LMFDB - Embedded newform 960.3.e.c.511.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [960,3,Mod(511,960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(960, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 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("960.511");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 960 = 2^{6} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	960.e (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:	$26.1581053786$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.85100625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 2x^{6} + x^{5} + 3x^{4} + 2x^{3} - 8x^{2} - 8x + 16 $ x^8 - x^7 - 2x^6 + x^5 + 3x^4 + 2x^3 - 8x^2 - 8*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{16} $
Twist minimal:	no (minimal twist has level 60)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			511.8
Root			$1.04064 + 0.957636i$ of defining polynomial
Character	$\chi$	$=$	960.511
Dual form			960.3.e.c.511.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{3} +2.23607 q^{5} +5.46770i q^{7} -3.00000 q^{9} +O(q^{10})$	$q+1.73205i q^{3} +2.23607 q^{5} +5.46770i q^{7} -3.00000 q^{9} -11.0403i q^{11} -10.1242 q^{13} +3.87298i q^{15} -24.4146 q^{17} -23.7757i q^{19} -9.47033 q^{21} -37.2526i q^{23} +5.00000 q^{25} -5.19615i q^{27} +25.7726 q^{29} -4.83647i q^{31} +19.1224 q^{33} +12.2261i q^{35} -35.6493 q^{37} -17.5356i q^{39} -9.30410 q^{41} +70.0287i q^{43} -6.70820 q^{45} -38.0223i q^{47} +19.1043 q^{49} -42.2873i q^{51} -55.7762 q^{53} -24.6869i q^{55} +41.1808 q^{57} -55.5411i q^{59} +82.2412 q^{61} -16.4031i q^{63} -22.6384 q^{65} -104.493i q^{67} +64.5233 q^{69} +76.7471i q^{71} -93.5215 q^{73} +8.66025i q^{75} +60.3651 q^{77} -49.3762i q^{79} +9.00000 q^{81} -72.3768i q^{83} -54.5927 q^{85} +44.6395i q^{87} +115.691 q^{89} -55.3560i q^{91} +8.37701 q^{93} -53.1641i q^{95} -72.9589 q^{97} +33.1209i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 24 q^{9}+O(q^{10}) $ 8 * q - 24 * q^9	$ 8 q - 24 q^{9} - 16 q^{13} + 48 q^{21} + 40 q^{25} - 64 q^{29} + 112 q^{37} - 16 q^{41} - 56 q^{49} - 352 q^{53} + 144 q^{57} + 176 q^{61} - 80 q^{65} + 96 q^{69} - 240 q^{73} + 288 q^{77} + 72 q^{81} - 160 q^{85} + 80 q^{89} - 144 q^{93} + 432 q^{97}+O(q^{100}) $ 8 * q - 24 * q^9 - 16 * q^13 + 48 * q^21 + 40 * q^25 - 64 * q^29 + 112 * q^37 - 16 * q^41 - 56 * q^49 - 352 * q^53 + 144 * q^57 + 176 * q^61 - 80 * q^65 + 96 * q^69 - 240 * q^73 + 288 * q^77 + 72 * q^81 - 160 * q^85 + 80 * q^89 - 144 * q^93 + 432 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/960\mathbb{Z}\right)^\times$.

$n$	$511$	$577$	$641$	$901$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$5$	2.23607	0.447214
$5$	2.23607	0.447214
$6$	0	0
$7$	5.46770i	0.781100i	0.920582	+	0.390550i	$0.127715\pi$
$7$	5.46770i	0.781100i	−0.920582	+	0.390550i	$0.872285\pi$
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	− 11.0403i	− 1.00366i	−0.864965	−	0.501832i	$-0.832659\pi$
$11$	− 11.0403i	− 1.00366i	0.864965	−	0.501832i	$-0.167341\pi$
$12$	0	0
$13$	−10.1242	−0.778784	−0.389392	−	0.921072i	$-0.627315\pi$
$13$	−10.1242	−0.778784	−0.389392	+	0.921072i	$0.627315\pi$
$14$	0	0
$15$	3.87298i	0.258199i
$16$	0	0
$17$	−24.4146	−1.43615	−0.718077	−	0.695964i	$-0.754977\pi$
$17$	−24.4146	−1.43615	−0.718077	+	0.695964i	$0.754977\pi$
$18$	0	0
$19$	− 23.7757i	− 1.25135i	−0.780082	−	0.625677i	$-0.784823\pi$
$19$	− 23.7757i	− 1.25135i	0.780082	−	0.625677i	$-0.215177\pi$
$20$	0	0
$21$	−9.47033	−0.450968
$22$	0	0
$23$	− 37.2526i	− 1.61968i	−0.586653	−	0.809838i	$-0.699555\pi$
$23$	− 37.2526i	− 1.61968i	0.586653	−	0.809838i	$-0.300445\pi$
$24$	0	0
$25$	5.00000	0.200000
$26$	0	0
$27$	− 5.19615i	− 0.192450i
$28$	0	0
$29$	25.7726	0.888712	0.444356	−	0.895850i	$-0.353433\pi$
$29$	25.7726	0.888712	0.444356	+	0.895850i	$0.353433\pi$
$30$	0	0
$31$	− 4.83647i	− 0.156015i	−0.996953	−	0.0780076i	$-0.975144\pi$
$31$	− 4.83647i	− 0.156015i	0.996953	−	0.0780076i	$-0.0248558\pi$
$32$	0	0
$33$	19.1224	0.579466
$34$	0	0
$35$	12.2261i	0.349319i
$36$	0	0
$37$	−35.6493	−0.963495	−0.481747	−	0.876310i	$-0.659998\pi$
$37$	−35.6493	−0.963495	−0.481747	+	0.876310i	$0.659998\pi$
$38$	0	0
$39$	− 17.5356i	− 0.449631i
$40$	0	0
$41$	−9.30410	−0.226929	−0.113465	−	0.993542i	$-0.536195\pi$
$41$	−9.30410	−0.226929	−0.113465	+	0.993542i	$0.536195\pi$
$42$	0	0
$43$	70.0287i	1.62857i	0.580462	+	0.814287i	$0.302872\pi$
$43$	70.0287i	1.62857i	−0.580462	+	0.814287i	$0.697128\pi$
$44$	0	0
$45$	−6.70820	−0.149071
$46$	0	0
$47$	− 38.0223i	− 0.808984i	−0.914542	−	0.404492i	$-0.867448\pi$
$47$	− 38.0223i	− 0.808984i	0.914542	−	0.404492i	$-0.132552\pi$
$48$	0	0
$49$	19.1043	0.389883
$50$	0	0
$51$	− 42.2873i	− 0.829164i
$52$	0	0
$53$	−55.7762	−1.05238	−0.526191	−	0.850366i	$-0.676380\pi$
$53$	−55.7762	−1.05238	−0.526191	+	0.850366i	$0.676380\pi$
$54$	0	0
$55$	− 24.6869i	− 0.448853i
$56$	0	0
$57$	41.1808	0.722470
$58$	0	0
$59$	− 55.5411i	− 0.941374i	−0.882300	−	0.470687i	$-0.844006\pi$
$59$	− 55.5411i	− 0.941374i	0.882300	−	0.470687i	$-0.155994\pi$
$60$	0	0
$61$	82.2412	1.34822	0.674108	−	0.738633i	$-0.264528\pi$
$61$	82.2412	1.34822	0.674108	+	0.738633i	$0.264528\pi$
$62$	0	0
$63$	− 16.4031i	− 0.260367i
$64$	0	0
$65$	−22.6384	−0.348283
$66$	0	0
$67$	− 104.493i	− 1.55960i	−0.626026	−	0.779802i	$-0.715320\pi$
$67$	− 104.493i	− 1.55960i	0.626026	−	0.779802i	$-0.284680\pi$
$68$	0	0
$69$	64.5233	0.935120
$70$	0	0
$71$	76.7471i	1.08094i	0.841362	+	0.540472i	$0.181754\pi$
$71$	76.7471i	1.08094i	−0.841362	+	0.540472i	$0.818246\pi$
$72$	0	0
$73$	−93.5215	−1.28112	−0.640558	−	0.767910i	$-0.721297\pi$
$73$	−93.5215	−1.28112	−0.640558	+	0.767910i	$0.721297\pi$
$74$	0	0
$75$	8.66025i	0.115470i
$76$	0	0
$77$	60.3651	0.783963
$78$	0	0
$79$	− 49.3762i	− 0.625016i	−0.949915	−	0.312508i	$-0.898831\pi$
$79$	− 49.3762i	− 0.625016i	0.949915	−	0.312508i	$-0.101169\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	− 72.3768i	− 0.872010i	−0.899944	−	0.436005i	$-0.856393\pi$
$83$	− 72.3768i	− 0.872010i	0.899944	−	0.436005i	$-0.143607\pi$
$84$	0	0
$85$	−54.5927	−0.642267
$86$	0	0
$87$	44.6395i	0.513098i
$88$	0	0
$89$	115.691	1.29990	0.649950	−	0.759977i	$-0.274790\pi$
$89$	115.691	1.29990	0.649950	+	0.759977i	$0.274790\pi$
$90$	0	0
$91$	− 55.3560i	− 0.608308i
$92$	0	0
$93$	8.37701	0.0900754
$94$	0	0
$95$	− 53.1641i	− 0.559623i
$96$	0	0
$97$	−72.9589	−0.752154	−0.376077	−	0.926588i	$-0.622727\pi$
$97$	−72.9589	−0.752154	−0.376077	+	0.926588i	$0.622727\pi$
$98$	0	0
$99$	33.1209i	0.334555i
$100$	0	0
$101$	−29.4092	−0.291180	−0.145590	−	0.989345i	$-0.546508\pi$
$101$	−29.4092	−0.291180	−0.145590	+	0.989345i	$0.546508\pi$
$102$	0	0
$103$	− 28.1884i	− 0.273673i	−0.990594	−	0.136837i	$-0.956306\pi$
$103$	− 28.1884i	− 0.273673i	0.990594	−	0.136837i	$-0.0436935\pi$
$104$	0	0
$105$	−21.1763	−0.201679
$106$	0	0
$107$	4.50700i	0.0421215i	0.999778	+	0.0210607i	$0.00670434\pi$
$107$	4.50700i	0.0421215i	−0.999778	+	0.0210607i	$0.993296\pi$
$108$	0	0
$109$	−193.315	−1.77353	−0.886767	−	0.462217i	$-0.847054\pi$
$109$	−193.315	−1.77353	−0.886767	+	0.462217i	$0.847054\pi$
$110$	0	0
$111$	− 61.7464i	− 0.556274i
$112$	0	0
$113$	75.5727	0.668785	0.334392	−	0.942434i	$-0.391469\pi$
$113$	75.5727	0.668785	0.334392	+	0.942434i	$0.391469\pi$
$114$	0	0
$115$	− 83.2992i	− 0.724341i
$116$	0	0
$117$	30.3726	0.259595
$118$	0	0
$119$	− 133.492i	− 1.12178i
$120$	0	0
$121$	−0.888544	−0.00734334
$122$	0	0
$123$	− 16.1152i	− 0.131018i
$124$	0	0
$125$	11.1803	0.0894427
$126$	0	0
$127$	− 131.306i	− 1.03390i	−0.856015	−	0.516951i	$-0.827067\pi$
$127$	− 131.306i	− 1.03390i	0.856015	−	0.516951i	$-0.172933\pi$
$128$	0	0
$129$	−121.293	−0.940258
$130$	0	0
$131$	75.7533i	0.578270i	0.957288	+	0.289135i	$0.0933676\pi$
$131$	75.7533i	0.578270i	−0.957288	+	0.289135i	$0.906632\pi$
$132$	0	0
$133$	129.999	0.977433
$134$	0	0
$135$	− 11.6190i	− 0.0860663i
$136$	0	0
$137$	66.7927	0.487538	0.243769	−	0.969833i	$-0.421616\pi$
$137$	66.7927	0.487538	0.243769	+	0.969833i	$0.421616\pi$
$138$	0	0
$139$	− 38.1214i	− 0.274255i	−0.990553	−	0.137127i	$-0.956213\pi$
$139$	− 38.1214i	− 0.274255i	0.990553	−	0.137127i	$-0.0437869\pi$
$140$	0	0
$141$	65.8565	0.467067
$142$	0	0
$143$	111.774i	0.781638i
$144$	0	0
$145$	57.6294	0.397444
$146$	0	0
$147$	33.0895i	0.225099i
$148$	0	0
$149$	126.717	0.850449	0.425225	−	0.905088i	$-0.360195\pi$
$149$	126.717	0.850449	0.425225	+	0.905088i	$0.360195\pi$
$150$	0	0
$151$	− 68.4403i	− 0.453247i	−0.973982	−	0.226623i	$-0.927231\pi$
$151$	− 68.4403i	− 0.453247i	0.973982	−	0.226623i	$-0.0727687\pi$
$152$	0	0
$153$	73.2438	0.478718
$154$	0	0
$155$	− 10.8147i	− 0.0697721i
$156$	0	0
$157$	−25.5777	−0.162915	−0.0814577	−	0.996677i	$-0.525958\pi$
$157$	−25.5777	−0.162915	−0.0814577	+	0.996677i	$0.525958\pi$
$158$	0	0
$159$	− 96.6073i	− 0.607593i
$160$	0	0
$161$	203.686	1.26513
$162$	0	0
$163$	− 63.4771i	− 0.389430i	−0.980860	−	0.194715i	$-0.937622\pi$
$163$	− 63.4771i	− 0.389430i	0.980860	−	0.194715i	$-0.0623782\pi$
$164$	0	0
$165$	42.7590	0.259145
$166$	0	0
$167$	12.3771i	0.0741144i	0.999313	+	0.0370572i	$0.0117984\pi$
$167$	12.3771i	0.0741144i	−0.999313	+	0.0370572i	$0.988202\pi$
$168$	0	0
$169$	−66.5008	−0.393496
$170$	0	0
$171$	71.3272i	0.417118i
$172$	0	0
$173$	−59.3729	−0.343196	−0.171598	−	0.985167i	$-0.554893\pi$
$173$	−59.3729	−0.343196	−0.171598	+	0.985167i	$0.554893\pi$
$174$	0	0
$175$	27.3385i	0.156220i
$176$	0	0
$177$	96.2000	0.543503
$178$	0	0
$179$	− 252.782i	− 1.41219i	−0.708118	−	0.706094i	$-0.750455\pi$
$179$	− 252.782i	− 1.41219i	0.708118	−	0.706094i	$-0.249545\pi$
$180$	0	0
$181$	−125.373	−0.692670	−0.346335	−	0.938111i	$-0.612574\pi$
$181$	−125.373	−0.692670	−0.346335	+	0.938111i	$0.612574\pi$
$182$	0	0
$183$	142.446i	0.778393i
$184$	0	0
$185$	−79.7143	−0.430888
$186$	0	0
$187$	269.545i	1.44142i
$188$	0	0
$189$	28.4110	0.150323
$190$	0	0
$191$	− 97.4640i	− 0.510283i	−0.966904	−	0.255141i	$-0.917878\pi$
$191$	− 97.4640i	− 0.510283i	0.966904	−	0.255141i	$-0.0821220\pi$
$192$	0	0
$193$	−342.376	−1.77397	−0.886985	−	0.461798i	$-0.847204\pi$
$193$	−342.376	−1.77397	−0.886985	+	0.461798i	$0.847204\pi$
$194$	0	0
$195$	− 39.2108i	− 0.201081i
$196$	0	0
$197$	−74.4829	−0.378086	−0.189043	−	0.981969i	$-0.560539\pi$
$197$	−74.4829	−0.378086	−0.189043	+	0.981969i	$0.560539\pi$
$198$	0	0
$199$	178.027i	0.894606i	0.894382	+	0.447303i	$0.147615\pi$
$199$	178.027i	0.894606i	−0.894382	+	0.447303i	$0.852385\pi$
$200$	0	0
$201$	180.988	0.900438
$202$	0	0
$203$	140.917i	0.694173i
$204$	0	0
$205$	−20.8046	−0.101486
$206$	0	0
$207$	111.758i	0.539892i
$208$	0	0
$209$	−262.491	−1.25594
$210$	0	0
$211$	185.893i	0.881008i	0.897751	+	0.440504i	$0.145200\pi$
$211$	185.893i	0.881008i	−0.897751	+	0.440504i	$0.854800\pi$
$212$	0	0
$213$	−132.930	−0.624084
$214$	0	0
$215$	156.589i	0.728321i
$216$	0	0
$217$	26.4444	0.121863
$218$	0	0
$219$	− 161.984i	− 0.739653i
$220$	0	0
$221$	247.178	1.11845
$222$	0	0
$223$	202.724i	0.909074i	0.890728	+	0.454537i	$0.150195\pi$
$223$	202.724i	0.909074i	−0.890728	+	0.454537i	$0.849805\pi$
$224$	0	0
$225$	−15.0000	−0.0666667
$226$	0	0
$227$	− 51.2708i	− 0.225863i	−0.993603	−	0.112931i	$-0.963976\pi$
$227$	− 51.2708i	− 0.225863i	0.993603	−	0.112931i	$-0.0360240\pi$
$228$	0	0
$229$	−337.056	−1.47186	−0.735930	−	0.677058i	$-0.763255\pi$
$229$	−337.056	−1.47186	−0.735930	+	0.677058i	$0.763255\pi$
$230$	0	0
$231$	104.555i	0.452621i
$232$	0	0
$233$	−80.2851	−0.344571	−0.172286	−	0.985047i	$-0.555115\pi$
$233$	−80.2851	−0.344571	−0.172286	+	0.985047i	$0.555115\pi$
$234$	0	0
$235$	− 85.0203i	− 0.361789i
$236$	0	0
$237$	85.5221	0.360853
$238$	0	0
$239$	− 330.808i	− 1.38413i	−0.721834	−	0.692066i	$-0.756701\pi$
$239$	− 330.808i	− 1.38413i	0.721834	−	0.692066i	$-0.243299\pi$
$240$	0	0
$241$	−359.914	−1.49342	−0.746710	−	0.665150i	$-0.768368\pi$
$241$	−359.914	−1.49342	−0.746710	+	0.665150i	$0.768368\pi$
$242$	0	0
$243$	15.5885i	0.0641500i
$244$	0	0
$245$	42.7184	0.174361
$246$	0	0
$247$	240.710i	0.974534i
$248$	0	0
$249$	125.360	0.503455
$250$	0	0
$251$	312.213i	1.24388i	0.783067	+	0.621938i	$0.213654\pi$
$251$	312.213i	1.24388i	−0.783067	+	0.621938i	$0.786346\pi$
$252$	0	0
$253$	−411.280	−1.62561
$254$	0	0
$255$	− 94.5574i	− 0.370813i
$256$	0	0
$257$	80.2592	0.312293	0.156146	−	0.987734i	$-0.450093\pi$
$257$	80.2592	0.312293	0.156146	+	0.987734i	$0.450093\pi$
$258$	0	0
$259$	− 194.920i	− 0.752586i
$260$	0	0
$261$	−77.3179	−0.296237
$262$	0	0
$263$	487.967i	1.85539i	0.373342	+	0.927694i	$0.378212\pi$
$263$	487.967i	1.85539i	−0.373342	+	0.927694i	$0.621788\pi$
$264$	0	0
$265$	−124.719	−0.470639
$266$	0	0
$267$	200.383i	0.750498i
$268$	0	0
$269$	309.553	1.15076	0.575378	−	0.817888i	$-0.304855\pi$
$269$	309.553	1.15076	0.575378	+	0.817888i	$0.304855\pi$
$270$	0	0
$271$	− 48.9693i	− 0.180698i	−0.995910	−	0.0903492i	$-0.971202\pi$
$271$	− 48.9693i	− 0.180698i	0.995910	−	0.0903492i	$-0.0287983\pi$
$272$	0	0
$273$	95.8794	0.351207
$274$	0	0
$275$	− 55.2016i	− 0.200733i
$276$	0	0
$277$	199.644	0.720736	0.360368	−	0.932810i	$-0.382651\pi$
$277$	199.644	0.720736	0.360368	+	0.932810i	$0.382651\pi$
$278$	0	0
$279$	14.5094i	0.0520051i
$280$	0	0
$281$	61.1598	0.217650	0.108825	−	0.994061i	$-0.465291\pi$
$281$	61.1598	0.217650	0.108825	+	0.994061i	$0.465291\pi$
$282$	0	0
$283$	432.506i	1.52829i	0.645044	+	0.764145i	$0.276839\pi$
$283$	432.506i	1.52829i	−0.645044	+	0.764145i	$0.723161\pi$
$284$	0	0
$285$	92.0830	0.323098
$286$	0	0
$287$	− 50.8720i	− 0.177254i
$288$	0	0
$289$	307.073	1.06254
$290$	0	0
$291$	− 126.369i	− 0.434256i
$292$	0	0
$293$	283.234	0.966668	0.483334	−	0.875436i	$-0.339426\pi$
$293$	283.234	0.966668	0.483334	+	0.875436i	$0.339426\pi$
$294$	0	0
$295$	− 124.194i	− 0.420995i
$296$	0	0
$297$	−57.3672	−0.193155
$298$	0	0
$299$	377.152i	1.26138i
$300$	0	0
$301$	−382.896	−1.27208
$302$	0	0
$303$	− 50.9382i	− 0.168113i
$304$	0	0
$305$	183.897	0.602940
$306$	0	0
$307$	− 100.077i	− 0.325983i	−0.986627	−	0.162992i	$-0.947886\pi$
$307$	− 100.077i	− 0.325983i	0.986627	−	0.162992i	$-0.0521144\pi$
$308$	0	0
$309$	48.8237	0.158005
$310$	0	0
$311$	404.185i	1.29963i	0.760092	+	0.649815i	$0.225154\pi$
$311$	404.185i	1.29963i	−0.760092	+	0.649815i	$0.774846\pi$
$312$	0	0
$313$	−128.579	−0.410795	−0.205398	−	0.978679i	$-0.565849\pi$
$313$	−128.579	−0.410795	−0.205398	+	0.978679i	$0.565849\pi$
$314$	0	0
$315$	− 36.6784i	− 0.116440i
$316$	0	0
$317$	−85.9315	−0.271077	−0.135539	−	0.990772i	$-0.543276\pi$
$317$	−85.9315	−0.271077	−0.135539	+	0.990772i	$0.543276\pi$
$318$	0	0
$319$	− 284.538i	− 0.891969i
$320$	0	0
$321$	−7.80635	−0.0243189
$322$	0	0
$323$	580.475i	1.79714i
$324$	0	0
$325$	−50.6209	−0.155757
$326$	0	0
$327$	− 334.832i	− 1.02395i
$328$	0	0
$329$	207.894	0.631898
$330$	0	0
$331$	183.391i	0.554052i	0.960862	+	0.277026i	$0.0893488\pi$
$331$	183.391i	0.554052i	−0.960862	+	0.277026i	$0.910651\pi$
$332$	0	0
$333$	106.948	0.321165
$334$	0	0
$335$	− 233.654i	− 0.697476i
$336$	0	0
$337$	168.130	0.498901	0.249451	−	0.968388i	$-0.419750\pi$
$337$	168.130	0.498901	0.249451	+	0.968388i	$0.419750\pi$
$338$	0	0
$339$	130.896i	0.386123i
$340$	0	0
$341$	−53.3962	−0.156587
$342$	0	0
$343$	372.374i	1.08564i
$344$	0	0
$345$	144.279	0.418199
$346$	0	0
$347$	− 137.414i	− 0.396006i	−0.980201	−	0.198003i	$-0.936554\pi$
$347$	− 137.414i	− 0.396006i	0.980201	−	0.198003i	$-0.0634455\pi$
$348$	0	0
$349$	13.4893	0.0386513	0.0193256	−	0.999813i	$-0.493848\pi$
$349$	13.4893	0.0386513	0.0193256	+	0.999813i	$0.493848\pi$
$350$	0	0
$351$	52.6068i	0.149877i
$352$	0	0
$353$	243.547	0.689935	0.344968	−	0.938615i	$-0.387890\pi$
$353$	243.547	0.689935	0.344968	+	0.938615i	$0.387890\pi$
$354$	0	0
$355$	171.612i	0.483413i
$356$	0	0
$357$	231.215	0.647660
$358$	0	0
$359$	− 17.9166i	− 0.0499068i	−0.999689	−	0.0249534i	$-0.992056\pi$
$359$	− 17.9166i	− 0.0499068i	0.999689	−	0.0249534i	$-0.00794374\pi$
$360$	0	0
$361$	−204.285	−0.565887
$362$	0	0
$363$	− 1.53900i	− 0.00423968i
$364$	0	0
$365$	−209.120	−0.572933
$366$	0	0
$367$	238.417i	0.649637i	0.945776	+	0.324818i	$0.105303\pi$
$367$	238.417i	0.649637i	−0.945776	+	0.324818i	$0.894697\pi$
$368$	0	0
$369$	27.9123	0.0756431
$370$	0	0
$371$	− 304.968i	− 0.822016i
$372$	0	0
$373$	181.271	0.485981	0.242990	−	0.970029i	$-0.421872\pi$
$373$	181.271	0.485981	0.242990	+	0.970029i	$0.421872\pi$
$374$	0	0
$375$	19.3649i	0.0516398i
$376$	0	0
$377$	−260.927	−0.692114
$378$	0	0
$379$	− 306.206i	− 0.807931i	−0.914774	−	0.403965i	$-0.867632\pi$
$379$	− 306.206i	− 0.807931i	0.914774	−	0.403965i	$-0.132368\pi$
$380$	0	0
$381$	227.428	0.596924
$382$	0	0
$383$	− 144.027i	− 0.376050i	−0.982164	−	0.188025i	$-0.939791\pi$
$383$	− 144.027i	− 0.376050i	0.982164	−	0.188025i	$-0.0602086\pi$
$384$	0	0
$385$	134.981	0.350599
$386$	0	0
$387$	− 210.086i	− 0.542858i
$388$	0	0
$389$	−14.0099	−0.0360152	−0.0180076	−	0.999838i	$-0.505732\pi$
$389$	−14.0099	−0.0360152	−0.0180076	+	0.999838i	$0.505732\pi$
$390$	0	0
$391$	909.506i	2.32610i
$392$	0	0
$393$	−131.209	−0.333864
$394$	0	0
$395$	− 110.409i	− 0.279515i
$396$	0	0
$397$	−39.1084	−0.0985098	−0.0492549	−	0.998786i	$-0.515685\pi$
$397$	−39.1084	−0.0985098	−0.0492549	+	0.998786i	$0.515685\pi$
$398$	0	0
$399$	225.164i	0.564321i
$400$	0	0
$401$	−121.067	−0.301913	−0.150957	−	0.988540i	$-0.548235\pi$
$401$	−121.067	−0.301913	−0.150957	+	0.988540i	$0.548235\pi$
$402$	0	0
$403$	48.9653i	0.121502i
$404$	0	0
$405$	20.1246	0.0496904
$406$	0	0
$407$	393.579i	0.967026i
$408$	0	0
$409$	−541.795	−1.32468	−0.662342	−	0.749202i	$-0.730437\pi$
$409$	−541.795	−1.32468	−0.662342	+	0.749202i	$0.730437\pi$
$410$	0	0
$411$	115.688i	0.281480i
$412$	0	0
$413$	303.682	0.735307
$414$	0	0
$415$	− 161.839i	− 0.389975i
$416$	0	0
$417$	66.0282	0.158341
$418$	0	0
$419$	− 687.825i	− 1.64159i	−0.571224	−	0.820794i	$-0.693531\pi$
$419$	− 687.825i	− 1.64159i	0.571224	−	0.820794i	$-0.306469\pi$
$420$	0	0
$421$	454.396	1.07932	0.539662	−	0.841882i	$-0.318552\pi$
$421$	454.396	1.07932	0.539662	+	0.841882i	$0.318552\pi$
$422$	0	0
$423$	114.067i	0.269661i
$424$	0	0
$425$	−122.073	−0.287231
$426$	0	0
$427$	449.670i	1.05309i
$428$	0	0
$429$	−193.599	−0.451279
$430$	0	0
$431$	466.145i	1.08154i	0.841169	+	0.540772i	$0.181868\pi$
$431$	466.145i	1.08154i	−0.841169	+	0.540772i	$0.818132\pi$
$432$	0	0
$433$	457.094	1.05565	0.527823	−	0.849355i	$-0.323009\pi$
$433$	457.094	1.05565	0.527823	+	0.849355i	$0.323009\pi$
$434$	0	0
$435$	99.8170i	0.229464i
$436$	0	0
$437$	−885.706	−2.02679
$438$	0	0
$439$	− 777.467i	− 1.77100i	−0.464644	−	0.885498i	$-0.653818\pi$
$439$	− 777.467i	− 1.77100i	0.464644	−	0.885498i	$-0.346182\pi$
$440$	0	0
$441$	−57.3128	−0.129961
$442$	0	0
$443$	247.484i	0.558654i	0.960196	+	0.279327i	$0.0901114\pi$
$443$	247.484i	0.558654i	−0.960196	+	0.279327i	$0.909889\pi$
$444$	0	0
$445$	258.693	0.581333
$446$	0	0
$447$	219.480i	0.491007i
$448$	0	0
$449$	412.508	0.918726	0.459363	−	0.888249i	$-0.348078\pi$
$449$	412.508	0.918726	0.459363	+	0.888249i	$0.348078\pi$
$450$	0	0
$451$	102.720i	0.227761i
$452$	0	0
$453$	118.542	0.261682
$454$	0	0
$455$	− 123.780i	− 0.272044i
$456$	0	0
$457$	−745.400	−1.63107	−0.815537	−	0.578706i	$-0.803558\pi$
$457$	−745.400	−1.63107	−0.815537	+	0.578706i	$0.803558\pi$
$458$	0	0
$459$	126.862i	0.276388i
$460$	0	0
$461$	−81.6151	−0.177039	−0.0885196	−	0.996074i	$-0.528214\pi$
$461$	−81.6151	−0.177039	−0.0885196	+	0.996074i	$0.528214\pi$
$462$	0	0
$463$	− 292.248i	− 0.631205i	−0.948891	−	0.315603i	$-0.897793\pi$
$463$	− 292.248i	− 0.631205i	0.948891	−	0.315603i	$-0.102207\pi$
$464$	0	0
$465$	18.7316	0.0402829
$466$	0	0
$467$	− 51.4163i	− 0.110099i	−0.998484	−	0.0550495i	$-0.982468\pi$
$467$	− 51.4163i	− 0.110099i	0.998484	−	0.0550495i	$-0.0175317\pi$
$468$	0	0
$469$	571.339	1.21821
$470$	0	0
$471$	− 44.3019i	− 0.0940592i
$472$	0	0
$473$	773.139	1.63454
$474$	0	0
$475$	− 118.879i	− 0.250271i
$476$	0	0
$477$	167.329	0.350794
$478$	0	0
$479$	− 122.593i	− 0.255935i	−0.991778	−	0.127967i	$-0.959155\pi$
$479$	− 122.593i	− 0.255935i	0.991778	−	0.127967i	$-0.0408453\pi$
$480$	0	0
$481$	360.920	0.750354
$482$	0	0
$483$	352.794i	0.730423i
$484$	0	0
$485$	−163.141	−0.336373
$486$	0	0
$487$	− 65.9859i	− 0.135495i	−0.997703	−	0.0677474i	$-0.978419\pi$
$487$	− 65.9859i	− 0.135495i	0.997703	−	0.0677474i	$-0.0215812\pi$
$488$	0	0
$489$	109.946	0.224837
$490$	0	0
$491$	361.163i	0.735567i	0.929911	+	0.367783i	$0.119883\pi$
$491$	361.163i	0.735567i	−0.929911	+	0.367783i	$0.880117\pi$
$492$	0	0
$493$	−629.229	−1.27633
$494$	0	0
$495$	74.0607i	0.149618i
$496$	0	0
$497$	−419.630	−0.844326
$498$	0	0
$499$	− 711.138i	− 1.42513i	−0.701608	−	0.712564i	$-0.747534\pi$
$499$	− 711.138i	− 1.42513i	0.701608	−	0.712564i	$-0.252466\pi$
$500$	0	0
$501$	−21.4378	−0.0427900
$502$	0	0
$503$	− 353.756i	− 0.703292i	−0.936133	−	0.351646i	$-0.885622\pi$
$503$	− 353.756i	− 0.703292i	0.936133	−	0.351646i	$-0.114378\pi$
$504$	0	0
$505$	−65.7610	−0.130220
$506$	0	0
$507$	− 115.183i	− 0.227185i
$508$	0	0
$509$	−478.049	−0.939192	−0.469596	−	0.882881i	$-0.655600\pi$
$509$	−478.049	−0.939192	−0.469596	+	0.882881i	$0.655600\pi$
$510$	0	0
$511$	− 511.348i	− 1.00068i
$512$	0	0
$513$	−123.542	−0.240823
$514$	0	0
$515$	− 63.0311i	− 0.122390i
$516$	0	0
$517$	−419.778	−0.811949
$518$	0	0
$519$	− 102.837i	− 0.198144i
$520$	0	0
$521$	−35.7365	−0.0685921	−0.0342960	−	0.999412i	$-0.510919\pi$
$521$	−35.7365	−0.0685921	−0.0342960	+	0.999412i	$0.510919\pi$
$522$	0	0
$523$	733.562i	1.40260i	0.712864	+	0.701302i	$0.247398\pi$
$523$	733.562i	1.40260i	−0.712864	+	0.701302i	$0.752602\pi$
$524$	0	0
$525$	−47.3517	−0.0901937
$526$	0	0
$527$	118.081i	0.224062i
$528$	0	0
$529$	−858.753	−1.62335
$530$	0	0
$531$	166.623i	0.313791i
$532$	0	0
$533$	94.1965	0.176729
$534$	0	0
$535$	10.0780i	0.0188373i
$536$	0	0
$537$	437.831	0.815327
$538$	0	0
$539$	− 210.917i	− 0.391312i
$540$	0	0
$541$	−608.939	−1.12558	−0.562790	−	0.826600i	$-0.690272\pi$
$541$	−608.939	−1.12558	−0.562790	+	0.826600i	$0.690272\pi$
$542$	0	0
$543$	− 217.153i	− 0.399913i
$544$	0	0
$545$	−432.266	−0.793148
$546$	0	0
$547$	− 78.5868i	− 0.143669i	−0.997417	−	0.0718344i	$-0.977115\pi$
$547$	− 78.5868i	− 0.143669i	0.997417	−	0.0718344i	$-0.0228853\pi$
$548$	0	0
$549$	−246.724	−0.449405
$550$	0	0
$551$	− 612.763i	− 1.11209i
$552$	0	0
$553$	269.974	0.488200
$554$	0	0
$555$	− 138.069i	− 0.248773i
$556$	0	0
$557$	928.488	1.66694	0.833472	−	0.552561i	$-0.186349\pi$
$557$	928.488	1.66694	0.833472	+	0.552561i	$0.186349\pi$
$558$	0	0
$559$	− 708.984i	− 1.26831i
$560$	0	0
$561$	−466.866	−0.832202
$562$	0	0
$563$	− 447.978i	− 0.795697i	−0.917451	−	0.397849i	$-0.869757\pi$
$563$	− 447.978i	− 0.795697i	0.917451	−	0.397849i	$-0.130243\pi$
$564$	0	0
$565$	168.986	0.299090
$566$	0	0
$567$	49.2093i	0.0867889i
$568$	0	0
$569$	571.441	1.00429	0.502145	−	0.864783i	$-0.332544\pi$
$569$	571.441	1.00429	0.502145	+	0.864783i	$0.332544\pi$
$570$	0	0
$571$	− 990.801i	− 1.73520i	−0.497260	−	0.867602i	$-0.665660\pi$
$571$	− 990.801i	− 1.73520i	0.497260	−	0.867602i	$-0.334340\pi$
$572$	0	0
$573$	168.813	0.294612
$574$	0	0
$575$	− 186.263i	− 0.323935i
$576$	0	0
$577$	826.638	1.43265	0.716324	−	0.697768i	$-0.245823\pi$
$577$	826.638	1.43265	0.716324	+	0.697768i	$0.245823\pi$
$578$	0	0
$579$	− 593.013i	− 1.02420i
$580$	0	0
$581$	395.735	0.681127
$582$	0	0
$583$	615.787i	1.05624i
$584$	0	0
$585$	67.9151	0.116094
$586$	0	0
$587$	900.009i	1.53323i	0.642104	+	0.766617i	$0.278062\pi$
$587$	900.009i	1.53323i	−0.642104	+	0.766617i	$0.721938\pi$
$588$	0	0
$589$	−114.991	−0.195230
$590$	0	0
$591$	− 129.008i	− 0.218288i
$592$	0	0
$593$	−704.088	−1.18733	−0.593666	−	0.804711i	$-0.702320\pi$
$593$	−704.088	−1.18733	−0.593666	+	0.804711i	$0.702320\pi$
$594$	0	0
$595$	− 298.497i	− 0.501675i
$596$	0	0
$597$	−308.351	−0.516501
$598$	0	0
$599$	376.098i	0.627876i	0.949444	+	0.313938i	$0.101648\pi$
$599$	376.098i	0.627876i	−0.949444	+	0.313938i	$0.898352\pi$
$600$	0	0
$601$	430.191	0.715791	0.357896	−	0.933762i	$-0.383494\pi$
$601$	430.191	0.715791	0.357896	+	0.933762i	$0.383494\pi$
$602$	0	0
$603$	313.480i	0.519868i
$604$	0	0
$605$	−1.98684	−0.00328404
$606$	0	0
$607$	93.4019i	0.153875i	0.997036	+	0.0769373i	$0.0245141\pi$
$607$	93.4019i	0.153875i	−0.997036	+	0.0769373i	$0.975486\pi$
$608$	0	0
$609$	−244.075	−0.400781
$610$	0	0
$611$	384.944i	0.630024i
$612$	0	0
$613$	156.506	0.255312	0.127656	−	0.991818i	$-0.459255\pi$
$613$	156.506	0.255312	0.127656	+	0.991818i	$0.459255\pi$
$614$	0	0
$615$	− 36.0346i	− 0.0585929i
$616$	0	0
$617$	−553.493	−0.897072	−0.448536	−	0.893765i	$-0.648054\pi$
$617$	−553.493	−0.897072	−0.448536	+	0.893765i	$0.648054\pi$
$618$	0	0
$619$	− 14.4398i	− 0.0233276i	−0.999932	−	0.0116638i	$-0.996287\pi$
$619$	− 14.4398i	− 0.0233276i	0.999932	−	0.0116638i	$-0.00371278\pi$
$620$	0	0
$621$	−193.570	−0.311707
$622$	0	0
$623$	632.564i	1.01535i
$624$	0	0
$625$	25.0000	0.0400000
$626$	0	0
$627$	− 454.649i	− 0.725117i
$628$	0	0
$629$	870.364	1.38373
$630$	0	0
$631$	352.389i	0.558460i	0.960224	+	0.279230i	$0.0900793\pi$
$631$	352.389i	0.558460i	−0.960224	+	0.279230i	$0.909921\pi$
$632$	0	0
$633$	−321.976	−0.508650
$634$	0	0
$635$	− 293.608i	− 0.462375i
$636$	0	0
$637$	−193.415	−0.303634
$638$	0	0
$639$	− 230.241i	− 0.360315i
$640$	0	0
$641$	−545.742	−0.851391	−0.425696	−	0.904866i	$-0.639971\pi$
$641$	−545.742	−0.851391	−0.425696	+	0.904866i	$0.639971\pi$
$642$	0	0
$643$	− 757.447i	− 1.17799i	−0.808137	−	0.588995i	$-0.799524\pi$
$643$	− 757.447i	− 1.17799i	0.808137	−	0.588995i	$-0.200476\pi$
$644$	0	0
$645$	−271.220	−0.420496
$646$	0	0
$647$	− 1161.36i	− 1.79500i	−0.441016	−	0.897499i	$-0.645382\pi$
$647$	− 1161.36i	− 1.79500i	0.441016	−	0.897499i	$-0.354618\pi$
$648$	0	0
$649$	−613.191	−0.944824
$650$	0	0
$651$	45.8030i	0.0703579i
$652$	0	0
$653$	621.231	0.951348	0.475674	−	0.879622i	$-0.342204\pi$
$653$	621.231	0.951348	0.475674	+	0.879622i	$0.342204\pi$
$654$	0	0
$655$	169.390i	0.258610i
$656$	0	0
$657$	280.565	0.427039
$658$	0	0
$659$	− 736.047i	− 1.11692i	−0.829533	−	0.558458i	$-0.811393\pi$
$659$	− 736.047i	− 1.11692i	0.829533	−	0.558458i	$-0.188607\pi$
$660$	0	0
$661$	−383.845	−0.580704	−0.290352	−	0.956920i	$-0.593772\pi$
$661$	−383.845	−0.580704	−0.290352	+	0.956920i	$0.593772\pi$
$662$	0	0
$663$	428.125i	0.645739i
$664$	0	0
$665$	290.686	0.437121
$666$	0	0
$667$	− 960.096i	− 1.43942i
$668$	0	0
$669$	−351.128	−0.524854
$670$	0	0
$671$	− 907.968i	− 1.35316i
$672$	0	0
$673$	984.464	1.46280	0.731400	−	0.681949i	$-0.238867\pi$
$673$	984.464	1.46280	0.731400	+	0.681949i	$0.238867\pi$
$674$	0	0
$675$	− 25.9808i	− 0.0384900i
$676$	0	0
$677$	−673.154	−0.994319	−0.497160	−	0.867659i	$-0.665624\pi$
$677$	−673.154	−0.994319	−0.497160	+	0.867659i	$0.665624\pi$
$678$	0	0
$679$	− 398.918i	− 0.587507i
$680$	0	0
$681$	88.8037	0.130402
$682$	0	0
$683$	− 291.192i	− 0.426343i	−0.977015	−	0.213171i	$-0.931621\pi$
$683$	− 291.192i	− 0.426343i	0.977015	−	0.213171i	$-0.0683793\pi$
$684$	0	0
$685$	149.353	0.218034
$686$	0	0
$687$	− 583.798i	− 0.849778i
$688$	0	0
$689$	564.689	0.819578
$690$	0	0
$691$	943.693i	1.36569i	0.730563	+	0.682846i	$0.239258\pi$
$691$	943.693i	1.36569i	−0.730563	+	0.682846i	$0.760742\pi$
$692$	0	0
$693$	−181.095	−0.261321
$694$	0	0
$695$	− 85.2420i	− 0.122650i
$696$	0	0
$697$	227.156	0.325905
$698$	0	0
$699$	− 139.058i	− 0.198938i
$700$	0	0
$701$	−885.681	−1.26345	−0.631727	−	0.775191i	$-0.717654\pi$
$701$	−885.681	−1.26345	−0.631727	+	0.775191i	$0.717654\pi$
$702$	0	0
$703$	847.588i	1.20567i
$704$	0	0
$705$	147.260	0.208879
$706$	0	0
$707$	− 160.801i	− 0.227441i
$708$	0	0
$709$	286.183	0.403644	0.201822	−	0.979422i	$-0.435314\pi$
$709$	286.183	0.403644	0.201822	+	0.979422i	$0.435314\pi$
$710$	0	0
$711$	148.129i	0.208339i
$712$	0	0
$713$	−180.171	−0.252694
$714$	0	0
$715$	249.935i	0.349559i
$716$	0	0
$717$	572.976	0.799129
$718$	0	0
$719$	− 666.163i	− 0.926513i	−0.886224	−	0.463257i	$-0.846681\pi$
$719$	− 666.163i	− 0.926513i	0.886224	−	0.463257i	$-0.153319\pi$
$720$	0	0
$721$	154.125	0.213766
$722$	0	0
$723$	− 623.389i	− 0.862226i
$724$	0	0
$725$	128.863	0.177742
$726$	0	0
$727$	− 856.270i	− 1.17781i	−0.808201	−	0.588907i	$-0.799559\pi$
$727$	− 856.270i	− 1.17781i	0.808201	−	0.588907i	$-0.200441\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	− 1709.72i	− 2.33888i
$732$	0	0
$733$	769.487	1.04978	0.524889	−	0.851171i	$-0.324107\pi$
$733$	769.487	1.04978	0.524889	+	0.851171i	$0.324107\pi$
$734$	0	0
$735$	73.9905i	0.100667i
$736$	0	0
$737$	−1153.64	−1.56532
$738$	0	0
$739$	1156.70i	1.56522i	0.622511	+	0.782611i	$0.286113\pi$
$739$	1156.70i	1.56522i	−0.622511	+	0.782611i	$0.713887\pi$
$740$	0	0
$741$	−416.922	−0.562647
$742$	0	0
$743$	426.794i	0.574421i	0.957868	+	0.287210i	$0.0927278\pi$
$743$	426.794i	0.574421i	−0.957868	+	0.287210i	$0.907272\pi$
$744$	0	0
$745$	283.348	0.380332
$746$	0	0
$747$	217.130i	0.290670i
$748$	0	0
$749$	−24.6429	−0.0329011
$750$	0	0
$751$	− 1222.03i	− 1.62721i	−0.581420	−	0.813604i	$-0.697503\pi$
$751$	− 1222.03i	− 1.62721i	0.581420	−	0.813604i	$-0.302497\pi$
$752$	0	0
$753$	−540.768	−0.718152
$754$	0	0
$755$	− 153.037i	− 0.202698i
$756$	0	0
$757$	1312.95	1.73442	0.867209	−	0.497945i	$-0.165912\pi$
$757$	1312.95	1.73442	0.867209	+	0.497945i	$0.165912\pi$
$758$	0	0
$759$	− 712.358i	− 0.938548i
$760$	0	0
$761$	−189.584	−0.249124	−0.124562	−	0.992212i	$-0.539753\pi$
$761$	−189.584	−0.249124	−0.124562	+	0.992212i	$0.539753\pi$
$762$	0	0
$763$	− 1056.99i	− 1.38531i
$764$	0	0
$765$	163.778	0.214089
$766$	0	0
$767$	562.308i	0.733127i
$768$	0	0
$769$	254.995	0.331594	0.165797	−	0.986160i	$-0.446980\pi$
$769$	254.995	0.331594	0.165797	+	0.986160i	$0.446980\pi$
$770$	0	0
$771$	139.013i	0.180302i
$772$	0	0
$773$	−23.2536	−0.0300823	−0.0150411	−	0.999887i	$-0.504788\pi$
$773$	−23.2536	−0.0300823	−0.0150411	+	0.999887i	$0.504788\pi$
$774$	0	0
$775$	− 24.1824i	− 0.0312030i
$776$	0	0
$777$	337.611	0.434506
$778$	0	0
$779$	221.212i	0.283969i
$780$	0	0
$781$	847.312	1.08491
$782$	0	0
$783$	− 133.919i	− 0.171033i
$784$	0	0
$785$	−57.1935	−0.0728580
$786$	0	0
$787$	− 220.593i	− 0.280296i	−0.990131	−	0.140148i	$-0.955242\pi$
$787$	− 220.593i	− 0.280296i	0.990131	−	0.140148i	$-0.0447579\pi$
$788$	0	0
$789$	−845.183	−1.07121
$790$	0	0
$791$	413.209i	0.522388i
$792$	0	0
$793$	−832.625	−1.04997
$794$	0	0
$795$	− 216.020i	− 0.271724i
$796$	0	0
$797$	−1010.38	−1.26773	−0.633867	−	0.773442i	$-0.718533\pi$
$797$	−1010.38	−1.26773	−0.633867	+	0.773442i	$0.718533\pi$
$798$	0	0
$799$	928.298i	1.16183i
$800$	0	0
$801$	−347.073	−0.433300
$802$	0	0
$803$	1032.51i	1.28581i
$804$	0	0
$805$	455.455	0.565783
$806$	0	0
$807$	536.162i	0.664389i
$808$	0	0
$809$	1410.37	1.74335	0.871674	−	0.490086i	$-0.163035\pi$
$809$	1410.37	1.74335	0.871674	+	0.490086i	$0.163035\pi$
$810$	0	0
$811$	950.157i	1.17159i	0.810460	+	0.585793i	$0.199217\pi$
$811$	950.157i	1.17159i	−0.810460	+	0.585793i	$0.800783\pi$
$812$	0	0
$813$	84.8173	0.104326
$814$	0	0
$815$	− 141.939i	− 0.174158i
$816$	0	0
$817$	1664.98	2.03792
$818$	0	0
$819$	166.068i	0.202769i
$820$	0	0
$821$	−77.3347	−0.0941957	−0.0470979	−	0.998890i	$-0.514997\pi$
$821$	−77.3347	−0.0941957	−0.0470979	+	0.998890i	$0.514997\pi$
$822$	0	0
$823$	− 1260.16i	− 1.53118i	−0.643328	−	0.765591i	$-0.722447\pi$
$823$	− 1260.16i	− 1.53118i	0.643328	−	0.765591i	$-0.277553\pi$
$824$	0	0
$825$	95.6119	0.115893
$826$	0	0
$827$	438.047i	0.529681i	0.964292	+	0.264841i	$0.0853194\pi$
$827$	438.047i	0.529681i	−0.964292	+	0.264841i	$0.914681\pi$
$828$	0	0
$829$	−361.388	−0.435933	−0.217966	−	0.975956i	$-0.569942\pi$
$829$	−361.388	−0.435933	−0.217966	+	0.975956i	$0.569942\pi$
$830$	0	0
$831$	345.793i	0.416117i
$832$	0	0
$833$	−466.423	−0.559931
$834$	0	0
$835$	27.6760i	0.0331450i
$836$	0	0
$837$	−25.1310	−0.0300251
$838$	0	0
$839$	− 785.017i	− 0.935658i	−0.883819	−	0.467829i	$-0.845036\pi$
$839$	− 785.017i	− 0.935658i	0.883819	−	0.467829i	$-0.154964\pi$
$840$	0	0
$841$	−176.771	−0.210192
$842$	0	0
$843$	105.932i	0.125661i
$844$	0	0
$845$	−148.700	−0.175977
$846$	0	0
$847$	− 4.85829i	− 0.00573588i
$848$	0	0
$849$	−749.123	−0.882359
$850$	0	0
$851$	1328.03i	1.56055i
$852$	0	0
$853$	−1113.79	−1.30573	−0.652865	−	0.757474i	$-0.726433\pi$
$853$	−1113.79	−1.30573	−0.652865	+	0.757474i	$0.726433\pi$
$854$	0	0
$855$	159.492i	0.186541i
$856$	0	0
$857$	−306.591	−0.357749	−0.178875	−	0.983872i	$-0.557246\pi$
$857$	−306.591	−0.357749	−0.178875	+	0.983872i	$0.557246\pi$
$858$	0	0
$859$	204.542i	0.238116i	0.992887	+	0.119058i	$0.0379875\pi$
$859$	204.542i	0.238116i	−0.992887	+	0.119058i	$0.962013\pi$
$860$	0	0
$861$	88.1130	0.102338
$862$	0	0
$863$	654.384i	0.758266i	0.925342	+	0.379133i	$0.123778\pi$
$863$	654.384i	0.758266i	−0.925342	+	0.379133i	$0.876222\pi$
$864$	0	0
$865$	−132.762	−0.153482
$866$	0	0
$867$	531.866i	0.613456i
$868$	0	0
$869$	−545.129	−0.627306
$870$	0	0
$871$	1057.91i	1.21459i
$872$	0	0
$873$	218.877	0.250718
$874$	0	0
$875$	61.1307i	0.0698637i
$876$	0	0
$877$	604.453	0.689228	0.344614	−	0.938744i	$-0.388010\pi$
$877$	604.453	0.689228	0.344614	+	0.938744i	$0.388010\pi$
$878$	0	0
$879$	490.575i	0.558106i
$880$	0	0
$881$	1436.81	1.63089	0.815445	−	0.578834i	$-0.196492\pi$
$881$	1436.81	1.63089	0.815445	+	0.578834i	$0.196492\pi$
$882$	0	0
$883$	120.993i	0.137025i	0.997650	+	0.0685123i	$0.0218252\pi$
$883$	120.993i	0.137025i	−0.997650	+	0.0685123i	$0.978175\pi$
$884$	0	0
$885$	215.110	0.243062
$886$	0	0
$887$	286.448i	0.322941i	0.986878	+	0.161470i	$0.0516236\pi$
$887$	286.448i	0.322941i	−0.986878	+	0.161470i	$0.948376\pi$
$888$	0	0
$889$	717.940	0.807582
$890$	0	0
$891$	− 99.3628i	− 0.111518i
$892$	0	0
$893$	−904.007	−1.01233
$894$	0	0
$895$	− 565.237i	− 0.631550i
$896$	0	0
$897$	−653.246	−0.728256
$898$	0	0
$899$	− 124.649i	− 0.138652i
$900$	0	0
$901$	1361.75	1.51138
$902$	0	0
$903$	− 663.195i	− 0.734436i
$904$	0	0
$905$	−280.343	−0.309771
$906$	0	0
$907$	234.706i	0.258772i	0.991594	+	0.129386i	$0.0413006\pi$
$907$	234.706i	0.258772i	−0.991594	+	0.129386i	$0.958699\pi$
$908$	0	0
$909$	88.2276	0.0970601
$910$	0	0
$911$	− 491.244i	− 0.539236i	−0.962967	−	0.269618i	$-0.913103\pi$
$911$	− 491.244i	− 0.539236i	0.962967	−	0.269618i	$-0.0868974\pi$
$912$	0	0
$913$	−799.063	−0.875206
$914$	0	0
$915$	318.519i	0.348108i
$916$	0	0
$917$	−414.197	−0.451687
$918$	0	0
$919$	356.091i	0.387477i	0.981053	+	0.193738i	$0.0620613\pi$
$919$	356.091i	0.387477i	−0.981053	+	0.193738i	$0.937939\pi$
$920$	0	0
$921$	173.338	0.188207
$922$	0	0
$923$	− 777.002i	− 0.841822i
$924$	0	0
$925$	−178.246	−0.192699
$926$	0	0
$927$	84.5651i	0.0912244i
$928$	0	0
$929$	−916.019	−0.986027	−0.493014	−	0.870022i	$-0.664105\pi$
$929$	−916.019	−0.986027	−0.493014	+	0.870022i	$0.664105\pi$
$930$	0	0
$931$	− 454.217i	− 0.487881i
$932$	0	0
$933$	−700.069	−0.750342
$934$	0	0
$935$	602.721i	0.644621i
$936$	0	0
$937$	143.818	0.153488	0.0767440	−	0.997051i	$-0.475548\pi$
$937$	143.818	0.153488	0.0767440	+	0.997051i	$0.475548\pi$
$938$	0	0
$939$	− 222.705i	− 0.237173i
$940$	0	0
$941$	1488.04	1.58133	0.790667	−	0.612246i	$-0.209734\pi$
$941$	1488.04	1.58133	0.790667	+	0.612246i	$0.209734\pi$
$942$	0	0
$943$	346.602i	0.367552i
$944$	0	0
$945$	63.5289	0.0672264
$946$	0	0
$947$	− 1095.51i	− 1.15682i	−0.815745	−	0.578411i	$-0.803673\pi$
$947$	− 1095.51i	− 1.15682i	0.815745	−	0.578411i	$-0.196327\pi$
$948$	0	0
$949$	946.829	0.997713
$950$	0	0
$951$	− 148.838i	− 0.156506i
$952$	0	0
$953$	1277.86	1.34089	0.670443	−	0.741961i	$-0.266104\pi$
$953$	1277.86	1.34089	0.670443	+	0.741961i	$0.266104\pi$
$954$	0	0
$955$	− 217.936i	− 0.228205i
$956$	0	0
$957$	492.834	0.514978
$958$	0	0
$959$	365.202i	0.380816i
$960$	0	0
$961$	937.609	0.975659
$962$	0	0
$963$	− 13.5210i	− 0.0140405i
$964$	0	0
$965$	−765.576	−0.793343
$966$	0	0
$967$	237.958i	0.246079i	0.992402	+	0.123039i	$0.0392641\pi$
$967$	237.958i	0.246079i	−0.992402	+	0.123039i	$0.960736\pi$
$968$	0	0
$969$	−1005.41	−1.03758
$970$	0	0
$971$	− 1602.10i	− 1.64995i	−0.565169	−	0.824975i	$-0.691189\pi$
$971$	− 1602.10i	− 1.64995i	0.565169	−	0.824975i	$-0.308811\pi$
$972$	0	0
$973$	208.436	0.214220
$974$	0	0
$975$	− 87.6780i	− 0.0899262i
$976$	0	0
$977$	918.977	0.940611	0.470305	−	0.882504i	$-0.344144\pi$
$977$	918.977	0.940611	0.470305	+	0.882504i	$0.344144\pi$
$978$	0	0
$979$	− 1277.27i	− 1.30466i
$980$	0	0
$981$	579.945	0.591178
$982$	0	0
$983$	− 1162.30i	− 1.18240i	−0.806525	−	0.591200i	$-0.798654\pi$
$983$	− 1162.30i	− 1.18240i	0.806525	−	0.591200i	$-0.201346\pi$
$984$	0	0
$985$	−166.549	−0.169085
$986$	0	0
$987$	360.083i	0.364826i
$988$	0	0
$989$	2608.75	2.63776
$990$	0	0
$991$	491.614i	0.496079i	0.968750	+	0.248040i	$0.0797863\pi$
$991$	491.614i	0.496079i	−0.968750	+	0.248040i	$0.920214\pi$
$992$	0	0
$993$	−317.643	−0.319882
$994$	0	0
$995$	398.080i	0.400080i
$996$	0	0
$997$	1262.51	1.26630	0.633152	−	0.774027i	$-0.281761\pi$
$997$	1262.51	1.26630	0.633152	+	0.774027i	$0.281761\pi$
$998$	0	0
$999$	185.239i	0.185425i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	960.3.e.c.511.8		8
3.2	odd	2		2880.3.e.j.2431.3		8
4.3	odd	2	inner	960.3.e.c.511.3		8
8.3	odd	2		60.3.c.a.31.8	yes	8
8.5	even	2		60.3.c.a.31.7	✓	8
12.11	even	2		2880.3.e.j.2431.2		8
24.5	odd	2		180.3.c.b.91.2		8
24.11	even	2		180.3.c.b.91.1		8
40.3	even	4		300.3.f.b.199.11		16
40.13	odd	4		300.3.f.b.199.5		16
40.19	odd	2		300.3.c.d.151.1		8
40.27	even	4		300.3.f.b.199.6		16
40.29	even	2		300.3.c.d.151.2		8
40.37	odd	4		300.3.f.b.199.12		16
120.29	odd	2		900.3.c.u.451.7		8
120.53	even	4		900.3.f.f.199.12		16
120.59	even	2		900.3.c.u.451.8		8
120.77	even	4		900.3.f.f.199.5		16
120.83	odd	4		900.3.f.f.199.6		16
120.107	odd	4		900.3.f.f.199.11		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.3.c.a.31.7	✓	8	8.5	even	2
60.3.c.a.31.8	yes	8	8.3	odd	2
180.3.c.b.91.1		8	24.11	even	2
180.3.c.b.91.2		8	24.5	odd	2
300.3.c.d.151.1		8	40.19	odd	2
300.3.c.d.151.2		8	40.29	even	2
300.3.f.b.199.5		16	40.13	odd	4
300.3.f.b.199.6		16	40.27	even	4
300.3.f.b.199.11		16	40.3	even	4
300.3.f.b.199.12		16	40.37	odd	4
900.3.c.u.451.7		8	120.29	odd	2
900.3.c.u.451.8		8	120.59	even	2
900.3.f.f.199.5		16	120.77	even	4
900.3.f.f.199.6		16	120.83	odd	4
900.3.f.f.199.11		16	120.107	odd	4
900.3.f.f.199.12		16	120.53	even	4
960.3.e.c.511.3		8	4.3	odd	2	inner
960.3.e.c.511.8		8	1.1	even	1	trivial
2880.3.e.j.2431.2		8	12.11	even	2
2880.3.e.j.2431.3		8	3.2	odd	2

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:	\( N \)	\(=\)	\( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	960.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} +2.23607 q^{5} +5.46770i q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q+1.73205i q^{3} +2.23607 q^{5} +5.46770i q^{7} -3.00000 q^{9} -11.0403i q^{11} -10.1242 q^{13} +3.87298i q^{15} -24.4146 q^{17} -23.7757i q^{19} -9.47033 q^{21} -37.2526i q^{23} +5.00000 q^{25} -5.19615i q^{27} +25.7726 q^{29} -4.83647i q^{31} +19.1224 q^{33} +12.2261i q^{35} -35.6493 q^{37} -17.5356i q^{39} -9.30410 q^{41} +70.0287i q^{43} -6.70820 q^{45} -38.0223i q^{47} +19.1043 q^{49} -42.2873i q^{51} -55.7762 q^{53} -24.6869i q^{55} +41.1808 q^{57} -55.5411i q^{59} +82.2412 q^{61} -16.4031i q^{63} -22.6384 q^{65} -104.493i q^{67} +64.5233 q^{69} +76.7471i q^{71} -93.5215 q^{73} +8.66025i q^{75} +60.3651 q^{77} -49.3762i q^{79} +9.00000 q^{81} -72.3768i q^{83} -54.5927 q^{85} +44.6395i q^{87} +115.691 q^{89} -55.3560i q^{91} +8.37701 q^{93} -53.1641i q^{95} -72.9589 q^{97} +33.1209i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 24 q^{9}+O(q^{10}) \) 8 * q - 24 * q^9	\( 8 q - 24 q^{9} - 16 q^{13} + 48 q^{21} + 40 q^{25} - 64 q^{29} + 112 q^{37} - 16 q^{41} - 56 q^{49} - 352 q^{53} + 144 q^{57} + 176 q^{61} - 80 q^{65} + 96 q^{69} - 240 q^{73} + 288 q^{77} + 72 q^{81} - 160 q^{85} + 80 q^{89} - 144 q^{93} + 432 q^{97}+O(q^{100}) \) 8 * q - 24 * q^9 - 16 * q^13 + 48 * q^21 + 40 * q^25 - 64 * q^29 + 112 * q^37 - 16 * q^41 - 56 * q^49 - 352 * q^53 + 144 * q^57 + 176 * q^61 - 80 * q^65 + 96 * q^69 - 240 * q^73 + 288 * q^77 + 72 * q^81 - 160 * q^85 + 80 * q^89 - 144 * q^93 + 432 * q^97

Introduction

L-functions

Modular forms

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