\pi$
$398$	194.049i	0.487560i
$399$	0	0
$400$	0	0
$401$	695.418i	1.73421i	0.498125	+	0.867105i	$0.334022\pi$
$401$	695.418i	1.73421i	−0.498125	+	0.867105i	$0.665978\pi$
$402$	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	$0.0203914\pi$
$419$	53.6468i	0.128035i	−0.997949	+	0.0640176i	$0.979609\pi$
$420$	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	$-0.612782\pi$
$431$	− 808.456i	− 1.87577i	0.346950	−	0.937884i	$-0.387218\pi$
$432$	0	0
$433$	632.706	1.46121	0.730607	−	0.682798i	$-0.239237\pi$
$433$	632.706	1.46121	0.730607	+	0.682798i	$0.239237\pi$
$434$	− 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	$-0.990504\pi$
$449$	− 26.7868i	− 0.0596588i	0.999555	−	0.0298294i	$-0.00949640\pi$
$450$	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	$-0.694076\pi$
$457$	−523.382	−1.14526	−0.572628	+	0.819815i	$0.694076\pi$
$458$	− 267.973i	− 0.585093i
$459$	0	0
$460$	0	0
$461$	− 471.058i	− 1.02182i	−0.859635	−	0.510909i	$-0.829309\pi$
$461$	− 471.058i	− 1.02182i	0.859635	−	0.510909i	$-0.170691\pi$
$462$	0	0
$463$	−238.912	−0.516008	−0.258004	−	0.966144i	$-0.583065\pi$
$463$	−238.912	−0.516008	−0.258004	+	0.966144i	$0.583065\pi$
$464$	− 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	$0.0600458\pi$
$479$	179.647i	0.375045i	−0.982260	+	0.187523i	$0.939954\pi$
$480$	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	$-0.623636\pi$
$487$	−368.874	−0.757442	−0.378721	+	0.925511i	$0.623636\pi$
$488$	269.740i	0.552746i
$489$	0	0
$490$	0	0
$491$	− 745.882i	− 1.51911i	−0.650444	−	0.759554i	$-0.725417\pi$
$491$	− 745.882i	− 1.51911i	0.650444	−	0.759554i	$-0.274583\pi$
$492$	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	$0.00993155\pi$
$509$	31.7574i	0.0623917i	−0.999513	+	0.0311958i	$0.990068\pi$
$510$	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	$0.154938\pi$
$521$	487.404i	0.935517i	−0.883856	+	0.467758i	$0.845062\pi$
$522$	0	0
$523$	304.353	0.581937	0.290969	−	0.956733i	$-0.406022\pi$
$523$	304.353	0.581937	0.290969	+	0.956733i	$0.406022\pi$
$524$	− 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	$-0.545977\pi$
$547$	−157.470	−0.287880	−0.143940	+	0.989586i	$0.545977\pi$
$548$	− 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	$0.138297\pi$
$569$	479.022i	0.841867i	−0.907092	+	0.420933i	$0.861703\pi$
$570$	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	$-0.194151\pi$
$577$	945.912	1.63936	0.819681	+	0.572820i	$0.194151\pi$
$578$	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	$0.238940\pi$
$599$	817.176i	1.36423i	−0.731243	+	0.682117i	$0.761060\pi$
$600$	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	$-0.744166\pi$
$607$	−842.551	−1.38806	−0.694029	+	0.719947i	$0.744166\pi$
$608$	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	$-0.513203\pi$
$613$	−50.8377	−0.0829326	−0.0414663	+	0.999140i	$0.513203\pi$
$614$	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	$0.343017\pi$
$641$	1129.23i	1.76167i	−0.473428	+	0.880833i	$0.656983\pi$
$642$	0	0
$643$	−1041.83	−1.62027	−0.810133	−	0.586247i	$-0.800605\pi$
$643$	−1041.83	−1.62027	−0.810133	+	0.586247i	$0.800605\pi$
$644$	− 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	$-0.661587\pi$
$659$	− 1151.79i	− 1.74779i	0.486115	−	0.873895i	$-0.338413\pi$
$660$	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	$-0.652398\pi$
$673$	−620.088	−0.921379	−0.460690	+	0.887561i	$0.652398\pi$
$674$	− 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	$0.331593\pi$
$701$	1210.32i	1.72656i	−0.504729	+	0.863278i	$0.668407\pi$
$702$	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	$0.00262056\pi$
$719$	11.8385i	0.0164653i	−0.999966	+	0.00823263i	$0.997379\pi$
$720$	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	$-0.462413\pi$
$727$	171.294	0.235617	0.117808	+	0.993036i	$0.462413\pi$
$728$	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	$-0.457054\pi$
$733$	197.191	0.269019	0.134509	+	0.990912i	$0.457054\pi$
$734$	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	$-0.331735\pi$
$757$	763.573	1.00868	0.504341	+	0.863504i	$0.331735\pi$
$758$	− 677.480i	− 0.893773i
$759$	0	0
$760$	0	0
$761$	212.214i	0.278862i	0.990232	+	0.139431i	$0.0445274\pi$
$761$	212.214i	0.278862i	−0.990232	+	0.139431i	$0.955473\pi$
$762$	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	$-0.432194\pi$
$787$	332.764	0.422825	0.211413	+	0.977397i	$0.432194\pi$
$788$	− 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	$0.0949956\pi$
$809$	475.736i	0.588054i	−0.955797	+	0.294027i	$0.905004\pi$
$810$	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	$0.261381\pi$
$821$	1201.83i	1.46386i	−0.681379	+	0.731931i	$0.738619\pi$
$822$	0	0
$823$	−423.383	−0.514438	−0.257219	−	0.966353i	$-0.582806\pi$
$823$	−423.383	−0.514438	−0.257219	+	0.966353i	$0.582806\pi$
$824$	− 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	$0.154805\pi$
$839$	784.279i	0.934779i	−0.884052	+	0.467389i	$0.845195\pi$
$840$	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	$-0.352396\pi$
$853$	763.043	0.894540	0.447270	+	0.894399i	$0.352396\pi$
$854$	− 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	$-0.589814\pi$
$877$	−488.368	−0.556862	−0.278431	+	0.960456i	$0.589814\pi$
$878$	− 59.1573i	− 0.0673773i
$879$	0	0
$880$	0	0
$881$	217.060i	0.246379i	0.992383	+	0.123189i	$0.0393123\pi$
$881$	217.060i	0.246379i	−0.992383	+	0.123189i	$0.960688\pi$
$882$	0	0
$883$	20.1392	0.0228077	0.0114038	−	0.999935i	$-0.496370\pi$
$883$	20.1392	0.0228077	0.0114038	+	0.999935i	$0.496370\pi$
$884$	− 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	$-0.395932\pi$
$907$	582.558	0.642292	0.321146	+	0.947030i	$0.395932\pi$
$908$	590.881i	0.650750i
$909$	0	0
$910$	0	0
$911$	1683.69i	1.84818i	0.382179	+	0.924088i	$0.375174\pi$
$911$	1683.69i	1.84818i	−0.382179	+	0.924088i	$0.624826\pi$
$912$	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	$0.343073\pi$
$929$	1636.74i	1.76183i	−0.473272	+	0.880916i	$0.656927\pi$
$930$	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	$-0.303582\pi$
$937$	1084.38	1.15729	0.578645	+	0.815579i	$0.303582\pi$
$938$	945.318i	1.00780i
$939$	0	0
$940$	0	0
$941$	− 773.138i	− 0.821614i	−0.911722	−	0.410807i	$-0.865247\pi$
$941$	− 773.138i	− 0.821614i	0.911722	−	0.410807i	$-0.134753\pi$
$942$	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	$-0.331957\pi$
$967$	974.234	1.00748	0.503740	+	0.863855i	$0.331957\pi$
$968$	− 316.784i	− 0.327256i
$969$	0	0
$970$	0	0
$971$	− 771.500i	− 0.794541i	−0.917702	−	0.397271i	$-0.869957\pi$
$971$	− 771.500i	− 0.794541i	0.917702	−	0.397271i	$-0.130043\pi$
$972$	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	$-0.516038\pi$
$997$	−100.427	−0.100729	−0.0503647	+	0.998731i	$0.516038\pi$
$998$	− 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.n.701.4	yes	4
3.2	odd	2	inner	1350.3.d.n.701.2	yes	4
5.2	odd	4		1350.3.b.f.1349.2		4
5.3	odd	4		1350.3.b.a.1349.3		4
5.4	even	2		1350.3.d.l.701.1	✓	4
15.2	even	4		1350.3.b.a.1349.4		4
15.8	even	4		1350.3.b.f.1349.1		4
15.14	odd	2		1350.3.d.l.701.3	yes	4

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

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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