LMFDB - Embedded newform 1350.3.b.f.1349.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1350,3,Mod(1349,1350)]

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, 1]))

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

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

chi := DirichletCharacter("1350.1349");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1350 = 2 \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1350.b (of order $2$, degree $1$, not 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 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1349.1
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1350.1349
Dual form			1350.3.b.f.1349.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} +2.00000 q^{4} -8.24264i q^{7} -2.82843 q^{8} +O(q^{10})$	$q-1.41421 q^{2} +2.00000 q^{4} -8.24264i q^{7} -2.82843 q^{8} +3.00000i q^{11} -13.4853i q^{13} +11.6569i q^{14} +4.00000 q^{16} +16.2426 q^{17} -24.9706 q^{19} -4.24264i q^{22} +23.4853 q^{23} +19.0711i q^{26} -16.4853i q^{28} -40.9706i q^{29} -13.2132 q^{31} -5.65685 q^{32} -22.9706 q^{34} -14.4558i q^{37} +35.3137 q^{38} +14.7868i q^{41} +44.9706i q^{43} +6.00000i q^{44} -33.2132 q^{46} +23.4853 q^{47} -18.9411 q^{49} -26.9706i q^{52} -74.9117 q^{53} +23.3137i q^{56} +57.9411i q^{58} -17.0589i q^{59} -95.3675 q^{61} +18.6863 q^{62} +8.00000 q^{64} -81.0955i q^{67} +32.4853 q^{68} +89.4853i q^{71} -5.08831i q^{73} +20.4437i q^{74} -49.9411 q^{76} +24.7279 q^{77} +1.81623 q^{79} -20.9117i q^{82} +109.882 q^{83} -63.5980i q^{86} -8.48528i q^{88} -40.2426i q^{89} -111.154 q^{91} +46.9706 q^{92} -33.2132 q^{94} -160.765i q^{97} +26.7868 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{4}+O(q^{10}) $ 4 * q + 8 * q^4	$ 4 q + 8 q^{4} + 16 q^{16} + 48 q^{17} - 32 q^{19} + 60 q^{23} + 32 q^{31} - 24 q^{34} + 96 q^{38} - 48 q^{46} + 60 q^{47} + 60 q^{49} - 96 q^{53} - 76 q^{61} + 120 q^{62} + 32 q^{64} + 96 q^{68} - 64 q^{76} + 48 q^{77} + 160 q^{79} + 168 q^{83} - 224 q^{91} + 120 q^{92} - 48 q^{94} + 192 q^{98}+O(q^{100}) $ 4 * q + 8 * q^4 + 16 * q^16 + 48 * q^17 - 32 * q^19 + 60 * q^23 + 32 * q^31 - 24 * q^34 + 96 * q^38 - 48 * q^46 + 60 * q^47 + 60 * q^49 - 96 * q^53 - 76 * q^61 + 120 * q^62 + 32 * q^64 + 96 * q^68 - 64 * q^76 + 48 * q^77 + 160 * q^79 + 168 * q^83 - 224 * q^91 + 120 * q^92 - 48 * q^94 + 192 * q^98

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 $

$2$	−1.41421	−0.707107
$2$	−1.41421	−0.707107
$3$	0	0
$3$	0	0
$4$	2.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 8.24264i	− 1.17752i	−0.808308	−	0.588760i	$-0.799616\pi$
$7$	− 8.24264i	− 1.17752i	0.808308	−	0.588760i	$-0.200384\pi$
$8$	−2.82843	−0.353553
$9$	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
$13$	− 13.4853i	− 1.03733i	−0.854978	−	0.518665i	$-0.826429\pi$
$13$	− 13.4853i	− 1.03733i	0.854978	−	0.518665i	$-0.173571\pi$
$14$	11.6569i	0.832632i
$15$	0	0
$16$	4.00000	0.250000
$17$	16.2426	0.955449	0.477725	−	0.878510i	$-0.341462\pi$
$17$	16.2426	0.955449	0.477725	+	0.878510i	$0.341462\pi$
$18$	0	0
$19$	−24.9706	−1.31424	−0.657120	−	0.753786i	$-0.728226\pi$
$19$	−24.9706	−1.31424	−0.657120	+	0.753786i	$0.728226\pi$
$20$	0	0
$21$	0	0
$22$	− 4.24264i	− 0.192847i
$23$	23.4853	1.02110	0.510550	−	0.859848i	$-0.329442\pi$
$23$	23.4853	1.02110	0.510550	+	0.859848i	$0.329442\pi$
$24$	0	0
$25$	0	0
$26$	19.0711i	0.733503i
$27$	0	0
$28$	− 16.4853i	− 0.588760i
$29$	− 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	$0.568361\pi$
$32$	−5.65685	−0.176777
$33$	0	0
$34$	−22.9706	−0.675605
$35$	0	0
$36$	0	0
$37$	− 14.4558i	− 0.390698i	−0.980734	−	0.195349i	$-0.937416\pi$
$37$	− 14.4558i	− 0.390698i	0.980734	−	0.195349i	$-0.0625840\pi$
$38$	35.3137	0.929308
$39$	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
$43$	44.9706i	1.04583i	0.852386	+	0.522914i	$0.175155\pi$
$43$	44.9706i	1.04583i	−0.852386	+	0.522914i	$0.824845\pi$
$44$	6.00000i	0.136364i
$45$	0	0
$46$	−33.2132	−0.722026
$47$	23.4853	0.499687	0.249843	−	0.968286i	$-0.419621\pi$
$47$	23.4853	0.499687	0.249843	+	0.968286i	$0.419621\pi$
$48$	0	0
$49$	−18.9411	−0.386554
$50$	0	0
$51$	0	0
$52$	− 26.9706i	− 0.518665i
$53$	−74.9117	−1.41343	−0.706714	−	0.707499i	$-0.749823\pi$
$53$	−74.9117	−1.41343	−0.706714	+	0.707499i	$0.749823\pi$
$54$	0	0
$55$	0	0
$56$	23.3137i	0.416316i
$57$	0	0
$58$	57.9411i	0.998985i
$59$	− 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	$0.785648\pi$
$62$	18.6863	0.301392
$63$	0	0
$64$	8.00000	0.125000
$65$	0	0
$66$	0	0
$67$	− 81.0955i	− 1.21038i	−0.796081	−	0.605190i	$-0.793097\pi$
$67$	− 81.0955i	− 1.21038i	0.796081	−	0.605190i	$-0.206903\pi$
$68$	32.4853	0.477725
$69$	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
$73$	− 5.08831i	− 0.0697029i	−0.999393	−	0.0348515i	$-0.988904\pi$
$73$	− 5.08831i	− 0.0697029i	0.999393	−	0.0348515i	$-0.0110958\pi$
$74$	20.4437i	0.276266i
$75$	0	0
$76$	−49.9411	−0.657120
$77$	24.7279	0.321142
$78$	0	0
$79$	1.81623	0.0229903	0.0114952	−	0.999934i	$-0.496341\pi$
$79$	1.81623	0.0229903	0.0114952	+	0.999934i	$0.496341\pi$
$80$	0	0
$81$	0	0
$82$	− 20.9117i	− 0.255021i
$83$	109.882	1.32388	0.661941	−	0.749556i	$-0.269733\pi$
$83$	109.882	1.32388	0.661941	+	0.749556i	$0.269733\pi$
$84$	0	0
$85$	0	0
$86$	− 63.5980i	− 0.739511i
$87$	0	0
$88$	− 8.48528i	− 0.0964237i
$89$	− 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
$92$	46.9706	0.510550
$93$	0	0
$94$	−33.2132	−0.353332
$95$	0	0
$96$	0	0
$97$	− 160.765i	− 1.65737i	−0.559718	−	0.828683i	$-0.689091\pi$
$97$	− 160.765i	− 1.65737i	0.559718	−	0.828683i	$-0.310909\pi$
$98$	26.7868	0.273335
$99$	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
$103$	117.095i	1.13685i	0.822735	+	0.568425i	$0.192447\pi$
$103$	117.095i	1.13685i	−0.822735	+	0.568425i	$0.807553\pi$
$104$	38.1421i	0.366751i
$105$	0	0
$106$	105.941	0.999445
$107$	−135.853	−1.26965	−0.634826	−	0.772655i	$-0.718928\pi$
$107$	−135.853	−1.26965	−0.634826	+	0.772655i	$0.718928\pi$
$108$	0	0
$109$	−188.853	−1.73259	−0.866297	−	0.499529i	$-0.833507\pi$
$109$	−188.853	−1.73259	−0.866297	+	0.499529i	$0.833507\pi$
$110$	0	0
$111$	0	0
$112$	− 32.9706i	− 0.294380i
$113$	−69.2132	−0.612506	−0.306253	−	0.951950i	$-0.599075\pi$
$113$	−69.2132	−0.612506	−0.306253	+	0.951950i	$0.599075\pi$
$114$	0	0
$115$	0	0
$116$	− 81.9411i	− 0.706389i
$117$	0	0
$118$	24.1249i	0.204448i
$119$	− 133.882i	− 1.12506i
$120$	0	0
$121$	112.000	0.925620
$122$	134.870	1.10549
$123$	0	0
$124$	−26.4264	−0.213116
$125$	0	0
$126$	0	0
$127$	− 206.919i	− 1.62928i	−0.579965	−	0.814641i	$-0.696934\pi$
$127$	− 206.919i	− 1.62928i	0.579965	−	0.814641i	$-0.303066\pi$
$128$	−11.3137	−0.0883883
$129$	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
$133$	205.823i	1.54754i
$134$	114.686i	0.855868i
$135$	0	0
$136$	−45.9411	−0.337802
$137$	−23.1472	−0.168958	−0.0844788	−	0.996425i	$-0.526923\pi$
$137$	−23.1472	−0.168958	−0.0844788	+	0.996425i	$0.526923\pi$
$138$	0	0
$139$	−61.0955	−0.439536	−0.219768	−	0.975552i	$-0.570530\pi$
$139$	−61.0955	−0.439536	−0.219768	+	0.975552i	$0.570530\pi$
$140$	0	0
$141$	0	0
$142$	− 126.551i	− 0.891206i
$143$	40.4558	0.282908
$144$	0	0
$145$	0	0
$146$	7.19596i	0.0492874i
$147$	0	0
$148$	− 28.9117i	− 0.195349i
$149$	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	$0.469293\pi$
$152$	70.6274	0.464654
$153$	0	0
$154$	−34.9706	−0.227082
$155$	0	0
$156$	0	0
$157$	− 185.706i	− 1.18284i	−0.806364	−	0.591419i	$-0.798568\pi$
$157$	− 185.706i	− 1.18284i	0.806364	−	0.591419i	$-0.201432\pi$
$158$	−2.56854	−0.0162566
$159$	0	0
$160$	0	0
$161$	− 193.581i	− 1.20236i
$162$	0	0
$163$	− 241.698i	− 1.48281i	−0.671056	−	0.741406i	$-0.734159\pi$
$163$	− 241.698i	− 1.48281i	0.671056	−	0.741406i	$-0.265841\pi$
$164$	29.5736i	0.180327i
$165$	0	0
$166$	−155.397	−0.936126
$167$	−286.279	−1.71425	−0.857123	−	0.515111i	$-0.827751\pi$
$167$	−286.279	−1.71425	−0.857123	+	0.515111i	$0.827751\pi$
$168$	0	0
$169$	−12.8528	−0.0760522
$170$	0	0
$171$	0	0
$172$	89.9411i	0.522914i
$173$	−220.846	−1.27656	−0.638282	−	0.769802i	$-0.720355\pi$
$173$	−220.846	−1.27656	−0.638282	+	0.769802i	$0.720355\pi$
$174$	0	0
$175$	0	0
$176$	12.0000i	0.0681818i
$177$	0	0
$178$	56.9117i	0.319729i
$179$	− 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	$0.764429\pi$
$182$	157.196	0.863714
$183$	0	0
$184$	−66.4264	−0.361013
$185$	0	0
$186$	0	0
$187$	48.7279i	0.260577i
$188$	46.9706	0.249843
$189$	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
$193$	− 34.9117i	− 0.180890i	−0.995901	−	0.0904448i	$-0.971171\pi$
$193$	− 34.9117i	− 0.180890i	0.995901	−	0.0904448i	$-0.0288289\pi$
$194$	227.355i	1.17193i
$195$	0	0
$196$	−37.8823	−0.193277
$197$	−189.941	−0.964168	−0.482084	−	0.876125i	$-0.660120\pi$
$197$	−189.941	−0.964168	−0.482084	+	0.876125i	$0.660120\pi$
$198$	0	0
$199$	−137.213	−0.689514	−0.344757	−	0.938692i	$-0.612039\pi$
$199$	−137.213	−0.689514	−0.344757	+	0.938692i	$0.612039\pi$
$200$	0	0
$201$	0	0
$202$	− 241.882i	− 1.19744i
$203$	−337.706	−1.66357
$204$	0	0
$205$	0	0
$206$	− 165.598i	− 0.803874i
$207$	0	0
$208$	− 53.9411i	− 0.259332i
$209$	− 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	$0.635039\pi$
$212$	−149.823	−0.706714
$213$	0	0
$214$	192.125	0.897780
$215$	0	0
$216$	0	0
$217$	108.912i	0.501897i
$218$	267.078	1.22513
$219$	0	0
$220$	0	0
$221$	− 219.037i	− 0.991116i
$222$	0	0
$223$	167.029i	0.749011i	0.927225	+	0.374505i	$0.122188\pi$
$223$	167.029i	0.749011i	−0.927225	+	0.374505i	$0.877812\pi$
$224$	46.6274i	0.208158i
$225$	0	0
$226$	97.8823	0.433107
$227$	295.441	1.30150	0.650750	−	0.759292i	$-0.274454\pi$
$227$	295.441	1.30150	0.650750	+	0.759292i	$0.274454\pi$
$228$	0	0
$229$	189.485	0.827447	0.413723	−	0.910403i	$-0.364228\pi$
$229$	189.485	0.827447	0.413723	+	0.910403i	$0.364228\pi$
$230$	0	0
$231$	0	0
$232$	115.882i	0.499492i
$233$	−139.757	−0.599817	−0.299908	−	0.953968i	$-0.596956\pi$
$233$	−139.757	−0.599817	−0.299908	+	0.953968i	$0.596956\pi$
$234$	0	0
$235$	0	0
$236$	− 34.1177i	− 0.144567i
$237$	0	0
$238$	189.338i	0.795538i
$239$	− 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	$0.557001\pi$
$242$	−158.392	−0.654512
$243$	0	0
$244$	−190.735	−0.781701
$245$	0	0
$246$	0	0
$247$	336.735i	1.36330i
$248$	37.3726	0.150696
$249$	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
$253$	70.4558i	0.278482i
$254$	292.627i	1.15208i
$255$	0	0
$256$	16.0000	0.0625000
$257$	−375.889	−1.46260	−0.731302	−	0.682053i	$-0.761087\pi$
$257$	−375.889	−1.46260	−0.731302	+	0.682053i	$0.761087\pi$
$258$	0	0
$259$	−119.154	−0.460055
$260$	0	0
$261$	0	0
$262$	220.243i	0.840621i
$263$	265.014	1.00766	0.503829	−	0.863803i	$-0.331924\pi$
$263$	265.014	1.00766	0.503829	+	0.863803i	$0.331924\pi$
$264$	0	0
$265$	0	0
$266$	− 291.078i	− 1.09428i
$267$	0	0
$268$	− 162.191i	− 0.605190i
$269$	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	$0.534232\pi$
$272$	64.9706	0.238862
$273$	0	0
$274$	32.7351	0.119471
$275$	0	0
$276$	0	0
$277$	− 170.118i	− 0.614143i	−0.951686	−	0.307072i	$-0.900651\pi$
$277$	− 170.118i	− 0.614143i	0.951686	−	0.307072i	$-0.0993492\pi$
$278$	86.4020	0.310799
$279$	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
$283$	− 24.9706i	− 0.0882352i	−0.999026	−	0.0441176i	$-0.985952\pi$
$283$	− 24.9706i	− 0.0882352i	0.999026	−	0.0441176i	$-0.0140476\pi$
$284$	178.971i	0.630178i
$285$	0	0
$286$	−57.2132	−0.200046
$287$	121.882	0.424677
$288$	0	0
$289$	−25.1766	−0.0871163
$290$	0	0
$291$	0	0
$292$	− 10.1766i	− 0.0348515i
$293$	−506.683	−1.72929	−0.864647	−	0.502379i	$-0.832458\pi$
$293$	−506.683	−1.72929	−0.864647	+	0.502379i	$0.832458\pi$
$294$	0	0
$295$	0	0
$296$	40.8873i	0.138133i
$297$	0	0
$298$	− 198.000i	− 0.664430i
$299$	− 316.706i	− 1.05922i
$300$	0	0
$301$	370.676	1.23148
$302$	−41.1371	−0.136216
$303$	0	0
$304$	−99.8823	−0.328560
$305$	0	0
$306$	0	0
$307$	− 216.360i	− 0.704757i	−0.935858	−	0.352378i	$-0.885373\pi$
$307$	− 216.360i	− 0.704757i	0.935858	−	0.352378i	$-0.114627\pi$
$308$	49.4558	0.160571
$309$	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
$313$	− 372.617i	− 1.19047i	−0.803551	−	0.595235i	$-0.797059\pi$
$313$	− 372.617i	− 1.19047i	0.803551	−	0.595235i	$-0.202941\pi$
$314$	262.627i	0.836393i
$315$	0	0
$316$	3.63247	0.0114952
$317$	60.4996	0.190850	0.0954252	−	0.995437i	$-0.469579\pi$
$317$	60.4996	0.190850	0.0954252	+	0.995437i	$0.469579\pi$
$318$	0	0
$319$	122.912	0.385303
$320$	0	0
$321$	0	0
$322$	273.765i	0.850200i
$323$	−405.588	−1.25569
$324$	0	0
$325$	0	0
$326$	341.813i	1.04851i
$327$	0	0
$328$	− 41.8234i	− 0.127510i
$329$	− 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	$0.246578\pi$
$332$	219.765	0.661941
$333$	0	0
$334$	404.860	1.21216
$335$	0	0
$336$	0	0
$337$	615.529i	1.82650i	0.407405	+	0.913248i	$0.366434\pi$
$337$	615.529i	1.82650i	−0.407405	+	0.913248i	$0.633566\pi$
$338$	18.1766	0.0537770
$339$	0	0
$340$	0	0
$341$	− 39.6396i	− 0.116245i
$342$	0	0
$343$	− 247.765i	− 0.722345i
$344$	− 127.196i	− 0.369756i
$345$	0	0
$346$	312.323	0.902667
$347$	190.294	0.548399	0.274199	−	0.961673i	$-0.411587\pi$
$347$	190.294	0.548399	0.274199	+	0.961673i	$0.411587\pi$
$348$	0	0
$349$	273.470	0.783582	0.391791	−	0.920054i	$-0.371856\pi$
$349$	273.470	0.783582	0.391791	+	0.920054i	$0.371856\pi$
$350$	0	0
$351$	0	0
$352$	− 16.9706i	− 0.0482118i
$353$	562.087	1.59232	0.796158	−	0.605089i	$-0.206862\pi$
$353$	562.087	1.59232	0.796158	+	0.605089i	$0.206862\pi$
$354$	0	0
$355$	0	0
$356$	− 80.4853i	− 0.226082i
$357$	0	0
$358$	347.772i	0.971429i
$359$	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
$362$	378.032	1.04429
$363$	0	0
$364$	−222.309	−0.610738
$365$	0	0
$366$	0	0
$367$	− 190.177i	− 0.518192i	−0.965852	−	0.259096i	$-0.916575\pi$
$367$	− 190.177i	− 0.518192i	0.965852	−	0.259096i	$-0.0834247\pi$
$368$	93.9411	0.255275
$369$	0	0
$370$	0	0
$371$	617.470i	1.66434i
$372$	0	0
$373$	− 225.882i	− 0.605582i	−0.953057	−	0.302791i	$-0.902082\pi$
$373$	− 225.882i	− 0.605582i	0.953057	−	0.302791i	$-0.0979185\pi$
$374$	− 68.9117i	− 0.184256i
$375$	0	0
$376$	−66.4264	−0.176666
$377$	−552.500	−1.46552
$378$	0	0
$379$	479.051	1.26399	0.631993	−	0.774974i	$-0.282237\pi$
$379$	479.051	1.26399	0.631993	+	0.774974i	$0.282237\pi$
$380$	0	0
$381$	0	0
$382$	− 489.941i	− 1.28257i
$383$	281.662	0.735410	0.367705	−	0.929943i	$-0.380144\pi$
$383$	281.662	0.735410	0.367705	+	0.929943i	$0.380144\pi$
$384$	0	0
$385$	0	0
$386$	49.3726i	0.127908i
$387$	0	0
$388$	− 321.529i	− 0.828683i
$389$	− 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
$392$	53.5736	0.136667
$393$	0	0
$394$	268.617	0.681770
$395$	0	0
$396$	0	0
$397$	511.632i	1.28874i	0.764712	+	0.644372i	$0.222881\pi$
$397$	511.632i	1.28874i	−0.764712	+	0.644372i	$0.777119\pi$
$398$	194.049	0.487560
$399$	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
$403$	178.184i	0.442143i
$404$	342.073i	0.846716i
$405$	0	0
$406$	477.588	1.17632
$407$	43.3675	0.106554
$408$	0	0
$409$	508.411	1.24306	0.621530	−	0.783391i	$-0.286512\pi$
$409$	508.411	1.24306	0.621530	+	0.783391i	$0.286512\pi$
$410$	0	0
$411$	0	0
$412$	234.191i	0.568425i
$413$	−140.610	−0.340460
$414$	0	0
$415$	0	0
$416$	76.2843i	0.183376i
$417$	0	0
$418$	105.941i	0.253448i
$419$	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	$0.422202\pi$
$422$	245.657	0.582125
$423$	0	0
$424$	211.882	0.499722
$425$	0	0
$426$	0	0
$427$	786.080i	1.84094i
$428$	−271.706	−0.634826
$429$	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
$433$	632.706i	1.46121i	0.682798	+	0.730607i	$0.260763\pi$
$433$	632.706i	1.46121i	−0.682798	+	0.730607i	$0.739237\pi$
$434$	− 154.024i	− 0.354895i
$435$	0	0
$436$	−377.706	−0.866297
$437$	−586.441	−1.34197
$438$	0	0
$439$	41.8305	0.0952859	0.0476430	−	0.998864i	$-0.484829\pi$
$439$	41.8305	0.0952859	0.0476430	+	0.998864i	$0.484829\pi$
$440$	0	0
$441$	0	0
$442$	309.765i	0.700825i
$443$	487.617	1.10072	0.550358	−	0.834929i	$-0.314491\pi$
$443$	487.617	1.10072	0.550358	+	0.834929i	$0.314491\pi$
$444$	0	0
$445$	0	0
$446$	− 236.215i	− 0.529631i
$447$	0	0
$448$	− 65.9411i	− 0.147190i
$449$	− 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
$452$	−138.426	−0.306253
$453$	0	0
$454$	−417.816	−0.920300
$455$	0	0
$456$	0	0
$457$	523.382i	1.14526i	0.819815	+	0.572628i	$0.194076\pi$
$457$	523.382i	1.14526i	−0.819815	+	0.572628i	$0.805924\pi$
$458$	−267.973	−0.585093
$459$	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
$463$	− 238.912i	− 0.516008i	−0.966144	−	0.258004i	$-0.916935\pi$
$463$	− 238.912i	− 0.516008i	0.966144	−	0.258004i	$-0.0830648\pi$
$464$	− 163.882i	− 0.353195i
$465$	0	0
$466$	197.647	0.424135
$467$	472.882	1.01260	0.506298	−	0.862359i	$-0.331014\pi$
$467$	472.882	1.01260	0.506298	+	0.862359i	$0.331014\pi$
$468$	0	0
$469$	−668.441	−1.42525
$470$	0	0
$471$	0	0
$472$	48.2498i	0.102224i
$473$	−134.912	−0.285226
$474$	0	0
$475$	0	0
$476$	− 267.765i	− 0.562530i
$477$	0	0
$478$	368.257i	0.770412i
$479$	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
$482$	121.414	0.251897
$483$	0	0
$484$	224.000	0.462810
$485$	0	0
$486$	0	0
$487$	368.874i	0.757442i	0.925511	+	0.378721i	$0.123636\pi$
$487$	368.874i	0.757442i	−0.925511	+	0.378721i	$0.876364\pi$
$488$	269.740	0.552746
$489$	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
$493$	− 665.470i	− 1.34984i
$494$	− 476.215i	− 0.963999i
$495$	0	0
$496$	−52.8528	−0.106558
$497$	737.595	1.48409
$498$	0	0
$499$	771.294	1.54568	0.772839	−	0.634602i	$-0.218836\pi$
$499$	771.294	1.54568	0.772839	+	0.634602i	$0.218836\pi$
$500$	0	0
$501$	0	0
$502$	481.581i	0.959324i
$503$	32.2355	0.0640865	0.0320432	−	0.999486i	$-0.489799\pi$
$503$	32.2355	0.0640865	0.0320432	+	0.999486i	$0.489799\pi$
$504$	0	0
$505$	0	0
$506$	− 99.6396i	− 0.196916i
$507$	0	0
$508$	− 413.838i	− 0.814641i
$509$	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
$512$	−22.6274	−0.0441942
$513$	0	0
$514$	531.588	1.03422
$515$	0	0
$516$	0	0
$517$	70.4558i	0.136278i
$518$	168.510	0.325308
$519$	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
$523$	304.353i	0.581937i	0.956733	+	0.290969i	$0.0939775\pi$
$523$	304.353i	0.581937i	−0.956733	+	0.290969i	$0.906022\pi$
$524$	− 311.470i	− 0.594409i
$525$	0	0
$526$	−374.787	−0.712522
$527$	−214.617	−0.407243
$528$	0	0
$529$	22.5584	0.0426436
$530$	0	0
$531$	0	0
$532$	411.647i	0.773772i
$533$	199.404	0.374117
$534$	0	0
$535$	0	0
$536$	229.373i	0.427934i
$537$	0	0
$538$	− 223.029i	− 0.414553i
$539$	− 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	$0.466822\pi$
$542$	82.2742	0.151797
$543$	0	0
$544$	−91.8823	−0.168901
$545$	0	0
$546$	0	0
$547$	157.470i	0.287880i	0.989586	+	0.143940i	$0.0459772\pi$
$547$	157.470i	0.287880i	−0.989586	+	0.143940i	$0.954023\pi$
$548$	−46.2944	−0.0844788
$549$	0	0
$550$	0	0
$551$	1023.06i	1.85673i
$552$	0	0
$553$	− 14.9706i	− 0.0270715i
$554$	240.583i	0.434265i
$555$	0	0
$556$	−122.191	−0.219768
$557$	156.978	0.281827	0.140914	−	0.990022i	$-0.454996\pi$
$557$	156.978	0.281827	0.140914	+	0.990022i	$0.454996\pi$
$558$	0	0
$559$	606.441	1.08487
$560$	0	0
$561$	0	0
$562$	− 314.412i	− 0.559452i
$563$	−81.3229	−0.144446	−0.0722229	−	0.997389i	$-0.523009\pi$
$563$	−81.3229	−0.144446	−0.0722229	+	0.997389i	$0.523009\pi$
$564$	0	0
$565$	0	0
$566$	35.3137i	0.0623917i
$567$	0	0
$568$	− 253.103i	− 0.445603i
$569$	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	$0.313519\pi$
$572$	80.9117	0.141454
$573$	0	0
$574$	−172.368	−0.300292
$575$	0	0
$576$	0	0
$577$	− 945.912i	− 1.63936i	−0.572820	−	0.819681i	$-0.694151\pi$
$577$	− 945.912i	− 1.63936i	0.572820	−	0.819681i	$-0.305849\pi$
$578$	35.6051	0.0616006
$579$	0	0
$580$	0	0
$581$	− 905.720i	− 1.55890i
$582$	0	0
$583$	− 224.735i	− 0.385480i
$584$	14.3919i	0.0246437i
$585$	0	0
$586$	716.558	1.22280
$587$	847.529	1.44383	0.721916	−	0.691981i	$-0.243262\pi$
$587$	847.529	1.44383	0.721916	+	0.691981i	$0.243262\pi$
$588$	0	0
$589$	329.941	0.560172
$590$	0	0
$591$	0	0
$592$	− 57.8234i	− 0.0976746i
$593$	268.118	0.452138	0.226069	−	0.974111i	$-0.427413\pi$
$593$	268.118	0.452138	0.226069	+	0.974111i	$0.427413\pi$
$594$	0	0
$595$	0	0
$596$	280.014i	0.469823i
$597$	0	0
$598$	447.889i	0.748979i
$599$	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	$0.667188\pi$
$602$	−524.215	−0.870790
$603$	0	0
$604$	58.1766	0.0963189
$605$	0	0
$606$	0	0
$607$	842.551i	1.38806i	0.719947	+	0.694029i	$0.244166\pi$
$607$	842.551i	1.38806i	−0.719947	+	0.694029i	$0.755834\pi$
$608$	141.255	0.232327
$609$	0	0
$610$	0	0
$611$	− 316.706i	− 0.518340i
$612$	0	0
$613$	− 50.8377i	− 0.0829326i	−0.999140	−	0.0414663i	$-0.986797\pi$
$613$	− 50.8377i	− 0.0829326i	0.999140	−	0.0414663i	$-0.0132029\pi$
$614$	305.980i	0.498338i
$615$	0	0
$616$	−69.9411	−0.113541
$617$	−167.272	−0.271105	−0.135553	−	0.990770i	$-0.543281\pi$
$617$	−167.272	−0.271105	−0.135553	+	0.990770i	$0.543281\pi$
$618$	0	0
$619$	−892.403	−1.44169	−0.720843	−	0.693099i	$-0.756245\pi$
$619$	−892.403	−1.44169	−0.720843	+	0.693099i	$0.756245\pi$
$620$	0	0
$621$	0	0
$622$	19.8608i	0.0319306i
$623$	−331.706	−0.532433
$624$	0	0
$625$	0	0
$626$	526.960i	0.841790i
$627$	0	0
$628$	− 371.411i	− 0.591419i
$629$	− 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	$0.523992\pi$
$632$	−5.13708	−0.00812830
$633$	0	0
$634$	−85.5593	−0.134952
$635$	0	0
$636$	0	0
$637$	255.426i	0.400983i
$638$	−173.823	−0.272450
$639$	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
$643$	− 1041.83i	− 1.62027i	−0.586247	−	0.810133i	$-0.699395\pi$
$643$	− 1041.83i	− 1.62027i	0.586247	−	0.810133i	$-0.300605\pi$
$644$	− 387.161i	− 0.601182i
$645$	0	0
$646$	573.588	0.887907
$647$	−588.131	−0.909013	−0.454506	−	0.890744i	$-0.650184\pi$
$647$	−588.131	−0.909013	−0.454506	+	0.890744i	$0.650184\pi$
$648$	0	0
$649$	51.1766	0.0788546
$650$	0	0
$651$	0	0
$652$	− 483.397i	− 0.741406i
$653$	−127.861	−0.195805	−0.0979026	−	0.995196i	$-0.531213\pi$
$653$	−127.861	−0.195805	−0.0979026	+	0.995196i	$0.531213\pi$
$654$	0	0
$655$	0	0
$656$	59.1472i	0.0901634i
$657$	0	0
$658$	273.765i	0.416055i
$659$	− 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	$0.378360\pi$
$662$	−669.078	−1.01069
$663$	0	0
$664$	−310.794	−0.468063
$665$	0	0
$666$	0	0
$667$	− 962.205i	− 1.44259i
$668$	−572.558	−0.857123
$669$	0	0
$670$	0	0
$671$	− 286.103i	− 0.426382i
$672$	0	0
$673$	− 620.088i	− 0.921379i	−0.887561	−	0.460690i	$-0.847602\pi$
$673$	− 620.088i	− 0.921379i	0.887561	−	0.460690i	$-0.152398\pi$
$674$	− 870.489i	− 1.29153i
$675$	0	0
$676$	−25.7056	−0.0380261
$677$	−733.581	−1.08358	−0.541788	−	0.840515i	$-0.682252\pi$
$677$	−733.581	−1.08358	−0.541788	+	0.840515i	$0.682252\pi$
$678$	0	0
$679$	−1325.12	−1.95158
$680$	0	0
$681$	0	0
$682$	56.0589i	0.0821978i
$683$	810.588	1.18681	0.593403	−	0.804906i	$-0.297784\pi$
$683$	810.588	1.18681	0.593403	+	0.804906i	$0.297784\pi$
$684$	0	0
$685$	0	0
$686$	350.392i	0.510775i
$687$	0	0
$688$	179.882i	0.261457i
$689$	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	$0.659919\pi$
$692$	−441.691	−0.638282
$693$	0	0
$694$	−269.117	−0.387776
$695$	0	0
$696$	0	0
$697$	240.177i	0.344586i
$698$	−386.745	−0.554076
$699$	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
$703$	360.971i	0.513472i
$704$	24.0000i	0.0340909i
$705$	0	0
$706$	−794.912	−1.12594
$707$	1409.79	1.99405
$708$	0	0
$709$	65.8671	0.0929014	0.0464507	−	0.998921i	$-0.485209\pi$
$709$	65.8671	0.0929014	0.0464507	+	0.998921i	$0.485209\pi$
$710$	0	0
$711$	0	0
$712$	113.823i	0.159864i
$713$	−310.316	−0.435226
$714$	0	0
$715$	0	0
$716$	− 491.823i	− 0.686904i
$717$	0	0
$718$	− 363.390i	− 0.506114i
$719$	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
$722$	−371.272	−0.514227
$723$	0	0
$724$	−534.617	−0.738422
$725$	0	0
$726$	0	0
$727$	− 171.294i	− 0.235617i	−0.993036	−	0.117808i	$-0.962413\pi$
$727$	− 171.294i	− 0.235617i	0.993036	−	0.117808i	$-0.0375869\pi$
$728$	314.392	0.431857
$729$	0	0
$730$	0	0
$731$	730.441i	0.999235i
$732$	0	0
$733$	197.191i	0.269019i	0.990912	+	0.134509i	$0.0429459\pi$
$733$	197.191i	0.269019i	−0.990912	+	0.134509i	$0.957054\pi$
$734$	268.950i	0.366417i
$735$	0	0
$736$	−132.853	−0.180507
$737$	243.286	0.330104
$738$	0	0
$739$	−140.728	−0.190430	−0.0952151	−	0.995457i	$-0.530354\pi$
$739$	−140.728	−0.190430	−0.0952151	+	0.995457i	$0.530354\pi$
$740$	0	0
$741$	0	0
$742$	− 873.235i	− 1.17687i
$743$	399.073	0.537111	0.268555	−	0.963264i	$-0.413454\pi$
$743$	399.073	0.537111	0.268555	+	0.963264i	$0.413454\pi$
$744$	0	0
$745$	0	0
$746$	319.446i	0.428211i
$747$	0	0
$748$	97.4558i	0.130289i
$749$	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	$0.519896\pi$
$752$	93.9411	0.124922
$753$	0	0
$754$	781.352	1.03628
$755$	0	0
$756$	0	0
$757$	− 763.573i	− 1.00868i	−0.863504	−	0.504341i	$-0.831735\pi$
$757$	− 763.573i	− 1.00868i	0.863504	−	0.504341i	$-0.168265\pi$
$758$	−677.480	−0.893773
$759$	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
$763$	1556.65i	2.04016i
$764$	692.881i	0.906913i
$765$	0	0
$766$	−398.330	−0.520013
$767$	−230.044	−0.299927
$768$	0	0
$769$	−7.79307	−0.0101340	−0.00506702	−	0.999987i	$-0.501613\pi$
$769$	−7.79307	−0.0101340	−0.00506702	+	0.999987i	$0.501613\pi$
$770$	0	0
$771$	0	0
$772$	− 69.8234i	− 0.0904448i
$773$	−549.442	−0.710791	−0.355396	−	0.934716i	$-0.615654\pi$
$773$	−549.442	−0.710791	−0.355396	+	0.934716i	$0.615654\pi$
$774$	0	0
$775$	0	0
$776$	454.711i	0.585967i
$777$	0	0
$778$	894.323i	1.14952i
$779$	− 369.235i	− 0.473985i
$780$	0	0
$781$	−268.456	−0.343733
$782$	−539.470	−0.689860
$783$	0	0
$784$	−75.7645	−0.0966384
$785$	0	0
$786$	0	0
$787$	− 332.764i	− 0.422825i	−0.977397	−	0.211413i	$-0.932194\pi$
$787$	− 332.764i	− 0.422825i	0.977397	−	0.211413i	$-0.0678064\pi$
$788$	−379.882	−0.482084
$789$	0	0
$790$	0	0
$791$	570.500i	0.721238i
$792$	0	0
$793$	1286.06i	1.62176i
$794$	− 723.556i	− 0.911280i
$795$	0	0
$796$	−274.426	−0.344757
$797$	656.029	0.823122	0.411561	−	0.911382i	$-0.364984\pi$
$797$	656.029	0.823122	0.411561	+	0.911382i	$0.364984\pi$
$798$	0	0
$799$	381.463	0.477426
$800$	0	0
$801$	0	0
$802$	983.470i	1.22627i
$803$	15.2649	0.0190099
$804$	0	0
$805$	0	0
$806$	− 251.990i	− 0.312643i
$807$	0	0
$808$	− 483.765i	− 0.598718i
$809$	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	$0.709329\pi$
$812$	−675.411	−0.831787
$813$	0	0
$814$	−61.3310	−0.0753452
$815$	0	0
$816$	0	0
$817$	− 1122.94i	− 1.37447i
$818$	−719.002	−0.878976
$819$	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
$823$	− 423.383i	− 0.514438i	−0.966353	−	0.257219i	$-0.917194\pi$
$823$	− 423.383i	− 0.514438i	0.966353	−	0.257219i	$-0.0828062\pi$
$824$	− 331.196i	− 0.401937i
$825$	0	0
$826$	198.853	0.240742
$827$	225.323	0.272458	0.136229	−	0.990677i	$-0.456502\pi$
$827$	225.323	0.272458	0.136229	+	0.990677i	$0.456502\pi$
$828$	0	0
$829$	602.279	0.726513	0.363256	−	0.931689i	$-0.381665\pi$
$829$	602.279	0.726513	0.363256	+	0.931689i	$0.381665\pi$
$830$	0	0
$831$	0	0
$832$	− 107.882i	− 0.129666i
$833$	−307.654	−0.369332
$834$	0	0
$835$	0	0
$836$	− 149.823i	− 0.179215i
$837$	0	0
$838$	− 75.8680i	− 0.0905346i
$839$	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
$842$	−288.145	−0.342215
$843$	0	0
$844$	−347.411	−0.411625
$845$	0	0
$846$	0	0
$847$	− 923.176i	− 1.08994i
$848$	−299.647	−0.353357
$849$	0	0
$850$	0	0
$851$	− 339.500i	− 0.398942i
$852$	0	0
$853$	763.043i	0.894540i	0.894399	+	0.447270i	$0.147604\pi$
$853$	763.043i	0.894540i	−0.894399	+	0.447270i	$0.852396\pi$
$854$	− 1111.69i	− 1.30174i
$855$	0	0
$856$	384.250	0.448890
$857$	−527.294	−0.615278	−0.307639	−	0.951503i	$-0.599539\pi$
$857$	−527.294	−0.615278	−0.307639	+	0.951503i	$0.599539\pi$
$858$	0	0
$859$	426.558	0.496576	0.248288	−	0.968686i	$-0.420132\pi$
$859$	426.558	0.496576	0.248288	+	0.968686i	$0.420132\pi$
$860$	0	0
$861$	0	0
$862$	− 1143.33i	− 1.32637i
$863$	705.442	0.817429	0.408715	−	0.912662i	$-0.365977\pi$
$863$	705.442	0.817429	0.408715	+	0.912662i	$0.365977\pi$
$864$	0	0
$865$	0	0
$866$	− 894.781i	− 1.03323i
$867$	0	0
$868$	217.823i	0.250949i
$869$	5.44870i	0.00627008i
$870$	0	0
$871$	−1093.60	−1.25556
$872$	534.156	0.612565
$873$	0	0
$874$	829.352	0.948916
$875$	0	0
$876$	0	0
$877$	488.368i	0.556862i	0.960456	+	0.278431i	$0.0898144\pi$
$877$	488.368i	0.556862i	−0.960456	+	0.278431i	$0.910186\pi$
$878$	−59.1573	−0.0673773
$879$	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
$883$	20.1392i	0.0228077i	0.999935	+	0.0114038i	$0.00363003\pi$
$883$	20.1392i	0.0228077i	−0.999935	+	0.0114038i	$0.996370\pi$
$884$	− 438.073i	− 0.495558i
$885$	0	0
$886$	−689.595	−0.778324
$887$	−63.5727	−0.0716716	−0.0358358	−	0.999358i	$-0.511409\pi$
$887$	−63.5727	−0.0716716	−0.0358358	+	0.999358i	$0.511409\pi$
$888$	0	0
$889$	−1705.56	−1.91851
$890$	0	0
$891$	0	0
$892$	334.059i	0.374505i
$893$	−586.441	−0.656709
$894$	0	0
$895$	0	0
$896$	93.2548i	0.104079i
$897$	0	0
$898$	37.8823i	0.0421851i
$899$	541.352i	0.602172i
$900$	0	0
$901$	−1216.76	−1.35046
$902$	62.7351	0.0695511
$903$	0	0
$904$	195.765	0.216554
$905$	0	0
$906$	0	0
$907$	− 582.558i	− 0.642292i	−0.947030	−	0.321146i	$-0.895932\pi$
$907$	− 582.558i	− 0.642292i	0.947030	−	0.321146i	$-0.104068\pi$
$908$	590.881	0.650750
$909$	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
$913$	329.647i	0.361059i
$914$	− 740.174i	− 0.809818i
$915$	0	0
$916$	378.971	0.413723
$917$	−1283.67	−1.39986
$918$	0	0
$919$	−1299.32	−1.41384	−0.706922	−	0.707292i	$-0.749917\pi$
$919$	−1299.32	−1.41384	−0.706922	+	0.707292i	$0.749917\pi$
$920$	0	0
$921$	0	0
$922$	− 666.177i	− 0.722534i
$923$	1206.73	1.30740
$924$	0	0
$925$	0	0
$926$	337.872i	0.364873i
$927$	0	0
$928$	231.765i	0.249746i
$929$	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
$932$	−279.515	−0.299908
$933$	0	0
$934$	−668.756	−0.716013
$935$	0	0
$936$	0	0
$937$	− 1084.38i	− 1.15729i	−0.815579	−	0.578645i	$-0.803582\pi$
$937$	− 1084.38i	− 1.15729i	0.815579	−	0.578645i	$-0.196418\pi$
$938$	945.318	1.00780
$939$	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
$943$	347.272i	0.368263i
$944$	− 68.2355i	− 0.0722834i
$945$	0	0
$946$	190.794	0.201685
$947$	849.442	0.896982	0.448491	−	0.893787i	$-0.351962\pi$
$947$	849.442	0.896982	0.448491	+	0.893787i	$0.351962\pi$
$948$	0	0
$949$	−68.6173	−0.0723049
$950$	0	0
$951$	0	0
$952$	378.676i	0.397769i
$953$	−1537.35	−1.61317	−0.806586	−	0.591117i	$-0.798687\pi$
$953$	−1537.35	−1.61317	−0.806586	+	0.591117i	$0.798687\pi$
$954$	0	0
$955$	0	0
$956$	− 520.794i	− 0.544764i
$957$	0	0
$958$	− 254.059i	− 0.265197i
$959$	190.794i	0.198951i
$960$	0	0
$961$	−786.411	−0.818326
$962$	275.688	0.286578
$963$	0	0
$964$	−171.706	−0.178118
$965$	0	0
$966$	0	0
$967$	− 974.234i	− 1.00748i	−0.863855	−	0.503740i	$-0.831957\pi$
$967$	− 974.234i	− 1.00748i	0.863855	−	0.503740i	$-0.168043\pi$
$968$	−316.784	−0.327256
$969$	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
$973$	503.588i	0.517562i
$974$	− 521.667i	− 0.535592i
$975$	0	0
$976$	−381.470	−0.390851
$977$	−580.118	−0.593775	−0.296887	−	0.954913i	$-0.595949\pi$
$977$	−580.118	−0.593775	−0.296887	+	0.954913i	$0.595949\pi$
$978$	0	0
$979$	120.728	0.123318
$980$	0	0
$981$	0	0
$982$	− 1054.84i	− 1.07417i
$983$	1086.02	1.10480	0.552398	−	0.833580i	$-0.313713\pi$
$983$	1086.02	1.10480	0.552398	+	0.833580i	$0.313713\pi$
$984$	0	0
$985$	0	0
$986$	941.117i	0.954480i
$987$	0	0
$988$	673.470i	0.681650i
$989$	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	$0.525610\pi$
$992$	74.7452	0.0753479
$993$	0	0
$994$	−1043.12	−1.04941
$995$	0	0
$996$	0	0
$997$	100.427i	0.100729i	0.998731	+	0.0503647i	$0.0160384\pi$
$997$	100.427i	0.100729i	−0.998731	+	0.0503647i	$0.983962\pi$
$998$	−1090.77	−1.09296
$999$	0	0

Twists

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

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

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.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +2.00000 q^{4} -8.24264i q^{7} -2.82843 q^{8} +O(q^{10})\)	\(q-1.41421 q^{2} +2.00000 q^{4} -8.24264i q^{7} -2.82843 q^{8} +3.00000i q^{11} -13.4853i q^{13} +11.6569i q^{14} +4.00000 q^{16} +16.2426 q^{17} -24.9706 q^{19} -4.24264i q^{22} +23.4853 q^{23} +19.0711i q^{26} -16.4853i q^{28} -40.9706i q^{29} -13.2132 q^{31} -5.65685 q^{32} -22.9706 q^{34} -14.4558i q^{37} +35.3137 q^{38} +14.7868i q^{41} +44.9706i q^{43} +6.00000i q^{44} -33.2132 q^{46} +23.4853 q^{47} -18.9411 q^{49} -26.9706i q^{52} -74.9117 q^{53} +23.3137i q^{56} +57.9411i q^{58} -17.0589i q^{59} -95.3675 q^{61} +18.6863 q^{62} +8.00000 q^{64} -81.0955i q^{67} +32.4853 q^{68} +89.4853i q^{71} -5.08831i q^{73} +20.4437i q^{74} -49.9411 q^{76} +24.7279 q^{77} +1.81623 q^{79} -20.9117i q^{82} +109.882 q^{83} -63.5980i q^{86} -8.48528i q^{88} -40.2426i q^{89} -111.154 q^{91} +46.9706 q^{92} -33.2132 q^{94} -160.765i q^{97} +26.7868 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{4}+O(q^{10}) \) 4 * q + 8 * q^4	\( 4 q + 8 q^{4} + 16 q^{16} + 48 q^{17} - 32 q^{19} + 60 q^{23} + 32 q^{31} - 24 q^{34} + 96 q^{38} - 48 q^{46} + 60 q^{47} + 60 q^{49} - 96 q^{53} - 76 q^{61} + 120 q^{62} + 32 q^{64} + 96 q^{68} - 64 q^{76} + 48 q^{77} + 160 q^{79} + 168 q^{83} - 224 q^{91} + 120 q^{92} - 48 q^{94} + 192 q^{98}+O(q^{100}) \) 4 * q + 8 * q^4 + 16 * q^16 + 48 * q^17 - 32 * q^19 + 60 * q^23 + 32 * q^31 - 24 * q^34 + 96 * q^38 - 48 * q^46 + 60 * q^47 + 60 * q^49 - 96 * q^53 - 76 * q^61 + 120 * q^62 + 32 * q^64 + 96 * q^68 - 64 * q^76 + 48 * q^77 + 160 * q^79 + 168 * q^83 - 224 * q^91 + 120 * q^92 - 48 * q^94 + 192 * q^98

Introduction

L-functions

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