LMFDB - Embedded newform 768.3.b.d.127.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,3,Mod(127,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("768.127");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	768.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:	$20.9264843029$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 96)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.3
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	768.127
Dual form			768.3.b.d.127.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{3} -8.92820i q^{5} -10.9282i q^{7} +3.00000 q^{9} +O(q^{10})$	$q+1.73205 q^{3} -8.92820i q^{5} -10.9282i q^{7} +3.00000 q^{9} +1.07180 q^{11} +3.85641i q^{13} -15.4641i q^{15} +7.85641 q^{17} +17.0718 q^{19} -18.9282i q^{21} -8.00000i q^{23} -54.7128 q^{25} +5.19615 q^{27} +3.07180i q^{29} -30.6410i q^{31} +1.85641 q^{33} -97.5692 q^{35} +45.7128i q^{37} +6.67949i q^{39} +35.8564 q^{41} -74.6410 q^{43} -26.7846i q^{45} +42.1436i q^{47} -70.4256 q^{49} +13.6077 q^{51} +12.9282i q^{53} -9.56922i q^{55} +29.5692 q^{57} -44.2102 q^{59} +14.0000i q^{61} -32.7846i q^{63} +34.4308 q^{65} -80.4974 q^{67} -13.8564i q^{69} -123.138i q^{71} +85.4256 q^{73} -94.7654 q^{75} -11.7128i q^{77} -55.2154i q^{79} +9.00000 q^{81} +49.0718 q^{83} -70.1436i q^{85} +5.32051i q^{87} +105.713 q^{89} +42.1436 q^{91} -53.0718i q^{93} -152.420i q^{95} +21.1384 q^{97} +3.21539 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{9}+O(q^{10}) $ 4 * q + 12 * q^9	$ 4 q + 12 q^{9} + 32 q^{11} - 24 q^{17} + 96 q^{19} - 108 q^{25} - 48 q^{33} - 224 q^{35} + 88 q^{41} - 160 q^{43} - 60 q^{49} + 96 q^{51} - 48 q^{57} + 128 q^{59} + 304 q^{65} - 128 q^{67} + 120 q^{73} - 192 q^{75} + 36 q^{81} + 224 q^{83} + 312 q^{89} + 224 q^{91} - 248 q^{97} + 96 q^{99}+O(q^{100}) $ 4 * q + 12 * q^9 + 32 * q^11 - 24 * q^17 + 96 * q^19 - 108 * q^25 - 48 * q^33 - 224 * q^35 + 88 * q^41 - 160 * q^43 - 60 * q^49 + 96 * q^51 - 48 * q^57 + 128 * q^59 + 304 * q^65 - 128 * q^67 + 120 * q^73 - 192 * q^75 + 36 * q^81 + 224 * q^83 + 312 * q^89 + 224 * q^91 - 248 * q^97 + 96 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$.

$n$	$257$	$511$	$517$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$	0	0
$2$	0	0
$3$	1.73205	0.577350
$3$	1.73205	0.577350
$4$	0	0
$5$	− 8.92820i	− 1.78564i	−0.450413	−	0.892820i	$-0.648723\pi$
$5$	− 8.92820i	− 1.78564i	0.450413	−	0.892820i	$-0.351277\pi$
$6$	0	0
$7$	− 10.9282i	− 1.56117i	−0.625048	−	0.780586i	$-0.714921\pi$
$7$	− 10.9282i	− 1.56117i	0.625048	−	0.780586i	$-0.285079\pi$
$8$	0	0
$9$	3.00000	0.333333
$10$	0	0
$11$	1.07180	0.0974361	0.0487180	−	0.998813i	$-0.484486\pi$
$11$	1.07180	0.0974361	0.0487180	+	0.998813i	$0.484486\pi$
$12$	0	0
$13$	3.85641i	0.296647i	0.988939	+	0.148323i	$0.0473876\pi$
$13$	3.85641i	0.296647i	−0.988939	+	0.148323i	$0.952612\pi$
$14$	0	0
$15$	− 15.4641i	− 1.03094i
$16$	0	0
$17$	7.85641	0.462142	0.231071	−	0.972937i	$-0.425777\pi$
$17$	7.85641	0.462142	0.231071	+	0.972937i	$0.425777\pi$
$18$	0	0
$19$	17.0718	0.898516	0.449258	−	0.893402i	$-0.351688\pi$
$19$	17.0718	0.898516	0.449258	+	0.893402i	$0.351688\pi$
$20$	0	0
$21$	− 18.9282i	− 0.901343i
$22$	0	0
$23$	− 8.00000i	− 0.347826i	−0.984761	−	0.173913i	$-0.944359\pi$
$23$	− 8.00000i	− 0.347826i	0.984761	−	0.173913i	$-0.0556412\pi$
$24$	0	0
$25$	−54.7128	−2.18851
$26$	0	0
$27$	5.19615	0.192450
$28$	0	0
$29$	3.07180i	0.105924i	0.998597	+	0.0529620i	$0.0168662\pi$
$29$	3.07180i	0.105924i	−0.998597	+	0.0529620i	$0.983134\pi$
$30$	0	0
$31$	− 30.6410i	− 0.988420i	−0.869343	−	0.494210i	$-0.835457\pi$
$31$	− 30.6410i	− 0.988420i	0.869343	−	0.494210i	$-0.164543\pi$
$32$	0	0
$33$	1.85641	0.0562547
$34$	0	0
$35$	−97.5692	−2.78769
$36$	0	0
$37$	45.7128i	1.23548i	0.786382	+	0.617741i	$0.211952\pi$
$37$	45.7128i	1.23548i	−0.786382	+	0.617741i	$0.788048\pi$
$38$	0	0
$39$	6.67949i	0.171269i
$40$	0	0
$41$	35.8564	0.874546	0.437273	−	0.899329i	$-0.355944\pi$
$41$	35.8564	0.874546	0.437273	+	0.899329i	$0.355944\pi$
$42$	0	0
$43$	−74.6410	−1.73584	−0.867919	−	0.496706i	$-0.834543\pi$
$43$	−74.6410	−1.73584	−0.867919	+	0.496706i	$0.834543\pi$
$44$	0	0
$45$	− 26.7846i	− 0.595214i
$46$	0	0
$47$	42.1436i	0.896672i	0.893865	+	0.448336i	$0.147983\pi$
$47$	42.1436i	0.896672i	−0.893865	+	0.448336i	$0.852017\pi$
$48$	0	0
$49$	−70.4256	−1.43726
$50$	0	0
$51$	13.6077	0.266818
$52$	0	0
$53$	12.9282i	0.243928i	0.992535	+	0.121964i	$0.0389193\pi$
$53$	12.9282i	0.243928i	−0.992535	+	0.121964i	$0.961081\pi$
$54$	0	0
$55$	− 9.56922i	− 0.173986i
$56$	0	0
$57$	29.5692	0.518758
$58$	0	0
$59$	−44.2102	−0.749326	−0.374663	−	0.927161i	$-0.622242\pi$
$59$	−44.2102	−0.749326	−0.374663	+	0.927161i	$0.622242\pi$
$60$	0	0
$61$	14.0000i	0.229508i	0.993394	+	0.114754i	$0.0366080\pi$
$61$	14.0000i	0.229508i	−0.993394	+	0.114754i	$0.963392\pi$
$62$	0	0
$63$	− 32.7846i	− 0.520391i
$64$	0	0
$65$	34.4308	0.529704
$66$	0	0
$67$	−80.4974	−1.20145	−0.600727	−	0.799454i	$-0.705122\pi$
$67$	−80.4974	−1.20145	−0.600727	+	0.799454i	$0.705122\pi$
$68$	0	0
$69$	− 13.8564i	− 0.200817i
$70$	0	0
$71$	− 123.138i	− 1.73434i	−0.498009	−	0.867172i	$-0.665935\pi$
$71$	− 123.138i	− 1.73434i	0.498009	−	0.867172i	$-0.334065\pi$
$72$	0	0
$73$	85.4256	1.17021	0.585107	−	0.810956i	$-0.301052\pi$
$73$	85.4256	1.17021	0.585107	+	0.810956i	$0.301052\pi$
$74$	0	0
$75$	−94.7654	−1.26354
$76$	0	0
$77$	− 11.7128i	− 0.152114i
$78$	0	0
$79$	− 55.2154i	− 0.698929i	−0.936950	−	0.349464i	$-0.886364\pi$
$79$	− 55.2154i	− 0.698929i	0.936950	−	0.349464i	$-0.113636\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	49.0718	0.591226	0.295613	−	0.955308i	$-0.404476\pi$
$83$	49.0718	0.591226	0.295613	+	0.955308i	$0.404476\pi$
$84$	0	0
$85$	− 70.1436i	− 0.825219i
$86$	0	0
$87$	5.32051i	0.0611553i
$88$	0	0
$89$	105.713	1.18778	0.593892	−	0.804545i	$-0.297591\pi$
$89$	105.713	1.18778	0.593892	+	0.804545i	$0.297591\pi$
$90$	0	0
$91$	42.1436	0.463116
$92$	0	0
$93$	− 53.0718i	− 0.570664i
$94$	0	0
$95$	− 152.420i	− 1.60443i
$96$	0	0
$97$	21.1384	0.217922	0.108961	−	0.994046i	$-0.465248\pi$
$97$	21.1384	0.217922	0.108961	+	0.994046i	$0.465248\pi$
$98$	0	0
$99$	3.21539	0.0324787
$100$	0	0
$101$	172.928i	1.71216i	0.516843	+	0.856080i	$0.327107\pi$
$101$	172.928i	1.71216i	−0.516843	+	0.856080i	$0.672893\pi$
$102$	0	0
$103$	− 16.7846i	− 0.162957i	−0.996675	−	0.0814787i	$-0.974036\pi$
$103$	− 16.7846i	− 0.162957i	0.996675	−	0.0814787i	$-0.0259643\pi$
$104$	0	0
$105$	−168.995	−1.60947
$106$	0	0
$107$	138.067	1.29034	0.645171	−	0.764038i	$-0.276786\pi$
$107$	138.067	1.29034	0.645171	+	0.764038i	$0.276786\pi$
$108$	0	0
$109$	− 7.85641i	− 0.0720771i	−0.999350	−	0.0360386i	$-0.988526\pi$
$109$	− 7.85641i	− 0.0720771i	0.999350	−	0.0360386i	$-0.0114739\pi$
$110$	0	0
$111$	79.1769i	0.713306i
$112$	0	0
$113$	165.138	1.46140	0.730701	−	0.682698i	$-0.239193\pi$
$113$	165.138	1.46140	0.730701	+	0.682698i	$0.239193\pi$
$114$	0	0
$115$	−71.4256	−0.621092
$116$	0	0
$117$	11.5692i	0.0988822i
$118$	0	0
$119$	− 85.8564i	− 0.721482i
$120$	0	0
$121$	−119.851	−0.990506
$122$	0	0
$123$	62.1051	0.504920
$124$	0	0
$125$	265.282i	2.12226i
$126$	0	0
$127$	− 147.349i	− 1.16023i	−0.814536	−	0.580113i	$-0.803008\pi$
$127$	− 147.349i	− 1.16023i	0.814536	−	0.580113i	$-0.196992\pi$
$128$	0	0
$129$	−129.282	−1.00219
$130$	0	0
$131$	70.9282	0.541437	0.270718	−	0.962659i	$-0.412739\pi$
$131$	70.9282	0.541437	0.270718	+	0.962659i	$0.412739\pi$
$132$	0	0
$133$	− 186.564i	− 1.40274i
$134$	0	0
$135$	− 46.3923i	− 0.343647i
$136$	0	0
$137$	−206.133	−1.50462	−0.752311	−	0.658808i	$-0.771061\pi$
$137$	−206.133	−1.50462	−0.752311	+	0.658808i	$0.771061\pi$
$138$	0	0
$139$	−39.9230	−0.287216	−0.143608	−	0.989635i	$-0.545870\pi$
$139$	−39.9230	−0.287216	−0.143608	+	0.989635i	$0.545870\pi$
$140$	0	0
$141$	72.9948i	0.517694i
$142$	0	0
$143$	4.13328i	0.0289041i
$144$	0	0
$145$	27.4256	0.189142
$146$	0	0
$147$	−121.981	−0.829801
$148$	0	0
$149$	− 0.353829i	− 0.00237469i	−0.999999	−	0.00118735i	$-0.999622\pi$
$149$	− 0.353829i	− 0.00237469i	0.999999	−	0.00118735i	$-0.000377944\pi$
$150$	0	0
$151$	104.210i	0.690134i	0.938578	+	0.345067i	$0.112144\pi$
$151$	104.210i	0.690134i	−0.938578	+	0.345067i	$0.887856\pi$
$152$	0	0
$153$	23.5692	0.154047
$154$	0	0
$155$	−273.569	−1.76496
$156$	0	0
$157$	− 221.713i	− 1.41218i	−0.708120	−	0.706092i	$-0.750457\pi$
$157$	− 221.713i	− 1.41218i	0.708120	−	0.706092i	$-0.249543\pi$
$158$	0	0
$159$	22.3923i	0.140832i
$160$	0	0
$161$	−87.4256	−0.543016
$162$	0	0
$163$	175.923	1.07928	0.539641	−	0.841895i	$-0.318560\pi$
$163$	175.923	1.07928	0.539641	+	0.841895i	$0.318560\pi$
$164$	0	0
$165$	− 16.5744i	− 0.100451i
$166$	0	0
$167$	237.856i	1.42429i	0.702032	+	0.712145i	$0.252276\pi$
$167$	237.856i	1.42429i	−0.702032	+	0.712145i	$0.747724\pi$
$168$	0	0
$169$	154.128	0.912001
$170$	0	0
$171$	51.2154	0.299505
$172$	0	0
$173$	71.7795i	0.414910i	0.978245	+	0.207455i	$0.0665181\pi$
$173$	71.7795i	0.414910i	−0.978245	+	0.207455i	$0.933482\pi$
$174$	0	0
$175$	597.913i	3.41664i
$176$	0	0
$177$	−76.5744	−0.432624
$178$	0	0
$179$	−5.93336	−0.0331473	−0.0165736	−	0.999863i	$-0.505276\pi$
$179$	−5.93336	−0.0331473	−0.0165736	+	0.999863i	$0.505276\pi$
$180$	0	0
$181$	− 150.995i	− 0.834226i	−0.908855	−	0.417113i	$-0.863042\pi$
$181$	− 150.995i	− 0.834226i	0.908855	−	0.417113i	$-0.136958\pi$
$182$	0	0
$183$	24.2487i	0.132507i
$184$	0	0
$185$	408.133	2.20613
$186$	0	0
$187$	8.42047	0.0450293
$188$	0	0
$189$	− 56.7846i	− 0.300448i
$190$	0	0
$191$	− 23.4256i	− 0.122647i	−0.998118	−	0.0613236i	$-0.980468\pi$
$191$	− 23.4256i	− 0.122647i	0.998118	−	0.0613236i	$-0.0195322\pi$
$192$	0	0
$193$	137.426	0.712050	0.356025	−	0.934476i	$-0.384132\pi$
$193$	137.426	0.712050	0.356025	+	0.934476i	$0.384132\pi$
$194$	0	0
$195$	59.6359	0.305825
$196$	0	0
$197$	39.4923i	0.200468i	0.994964	+	0.100234i	$0.0319592\pi$
$197$	39.4923i	0.200468i	−0.994964	+	0.100234i	$0.968041\pi$
$198$	0	0
$199$	− 275.923i	− 1.38655i	−0.720674	−	0.693274i	$-0.756168\pi$
$199$	− 275.923i	− 1.38655i	0.720674	−	0.693274i	$-0.243832\pi$
$200$	0	0
$201$	−139.426	−0.693660
$202$	0	0
$203$	33.5692	0.165366
$204$	0	0
$205$	− 320.133i	− 1.56163i
$206$	0	0
$207$	− 24.0000i	− 0.115942i
$208$	0	0
$209$	18.2975	0.0875478
$210$	0	0
$211$	−189.359	−0.897436	−0.448718	−	0.893673i	$-0.648119\pi$
$211$	−189.359	−0.897436	−0.448718	+	0.893673i	$0.648119\pi$
$212$	0	0
$213$	− 213.282i	− 1.00132i
$214$	0	0
$215$	666.410i	3.09958i
$216$	0	0
$217$	−334.851	−1.54309
$218$	0	0
$219$	147.962	0.675623
$220$	0	0
$221$	30.2975i	0.137093i
$222$	0	0
$223$	− 144.210i	− 0.646683i	−0.946282	−	0.323341i	$-0.895194\pi$
$223$	− 144.210i	− 0.646683i	0.946282	−	0.323341i	$-0.104806\pi$
$224$	0	0
$225$	−164.138	−0.729504
$226$	0	0
$227$	225.072	0.991506	0.495753	−	0.868464i	$-0.334892\pi$
$227$	225.072	0.991506	0.495753	+	0.868464i	$0.334892\pi$
$228$	0	0
$229$	233.846i	1.02116i	0.859830	+	0.510581i	$0.170570\pi$
$229$	233.846i	1.02116i	−0.859830	+	0.510581i	$0.829430\pi$
$230$	0	0
$231$	− 20.2872i	− 0.0878233i
$232$	0	0
$233$	218.862	0.939320	0.469660	−	0.882847i	$-0.344377\pi$
$233$	218.862	0.939320	0.469660	+	0.882847i	$0.344377\pi$
$234$	0	0
$235$	376.267	1.60113
$236$	0	0
$237$	− 95.6359i	− 0.403527i
$238$	0	0
$239$	371.979i	1.55640i	0.628017	+	0.778200i	$0.283867\pi$
$239$	371.979i	1.55640i	−0.628017	+	0.778200i	$0.716133\pi$
$240$	0	0
$241$	−328.564	−1.36334	−0.681668	−	0.731661i	$-0.738745\pi$
$241$	−328.564	−1.36334	−0.681668	+	0.731661i	$0.738745\pi$
$242$	0	0
$243$	15.5885	0.0641500
$244$	0	0
$245$	628.774i	2.56643i
$246$	0	0
$247$	65.8358i	0.266542i
$248$	0	0
$249$	84.9948	0.341345
$250$	0	0
$251$	148.210	0.590479	0.295240	−	0.955423i	$-0.404601\pi$
$251$	148.210	0.590479	0.295240	+	0.955423i	$0.404601\pi$
$252$	0	0
$253$	− 8.57437i	− 0.0338908i
$254$	0	0
$255$	− 121.492i	− 0.476440i
$256$	0	0
$257$	351.703	1.36849	0.684246	−	0.729251i	$-0.260131\pi$
$257$	351.703	1.36849	0.684246	+	0.729251i	$0.260131\pi$
$258$	0	0
$259$	499.559	1.92880
$260$	0	0
$261$	9.21539i	0.0353080i
$262$	0	0
$263$	− 242.144i	− 0.920698i	−0.887738	−	0.460349i	$-0.847724\pi$
$263$	− 242.144i	− 0.920698i	0.887738	−	0.460349i	$-0.152276\pi$
$264$	0	0
$265$	115.426	0.435568
$266$	0	0
$267$	183.100	0.685768
$268$	0	0
$269$	75.6462i	0.281213i	0.990066	+	0.140606i	$0.0449052\pi$
$269$	75.6462i	0.281213i	−0.990066	+	0.140606i	$0.955095\pi$
$270$	0	0
$271$	− 150.795i	− 0.556439i	−0.960518	−	0.278219i	$-0.910256\pi$
$271$	− 150.795i	− 0.556439i	0.960518	−	0.278219i	$-0.0897442\pi$
$272$	0	0
$273$	72.9948	0.267380
$274$	0	0
$275$	−58.6410	−0.213240
$276$	0	0
$277$	213.559i	0.770971i	0.922714	+	0.385485i	$0.125966\pi$
$277$	213.559i	0.770971i	−0.922714	+	0.385485i	$0.874034\pi$
$278$	0	0
$279$	− 91.9230i	− 0.329473i
$280$	0	0
$281$	−194.000	−0.690391	−0.345196	−	0.938531i	$-0.612187\pi$
$281$	−194.000	−0.690391	−0.345196	+	0.938531i	$0.612187\pi$
$282$	0	0
$283$	179.790	0.635300	0.317650	−	0.948208i	$-0.397106\pi$
$283$	179.790	0.635300	0.317650	+	0.948208i	$0.397106\pi$
$284$	0	0
$285$	− 264.000i	− 0.926316i
$286$	0	0
$287$	− 391.846i	− 1.36532i
$288$	0	0
$289$	−227.277	−0.786425
$290$	0	0
$291$	36.6128	0.125817
$292$	0	0
$293$	− 517.902i	− 1.76759i	−0.467879	−	0.883793i	$-0.654982\pi$
$293$	− 517.902i	− 1.76759i	0.467879	−	0.883793i	$-0.345018\pi$
$294$	0	0
$295$	394.718i	1.33803i
$296$	0	0
$297$	5.56922	0.0187516
$298$	0	0
$299$	30.8513	0.103181
$300$	0	0
$301$	815.692i	2.70994i
$302$	0	0
$303$	299.520i	0.988516i
$304$	0	0
$305$	124.995	0.409819
$306$	0	0
$307$	407.769	1.32824	0.664119	−	0.747627i	$-0.268807\pi$
$307$	407.769	1.32824	0.664119	+	0.747627i	$0.268807\pi$
$308$	0	0
$309$	− 29.0718i	− 0.0940835i
$310$	0	0
$311$	− 368.995i	− 1.18648i	−0.805026	−	0.593239i	$-0.797849\pi$
$311$	− 368.995i	− 1.18648i	0.805026	−	0.593239i	$-0.202151\pi$
$312$	0	0
$313$	−341.979	−1.09259	−0.546293	−	0.837594i	$-0.683961\pi$
$313$	−341.979	−1.09259	−0.546293	+	0.837594i	$0.683961\pi$
$314$	0	0
$315$	−292.708	−0.929231
$316$	0	0
$317$	396.487i	1.25075i	0.780325	+	0.625374i	$0.215054\pi$
$317$	396.487i	1.25075i	−0.780325	+	0.625374i	$0.784946\pi$
$318$	0	0
$319$	3.29234i	0.0103208i
$320$	0	0
$321$	239.138	0.744980
$322$	0	0
$323$	134.123	0.415241
$324$	0	0
$325$	− 210.995i	− 0.649215i
$326$	0	0
$327$	− 13.6077i	− 0.0416137i
$328$	0	0
$329$	460.554	1.39986
$330$	0	0
$331$	−267.061	−0.806832	−0.403416	−	0.915017i	$-0.632177\pi$
$331$	−267.061	−0.806832	−0.403416	+	0.915017i	$0.632177\pi$
$332$	0	0
$333$	137.138i	0.411827i
$334$	0	0
$335$	718.697i	2.14537i
$336$	0	0
$337$	−149.426	−0.443399	−0.221700	−	0.975115i	$-0.571160\pi$
$337$	−149.426	−0.443399	−0.221700	+	0.975115i	$0.571160\pi$
$338$	0	0
$339$	286.028	0.843741
$340$	0	0
$341$	− 32.8409i	− 0.0963077i
$342$	0	0
$343$	234.144i	0.682634i
$344$	0	0
$345$	−123.713	−0.358588
$346$	0	0
$347$	529.913	1.52713	0.763563	−	0.645733i	$-0.223448\pi$
$347$	529.913	1.52713	0.763563	+	0.645733i	$0.223448\pi$
$348$	0	0
$349$	75.7025i	0.216913i	0.994101	+	0.108456i	$0.0345908\pi$
$349$	75.7025i	0.216913i	−0.994101	+	0.108456i	$0.965409\pi$
$350$	0	0
$351$	20.0385i	0.0570897i
$352$	0	0
$353$	325.138	0.921072	0.460536	−	0.887641i	$-0.347657\pi$
$353$	325.138	0.921072	0.460536	+	0.887641i	$0.347657\pi$
$354$	0	0
$355$	−1099.41	−3.09692
$356$	0	0
$357$	− 148.708i	− 0.416548i
$358$	0	0
$359$	− 195.713i	− 0.545161i	−0.962133	−	0.272581i	$-0.912123\pi$
$359$	− 195.713i	− 0.545161i	0.962133	−	0.272581i	$-0.0878771\pi$
$360$	0	0
$361$	−69.5538	−0.192670
$362$	0	0
$363$	−207.588	−0.571869
$364$	0	0
$365$	− 762.697i	− 2.08958i
$366$	0	0
$367$	257.051i	0.700412i	0.936673	+	0.350206i	$0.113888\pi$
$367$	257.051i	0.700412i	−0.936673	+	0.350206i	$0.886112\pi$
$368$	0	0
$369$	107.569	0.291515
$370$	0	0
$371$	141.282	0.380814
$372$	0	0
$373$	− 339.128i	− 0.909191i	−0.890698	−	0.454595i	$-0.849784\pi$
$373$	− 339.128i	− 0.909191i	0.890698	−	0.454595i	$-0.150216\pi$
$374$	0	0
$375$	459.482i	1.22529i
$376$	0	0
$377$	−11.8461	−0.0314220
$378$	0	0
$379$	−163.215	−0.430647	−0.215324	−	0.976543i	$-0.569081\pi$
$379$	−163.215	−0.430647	−0.215324	+	0.976543i	$0.569081\pi$
$380$	0	0
$381$	− 255.215i	− 0.669857i
$382$	0	0
$383$	− 76.5538i	− 0.199879i	−0.994994	−	0.0999396i	$-0.968135\pi$
$383$	− 76.5538i	− 0.199879i	0.994994	−	0.0999396i	$-0.0318650\pi$
$384$	0	0
$385$	−104.574	−0.271622
$386$	0	0
$387$	−223.923	−0.578613
$388$	0	0
$389$	338.477i	0.870120i	0.900401	+	0.435060i	$0.143273\pi$
$389$	338.477i	0.870120i	−0.900401	+	0.435060i	$0.856727\pi$
$390$	0	0
$391$	− 62.8513i	− 0.160745i
$392$	0	0
$393$	122.851	0.312599
$394$	0	0
$395$	−492.974	−1.24804
$396$	0	0
$397$	7.72312i	0.0194537i	0.999953	+	0.00972685i	$0.00309620\pi$
$397$	7.72312i	0.0194537i	−0.999953	+	0.00972685i	$0.996904\pi$
$398$	0	0
$399$	− 323.138i	− 0.809871i
$400$	0	0
$401$	448.123	1.11751	0.558757	−	0.829332i	$-0.311279\pi$
$401$	448.123	1.11751	0.558757	+	0.829332i	$0.311279\pi$
$402$	0	0
$403$	118.164	0.293211
$404$	0	0
$405$	− 80.3538i	− 0.198405i
$406$	0	0
$407$	48.9948i	0.120380i
$408$	0	0
$409$	317.692	0.776754	0.388377	−	0.921501i	$-0.373036\pi$
$409$	317.692	0.776754	0.388377	+	0.921501i	$0.373036\pi$
$410$	0	0
$411$	−357.033	−0.868694
$412$	0	0
$413$	483.138i	1.16983i
$414$	0	0
$415$	− 438.123i	− 1.05572i
$416$	0	0
$417$	−69.1487	−0.165824
$418$	0	0
$419$	−660.056	−1.57531	−0.787657	−	0.616114i	$-0.788706\pi$
$419$	−660.056	−1.57531	−0.787657	+	0.616114i	$0.788706\pi$
$420$	0	0
$421$	− 42.4411i	− 0.100810i	−0.998729	−	0.0504051i	$-0.983949\pi$
$421$	− 42.4411i	− 0.100810i	0.998729	−	0.0504051i	$-0.0160512\pi$
$422$	0	0
$423$	126.431i	0.298891i
$424$	0	0
$425$	−429.846	−1.01140
$426$	0	0
$427$	152.995	0.358302
$428$	0	0
$429$	7.15906i	0.0166878i
$430$	0	0
$431$	− 302.123i	− 0.700981i	−0.936566	−	0.350491i	$-0.886015\pi$
$431$	− 302.123i	− 0.700981i	0.936566	−	0.350491i	$-0.113985\pi$
$432$	0	0
$433$	376.277	0.869000	0.434500	−	0.900672i	$-0.356925\pi$
$433$	376.277	0.869000	0.434500	+	0.900672i	$0.356925\pi$
$434$	0	0
$435$	47.5026	0.109201
$436$	0	0
$437$	− 136.574i	− 0.312527i
$438$	0	0
$439$	394.928i	0.899609i	0.893127	+	0.449804i	$0.148506\pi$
$439$	394.928i	0.899609i	−0.893127	+	0.449804i	$0.851494\pi$
$440$	0	0
$441$	−211.277	−0.479086
$442$	0	0
$443$	666.487	1.50449	0.752243	−	0.658886i	$-0.228972\pi$
$443$	666.487	1.50449	0.752243	+	0.658886i	$0.228972\pi$
$444$	0	0
$445$	− 943.825i	− 2.12096i
$446$	0	0
$447$	− 0.612850i	− 0.00137103i
$448$	0	0
$449$	−793.825	−1.76799	−0.883993	−	0.467501i	$-0.845155\pi$
$449$	−793.825	−1.76799	−0.883993	+	0.467501i	$0.845155\pi$
$450$	0	0
$451$	38.4308	0.0852124
$452$	0	0
$453$	180.497i	0.398449i
$454$	0	0
$455$	− 376.267i	− 0.826959i
$456$	0	0
$457$	−693.138	−1.51671	−0.758357	−	0.651839i	$-0.773998\pi$
$457$	−693.138	−1.51671	−0.758357	+	0.651839i	$0.773998\pi$
$458$	0	0
$459$	40.8231	0.0889392
$460$	0	0
$461$	− 99.0924i	− 0.214951i	−0.994208	−	0.107476i	$-0.965723\pi$
$461$	− 99.0924i	− 0.214951i	0.994208	−	0.107476i	$-0.0342767\pi$
$462$	0	0
$463$	308.077i	0.665393i	0.943034	+	0.332696i	$0.107958\pi$
$463$	308.077i	0.665393i	−0.943034	+	0.332696i	$0.892042\pi$
$464$	0	0
$465$	−473.836	−1.01900
$466$	0	0
$467$	−295.195	−0.632109	−0.316054	−	0.948741i	$-0.602358\pi$
$467$	−295.195	−0.632109	−0.316054	+	0.948741i	$0.602358\pi$
$468$	0	0
$469$	879.692i	1.87568i
$470$	0	0
$471$	− 384.018i	− 0.815325i
$472$	0	0
$473$	−80.0000	−0.169133
$474$	0	0
$475$	−934.046	−1.96641
$476$	0	0
$477$	38.7846i	0.0813095i
$478$	0	0
$479$	− 291.559i	− 0.608682i	−0.952563	−	0.304341i	$-0.901564\pi$
$479$	− 291.559i	− 0.608682i	0.952563	−	0.304341i	$-0.0984363\pi$
$480$	0	0
$481$	−176.287	−0.366501
$482$	0	0
$483$	−151.426	−0.313511
$484$	0	0
$485$	− 188.728i	− 0.389130i
$486$	0	0
$487$	− 729.779i	− 1.49852i	−0.662276	−	0.749260i	$-0.730409\pi$
$487$	− 729.779i	− 1.49852i	0.662276	−	0.749260i	$-0.269591\pi$
$488$	0	0
$489$	304.708	0.623124
$490$	0	0
$491$	−258.487	−0.526450	−0.263225	−	0.964734i	$-0.584786\pi$
$491$	−258.487	−0.526450	−0.263225	+	0.964734i	$0.584786\pi$
$492$	0	0
$493$	24.1333i	0.0489519i
$494$	0	0
$495$	− 28.7077i	− 0.0579953i
$496$	0	0
$497$	−1345.68	−2.70761
$498$	0	0
$499$	495.195	0.992374	0.496187	−	0.868216i	$-0.334733\pi$
$499$	495.195	0.992374	0.496187	+	0.868216i	$0.334733\pi$
$500$	0	0
$501$	411.979i	0.822314i
$502$	0	0
$503$	− 698.831i	− 1.38933i	−0.719336	−	0.694663i	$-0.755554\pi$
$503$	− 698.831i	− 1.38933i	0.719336	−	0.694663i	$-0.244446\pi$
$504$	0	0
$505$	1543.94	3.05730
$506$	0	0
$507$	266.958	0.526544
$508$	0	0
$509$	− 21.1948i	− 0.0416400i	−0.999783	−	0.0208200i	$-0.993372\pi$
$509$	− 21.1948i	− 0.0416400i	0.999783	−	0.0208200i	$-0.00662769\pi$
$510$	0	0
$511$	− 933.549i	− 1.82691i
$512$	0	0
$513$	88.7077	0.172919
$514$	0	0
$515$	−149.856	−0.290983
$516$	0	0
$517$	45.1694i	0.0873682i
$518$	0	0
$519$	124.326i	0.239548i
$520$	0	0
$521$	−167.015	−0.320567	−0.160284	−	0.987071i	$-0.551241\pi$
$521$	−167.015	−0.320567	−0.160284	+	0.987071i	$0.551241\pi$
$522$	0	0
$523$	−450.908	−0.862156	−0.431078	−	0.902315i	$-0.641867\pi$
$523$	−450.908	−0.862156	−0.431078	+	0.902315i	$0.641867\pi$
$524$	0	0
$525$	1035.62i	1.97260i
$526$	0	0
$527$	− 240.728i	− 0.456790i
$528$	0	0
$529$	465.000	0.879017
$530$	0	0
$531$	−132.631	−0.249775
$532$	0	0
$533$	138.277i	0.259431i
$534$	0	0
$535$	− 1232.69i	− 2.30409i
$536$	0	0
$537$	−10.2769	−0.0191376
$538$	0	0
$539$	−75.4820	−0.140041
$540$	0	0
$541$	79.5692i	0.147078i	0.997292	+	0.0735390i	$0.0234293\pi$
$541$	79.5692i	0.147078i	−0.997292	+	0.0735390i	$0.976571\pi$
$542$	0	0
$543$	− 261.531i	− 0.481640i
$544$	0	0
$545$	−70.1436	−0.128704
$546$	0	0
$547$	349.933	0.639732	0.319866	−	0.947463i	$-0.396362\pi$
$547$	349.933	0.639732	0.319866	+	0.947463i	$0.396362\pi$
$548$	0	0
$549$	42.0000i	0.0765027i
$550$	0	0
$551$	52.4411i	0.0951744i
$552$	0	0
$553$	−603.405	−1.09115
$554$	0	0
$555$	706.908	1.27371
$556$	0	0
$557$	− 874.210i	− 1.56950i	−0.619814	−	0.784749i	$-0.712792\pi$
$557$	− 874.210i	− 1.56950i	0.619814	−	0.784749i	$-0.287208\pi$
$558$	0	0
$559$	− 287.846i	− 0.514930i
$560$	0	0
$561$	14.5847	0.0259977
$562$	0	0
$563$	104.497	0.185608	0.0928041	−	0.995684i	$-0.470417\pi$
$563$	104.497	0.185608	0.0928041	+	0.995684i	$0.470417\pi$
$564$	0	0
$565$	− 1474.39i	− 2.60954i
$566$	0	0
$567$	− 98.3538i	− 0.173464i
$568$	0	0
$569$	−136.697	−0.240241	−0.120121	−	0.992759i	$-0.538328\pi$
$569$	−136.697	−0.240241	−0.120121	+	0.992759i	$0.538328\pi$
$570$	0	0
$571$	566.928	0.992869	0.496435	−	0.868074i	$-0.334642\pi$
$571$	566.928	0.992869	0.496435	+	0.868074i	$0.334642\pi$
$572$	0	0
$573$	− 40.5744i	− 0.0708104i
$574$	0	0
$575$	437.703i	0.761222i
$576$	0	0
$577$	778.574	1.34935	0.674675	−	0.738115i	$-0.264284\pi$
$577$	778.574	1.34935	0.674675	+	0.738115i	$0.264284\pi$
$578$	0	0
$579$	238.028	0.411102
$580$	0	0
$581$	− 536.267i	− 0.923006i
$582$	0	0
$583$	13.8564i	0.0237674i
$584$	0	0
$585$	103.292	0.176568
$586$	0	0
$587$	−1053.89	−1.79539	−0.897693	−	0.440621i	$-0.854758\pi$
$587$	−1053.89	−1.79539	−0.897693	+	0.440621i	$0.854758\pi$
$588$	0	0
$589$	− 523.097i	− 0.888111i
$590$	0	0
$591$	68.4026i	0.115740i
$592$	0	0
$593$	544.543	0.918286	0.459143	−	0.888362i	$-0.348157\pi$
$593$	544.543	0.918286	0.459143	+	0.888362i	$0.348157\pi$
$594$	0	0
$595$	−766.543	−1.28831
$596$	0	0
$597$	− 477.913i	− 0.800524i
$598$	0	0
$599$	948.246i	1.58305i	0.611138	+	0.791524i	$0.290712\pi$
$599$	948.246i	1.58305i	−0.611138	+	0.791524i	$0.709288\pi$
$600$	0	0
$601$	542.000	0.901830	0.450915	−	0.892567i	$-0.351098\pi$
$601$	542.000	0.901830	0.450915	+	0.892567i	$0.351098\pi$
$602$	0	0
$603$	−241.492	−0.400485
$604$	0	0
$605$	1070.06i	1.76869i
$606$	0	0
$607$	1096.06i	1.80569i	0.429962	+	0.902847i	$0.358527\pi$
$607$	1096.06i	1.80569i	−0.429962	+	0.902847i	$0.641473\pi$
$608$	0	0
$609$	58.1436	0.0954739
$610$	0	0
$611$	−162.523	−0.265995
$612$	0	0
$613$	738.000i	1.20392i	0.798528	+	0.601958i	$0.205612\pi$
$613$	738.000i	1.20392i	−0.798528	+	0.601958i	$0.794388\pi$
$614$	0	0
$615$	− 554.487i	− 0.901605i
$616$	0	0
$617$	385.979	0.625574	0.312787	−	0.949823i	$-0.398737\pi$
$617$	385.979	0.625574	0.312787	+	0.949823i	$0.398737\pi$
$618$	0	0
$619$	558.887	0.902887	0.451443	−	0.892300i	$-0.350909\pi$
$619$	558.887	0.902887	0.451443	+	0.892300i	$0.350909\pi$
$620$	0	0
$621$	− 41.5692i	− 0.0669392i
$622$	0	0
$623$	− 1155.25i	− 1.85434i
$624$	0	0
$625$	1000.67	1.60107
$626$	0	0
$627$	31.6922	0.0505458
$628$	0	0
$629$	359.138i	0.570967i
$630$	0	0
$631$	420.231i	0.665976i	0.942931	+	0.332988i	$0.108057\pi$
$631$	420.231i	0.665976i	−0.942931	+	0.332988i	$0.891943\pi$
$632$	0	0
$633$	−327.979	−0.518135
$634$	0	0
$635$	−1315.56	−2.07175
$636$	0	0
$637$	− 271.590i	− 0.426358i
$638$	0	0
$639$	− 369.415i	− 0.578115i
$640$	0	0
$641$	256.964	0.400880	0.200440	−	0.979706i	$-0.435763\pi$
$641$	256.964	0.400880	0.200440	+	0.979706i	$0.435763\pi$
$642$	0	0
$643$	−467.215	−0.726618	−0.363309	−	0.931669i	$-0.618353\pi$
$643$	−467.215	−0.726618	−0.363309	+	0.931669i	$0.618353\pi$
$644$	0	0
$645$	1154.26i	1.78954i
$646$	0	0
$647$	971.138i	1.50099i	0.660878	+	0.750493i	$0.270184\pi$
$647$	971.138i	1.50099i	−0.660878	+	0.750493i	$0.729816\pi$
$648$	0	0
$649$	−47.3844	−0.0730114
$650$	0	0
$651$	−579.979	−0.890905
$652$	0	0
$653$	− 439.380i	− 0.672863i	−0.941708	−	0.336432i	$-0.890780\pi$
$653$	− 439.380i	− 0.672863i	0.941708	−	0.336432i	$-0.109220\pi$
$654$	0	0
$655$	− 633.261i	− 0.966811i
$656$	0	0
$657$	256.277	0.390071
$658$	0	0
$659$	−1092.48	−1.65778	−0.828890	−	0.559412i	$-0.811027\pi$
$659$	−1092.48	−1.65778	−0.828890	+	0.559412i	$0.811027\pi$
$660$	0	0
$661$	122.267i	0.184972i	0.995714	+	0.0924861i	$0.0294814\pi$
$661$	122.267i	0.184972i	−0.995714	+	0.0924861i	$0.970519\pi$
$662$	0	0
$663$	52.4768i	0.0791505i
$664$	0	0
$665$	−1665.68	−2.50478
$666$	0	0
$667$	24.5744	0.0368431
$668$	0	0
$669$	− 249.779i	− 0.373362i
$670$	0	0
$671$	15.0052i	0.0223624i
$672$	0	0
$673$	−796.851	−1.18403	−0.592014	−	0.805927i	$-0.701667\pi$
$673$	−796.851	−1.18403	−0.592014	+	0.805927i	$0.701667\pi$
$674$	0	0
$675$	−284.296	−0.421179
$676$	0	0
$677$	− 1185.19i	− 1.75066i	−0.483529	−	0.875328i	$-0.660645\pi$
$677$	− 1185.19i	− 1.75066i	0.483529	−	0.875328i	$-0.339355\pi$
$678$	0	0
$679$	− 231.005i	− 0.340214i
$680$	0	0
$681$	389.836	0.572446
$682$	0	0
$683$	850.446	1.24516	0.622581	−	0.782555i	$-0.286084\pi$
$683$	850.446	1.24516	0.622581	+	0.782555i	$0.286084\pi$
$684$	0	0
$685$	1840.40i	2.68672i
$686$	0	0
$687$	405.033i	0.589568i
$688$	0	0
$689$	−49.8564	−0.0723605
$690$	0	0
$691$	−1161.49	−1.68089	−0.840443	−	0.541900i	$-0.817705\pi$
$691$	−1161.49	−1.68089	−0.840443	+	0.541900i	$0.817705\pi$
$692$	0	0
$693$	− 35.1384i	− 0.0507048i
$694$	0	0
$695$	356.441i	0.512865i
$696$	0	0
$697$	281.703	0.404164
$698$	0	0
$699$	379.079	0.542317
$700$	0	0
$701$	− 177.215i	− 0.252804i	−0.991979	−	0.126402i	$-0.959657\pi$
$701$	− 177.215i	− 0.252804i	0.991979	−	0.126402i	$-0.0403429\pi$
$702$	0	0
$703$	780.400i	1.11010i
$704$	0	0
$705$	651.713	0.924415
$706$	0	0
$707$	1889.79	2.67298
$708$	0	0
$709$	329.846i	0.465227i	0.972569	+	0.232614i	$0.0747277\pi$
$709$	329.846i	0.465227i	−0.972569	+	0.232614i	$0.925272\pi$
$710$	0	0
$711$	− 165.646i	− 0.232976i
$712$	0	0
$713$	−245.128	−0.343798
$714$	0	0
$715$	36.9028	0.0516123
$716$	0	0
$717$	644.287i	0.898587i
$718$	0	0
$719$	1343.27i	1.86825i	0.356946	+	0.934125i	$0.383818\pi$
$719$	1343.27i	1.86825i	−0.356946	+	0.934125i	$0.616182\pi$
$720$	0	0
$721$	−183.426	−0.254404
$722$	0	0
$723$	−569.090	−0.787123
$724$	0	0
$725$	− 168.067i	− 0.231816i
$726$	0	0
$727$	− 6.64102i	− 0.00913482i	−0.999990	−	0.00456741i	$-0.998546\pi$
$727$	− 6.64102i	− 0.00913482i	0.999990	−	0.00456741i	$-0.00145386\pi$
$728$	0	0
$729$	27.0000	0.0370370
$730$	0	0
$731$	−586.410	−0.802203
$732$	0	0
$733$	1396.39i	1.90503i	0.304486	+	0.952517i	$0.401515\pi$
$733$	1396.39i	1.90503i	−0.304486	+	0.952517i	$0.598485\pi$
$734$	0	0
$735$	1089.07i	1.48173i
$736$	0	0
$737$	−86.2769	−0.117065
$738$	0	0
$739$	276.939	0.374748	0.187374	−	0.982289i	$-0.440002\pi$
$739$	276.939	0.374748	0.187374	+	0.982289i	$0.440002\pi$
$740$	0	0
$741$	114.031i	0.153888i
$742$	0	0
$743$	− 424.728i	− 0.571640i	−0.958283	−	0.285820i	$-0.907734\pi$
$743$	− 424.728i	− 0.571640i	0.958283	−	0.285820i	$-0.0922659\pi$
$744$	0	0
$745$	−3.15906	−0.00424035
$746$	0	0
$747$	147.215	0.197075
$748$	0	0
$749$	− 1508.82i	− 2.01445i
$750$	0	0
$751$	− 1008.32i	− 1.34264i	−0.741167	−	0.671320i	$-0.765728\pi$
$751$	− 1008.32i	− 1.34264i	0.741167	−	0.671320i	$-0.234272\pi$
$752$	0	0
$753$	256.708	0.340913
$754$	0	0
$755$	930.410	1.23233
$756$	0	0
$757$	− 2.70766i	− 0.00357683i	−0.999998	−	0.00178841i	$-0.999431\pi$
$757$	− 2.70766i	− 0.00357683i	0.999998	−	0.00178841i	$-0.000569270\pi$
$758$	0	0
$759$	− 14.8513i	− 0.0195669i
$760$	0	0
$761$	−656.431	−0.862590	−0.431295	−	0.902211i	$-0.641943\pi$
$761$	−656.431	−0.862590	−0.431295	+	0.902211i	$0.641943\pi$
$762$	0	0
$763$	−85.8564	−0.112525
$764$	0	0
$765$	− 210.431i	− 0.275073i
$766$	0	0
$767$	− 170.493i	− 0.222285i
$768$	0	0
$769$	−533.959	−0.694355	−0.347177	−	0.937799i	$-0.612860\pi$
$769$	−533.959	−0.694355	−0.347177	+	0.937799i	$0.612860\pi$
$770$	0	0
$771$	609.167	0.790099
$772$	0	0
$773$	350.918i	0.453969i	0.973898	+	0.226984i	$0.0728866\pi$
$773$	350.918i	0.453969i	−0.973898	+	0.226984i	$0.927113\pi$
$774$	0	0
$775$	1676.46i	2.16317i
$776$	0	0
$777$	865.261	1.11359
$778$	0	0
$779$	612.133	0.785794
$780$	0	0
$781$	− 131.979i	− 0.168988i
$782$	0	0
$783$	15.9615i	0.0203851i
$784$	0	0
$785$	−1979.50	−2.52165
$786$	0	0
$787$	391.041	0.496875	0.248438	−	0.968648i	$-0.420083\pi$
$787$	391.041	0.496875	0.248438	+	0.968648i	$0.420083\pi$
$788$	0	0
$789$	− 419.405i	− 0.531565i
$790$	0	0
$791$	− 1804.67i	− 2.28150i
$792$	0	0
$793$	−53.9897	−0.0680828
$794$	0	0
$795$	199.923	0.251476
$796$	0	0
$797$	− 138.210i	− 0.173413i	−0.996234	−	0.0867065i	$-0.972366\pi$
$797$	− 138.210i	− 0.173413i	0.996234	−	0.0867065i	$-0.0276343\pi$
$798$	0	0
$799$	331.097i	0.414389i
$800$	0	0
$801$	317.138	0.395928
$802$	0	0
$803$	91.5589	0.114021
$804$	0	0
$805$	780.554i	0.969632i
$806$	0	0
$807$	131.023i	0.162358i
$808$	0	0
$809$	919.528	1.13662	0.568311	−	0.822814i	$-0.307597\pi$
$809$	919.528	1.13662	0.568311	+	0.822814i	$0.307597\pi$
$810$	0	0
$811$	181.359	0.223624	0.111812	−	0.993729i	$-0.464335\pi$
$811$	181.359	0.223624	0.111812	+	0.993729i	$0.464335\pi$
$812$	0	0
$813$	− 261.184i	− 0.321260i
$814$	0	0
$815$	− 1570.68i	− 1.92721i
$816$	0	0
$817$	−1274.26	−1.55968
$818$	0	0
$819$	126.431	0.154372
$820$	0	0
$821$	− 839.359i	− 1.02236i	−0.859473	−	0.511181i	$-0.829208\pi$
$821$	− 839.359i	− 1.02236i	0.859473	−	0.511181i	$-0.170792\pi$
$822$	0	0
$823$	26.1999i	0.0318347i	0.999873	+	0.0159173i	$0.00506686\pi$
$823$	26.1999i	0.0318347i	−0.999873	+	0.0159173i	$0.994933\pi$
$824$	0	0
$825$	−101.569	−0.123114
$826$	0	0
$827$	−7.38991	−0.00893581	−0.00446790	−	0.999990i	$-0.501422\pi$
$827$	−7.38991	−0.00893581	−0.00446790	+	0.999990i	$0.501422\pi$
$828$	0	0
$829$	− 340.410i	− 0.410627i	−0.978696	−	0.205314i	$-0.934179\pi$
$829$	− 340.410i	− 0.410627i	0.978696	−	0.205314i	$-0.0658215\pi$
$830$	0	0
$831$	369.895i	0.445120i
$832$	0	0
$833$	−553.292	−0.664216
$834$	0	0
$835$	2123.63	2.54327
$836$	0	0
$837$	− 159.215i	− 0.190221i
$838$	0	0
$839$	734.810i	0.875816i	0.899020	+	0.437908i	$0.144281\pi$
$839$	734.810i	0.875816i	−0.899020	+	0.437908i	$0.855719\pi$
$840$	0	0
$841$	831.564	0.988780
$842$	0	0
$843$	−336.018	−0.398598
$844$	0	0
$845$	− 1376.09i	− 1.62851i
$846$	0	0
$847$	1309.76i	1.54635i
$848$	0	0
$849$	311.405	0.366790
$850$	0	0
$851$	365.703	0.429733
$852$	0	0
$853$	778.800i	0.913013i	0.889720	+	0.456506i	$0.150899\pi$
$853$	778.800i	0.913013i	−0.889720	+	0.456506i	$0.849101\pi$
$854$	0	0
$855$	− 457.261i	− 0.534809i
$856$	0	0
$857$	1129.52	1.31799	0.658995	−	0.752147i	$-0.270982\pi$
$857$	1129.52	1.31799	0.658995	+	0.752147i	$0.270982\pi$
$858$	0	0
$859$	1373.85	1.59936	0.799680	−	0.600426i	$-0.205002\pi$
$859$	1373.85	1.59936	0.799680	+	0.600426i	$0.205002\pi$
$860$	0	0
$861$	− 678.697i	− 0.788266i
$862$	0	0
$863$	539.405i	0.625035i	0.949912	+	0.312517i	$0.101172\pi$
$863$	539.405i	0.625035i	−0.949912	+	0.312517i	$0.898828\pi$
$864$	0	0
$865$	640.862	0.740880
$866$	0	0
$867$	−393.655	−0.454043
$868$	0	0
$869$	− 59.1797i	− 0.0681009i
$870$	0	0
$871$	− 310.431i	− 0.356407i
$872$	0	0
$873$	63.4153	0.0726407
$874$	0	0
$875$	2899.06	3.31321
$876$	0	0
$877$	668.236i	0.761956i	0.924584	+	0.380978i	$0.124413\pi$
$877$	668.236i	0.761956i	−0.924584	+	0.380978i	$0.875587\pi$
$878$	0	0
$879$	− 897.033i	− 1.02052i
$880$	0	0
$881$	1170.53	1.32864	0.664321	−	0.747448i	$-0.268721\pi$
$881$	1170.53	1.32864	0.664321	+	0.747448i	$0.268721\pi$
$882$	0	0
$883$	−32.9179	−0.0372796	−0.0186398	−	0.999826i	$-0.505934\pi$
$883$	−32.9179	−0.0372796	−0.0186398	+	0.999826i	$0.505934\pi$
$884$	0	0
$885$	683.672i	0.772510i
$886$	0	0
$887$	655.846i	0.739398i	0.929152	+	0.369699i	$0.120539\pi$
$887$	655.846i	0.739398i	−0.929152	+	0.369699i	$0.879461\pi$
$888$	0	0
$889$	−1610.26	−1.81131
$890$	0	0
$891$	9.64617	0.0108262
$892$	0	0
$893$	719.467i	0.805674i
$894$	0	0
$895$	52.9742i	0.0591891i
$896$	0	0
$897$	53.4359	0.0595718
$898$	0	0
$899$	94.1230	0.104697
$900$	0	0
$901$	101.569i	0.112729i
$902$	0	0
$903$	1412.82i	1.56459i
$904$	0	0
$905$	−1348.11	−1.48963
$906$	0	0
$907$	−378.949	−0.417805	−0.208902	−	0.977937i	$-0.566989\pi$
$907$	−378.949	−0.417805	−0.208902	+	0.977937i	$0.566989\pi$
$908$	0	0
$909$	518.785i	0.570720i
$910$	0	0
$911$	− 571.938i	− 0.627814i	−0.949454	−	0.313907i	$-0.898362\pi$
$911$	− 571.938i	− 0.627814i	0.949454	−	0.313907i	$-0.101638\pi$
$912$	0	0
$913$	52.5950	0.0576068
$914$	0	0
$915$	216.497	0.236609
$916$	0	0
$917$	− 775.118i	− 0.845276i
$918$	0	0
$919$	− 97.8207i	− 0.106443i	−0.998583	−	0.0532213i	$-0.983051\pi$
$919$	− 97.8207i	− 0.106443i	0.998583	−	0.0532213i	$-0.0169489\pi$
$920$	0	0
$921$	706.277	0.766859
$922$	0	0
$923$	474.872	0.514487
$924$	0	0
$925$	− 2501.08i	− 2.70387i
$926$	0	0
$927$	− 50.3538i	− 0.0543191i
$928$	0	0
$929$	1228.06	1.32192	0.660959	−	0.750422i	$-0.270150\pi$
$929$	1228.06	1.32192	0.660959	+	0.750422i	$0.270150\pi$
$930$	0	0
$931$	−1202.29	−1.29140
$932$	0	0
$933$	− 639.118i	− 0.685014i
$934$	0	0
$935$	− 75.1797i	− 0.0804061i
$936$	0	0
$937$	−494.554	−0.527806	−0.263903	−	0.964549i	$-0.585010\pi$
$937$	−494.554	−0.527806	−0.263903	+	0.964549i	$0.585010\pi$
$938$	0	0
$939$	−592.326	−0.630805
$940$	0	0
$941$	− 1154.48i	− 1.22686i	−0.789748	−	0.613431i	$-0.789789\pi$
$941$	− 1154.48i	− 1.22686i	0.789748	−	0.613431i	$-0.210211\pi$
$942$	0	0
$943$	− 286.851i	− 0.304190i
$944$	0	0
$945$	−506.985	−0.536492
$946$	0	0
$947$	1085.43	1.14618	0.573089	−	0.819493i	$-0.305745\pi$
$947$	1085.43	1.14618	0.573089	+	0.819493i	$0.305745\pi$
$948$	0	0
$949$	329.436i	0.347140i
$950$	0	0
$951$	686.736i	0.722120i
$952$	0	0
$953$	−53.8667	−0.0565233	−0.0282617	−	0.999601i	$-0.508997\pi$
$953$	−53.8667	−0.0565233	−0.0282617	+	0.999601i	$0.508997\pi$
$954$	0	0
$955$	−209.149	−0.219004
$956$	0	0
$957$	5.70250i	0.00595873i
$958$	0	0
$959$	2252.67i	2.34897i
$960$	0	0
$961$	22.1281	0.0230261
$962$	0	0
$963$	414.200	0.430114
$964$	0	0
$965$	− 1226.96i	− 1.27147i
$966$	0	0
$967$	172.918i	0.178819i	0.995995	+	0.0894095i	$0.0284980\pi$
$967$	172.918i	0.178819i	−0.995995	+	0.0894095i	$0.971502\pi$
$968$	0	0
$969$	232.308	0.239740
$970$	0	0
$971$	−984.877	−1.01429	−0.507146	−	0.861860i	$-0.669299\pi$
$971$	−984.877	−1.01429	−0.507146	+	0.861860i	$0.669299\pi$
$972$	0	0
$973$	436.287i	0.448394i
$974$	0	0
$975$	− 365.454i	− 0.374824i
$976$	0	0
$977$	−675.856	−0.691767	−0.345884	−	0.938277i	$-0.612421\pi$
$977$	−675.856	−0.691767	−0.345884	+	0.938277i	$0.612421\pi$
$978$	0	0
$979$	113.303	0.115733
$980$	0	0
$981$	− 23.5692i	− 0.0240257i
$982$	0	0
$983$	564.441i	0.574203i	0.957900	+	0.287101i	$0.0926916\pi$
$983$	564.441i	0.574203i	−0.957900	+	0.287101i	$0.907308\pi$
$984$	0	0
$985$	352.595	0.357964
$986$	0	0
$987$	797.703	0.808209
$988$	0	0
$989$	597.128i	0.603770i
$990$	0	0
$991$	1392.44i	1.40508i	0.711644	+	0.702541i	$0.247951\pi$
$991$	1392.44i	1.40508i	−0.711644	+	0.702541i	$0.752049\pi$
$992$	0	0
$993$	−462.564	−0.465825
$994$	0	0
$995$	−2463.50	−2.47588
$996$	0	0
$997$	123.723i	0.124095i	0.998073	+	0.0620477i	$0.0197631\pi$
$997$	123.723i	0.124095i	−0.998073	+	0.0620477i	$0.980237\pi$
$998$	0	0
$999$	237.531i	0.237769i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.3.b.d.127.3		4
3.2	odd	2		2304.3.b.k.127.4		4
4.3	odd	2		768.3.b.a.127.1		4
8.3	odd	2	inner	768.3.b.d.127.4		4
8.5	even	2		768.3.b.a.127.2		4
12.11	even	2		2304.3.b.o.127.4		4
16.3	odd	4		96.3.g.a.31.1	✓	4
16.5	even	4		192.3.g.c.127.2		4
16.11	odd	4		192.3.g.c.127.4		4
16.13	even	4		96.3.g.a.31.3	yes	4
24.5	odd	2		2304.3.b.o.127.1		4
24.11	even	2		2304.3.b.k.127.1		4
48.5	odd	4		576.3.g.j.127.2		4
48.11	even	4		576.3.g.j.127.1		4
48.29	odd	4		288.3.g.d.127.4		4
48.35	even	4		288.3.g.d.127.3		4
80.3	even	4		2400.3.j.a.799.4		4
80.13	odd	4		2400.3.j.b.799.1		4
80.19	odd	4		2400.3.e.a.1951.4		4
80.29	even	4		2400.3.e.a.1951.1		4
80.67	even	4		2400.3.j.b.799.2		4
80.77	odd	4		2400.3.j.a.799.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.3.g.a.31.1	✓	4	16.3	odd	4
96.3.g.a.31.3	yes	4	16.13	even	4
192.3.g.c.127.2		4	16.5	even	4
192.3.g.c.127.4		4	16.11	odd	4
288.3.g.d.127.3		4	48.35	even	4
288.3.g.d.127.4		4	48.29	odd	4
576.3.g.j.127.1		4	48.11	even	4
576.3.g.j.127.2		4	48.5	odd	4
768.3.b.a.127.1		4	4.3	odd	2
768.3.b.a.127.2		4	8.5	even	2
768.3.b.d.127.3		4	1.1	even	1	trivial
768.3.b.d.127.4		4	8.3	odd	2	inner
2304.3.b.k.127.1		4	24.11	even	2
2304.3.b.k.127.4		4	3.2	odd	2
2304.3.b.o.127.1		4	24.5	odd	2
2304.3.b.o.127.4		4	12.11	even	2
2400.3.e.a.1951.1		4	80.29	even	4
2400.3.e.a.1951.4		4	80.19	odd	4
2400.3.j.a.799.3		4	80.77	odd	4
2400.3.j.a.799.4		4	80.3	even	4
2400.3.j.b.799.1		4	80.13	odd	4
2400.3.j.b.799.2		4	80.67	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:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	768.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.73205 q^{3} -8.92820i q^{5} -10.9282i q^{7} +3.00000 q^{9} +O(q^{10})\)	\(q+1.73205 q^{3} -8.92820i q^{5} -10.9282i q^{7} +3.00000 q^{9} +1.07180 q^{11} +3.85641i q^{13} -15.4641i q^{15} +7.85641 q^{17} +17.0718 q^{19} -18.9282i q^{21} -8.00000i q^{23} -54.7128 q^{25} +5.19615 q^{27} +3.07180i q^{29} -30.6410i q^{31} +1.85641 q^{33} -97.5692 q^{35} +45.7128i q^{37} +6.67949i q^{39} +35.8564 q^{41} -74.6410 q^{43} -26.7846i q^{45} +42.1436i q^{47} -70.4256 q^{49} +13.6077 q^{51} +12.9282i q^{53} -9.56922i q^{55} +29.5692 q^{57} -44.2102 q^{59} +14.0000i q^{61} -32.7846i q^{63} +34.4308 q^{65} -80.4974 q^{67} -13.8564i q^{69} -123.138i q^{71} +85.4256 q^{73} -94.7654 q^{75} -11.7128i q^{77} -55.2154i q^{79} +9.00000 q^{81} +49.0718 q^{83} -70.1436i q^{85} +5.32051i q^{87} +105.713 q^{89} +42.1436 q^{91} -53.0718i q^{93} -152.420i q^{95} +21.1384 q^{97} +3.21539 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{9}+O(q^{10}) \) 4 * q + 12 * q^9	\( 4 q + 12 q^{9} + 32 q^{11} - 24 q^{17} + 96 q^{19} - 108 q^{25} - 48 q^{33} - 224 q^{35} + 88 q^{41} - 160 q^{43} - 60 q^{49} + 96 q^{51} - 48 q^{57} + 128 q^{59} + 304 q^{65} - 128 q^{67} + 120 q^{73} - 192 q^{75} + 36 q^{81} + 224 q^{83} + 312 q^{89} + 224 q^{91} - 248 q^{97} + 96 q^{99}+O(q^{100}) \) 4 * q + 12 * q^9 + 32 * q^11 - 24 * q^17 + 96 * q^19 - 108 * q^25 - 48 * q^33 - 224 * q^35 + 88 * q^41 - 160 * q^43 - 60 * q^49 + 96 * q^51 - 48 * q^57 + 128 * q^59 + 304 * q^65 - 128 * q^67 + 120 * q^73 - 192 * q^75 + 36 * q^81 + 224 * q^83 + 312 * q^89 + 224 * q^91 - 248 * q^97 + 96 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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