LMFDB - Embedded newform 768.3.b.d.127.1

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			127.1
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	768.127
Dual form			768.3.b.d.127.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205 q^{3} -4.92820i q^{5} -2.92820i q^{7} +3.00000 q^{9} +O(q^{10})$	$q-1.73205 q^{3} -4.92820i q^{5} -2.92820i q^{7} +3.00000 q^{9} +14.9282 q^{11} +23.8564i q^{13} +8.53590i q^{15} -19.8564 q^{17} +30.9282 q^{19} +5.07180i q^{21} +8.00000i q^{23} +0.712813 q^{25} -5.19615 q^{27} -16.9282i q^{29} -38.6410i q^{31} -25.8564 q^{33} -14.4308 q^{35} +9.71281i q^{37} -41.3205i q^{39} +8.14359 q^{41} -5.35898 q^{43} -14.7846i q^{45} -69.8564i q^{47} +40.4256 q^{49} +34.3923 q^{51} +0.928203i q^{53} -73.5692i q^{55} -53.5692 q^{57} +108.210 q^{59} -14.0000i q^{61} -8.78461i q^{63} +117.569 q^{65} +16.4974 q^{67} -13.8564i q^{69} -43.1384i q^{71} -25.4256 q^{73} -1.23463 q^{75} -43.7128i q^{77} +96.7846i q^{79} +9.00000 q^{81} +62.9282 q^{83} +97.8564i q^{85} +29.3205i q^{87} +50.2872 q^{89} +69.8564 q^{91} +66.9282i q^{93} -152.420i q^{95} -145.138 q^{97} +44.7846 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	− 4.92820i	− 0.985641i	−0.870131	−	0.492820i	$-0.835966\pi$
$5$	− 4.92820i	− 0.985641i	0.870131	−	0.492820i	$-0.164034\pi$
$6$	0	0
$7$	− 2.92820i	− 0.418315i	−0.977882	−	0.209157i	$-0.932928\pi$
$7$	− 2.92820i	− 0.418315i	0.977882	−	0.209157i	$-0.0670721\pi$
$8$	0	0
$9$	3.00000	0.333333
$10$	0	0
$11$	14.9282	1.35711	0.678555	−	0.734550i	$-0.262607\pi$
$11$	14.9282	1.35711	0.678555	+	0.734550i	$0.262607\pi$
$12$	0	0
$13$	23.8564i	1.83511i	0.397611	+	0.917554i	$0.369839\pi$
$13$	23.8564i	1.83511i	−0.397611	+	0.917554i	$0.630161\pi$
$14$	0	0
$15$	8.53590i	0.569060i
$16$	0	0
$17$	−19.8564	−1.16802	−0.584012	−	0.811745i	$-0.698518\pi$
$17$	−19.8564	−1.16802	−0.584012	+	0.811745i	$0.698518\pi$
$18$	0	0
$19$	30.9282	1.62780	0.813900	−	0.581005i	$-0.197340\pi$
$19$	30.9282	1.62780	0.813900	+	0.581005i	$0.197340\pi$
$20$	0	0
$21$	5.07180i	0.241514i
$22$	0	0
$23$	8.00000i	0.347826i	0.984761	+	0.173913i	$0.0556412\pi$
$23$	8.00000i	0.347826i	−0.984761	+	0.173913i	$0.944359\pi$
$24$	0	0
$25$	0.712813	0.0285125
$26$	0	0
$27$	−5.19615	−0.192450
$28$	0	0
$29$	− 16.9282i	− 0.583731i	−0.956459	−	0.291866i	$-0.905724\pi$
$29$	− 16.9282i	− 0.583731i	0.956459	−	0.291866i	$-0.0942760\pi$
$30$	0	0
$31$	− 38.6410i	− 1.24648i	−0.782029	−	0.623242i	$-0.785815\pi$
$31$	− 38.6410i	− 1.24648i	0.782029	−	0.623242i	$-0.214185\pi$
$32$	0	0
$33$	−25.8564	−0.783527
$34$	0	0
$35$	−14.4308	−0.412308
$36$	0	0
$37$	9.71281i	0.262508i	0.991349	+	0.131254i	$0.0419004\pi$
$37$	9.71281i	0.262508i	−0.991349	+	0.131254i	$0.958100\pi$
$38$	0	0
$39$	− 41.3205i	− 1.05950i
$40$	0	0
$41$	8.14359	0.198624	0.0993121	−	0.995056i	$-0.468336\pi$
$41$	8.14359	0.198624	0.0993121	+	0.995056i	$0.468336\pi$
$42$	0	0
$43$	−5.35898	−0.124628	−0.0623138	−	0.998057i	$-0.519848\pi$
$43$	−5.35898	−0.124628	−0.0623138	+	0.998057i	$0.519848\pi$
$44$	0	0
$45$	− 14.7846i	− 0.328547i
$46$	0	0
$47$	− 69.8564i	− 1.48631i	−0.669121	−	0.743153i	$-0.733329\pi$
$47$	− 69.8564i	− 1.48631i	0.669121	−	0.743153i	$-0.266671\pi$
$48$	0	0
$49$	40.4256	0.825013
$50$	0	0
$51$	34.3923	0.674359
$52$	0	0
$53$	0.928203i	0.0175133i	0.999962	+	0.00875663i	$0.00278736\pi$
$53$	0.928203i	0.0175133i	−0.999962	+	0.00875663i	$0.997213\pi$
$54$	0	0
$55$	− 73.5692i	− 1.33762i
$56$	0	0
$57$	−53.5692	−0.939811
$58$	0	0
$59$	108.210	1.83407	0.917036	−	0.398805i	$-0.130575\pi$
$59$	108.210	1.83407	0.917036	+	0.398805i	$0.130575\pi$
$60$	0	0
$61$	− 14.0000i	− 0.229508i	−0.993394	−	0.114754i	$-0.963392\pi$
$61$	− 14.0000i	− 0.229508i	0.993394	−	0.114754i	$-0.0366080\pi$
$62$	0	0
$63$	− 8.78461i	− 0.139438i
$64$	0	0
$65$	117.569	1.80876
$66$	0	0
$67$	16.4974	0.246230	0.123115	−	0.992392i	$-0.460712\pi$
$67$	16.4974	0.246230	0.123115	+	0.992392i	$0.460712\pi$
$68$	0	0
$69$	− 13.8564i	− 0.200817i
$70$	0	0
$71$	− 43.1384i	− 0.607584i	−0.952738	−	0.303792i	$-0.901747\pi$
$71$	− 43.1384i	− 0.607584i	0.952738	−	0.303792i	$-0.0982527\pi$
$72$	0	0
$73$	−25.4256	−0.348296	−0.174148	−	0.984719i	$-0.555717\pi$
$73$	−25.4256	−0.348296	−0.174148	+	0.984719i	$0.555717\pi$
$74$	0	0
$75$	−1.23463	−0.0164617
$76$	0	0
$77$	− 43.7128i	− 0.567699i
$78$	0	0
$79$	96.7846i	1.22512i	0.790423	+	0.612561i	$0.209861\pi$
$79$	96.7846i	1.22512i	−0.790423	+	0.612561i	$0.790139\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	62.9282	0.758171	0.379086	−	0.925362i	$-0.376239\pi$
$83$	62.9282	0.758171	0.379086	+	0.925362i	$0.376239\pi$
$84$	0	0
$85$	97.8564i	1.15125i
$86$	0	0
$87$	29.3205i	0.337017i
$88$	0	0
$89$	50.2872	0.565025	0.282512	−	0.959264i	$-0.408832\pi$
$89$	50.2872	0.565025	0.282512	+	0.959264i	$0.408832\pi$
$90$	0	0
$91$	69.8564	0.767653
$92$	0	0
$93$	66.9282i	0.719658i
$94$	0	0
$95$	− 152.420i	− 1.60443i
$96$	0	0
$97$	−145.138	−1.49627	−0.748136	−	0.663545i	$-0.769051\pi$
$97$	−145.138	−1.49627	−0.748136	+	0.663545i	$0.769051\pi$
$98$	0	0
$99$	44.7846	0.452370
$100$	0	0
$101$	− 159.072i	− 1.57497i	−0.616335	−	0.787484i	$-0.711383\pi$
$101$	− 159.072i	− 1.57497i	0.616335	−	0.787484i	$-0.288617\pi$
$102$	0	0
$103$	− 24.7846i	− 0.240627i	−0.992736	−	0.120314i	$-0.961610\pi$
$103$	− 24.7846i	− 0.240627i	0.992736	−	0.120314i	$-0.0383900\pi$
$104$	0	0
$105$	24.9948	0.238046
$106$	0	0
$107$	−42.0666	−0.393146	−0.196573	−	0.980489i	$-0.562981\pi$
$107$	−42.0666	−0.393146	−0.196573	+	0.980489i	$0.562981\pi$
$108$	0	0
$109$	− 19.8564i	− 0.182169i	−0.995843	−	0.0910844i	$-0.970967\pi$
$109$	− 19.8564i	− 0.182169i	0.995843	−	0.0910844i	$-0.0290333\pi$
$110$	0	0
$111$	− 16.8231i	− 0.151559i
$112$	0	0
$113$	−1.13844	−0.0100747	−0.00503734	−	0.999987i	$-0.501603\pi$
$113$	−1.13844	−0.0100747	−0.00503734	+	0.999987i	$0.501603\pi$
$114$	0	0
$115$	39.4256	0.342832
$116$	0	0
$117$	71.5692i	0.611703i
$118$	0	0
$119$	58.1436i	0.488602i
$120$	0	0
$121$	101.851	0.841746
$122$	0	0
$123$	−14.1051	−0.114676
$124$	0	0
$125$	− 126.718i	− 1.01374i
$126$	0	0
$127$	− 171.349i	− 1.34920i	−0.738182	−	0.674601i	$-0.764316\pi$
$127$	− 171.349i	− 1.34920i	0.738182	−	0.674601i	$-0.235684\pi$
$128$	0	0
$129$	9.28203	0.0719537
$130$	0	0
$131$	57.0718	0.435663	0.217831	−	0.975986i	$-0.430102\pi$
$131$	57.0718	0.435663	0.217831	+	0.975986i	$0.430102\pi$
$132$	0	0
$133$	− 90.5641i	− 0.680933i
$134$	0	0
$135$	25.6077i	0.189687i
$136$	0	0
$137$	154.133	1.12506	0.562530	−	0.826777i	$-0.309828\pi$
$137$	154.133	1.12506	0.562530	+	0.826777i	$0.309828\pi$
$138$	0	0
$139$	167.923	1.20808	0.604040	−	0.796954i	$-0.293557\pi$
$139$	167.923	1.20808	0.604040	+	0.796954i	$0.293557\pi$
$140$	0	0
$141$	120.995i	0.858119i
$142$	0	0
$143$	356.133i	2.49044i
$144$	0	0
$145$	−83.4256	−0.575349
$146$	0	0
$147$	−70.0192	−0.476321
$148$	0	0
$149$	− 124.354i	− 0.834589i	−0.908771	−	0.417295i	$-0.862978\pi$
$149$	− 124.354i	− 0.834589i	0.908771	−	0.417295i	$-0.137022\pi$
$150$	0	0
$151$	48.2102i	0.319273i	0.987176	+	0.159637i	$0.0510322\pi$
$151$	48.2102i	0.319273i	−0.987176	+	0.159637i	$0.948968\pi$
$152$	0	0
$153$	−59.5692	−0.389341
$154$	0	0
$155$	−190.431	−1.22859
$156$	0	0
$157$	166.287i	1.05915i	0.848262	+	0.529577i	$0.177649\pi$
$157$	166.287i	1.05915i	−0.848262	+	0.529577i	$0.822351\pi$
$158$	0	0
$159$	− 1.60770i	− 0.0101113i
$160$	0	0
$161$	23.4256	0.145501
$162$	0	0
$163$	−31.9230	−0.195847	−0.0979235	−	0.995194i	$-0.531220\pi$
$163$	−31.9230	−0.195847	−0.0979235	+	0.995194i	$0.531220\pi$
$164$	0	0
$165$	127.426i	0.772277i
$166$	0	0
$167$	− 210.144i	− 1.25834i	−0.777266	−	0.629172i	$-0.783394\pi$
$167$	− 210.144i	− 1.25834i	0.777266	−	0.629172i	$-0.216606\pi$
$168$	0	0
$169$	−400.128	−2.36762
$170$	0	0
$171$	92.7846	0.542600
$172$	0	0
$173$	163.779i	0.946702i	0.880874	+	0.473351i	$0.156956\pi$
$173$	163.779i	0.946702i	−0.880874	+	0.473351i	$0.843044\pi$
$174$	0	0
$175$	− 2.08726i	− 0.0119272i
$176$	0	0
$177$	−187.426	−1.05890
$178$	0	0
$179$	−186.067	−1.03948	−0.519739	−	0.854325i	$-0.673971\pi$
$179$	−186.067	−1.03948	−0.519739	+	0.854325i	$0.673971\pi$
$180$	0	0
$181$	− 42.9948i	− 0.237541i	−0.992922	−	0.118770i	$-0.962105\pi$
$181$	− 42.9948i	− 0.237541i	0.992922	−	0.118770i	$-0.0378952\pi$
$182$	0	0
$183$	24.2487i	0.132507i
$184$	0	0
$185$	47.8667	0.258739
$186$	0	0
$187$	−296.420	−1.58514
$188$	0	0
$189$	15.2154i	0.0805047i
$190$	0	0
$191$	− 87.4256i	− 0.457726i	−0.973459	−	0.228863i	$-0.926499\pi$
$191$	− 87.4256i	− 0.457726i	0.973459	−	0.228863i	$-0.0735007\pi$
$192$	0	0
$193$	26.5744	0.137691	0.0688455	−	0.997627i	$-0.478068\pi$
$193$	26.5744	0.137691	0.0688455	+	0.997627i	$0.478068\pi$
$194$	0	0
$195$	−203.636	−1.04429
$196$	0	0
$197$	251.492i	1.27661i	0.769783	+	0.638305i	$0.220364\pi$
$197$	251.492i	1.27661i	−0.769783	+	0.638305i	$0.779636\pi$
$198$	0	0
$199$	68.0770i	0.342095i	0.985263	+	0.171048i	$0.0547152\pi$
$199$	68.0770i	0.342095i	−0.985263	+	0.171048i	$0.945285\pi$
$200$	0	0
$201$	−28.5744	−0.142161
$202$	0	0
$203$	−49.5692	−0.244183
$204$	0	0
$205$	− 40.1333i	− 0.195772i
$206$	0	0
$207$	24.0000i	0.115942i
$208$	0	0
$209$	461.703	2.20910
$210$	0	0
$211$	−258.641	−1.22579	−0.612893	−	0.790166i	$-0.709995\pi$
$211$	−258.641	−1.22579	−0.612893	+	0.790166i	$0.709995\pi$
$212$	0	0
$213$	74.7180i	0.350789i
$214$	0	0
$215$	26.4102i	0.122838i
$216$	0	0
$217$	−113.149	−0.521423
$218$	0	0
$219$	44.0385	0.201089
$220$	0	0
$221$	− 473.703i	− 2.14345i
$222$	0	0
$223$	− 8.21024i	− 0.0368172i	−0.999831	−	0.0184086i	$-0.994140\pi$
$223$	− 8.21024i	− 0.0368172i	0.999831	−	0.0184086i	$-0.00585997\pi$
$224$	0	0
$225$	2.13844	0.00950417
$226$	0	0
$227$	238.928	1.05255	0.526274	−	0.850315i	$-0.323589\pi$
$227$	238.928	1.05255	0.526274	+	0.850315i	$0.323589\pi$
$228$	0	0
$229$	181.846i	0.794088i	0.917800	+	0.397044i	$0.129964\pi$
$229$	181.846i	0.794088i	−0.917800	+	0.397044i	$0.870036\pi$
$230$	0	0
$231$	75.7128i	0.327761i
$232$	0	0
$233$	385.138	1.65295	0.826477	−	0.562970i	$-0.190341\pi$
$233$	385.138	1.65295	0.826477	+	0.562970i	$0.190341\pi$
$234$	0	0
$235$	−344.267	−1.46496
$236$	0	0
$237$	− 167.636i	− 0.707324i
$238$	0	0
$239$	403.979i	1.69029i	0.534537	+	0.845145i	$0.320486\pi$
$239$	403.979i	1.69029i	−0.534537	+	0.845145i	$0.679514\pi$
$240$	0	0
$241$	−51.4359	−0.213427	−0.106714	−	0.994290i	$-0.534033\pi$
$241$	−51.4359	−0.213427	−0.106714	+	0.994290i	$0.534033\pi$
$242$	0	0
$243$	−15.5885	−0.0641500
$244$	0	0
$245$	− 199.226i	− 0.813166i
$246$	0	0
$247$	737.836i	2.98719i
$248$	0	0
$249$	−108.995	−0.437730
$250$	0	0
$251$	−4.21024	−0.0167738	−0.00838692	−	0.999965i	$-0.502670\pi$
$251$	−4.21024	−0.0167738	−0.00838692	+	0.999965i	$0.502670\pi$
$252$	0	0
$253$	119.426i	0.472038i
$254$	0	0
$255$	− 169.492i	− 0.664676i
$256$	0	0
$257$	−91.7025	−0.356819	−0.178410	−	0.983956i	$-0.557095\pi$
$257$	−91.7025	−0.356819	−0.178410	+	0.983956i	$0.557095\pi$
$258$	0	0
$259$	28.4411	0.109811
$260$	0	0
$261$	− 50.7846i	− 0.194577i
$262$	0	0
$263$	269.856i	1.02607i	0.858368	+	0.513035i	$0.171479\pi$
$263$	269.856i	1.02607i	−0.858368	+	0.513035i	$0.828521\pi$
$264$	0	0
$265$	4.57437	0.0172618
$266$	0	0
$267$	−87.1000	−0.326217
$268$	0	0
$269$	− 200.354i	− 0.744810i	−0.928070	−	0.372405i	$-0.878533\pi$
$269$	− 200.354i	− 0.744810i	0.928070	−	0.372405i	$-0.121467\pi$
$270$	0	0
$271$	497.205i	1.83471i	0.398076	+	0.917353i	$0.369678\pi$
$271$	497.205i	1.83471i	−0.398076	+	0.917353i	$0.630322\pi$
$272$	0	0
$273$	−120.995	−0.443205
$274$	0	0
$275$	10.6410	0.0386946
$276$	0	0
$277$	257.559i	0.929816i	0.885359	+	0.464908i	$0.153913\pi$
$277$	257.559i	0.929816i	−0.885359	+	0.464908i	$0.846087\pi$
$278$	0	0
$279$	− 115.923i	− 0.415495i
$280$	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
$283$	332.210	1.17389	0.586944	−	0.809628i	$-0.300331\pi$
$283$	332.210	1.17389	0.586944	+	0.809628i	$0.300331\pi$
$284$	0	0
$285$	264.000i	0.926316i
$286$	0	0
$287$	− 23.8461i	− 0.0830874i
$288$	0	0
$289$	105.277	0.364280
$290$	0	0
$291$	251.387	0.863873
$292$	0	0
$293$	− 465.902i	− 1.59011i	−0.606537	−	0.795055i	$-0.707442\pi$
$293$	− 465.902i	− 1.59011i	0.606537	−	0.795055i	$-0.292558\pi$
$294$	0	0
$295$	− 533.282i	− 1.80774i
$296$	0	0
$297$	−77.5692	−0.261176
$298$	0	0
$299$	−190.851	−0.638299
$300$	0	0
$301$	15.6922i	0.0521335i
$302$	0	0
$303$	275.520i	0.909308i
$304$	0	0
$305$	−68.9948	−0.226213
$306$	0	0
$307$	−215.769	−0.702831	−0.351416	−	0.936220i	$-0.614300\pi$
$307$	−215.769	−0.702831	−0.351416	+	0.936220i	$0.614300\pi$
$308$	0	0
$309$	42.9282i	0.138926i
$310$	0	0
$311$	175.005i	0.562718i	0.959603	+	0.281359i	$0.0907851\pi$
$311$	175.005i	0.562718i	−0.959603	+	0.281359i	$0.909215\pi$
$312$	0	0
$313$	433.979	1.38652	0.693258	−	0.720690i	$-0.256175\pi$
$313$	433.979	1.38652	0.693258	+	0.720690i	$0.256175\pi$
$314$	0	0
$315$	−43.2923	−0.137436
$316$	0	0
$317$	88.4871i	0.279139i	0.990212	+	0.139570i	$0.0445719\pi$
$317$	88.4871i	0.279139i	−0.990212	+	0.139570i	$0.955428\pi$
$318$	0	0
$319$	− 252.708i	− 0.792187i
$320$	0	0
$321$	72.8616	0.226983
$322$	0	0
$323$	−614.123	−1.90131
$324$	0	0
$325$	17.0052i	0.0523236i
$326$	0	0
$327$	34.3923i	0.105175i
$328$	0	0
$329$	−204.554	−0.621744
$330$	0	0
$331$	107.061	0.323449	0.161724	−	0.986836i	$-0.448294\pi$
$331$	107.061	0.323449	0.161724	+	0.986836i	$0.448294\pi$
$332$	0	0
$333$	29.1384i	0.0875028i
$334$	0	0
$335$	− 81.3027i	− 0.242694i
$336$	0	0
$337$	−38.5744	−0.114464	−0.0572320	−	0.998361i	$-0.518227\pi$
$337$	−38.5744	−0.114464	−0.0572320	+	0.998361i	$0.518227\pi$
$338$	0	0
$339$	1.97183	0.00581662
$340$	0	0
$341$	− 576.841i	− 1.69162i
$342$	0	0
$343$	− 261.856i	− 0.763430i
$344$	0	0
$345$	−68.2872	−0.197934
$346$	0	0
$347$	−65.9127	−0.189950	−0.0949751	−	0.995480i	$-0.530277\pi$
$347$	−65.9127	−0.189950	−0.0949751	+	0.995480i	$0.530277\pi$
$348$	0	0
$349$	367.703i	1.05359i	0.849993	+	0.526794i	$0.176606\pi$
$349$	367.703i	1.05359i	−0.849993	+	0.526794i	$0.823394\pi$
$350$	0	0
$351$	− 123.962i	− 0.353167i
$352$	0	0
$353$	158.862	0.450033	0.225016	−	0.974355i	$-0.427756\pi$
$353$	158.862	0.450033	0.225016	+	0.974355i	$0.427756\pi$
$354$	0	0
$355$	−212.595	−0.598859
$356$	0	0
$357$	− 100.708i	− 0.282094i
$358$	0	0
$359$	140.287i	0.390772i	0.980726	+	0.195386i	$0.0625960\pi$
$359$	140.287i	0.390772i	−0.980726	+	0.195386i	$0.937404\pi$
$360$	0	0
$361$	595.554	1.64973
$362$	0	0
$363$	−176.412	−0.485982
$364$	0	0
$365$	125.303i	0.343295i
$366$	0	0
$367$	505.051i	1.37616i	0.725634	+	0.688081i	$0.241546\pi$
$367$	505.051i	1.37616i	−0.725634	+	0.688081i	$0.758454\pi$
$368$	0	0
$369$	24.4308	0.0662081
$370$	0	0
$371$	2.71797	0.00732606
$372$	0	0
$373$	− 215.128i	− 0.576751i	−0.957517	−	0.288376i	$-0.906885\pi$
$373$	− 215.128i	− 0.576751i	0.957517	−	0.288376i	$-0.0931151\pi$
$374$	0	0
$375$	219.482i	0.585285i
$376$	0	0
$377$	403.846	1.07121
$378$	0	0
$379$	−204.785	−0.540329	−0.270164	−	0.962814i	$-0.587078\pi$
$379$	−204.785	−0.540329	−0.270164	+	0.962814i	$0.587078\pi$
$380$	0	0
$381$	296.785i	0.778962i
$382$	0	0
$383$	− 588.554i	− 1.53669i	−0.640034	−	0.768347i	$-0.721080\pi$
$383$	− 588.554i	− 1.53669i	0.640034	−	0.768347i	$-0.278920\pi$
$384$	0	0
$385$	−215.426	−0.559547
$386$	0	0
$387$	−16.0770	−0.0415425
$388$	0	0
$389$	534.477i	1.37398i	0.726669	+	0.686988i	$0.241068\pi$
$389$	534.477i	1.37398i	−0.726669	+	0.686988i	$0.758932\pi$
$390$	0	0
$391$	− 158.851i	− 0.406269i
$392$	0	0
$393$	−98.8513	−0.251530
$394$	0	0
$395$	476.974	1.20753
$396$	0	0
$397$	− 340.277i	− 0.857121i	−0.903513	−	0.428560i	$-0.859021\pi$
$397$	− 340.277i	− 0.857121i	0.903513	−	0.428560i	$-0.140979\pi$
$398$	0	0
$399$	156.862i	0.393137i
$400$	0	0
$401$	−300.123	−0.748436	−0.374218	−	0.927341i	$-0.622089\pi$
$401$	−300.123	−0.748436	−0.374218	+	0.927341i	$0.622089\pi$
$402$	0	0
$403$	921.836	2.28743
$404$	0	0
$405$	− 44.3538i	− 0.109516i
$406$	0	0
$407$	144.995i	0.356253i
$408$	0	0
$409$	−513.692	−1.25597	−0.627986	−	0.778225i	$-0.716120\pi$
$409$	−513.692	−1.25597	−0.627986	+	0.778225i	$0.716120\pi$
$410$	0	0
$411$	−266.967	−0.649554
$412$	0	0
$413$	− 316.862i	− 0.767219i
$414$	0	0
$415$	− 310.123i	− 0.747284i
$416$	0	0
$417$	−290.851	−0.697485
$418$	0	0
$419$	−91.9437	−0.219436	−0.109718	−	0.993963i	$-0.534995\pi$
$419$	−91.9437	−0.219436	−0.109718	+	0.993963i	$0.534995\pi$
$420$	0	0
$421$	513.559i	1.21985i	0.792457	+	0.609927i	$0.208801\pi$
$421$	513.559i	1.21985i	−0.792457	+	0.609927i	$0.791199\pi$
$422$	0	0
$423$	− 209.569i	− 0.495436i
$424$	0	0
$425$	−14.1539	−0.0333033
$426$	0	0
$427$	−40.9948	−0.0960067
$428$	0	0
$429$	− 616.841i	− 1.43786i
$430$	0	0
$431$	− 446.123i	− 1.03509i	−0.855657	−	0.517544i	$-0.826846\pi$
$431$	− 446.123i	− 1.03509i	0.855657	−	0.517544i	$-0.173154\pi$
$432$	0	0
$433$	43.7231	0.100977	0.0504886	−	0.998725i	$-0.483922\pi$
$433$	43.7231	0.100977	0.0504886	+	0.998725i	$0.483922\pi$
$434$	0	0
$435$	144.497	0.332178
$436$	0	0
$437$	247.426i	0.566191i
$438$	0	0
$439$	− 381.072i	− 0.868045i	−0.900902	−	0.434023i	$-0.857094\pi$
$439$	− 381.072i	− 0.868045i	0.900902	−	0.434023i	$-0.142906\pi$
$440$	0	0
$441$	121.277	0.275004
$442$	0	0
$443$	181.513	0.409736	0.204868	−	0.978790i	$-0.434324\pi$
$443$	181.513	0.409736	0.204868	+	0.978790i	$0.434324\pi$
$444$	0	0
$445$	− 247.825i	− 0.556911i
$446$	0	0
$447$	215.387i	0.481850i
$448$	0	0
$449$	397.825	0.886026	0.443013	−	0.896515i	$-0.353910\pi$
$449$	397.825	0.886026	0.443013	+	0.896515i	$0.353910\pi$
$450$	0	0
$451$	121.569	0.269555
$452$	0	0
$453$	− 83.5026i	− 0.184332i
$454$	0	0
$455$	− 344.267i	− 0.756630i
$456$	0	0
$457$	−526.862	−1.15287	−0.576435	−	0.817143i	$-0.695557\pi$
$457$	−526.862	−1.15287	−0.576435	+	0.817143i	$0.695557\pi$
$458$	0	0
$459$	103.177	0.224786
$460$	0	0
$461$	888.908i	1.92822i	0.265510	+	0.964108i	$0.414460\pi$
$461$	888.908i	1.92822i	−0.265510	+	0.964108i	$0.585540\pi$
$462$	0	0
$463$	− 515.923i	− 1.11430i	−0.830410	−	0.557152i	$-0.811894\pi$
$463$	− 515.923i	− 1.11430i	0.830410	−	0.557152i	$-0.188106\pi$
$464$	0	0
$465$	329.836	0.709324
$466$	0	0
$467$	439.195	0.940460	0.470230	−	0.882544i	$-0.344171\pi$
$467$	439.195	0.940460	0.470230	+	0.882544i	$0.344171\pi$
$468$	0	0
$469$	− 48.3078i	− 0.103002i
$470$	0	0
$471$	− 288.018i	− 0.611503i
$472$	0	0
$473$	−80.0000	−0.169133
$474$	0	0
$475$	22.0460	0.0464127
$476$	0	0
$477$	2.78461i	0.00583776i
$478$	0	0
$479$	− 179.559i	− 0.374862i	−0.982278	−	0.187431i	$-0.939984\pi$
$479$	− 179.559i	− 0.374862i	0.982278	−	0.187431i	$-0.0600161\pi$
$480$	0	0
$481$	−231.713	−0.481731
$482$	0	0
$483$	−40.5744	−0.0840049
$484$	0	0
$485$	715.272i	1.47479i
$486$	0	0
$487$	494.221i	1.01483i	0.861703	+	0.507413i	$0.169398\pi$
$487$	494.221i	1.01483i	−0.861703	+	0.507413i	$0.830602\pi$
$488$	0	0
$489$	55.2923	0.113072
$490$	0	0
$491$	226.487	0.461277	0.230639	−	0.973039i	$-0.425918\pi$
$491$	226.487	0.461277	0.230639	+	0.973039i	$0.425918\pi$
$492$	0	0
$493$	336.133i	0.681812i
$494$	0	0
$495$	− 220.708i	− 0.445874i
$496$	0	0
$497$	−126.318	−0.254161
$498$	0	0
$499$	−239.195	−0.479348	−0.239674	−	0.970853i	$-0.577041\pi$
$499$	−239.195	−0.479348	−0.239674	+	0.970853i	$0.577041\pi$
$500$	0	0
$501$	363.979i	0.726506i
$502$	0	0
$503$	− 298.831i	− 0.594097i	−0.954862	−	0.297048i	$-0.903998\pi$
$503$	− 298.831i	− 0.594097i	0.954862	−	0.297048i	$-0.0960023\pi$
$504$	0	0
$505$	−783.938	−1.55235
$506$	0	0
$507$	693.042	1.36695
$508$	0	0
$509$	− 713.195i	− 1.40117i	−0.713570	−	0.700584i	$-0.752923\pi$
$509$	− 713.195i	− 1.40117i	0.713570	−	0.700584i	$-0.247077\pi$
$510$	0	0
$511$	74.4514i	0.145697i
$512$	0	0
$513$	−160.708	−0.313270
$514$	0	0
$515$	−122.144	−0.237172
$516$	0	0
$517$	− 1042.83i	− 2.01708i
$518$	0	0
$519$	− 283.674i	− 0.546579i
$520$	0	0
$521$	−748.985	−1.43759	−0.718795	−	0.695222i	$-0.755306\pi$
$521$	−748.985	−1.43759	−0.718795	+	0.695222i	$0.755306\pi$
$522$	0	0
$523$	338.908	0.648007	0.324003	−	0.946056i	$-0.394971\pi$
$523$	338.908	0.648007	0.324003	+	0.946056i	$0.394971\pi$
$524$	0	0
$525$	3.61524i	0.00688618i
$526$	0	0
$527$	767.272i	1.45592i
$528$	0	0
$529$	465.000	0.879017
$530$	0	0
$531$	324.631	0.611357
$532$	0	0
$533$	194.277i	0.364497i
$534$	0	0
$535$	207.313i	0.387501i
$536$	0	0
$537$	322.277	0.600143
$538$	0	0
$539$	603.482	1.11963
$540$	0	0
$541$	3.56922i	0.00659745i	0.999995	+	0.00329872i	$0.00105002\pi$
$541$	3.56922i	0.00659745i	−0.999995	+	0.00329872i	$0.998950\pi$
$542$	0	0
$543$	74.4693i	0.137144i
$544$	0	0
$545$	−97.8564	−0.179553
$546$	0	0
$547$	530.067	0.969043	0.484522	−	0.874779i	$-0.338994\pi$
$547$	530.067	0.969043	0.484522	+	0.874779i	$0.338994\pi$
$548$	0	0
$549$	− 42.0000i	− 0.0765027i
$550$	0	0
$551$	− 523.559i	− 0.950198i
$552$	0	0
$553$	283.405	0.512486
$554$	0	0
$555$	−82.9076	−0.149383
$556$	0	0
$557$	721.790i	1.29585i	0.761703	+	0.647926i	$0.224363\pi$
$557$	721.790i	1.29585i	−0.761703	+	0.647926i	$0.775637\pi$
$558$	0	0
$559$	− 127.846i	− 0.228705i
$560$	0	0
$561$	513.415	0.915179
$562$	0	0
$563$	7.50258	0.0133261	0.00666303	−	0.999978i	$-0.497879\pi$
$563$	7.50258	0.0133261	0.00666303	+	0.999978i	$0.497879\pi$
$564$	0	0
$565$	5.61046i	0.00993001i
$566$	0	0
$567$	− 26.3538i	− 0.0464794i
$568$	0	0
$569$	500.697	0.879960	0.439980	−	0.898008i	$-0.354985\pi$
$569$	500.697	0.879960	0.439980	+	0.898008i	$0.354985\pi$
$570$	0	0
$571$	553.072	0.968602	0.484301	−	0.874901i	$-0.339074\pi$
$571$	553.072	0.968602	0.484301	+	0.874901i	$0.339074\pi$
$572$	0	0
$573$	151.426i	0.264268i
$574$	0	0
$575$	5.70250i	0.00991740i
$576$	0	0
$577$	889.426	1.54147	0.770733	−	0.637159i	$-0.219890\pi$
$577$	889.426	1.54147	0.770733	+	0.637159i	$0.219890\pi$
$578$	0	0
$579$	−46.0282	−0.0794960
$580$	0	0
$581$	− 184.267i	− 0.317154i
$582$	0	0
$583$	13.8564i	0.0237674i
$584$	0	0
$585$	352.708	0.602919
$586$	0	0
$587$	317.892	0.541554	0.270777	−	0.962642i	$-0.412719\pi$
$587$	317.892	0.541554	0.270777	+	0.962642i	$0.412719\pi$
$588$	0	0
$589$	− 1195.10i	− 2.02903i
$590$	0	0
$591$	− 435.597i	− 0.737051i
$592$	0	0
$593$	−508.543	−0.857577	−0.428789	−	0.903405i	$-0.641060\pi$
$593$	−508.543	−0.857577	−0.428789	+	0.903405i	$0.641060\pi$
$594$	0	0
$595$	286.543	0.481586
$596$	0	0
$597$	− 117.913i	− 0.197509i
$598$	0	0
$599$	548.246i	0.915269i	0.889140	+	0.457634i	$0.151303\pi$
$599$	548.246i	0.915269i	−0.889140	+	0.457634i	$0.848697\pi$
$600$	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
$603$	49.4923	0.0820767
$604$	0	0
$605$	− 501.944i	− 0.829659i
$606$	0	0
$607$	− 527.944i	− 0.869759i	−0.900489	−	0.434879i	$-0.856791\pi$
$607$	− 527.944i	− 0.869759i	0.900489	−	0.434879i	$-0.143209\pi$
$608$	0	0
$609$	85.8564	0.140979
$610$	0	0
$611$	1666.52	2.72753
$612$	0	0
$613$	− 738.000i	− 1.20392i	−0.798528	−	0.601958i	$-0.794388\pi$
$613$	− 738.000i	− 1.20392i	0.798528	−	0.601958i	$-0.205612\pi$
$614$	0	0
$615$	69.5129i	0.113029i
$616$	0	0
$617$	−389.979	−0.632057	−0.316029	−	0.948750i	$-0.602350\pi$
$617$	−389.979	−0.632057	−0.316029	+	0.948750i	$0.602350\pi$
$618$	0	0
$619$	−1006.89	−1.62663	−0.813317	−	0.581820i	$-0.802341\pi$
$619$	−1006.89	−1.62663	−0.813317	+	0.581820i	$0.802341\pi$
$620$	0	0
$621$	− 41.5692i	− 0.0669392i
$622$	0	0
$623$	− 147.251i	− 0.236358i
$624$	0	0
$625$	−606.672	−0.970675
$626$	0	0
$627$	−799.692	−1.27543
$628$	0	0
$629$	− 192.862i	− 0.306616i
$630$	0	0
$631$	− 1043.77i	− 1.65415i	−0.562091	−	0.827075i	$-0.690003\pi$
$631$	− 1043.77i	− 1.65415i	0.562091	−	0.827075i	$-0.309997\pi$
$632$	0	0
$633$	447.979	0.707708
$634$	0	0
$635$	−844.441	−1.32983
$636$	0	0
$637$	964.410i	1.51399i
$638$	0	0
$639$	− 129.415i	− 0.202528i
$640$	0	0
$641$	−1100.96	−1.71757	−0.858786	−	0.512334i	$-0.828781\pi$
$641$	−1100.96	−1.71757	−0.858786	+	0.512334i	$0.828781\pi$
$642$	0	0
$643$	−508.785	−0.791267	−0.395633	−	0.918409i	$-0.629475\pi$
$643$	−508.785	−0.791267	−0.395633	+	0.918409i	$0.629475\pi$
$644$	0	0
$645$	− 45.7437i	− 0.0709205i
$646$	0	0
$647$	− 804.862i	− 1.24399i	−0.783021	−	0.621995i	$-0.786322\pi$
$647$	− 804.862i	− 1.24399i	0.783021	−	0.621995i	$-0.213678\pi$
$648$	0	0
$649$	1615.38	2.48904
$650$	0	0
$651$	195.979	0.301044
$652$	0	0
$653$	1284.62i	1.96726i	0.180202	+	0.983630i	$0.442325\pi$
$653$	1284.62i	1.96726i	−0.180202	+	0.983630i	$0.557675\pi$
$654$	0	0
$655$	− 281.261i	− 0.429407i
$656$	0	0
$657$	−76.2769	−0.116099
$658$	0	0
$659$	−219.523	−0.333116	−0.166558	−	0.986032i	$-0.553265\pi$
$659$	−219.523	−0.333116	−0.166558	+	0.986032i	$0.553265\pi$
$660$	0	0
$661$	598.267i	0.905093i	0.891741	+	0.452547i	$0.149484\pi$
$661$	598.267i	0.905093i	−0.891741	+	0.452547i	$0.850516\pi$
$662$	0	0
$663$	820.477i	1.23752i
$664$	0	0
$665$	−446.318	−0.671155
$666$	0	0
$667$	135.426	0.203037
$668$	0	0
$669$	14.2205i	0.0212564i
$670$	0	0
$671$	− 208.995i	− 0.311468i
$672$	0	0
$673$	−575.149	−0.854604	−0.427302	−	0.904109i	$-0.640536\pi$
$673$	−575.149	−0.854604	−0.427302	+	0.904109i	$0.640536\pi$
$674$	0	0
$675$	−3.70388	−0.00548724
$676$	0	0
$677$	450.805i	0.665887i	0.942947	+	0.332943i	$0.108042\pi$
$677$	450.805i	0.665887i	−0.942947	+	0.332943i	$0.891958\pi$
$678$	0	0
$679$	424.995i	0.625913i
$680$	0	0
$681$	−413.836	−0.607688
$682$	0	0
$683$	−1186.45	−1.73711	−0.868555	−	0.495593i	$-0.834951\pi$
$683$	−1186.45	−1.73711	−0.868555	+	0.495593i	$0.834951\pi$
$684$	0	0
$685$	− 759.600i	− 1.10891i
$686$	0	0
$687$	− 314.967i	− 0.458467i
$688$	0	0
$689$	−22.1436	−0.0321387
$690$	0	0
$691$	−870.508	−1.25978	−0.629890	−	0.776685i	$-0.716900\pi$
$691$	−870.508	−1.25978	−0.629890	+	0.776685i	$0.716900\pi$
$692$	0	0
$693$	− 131.138i	− 0.189233i
$694$	0	0
$695$	− 827.559i	− 1.19073i
$696$	0	0
$697$	−161.703	−0.231998
$698$	0	0
$699$	−667.079	−0.954334
$700$	0	0
$701$	218.785i	0.312104i	0.987749	+	0.156052i	$0.0498767\pi$
$701$	218.785i	0.312104i	−0.987749	+	0.156052i	$0.950123\pi$
$702$	0	0
$703$	300.400i	0.427311i
$704$	0	0
$705$	596.287	0.845797
$706$	0	0
$707$	−465.795	−0.658832
$708$	0	0
$709$	85.8461i	0.121081i	0.998166	+	0.0605403i	$0.0192824\pi$
$709$	85.8461i	0.121081i	−0.998166	+	0.0605403i	$0.980718\pi$
$710$	0	0
$711$	290.354i	0.408374i
$712$	0	0
$713$	309.128	0.433560
$714$	0	0
$715$	1755.10	2.45468
$716$	0	0
$717$	− 699.713i	− 0.975890i
$718$	0	0
$719$	− 816.728i	− 1.13592i	−0.823055	−	0.567961i	$-0.807732\pi$
$719$	− 816.728i	− 1.13592i	0.823055	−	0.567961i	$-0.192268\pi$
$720$	0	0
$721$	−72.5744	−0.100658
$722$	0	0
$723$	89.0897	0.123222
$724$	0	0
$725$	− 12.0666i	− 0.0166436i
$726$	0	0
$727$	− 62.6410i	− 0.0861637i	−0.999072	−	0.0430819i	$-0.986282\pi$
$727$	− 62.6410i	− 0.0861637i	0.999072	−	0.0430819i	$-0.0137176\pi$
$728$	0	0
$729$	27.0000	0.0370370
$730$	0	0
$731$	106.410	0.145568
$732$	0	0
$733$	72.3895i	0.0987579i	0.998780	+	0.0493790i	$0.0157242\pi$
$733$	72.3895i	0.0987579i	−0.998780	+	0.0493790i	$0.984276\pi$
$734$	0	0
$735$	345.069i	0.469482i
$736$	0	0
$737$	246.277	0.334161
$738$	0	0
$739$	651.061	0.881003	0.440502	−	0.897752i	$-0.354801\pi$
$739$	651.061	0.881003	0.440502	+	0.897752i	$0.354801\pi$
$740$	0	0
$741$	− 1277.97i	− 1.72465i
$742$	0	0
$743$	951.272i	1.28031i	0.768245	+	0.640156i	$0.221130\pi$
$743$	951.272i	1.28031i	−0.768245	+	0.640156i	$0.778870\pi$
$744$	0	0
$745$	−612.841	−0.822605
$746$	0	0
$747$	188.785	0.252724
$748$	0	0
$749$	123.180i	0.164459i
$750$	0	0
$751$	− 280.323i	− 0.373266i	−0.982430	−	0.186633i	$-0.940242\pi$
$751$	− 280.323i	− 0.373266i	0.982430	−	0.186633i	$-0.0597576\pi$
$752$	0	0
$753$	7.29234	0.00968438
$754$	0	0
$755$	237.590	0.314689
$756$	0	0
$757$	− 246.708i	− 0.325902i	−0.986634	−	0.162951i	$-0.947899\pi$
$757$	− 246.708i	− 0.325902i	0.986634	−	0.162951i	$-0.0521012\pi$
$758$	0	0
$759$	− 206.851i	− 0.272531i
$760$	0	0
$761$	−739.569	−0.971839	−0.485919	−	0.874004i	$-0.661515\pi$
$761$	−739.569	−0.971839	−0.485919	+	0.874004i	$0.661515\pi$
$762$	0	0
$763$	−58.1436	−0.0762039
$764$	0	0
$765$	293.569i	0.383751i
$766$	0	0
$767$	2581.51i	3.36572i
$768$	0	0
$769$	1017.96	1.32374	0.661872	−	0.749617i	$-0.269762\pi$
$769$	1017.96	1.32374	0.661872	+	0.749617i	$0.269762\pi$
$770$	0	0
$771$	158.833	0.206010
$772$	0	0
$773$	50.9179i	0.0658705i	0.999457	+	0.0329352i	$0.0104855\pi$
$773$	50.9179i	0.0658705i	−0.999457	+	0.0329352i	$0.989514\pi$
$774$	0	0
$775$	− 27.5438i	− 0.0355404i
$776$	0	0
$777$	−49.2614	−0.0633995
$778$	0	0
$779$	251.867	0.323321
$780$	0	0
$781$	− 643.979i	− 0.824557i
$782$	0	0
$783$	87.9615i	0.112339i
$784$	0	0
$785$	819.497	1.04395
$786$	0	0
$787$	−759.041	−0.964474	−0.482237	−	0.876041i	$-0.660176\pi$
$787$	−759.041	−0.964474	−0.482237	+	0.876041i	$0.660176\pi$
$788$	0	0
$789$	− 467.405i	− 0.592402i
$790$	0	0
$791$	3.33358i	0.00421439i
$792$	0	0
$793$	333.990	0.421172
$794$	0	0
$795$	−7.92305	−0.00996610
$796$	0	0
$797$	− 14.2102i	− 0.0178297i	−0.999960	−	0.00891483i	$-0.997162\pi$
$797$	− 14.2102i	− 0.0178297i	0.999960	−	0.00891483i	$-0.00283772\pi$
$798$	0	0
$799$	1387.10i	1.73604i
$800$	0	0
$801$	150.862	0.188342
$802$	0	0
$803$	−379.559	−0.472676
$804$	0	0
$805$	− 115.446i	− 0.143411i
$806$	0	0
$807$	347.023i	0.430016i
$808$	0	0
$809$	−715.528	−0.884460	−0.442230	−	0.896902i	$-0.645812\pi$
$809$	−715.528	−0.884460	−0.442230	+	0.896902i	$0.645812\pi$
$810$	0	0
$811$	250.641	0.309052	0.154526	−	0.987989i	$-0.450615\pi$
$811$	250.641	0.309052	0.154526	+	0.987989i	$0.450615\pi$
$812$	0	0
$813$	− 861.184i	− 1.05927i
$814$	0	0
$815$	157.323i	0.193035i
$816$	0	0
$817$	−165.744	−0.202869
$818$	0	0
$819$	209.569	0.255884
$820$	0	0
$821$	908.641i	1.10675i	0.832933	+	0.553375i	$0.186660\pi$
$821$	908.641i	1.10675i	−0.832933	+	0.553375i	$0.813340\pi$
$822$	0	0
$823$	514.200i	0.624787i	0.949953	+	0.312394i	$0.101131\pi$
$823$	514.200i	0.624787i	−0.949953	+	0.312394i	$0.898869\pi$
$824$	0	0
$825$	−18.4308	−0.0223403
$826$	0	0
$827$	−1240.61	−1.50013	−0.750067	−	0.661362i	$-0.769979\pi$
$827$	−1240.61	−1.50013	−0.750067	+	0.661362i	$0.769979\pi$
$828$	0	0
$829$	− 352.410i	− 0.425103i	−0.977150	−	0.212551i	$-0.931823\pi$
$829$	− 352.410i	− 0.425103i	0.977150	−	0.212551i	$-0.0681773\pi$
$830$	0	0
$831$	− 446.105i	− 0.536829i
$832$	0	0
$833$	−802.708	−0.963635
$834$	0	0
$835$	−1035.63	−1.24028
$836$	0	0
$837$	200.785i	0.239886i
$838$	0	0
$839$	1038.81i	1.23815i	0.785331	+	0.619076i	$0.212493\pi$
$839$	1038.81i	1.23815i	−0.785331	+	0.619076i	$0.787507\pi$
$840$	0	0
$841$	554.436	0.659258
$842$	0	0
$843$	336.018	0.398598
$844$	0	0
$845$	1971.91i	2.33362i
$846$	0	0
$847$	− 298.241i	− 0.352115i
$848$	0	0
$849$	−575.405	−0.677744
$850$	0	0
$851$	−77.7025	−0.0913073
$852$	0	0
$853$	1382.80i	1.62110i	0.585668	+	0.810551i	$0.300832\pi$
$853$	1382.80i	1.62110i	−0.585668	+	0.810551i	$0.699168\pi$
$854$	0	0
$855$	− 457.261i	− 0.534809i
$856$	0	0
$857$	−893.518	−1.04261	−0.521306	−	0.853370i	$-0.674555\pi$
$857$	−893.518	−1.04261	−0.521306	+	0.853370i	$0.674555\pi$
$858$	0	0
$859$	−1549.85	−1.80425	−0.902125	−	0.431475i	$-0.857993\pi$
$859$	−1549.85	−1.80425	−0.902125	+	0.431475i	$0.857993\pi$
$860$	0	0
$861$	41.3027i	0.0479706i
$862$	0	0
$863$	347.405i	0.402555i	0.979534	+	0.201278i	$0.0645093\pi$
$863$	347.405i	0.402555i	−0.979534	+	0.201278i	$0.935491\pi$
$864$	0	0
$865$	807.138	0.933108
$866$	0	0
$867$	−182.345	−0.210317
$868$	0	0
$869$	1444.82i	1.66262i
$870$	0	0
$871$	393.569i	0.451859i
$872$	0	0
$873$	−435.415	−0.498758
$874$	0	0
$875$	−371.056	−0.424064
$876$	0	0
$877$	1216.24i	1.38681i	0.720546	+	0.693407i	$0.243891\pi$
$877$	1216.24i	1.38681i	−0.720546	+	0.693407i	$0.756109\pi$
$878$	0	0
$879$	806.967i	0.918051i
$880$	0	0
$881$	−270.533	−0.307075	−0.153538	−	0.988143i	$-0.549067\pi$
$881$	−270.533	−0.307075	−0.153538	+	0.988143i	$0.549067\pi$
$882$	0	0
$883$	368.918	0.417801	0.208900	−	0.977937i	$-0.433012\pi$
$883$	368.918	0.417801	0.208900	+	0.977937i	$0.433012\pi$
$884$	0	0
$885$	923.672i	1.04370i
$886$	0	0
$887$	− 240.154i	− 0.270748i	−0.990795	−	0.135374i	$-0.956776\pi$
$887$	− 240.154i	− 0.270748i	0.990795	−	0.135374i	$-0.0432237\pi$
$888$	0	0
$889$	−501.744	−0.564391
$890$	0	0
$891$	134.354	0.150790
$892$	0	0
$893$	− 2160.53i	− 2.41941i
$894$	0	0
$895$	916.974i	1.02455i
$896$	0	0
$897$	330.564	0.368522
$898$	0	0
$899$	−654.123	−0.727612
$900$	0	0
$901$	− 18.4308i	− 0.0204559i
$902$	0	0
$903$	− 27.1797i	− 0.0300993i
$904$	0	0
$905$	−211.887	−0.234130
$906$	0	0
$907$	−1141.05	−1.25805	−0.629025	−	0.777385i	$-0.716546\pi$
$907$	−1141.05	−1.25805	−0.629025	+	0.777385i	$0.716546\pi$
$908$	0	0
$909$	− 477.215i	− 0.524989i
$910$	0	0
$911$	− 1755.94i	− 1.92748i	−0.266835	−	0.963742i	$-0.585978\pi$
$911$	− 1755.94i	− 1.92748i	0.266835	−	0.963742i	$-0.414022\pi$
$912$	0	0
$913$	939.405	1.02892
$914$	0	0
$915$	119.503	0.130604
$916$	0	0
$917$	− 167.118i	− 0.182244i
$918$	0	0
$919$	1414.18i	1.53882i	0.638753	+	0.769412i	$0.279451\pi$
$919$	1414.18i	1.53882i	−0.638753	+	0.769412i	$0.720549\pi$
$920$	0	0
$921$	373.723	0.405780
$922$	0	0
$923$	1029.13	1.11498
$924$	0	0
$925$	6.92342i	0.00748478i
$926$	0	0
$927$	− 74.3538i	− 0.0802091i
$928$	0	0
$929$	−1848.06	−1.98930	−0.994651	−	0.103295i	$-0.967061\pi$
$929$	−1848.06	−1.98930	−0.994651	+	0.103295i	$0.967061\pi$
$930$	0	0
$931$	1250.29	1.34296
$932$	0	0
$933$	− 303.118i	− 0.324885i
$934$	0	0
$935$	1460.82i	1.56237i
$936$	0	0
$937$	170.554	0.182021	0.0910105	−	0.995850i	$-0.470990\pi$
$937$	170.554	0.182021	0.0910105	+	0.995850i	$0.470990\pi$
$938$	0	0
$939$	−751.674	−0.800505
$940$	0	0
$941$	281.523i	0.299174i	0.988749	+	0.149587i	$0.0477945\pi$
$941$	281.523i	0.299174i	−0.988749	+	0.149587i	$0.952206\pi$
$942$	0	0
$943$	65.1487i	0.0690867i
$944$	0	0
$945$	74.9845	0.0793487
$946$	0	0
$947$	−1533.43	−1.61925	−0.809625	−	0.586947i	$-0.800330\pi$
$947$	−1533.43	−1.61925	−0.809625	+	0.586947i	$0.800330\pi$
$948$	0	0
$949$	− 606.564i	− 0.639161i
$950$	0	0
$951$	− 153.264i	− 0.161161i
$952$	0	0
$953$	−414.133	−0.434557	−0.217279	−	0.976110i	$-0.569718\pi$
$953$	−414.133	−0.434557	−0.217279	+	0.976110i	$0.569718\pi$
$954$	0	0
$955$	−430.851	−0.451153
$956$	0	0
$957$	437.703i	0.457369i
$958$	0	0
$959$	− 451.334i	− 0.470629i
$960$	0	0
$961$	−532.128	−0.553723
$962$	0	0
$963$	−126.200	−0.131049
$964$	0	0
$965$	− 130.964i	− 0.135714i
$966$	0	0
$967$	228.918i	0.236730i	0.992970	+	0.118365i	$0.0377653\pi$
$967$	228.918i	0.236730i	−0.992970	+	0.118365i	$0.962235\pi$
$968$	0	0
$969$	1063.69	1.09772
$970$	0	0
$971$	968.877	0.997813	0.498907	−	0.866656i	$-0.333735\pi$
$971$	968.877	0.997813	0.498907	+	0.866656i	$0.333735\pi$
$972$	0	0
$973$	− 491.713i	− 0.505357i
$974$	0	0
$975$	− 29.4538i	− 0.0302090i
$976$	0	0
$977$	−648.144	−0.663402	−0.331701	−	0.943385i	$-0.607622\pi$
$977$	−648.144	−0.663402	−0.331701	+	0.943385i	$0.607622\pi$
$978$	0	0
$979$	750.697	0.766800
$980$	0	0
$981$	− 59.5692i	− 0.0607230i
$982$	0	0
$983$	− 1035.56i	− 1.05347i	−0.850030	−	0.526734i	$-0.823416\pi$
$983$	− 1035.56i	− 1.05347i	0.850030	−	0.526734i	$-0.176584\pi$
$984$	0	0
$985$	1239.41	1.25828
$986$	0	0
$987$	354.297	0.358964
$988$	0	0
$989$	− 42.8719i	− 0.0433487i
$990$	0	0
$991$	1032.44i	1.04181i	0.853614	+	0.520906i	$0.174406\pi$
$991$	1032.44i	1.04181i	−0.853614	+	0.520906i	$0.825594\pi$
$992$	0	0
$993$	−185.436	−0.186743
$994$	0	0
$995$	335.497	0.337183
$996$	0	0
$997$	− 456.277i	− 0.457650i	−0.973468	−	0.228825i	$-0.926512\pi$
$997$	− 456.277i	− 0.457650i	0.973468	−	0.228825i	$-0.0734883\pi$
$998$	0	0
$999$	− 50.4693i	− 0.0505198i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.3.b.d.127.1		4
3.2	odd	2		2304.3.b.k.127.3		4
4.3	odd	2		768.3.b.a.127.3		4
8.3	odd	2	inner	768.3.b.d.127.2		4
8.5	even	2		768.3.b.a.127.4		4
12.11	even	2		2304.3.b.o.127.3		4
16.3	odd	4		192.3.g.c.127.3		4
16.5	even	4		96.3.g.a.31.4	yes	4
16.11	odd	4		96.3.g.a.31.2	✓	4
16.13	even	4		192.3.g.c.127.1		4
24.5	odd	2		2304.3.b.o.127.2		4
24.11	even	2		2304.3.b.k.127.2		4
48.5	odd	4		288.3.g.d.127.2		4
48.11	even	4		288.3.g.d.127.1		4
48.29	odd	4		576.3.g.j.127.4		4
48.35	even	4		576.3.g.j.127.3		4
80.27	even	4		2400.3.j.a.799.1		4
80.37	odd	4		2400.3.j.b.799.4		4
80.43	even	4		2400.3.j.b.799.3		4
80.53	odd	4		2400.3.j.a.799.2		4
80.59	odd	4		2400.3.e.a.1951.3		4
80.69	even	4		2400.3.e.a.1951.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.3.g.a.31.2	✓	4	16.11	odd	4
96.3.g.a.31.4	yes	4	16.5	even	4
192.3.g.c.127.1		4	16.13	even	4
192.3.g.c.127.3		4	16.3	odd	4
288.3.g.d.127.1		4	48.11	even	4
288.3.g.d.127.2		4	48.5	odd	4
576.3.g.j.127.3		4	48.35	even	4
576.3.g.j.127.4		4	48.29	odd	4
768.3.b.a.127.3		4	4.3	odd	2
768.3.b.a.127.4		4	8.5	even	2
768.3.b.d.127.1		4	1.1	even	1	trivial
768.3.b.d.127.2		4	8.3	odd	2	inner
2304.3.b.k.127.2		4	24.11	even	2
2304.3.b.k.127.3		4	3.2	odd	2
2304.3.b.o.127.2		4	24.5	odd	2
2304.3.b.o.127.3		4	12.11	even	2
2400.3.e.a.1951.2		4	80.69	even	4
2400.3.e.a.1951.3		4	80.59	odd	4
2400.3.j.a.799.1		4	80.27	even	4
2400.3.j.a.799.2		4	80.53	odd	4
2400.3.j.b.799.3		4	80.43	even	4
2400.3.j.b.799.4		4	80.37	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:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	768.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.73205 q^{3} -4.92820i q^{5} -2.92820i q^{7} +3.00000 q^{9} +O(q^{10})\)	\(q-1.73205 q^{3} -4.92820i q^{5} -2.92820i q^{7} +3.00000 q^{9} +14.9282 q^{11} +23.8564i q^{13} +8.53590i q^{15} -19.8564 q^{17} +30.9282 q^{19} +5.07180i q^{21} +8.00000i q^{23} +0.712813 q^{25} -5.19615 q^{27} -16.9282i q^{29} -38.6410i q^{31} -25.8564 q^{33} -14.4308 q^{35} +9.71281i q^{37} -41.3205i q^{39} +8.14359 q^{41} -5.35898 q^{43} -14.7846i q^{45} -69.8564i q^{47} +40.4256 q^{49} +34.3923 q^{51} +0.928203i q^{53} -73.5692i q^{55} -53.5692 q^{57} +108.210 q^{59} -14.0000i q^{61} -8.78461i q^{63} +117.569 q^{65} +16.4974 q^{67} -13.8564i q^{69} -43.1384i q^{71} -25.4256 q^{73} -1.23463 q^{75} -43.7128i q^{77} +96.7846i q^{79} +9.00000 q^{81} +62.9282 q^{83} +97.8564i q^{85} +29.3205i q^{87} +50.2872 q^{89} +69.8564 q^{91} +66.9282i q^{93} -152.420i q^{95} -145.138 q^{97} +44.7846 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{9}+O(q^{10}) \) 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

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