LMFDB - Embedded newform 1350.3.b.a.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.a.1349.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} +2.00000 q^{4} -0.242641i q^{7} -2.82843 q^{8} +O(q^{10})$	$q-1.41421 q^{2} +2.00000 q^{4} -0.242641i q^{7} -2.82843 q^{8} +3.00000i q^{11} -3.48528i q^{13} +0.343146i q^{14} +4.00000 q^{16} -7.75736 q^{17} +8.97056 q^{19} -4.24264i q^{22} -6.51472 q^{23} +4.92893i q^{26} -0.485281i q^{28} -7.02944i q^{29} +29.2132 q^{31} -5.65685 q^{32} +10.9706 q^{34} -36.4558i q^{37} -12.6863 q^{38} +57.2132i q^{41} -11.0294i q^{43} +6.00000i q^{44} +9.21320 q^{46} -6.51472 q^{47} +48.9411 q^{49} -6.97056i q^{52} -26.9117 q^{53} +0.686292i q^{56} +9.94113i q^{58} -84.9411i q^{59} +57.3675 q^{61} -41.3137 q^{62} +8.00000 q^{64} -97.0955i q^{67} -15.5147 q^{68} +72.5147i q^{71} +106.912i q^{73} +51.5563i q^{74} +17.9411 q^{76} +0.727922 q^{77} +78.1838 q^{79} -80.9117i q^{82} +25.8823 q^{83} +15.5980i q^{86} -8.48528i q^{88} -31.7574i q^{89} -0.845671 q^{91} -13.0294 q^{92} +9.21320 q^{94} -110.765i q^{97} -69.2132 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	− 0.242641i	− 0.0346630i	−0.999850	−	0.0173315i	$-0.994483\pi$
$7$	− 0.242641i	− 0.0346630i	0.999850	−	0.0173315i	$-0.00551706\pi$
$8$	−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
$13$	− 3.48528i	− 0.268099i	−0.990975	−	0.134049i	$-0.957202\pi$
$13$	− 3.48528i	− 0.268099i	0.990975	−	0.134049i	$-0.0427980\pi$
$14$	0.343146i	0.0245104i
$15$	0	0
$16$	4.00000	0.250000
$17$	−7.75736	−0.456315	−0.228158	−	0.973624i	$-0.573270\pi$
$17$	−7.75736	−0.456315	−0.228158	+	0.973624i	$0.573270\pi$
$18$	0	0
$19$	8.97056	0.472135	0.236067	−	0.971737i	$-0.424141\pi$
$19$	8.97056	0.472135	0.236067	+	0.971737i	$0.424141\pi$
$20$	0	0
$21$	0	0
$22$	− 4.24264i	− 0.192847i
$23$	−6.51472	−0.283249	−0.141624	−	0.989920i	$-0.545232\pi$
$23$	−6.51472	−0.283249	−0.141624	+	0.989920i	$0.545232\pi$
$24$	0	0
$25$	0	0
$26$	4.92893i	0.189574i
$27$	0	0
$28$	− 0.485281i	− 0.0173315i
$29$	− 7.02944i	− 0.242394i	−0.992628	−	0.121197i	$-0.961327\pi$
$29$	− 7.02944i	− 0.242394i	0.992628	−	0.121197i	$-0.0386733\pi$
$30$	0	0
$31$	29.2132	0.942361	0.471181	−	0.882037i	$-0.343828\pi$
$31$	29.2132	0.942361	0.471181	+	0.882037i	$0.343828\pi$
$32$	−5.65685	−0.176777
$33$	0	0
$34$	10.9706	0.322664
$35$	0	0
$36$	0	0
$37$	− 36.4558i	− 0.985293i	−0.870230	−	0.492647i	$-0.836030\pi$
$37$	− 36.4558i	− 0.985293i	0.870230	−	0.492647i	$-0.163970\pi$
$38$	−12.6863	−0.333850
$39$	0	0
$40$	0	0
$41$	57.2132i	1.39544i	0.716369	+	0.697722i	$0.245803\pi$
$41$	57.2132i	1.39544i	−0.716369	+	0.697722i	$0.754197\pi$
$42$	0	0
$43$	− 11.0294i	− 0.256499i	−0.991742	−	0.128249i	$-0.959064\pi$
$43$	− 11.0294i	− 0.256499i	0.991742	−	0.128249i	$-0.0409358\pi$
$44$	6.00000i	0.136364i
$45$	0	0
$46$	9.21320	0.200287
$47$	−6.51472	−0.138611	−0.0693055	−	0.997595i	$-0.522078\pi$
$47$	−6.51472	−0.138611	−0.0693055	+	0.997595i	$0.522078\pi$
$48$	0	0
$49$	48.9411	0.998798
$50$	0	0
$51$	0	0
$52$	− 6.97056i	− 0.134049i
$53$	−26.9117	−0.507768	−0.253884	−	0.967235i	$-0.581708\pi$
$53$	−26.9117	−0.507768	−0.253884	+	0.967235i	$0.581708\pi$
$54$	0	0
$55$	0	0
$56$	0.686292i	0.0122552i
$57$	0	0
$58$	9.94113i	0.171399i
$59$	− 84.9411i	− 1.43968i	−0.694140	−	0.719840i	$-0.744215\pi$
$59$	− 84.9411i	− 1.43968i	0.694140	−	0.719840i	$-0.255785\pi$
$60$	0	0
$61$	57.3675	0.940451	0.470226	−	0.882546i	$-0.344173\pi$
$61$	57.3675	0.940451	0.470226	+	0.882546i	$0.344173\pi$
$62$	−41.3137	−0.666350
$63$	0	0
$64$	8.00000	0.125000
$65$	0	0
$66$	0	0
$67$	− 97.0955i	− 1.44919i	−0.689177	−	0.724593i	$-0.742028\pi$
$67$	− 97.0955i	− 1.44919i	0.689177	−	0.724593i	$-0.257972\pi$
$68$	−15.5147	−0.228158
$69$	0	0
$70$	0	0
$71$	72.5147i	1.02133i	0.859779	+	0.510667i	$0.170602\pi$
$71$	72.5147i	1.02133i	−0.859779	+	0.510667i	$0.829398\pi$
$72$	0	0
$73$	106.912i	1.46454i	0.681012	+	0.732272i	$0.261540\pi$
$73$	106.912i	1.46454i	−0.681012	+	0.732272i	$0.738460\pi$
$74$	51.5563i	0.696707i
$75$	0	0
$76$	17.9411	0.236067
$77$	0.727922	0.00945353
$78$	0	0
$79$	78.1838	0.989668	0.494834	−	0.868988i	$-0.335229\pi$
$79$	78.1838	0.989668	0.494834	+	0.868988i	$0.335229\pi$
$80$	0	0
$81$	0	0
$82$	− 80.9117i	− 0.986728i
$83$	25.8823	0.311834	0.155917	−	0.987770i	$-0.450167\pi$
$83$	25.8823	0.311834	0.155917	+	0.987770i	$0.450167\pi$
$84$	0	0
$85$	0	0
$86$	15.5980i	0.181372i
$87$	0	0
$88$	− 8.48528i	− 0.0964237i
$89$	− 31.7574i	− 0.356824i	−0.983956	−	0.178412i	$-0.942904\pi$
$89$	− 31.7574i	− 0.356824i	0.983956	−	0.178412i	$-0.0570960\pi$
$90$	0	0
$91$	−0.845671	−0.00929309
$92$	−13.0294	−0.141624
$93$	0	0
$94$	9.21320	0.0980128
$95$	0	0
$96$	0	0
$97$	− 110.765i	− 1.14190i	−0.820984	−	0.570951i	$-0.806575\pi$
$97$	− 110.765i	− 1.14190i	0.820984	−	0.570951i	$-0.193425\pi$
$98$	−69.2132	−0.706257
$99$	0	0
$100$	0	0
$101$	− 75.0366i	− 0.742936i	−0.928446	−	0.371468i	$-0.878854\pi$
$101$	− 75.0366i	− 0.742936i	0.928446	−	0.371468i	$-0.121146\pi$
$102$	0	0
$103$	61.0955i	0.593160i	0.955008	+	0.296580i	$0.0958461\pi$
$103$	61.0955i	0.593160i	−0.955008	+	0.296580i	$0.904154\pi$
$104$	9.85786i	0.0947872i
$105$	0	0
$106$	38.0589	0.359046
$107$	−33.8528	−0.316381	−0.158191	−	0.987409i	$-0.550566\pi$
$107$	−33.8528	−0.316381	−0.158191	+	0.987409i	$0.550566\pi$
$108$	0	0
$109$	−19.1472	−0.175662	−0.0878311	−	0.996135i	$-0.527994\pi$
$109$	−19.1472	−0.175662	−0.0878311	+	0.996135i	$0.527994\pi$
$110$	0	0
$111$	0	0
$112$	− 0.970563i	− 0.00866574i
$113$	26.7868	0.237051	0.118526	−	0.992951i	$-0.462183\pi$
$113$	26.7868	0.237051	0.118526	+	0.992951i	$0.462183\pi$
$114$	0	0
$115$	0	0
$116$	− 14.0589i	− 0.121197i
$117$	0	0
$118$	120.125i	1.01801i
$119$	1.88225i	0.0158172i
$120$	0	0
$121$	112.000	0.925620
$122$	−81.1299	−0.665000
$123$	0	0
$124$	58.4264	0.471181
$125$	0	0
$126$	0	0
$127$	− 174.919i	− 1.37731i	−0.725087	−	0.688657i	$-0.758201\pi$
$127$	− 174.919i	− 1.37731i	0.725087	−	0.688657i	$-0.241799\pi$
$128$	−11.3137	−0.0883883
$129$	0	0
$130$	0	0
$131$	149.735i	1.14302i	0.820597	+	0.571508i	$0.193641\pi$
$131$	149.735i	1.14302i	−0.820597	+	0.571508i	$0.806359\pi$
$132$	0	0
$133$	− 2.17662i	− 0.0163656i
$134$	137.314i	1.02473i
$135$	0	0
$136$	21.9411	0.161332
$137$	192.853	1.40768	0.703842	−	0.710356i	$-0.251466\pi$
$137$	192.853	1.40768	0.703842	+	0.710356i	$0.251466\pi$
$138$	0	0
$139$	117.095	0.842413	0.421207	−	0.906965i	$-0.361607\pi$
$139$	117.095	0.842413	0.421207	+	0.906965i	$0.361607\pi$
$140$	0	0
$141$	0	0
$142$	− 102.551i	− 0.722192i
$143$	10.4558	0.0731178
$144$	0	0
$145$	0	0
$146$	− 151.196i	− 1.03559i
$147$	0	0
$148$	− 72.9117i	− 0.492647i
$149$	− 140.007i	− 0.939645i	−0.882761	−	0.469823i	$-0.844318\pi$
$149$	− 140.007i	− 0.939645i	0.882761	−	0.469823i	$-0.155682\pi$
$150$	0	0
$151$	130.912	0.866965	0.433482	−	0.901162i	$-0.357285\pi$
$151$	130.912	0.866965	0.433482	+	0.901162i	$0.357285\pi$
$152$	−25.3726	−0.166925
$153$	0	0
$154$	−1.02944	−0.00668466
$155$	0	0
$156$	0	0
$157$	− 153.706i	− 0.979017i	−0.871999	−	0.489508i	$-0.837176\pi$
$157$	− 153.706i	− 0.979017i	0.871999	−	0.489508i	$-0.162824\pi$
$158$	−110.569	−0.699801
$159$	0	0
$160$	0	0
$161$	1.58074i	0.00981823i
$162$	0	0
$163$	182.302i	1.11841i	0.829028	+	0.559207i	$0.188894\pi$
$163$	182.302i	1.11841i	−0.829028	+	0.559207i	$0.811106\pi$
$164$	114.426i	0.697722i
$165$	0	0
$166$	−36.6030	−0.220500
$167$	31.7208	0.189945	0.0949724	−	0.995480i	$-0.469724\pi$
$167$	31.7208	0.189945	0.0949724	+	0.995480i	$0.469724\pi$
$168$	0	0
$169$	156.853	0.928123
$170$	0	0
$171$	0	0
$172$	− 22.0589i	− 0.128249i
$173$	331.154	1.91419	0.957093	−	0.289779i	$-0.0935820\pi$
$173$	331.154	1.91419	0.957093	+	0.289779i	$0.0935820\pi$
$174$	0	0
$175$	0	0
$176$	12.0000i	0.0681818i
$177$	0	0
$178$	44.9117i	0.252313i
$179$	− 144.088i	− 0.804963i	−0.915428	−	0.402481i	$-0.868148\pi$
$179$	− 144.088i	− 0.804963i	0.915428	−	0.402481i	$-0.131852\pi$
$180$	0	0
$181$	−46.6913	−0.257963	−0.128982	−	0.991647i	$-0.541171\pi$
$181$	−46.6913	−0.257963	−0.128982	+	0.991647i	$0.541171\pi$
$182$	1.19596	0.00657121
$183$	0	0
$184$	18.4264	0.100144
$185$	0	0
$186$	0	0
$187$	− 23.2721i	− 0.124450i
$188$	−13.0294	−0.0693055
$189$	0	0
$190$	0	0
$191$	− 298.441i	− 1.56252i	−0.624208	−	0.781258i	$-0.714578\pi$
$191$	− 298.441i	− 1.56252i	0.624208	−	0.781258i	$-0.285422\pi$
$192$	0	0
$193$	− 66.9117i	− 0.346693i	−0.984861	−	0.173346i	$-0.944542\pi$
$193$	− 66.9117i	− 0.346693i	0.984861	−	0.173346i	$-0.0554580\pi$
$194$	156.645i	0.807447i
$195$	0	0
$196$	97.8823	0.499399
$197$	122.059	0.619588	0.309794	−	0.950804i	$-0.399740\pi$
$197$	122.059	0.619588	0.309794	+	0.950804i	$0.399740\pi$
$198$	0	0
$199$	−94.7868	−0.476316	−0.238158	−	0.971226i	$-0.576544\pi$
$199$	−94.7868	−0.476316	−0.238158	+	0.971226i	$0.576544\pi$
$200$	0	0
$201$	0	0
$202$	106.118i	0.525335i
$203$	−1.70563	−0.00840211
$204$	0	0
$205$	0	0
$206$	− 86.4020i	− 0.419427i
$207$	0	0
$208$	− 13.9411i	− 0.0670246i
$209$	26.9117i	0.128764i
$210$	0	0
$211$	165.706	0.785335	0.392667	−	0.919681i	$-0.371552\pi$
$211$	165.706	0.785335	0.392667	+	0.919681i	$0.371552\pi$
$212$	−53.8234	−0.253884
$213$	0	0
$214$	47.8751	0.223715
$215$	0	0
$216$	0	0
$217$	− 7.08831i	− 0.0326650i
$218$	27.0782	0.124212
$219$	0	0
$220$	0	0
$221$	27.0366i	0.122337i
$222$	0	0
$223$	− 200.971i	− 0.901213i	−0.892723	−	0.450607i	$-0.851208\pi$
$223$	− 200.971i	− 0.901213i	0.892723	−	0.450607i	$-0.148792\pi$
$224$	1.37258i	0.00612760i
$225$	0	0
$226$	−37.8823	−0.167621
$227$	349.441	1.53939	0.769693	−	0.638414i	$-0.220409\pi$
$227$	349.441	1.53939	0.769693	+	0.638414i	$0.220409\pi$
$228$	0	0
$229$	172.515	0.753339	0.376670	−	0.926348i	$-0.377069\pi$
$229$	172.515	0.753339	0.376670	+	0.926348i	$0.377069\pi$
$230$	0	0
$231$	0	0
$232$	19.8823i	0.0856994i
$233$	148.243	0.636235	0.318117	−	0.948051i	$-0.396949\pi$
$233$	148.243	0.636235	0.318117	+	0.948051i	$0.396949\pi$
$234$	0	0
$235$	0	0
$236$	− 169.882i	− 0.719840i
$237$	0	0
$238$	− 2.66190i	− 0.0111845i
$239$	− 141.603i	− 0.592481i	−0.955113	−	0.296241i	$-0.904267\pi$
$239$	− 141.603i	− 0.592481i	0.955113	−	0.296241i	$-0.0957330\pi$
$240$	0	0
$241$	83.8528	0.347937	0.173968	−	0.984751i	$-0.444341\pi$
$241$	83.8528	0.347937	0.173968	+	0.984751i	$0.444341\pi$
$242$	−158.392	−0.654512
$243$	0	0
$244$	114.735	0.470226
$245$	0	0
$246$	0	0
$247$	− 31.2649i	− 0.126579i
$248$	−82.6274	−0.333175
$249$	0	0
$250$	0	0
$251$	202.529i	0.806888i	0.915004	+	0.403444i	$0.132187\pi$
$251$	202.529i	0.806888i	−0.915004	+	0.403444i	$0.867813\pi$
$252$	0	0
$253$	− 19.5442i	− 0.0772496i
$254$	247.373i	0.973908i
$255$	0	0
$256$	16.0000	0.0625000
$257$	−39.8894	−0.155212	−0.0776058	−	0.996984i	$-0.524728\pi$
$257$	−39.8894	−0.155212	−0.0776058	+	0.996984i	$0.524728\pi$
$258$	0	0
$259$	−8.84567	−0.0341532
$260$	0	0
$261$	0	0
$262$	− 211.757i	− 0.808234i
$263$	295.014	1.12173	0.560864	−	0.827908i	$-0.310469\pi$
$263$	295.014	1.12173	0.560864	+	0.827908i	$0.310469\pi$
$264$	0	0
$265$	0	0
$266$	3.07821i	0.0115722i
$267$	0	0
$268$	− 194.191i	− 0.724593i
$269$	− 181.706i	− 0.675486i	−0.941238	−	0.337743i	$-0.890337\pi$
$269$	− 181.706i	− 0.675486i	0.941238	−	0.337743i	$-0.109663\pi$
$270$	0	0
$271$	−261.823	−0.966138	−0.483069	−	0.875582i	$-0.660478\pi$
$271$	−261.823	−0.966138	−0.483069	+	0.875582i	$0.660478\pi$
$272$	−31.0294	−0.114079
$273$	0	0
$274$	−272.735	−0.995383
$275$	0	0
$276$	0	0
$277$	305.882i	1.10427i	0.833755	+	0.552134i	$0.186186\pi$
$277$	305.882i	1.10427i	−0.833755	+	0.552134i	$0.813814\pi$
$278$	−165.598	−0.595676
$279$	0	0
$280$	0	0
$281$	− 558.323i	− 1.98691i	−0.114205	−	0.993457i	$-0.536432\pi$
$281$	− 558.323i	− 1.98691i	0.114205	−	0.993457i	$-0.463568\pi$
$282$	0	0
$283$	− 8.97056i	− 0.0316981i	−0.999874	−	0.0158491i	$-0.994955\pi$
$283$	− 8.97056i	− 0.0316981i	0.999874	−	0.0158491i	$-0.00504512\pi$
$284$	145.029i	0.510667i
$285$	0	0
$286$	−14.7868	−0.0517021
$287$	13.8823	0.0483702
$288$	0	0
$289$	−228.823	−0.791776
$290$	0	0
$291$	0	0
$292$	213.823i	0.732272i
$293$	−146.683	−0.500626	−0.250313	−	0.968165i	$-0.580533\pi$
$293$	−146.683	−0.500626	−0.250313	+	0.968165i	$0.580533\pi$
$294$	0	0
$295$	0	0
$296$	103.113i	0.348354i
$297$	0	0
$298$	198.000i	0.664430i
$299$	22.7056i	0.0759386i
$300$	0	0
$301$	−2.67619	−0.00889100
$302$	−185.137	−0.613037
$303$	0	0
$304$	35.8823	0.118034
$305$	0	0
$306$	0	0
$307$	343.640i	1.11935i	0.828713	+	0.559674i	$0.189074\pi$
$307$	343.640i	1.11935i	−0.828713	+	0.559674i	$0.810926\pi$
$308$	1.45584	0.00472677
$309$	0	0
$310$	0	0
$311$	512.044i	1.64644i	0.567720	+	0.823221i	$0.307825\pi$
$311$	512.044i	1.64644i	−0.567720	+	0.823221i	$0.692175\pi$
$312$	0	0
$313$	− 68.6173i	− 0.219225i	−0.993974	−	0.109612i	$-0.965039\pi$
$313$	− 68.6173i	− 0.219225i	0.993974	−	0.109612i	$-0.0349609\pi$
$314$	217.373i	0.692269i
$315$	0	0
$316$	156.368	0.494834
$317$	516.500	1.62934	0.814668	−	0.579928i	$-0.196919\pi$
$317$	516.500	1.62934	0.814668	+	0.579928i	$0.196919\pi$
$318$	0	0
$319$	21.0883	0.0661076
$320$	0	0
$321$	0	0
$322$	− 2.23550i	− 0.00694254i
$323$	−69.5879	−0.215442
$324$	0	0
$325$	0	0
$326$	− 257.813i	− 0.790838i
$327$	0	0
$328$	− 161.823i	− 0.493364i
$329$	1.58074i	0.00480467i
$330$	0	0
$331$	−265.110	−0.800936	−0.400468	−	0.916311i	$-0.631152\pi$
$331$	−265.110	−0.800936	−0.400468	+	0.916311i	$0.631152\pi$
$332$	51.7645	0.155917
$333$	0	0
$334$	−44.8600	−0.134311
$335$	0	0
$336$	0	0
$337$	− 72.4710i	− 0.215047i	−0.994203	−	0.107524i	$-0.965708\pi$
$337$	− 72.4710i	− 0.215047i	0.994203	−	0.107524i	$-0.0342922\pi$
$338$	−221.823	−0.656282
$339$	0	0
$340$	0	0
$341$	87.6396i	0.257008i
$342$	0	0
$343$	− 23.7645i	− 0.0692843i
$344$	31.1960i	0.0906859i
$345$	0	0
$346$	−468.323	−1.35353
$347$	−529.706	−1.52653	−0.763265	−	0.646086i	$-0.776405\pi$
$347$	−529.706	−1.52653	−0.763265	+	0.646086i	$0.776405\pi$
$348$	0	0
$349$	−337.470	−0.966963	−0.483482	−	0.875355i	$-0.660628\pi$
$349$	−337.470	−0.966963	−0.483482	+	0.875355i	$0.660628\pi$
$350$	0	0
$351$	0	0
$352$	− 16.9706i	− 0.0482118i
$353$	490.087	1.38835	0.694175	−	0.719806i	$-0.255769\pi$
$353$	490.087	1.38835	0.694175	+	0.719806i	$0.255769\pi$
$354$	0	0
$355$	0	0
$356$	− 63.5147i	− 0.178412i
$357$	0	0
$358$	203.772i	0.569195i
$359$	− 370.955i	− 1.03330i	−0.856196	−	0.516651i	$-0.827178\pi$
$359$	− 370.955i	− 1.03330i	0.856196	−	0.516651i	$-0.172822\pi$
$360$	0	0
$361$	−280.529	−0.777089
$362$	66.0315	0.182408
$363$	0	0
$364$	−1.69134	−0.00464654
$365$	0	0
$366$	0	0
$367$	393.823i	1.07309i	0.843872	+	0.536544i	$0.180270\pi$
$367$	393.823i	1.07309i	−0.843872	+	0.536544i	$0.819730\pi$
$368$	−26.0589	−0.0708122
$369$	0	0
$370$	0	0
$371$	6.52987i	0.0176007i
$372$	0	0
$373$	90.1177i	0.241603i	0.992677	+	0.120801i	$0.0385464\pi$
$373$	90.1177i	0.241603i	−0.992677	+	0.120801i	$0.961454\pi$
$374$	32.9117i	0.0879992i
$375$	0	0
$376$	18.4264	0.0490064
$377$	−24.4996	−0.0649856
$378$	0	0
$379$	−327.051	−0.862931	−0.431466	−	0.902129i	$-0.642003\pi$
$379$	−327.051	−0.862931	−0.431466	+	0.902129i	$0.642003\pi$
$380$	0	0
$381$	0	0
$382$	422.059i	1.10487i
$383$	−468.338	−1.22281	−0.611407	−	0.791316i	$-0.709396\pi$
$383$	−468.338	−1.22281	−0.611407	+	0.791316i	$0.709396\pi$
$384$	0	0
$385$	0	0
$386$	94.6274i	0.245149i
$387$	0	0
$388$	− 221.529i	− 0.570951i
$389$	80.3818i	0.206637i	0.994648	+	0.103319i	$0.0329461\pi$
$389$	80.3818i	0.206637i	−0.994648	+	0.103319i	$0.967054\pi$
$390$	0	0
$391$	50.5370	0.129251
$392$	−138.426	−0.353129
$393$	0	0
$394$	−172.617	−0.438115
$395$	0	0
$396$	0	0
$397$	489.632i	1.23333i	0.787226	+	0.616664i	$0.211516\pi$
$397$	489.632i	1.23333i	−0.787226	+	0.616664i	$0.788484\pi$
$398$	134.049	0.336806
$399$	0	0
$400$	0	0
$401$	263.418i	0.656904i	0.944521	+	0.328452i	$0.106527\pi$
$401$	263.418i	0.656904i	−0.944521	+	0.328452i	$0.893473\pi$
$402$	0	0
$403$	− 101.816i	− 0.252646i
$404$	− 150.073i	− 0.371468i
$405$	0	0
$406$	2.41212	0.00594119
$407$	109.368	0.268716
$408$	0	0
$409$	−170.411	−0.416653	−0.208327	−	0.978059i	$-0.566802\pi$
$409$	−170.411	−0.416653	−0.208327	+	0.978059i	$0.566802\pi$
$410$	0	0
$411$	0	0
$412$	122.191i	0.296580i
$413$	−20.6102	−0.0499036
$414$	0	0
$415$	0	0
$416$	19.7157i	0.0473936i
$417$	0	0
$418$	− 38.0589i	− 0.0910499i
$419$	− 353.647i	− 0.844026i	−0.906590	−	0.422013i	$-0.861324\pi$
$419$	− 353.647i	− 0.844026i	0.906590	−	0.422013i	$-0.138676\pi$
$420$	0	0
$421$	−661.749	−1.57185	−0.785926	−	0.618321i	$-0.787813\pi$
$421$	−661.749	−1.57185	−0.785926	+	0.618321i	$0.787813\pi$
$422$	−234.343	−0.555316
$423$	0	0
$424$	76.1177	0.179523
$425$	0	0
$426$	0	0
$427$	− 13.9197i	− 0.0325988i
$428$	−67.7056	−0.158191
$429$	0	0
$430$	0	0
$431$	757.544i	1.75764i	0.477151	+	0.878822i	$0.341670\pi$
$431$	757.544i	1.75764i	−0.477151	+	0.878822i	$0.658330\pi$
$432$	0	0
$433$	− 293.294i	− 0.677354i	−0.940903	−	0.338677i	$-0.890021\pi$
$433$	− 293.294i	− 0.677354i	0.940903	−	0.338677i	$-0.109979\pi$
$434$	10.0244i	0.0230977i
$435$	0	0
$436$	−38.2944	−0.0878311
$437$	−58.4407	−0.133732
$438$	0	0
$439$	−441.831	−1.00645	−0.503224	−	0.864156i	$-0.667853\pi$
$439$	−441.831	−1.00645	−0.503224	+	0.864156i	$0.667853\pi$
$440$	0	0
$441$	0	0
$442$	− 38.2355i	− 0.0865057i
$443$	−46.3827	−0.104701	−0.0523507	−	0.998629i	$-0.516671\pi$
$443$	−46.3827	−0.104701	−0.0523507	+	0.998629i	$0.516671\pi$
$444$	0	0
$445$	0	0
$446$	284.215i	0.637254i
$447$	0	0
$448$	− 1.94113i	− 0.00433287i
$449$	− 69.2132i	− 0.154150i	−0.997025	−	0.0770748i	$-0.975442\pi$
$449$	− 69.2132i	− 0.154150i	0.997025	−	0.0770748i	$-0.0245580\pi$
$450$	0	0
$451$	−171.640	−0.380576
$452$	53.5736	0.118526
$453$	0	0
$454$	−494.184	−1.08851
$455$	0	0
$456$	0	0
$457$	189.382i	0.414402i	0.978298	+	0.207201i	$0.0664354\pi$
$457$	189.382i	0.414402i	−0.978298	+	0.207201i	$0.933565\pi$
$458$	−243.973	−0.532691
$459$	0	0
$460$	0	0
$461$	− 615.058i	− 1.33418i	−0.744976	−	0.667091i	$-0.767539\pi$
$461$	− 615.058i	− 1.33418i	0.744976	−	0.667091i	$-0.232461\pi$
$462$	0	0
$463$	137.088i	0.296087i	0.988981	+	0.148044i	$0.0472976\pi$
$463$	137.088i	0.296087i	−0.988981	+	0.148044i	$0.952702\pi$
$464$	− 28.1177i	− 0.0605986i
$465$	0	0
$466$	−209.647	−0.449886
$467$	−337.118	−0.721880	−0.360940	−	0.932589i	$-0.617544\pi$
$467$	−337.118	−0.721880	−0.360940	+	0.932589i	$0.617544\pi$
$468$	0	0
$469$	−23.5593	−0.0502331
$470$	0	0
$471$	0	0
$472$	240.250i	0.509004i
$473$	33.0883	0.0699541
$474$	0	0
$475$	0	0
$476$	3.76450i	0.00790862i
$477$	0	0
$478$	200.257i	0.418948i
$479$	− 227.647i	− 0.475254i	−0.971356	−	0.237627i	$-0.923630\pi$
$479$	− 227.647i	− 0.475254i	0.971356	−	0.237627i	$-0.0763696\pi$
$480$	0	0
$481$	−127.059	−0.264156
$482$	−118.586	−0.246029
$483$	0	0
$484$	224.000	0.462810
$485$	0	0
$486$	0	0
$487$	640.874i	1.31596i	0.753034	+	0.657982i	$0.228590\pi$
$487$	640.874i	1.31596i	−0.753034	+	0.657982i	$0.771410\pi$
$488$	−162.260	−0.332500
$489$	0	0
$490$	0	0
$491$	610.118i	1.24260i	0.783572	+	0.621301i	$0.213396\pi$
$491$	610.118i	1.24260i	−0.783572	+	0.621301i	$0.786604\pi$
$492$	0	0
$493$	54.5299i	0.110608i
$494$	44.2153i	0.0895046i
$495$	0	0
$496$	116.853	0.235590
$497$	17.5950	0.0354025
$498$	0	0
$499$	−43.2935	−0.0867605	−0.0433803	−	0.999059i	$-0.513813\pi$
$499$	−43.2935	−0.0867605	−0.0433803	+	0.999059i	$0.513813\pi$
$500$	0	0
$501$	0	0
$502$	− 286.419i	− 0.570556i
$503$	−303.765	−0.603906	−0.301953	−	0.953323i	$-0.597638\pi$
$503$	−303.765	−0.603906	−0.301953	+	0.953323i	$0.597638\pi$
$504$	0	0
$505$	0	0
$506$	27.6396i	0.0546237i
$507$	0	0
$508$	− 349.838i	− 0.688657i
$509$	40.2426i	0.0790622i	0.999218	+	0.0395311i	$0.0125864\pi$
$509$	40.2426i	0.0790622i	−0.999218	+	0.0395311i	$0.987414\pi$
$510$	0	0
$511$	25.9411	0.0507654
$512$	−22.6274	−0.0441942
$513$	0	0
$514$	56.4121	0.109751
$515$	0	0
$516$	0	0
$517$	− 19.5442i	− 0.0378030i
$518$	12.5097	0.0241499
$519$	0	0
$520$	0	0
$521$	− 88.5959i	− 0.170050i	−0.996379	−	0.0850248i	$-0.972903\pi$
$521$	− 88.5959i	− 0.170050i	0.996379	−	0.0850248i	$-0.0270970\pi$
$522$	0	0
$523$	− 711.647i	− 1.36070i	−0.732887	−	0.680351i	$-0.761828\pi$
$523$	− 711.647i	− 1.36070i	0.732887	−	0.680351i	$-0.238172\pi$
$524$	299.470i	0.571508i
$525$	0	0
$526$	−417.213	−0.793181
$527$	−226.617	−0.430014
$528$	0	0
$529$	−486.558	−0.919770
$530$	0	0
$531$	0	0
$532$	− 4.35325i	− 0.00818280i
$533$	199.404	0.374117
$534$	0	0
$535$	0	0
$536$	274.627i	0.512365i
$537$	0	0
$538$	256.971i	0.477640i
$539$	146.823i	0.272400i
$540$	0	0
$541$	197.426	0.364929	0.182464	−	0.983212i	$-0.441593\pi$
$541$	197.426	0.364929	0.182464	+	0.983212i	$0.441593\pi$
$542$	370.274	0.683163
$543$	0	0
$544$	43.8823	0.0806659
$545$	0	0
$546$	0	0
$547$	453.470i	0.829013i	0.910046	+	0.414507i	$0.136046\pi$
$547$	453.470i	0.829013i	−0.910046	+	0.414507i	$0.863954\pi$
$548$	385.706	0.703842
$549$	0	0
$550$	0	0
$551$	− 63.0580i	− 0.114443i
$552$	0	0
$553$	− 18.9706i	− 0.0343048i
$554$	− 432.583i	− 0.780835i
$555$	0	0
$556$	234.191	0.421207
$557$	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
$559$	−38.4407	−0.0687669
$560$	0	0
$561$	0	0
$562$	789.588i	1.40496i
$563$	−699.323	−1.24214	−0.621068	−	0.783756i	$-0.713301\pi$
$563$	−699.323	−1.24214	−0.621068	+	0.783756i	$0.713301\pi$
$564$	0	0
$565$	0	0
$566$	12.6863i	0.0224139i
$567$	0	0
$568$	− 205.103i	− 0.361096i
$569$	792.978i	1.39363i	0.717249	+	0.696817i	$0.245401\pi$
$569$	792.978i	1.39363i	−0.717249	+	0.696817i	$0.754599\pi$
$570$	0	0
$571$	−327.418	−0.573412	−0.286706	−	0.958019i	$-0.592560\pi$
$571$	−327.418	−0.573412	−0.286706	+	0.958019i	$0.592560\pi$
$572$	20.9117	0.0365589
$573$	0	0
$574$	−19.6325	−0.0342029
$575$	0	0
$576$	0	0
$577$	844.088i	1.46289i	0.681900	+	0.731446i	$0.261154\pi$
$577$	844.088i	1.46289i	−0.681900	+	0.731446i	$0.738846\pi$
$578$	323.605	0.559870
$579$	0	0
$580$	0	0
$581$	− 6.28009i	− 0.0108091i
$582$	0	0
$583$	− 80.7351i	− 0.138482i
$584$	− 302.392i	− 0.517794i
$585$	0	0
$586$	207.442	0.353996
$587$	−304.471	−0.518690	−0.259345	−	0.965785i	$-0.583507\pi$
$587$	−304.471	−0.518690	−0.259345	+	0.965785i	$0.583507\pi$
$588$	0	0
$589$	262.059	0.444922
$590$	0	0
$591$	0	0
$592$	− 145.823i	− 0.246323i
$593$	−403.882	−0.681083	−0.340542	−	0.940229i	$-0.610610\pi$
$593$	−403.882	−0.681083	−0.340542	+	0.940229i	$0.610610\pi$
$594$	0	0
$595$	0	0
$596$	− 280.014i	− 0.469823i
$597$	0	0
$598$	− 32.1106i	− 0.0536967i
$599$	− 133.176i	− 0.222330i	−0.993802	−	0.111165i	$-0.964542\pi$
$599$	− 133.176i	− 0.222330i	0.993802	−	0.111165i	$-0.0354582\pi$
$600$	0	0
$601$	−263.294	−0.438094	−0.219047	−	0.975714i	$-0.570295\pi$
$601$	−263.294	−0.438094	−0.219047	+	0.975714i	$0.570295\pi$
$602$	3.78470	0.00628688
$603$	0	0
$604$	261.823	0.433482
$605$	0	0
$606$	0	0
$607$	− 613.449i	− 1.01062i	−0.862937	−	0.505312i	$-0.831377\pi$
$607$	− 613.449i	− 1.01062i	0.862937	−	0.505312i	$-0.168623\pi$
$608$	−50.7452	−0.0834624
$609$	0	0
$610$	0	0
$611$	22.7056i	0.0371614i
$612$	0	0
$613$	− 712.838i	− 1.16287i	−0.813594	−	0.581434i	$-0.802492\pi$
$613$	− 712.838i	− 1.16287i	0.813594	−	0.581434i	$-0.197508\pi$
$614$	− 485.980i	− 0.791498i
$615$	0	0
$616$	−2.05887	−0.00334233
$617$	192.728	0.312363	0.156181	−	0.987728i	$-0.450082\pi$
$617$	192.728	0.312363	0.156181	+	0.987728i	$0.450082\pi$
$618$	0	0
$619$	660.403	1.06689	0.533444	−	0.845836i	$-0.320898\pi$
$619$	660.403	1.06689	0.533444	+	0.845836i	$0.320898\pi$
$620$	0	0
$621$	0	0
$622$	− 724.139i	− 1.16421i
$623$	−7.70563	−0.0123686
$624$	0	0
$625$	0	0
$626$	97.0395i	0.155015i
$627$	0	0
$628$	− 307.411i	− 0.489508i
$629$	282.801i	0.449604i
$630$	0	0
$631$	−128.971	−0.204391	−0.102195	−	0.994764i	$-0.532587\pi$
$631$	−128.971	−0.204391	−0.102195	+	0.994764i	$0.532587\pi$
$632$	−221.137	−0.349900
$633$	0	0
$634$	−730.441	−1.15211
$635$	0	0
$636$	0	0
$637$	− 170.574i	− 0.267776i
$638$	−29.8234	−0.0467451
$639$	0	0
$640$	0	0
$641$	− 526.773i	− 0.821798i	−0.911681	−	0.410899i	$-0.865215\pi$
$641$	− 526.773i	− 0.821798i	0.911681	−	0.410899i	$-0.134785\pi$
$642$	0	0
$643$	558.169i	0.868071i	0.900896	+	0.434035i	$0.142911\pi$
$643$	558.169i	0.868071i	−0.900896	+	0.434035i	$0.857089\pi$
$644$	3.16147i	0.00490912i
$645$	0	0
$646$	98.4121	0.152341
$647$	−990.131	−1.53034	−0.765171	−	0.643827i	$-0.777346\pi$
$647$	−990.131	−1.53034	−0.765171	+	0.643827i	$0.777346\pi$
$648$	0	0
$649$	254.823	0.392640
$650$	0	0
$651$	0	0
$652$	364.603i	0.559207i
$653$	832.139	1.27433	0.637166	−	0.770726i	$-0.280106\pi$
$653$	832.139	1.27433	0.637166	+	0.770726i	$0.280106\pi$
$654$	0	0
$655$	0	0
$656$	228.853i	0.348861i
$657$	0	0
$658$	− 2.23550i	− 0.00339741i
$659$	239.793i	0.363874i	0.983310	+	0.181937i	$0.0582367\pi$
$659$	239.793i	0.363874i	−0.983310	+	0.181937i	$0.941763\pi$
$660$	0	0
$661$	1053.01	1.59306	0.796531	−	0.604597i	$-0.206666\pi$
$661$	1053.01	1.59306	0.796531	+	0.604597i	$0.206666\pi$
$662$	374.922	0.566347
$663$	0	0
$664$	−73.2061	−0.110250
$665$	0	0
$666$	0	0
$667$	45.7948i	0.0686579i
$668$	63.4416	0.0949724
$669$	0	0
$670$	0	0
$671$	172.103i	0.256487i
$672$	0	0
$673$	721.912i	1.07268i	0.844003	+	0.536339i	$0.180193\pi$
$673$	721.912i	1.07268i	−0.844003	+	0.536339i	$0.819807\pi$
$674$	102.489i	0.152062i
$675$	0	0
$676$	313.706	0.464062
$677$	538.419	0.795302	0.397651	−	0.917537i	$-0.369826\pi$
$677$	538.419	0.795302	0.397651	+	0.917537i	$0.369826\pi$
$678$	0	0
$679$	−26.8760	−0.0395817
$680$	0	0
$681$	0	0
$682$	− 123.941i	− 0.181732i
$683$	−335.412	−0.491087	−0.245543	−	0.969386i	$-0.578966\pi$
$683$	−335.412	−0.491087	−0.245543	+	0.969386i	$0.578966\pi$
$684$	0	0
$685$	0	0
$686$	33.6081i	0.0489914i
$687$	0	0
$688$	− 44.1177i	− 0.0641246i
$689$	93.7948i	0.136132i
$690$	0	0
$691$	225.477	0.326306	0.163153	−	0.986601i	$-0.447834\pi$
$691$	225.477	0.326306	0.163153	+	0.986601i	$0.447834\pi$
$692$	662.309	0.957093
$693$	0	0
$694$	749.117	1.07942
$695$	0	0
$696$	0	0
$697$	− 443.823i	− 0.636762i
$698$	477.255	0.683746
$699$	0	0
$700$	0	0
$701$	− 709.684i	− 1.01239i	−0.862420	−	0.506194i	$-0.831052\pi$
$701$	− 709.684i	− 1.01239i	0.862420	−	0.506194i	$-0.168948\pi$
$702$	0	0
$703$	− 327.029i	− 0.465191i
$704$	24.0000i	0.0340909i
$705$	0	0
$706$	−693.088	−0.981711
$707$	−18.2069	−0.0257524
$708$	0	0
$709$	−663.867	−0.936343	−0.468171	−	0.883638i	$-0.655087\pi$
$709$	−663.867	−0.936343	−0.468171	+	0.883638i	$0.655087\pi$
$710$	0	0
$711$	0	0
$712$	89.8234i	0.126156i
$713$	−190.316	−0.266923
$714$	0	0
$715$	0	0
$716$	− 288.177i	− 0.402481i
$717$	0	0
$718$	524.610i	0.730655i
$719$	402.161i	0.559334i	0.960097	+	0.279667i	$0.0902241\pi$
$719$	402.161i	0.559334i	−0.960097	+	0.279667i	$0.909776\pi$
$720$	0	0
$721$	14.8242	0.0205607
$722$	396.728	0.549485
$723$	0	0
$724$	−93.3827	−0.128982
$725$	0	0
$726$	0	0
$727$	− 643.294i	− 0.884860i	−0.896803	−	0.442430i	$-0.854116\pi$
$727$	− 643.294i	− 0.884860i	0.896803	−	0.442430i	$-0.145884\pi$
$728$	2.39192	0.00328560
$729$	0	0
$730$	0	0
$731$	85.5593i	0.117044i
$732$	0	0
$733$	159.191i	0.217177i	0.994087	+	0.108589i	$0.0346331\pi$
$733$	159.191i	0.217177i	−0.994087	+	0.108589i	$0.965367\pi$
$734$	− 556.950i	− 0.758788i
$735$	0	0
$736$	36.8528	0.0500718
$737$	291.286	0.395233
$738$	0	0
$739$	−115.272	−0.155984	−0.0779919	−	0.996954i	$-0.524851\pi$
$739$	−115.272	−0.155984	−0.0779919	+	0.996954i	$0.524851\pi$
$740$	0	0
$741$	0	0
$742$	− 9.23463i	− 0.0124456i
$743$	93.0732	0.125267	0.0626334	−	0.998037i	$-0.480050\pi$
$743$	93.0732	0.125267	0.0626334	+	0.998037i	$0.480050\pi$
$744$	0	0
$745$	0	0
$746$	− 127.446i	− 0.170839i
$747$	0	0
$748$	− 46.5442i	− 0.0622248i
$749$	8.21407i	0.0109667i
$750$	0	0
$751$	109.823	0.146236	0.0731181	−	0.997323i	$-0.476705\pi$
$751$	109.823	0.146236	0.0731181	+	0.997323i	$0.476705\pi$
$752$	−26.0589	−0.0346528
$753$	0	0
$754$	34.6476	0.0459518
$755$	0	0
$756$	0	0
$757$	− 305.573i	− 0.403663i	−0.979420	−	0.201831i	$-0.935311\pi$
$757$	− 305.573i	− 0.403663i	0.979420	−	0.201831i	$-0.0646893\pi$
$758$	462.520	0.610184
$759$	0	0
$760$	0	0
$761$	− 1323.79i	− 1.73953i	−0.493462	−	0.869767i	$-0.664269\pi$
$761$	− 1323.79i	− 1.73953i	0.493462	−	0.869767i	$-0.335731\pi$
$762$	0	0
$763$	4.64589i	0.00608897i
$764$	− 596.881i	− 0.781258i
$765$	0	0
$766$	662.330	0.864661
$767$	−296.044	−0.385976
$768$	0	0
$769$	1383.79	1.79947	0.899735	−	0.436436i	$-0.143759\pi$
$769$	1383.79	1.79947	0.899735	+	0.436436i	$0.143759\pi$
$770$	0	0
$771$	0	0
$772$	− 133.823i	− 0.173346i
$773$	1058.56	1.36942	0.684708	−	0.728818i	$-0.259930\pi$
$773$	1058.56	1.36942	0.684708	+	0.728818i	$0.259930\pi$
$774$	0	0
$775$	0	0
$776$	313.289i	0.403723i
$777$	0	0
$778$	− 113.677i	− 0.146114i
$779$	513.235i	0.658838i
$780$	0	0
$781$	−217.544	−0.278546
$782$	−71.4701	−0.0913940
$783$	0	0
$784$	195.765	0.249700
$785$	0	0
$786$	0	0
$787$	− 1092.76i	− 1.38852i	−0.719725	−	0.694259i	$-0.755732\pi$
$787$	− 1092.76i	− 1.38852i	0.719725	−	0.694259i	$-0.244268\pi$
$788$	244.118	0.309794
$789$	0	0
$790$	0	0
$791$	− 6.49957i	− 0.00821690i
$792$	0	0
$793$	− 199.942i	− 0.252134i
$794$	− 692.444i	− 0.872095i
$795$	0	0
$796$	−189.574	−0.238158
$797$	464.029	0.582219	0.291110	−	0.956690i	$-0.405976\pi$
$797$	464.029	0.582219	0.291110	+	0.956690i	$0.405976\pi$
$798$	0	0
$799$	50.5370	0.0632503
$800$	0	0
$801$	0	0
$802$	− 372.530i	− 0.464501i
$803$	−320.735	−0.399421
$804$	0	0
$805$	0	0
$806$	143.990i	0.178648i
$807$	0	0
$808$	212.235i	0.262668i
$809$	1324.26i	1.63691i	0.574567	+	0.818457i	$0.305170\pi$
$809$	1324.26i	1.63691i	−0.574567	+	0.818457i	$0.694830\pi$
$810$	0	0
$811$	527.433	0.650349	0.325174	−	0.945654i	$-0.394577\pi$
$811$	527.433	0.650349	0.325174	+	0.945654i	$0.394577\pi$
$812$	−3.41125	−0.00420105
$813$	0	0
$814$	−154.669	−0.190011
$815$	0	0
$816$	0	0
$817$	− 98.9403i	− 0.121102i
$818$	240.998	0.294618
$819$	0	0
$820$	0	0
$821$	− 718.169i	− 0.874750i	−0.899279	−	0.437375i	$-0.855908\pi$
$821$	− 718.169i	− 0.874750i	0.899279	−	0.437375i	$-0.144092\pi$
$822$	0	0
$823$	864.617i	1.05057i	0.850927	+	0.525284i	$0.176041\pi$
$823$	864.617i	1.05057i	−0.850927	+	0.525284i	$0.823959\pi$
$824$	− 172.804i	− 0.209714i
$825$	0	0
$826$	29.1472	0.0352872
$827$	555.323	0.671491	0.335745	−	0.941953i	$-0.391012\pi$
$827$	555.323	0.671491	0.335745	+	0.941953i	$0.391012\pi$
$828$	0	0
$829$	347.721	0.419446	0.209723	−	0.977761i	$-0.432744\pi$
$829$	347.721	0.419446	0.209723	+	0.977761i	$0.432744\pi$
$830$	0	0
$831$	0	0
$832$	− 27.8823i	− 0.0335123i
$833$	−379.654	−0.455767
$834$	0	0
$835$	0	0
$836$	53.8234i	0.0643820i
$837$	0	0
$838$	500.132i	0.596816i
$839$	529.721i	0.631372i	0.948864	+	0.315686i	$0.102235\pi$
$839$	529.721i	0.631372i	−0.948864	+	0.315686i	$0.897765\pi$
$840$	0	0
$841$	791.587	0.941245
$842$	935.855	1.11147
$843$	0	0
$844$	331.411	0.392667
$845$	0	0
$846$	0	0
$847$	− 27.1758i	− 0.0320847i
$848$	−107.647	−0.126942
$849$	0	0
$850$	0	0
$851$	237.500i	0.279083i
$852$	0	0
$853$	917.043i	1.07508i	0.843238	+	0.537540i	$0.180646\pi$
$853$	917.043i	1.07508i	−0.843238	+	0.537540i	$0.819354\pi$
$854$	19.6854i	0.0230508i
$855$	0	0
$856$	95.7502	0.111858
$857$	−287.294	−0.335232	−0.167616	−	0.985852i	$-0.553607\pi$
$857$	−287.294	−0.335232	−0.167616	+	0.985852i	$0.553607\pi$
$858$	0	0
$859$	−82.5584	−0.0961099	−0.0480550	−	0.998845i	$-0.515302\pi$
$859$	−82.5584	−0.0961099	−0.0480550	+	0.998845i	$0.515302\pi$
$860$	0	0
$861$	0	0
$862$	− 1071.33i	− 1.24284i
$863$	−1214.56	−1.40737	−0.703684	−	0.710513i	$-0.748463\pi$
$863$	−1214.56	−1.40737	−0.703684	+	0.710513i	$0.748463\pi$
$864$	0	0
$865$	0	0
$866$	414.781i	0.478962i
$867$	0	0
$868$	− 14.1766i	− 0.0163325i
$869$	234.551i	0.269909i
$870$	0	0
$871$	−338.405	−0.388525
$872$	54.1564	0.0621060
$873$	0	0
$874$	82.6476	0.0945625
$875$	0	0
$876$	0	0
$877$	− 1489.63i	− 1.69855i	−0.527948	−	0.849277i	$-0.677038\pi$
$877$	− 1489.63i	− 1.69855i	0.527948	−	0.849277i	$-0.322962\pi$
$878$	624.843	0.711666
$879$	0	0
$880$	0	0
$881$	− 1438.94i	− 1.63330i	−0.577131	−	0.816652i	$-0.695828\pi$
$881$	− 1438.94i	− 1.63330i	0.577131	−	0.816652i	$-0.304172\pi$
$882$	0	0
$883$	684.139i	0.774790i	0.921914	+	0.387395i	$0.126625\pi$
$883$	684.139i	0.774790i	−0.921914	+	0.387395i	$0.873375\pi$
$884$	54.0732i	0.0611687i
$885$	0	0
$886$	65.5950	0.0740350
$887$	−1005.57	−1.13368	−0.566839	−	0.823828i	$-0.691834\pi$
$887$	−1005.57	−1.13368	−0.566839	+	0.823828i	$0.691834\pi$
$888$	0	0
$889$	−42.4424	−0.0477418
$890$	0	0
$891$	0	0
$892$	− 401.941i	− 0.450607i
$893$	−58.4407	−0.0654431
$894$	0	0
$895$	0	0
$896$	2.74517i	0.00306380i
$897$	0	0
$898$	97.8823i	0.109000i
$899$	− 205.352i	− 0.228423i
$900$	0	0
$901$	208.764	0.231702
$902$	242.735	0.269108
$903$	0	0
$904$	−75.7645	−0.0838103
$905$	0	0
$906$	0	0
$907$	73.4416i	0.0809720i	0.999180	+	0.0404860i	$0.0128906\pi$
$907$	73.4416i	0.0809720i	−0.999180	+	0.0404860i	$0.987109\pi$
$908$	698.881	0.769693
$909$	0	0
$910$	0	0
$911$	1557.69i	1.70987i	0.518738	+	0.854933i	$0.326402\pi$
$911$	1557.69i	1.70987i	−0.518738	+	0.854933i	$0.673598\pi$
$912$	0	0
$913$	77.6468i	0.0850457i
$914$	− 267.826i	− 0.293027i
$915$	0	0
$916$	345.029	0.376670
$917$	36.3318	0.0396203
$918$	0	0
$919$	635.322	0.691319	0.345659	−	0.938360i	$-0.387655\pi$
$919$	635.322	0.691319	0.345659	+	0.938360i	$0.387655\pi$
$920$	0	0
$921$	0	0
$922$	869.823i	0.943409i
$923$	252.734	0.273818
$924$	0	0
$925$	0	0
$926$	− 193.872i	− 0.209365i
$927$	0	0
$928$	39.7645i	0.0428497i
$929$	1051.26i	1.13160i	0.824542	+	0.565801i	$0.191433\pi$
$929$	1051.26i	1.13160i	−0.824542	+	0.565801i	$0.808567\pi$
$930$	0	0
$931$	439.029	0.471568
$932$	296.485	0.318117
$933$	0	0
$934$	476.756	0.510446
$935$	0	0
$936$	0	0
$937$	− 782.381i	− 0.834985i	−0.908680	−	0.417493i	$-0.862909\pi$
$937$	− 782.381i	− 0.834985i	0.908680	−	0.417493i	$-0.137091\pi$
$938$	33.3179	0.0355201
$939$	0	0
$940$	0	0
$941$	− 1085.14i	− 1.15318i	−0.817035	−	0.576588i	$-0.804384\pi$
$941$	− 1085.14i	− 1.15318i	0.817035	−	0.576588i	$-0.195616\pi$
$942$	0	0
$943$	− 372.728i	− 0.395258i
$944$	− 339.765i	− 0.359920i
$945$	0	0
$946$	−46.7939	−0.0494651
$947$	−1358.56	−1.43459	−0.717296	−	0.696769i	$-0.754620\pi$
$947$	−1358.56	−1.43459	−0.717296	+	0.696769i	$0.754620\pi$
$948$	0	0
$949$	372.617	0.392642
$950$	0	0
$951$	0	0
$952$	− 5.32381i	− 0.00559224i
$953$	790.648	0.829641	0.414820	−	0.909903i	$-0.363844\pi$
$953$	790.648	0.829641	0.414820	+	0.909903i	$0.363844\pi$
$954$	0	0
$955$	0	0
$956$	− 283.206i	− 0.296241i
$957$	0	0
$958$	321.941i	0.336055i
$959$	− 46.7939i	− 0.0487945i
$960$	0	0
$961$	−107.589	−0.111955
$962$	179.688	0.186786
$963$	0	0
$964$	167.706	0.173968
$965$	0	0
$966$	0	0
$967$	− 1062.23i	− 1.09848i	−0.835663	−	0.549242i	$-0.814917\pi$
$967$	− 1062.23i	− 1.09848i	0.835663	−	0.549242i	$-0.185083\pi$
$968$	−316.784	−0.327256
$969$	0	0
$970$	0	0
$971$	194.500i	0.200309i	0.994972	+	0.100155i	$0.0319338\pi$
$971$	194.500i	0.200309i	−0.994972	+	0.100155i	$0.968066\pi$
$972$	0	0
$973$	− 28.4121i	− 0.0292005i
$974$	− 906.333i	− 0.930527i
$975$	0	0
$976$	229.470	0.235113
$977$	715.882	0.732735	0.366368	−	0.930470i	$-0.380601\pi$
$977$	715.882	0.732735	0.366368	+	0.930470i	$0.380601\pi$
$978$	0	0
$979$	95.2721	0.0973157
$980$	0	0
$981$	0	0
$982$	− 862.837i	− 0.878653i
$983$	−1679.98	−1.70904	−0.854519	−	0.519420i	$-0.826148\pi$
$983$	−1679.98	−1.70904	−0.854519	+	0.519420i	$0.826148\pi$
$984$	0	0
$985$	0	0
$986$	− 77.1169i	− 0.0782118i
$987$	0	0
$988$	− 62.5299i	− 0.0632893i
$989$	71.8537i	0.0726529i
$990$	0	0
$991$	655.294	0.661245	0.330622	−	0.943763i	$-0.392741\pi$
$991$	655.294	0.661245	0.330622	+	0.943763i	$0.392741\pi$
$992$	−165.255	−0.166588
$993$	0	0
$994$	−24.8831	−0.0250333
$995$	0	0
$996$	0	0
$997$	− 1169.57i	− 1.17309i	−0.809916	−	0.586546i	$-0.800487\pi$
$997$	− 1169.57i	− 1.17309i	0.809916	−	0.586546i	$-0.199513\pi$
$998$	61.2263	0.0613490
$999$	0	0

Twists

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

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

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