LMFDB - Embedded newform 192.3.g.c.127.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,3,Mod(127,192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(192, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0]))

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

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

chi := DirichletCharacter("192.127");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 192 = 2^{6} \cdot 3 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	192.g (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:	$5.23162107572$
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^{6} $
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	$\chi$	$=$	192.127
Dual form			192.3.g.c.127.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{3} -4.92820 q^{5} -2.92820i q^{7} -3.00000 q^{9} +O(q^{10})$	$q+1.73205i q^{3} -4.92820 q^{5} -2.92820i q^{7} -3.00000 q^{9} +14.9282i q^{11} -23.8564 q^{13} -8.53590i q^{15} -19.8564 q^{17} -30.9282i q^{19} +5.07180 q^{21} +8.00000i q^{23} -0.712813 q^{25} -5.19615i q^{27} +16.9282 q^{29} +38.6410i q^{31} -25.8564 q^{33} +14.4308i q^{35} +9.71281 q^{37} -41.3205i q^{39} -8.14359 q^{41} -5.35898i q^{43} +14.7846 q^{45} +69.8564i q^{47} +40.4256 q^{49} -34.3923i q^{51} +0.928203 q^{53} -73.5692i q^{55} +53.5692 q^{57} +108.210i q^{59} +14.0000 q^{61} +8.78461i q^{63} +117.569 q^{65} -16.4974i q^{67} -13.8564 q^{69} -43.1384i q^{71} +25.4256 q^{73} -1.23463i q^{75} +43.7128 q^{77} -96.7846i q^{79} +9.00000 q^{81} -62.9282i q^{83} +97.8564 q^{85} +29.3205i q^{87} -50.2872 q^{89} +69.8564i q^{91} -66.9282 q^{93} +152.420i q^{95} -145.138 q^{97} -44.7846i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{5} - 12 q^{9}+O(q^{10}) $ 4 * q + 8 * q^5 - 12 * q^9	$ 4 q + 8 q^{5} - 12 q^{9} - 40 q^{13} - 24 q^{17} + 48 q^{21} + 108 q^{25} + 40 q^{29} - 48 q^{33} - 72 q^{37} - 88 q^{41} - 24 q^{45} - 60 q^{49} - 24 q^{53} + 48 q^{57} + 56 q^{61} + 304 q^{65} - 120 q^{73} + 64 q^{77} + 36 q^{81} + 336 q^{85} - 312 q^{89} - 240 q^{93} - 248 q^{97}+O(q^{100}) $ 4 * q + 8 * q^5 - 12 * q^9 - 40 * q^13 - 24 * q^17 + 48 * q^21 + 108 * q^25 + 40 * q^29 - 48 * q^33 - 72 * q^37 - 88 * q^41 - 24 * q^45 - 60 * q^49 - 24 * q^53 + 48 * q^57 + 56 * q^61 + 304 * q^65 - 120 * q^73 + 64 * q^77 + 36 * q^81 + 336 * q^85 - 312 * q^89 - 240 * q^93 - 248 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/192\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$133$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	1.73205i	0.577350i
$3$	1.73205i	0.577350i
$4$	0	0
$5$	−4.92820	−0.985641	−0.492820	−	0.870131i	$-0.664034\pi$
$5$	−4.92820	−0.985641	−0.492820	+	0.870131i	$0.664034\pi$
$6$	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
$11$	14.9282i	1.35711i	0.734550	+	0.678555i	$0.237393\pi$
$11$	14.9282i	1.35711i	−0.734550	+	0.678555i	$0.762607\pi$
$12$	0	0
$13$	−23.8564	−1.83511	−0.917554	−	0.397611i	$-0.869839\pi$
$13$	−23.8564	−1.83511	−0.917554	+	0.397611i	$0.869839\pi$
$14$	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
$19$	− 30.9282i	− 1.62780i	−0.581005	−	0.813900i	$-0.697340\pi$
$19$	− 30.9282i	− 1.62780i	0.581005	−	0.813900i	$-0.302660\pi$
$20$	0	0
$21$	5.07180	0.241514
$22$	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
$27$	− 5.19615i	− 0.192450i
$28$	0	0
$29$	16.9282	0.583731	0.291866	−	0.956459i	$-0.405724\pi$
$29$	16.9282	0.583731	0.291866	+	0.956459i	$0.405724\pi$
$30$	0	0
$31$	38.6410i	1.24648i	0.782029	+	0.623242i	$0.214185\pi$
$31$	38.6410i	1.24648i	−0.782029	+	0.623242i	$0.785815\pi$
$32$	0	0
$33$	−25.8564	−0.783527
$34$	0	0
$35$	14.4308i	0.412308i
$36$	0	0
$37$	9.71281	0.262508	0.131254	−	0.991349i	$-0.458100\pi$
$37$	9.71281	0.262508	0.131254	+	0.991349i	$0.458100\pi$
$38$	0	0
$39$	− 41.3205i	− 1.05950i
$40$	0	0
$41$	−8.14359	−0.198624	−0.0993121	−	0.995056i	$-0.531664\pi$
$41$	−8.14359	−0.198624	−0.0993121	+	0.995056i	$0.531664\pi$
$42$	0	0
$43$	− 5.35898i	− 0.124628i	−0.998057	−	0.0623138i	$-0.980152\pi$
$43$	− 5.35898i	− 0.124628i	0.998057	−	0.0623138i	$-0.0198479\pi$
$44$	0	0
$45$	14.7846	0.328547
$46$	0	0
$47$	69.8564i	1.48631i	0.669121	+	0.743153i	$0.266671\pi$
$47$	69.8564i	1.48631i	−0.669121	+	0.743153i	$0.733329\pi$
$48$	0	0
$49$	40.4256	0.825013
$50$	0	0
$51$	− 34.3923i	− 0.674359i
$52$	0	0
$53$	0.928203	0.0175133	0.00875663	−	0.999962i	$-0.497213\pi$
$53$	0.928203	0.0175133	0.00875663	+	0.999962i	$0.497213\pi$
$54$	0	0
$55$	− 73.5692i	− 1.33762i
$56$	0	0
$57$	53.5692	0.939811
$58$	0	0
$59$	108.210i	1.83407i	0.398805	+	0.917036i	$0.369425\pi$
$59$	108.210i	1.83407i	−0.398805	+	0.917036i	$0.630575\pi$
$60$	0	0
$61$	14.0000	0.229508	0.114754	−	0.993394i	$-0.463392\pi$
$61$	14.0000	0.229508	0.114754	+	0.993394i	$0.463392\pi$
$62$	0	0
$63$	8.78461i	0.139438i
$64$	0	0
$65$	117.569	1.80876
$66$	0	0
$67$	− 16.4974i	− 0.246230i	−0.992392	−	0.123115i	$-0.960712\pi$
$67$	− 16.4974i	− 0.246230i	0.992392	−	0.123115i	$-0.0392884\pi$
$68$	0	0
$69$	−13.8564	−0.200817
$70$	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	$-0.444283\pi$
$73$	25.4256	0.348296	0.174148	+	0.984719i	$0.444283\pi$
$74$	0	0
$75$	− 1.23463i	− 0.0164617i
$76$	0	0
$77$	43.7128	0.567699
$78$	0	0
$79$	− 96.7846i	− 1.22512i	−0.790423	−	0.612561i	$-0.790139\pi$
$79$	− 96.7846i	− 1.22512i	0.790423	−	0.612561i	$-0.209861\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	− 62.9282i	− 0.758171i	−0.925362	−	0.379086i	$-0.876239\pi$
$83$	− 62.9282i	− 0.758171i	0.925362	−	0.379086i	$-0.123761\pi$
$84$	0	0
$85$	97.8564	1.15125
$86$	0	0
$87$	29.3205i	0.337017i
$88$	0	0
$89$	−50.2872	−0.565025	−0.282512	−	0.959264i	$-0.591168\pi$
$89$	−50.2872	−0.565025	−0.282512	+	0.959264i	$0.591168\pi$
$90$	0	0
$91$	69.8564i	0.767653i
$92$	0	0
$93$	−66.9282	−0.719658
$94$	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
$99$	− 44.7846i	− 0.452370i
$100$	0	0
$101$	−159.072	−1.57497	−0.787484	−	0.616335i	$-0.788617\pi$
$101$	−159.072	−1.57497	−0.787484	+	0.616335i	$0.788617\pi$
$102$	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
$107$	− 42.0666i	− 0.393146i	−0.980489	−	0.196573i	$-0.937019\pi$
$107$	− 42.0666i	− 0.393146i	0.980489	−	0.196573i	$-0.0629813\pi$
$108$	0	0
$109$	19.8564	0.182169	0.0910844	−	0.995843i	$-0.470967\pi$
$109$	19.8564	0.182169	0.0910844	+	0.995843i	$0.470967\pi$
$110$	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
$115$	− 39.4256i	− 0.342832i
$116$	0	0
$117$	71.5692	0.611703
$118$	0	0
$119$	58.1436i	0.488602i
$120$	0	0
$121$	−101.851	−0.841746
$122$	0	0
$123$	− 14.1051i	− 0.114676i
$124$	0	0
$125$	126.718	1.01374
$126$	0	0
$127$	171.349i	1.34920i	0.738182	+	0.674601i	$0.235684\pi$
$127$	171.349i	1.34920i	−0.738182	+	0.674601i	$0.764316\pi$
$128$	0	0
$129$	9.28203	0.0719537
$130$	0	0
$131$	− 57.0718i	− 0.435663i	−0.975986	−	0.217831i	$-0.930102\pi$
$131$	− 57.0718i	− 0.435663i	0.975986	−	0.217831i	$-0.0698983\pi$
$132$	0	0
$133$	−90.5641	−0.680933
$134$	0	0
$135$	25.6077i	0.189687i
$136$	0	0
$137$	−154.133	−1.12506	−0.562530	−	0.826777i	$-0.690172\pi$
$137$	−154.133	−1.12506	−0.562530	+	0.826777i	$0.690172\pi$
$138$	0	0
$139$	167.923i	1.20808i	0.796954	+	0.604040i	$0.206443\pi$
$139$	167.923i	1.20808i	−0.796954	+	0.604040i	$0.793557\pi$
$140$	0	0
$141$	−120.995	−0.858119
$142$	0	0
$143$	− 356.133i	− 2.49044i
$144$	0	0
$145$	−83.4256	−0.575349
$146$	0	0
$147$	70.0192i	0.476321i
$148$	0	0
$149$	−124.354	−0.834589	−0.417295	−	0.908771i	$-0.637022\pi$
$149$	−124.354	−0.834589	−0.417295	+	0.908771i	$0.637022\pi$
$150$	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
$155$	− 190.431i	− 1.22859i
$156$	0	0
$157$	−166.287	−1.05915	−0.529577	−	0.848262i	$-0.677649\pi$
$157$	−166.287	−1.05915	−0.529577	+	0.848262i	$0.677649\pi$
$158$	0	0
$159$	1.60770i	0.0101113i
$160$	0	0
$161$	23.4256	0.145501
$162$	0	0
$163$	31.9230i	0.195847i	0.995194	+	0.0979235i	$0.0312200\pi$
$163$	31.9230i	0.195847i	−0.995194	+	0.0979235i	$0.968780\pi$
$164$	0	0
$165$	127.426	0.772277
$166$	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
$171$	92.7846i	0.542600i
$172$	0	0
$173$	−163.779	−0.946702	−0.473351	−	0.880874i	$-0.656956\pi$
$173$	−163.779	−0.946702	−0.473351	+	0.880874i	$0.656956\pi$
$174$	0	0
$175$	2.08726i	0.0119272i
$176$	0	0
$177$	−187.426	−1.05890
$178$	0	0
$179$	186.067i	1.03948i	0.854325	+	0.519739i	$0.173971\pi$
$179$	186.067i	1.03948i	−0.854325	+	0.519739i	$0.826029\pi$
$180$	0	0
$181$	−42.9948	−0.237541	−0.118770	−	0.992922i	$-0.537895\pi$
$181$	−42.9948	−0.237541	−0.118770	+	0.992922i	$0.537895\pi$
$182$	0	0
$183$	24.2487i	0.132507i
$184$	0	0
$185$	−47.8667	−0.258739
$186$	0	0
$187$	− 296.420i	− 1.58514i
$188$	0	0
$189$	−15.2154	−0.0805047
$190$	0	0
$191$	87.4256i	0.457726i	0.973459	+	0.228863i	$0.0735007\pi$
$191$	87.4256i	0.457726i	−0.973459	+	0.228863i	$0.926499\pi$
$192$	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
$195$	203.636i	1.04429i
$196$	0	0
$197$	251.492	1.27661	0.638305	−	0.769783i	$-0.279636\pi$
$197$	251.492	1.27661	0.638305	+	0.769783i	$0.279636\pi$
$198$	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
$203$	− 49.5692i	− 0.244183i
$204$	0	0
$205$	40.1333	0.195772
$206$	0	0
$207$	− 24.0000i	− 0.115942i
$208$	0	0
$209$	461.703	2.20910
$210$	0	0
$211$	258.641i	1.22579i	0.790166	+	0.612893i	$0.209995\pi$
$211$	258.641i	1.22579i	−0.790166	+	0.612893i	$0.790005\pi$
$212$	0	0
$213$	74.7180	0.350789
$214$	0	0
$215$	26.4102i	0.122838i
$216$	0	0
$217$	113.149	0.521423
$218$	0	0
$219$	44.0385i	0.201089i
$220$	0	0
$221$	473.703	2.14345
$222$	0	0
$223$	8.21024i	0.0368172i	0.999831	+	0.0184086i	$0.00585997\pi$
$223$	8.21024i	0.0368172i	−0.999831	+	0.0184086i	$0.994140\pi$
$224$	0	0
$225$	2.13844	0.00950417
$226$	0	0
$227$	− 238.928i	− 1.05255i	−0.850315	−	0.526274i	$-0.823589\pi$
$227$	− 238.928i	− 1.05255i	0.850315	−	0.526274i	$-0.176411\pi$
$228$	0	0
$229$	181.846	0.794088	0.397044	−	0.917800i	$-0.370036\pi$
$229$	181.846	0.794088	0.397044	+	0.917800i	$0.370036\pi$
$230$	0	0
$231$	75.7128i	0.327761i
$232$	0	0
$233$	−385.138	−1.65295	−0.826477	−	0.562970i	$-0.809659\pi$
$233$	−385.138	−1.65295	−0.826477	+	0.562970i	$0.809659\pi$
$234$	0	0
$235$	− 344.267i	− 1.46496i
$236$	0	0
$237$	167.636	0.707324
$238$	0	0
$239$	− 403.979i	− 1.69029i	−0.534537	−	0.845145i	$-0.679514\pi$
$239$	− 403.979i	− 1.69029i	0.534537	−	0.845145i	$-0.320486\pi$
$240$	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
$243$	15.5885i	0.0641500i
$244$	0	0
$245$	−199.226	−0.813166
$246$	0	0
$247$	737.836i	2.98719i
$248$	0	0
$249$	108.995	0.437730
$250$	0	0
$251$	− 4.21024i	− 0.0167738i	−0.999965	−	0.00838692i	$-0.997330\pi$
$251$	− 4.21024i	− 0.0167738i	0.999965	−	0.00838692i	$-0.00266967\pi$
$252$	0	0
$253$	−119.426	−0.472038
$254$	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
$259$	− 28.4411i	− 0.109811i
$260$	0	0
$261$	−50.7846	−0.194577
$262$	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
$267$	− 87.1000i	− 0.326217i
$268$	0	0
$269$	200.354	0.744810	0.372405	−	0.928070i	$-0.378533\pi$
$269$	200.354	0.744810	0.372405	+	0.928070i	$0.378533\pi$
$270$	0	0
$271$	− 497.205i	− 1.83471i	−0.398076	−	0.917353i	$-0.630322\pi$
$271$	− 497.205i	− 1.83471i	0.398076	−	0.917353i	$-0.369678\pi$
$272$	0	0
$273$	−120.995	−0.443205
$274$	0	0
$275$	− 10.6410i	− 0.0386946i
$276$	0	0
$277$	257.559	0.929816	0.464908	−	0.885359i	$-0.346087\pi$
$277$	257.559	0.929816	0.464908	+	0.885359i	$0.346087\pi$
$278$	0	0
$279$	− 115.923i	− 0.415495i
$280$	0	0
$281$	194.000	0.690391	0.345196	−	0.938531i	$-0.387813\pi$
$281$	194.000	0.690391	0.345196	+	0.938531i	$0.387813\pi$
$282$	0	0
$283$	332.210i	1.17389i	0.809628	+	0.586944i	$0.199669\pi$
$283$	332.210i	1.17389i	−0.809628	+	0.586944i	$0.800331\pi$
$284$	0	0
$285$	−264.000	−0.926316
$286$	0	0
$287$	23.8461i	0.0830874i
$288$	0	0
$289$	105.277	0.364280
$290$	0	0
$291$	− 251.387i	− 0.863873i
$292$	0	0
$293$	−465.902	−1.59011	−0.795055	−	0.606537i	$-0.792558\pi$
$293$	−465.902	−1.59011	−0.795055	+	0.606537i	$0.792558\pi$
$294$	0	0
$295$	− 533.282i	− 1.80774i
$296$	0	0
$297$	77.5692	0.261176
$298$	0	0
$299$	− 190.851i	− 0.638299i
$300$	0	0
$301$	−15.6922	−0.0521335
$302$	0	0
$303$	− 275.520i	− 0.909308i
$304$	0	0
$305$	−68.9948	−0.226213
$306$	0	0
$307$	215.769i	0.702831i	0.936220	+	0.351416i	$0.114300\pi$
$307$	215.769i	0.702831i	−0.936220	+	0.351416i	$0.885700\pi$
$308$	0	0
$309$	42.9282	0.138926
$310$	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	$-0.743825\pi$
$313$	−433.979	−1.38652	−0.693258	+	0.720690i	$0.743825\pi$
$314$	0	0
$315$	− 43.2923i	− 0.137436i
$316$	0	0
$317$	−88.4871	−0.279139	−0.139570	−	0.990212i	$-0.544572\pi$
$317$	−88.4871	−0.279139	−0.139570	+	0.990212i	$0.544572\pi$
$318$	0	0
$319$	252.708i	0.792187i
$320$	0	0
$321$	72.8616	0.226983
$322$	0	0
$323$	614.123i	1.90131i
$324$	0	0
$325$	17.0052	0.0523236
$326$	0	0
$327$	34.3923i	0.105175i
$328$	0	0
$329$	204.554	0.621744
$330$	0	0
$331$	107.061i	0.323449i	0.986836	+	0.161724i	$0.0517055\pi$
$331$	107.061i	0.323449i	−0.986836	+	0.161724i	$0.948294\pi$
$332$	0	0
$333$	−29.1384	−0.0875028
$334$	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
$339$	− 1.97183i	− 0.00581662i
$340$	0	0
$341$	−576.841	−1.69162
$342$	0	0
$343$	− 261.856i	− 0.763430i
$344$	0	0
$345$	68.2872	0.197934
$346$	0	0
$347$	− 65.9127i	− 0.189950i	−0.995480	−	0.0949751i	$-0.969723\pi$
$347$	− 65.9127i	− 0.189950i	0.995480	−	0.0949751i	$-0.0302772\pi$
$348$	0	0
$349$	−367.703	−1.05359	−0.526794	−	0.849993i	$-0.676606\pi$
$349$	−367.703	−1.05359	−0.526794	+	0.849993i	$0.676606\pi$
$350$	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
$355$	212.595i	0.598859i
$356$	0	0
$357$	−100.708	−0.282094
$358$	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
$363$	− 176.412i	− 0.485982i
$364$	0	0
$365$	−125.303	−0.343295
$366$	0	0
$367$	− 505.051i	− 1.37616i	−0.725634	−	0.688081i	$-0.758454\pi$
$367$	− 505.051i	− 1.37616i	0.725634	−	0.688081i	$-0.241546\pi$
$368$	0	0
$369$	24.4308	0.0662081
$370$	0	0
$371$	− 2.71797i	− 0.00732606i
$372$	0	0
$373$	−215.128	−0.576751	−0.288376	−	0.957517i	$-0.593115\pi$
$373$	−215.128	−0.576751	−0.288376	+	0.957517i	$0.593115\pi$
$374$	0	0
$375$	219.482i	0.585285i
$376$	0	0
$377$	−403.846	−1.07121
$378$	0	0
$379$	− 204.785i	− 0.540329i	−0.962814	−	0.270164i	$-0.912922\pi$
$379$	− 204.785i	− 0.540329i	0.962814	−	0.270164i	$-0.0870781\pi$
$380$	0	0
$381$	−296.785	−0.778962
$382$	0	0
$383$	588.554i	1.53669i	0.640034	+	0.768347i	$0.278920\pi$
$383$	588.554i	1.53669i	−0.640034	+	0.768347i	$0.721080\pi$
$384$	0	0
$385$	−215.426	−0.559547
$386$	0	0
$387$	16.0770i	0.0415425i
$388$	0	0
$389$	534.477	1.37398	0.686988	−	0.726669i	$-0.258932\pi$
$389$	534.477	1.37398	0.686988	+	0.726669i	$0.258932\pi$
$390$	0	0
$391$	− 158.851i	− 0.406269i
$392$	0	0
$393$	98.8513	0.251530
$394$	0	0
$395$	476.974i	1.20753i
$396$	0	0
$397$	340.277	0.857121	0.428560	−	0.903513i	$-0.359021\pi$
$397$	340.277	0.857121	0.428560	+	0.903513i	$0.359021\pi$
$398$	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
$403$	− 921.836i	− 2.28743i
$404$	0	0
$405$	−44.3538	−0.109516
$406$	0	0
$407$	144.995i	0.356253i
$408$	0	0
$409$	513.692	1.25597	0.627986	−	0.778225i	$-0.283880\pi$
$409$	513.692	1.25597	0.627986	+	0.778225i	$0.283880\pi$
$410$	0	0
$411$	− 266.967i	− 0.649554i
$412$	0	0
$413$	316.862	0.767219
$414$	0	0
$415$	310.123i	0.747284i
$416$	0	0
$417$	−290.851	−0.697485
$418$	0	0
$419$	91.9437i	0.219436i	0.993963	+	0.109718i	$0.0349948\pi$
$419$	91.9437i	0.219436i	−0.993963	+	0.109718i	$0.965005\pi$
$420$	0	0
$421$	513.559	1.21985	0.609927	−	0.792457i	$-0.291199\pi$
$421$	513.559	1.21985	0.609927	+	0.792457i	$0.291199\pi$
$422$	0	0
$423$	− 209.569i	− 0.495436i
$424$	0	0
$425$	14.1539	0.0333033
$426$	0	0
$427$	− 40.9948i	− 0.0960067i
$428$	0	0
$429$	616.841	1.43786
$430$	0	0
$431$	446.123i	1.03509i	0.855657	+	0.517544i	$0.173154\pi$
$431$	446.123i	1.03509i	−0.855657	+	0.517544i	$0.826846\pi$
$432$	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
$435$	− 144.497i	− 0.332178i
$436$	0	0
$437$	247.426	0.566191
$438$	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
$443$	181.513i	0.409736i	0.978790	+	0.204868i	$0.0656765\pi$
$443$	181.513i	0.409736i	−0.978790	+	0.204868i	$0.934324\pi$
$444$	0	0
$445$	247.825	0.556911
$446$	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
$451$	− 121.569i	− 0.269555i
$452$	0	0
$453$	−83.5026	−0.184332
$454$	0	0
$455$	− 344.267i	− 0.756630i
$456$	0	0
$457$	526.862	1.15287	0.576435	−	0.817143i	$-0.304443\pi$
$457$	526.862	1.15287	0.576435	+	0.817143i	$0.304443\pi$
$458$	0	0
$459$	103.177i	0.224786i
$460$	0	0
$461$	−888.908	−1.92822	−0.964108	−	0.265510i	$-0.914460\pi$
$461$	−888.908	−1.92822	−0.964108	+	0.265510i	$0.914460\pi$
$462$	0	0
$463$	515.923i	1.11430i	0.830410	+	0.557152i	$0.188106\pi$
$463$	515.923i	1.11430i	−0.830410	+	0.557152i	$0.811894\pi$
$464$	0	0
$465$	329.836	0.709324
$466$	0	0
$467$	− 439.195i	− 0.940460i	−0.882544	−	0.470230i	$-0.844171\pi$
$467$	− 439.195i	− 0.940460i	0.882544	−	0.470230i	$-0.155829\pi$
$468$	0	0
$469$	−48.3078	−0.103002
$470$	0	0
$471$	− 288.018i	− 0.611503i
$472$	0	0
$473$	80.0000	0.169133
$474$	0	0
$475$	22.0460i	0.0464127i
$476$	0	0
$477$	−2.78461	−0.00583776
$478$	0	0
$479$	179.559i	0.374862i	0.982278	+	0.187431i	$0.0600161\pi$
$479$	179.559i	0.374862i	−0.982278	+	0.187431i	$0.939984\pi$
$480$	0	0
$481$	−231.713	−0.481731
$482$	0	0
$483$	40.5744i	0.0840049i
$484$	0	0
$485$	715.272	1.47479
$486$	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
$491$	226.487i	0.461277i	0.973039	+	0.230639i	$0.0740815\pi$
$491$	226.487i	0.461277i	−0.973039	+	0.230639i	$0.925918\pi$
$492$	0	0
$493$	−336.133	−0.681812
$494$	0	0
$495$	220.708i	0.445874i
$496$	0	0
$497$	−126.318	−0.254161
$498$	0	0
$499$	239.195i	0.479348i	0.970853	+	0.239674i	$0.0770406\pi$
$499$	239.195i	0.479348i	−0.970853	+	0.239674i	$0.922959\pi$
$500$	0	0
$501$	363.979	0.726506
$502$	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
$507$	693.042i	1.36695i
$508$	0	0
$509$	713.195	1.40117	0.700584	−	0.713570i	$-0.252923\pi$
$509$	713.195	1.40117	0.700584	+	0.713570i	$0.252923\pi$
$510$	0	0
$511$	− 74.4514i	− 0.145697i
$512$	0	0
$513$	−160.708	−0.313270
$514$	0	0
$515$	122.144i	0.237172i
$516$	0	0
$517$	−1042.83	−2.01708
$518$	0	0
$519$	− 283.674i	− 0.546579i
$520$	0	0
$521$	748.985	1.43759	0.718795	−	0.695222i	$-0.244694\pi$
$521$	748.985	1.43759	0.718795	+	0.695222i	$0.244694\pi$
$522$	0	0
$523$	338.908i	0.648007i	0.946056	+	0.324003i	$0.105029\pi$
$523$	338.908i	0.648007i	−0.946056	+	0.324003i	$0.894971\pi$
$524$	0	0
$525$	−3.61524	−0.00688618
$526$	0	0
$527$	− 767.272i	− 1.45592i
$528$	0	0
$529$	465.000	0.879017
$530$	0	0
$531$	− 324.631i	− 0.611357i
$532$	0	0
$533$	194.277	0.364497
$534$	0	0
$535$	207.313i	0.387501i
$536$	0	0
$537$	−322.277	−0.600143
$538$	0	0
$539$	603.482i	1.11963i
$540$	0	0
$541$	−3.56922	−0.00659745	−0.00329872	−	0.999995i	$-0.501050\pi$
$541$	−3.56922	−0.00659745	−0.00329872	+	0.999995i	$0.501050\pi$
$542$	0	0
$543$	− 74.4693i	− 0.137144i
$544$	0	0
$545$	−97.8564	−0.179553
$546$	0	0
$547$	− 530.067i	− 0.969043i	−0.874779	−	0.484522i	$-0.838994\pi$
$547$	− 530.067i	− 0.969043i	0.874779	−	0.484522i	$-0.161006\pi$
$548$	0	0
$549$	−42.0000	−0.0765027
$550$	0	0
$551$	− 523.559i	− 0.950198i
$552$	0	0
$553$	−283.405	−0.512486
$554$	0	0
$555$	− 82.9076i	− 0.149383i
$556$	0	0
$557$	−721.790	−1.29585	−0.647926	−	0.761703i	$-0.724363\pi$
$557$	−721.790	−1.29585	−0.647926	+	0.761703i	$0.724363\pi$
$558$	0	0
$559$	127.846i	0.228705i
$560$	0	0
$561$	513.415	0.915179
$562$	0	0
$563$	− 7.50258i	− 0.0133261i	−0.999978	−	0.00666303i	$-0.997879\pi$
$563$	− 7.50258i	− 0.0133261i	0.999978	−	0.00666303i	$-0.00212093\pi$
$564$	0	0
$565$	5.61046	0.00993001
$566$	0	0
$567$	− 26.3538i	− 0.0464794i
$568$	0	0
$569$	−500.697	−0.879960	−0.439980	−	0.898008i	$-0.645015\pi$
$569$	−500.697	−0.879960	−0.439980	+	0.898008i	$0.645015\pi$
$570$	0	0
$571$	553.072i	0.968602i	0.874901	+	0.484301i	$0.160926\pi$
$571$	553.072i	0.968602i	−0.874901	+	0.484301i	$0.839074\pi$
$572$	0	0
$573$	−151.426	−0.264268
$574$	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
$579$	46.0282i	0.0794960i
$580$	0	0
$581$	−184.267	−0.317154
$582$	0	0
$583$	13.8564i	0.0237674i
$584$	0	0
$585$	−352.708	−0.602919
$586$	0	0
$587$	317.892i	0.541554i	0.962642	+	0.270777i	$0.0872806\pi$
$587$	317.892i	0.541554i	−0.962642	+	0.270777i	$0.912719\pi$
$588$	0	0
$589$	1195.10	2.02903
$590$	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
$595$	− 286.543i	− 0.481586i
$596$	0	0
$597$	−117.913	−0.197509
$598$	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	$-0.648902\pi$
$601$	−542.000	−0.901830	−0.450915	+	0.892567i	$0.648902\pi$
$602$	0	0
$603$	49.4923i	0.0820767i
$604$	0	0
$605$	501.944	0.829659
$606$	0	0
$607$	527.944i	0.869759i	0.900489	+	0.434879i	$0.143209\pi$
$607$	527.944i	0.869759i	−0.900489	+	0.434879i	$0.856791\pi$
$608$	0	0
$609$	85.8564	0.140979
$610$	0	0
$611$	− 1666.52i	− 2.72753i
$612$	0	0
$613$	−738.000	−1.20392	−0.601958	−	0.798528i	$-0.705612\pi$
$613$	−738.000	−1.20392	−0.601958	+	0.798528i	$0.705612\pi$
$614$	0	0
$615$	69.5129i	0.113029i
$616$	0	0
$617$	389.979	0.632057	0.316029	−	0.948750i	$-0.397650\pi$
$617$	389.979	0.632057	0.316029	+	0.948750i	$0.397650\pi$
$618$	0	0
$619$	− 1006.89i	− 1.62663i	−0.581820	−	0.813317i	$-0.697659\pi$
$619$	− 1006.89i	− 1.62663i	0.581820	−	0.813317i	$-0.302341\pi$
$620$	0	0
$621$	41.5692	0.0669392
$622$	0	0
$623$	147.251i	0.236358i
$624$	0	0
$625$	−606.672	−0.970675
$626$	0	0
$627$	799.692i	1.27543i
$628$	0	0
$629$	−192.862	−0.306616
$630$	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
$635$	− 844.441i	− 1.32983i
$636$	0	0
$637$	−964.410	−1.51399
$638$	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
$643$	508.785i	0.791267i	0.918409	+	0.395633i	$0.129475\pi$
$643$	508.785i	0.791267i	−0.918409	+	0.395633i	$0.870525\pi$
$644$	0	0
$645$	−45.7437	−0.0709205
$646$	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
$651$	195.979i	0.301044i
$652$	0	0
$653$	−1284.62	−1.96726	−0.983630	−	0.180202i	$-0.942325\pi$
$653$	−1284.62	−1.96726	−0.983630	+	0.180202i	$0.942325\pi$
$654$	0	0
$655$	281.261i	0.429407i
$656$	0	0
$657$	−76.2769	−0.116099
$658$	0	0
$659$	219.523i	0.333116i	0.986032	+	0.166558i	$0.0532652\pi$
$659$	219.523i	0.333116i	−0.986032	+	0.166558i	$0.946735\pi$
$660$	0	0
$661$	598.267	0.905093	0.452547	−	0.891741i	$-0.350516\pi$
$661$	598.267	0.905093	0.452547	+	0.891741i	$0.350516\pi$
$662$	0	0
$663$	820.477i	1.23752i
$664$	0	0
$665$	446.318	0.671155
$666$	0	0
$667$	135.426i	0.203037i
$668$	0	0
$669$	−14.2205	−0.0212564
$670$	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
$675$	3.70388i	0.00548724i
$676$	0	0
$677$	450.805	0.665887	0.332943	−	0.942947i	$-0.391958\pi$
$677$	450.805	0.665887	0.332943	+	0.942947i	$0.391958\pi$
$678$	0	0
$679$	424.995i	0.625913i
$680$	0	0
$681$	413.836	0.607688
$682$	0	0
$683$	− 1186.45i	− 1.73711i	−0.495593	−	0.868555i	$-0.665049\pi$
$683$	− 1186.45i	− 1.73711i	0.495593	−	0.868555i	$-0.334951\pi$
$684$	0	0
$685$	759.600	1.10891
$686$	0	0
$687$	314.967i	0.458467i
$688$	0	0
$689$	−22.1436	−0.0321387
$690$	0	0
$691$	870.508i	1.25978i	0.776685	+	0.629890i	$0.216900\pi$
$691$	870.508i	1.25978i	−0.776685	+	0.629890i	$0.783100\pi$
$692$	0	0
$693$	−131.138	−0.189233
$694$	0	0
$695$	− 827.559i	− 1.19073i
$696$	0	0
$697$	161.703	0.231998
$698$	0	0
$699$	− 667.079i	− 0.954334i
$700$	0	0
$701$	−218.785	−0.312104	−0.156052	−	0.987749i	$-0.549877\pi$
$701$	−218.785	−0.312104	−0.156052	+	0.987749i	$0.549877\pi$
$702$	0	0
$703$	− 300.400i	− 0.427311i
$704$	0	0
$705$	596.287	0.845797
$706$	0	0
$707$	465.795i	0.658832i
$708$	0	0
$709$	85.8461	0.121081	0.0605403	−	0.998166i	$-0.480718\pi$
$709$	85.8461	0.121081	0.0605403	+	0.998166i	$0.480718\pi$
$710$	0	0
$711$	290.354i	0.408374i
$712$	0	0
$713$	−309.128	−0.433560
$714$	0	0
$715$	1755.10i	2.45468i
$716$	0	0
$717$	699.713	0.975890
$718$	0	0
$719$	816.728i	1.13592i	0.823055	+	0.567961i	$0.192268\pi$
$719$	816.728i	1.13592i	−0.823055	+	0.567961i	$0.807732\pi$
$720$	0	0
$721$	−72.5744	−0.100658
$722$	0	0
$723$	− 89.0897i	− 0.123222i
$724$	0	0
$725$	−12.0666	−0.0166436
$726$	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
$731$	106.410i	0.145568i
$732$	0	0
$733$	−72.3895	−0.0987579	−0.0493790	−	0.998780i	$-0.515724\pi$
$733$	−72.3895	−0.0987579	−0.0493790	+	0.998780i	$0.515724\pi$
$734$	0	0
$735$	− 345.069i	− 0.469482i
$736$	0	0
$737$	246.277	0.334161
$738$	0	0
$739$	− 651.061i	− 0.881003i	−0.897752	−	0.440502i	$-0.854801\pi$
$739$	− 651.061i	− 0.881003i	0.897752	−	0.440502i	$-0.145199\pi$
$740$	0	0
$741$	−1277.97	−1.72465
$742$	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
$747$	188.785i	0.252724i
$748$	0	0
$749$	−123.180	−0.164459
$750$	0	0
$751$	280.323i	0.373266i	0.982430	+	0.186633i	$0.0597576\pi$
$751$	280.323i	0.373266i	−0.982430	+	0.186633i	$0.940242\pi$
$752$	0	0
$753$	7.29234	0.00968438
$754$	0	0
$755$	− 237.590i	− 0.314689i
$756$	0	0
$757$	−246.708	−0.325902	−0.162951	−	0.986634i	$-0.552101\pi$
$757$	−246.708	−0.325902	−0.162951	+	0.986634i	$0.552101\pi$
$758$	0	0
$759$	− 206.851i	− 0.272531i
$760$	0	0
$761$	739.569	0.971839	0.485919	−	0.874004i	$-0.338485\pi$
$761$	739.569	0.971839	0.485919	+	0.874004i	$0.338485\pi$
$762$	0	0
$763$	− 58.1436i	− 0.0762039i
$764$	0	0
$765$	−293.569	−0.383751
$766$	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
$771$	− 158.833i	− 0.206010i
$772$	0	0
$773$	50.9179	0.0658705	0.0329352	−	0.999457i	$-0.489514\pi$
$773$	50.9179	0.0658705	0.0329352	+	0.999457i	$0.489514\pi$
$774$	0	0
$775$	− 27.5438i	− 0.0355404i
$776$	0	0
$777$	49.2614	0.0633995
$778$	0	0
$779$	251.867i	0.323321i
$780$	0	0
$781$	643.979	0.824557
$782$	0	0
$783$	− 87.9615i	− 0.112339i
$784$	0	0
$785$	819.497	1.04395
$786$	0	0
$787$	759.041i	0.964474i	0.876041	+	0.482237i	$0.160176\pi$
$787$	759.041i	0.964474i	−0.876041	+	0.482237i	$0.839824\pi$
$788$	0	0
$789$	−467.405	−0.592402
$790$	0	0
$791$	3.33358i	0.00421439i
$792$	0	0
$793$	−333.990	−0.421172
$794$	0	0
$795$	− 7.92305i	− 0.00996610i
$796$	0	0
$797$	14.2102	0.0178297	0.00891483	−	0.999960i	$-0.497162\pi$
$797$	14.2102	0.0178297	0.00891483	+	0.999960i	$0.497162\pi$
$798$	0	0
$799$	− 1387.10i	− 1.73604i
$800$	0	0
$801$	150.862	0.188342
$802$	0	0
$803$	379.559i	0.472676i
$804$	0	0
$805$	−115.446	−0.143411
$806$	0	0
$807$	347.023i	0.430016i
$808$	0	0
$809$	715.528	0.884460	0.442230	−	0.896902i	$-0.354188\pi$
$809$	715.528	0.884460	0.442230	+	0.896902i	$0.354188\pi$
$810$	0	0
$811$	250.641i	0.309052i	0.987989	+	0.154526i	$0.0493850\pi$
$811$	250.641i	0.309052i	−0.987989	+	0.154526i	$0.950615\pi$
$812$	0	0
$813$	861.184	1.05927
$814$	0	0
$815$	− 157.323i	− 0.193035i
$816$	0	0
$817$	−165.744	−0.202869
$818$	0	0
$819$	− 209.569i	− 0.255884i
$820$	0	0
$821$	908.641	1.10675	0.553375	−	0.832933i	$-0.313340\pi$
$821$	908.641	1.10675	0.553375	+	0.832933i	$0.313340\pi$
$822$	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
$827$	− 1240.61i	− 1.50013i	−0.661362	−	0.750067i	$-0.730021\pi$
$827$	− 1240.61i	− 1.50013i	0.661362	−	0.750067i	$-0.269979\pi$
$828$	0	0
$829$	352.410	0.425103	0.212551	−	0.977150i	$-0.431823\pi$
$829$	352.410	0.425103	0.212551	+	0.977150i	$0.431823\pi$
$830$	0	0
$831$	446.105i	0.536829i
$832$	0	0
$833$	−802.708	−0.963635
$834$	0	0
$835$	1035.63i	1.24028i
$836$	0	0
$837$	200.785	0.239886
$838$	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
$843$	336.018i	0.398598i
$844$	0	0
$845$	−1971.91	−2.33362
$846$	0	0
$847$	298.241i	0.352115i
$848$	0	0
$849$	−575.405	−0.677744
$850$	0	0
$851$	77.7025i	0.0913073i
$852$	0	0
$853$	1382.80	1.62110	0.810551	−	0.585668i	$-0.199168\pi$
$853$	1382.80	1.62110	0.810551	+	0.585668i	$0.199168\pi$
$854$	0	0
$855$	− 457.261i	− 0.534809i
$856$	0	0
$857$	893.518	1.04261	0.521306	−	0.853370i	$-0.325445\pi$
$857$	893.518	1.04261	0.521306	+	0.853370i	$0.325445\pi$
$858$	0	0
$859$	− 1549.85i	− 1.80425i	−0.431475	−	0.902125i	$-0.642007\pi$
$859$	− 1549.85i	− 1.80425i	0.431475	−	0.902125i	$-0.357993\pi$
$860$	0	0
$861$	−41.3027	−0.0479706
$862$	0	0
$863$	− 347.405i	− 0.402555i	−0.979534	−	0.201278i	$-0.935491\pi$
$863$	− 347.405i	− 0.402555i	0.979534	−	0.201278i	$-0.0645093\pi$
$864$	0	0
$865$	807.138	0.933108
$866$	0	0
$867$	182.345i	0.210317i
$868$	0	0
$869$	1444.82	1.66262
$870$	0	0
$871$	393.569i	0.451859i
$872$	0	0
$873$	435.415	0.498758
$874$	0	0
$875$	− 371.056i	− 0.424064i
$876$	0	0
$877$	−1216.24	−1.38681	−0.693407	−	0.720546i	$-0.743891\pi$
$877$	−1216.24	−1.38681	−0.693407	+	0.720546i	$0.743891\pi$
$878$	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
$883$	− 368.918i	− 0.417801i	−0.977937	−	0.208900i	$-0.933012\pi$
$883$	− 368.918i	− 0.417801i	0.977937	−	0.208900i	$-0.0669884\pi$
$884$	0	0
$885$	923.672	1.04370
$886$	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
$891$	134.354i	0.150790i
$892$	0	0
$893$	2160.53	2.41941
$894$	0	0
$895$	− 916.974i	− 1.02455i
$896$	0	0
$897$	330.564	0.368522
$898$	0	0
$899$	654.123i	0.727612i
$900$	0	0
$901$	−18.4308	−0.0204559
$902$	0	0
$903$	− 27.1797i	− 0.0300993i
$904$	0	0
$905$	211.887	0.234130
$906$	0	0
$907$	− 1141.05i	− 1.25805i	−0.777385	−	0.629025i	$-0.783454\pi$
$907$	− 1141.05i	− 1.25805i	0.777385	−	0.629025i	$-0.216546\pi$
$908$	0	0
$909$	477.215	0.524989
$910$	0	0
$911$	1755.94i	1.92748i	0.266835	+	0.963742i	$0.414022\pi$
$911$	1755.94i	1.92748i	−0.266835	+	0.963742i	$0.585978\pi$
$912$	0	0
$913$	939.405	1.02892
$914$	0	0
$915$	− 119.503i	− 0.130604i
$916$	0	0
$917$	−167.118	−0.182244
$918$	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
$923$	1029.13i	1.11498i
$924$	0	0
$925$	−6.92342	−0.00748478
$926$	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
$931$	− 1250.29i	− 1.34296i
$932$	0	0
$933$	−303.118	−0.324885
$934$	0	0
$935$	1460.82i	1.56237i
$936$	0	0
$937$	−170.554	−0.182021	−0.0910105	−	0.995850i	$-0.529010\pi$
$937$	−170.554	−0.182021	−0.0910105	+	0.995850i	$0.529010\pi$
$938$	0	0
$939$	− 751.674i	− 0.800505i
$940$	0	0
$941$	−281.523	−0.299174	−0.149587	−	0.988749i	$-0.547794\pi$
$941$	−281.523	−0.299174	−0.149587	+	0.988749i	$0.547794\pi$
$942$	0	0
$943$	− 65.1487i	− 0.0690867i
$944$	0	0
$945$	74.9845	0.0793487
$946$	0	0
$947$	1533.43i	1.61925i	0.586947	+	0.809625i	$0.300330\pi$
$947$	1533.43i	1.61925i	−0.586947	+	0.809625i	$0.699670\pi$
$948$	0	0
$949$	−606.564	−0.639161
$950$	0	0
$951$	− 153.264i	− 0.161161i
$952$	0	0
$953$	414.133	0.434557	0.217279	−	0.976110i	$-0.430282\pi$
$953$	414.133	0.434557	0.217279	+	0.976110i	$0.430282\pi$
$954$	0	0
$955$	− 430.851i	− 0.451153i
$956$	0	0
$957$	−437.703	−0.457369
$958$	0	0
$959$	451.334i	0.470629i
$960$	0	0
$961$	−532.128	−0.553723
$962$	0	0
$963$	126.200i	0.131049i
$964$	0	0
$965$	−130.964	−0.135714
$966$	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
$971$	968.877i	0.997813i	0.866656	+	0.498907i	$0.166265\pi$
$971$	968.877i	0.997813i	−0.866656	+	0.498907i	$0.833735\pi$
$972$	0	0
$973$	491.713	0.505357
$974$	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
$979$	− 750.697i	− 0.766800i
$980$	0	0
$981$	−59.5692	−0.0607230
$982$	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
$987$	354.297i	0.358964i
$988$	0	0
$989$	42.8719	0.0433487
$990$	0	0
$991$	− 1032.44i	− 1.04181i	−0.853614	−	0.520906i	$-0.825594\pi$
$991$	− 1032.44i	− 1.04181i	0.853614	−	0.520906i	$-0.174406\pi$
$992$	0	0
$993$	−185.436	−0.186743
$994$	0	0
$995$	− 335.497i	− 0.337183i
$996$	0	0
$997$	−456.277	−0.457650	−0.228825	−	0.973468i	$-0.573488\pi$
$997$	−456.277	−0.457650	−0.228825	+	0.973468i	$0.573488\pi$
$998$	0	0
$999$	− 50.4693i	− 0.0505198i

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.3.g.a.31.2	✓	4	8.5	even	2
96.3.g.a.31.4	yes	4	8.3	odd	2
192.3.g.c.127.1		4	4.3	odd	2	inner
192.3.g.c.127.3		4	1.1	even	1	trivial
288.3.g.d.127.1		4	24.5	odd	2
288.3.g.d.127.2		4	24.11	even	2
576.3.g.j.127.3		4	3.2	odd	2
576.3.g.j.127.4		4	12.11	even	2
768.3.b.a.127.3		4	16.5	even	4
768.3.b.a.127.4		4	16.3	odd	4
768.3.b.d.127.1		4	16.11	odd	4
768.3.b.d.127.2		4	16.13	even	4
2304.3.b.k.127.2		4	48.29	odd	4
2304.3.b.k.127.3		4	48.11	even	4
2304.3.b.o.127.2		4	48.35	even	4
2304.3.b.o.127.3		4	48.5	odd	4
2400.3.e.a.1951.2		4	40.19	odd	2
2400.3.e.a.1951.3		4	40.29	even	2
2400.3.j.a.799.1		4	40.37	odd	4
2400.3.j.a.799.2		4	40.3	even	4
2400.3.j.b.799.3		4	40.13	odd	4
2400.3.j.b.799.4		4	40.27	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:	\( N \)	\(=\)	\( 192 = 2^{6} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	192.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} -4.92820 q^{5} -2.92820i q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q+1.73205i q^{3} -4.92820 q^{5} -2.92820i q^{7} -3.00000 q^{9} +14.9282i q^{11} -23.8564 q^{13} -8.53590i q^{15} -19.8564 q^{17} -30.9282i q^{19} +5.07180 q^{21} +8.00000i q^{23} -0.712813 q^{25} -5.19615i q^{27} +16.9282 q^{29} +38.6410i q^{31} -25.8564 q^{33} +14.4308i q^{35} +9.71281 q^{37} -41.3205i q^{39} -8.14359 q^{41} -5.35898i q^{43} +14.7846 q^{45} +69.8564i q^{47} +40.4256 q^{49} -34.3923i q^{51} +0.928203 q^{53} -73.5692i q^{55} +53.5692 q^{57} +108.210i q^{59} +14.0000 q^{61} +8.78461i q^{63} +117.569 q^{65} -16.4974i q^{67} -13.8564 q^{69} -43.1384i q^{71} +25.4256 q^{73} -1.23463i q^{75} +43.7128 q^{77} -96.7846i q^{79} +9.00000 q^{81} -62.9282i q^{83} +97.8564 q^{85} +29.3205i q^{87} -50.2872 q^{89} +69.8564i q^{91} -66.9282 q^{93} +152.420i q^{95} -145.138 q^{97} -44.7846i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{5} - 12 q^{9}+O(q^{10}) \) 4 * q + 8 * q^5 - 12 * q^9	\( 4 q + 8 q^{5} - 12 q^{9} - 40 q^{13} - 24 q^{17} + 48 q^{21} + 108 q^{25} + 40 q^{29} - 48 q^{33} - 72 q^{37} - 88 q^{41} - 24 q^{45} - 60 q^{49} - 24 q^{53} + 48 q^{57} + 56 q^{61} + 304 q^{65} - 120 q^{73} + 64 q^{77} + 36 q^{81} + 336 q^{85} - 312 q^{89} - 240 q^{93} - 248 q^{97}+O(q^{100}) \) 4 * q + 8 * q^5 - 12 * q^9 - 40 * q^13 - 24 * q^17 + 48 * q^21 + 108 * q^25 + 40 * q^29 - 48 * q^33 - 72 * q^37 - 88 * q^41 - 24 * q^45 - 60 * q^49 - 24 * q^53 + 48 * q^57 + 56 * q^61 + 304 * q^65 - 120 * q^73 + 64 * q^77 + 36 * q^81 + 336 * q^85 - 312 * q^89 - 240 * q^93 - 248 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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