LMFDB - Embedded newform 960.3.i.b.929.30

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [960,3,Mod(929,960)]

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([0, 1, 1, 1]))

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

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

chi := DirichletCharacter("960.929");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 960 = 2^{6} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	960.i (of order $2$, degree $1$, 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:	$64$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			929.30
Character	$\chi$	$=$	960.929
Dual form			960.3.i.b.929.25

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.619753 + 2.93529i) q^{3} +(-4.48354 + 2.21312i) q^{5} +4.63217i q^{7} +(-8.23181 - 3.63830i) q^{9} +O(q^{10})$	\(q+(-0.619753 + 2.93529i) q^{3} +(-4.48354 + 2.21312i) q^{5} +4.63217i q^{7} +(-8.23181 - 3.63830i) q^{9} -15.0818 q^{11} +3.35922 q^{13} +(-3.71746 - 14.5320i) q^{15} +11.1429 q^{17} +22.7628i q^{19} +(-13.5967 - 2.87080i) q^{21} -36.1566 q^{23} +(15.2042 - 19.8452i) q^{25} +(15.7811 - 21.9079i) q^{27} -21.5705 q^{29} +28.3890 q^{31} +(9.34696 - 44.2693i) q^{33} +(-10.2516 - 20.7685i) q^{35} -69.5592 q^{37} +(-2.08189 + 9.86027i) q^{39} -62.5531i q^{41} +55.2064 q^{43} +(44.9596 - 1.90555i) q^{45} +36.5257 q^{47} +27.5430 q^{49} +(-6.90582 + 32.7075i) q^{51} +29.2870i q^{53} +(67.6196 - 33.3778i) q^{55} +(-66.8152 - 14.1073i) q^{57} +8.07476 q^{59} +10.5504i q^{61} +(16.8532 - 38.1312i) q^{63} +(-15.0612 + 7.43436i) q^{65} +91.8695 q^{67} +(22.4082 - 106.130i) q^{69} +79.2826i q^{71} -59.0857i q^{73} +(48.8286 + 56.9278i) q^{75} -69.8613i q^{77} -86.3194 q^{79} +(54.5255 + 59.8997i) q^{81} -10.0197i q^{83} +(-49.9594 + 24.6605i) q^{85} +(13.3684 - 63.3157i) q^{87} +17.8051i q^{89} +15.5605i q^{91} +(-17.5941 + 83.3297i) q^{93} +(-50.3767 - 102.058i) q^{95} -139.268i q^{97} +(124.150 + 54.8720i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 64 q - 32 q^{9}+O(q^{10}) $ 64 * q - 32 * q^9	$ 64 q - 32 q^{9} - 32 q^{25} - 320 q^{49} + 448 q^{81}+O(q^{100}) $ 64 * q - 32 * q^9 - 32 * q^25 - 320 * q^49 + 448 * q^81

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
$3$	−0.619753	+	2.93529i	−0.206584	+	0.978429i
$3$	−0.619753	+	2.93529i	−0.206584	+	0.978429i
$4$		0			0
$5$	−4.48354	+	2.21312i	−0.896707	+	0.442624i
$5$	−4.48354	+	2.21312i	−0.896707	+	0.442624i
$6$		0			0
$7$			4.63217i			0.661739i	0.943677	+	0.330869i	$0.107342\pi$
$7$			4.63217i			0.661739i	−0.943677	+	0.330869i	$0.892658\pi$
$8$		0			0
$9$	−8.23181	−	3.63830i	−0.914646	−	0.404256i
$10$		0			0
$11$	−15.0818			−1.37107			−0.685534	−	0.728040i	$-0.740431\pi$
$11$	−15.0818			−1.37107			−0.685534	+	0.728040i	$0.740431\pi$
$12$		0			0
$13$	3.35922			0.258402			0.129201	−	0.991618i	$-0.458759\pi$
$13$	3.35922			0.258402			0.129201	+	0.991618i	$0.458759\pi$
$14$		0			0
$15$	−3.71746	−	14.5320i	−0.247831	−	0.968803i
$16$		0			0
$17$	11.1429			0.655463			0.327731	−	0.944771i	$-0.393716\pi$
$17$	11.1429			0.655463			0.327731	+	0.944771i	$0.393716\pi$
$18$		0			0
$19$			22.7628i			1.19804i	0.800734	+	0.599020i	$0.204443\pi$
$19$			22.7628i			1.19804i	−0.800734	+	0.599020i	$0.795557\pi$
$20$		0			0
$21$	−13.5967	−	2.87080i	−0.647464	−	0.136705i
$22$		0			0
$23$	−36.1566			−1.57203			−0.786014	−	0.618209i	$-0.787858\pi$
$23$	−36.1566			−1.57203			−0.786014	+	0.618209i	$0.787858\pi$
$24$		0			0
$25$	15.2042	−	19.8452i	0.608167	−	0.793809i
$26$		0			0
$27$	15.7811	−	21.9079i	0.584487	−	0.811403i
$28$		0			0
$29$	−21.5705			−0.743811			−0.371906	−	0.928271i	$-0.621295\pi$
$29$	−21.5705			−0.743811			−0.371906	+	0.928271i	$0.621295\pi$
$30$		0			0
$31$	28.3890			0.915773			0.457886	−	0.889011i	$-0.348607\pi$
$31$	28.3890			0.915773			0.457886	+	0.889011i	$0.348607\pi$
$32$		0			0
$33$	9.34696	−	44.2693i	0.283241	−	1.34149i
$34$		0			0
$35$	−10.2516	−	20.7685i	−0.292902	−	0.593386i
$36$		0			0
$37$	−69.5592			−1.87998			−0.939989	−	0.341205i	$-0.889165\pi$
$37$	−69.5592			−1.87998			−0.939989	+	0.341205i	$0.889165\pi$
$38$		0			0
$39$	−2.08189	+	9.86027i	−0.0533817	+	0.252828i
$40$		0			0
$41$		−	62.5531i		−	1.52568i	−0.646585	−	0.762842i	$-0.723803\pi$
$41$		−	62.5531i		−	1.52568i	0.646585	−	0.762842i	$-0.276197\pi$
$42$		0			0
$43$	55.2064			1.28387			0.641935	−	0.766759i	$-0.278132\pi$
$43$	55.2064			1.28387			0.641935	+	0.766759i	$0.278132\pi$
$44$		0			0
$45$	44.9596	−	1.90555i	0.999103	−	0.0423455i
$46$		0			0
$47$	36.5257			0.777142			0.388571	−	0.921419i	$-0.372969\pi$
$47$	36.5257			0.777142			0.388571	+	0.921419i	$0.372969\pi$
$48$		0			0
$49$	27.5430			0.562102
$50$		0			0
$51$	−6.90582	+	32.7075i	−0.135408	+	0.641323i
$52$		0			0
$53$			29.2870i			0.552585i	0.961074	+	0.276292i	$0.0891058\pi$
$53$			29.2870i			0.552585i	−0.961074	+	0.276292i	$0.910894\pi$
$54$		0			0
$55$	67.6196	−	33.3778i	1.22945	−	0.606868i
$56$		0			0
$57$	−66.8152	−	14.1073i	−1.17220	−	0.247496i
$58$		0			0
$59$	8.07476			0.136860			0.0684302	−	0.997656i	$-0.478201\pi$
$59$	8.07476			0.136860			0.0684302	+	0.997656i	$0.478201\pi$
$60$		0			0
$61$			10.5504i			0.172957i	0.996254	+	0.0864783i	$0.0275613\pi$
$61$			10.5504i			0.172957i	−0.996254	+	0.0864783i	$0.972439\pi$
$62$		0			0
$63$	16.8532	−	38.1312i	0.267512	−	0.605257i
$64$		0			0
$65$	−15.0612	+	7.43436i	−0.231711	+	0.114375i
$66$		0			0
$67$	91.8695			1.37119			0.685593	−	0.727985i	$-0.259543\pi$
$67$	91.8695			1.37119			0.685593	+	0.727985i	$0.259543\pi$
$68$		0			0
$69$	22.4082	−	106.130i	0.324756	−	1.53812i
$70$		0			0
$71$			79.2826i			1.11666i	0.829620	+	0.558328i	$0.188557\pi$
$71$			79.2826i			1.11666i	−0.829620	+	0.558328i	$0.811443\pi$
$72$		0			0
$73$		−	59.0857i		−	0.809394i	−0.914451	−	0.404697i	$-0.867377\pi$
$73$		−	59.0857i		−	0.809394i	0.914451	−	0.404697i	$-0.132623\pi$
$74$		0			0
$75$	48.8286	+	56.9278i	0.651048	+	0.759037i
$76$		0			0
$77$		−	69.8613i		−	0.907289i
$78$		0			0
$79$	−86.3194			−1.09265			−0.546325	−	0.837573i	$-0.683974\pi$
$79$	−86.3194			−1.09265			−0.546325	+	0.837573i	$0.683974\pi$
$80$		0			0
$81$	54.5255	+	59.8997i	0.673154	+	0.739502i
$82$		0			0
$83$		−	10.0197i		−	0.120720i	−0.998177	−	0.0603599i	$-0.980775\pi$
$83$		−	10.0197i		−	0.120720i	0.998177	−	0.0603599i	$-0.0192248\pi$
$84$		0			0
$85$	−49.9594	+	24.6605i	−0.587758	+	0.290124i
$86$		0			0
$87$	13.3684	−	63.3157i	0.153660	−	0.727766i
$88$		0			0
$89$			17.8051i			0.200057i	0.994985	+	0.100029i	$0.0318935\pi$
$89$			17.8051i			0.200057i	−0.994985	+	0.100029i	$0.968107\pi$
$90$		0			0
$91$			15.5605i			0.170994i
$92$		0			0
$93$	−17.5941	+	83.3297i	−0.189184	+	0.896019i
$94$		0			0
$95$	−50.3767	−	102.058i	−0.530282	−	1.07429i
$96$		0			0
$97$		−	139.268i		−	1.43575i	−0.696170	−	0.717877i	$-0.745114\pi$
$97$		−	139.268i		−	1.43575i	0.696170	−	0.717877i	$-0.254886\pi$
$98$		0			0
$99$	124.150	+	54.8720i	1.25404	+	0.554263i
$100$		0			0
$101$	46.6948			0.462325			0.231162	−	0.972915i	$-0.425747\pi$
$101$	46.6948			0.462325			0.231162	+	0.972915i	$0.425747\pi$
$102$		0			0
$103$		−	11.6291i		−	0.112904i	−0.998405	−	0.0564520i	$-0.982021\pi$
$103$		−	11.6291i		−	0.112904i	0.998405	−	0.0564520i	$-0.0179788\pi$
$104$		0			0
$105$	67.3149	−	17.2199i	0.641095	−	0.163999i
$106$		0			0
$107$		−	95.7996i		−	0.895323i	−0.894203	−	0.447662i	$-0.852257\pi$
$107$		−	95.7996i		−	0.895323i	0.894203	−	0.447662i	$-0.147743\pi$
$108$		0			0
$109$		−	124.614i		−	1.14324i	−0.820517	−	0.571622i	$-0.806314\pi$
$109$		−	124.614i		−	1.14324i	0.820517	−	0.571622i	$-0.193686\pi$
$110$		0			0
$111$	43.1095	−	204.176i	0.388374	−	1.83942i
$112$		0			0
$113$	9.29116			0.0822227			0.0411113	−	0.999155i	$-0.486910\pi$
$113$	9.29116			0.0822227			0.0411113	+	0.999155i	$0.486910\pi$
$114$		0			0
$115$	162.110	−	80.0190i	1.40965	−	0.695817i
$116$		0			0
$117$	−27.6525	−	12.2219i	−0.236346	−	0.104460i
$118$		0			0
$119$			51.6156i			0.433745i
$120$		0			0
$121$	106.459			0.879830
$122$		0			0
$123$	183.611	+	38.7674i	1.49277	+	0.315182i
$124$		0			0
$125$	−24.2486	+	122.625i	−0.193989	+	0.981004i
$126$		0			0
$127$		−	167.887i		−	1.32194i	−0.750412	−	0.660971i	$-0.770145\pi$
$127$		−	167.887i		−	1.32194i	0.750412	−	0.660971i	$-0.229855\pi$
$128$		0			0
$129$	−34.2143	+	162.047i	−0.265227	+	1.25618i
$130$		0			0
$131$	6.27609			0.0479091			0.0239546	−	0.999713i	$-0.492374\pi$
$131$	6.27609			0.0479091			0.0239546	+	0.999713i	$0.492374\pi$
$132$		0			0
$133$	−105.441			−0.792789
$134$		0			0
$135$	−22.2705	+	133.150i	−0.164967	+	0.986299i
$136$		0			0
$137$	−100.937			−0.736764			−0.368382	−	0.929675i	$-0.620088\pi$
$137$	−100.937			−0.736764			−0.368382	+	0.929675i	$0.620088\pi$
$138$		0			0
$139$			182.361i			1.31195i	0.754782	+	0.655976i	$0.227743\pi$
$139$			182.361i			1.31195i	−0.754782	+	0.655976i	$0.772257\pi$
$140$		0			0
$141$	−22.6369	+	107.213i	−0.160545	+	0.760378i
$142$		0			0
$143$	−50.6629			−0.354286
$144$		0			0
$145$	96.7122	−	47.7382i	0.666981	−	0.329229i
$146$		0			0
$147$	−17.0698	+	80.8466i	−0.116121	+	0.549977i
$148$		0			0
$149$	−207.094			−1.38989			−0.694946	−	0.719062i	$-0.744572\pi$
$149$	−207.094			−1.38989			−0.694946	+	0.719062i	$0.744572\pi$
$150$		0			0
$151$	51.4246			0.340560			0.170280	−	0.985396i	$-0.445533\pi$
$151$	51.4246			0.340560			0.170280	+	0.985396i	$0.445533\pi$
$152$		0			0
$153$	−91.7260	−	40.5411i	−0.599516	−	0.264975i
$154$		0			0
$155$	−127.283	+	62.8282i	−0.821180	+	0.405343i
$156$		0			0
$157$	143.715			0.915385			0.457692	−	0.889111i	$-0.348676\pi$
$157$	143.715			0.915385			0.457692	+	0.889111i	$0.348676\pi$
$158$		0			0
$159$	−85.9657	−	18.1507i	−0.540665	−	0.114155i
$160$		0			0
$161$		−	167.484i		−	1.04027i
$162$		0			0
$163$	−120.403			−0.738669			−0.369335	−	0.929296i	$-0.620414\pi$
$163$	−120.403			−0.738669			−0.369335	+	0.929296i	$0.620414\pi$
$164$		0			0
$165$	56.0659	+	219.169i	0.339793	+	1.32830i
$166$		0			0
$167$	117.491			0.703540			0.351770	−	0.936086i	$-0.385580\pi$
$167$	117.491			0.703540			0.351770	+	0.936086i	$0.385580\pi$
$168$		0			0
$169$	−157.716			−0.933229
$170$		0			0
$171$	82.8178	−	187.379i	0.484315	−	1.09578i
$172$		0			0
$173$		−	138.724i		−	0.801873i	−0.916106	−	0.400937i	$-0.868685\pi$
$173$		−	138.724i		−	0.801873i	0.916106	−	0.400937i	$-0.131315\pi$
$174$		0			0
$175$	91.9264	+	70.4284i	0.525294	+	0.402448i
$176$		0			0
$177$	−5.00435	+	23.7017i	−0.0282732	+	0.133908i
$178$		0			0
$179$	−241.283			−1.34795			−0.673974	−	0.738755i	$-0.735414\pi$
$179$	−241.283			−1.34795			−0.673974	+	0.738755i	$0.735414\pi$
$180$		0			0
$181$		−	110.406i		−	0.609978i	−0.952356	−	0.304989i	$-0.901347\pi$
$181$		−	110.406i		−	0.609978i	0.952356	−	0.304989i	$-0.0986528\pi$
$182$		0			0
$183$	−30.9683	−	6.53861i	−0.169226	−	0.0357301i
$184$		0			0
$185$	311.871	−	153.943i	1.68579	−	0.832124i
$186$		0			0
$187$	−168.054			−0.898684
$188$		0			0
$189$	101.481	+	73.1010i	0.536937	+	0.386778i
$190$		0			0
$191$			119.468i			0.625487i	0.949838	+	0.312744i	$0.101248\pi$
$191$			119.468i			0.625487i	−0.949838	+	0.312744i	$0.898752\pi$
$192$		0			0
$193$			148.889i			0.771445i	0.922615	+	0.385722i	$0.126048\pi$
$193$			148.889i			0.771445i	−0.922615	+	0.385722i	$0.873952\pi$
$194$		0			0
$195$	−12.4878	−	48.8164i	−0.0640399	−	0.250340i
$196$		0			0
$197$		−	277.454i		−	1.40839i	−0.710005	−	0.704197i	$-0.751307\pi$
$197$		−	277.454i		−	1.40839i	0.710005	−	0.704197i	$-0.248693\pi$
$198$		0			0
$199$	−141.225			−0.709672			−0.354836	−	0.934929i	$-0.615463\pi$
$199$	−141.225			−0.709672			−0.354836	+	0.934929i	$0.615463\pi$
$200$		0			0
$201$	−56.9363	+	269.663i	−0.283265	+	1.34161i
$202$		0			0
$203$		−	99.9183i		−	0.492209i
$204$		0			0
$205$	138.438	+	280.459i	0.675305	+	1.36809i
$206$		0			0
$207$	297.635	+	131.549i	1.43785	+	0.635501i
$208$		0			0
$209$		−	343.302i		−	1.64259i
$210$		0			0
$211$			109.167i			0.517381i	0.965960	+	0.258690i	$0.0832909\pi$
$211$			109.167i			0.517381i	−0.965960	+	0.258690i	$0.916709\pi$
$212$		0			0
$213$	−232.717	−	49.1356i	−1.09257	−	0.230683i
$214$		0			0
$215$	−247.520	+	122.179i	−1.15126	+	0.568272i
$216$		0			0
$217$			131.503i			0.606002i
$218$		0			0
$219$	173.434	+	36.6185i	0.791934	+	0.167208i
$220$		0			0
$221$	37.4313			0.169373
$222$		0			0
$223$			40.5784i			0.181966i	0.995852	+	0.0909830i	$0.0290009\pi$
$223$			40.5784i			0.181966i	−0.995852	+	0.0909830i	$0.970999\pi$
$224$		0			0
$225$	−197.361	+	108.045i	−0.877160	+	0.480199i
$226$		0			0
$227$		−	439.331i		−	1.93538i	−0.252141	−	0.967691i	$-0.581135\pi$
$227$		−	439.331i		−	1.93538i	0.252141	−	0.967691i	$-0.418865\pi$
$228$		0			0
$229$		−	250.730i		−	1.09489i	−0.836841	−	0.547445i	$-0.815600\pi$
$229$		−	250.730i		−	1.09489i	0.836841	−	0.547445i	$-0.184400\pi$
$230$		0			0
$231$	205.063	+	43.2967i	0.887718	+	0.187432i
$232$		0			0
$233$	269.289			1.15575			0.577873	−	0.816127i	$-0.303883\pi$
$233$	269.289			1.15575			0.577873	+	0.816127i	$0.303883\pi$
$234$		0			0
$235$	−163.764	+	80.8357i	−0.696869	+	0.343982i
$236$		0			0
$237$	53.4967	−	253.372i	0.225724	−	1.06908i
$238$		0			0
$239$			384.168i			1.60740i	0.595035	+	0.803699i	$0.297138\pi$
$239$			384.168i			1.60740i	−0.595035	+	0.803699i	$0.702862\pi$
$240$		0			0
$241$	−250.237			−1.03833			−0.519163	−	0.854675i	$-0.673756\pi$
$241$	−250.237			−1.03833			−0.519163	+	0.854675i	$0.673756\pi$
$242$		0			0
$243$	−209.615	+	122.925i	−0.862613	+	0.505864i
$244$		0			0
$245$	−123.490	+	60.9560i	−0.504041	+	0.248800i
$246$		0			0
$247$			76.4651i			0.309575i
$248$		0			0
$249$	29.4108	+	6.20976i	0.118116	+	0.0249388i
$250$		0			0
$251$	−319.368			−1.27238			−0.636191	−	0.771532i	$-0.719491\pi$
$251$	−319.368			−1.27238			−0.636191	+	0.771532i	$0.719491\pi$
$252$		0			0
$253$	545.305			2.15536
$254$		0			0
$255$	−41.4232	−	161.929i	−0.162444	−	0.635014i
$256$		0			0
$257$	−112.417			−0.437421			−0.218711	−	0.975790i	$-0.570185\pi$
$257$	−112.417			−0.437421			−0.218711	+	0.975790i	$0.570185\pi$
$258$		0			0
$259$		−	322.210i		−	1.24405i
$260$		0			0
$261$	177.565	+	78.4801i	0.680324	+	0.300690i
$262$		0			0
$263$	−65.7040			−0.249825			−0.124913	−	0.992168i	$-0.539865\pi$
$263$	−65.7040			−0.249825			−0.124913	+	0.992168i	$0.539865\pi$
$264$		0			0
$265$	−64.8157	−	131.309i	−0.244587	−	0.495507i
$266$		0			0
$267$	−52.2631	−	11.0348i	−0.195742	−	0.0413287i
$268$		0			0
$269$	−162.581			−0.604389			−0.302194	−	0.953246i	$-0.597719\pi$
$269$	−162.581			−0.604389			−0.302194	+	0.953246i	$0.597719\pi$
$270$		0			0
$271$	−351.281			−1.29624			−0.648119	−	0.761539i	$-0.724444\pi$
$271$	−351.281			−1.29624			−0.648119	+	0.761539i	$0.724444\pi$
$272$		0			0
$273$	−45.6745	−	9.64365i	−0.167306	−	0.0353247i
$274$		0			0
$275$	−229.306	+	299.301i	−0.833839	+	1.08837i
$276$		0			0
$277$	−472.325			−1.70514			−0.852572	−	0.522610i	$-0.824958\pi$
$277$	−472.325			−1.70514			−0.852572	+	0.522610i	$0.824958\pi$
$278$		0			0
$279$	−233.693	−	103.288i	−0.837608	−	0.370207i
$280$		0			0
$281$		−	97.1148i		−	0.345604i	−0.984957	−	0.172802i	$-0.944718\pi$
$281$		−	97.1148i		−	0.345604i	0.984957	−	0.172802i	$-0.0552821\pi$
$282$		0			0
$283$	−256.164			−0.905173			−0.452586	−	0.891721i	$-0.649499\pi$
$283$	−256.164			−0.905173			−0.452586	+	0.891721i	$0.649499\pi$
$284$		0			0
$285$	330.789	−	84.6197i	1.16066	−	0.296911i
$286$		0			0
$287$	289.756			1.00960
$288$		0			0
$289$	−164.837			−0.570369
$290$		0			0
$291$	408.792	+	86.3118i	1.40478	+	0.296604i
$292$		0			0
$293$			533.766i			1.82173i	0.412708	+	0.910863i	$0.364583\pi$
$293$			533.766i			1.82173i	−0.412708	+	0.910863i	$0.635417\pi$
$294$		0			0
$295$	−36.2035	+	17.8704i	−0.122724	+	0.0605777i
$296$		0			0
$297$	−238.007	+	330.409i	−0.801372	+	1.11249i
$298$		0			0
$299$	−121.458			−0.406214
$300$		0			0
$301$			255.726i			0.849587i
$302$		0			0
$303$	−28.9392	+	137.063i	−0.0955090	+	0.452352i
$304$		0			0
$305$	−23.3492	−	47.3029i	−0.0765548	−	0.155091i
$306$		0			0
$307$	−123.154			−0.401152			−0.200576	−	0.979678i	$-0.564281\pi$
$307$	−123.154			−0.401152			−0.200576	+	0.979678i	$0.564281\pi$
$308$		0			0
$309$	34.1348	+	7.20717i	0.110468	+	0.0233242i
$310$		0			0
$311$		−	265.458i		−	0.853564i	−0.904355	−	0.426782i	$-0.859647\pi$
$311$		−	265.458i		−	0.853564i	0.904355	−	0.426782i	$-0.140353\pi$
$312$		0			0
$313$			372.410i			1.18981i	0.803797	+	0.594904i	$0.202810\pi$
$313$			372.410i			1.18981i	−0.803797	+	0.594904i	$0.797190\pi$
$314$		0			0
$315$	8.82681	+	208.261i	0.0280216	+	0.661145i
$316$		0			0
$317$		−	585.780i		−	1.84789i	−0.382530	−	0.923943i	$-0.624947\pi$
$317$		−	585.780i		−	1.84789i	0.382530	−	0.923943i	$-0.375053\pi$
$318$		0			0
$319$	325.321			1.01982
$320$		0			0
$321$	281.199	+	59.3720i	0.876010	+	0.184960i
$322$		0			0
$323$			253.642i			0.785270i
$324$		0			0
$325$	51.0742	−	66.6645i	0.157151	−	0.205121i
$326$		0			0
$327$	365.777	+	77.2297i	1.11858	+	0.236176i
$328$		0			0
$329$			169.193i			0.514265i
$330$		0			0
$331$		−	481.587i		−	1.45495i	−0.686136	−	0.727473i	$-0.740695\pi$
$331$		−	481.587i		−	1.45495i	0.686136	−	0.727473i	$-0.259305\pi$
$332$		0			0
$333$	572.598	+	253.077i	1.71951	+	0.759992i
$334$		0			0
$335$	−411.900	+	203.318i	−1.22955	+	0.606920i
$336$		0			0
$337$			581.107i			1.72435i	0.506608	+	0.862176i	$0.330899\pi$
$337$			581.107i			1.72435i	−0.506608	+	0.862176i	$0.669101\pi$
$338$		0			0
$339$	−5.75822	+	27.2722i	−0.0169859	+	0.0804490i
$340$		0			0
$341$	−428.155			−1.25559
$342$		0			0
$343$			354.560i			1.03370i
$344$		0			0
$345$	134.411	+	525.430i	0.389597	+	1.52299i
$346$		0			0
$347$		−	7.03316i		−	0.0202685i	−0.999949	−	0.0101342i	$-0.996774\pi$
$347$		−	7.03316i		−	0.0202685i	0.999949	−	0.0101342i	$-0.00322588\pi$
$348$		0			0
$349$			678.568i			1.94432i	0.234315	+	0.972161i	$0.424715\pi$
$349$			678.568i			1.94432i	−0.234315	+	0.972161i	$0.575285\pi$
$350$		0			0
$351$	53.0124	−	73.5934i	0.151032	−	0.209668i
$352$		0			0
$353$	552.271			1.56451			0.782253	−	0.622961i	$-0.214070\pi$
$353$	552.271			1.56451			0.782253	+	0.622961i	$0.214070\pi$
$354$		0			0
$355$	−175.462	−	355.466i	−0.494259	−	1.00131i
$356$		0			0
$357$	−151.507	−	31.9889i	−0.424389	−	0.0896048i
$358$		0			0
$359$		−	510.038i		−	1.42072i	−0.703840	−	0.710359i	$-0.748533\pi$
$359$		−	510.038i		−	1.42072i	0.703840	−	0.710359i	$-0.251467\pi$
$360$		0			0
$361$	−157.143			−0.435299
$362$		0			0
$363$	−65.9785	+	312.489i	−0.181759	+	0.860851i
$364$		0			0
$365$	130.764	+	264.913i	0.358257	+	0.725789i
$366$		0			0
$367$			397.074i			1.08194i	0.841040	+	0.540972i	$0.181944\pi$
$367$			397.074i			1.08194i	−0.841040	+	0.540972i	$0.818056\pi$
$368$		0			0
$369$	−227.587	+	514.925i	−0.616767	+	1.39546i
$370$		0			0
$371$	−135.662			−0.365667
$372$		0			0
$373$	−192.029			−0.514823			−0.257411	−	0.966302i	$-0.582870\pi$
$373$	−192.029			−0.514823			−0.257411	+	0.966302i	$0.582870\pi$
$374$		0			0
$375$	−344.913	−	147.174i	−0.919767	−	0.392464i
$376$		0			0
$377$	−72.4601			−0.192202
$378$		0			0
$379$		−	150.900i		−	0.398152i	−0.979984	−	0.199076i	$-0.936206\pi$
$379$		−	150.900i		−	0.398152i	0.979984	−	0.199076i	$-0.0637941\pi$
$380$		0			0
$381$	492.795	+	104.048i	1.29343	+	0.273092i
$382$		0			0
$383$	−601.296			−1.56996			−0.784981	−	0.619520i	$-0.787327\pi$
$383$	−601.296			−1.56996			−0.784981	+	0.619520i	$0.787327\pi$
$384$		0			0
$385$	154.611	+	313.226i	0.401588	+	0.813573i
$386$		0			0
$387$	−454.449	−	200.858i	−1.17429	−	0.519012i
$388$		0			0
$389$	437.370			1.12434			0.562172	−	0.827020i	$-0.309966\pi$
$389$	437.370			1.12434			0.562172	+	0.827020i	$0.309966\pi$
$390$		0			0
$391$	−402.888			−1.03040
$392$		0			0
$393$	−3.88963	+	18.4221i	−0.00989727	+	0.0468757i
$394$		0			0
$395$	387.016	−	191.035i	0.979787	−	0.483634i
$396$		0			0
$397$	561.277			1.41380			0.706898	−	0.707315i	$-0.250094\pi$
$397$	561.277			1.41380			0.706898	+	0.707315i	$0.250094\pi$
$398$		0			0
$399$	65.3473	−	309.499i	0.163778	−	0.775688i
$400$		0			0
$401$			291.501i			0.726936i	0.931607	+	0.363468i	$0.118407\pi$
$401$			291.501i			0.726936i	−0.931607	+	0.363468i	$0.881593\pi$
$402$		0			0
$403$	95.3648			0.236637
$404$		0			0
$405$	−377.032	−	147.891i	−0.930944	−	0.365162i
$406$		0			0
$407$	1049.07			2.57758
$408$		0			0
$409$	432.727			1.05801			0.529006	−	0.848618i	$-0.322565\pi$
$409$	432.727			1.05801			0.529006	+	0.848618i	$0.322565\pi$
$410$		0			0
$411$	62.5558	−	296.278i	0.152204	−	0.720871i
$412$		0			0
$413$			37.4037i			0.0905658i
$414$		0			0
$415$	22.1749	+	44.9239i	0.0534335	+	0.108250i
$416$		0			0
$417$	−535.283	−	113.019i	−1.28365	−	0.271028i
$418$		0			0
$419$	133.034			0.317504			0.158752	−	0.987318i	$-0.449253\pi$
$419$	133.034			0.317504			0.158752	+	0.987318i	$0.449253\pi$
$420$		0			0
$421$		−	691.387i		−	1.64225i	−0.570749	−	0.821124i	$-0.693347\pi$
$421$		−	691.387i		−	1.64225i	0.570749	−	0.821124i	$-0.306653\pi$
$422$		0			0
$423$	−300.672	−	132.891i	−0.710810	−	0.314164i
$424$		0			0
$425$	169.418	−	221.133i	0.398631	−	0.520312i
$426$		0			0
$427$	−48.8710			−0.114452
$428$		0			0
$429$	31.3985	−	148.710i	0.0731900	−	0.346644i
$430$		0			0
$431$			467.222i			1.08404i	0.840365	+	0.542020i	$0.182340\pi$
$431$			467.222i			1.08404i	−0.840365	+	0.542020i	$0.817660\pi$
$432$		0			0
$433$		−	336.900i		−	0.778060i	−0.921225	−	0.389030i	$-0.872810\pi$
$433$		−	336.900i		−	0.778060i	0.921225	−	0.389030i	$-0.127190\pi$
$434$		0			0
$435$	80.1876	+	313.464i	0.184339	+	0.720607i
$436$		0			0
$437$		−	823.024i		−	1.88335i
$438$		0			0
$439$	370.466			0.843887			0.421944	−	0.906622i	$-0.361348\pi$
$439$	370.466			0.843887			0.421944	+	0.906622i	$0.361348\pi$
$440$		0			0
$441$	−226.729	−	100.210i	−0.514124	−	0.227233i
$442$		0			0
$443$			126.412i			0.285355i	0.989769	+	0.142677i	$0.0455711\pi$
$443$			126.412i			0.285355i	−0.989769	+	0.142677i	$0.954429\pi$
$444$		0			0
$445$	−39.4049	−	79.8299i	−0.0885503	−	0.179393i
$446$		0			0
$447$	128.347	−	607.880i	0.287130	−	1.35991i
$448$		0			0
$449$		−	372.521i		−	0.829667i	−0.909897	−	0.414834i	$-0.863840\pi$
$449$		−	372.521i		−	0.829667i	0.909897	−	0.414834i	$-0.136160\pi$
$450$		0			0
$451$			943.410i			2.09182i
$452$		0			0
$453$	−31.8705	+	150.946i	−0.0703544	+	0.333214i
$454$		0			0
$455$	−34.4372	−	69.7660i	−0.0756862	−	0.153332i
$456$		0			0
$457$			46.3235i			0.101364i	0.998715	+	0.0506822i	$0.0161396\pi$
$457$			46.3235i			0.101364i	−0.998715	+	0.0506822i	$0.983860\pi$
$458$		0			0
$459$	175.847	−	244.117i	0.383109	−	0.531844i
$460$		0			0
$461$	−523.953			−1.13656			−0.568279	−	0.822836i	$-0.692391\pi$
$461$	−523.953			−1.13656			−0.568279	+	0.822836i	$0.692391\pi$
$462$		0			0
$463$			233.896i			0.505175i	0.967574	+	0.252588i	$0.0812816\pi$
$463$			233.896i			0.505175i	−0.967574	+	0.252588i	$0.918718\pi$
$464$		0			0
$465$	−105.535	−	412.550i	−0.226957	−	0.887204i
$466$		0			0
$467$		−	310.079i		−	0.663981i	−0.943283	−	0.331990i	$-0.892280\pi$
$467$		−	310.079i		−	0.663981i	0.943283	−	0.331990i	$-0.107720\pi$
$468$		0			0
$469$			425.555i			0.907367i
$470$		0			0
$471$	−89.0680	+	421.846i	−0.189104	+	0.895639i
$472$		0			0
$473$	−832.610			−1.76027
$474$		0			0
$475$	451.732	+	346.089i	0.951014	+	0.728609i
$476$		0			0
$477$	106.555	−	241.085i	0.223386	−	0.505419i
$478$		0			0
$479$			618.890i			1.29205i	0.763318	+	0.646023i	$0.223569\pi$
$479$			618.890i			1.29205i	−0.763318	+	0.646023i	$0.776431\pi$
$480$		0			0
$481$	−233.665			−0.485789
$482$		0			0
$483$	491.612	+	103.798i	1.01783	+	0.214904i
$484$		0			0
$485$	308.217	+	624.413i	0.635499	+	1.28745i
$486$		0			0
$487$		−	884.108i		−	1.81542i	−0.419602	−	0.907708i	$-0.637830\pi$
$487$		−	884.108i		−	1.81542i	0.419602	−	0.907708i	$-0.362170\pi$
$488$		0			0
$489$	74.6201	−	353.418i	0.152597	−	0.722736i
$490$		0			0
$491$	−316.866			−0.645348			−0.322674	−	0.946510i	$-0.604582\pi$
$491$	−316.866			−0.645348			−0.322674	+	0.946510i	$0.604582\pi$
$492$		0			0
$493$	−240.357			−0.487540
$494$		0			0
$495$	−678.070	+	28.7390i	−1.36984	+	0.0580585i
$496$		0			0
$497$	−367.250			−0.738934
$498$		0			0
$499$		−	334.702i		−	0.670745i	−0.942086	−	0.335373i	$-0.891138\pi$
$499$		−	334.702i		−	0.670745i	0.942086	−	0.335373i	$-0.108862\pi$
$500$		0			0
$501$	−72.8155	+	344.870i	−0.145340	+	0.688364i
$502$		0			0
$503$	−16.1482			−0.0321037			−0.0160519	−	0.999871i	$-0.505110\pi$
$503$	−16.1482			−0.0321037			−0.0160519	+	0.999871i	$0.505110\pi$
$504$		0			0
$505$	−209.358	+	103.341i	−0.414570	+	0.204636i
$506$		0			0
$507$	97.7447	−	462.941i	0.192790	−	0.913098i
$508$		0			0
$509$	442.949			0.870233			0.435117	−	0.900374i	$-0.356707\pi$
$509$	442.949			0.870233			0.435117	+	0.900374i	$0.356707\pi$
$510$		0			0
$511$	273.695			0.535607
$512$		0			0
$513$	498.684	+	359.222i	0.972093	+	0.700239i
$514$		0			0
$515$	25.7366	+	52.1395i	0.0499740	+	0.101242i
$516$		0			0
$517$	−550.871			−1.06551
$518$		0			0
$519$	407.195	+	85.9746i	0.784576	+	0.165654i
$520$		0			0
$521$		−	601.758i		−	1.15501i	−0.816389	−	0.577503i	$-0.804027\pi$
$521$		−	601.758i		−	1.15501i	0.816389	−	0.577503i	$-0.195973\pi$
$522$		0			0
$523$	64.3965			0.123129			0.0615645	−	0.998103i	$-0.480391\pi$
$523$	64.3965			0.123129			0.0615645	+	0.998103i	$0.480391\pi$
$524$		0			0
$525$	−263.699	+	226.182i	−0.502284	+	0.430823i
$526$		0			0
$527$	316.334			0.600255
$528$		0			0
$529$	778.301			1.47127
$530$		0			0
$531$	−66.4699	−	29.3784i	−0.125179	−	0.0553266i
$532$		0			0
$533$		−	210.130i		−	0.394239i
$534$		0			0
$535$	212.016	+	429.521i	0.396292	+	0.802843i
$536$		0			0
$537$	149.536	−	708.234i	0.278465	−	1.31887i
$538$		0			0
$539$	−415.397			−0.770680
$540$		0			0
$541$		−	762.101i		−	1.40869i	−0.709858	−	0.704345i	$-0.751241\pi$
$541$		−	762.101i		−	1.40869i	0.709858	−	0.704345i	$-0.248759\pi$
$542$		0			0
$543$	324.073	+	68.4244i	0.596820	+	0.126012i
$544$		0			0
$545$	275.785	+	558.710i	0.506028	+	1.02516i
$546$		0			0
$547$	−238.920			−0.436783			−0.218391	−	0.975861i	$-0.570081\pi$
$547$	−238.920			−0.436783			−0.218391	+	0.975861i	$0.570081\pi$
$548$		0			0
$549$	38.3854	−	86.8485i	0.0699187	−	0.158194i
$550$		0			0
$551$		−	491.004i		−	0.891115i
$552$		0			0
$553$		−	399.846i		−	0.723049i
$554$		0			0
$555$	258.584	+	1010.84i	0.465917	+	1.82133i
$556$		0			0
$557$		−	525.434i		−	0.943329i	−0.881778	−	0.471664i	$-0.843653\pi$
$557$		−	525.434i		−	0.943329i	0.881778	−	0.471664i	$-0.156347\pi$
$558$		0			0
$559$	185.451			0.331754
$560$		0			0
$561$	104.152	−	493.286i	0.185654	−	0.879299i
$562$		0			0
$563$		−	465.978i		−	0.827669i	−0.910352	−	0.413835i	$-0.864189\pi$
$563$		−	465.978i		−	0.827669i	0.910352	−	0.413835i	$-0.135811\pi$
$564$		0			0
$565$	−41.6573	+	20.5625i	−0.0737297	+	0.0363938i
$566$		0			0
$567$	−277.465	+	252.571i	−0.489357	+	0.445452i
$568$		0			0
$569$		−	217.779i		−	0.382740i	−0.981518	−	0.191370i	$-0.938707\pi$
$569$		−	217.779i		−	0.382740i	0.981518	−	0.191370i	$-0.0612930\pi$
$570$		0			0
$571$			45.9447i			0.0804636i	0.999190	+	0.0402318i	$0.0128096\pi$
$571$			45.9447i			0.0804636i	−0.999190	+	0.0402318i	$0.987190\pi$
$572$		0			0
$573$	−350.673	−	74.0407i	−0.611995	−	0.129216i
$574$		0			0
$575$	−549.732	+	717.536i	−0.956056	+	1.24789i
$576$		0			0
$577$		−	810.365i		−	1.40444i	−0.711958	−	0.702222i	$-0.752191\pi$
$577$		−	810.365i		−	1.40444i	0.711958	−	0.702222i	$-0.247809\pi$
$578$		0			0
$579$	−437.031	−	92.2742i	−0.754804	−	0.159368i
$580$		0			0
$581$	46.4132			0.0798850
$582$		0			0
$583$		−	441.699i		−	0.757632i
$584$		0			0
$585$	151.029	−	6.40115i	0.258170	−	0.0109421i
$586$		0			0
$587$		−	627.562i		−	1.06910i	−0.845137	−	0.534550i	$-0.820481\pi$
$587$		−	627.562i		−	1.06910i	0.845137	−	0.534550i	$-0.179519\pi$
$588$		0			0
$589$			646.211i			1.09713i
$590$		0			0
$591$	814.406	+	171.953i	1.37801	+	0.290952i
$592$		0			0
$593$	913.788			1.54096			0.770479	−	0.637466i	$-0.220017\pi$
$593$	913.788			1.54096			0.770479	+	0.637466i	$0.220017\pi$
$594$		0			0
$595$	−114.232	−	231.421i	−0.191986	−	0.388942i
$596$		0			0
$597$	87.5244	−	414.535i	0.146607	−	0.694363i
$598$		0			0
$599$		−	556.341i		−	0.928782i	−0.885630	−	0.464391i	$-0.846273\pi$
$599$		−	556.341i		−	0.928782i	0.885630	−	0.464391i	$-0.153727\pi$
$600$		0			0
$601$	−265.244			−0.441338			−0.220669	−	0.975349i	$-0.570824\pi$
$601$	−265.244			−0.441338			−0.220669	+	0.975349i	$0.570824\pi$
$602$		0			0
$603$	−756.252	−	334.249i	−1.25415	−	0.554310i
$604$		0			0
$605$	−477.314	+	235.608i	−0.788949	+	0.389434i
$606$		0			0
$607$		−	496.303i		−	0.817633i	−0.912617	−	0.408816i	$-0.865942\pi$
$607$		−	496.303i		−	0.817633i	0.912617	−	0.408816i	$-0.134058\pi$
$608$		0			0
$609$	293.289	+	61.9246i	0.481591	+	0.101683i
$610$		0			0
$611$	122.698			0.200815
$612$		0			0
$613$	−335.215			−0.546843			−0.273421	−	0.961894i	$-0.588155\pi$
$613$	−335.215			−0.546843			−0.273421	+	0.961894i	$0.588155\pi$
$614$		0			0
$615$	−909.024	+	232.539i	−1.47809	+	0.378112i
$616$		0			0
$617$	−157.477			−0.255231			−0.127615	−	0.991824i	$-0.540732\pi$
$617$	−157.477			−0.255231			−0.127615	+	0.991824i	$0.540732\pi$
$618$		0			0
$619$			478.556i			0.773112i	0.922266	+	0.386556i	$0.126335\pi$
$619$			478.556i			0.773112i	−0.922266	+	0.386556i	$0.873665\pi$
$620$		0			0
$621$	−570.593	+	792.115i	−0.918829	+	1.27555i
$622$		0			0
$623$	−82.4763			−0.132386
$624$		0			0
$625$	−162.666	−	603.461i	−0.260265	−	0.965537i
$626$		0			0
$627$	1007.69	+	212.762i	1.60716	+	0.339334i
$628$		0			0
$629$	−775.088			−1.23226
$630$		0			0
$631$	−902.124			−1.42967			−0.714837	−	0.699291i	$-0.753499\pi$
$631$	−902.124			−1.42967			−0.714837	+	0.699291i	$0.753499\pi$
$632$		0			0
$633$	−320.437	−	67.6567i	−0.506220	−	0.106883i
$634$		0			0
$635$	371.553	+	752.725i	0.585123	+	1.18539i
$636$		0			0
$637$	92.5230			0.145248
$638$		0			0
$639$	288.454	−	652.639i	0.451415	−	1.02134i
$640$		0			0
$641$			354.712i			0.553373i	0.960960	+	0.276687i	$0.0892364\pi$
$641$			354.712i			0.553373i	−0.960960	+	0.276687i	$0.910764\pi$
$642$		0			0
$643$	1223.04			1.90208			0.951042	−	0.309062i	$-0.100015\pi$
$643$	1223.04			1.90208			0.951042	+	0.309062i	$0.100015\pi$
$644$		0			0
$645$	−205.228	−	802.263i	−0.318183	−	1.24382i
$646$		0			0
$647$	−82.6988			−0.127819			−0.0639094	−	0.997956i	$-0.520357\pi$
$647$	−82.6988			−0.127819			−0.0639094	+	0.997956i	$0.520357\pi$
$648$		0			0
$649$	−121.782			−0.187645
$650$		0			0
$651$	−385.998	−	81.4990i	−0.592930	−	0.125191i
$652$		0			0
$653$			605.030i			0.926539i	0.886217	+	0.463270i	$0.153324\pi$
$653$			605.030i			0.926539i	−0.886217	+	0.463270i	$0.846676\pi$
$654$		0			0
$655$	−28.1391	+	13.8898i	−0.0429604	+	0.0212057i
$656$		0			0
$657$	−214.972	+	486.383i	−0.327202	+	0.740309i
$658$		0			0
$659$	132.704			0.201372			0.100686	−	0.994918i	$-0.467896\pi$
$659$	132.704			0.201372			0.100686	+	0.994918i	$0.467896\pi$
$660$		0			0
$661$		−	412.823i		−	0.624544i	−0.949993	−	0.312272i	$-0.898910\pi$
$661$		−	412.823i		−	0.624544i	0.949993	−	0.312272i	$-0.101090\pi$
$662$		0			0
$663$	−23.1982	+	109.872i	−0.0349897	+	0.165719i
$664$		0			0
$665$	472.748	−	233.354i	0.710900	−	0.350908i
$666$		0			0
$667$	779.917			1.16929
$668$		0			0
$669$	−119.109	−	25.1486i	−0.178041	−	0.0375913i
$670$		0			0
$671$		−	159.118i		−	0.237135i
$672$		0			0
$673$		−	813.284i		−	1.20845i	−0.796816	−	0.604223i	$-0.793484\pi$
$673$		−	813.284i		−	1.20845i	0.796816	−	0.604223i	$-0.206516\pi$
$674$		0			0
$675$	−194.827	−	646.272i	−0.288633	−	0.957440i
$676$		0			0
$677$			671.082i			0.991258i	0.868534	+	0.495629i	$0.165062\pi$
$677$			671.082i			0.991258i	−0.868534	+	0.495629i	$0.834938\pi$
$678$		0			0
$679$	645.114			0.950094
$680$		0			0
$681$	1289.56	+	272.277i	1.89363	+	0.399819i
$682$		0			0
$683$			473.636i			0.693464i	0.937964	+	0.346732i	$0.112709\pi$
$683$			473.636i			0.693464i	−0.937964	+	0.346732i	$0.887291\pi$
$684$		0			0
$685$	452.553	−	223.385i	0.660662	−	0.326110i
$686$		0			0
$687$	735.964	+	155.391i	1.07127	+	0.226187i
$688$		0			0
$689$			98.3815i			0.142789i
$690$		0			0
$691$			1191.67i			1.72456i	0.506434	+	0.862279i	$0.330963\pi$
$691$			1191.67i			1.72456i	−0.506434	+	0.862279i	$0.669037\pi$
$692$		0			0
$693$	−254.176	+	575.085i	−0.366777	+	0.829848i
$694$		0			0
$695$	−403.588	−	817.623i	−0.580702	−	1.17644i
$696$		0			0
$697$		−	697.020i		−	1.00003i
$698$		0			0
$699$	−166.892	+	790.439i	−0.238759	+	1.13081i
$700$		0			0
$701$	837.233			1.19434			0.597171	−	0.802114i	$-0.296291\pi$
$701$	837.233			1.19434			0.597171	+	0.802114i	$0.296291\pi$
$702$		0			0
$703$		−	1583.36i		−	2.25229i
$704$		0			0
$705$	−135.783	−	530.793i	−0.192600	−	0.752898i
$706$		0			0
$707$			216.298i			0.305938i
$708$		0			0
$709$			119.868i			0.169067i	0.996421	+	0.0845334i	$0.0269400\pi$
$709$			119.868i			0.169067i	−0.996421	+	0.0845334i	$0.973060\pi$
$710$		0			0
$711$	710.565	+	314.056i	0.999388	+	0.441710i
$712$		0			0
$713$	−1026.45			−1.43962
$714$		0			0
$715$	227.149	−	112.123i	0.317691	−	0.156816i
$716$		0			0
$717$	−1127.64	−	238.089i	−1.57273	−	0.332063i
$718$		0			0
$719$		−	705.188i		−	0.980790i	−0.871500	−	0.490395i	$-0.836853\pi$
$719$		−	705.188i		−	0.980790i	0.871500	−	0.490395i	$-0.163147\pi$
$720$		0			0
$721$	53.8680			0.0747129
$722$		0			0
$723$	155.085	−	734.516i	0.214502	−	1.01593i
$724$		0			0
$725$	−327.962	+	428.072i	−0.452362	+	0.590444i
$726$		0			0
$727$			503.263i			0.692246i	0.938189	+	0.346123i	$0.112502\pi$
$727$			503.263i			0.692246i	−0.938189	+	0.346123i	$0.887498\pi$
$728$		0			0
$729$	−230.911	−	691.463i	−0.316750	−	0.948509i
$730$		0			0
$731$	615.158			0.841529
$732$		0			0
$733$	−48.5847			−0.0662820			−0.0331410	−	0.999451i	$-0.510551\pi$
$733$	−48.5847			−0.0662820			−0.0331410	+	0.999451i	$0.510551\pi$
$734$		0			0
$735$	−102.390	−	400.256i	−0.139306	−	0.544566i
$736$		0			0
$737$	−1385.55			−1.87999
$738$		0			0
$739$		−	281.321i		−	0.380678i	−0.981718	−	0.190339i	$-0.939041\pi$
$739$		−	281.321i		−	0.380678i	0.981718	−	0.190339i	$-0.0609587\pi$
$740$		0			0
$741$	−224.447	−	47.3894i	−0.302897	−	0.0639534i
$742$		0			0
$743$	237.515			0.319671			0.159835	−	0.987144i	$-0.448904\pi$
$743$	237.515			0.319671			0.159835	+	0.987144i	$0.448904\pi$
$744$		0			0
$745$	928.513	−	458.324i	1.24633	−	0.615200i
$746$		0			0
$747$	−36.4549	+	82.4807i	−0.0488017	+	0.110416i
$748$		0			0
$749$	443.760			0.592470
$750$		0			0
$751$	−1417.74			−1.88780			−0.943902	−	0.330225i	$-0.892875\pi$
$751$	−1417.74			−1.88780			−0.943902	+	0.330225i	$0.892875\pi$
$752$		0			0
$753$	197.929	−	937.436i	0.262854	−	1.24493i
$754$		0			0
$755$	−230.564	+	113.809i	−0.305383	+	0.150740i
$756$		0			0
$757$	−685.674			−0.905778			−0.452889	−	0.891567i	$-0.649607\pi$
$757$	−685.674			−0.905778			−0.452889	+	0.891567i	$0.649607\pi$
$758$		0			0
$759$	−337.954	+	1600.63i	−0.445263	+	2.10886i
$760$		0			0
$761$			1255.49i			1.64980i	0.565282	+	0.824898i	$0.308767\pi$
$761$			1255.49i			1.64980i	−0.565282	+	0.824898i	$0.691233\pi$
$762$		0			0
$763$	577.232			0.756529
$764$		0			0
$765$	500.979	−	21.2332i	0.654875	−	0.0277559i
$766$		0			0
$767$	27.1249			0.0353649
$768$		0			0
$769$	−111.595			−0.145117			−0.0725583	−	0.997364i	$-0.523116\pi$
$769$	−111.595			−0.145117			−0.0725583	+	0.997364i	$0.523116\pi$
$770$		0			0
$771$	69.6709	−	329.977i	0.0903643	−	0.427985i
$772$		0			0
$773$		−	496.659i		−	0.642509i	−0.946993	−	0.321254i	$-0.895896\pi$
$773$		−	496.659i		−	0.642509i	0.946993	−	0.321254i	$-0.104104\pi$
$774$		0			0
$775$	431.631	−	563.385i	0.556943	−	0.726949i
$776$		0			0
$777$	945.779	+	199.690i	1.21722	+	0.257002i
$778$		0			0
$779$	1423.88			1.82783
$780$		0			0
$781$		−	1195.72i		−	1.53101i
$782$		0			0
$783$	−340.408	+	472.564i	−0.434748	+	0.603531i
$784$		0			0
$785$	−644.353	+	318.060i	−0.820832	+	0.405172i
$786$		0			0
$787$	−122.847			−0.156096			−0.0780478	−	0.996950i	$-0.524869\pi$
$787$	−122.847			−0.156096			−0.0780478	+	0.996950i	$0.524869\pi$
$788$		0			0
$789$	40.7202	−	192.860i	0.0516099	−	0.244436i
$790$		0			0
$791$			43.0382i			0.0544099i
$792$		0			0
$793$			35.4410i			0.0446922i
$794$		0			0
$795$	425.600	−	108.873i	0.535346	−	0.136948i
$796$		0			0
$797$			159.677i			0.200348i	0.994970	+	0.100174i	$0.0319399\pi$
$797$			159.677i			0.200348i	−0.994970	+	0.100174i	$0.968060\pi$
$798$		0			0
$799$	407.001			0.509387
$800$		0			0
$801$	64.7804	−	146.568i	0.0808744	−	0.182982i
$802$		0			0
$803$			891.117i			1.10973i
$804$		0			0
$805$	370.662	+	750.919i	0.460449	+	0.932819i
$806$		0			0
$807$	100.760	−	477.220i	0.124857	−	0.591351i
$808$		0			0
$809$			1458.65i			1.80303i	0.432753	+	0.901513i	$0.357542\pi$
$809$			1458.65i			1.80303i	−0.432753	+	0.901513i	$0.642458\pi$
$810$		0			0
$811$		−	607.268i		−	0.748789i	−0.927270	−	0.374394i	$-0.877851\pi$
$811$		−	607.268i		−	0.748789i	0.927270	−	0.374394i	$-0.122149\pi$
$812$		0			0
$813$	217.707	−	1031.11i	0.267782	−	1.26828i
$814$		0			0
$815$	539.832	−	266.467i	0.662370	−	0.326953i
$816$		0			0
$817$			1256.65i			1.53813i
$818$		0			0
$819$	56.6137	−	128.091i	0.0691255	−	0.156399i
$820$		0			0
$821$	706.253			0.860236			0.430118	−	0.902773i	$-0.358472\pi$
$821$	706.253			0.860236			0.430118	+	0.902773i	$0.358472\pi$
$822$		0			0
$823$		−	101.874i		−	0.123784i	−0.998083	−	0.0618920i	$-0.980287\pi$
$823$		−	101.874i		−	0.123784i	0.998083	−	0.0618920i	$-0.0197134\pi$
$824$		0			0
$825$	−736.421	−	858.571i	−0.892631	−	1.04069i
$826$		0			0
$827$		−	565.412i		−	0.683691i	−0.939756	−	0.341845i	$-0.888948\pi$
$827$		−	565.412i		−	0.683691i	0.939756	−	0.341845i	$-0.111052\pi$
$828$		0			0
$829$		−	1107.57i		−	1.33603i	−0.744146	−	0.668017i	$-0.767143\pi$
$829$		−	1107.57i		−	1.33603i	0.744146	−	0.668017i	$-0.232857\pi$
$830$		0			0
$831$	292.724	−	1386.41i	0.352256	−	1.66836i
$832$		0			0
$833$	306.908			0.368437
$834$		0			0
$835$	−526.776	+	260.022i	−0.630870	+	0.311404i
$836$		0			0
$837$	448.010	−	621.942i	0.535257	−	0.743061i
$838$		0			0
$839$			220.914i			0.263307i	0.991296	+	0.131653i	$0.0420286\pi$
$839$			220.914i			0.263307i	−0.991296	+	0.131653i	$0.957971\pi$
$840$		0			0
$841$	−375.713			−0.446745
$842$		0			0
$843$	285.060	+	60.1871i	0.338149	+	0.0713964i
$844$		0			0
$845$	707.124	−	349.044i	0.836833	−	0.413070i
$846$		0			0
$847$			493.138i			0.582217i
$848$		0			0
$849$	158.758	−	751.914i	0.186994	−	0.885647i
$850$		0			0
$851$	2515.03			2.95538
$852$		0			0
$853$	666.300			0.781125			0.390562	−	0.920576i	$-0.372281\pi$
$853$	666.300			0.781125			0.390562	+	0.920576i	$0.372281\pi$
$854$		0			0
$855$	43.3755	+	1023.41i	0.0507315	+	1.19697i
$856$		0			0
$857$	696.570			0.812800			0.406400	−	0.913695i	$-0.366784\pi$
$857$	696.570			0.812800			0.406400	+	0.913695i	$0.366784\pi$
$858$		0			0
$859$		−	179.165i		−	0.208574i	−0.994547	−	0.104287i	$-0.966744\pi$
$859$		−	179.165i		−	0.208574i	0.994547	−	0.104287i	$-0.0332560\pi$
$860$		0			0
$861$	−179.577	+	850.518i	−0.208568	+	0.987826i
$862$		0			0
$863$	−216.695			−0.251095			−0.125548	−	0.992088i	$-0.540069\pi$
$863$	−216.695			−0.251095			−0.125548	+	0.992088i	$0.540069\pi$
$864$		0			0
$865$	307.013	+	621.974i	0.354929	+	0.719045i
$866$		0			0
$867$	102.158	−	483.843i	0.117829	−	0.558065i
$868$		0			0
$869$	1301.85			1.49810
$870$		0			0
$871$	308.610			0.354317
$872$		0			0
$873$	−506.699	+	1146.43i	−0.580412	+	1.31321i
$874$		0			0
$875$	−568.022	−	112.324i	−0.649168	−	0.128370i
$876$		0			0
$877$	−364.319			−0.415415			−0.207707	−	0.978191i	$-0.566600\pi$
$877$	−364.319			−0.415415			−0.207707	+	0.978191i	$0.566600\pi$
$878$		0			0
$879$	−1566.76	−	330.803i	−1.78243	−	0.376340i
$880$		0			0
$881$		−	77.6990i		−	0.0881941i	−0.999027	−	0.0440970i	$-0.985959\pi$
$881$		−	77.6990i		−	0.0881941i	0.999027	−	0.0440970i	$-0.0140411\pi$
$882$		0			0
$883$	−1139.19			−1.29014			−0.645068	−	0.764125i	$-0.723171\pi$
$883$	−1139.19			−1.29014			−0.645068	+	0.764125i	$0.723171\pi$
$884$		0			0
$885$	−30.0176	−	117.343i	−0.0339182	−	0.132591i
$886$		0			0
$887$	862.886			0.972814			0.486407	−	0.873732i	$-0.338307\pi$
$887$	862.886			0.972814			0.486407	+	0.873732i	$0.338307\pi$
$888$		0			0
$889$	777.679			0.874780
$890$		0			0
$891$	−822.340	−	903.392i	−0.922941	−	1.01391i
$892$		0			0
$893$			831.425i			0.931047i
$894$		0			0
$895$	1081.80	−	533.988i	1.20871	−	0.596635i
$896$		0			0
$897$	75.2740	−	356.514i	0.0839174	−	0.397452i
$898$		0			0
$899$	−612.365			−0.681162
$900$		0			0
$901$			326.341i			0.362199i
$902$		0			0
$903$	−750.628	−	158.487i	−0.831260	−	0.175511i
$904$		0			0
$905$	244.342	+	495.010i	0.269991	+	0.546972i
$906$		0			0
$907$	−22.0873			−0.0243520			−0.0121760	−	0.999926i	$-0.503876\pi$
$907$	−22.0873			−0.0243520			−0.0121760	+	0.999926i	$0.503876\pi$
$908$		0			0
$909$	−384.383	−	169.890i	−0.422864	−	0.186898i
$910$		0			0
$911$			565.531i			0.620781i	0.950609	+	0.310390i	$0.100460\pi$
$911$			565.531i			0.620781i	−0.950609	+	0.310390i	$0.899540\pi$
$912$		0			0
$913$			151.115i			0.165515i
$914$		0			0
$915$	153.318	−	39.2205i	0.167561	−	0.0428640i
$916$		0			0
$917$			29.0719i			0.0317033i
$918$		0			0
$919$	−105.968			−0.115308			−0.0576539	−	0.998337i	$-0.518362\pi$
$919$	−105.968			−0.115308			−0.0576539	+	0.998337i	$0.518362\pi$
$920$		0			0
$921$	76.3248	−	361.491i	0.0828716	−	0.392499i
$922$		0			0
$923$			266.328i			0.288546i
$924$		0			0
$925$	−1057.59	+	1380.42i	−1.14334	+	1.49234i
$926$		0			0
$927$	−42.3102	+	95.7287i	−0.0456421	+	0.103267i
$928$		0			0
$929$			339.002i			0.364911i	0.983214	+	0.182455i	$0.0584045\pi$
$929$			339.002i			0.364911i	−0.983214	+	0.182455i	$0.941595\pi$
$930$		0			0
$931$			626.954i			0.673420i
$932$		0			0
$933$	779.196	+	164.518i	0.835151	+	0.176333i
$934$		0			0
$935$	753.476	−	371.924i	0.805857	−	0.397780i
$936$		0			0
$937$		−	216.693i		−	0.231263i	−0.993292	−	0.115631i	$-0.963111\pi$
$937$		−	216.693i		−	0.231263i	0.993292	−	0.115631i	$-0.0368891\pi$
$938$		0			0
$939$	−1093.13	−	230.802i	−1.16414	−	0.245795i
$940$		0			0
$941$	575.135			0.611196			0.305598	−	0.952161i	$-0.401144\pi$
$941$	575.135			0.611196			0.305598	+	0.952161i	$0.401144\pi$
$942$		0			0
$943$			2261.71i			2.39842i
$944$		0			0
$945$	−616.775	−	103.161i	−0.652672	−	0.109165i
$946$		0			0
$947$			881.538i			0.930875i	0.885081	+	0.465437i	$0.154103\pi$
$947$			881.538i			0.930875i	−0.885081	+	0.465437i	$0.845897\pi$
$948$		0			0
$949$		−	198.482i		−	0.209149i
$950$		0			0
$951$	1719.43	+	363.039i	1.80803	+	0.381744i
$952$		0			0
$953$	−563.303			−0.591084			−0.295542	−	0.955330i	$-0.595500\pi$
$953$	−563.303			−0.591084			−0.295542	+	0.955330i	$0.595500\pi$
$954$		0			0
$955$	−264.397	−	535.639i	−0.276856	−	0.560879i
$956$		0			0
$957$	−201.619	+	954.911i	−0.210678	+	0.997818i
$958$		0			0
$959$		−	467.556i		−	0.487545i
$960$		0			0
$961$	−155.067			−0.161360
$962$		0			0
$963$	−348.548	+	788.604i	−0.361940	+	0.818904i
$964$		0			0
$965$	−329.509	−	667.548i	−0.341460	−	0.691760i
$966$		0			0
$967$			607.946i			0.628693i	0.949308	+	0.314347i	$0.101785\pi$
$967$			607.946i			0.628693i	−0.949308	+	0.314347i	$0.898215\pi$
$968$		0			0
$969$	−744.513	−	157.195i	−0.768331	−	0.162224i
$970$		0			0
$971$	−219.086			−0.225629			−0.112815	−	0.993616i	$-0.535987\pi$
$971$	−219.086			−0.225629			−0.112815	+	0.993616i	$0.535987\pi$
$972$		0			0
$973$	−844.728			−0.868169
$974$		0			0
$975$	164.026	+	191.233i	0.168232	+	0.196136i
$976$		0			0
$977$	−905.682			−0.927003			−0.463501	−	0.886096i	$-0.653407\pi$
$977$	−905.682			−0.927003			−0.463501	+	0.886096i	$0.653407\pi$
$978$		0			0
$979$		−	268.532i		−	0.274293i
$980$		0			0
$981$	−453.382	+	1025.80i	−0.462163	+	1.04566i
$982$		0			0
$983$	30.0755			0.0305956			0.0152978	−	0.999883i	$-0.495130\pi$
$983$	30.0755			0.0305956			0.0152978	+	0.999883i	$0.495130\pi$
$984$		0			0
$985$	614.039	+	1243.97i	0.623390	+	1.26292i
$986$		0			0
$987$	−496.630	−	104.858i	−0.503172	−	0.106239i
$988$		0			0
$989$	−1996.08			−2.01828
$990$		0			0
$991$	−379.273			−0.382717			−0.191358	−	0.981520i	$-0.561289\pi$
$991$	−379.273			−0.382717			−0.191358	+	0.981520i	$0.561289\pi$
$992$		0			0
$993$	1413.60	+	298.465i	1.42356	+	0.300569i
$994$		0			0
$995$	633.186	−	312.547i	0.636368	−	0.314118i
$996$		0			0
$997$	1676.89			1.68194			0.840970	−	0.541082i	$-0.181985\pi$
$997$	1676.89			1.68194			0.840970	+	0.541082i	$0.181985\pi$
$998$		0			0
$999$	−1097.72	+	1523.89i	−1.09882	+	1.52542i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	960.3.i.b.929.30	yes	64
3.2	odd	2	inner	960.3.i.b.929.27	yes	64
4.3	odd	2	inner	960.3.i.b.929.34	yes	64
5.4	even	2	inner	960.3.i.b.929.33	yes	64
8.3	odd	2	inner	960.3.i.b.929.31	yes	64
8.5	even	2	inner	960.3.i.b.929.35	yes	64
12.11	even	2	inner	960.3.i.b.929.39	yes	64
15.14	odd	2	inner	960.3.i.b.929.40	yes	64
20.19	odd	2	inner	960.3.i.b.929.29	yes	64
24.5	odd	2	inner	960.3.i.b.929.38	yes	64
24.11	even	2	inner	960.3.i.b.929.26	yes	64
40.19	odd	2	inner	960.3.i.b.929.36	yes	64
40.29	even	2	inner	960.3.i.b.929.32	yes	64
60.59	even	2	inner	960.3.i.b.929.28	yes	64
120.29	odd	2	inner	960.3.i.b.929.25	✓	64
120.59	even	2	inner	960.3.i.b.929.37	yes	64

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
960.3.i.b.929.25	✓	64	120.29	odd	2	inner
960.3.i.b.929.26	yes	64	24.11	even	2	inner
960.3.i.b.929.27	yes	64	3.2	odd	2	inner
960.3.i.b.929.28	yes	64	60.59	even	2	inner
960.3.i.b.929.29	yes	64	20.19	odd	2	inner
960.3.i.b.929.30	yes	64	1.1	even	1	trivial
960.3.i.b.929.31	yes	64	8.3	odd	2	inner
960.3.i.b.929.32	yes	64	40.29	even	2	inner
960.3.i.b.929.33	yes	64	5.4	even	2	inner
960.3.i.b.929.34	yes	64	4.3	odd	2	inner
960.3.i.b.929.35	yes	64	8.5	even	2	inner
960.3.i.b.929.36	yes	64	40.19	odd	2	inner
960.3.i.b.929.37	yes	64	120.59	even	2	inner
960.3.i.b.929.38	yes	64	24.5	odd	2	inner
960.3.i.b.929.39	yes	64	12.11	even	2	inner
960.3.i.b.929.40	yes	64	15.14	odd	2	inner

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.i (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.619753 + 2.93529i) q^{3} +(-4.48354 + 2.21312i) q^{5} +4.63217i q^{7} +(-8.23181 - 3.63830i) q^{9} +O(q^{10})\)	\(q+(-0.619753 + 2.93529i) q^{3} +(-4.48354 + 2.21312i) q^{5} +4.63217i q^{7} +(-8.23181 - 3.63830i) q^{9} -15.0818 q^{11} +3.35922 q^{13} +(-3.71746 - 14.5320i) q^{15} +11.1429 q^{17} +22.7628i q^{19} +(-13.5967 - 2.87080i) q^{21} -36.1566 q^{23} +(15.2042 - 19.8452i) q^{25} +(15.7811 - 21.9079i) q^{27} -21.5705 q^{29} +28.3890 q^{31} +(9.34696 - 44.2693i) q^{33} +(-10.2516 - 20.7685i) q^{35} -69.5592 q^{37} +(-2.08189 + 9.86027i) q^{39} -62.5531i q^{41} +55.2064 q^{43} +(44.9596 - 1.90555i) q^{45} +36.5257 q^{47} +27.5430 q^{49} +(-6.90582 + 32.7075i) q^{51} +29.2870i q^{53} +(67.6196 - 33.3778i) q^{55} +(-66.8152 - 14.1073i) q^{57} +8.07476 q^{59} +10.5504i q^{61} +(16.8532 - 38.1312i) q^{63} +(-15.0612 + 7.43436i) q^{65} +91.8695 q^{67} +(22.4082 - 106.130i) q^{69} +79.2826i q^{71} -59.0857i q^{73} +(48.8286 + 56.9278i) q^{75} -69.8613i q^{77} -86.3194 q^{79} +(54.5255 + 59.8997i) q^{81} -10.0197i q^{83} +(-49.9594 + 24.6605i) q^{85} +(13.3684 - 63.3157i) q^{87} +17.8051i q^{89} +15.5605i q^{91} +(-17.5941 + 83.3297i) q^{93} +(-50.3767 - 102.058i) q^{95} -139.268i q^{97} +(124.150 + 54.8720i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 32 q^{9}+O(q^{10}) \) 64 * q - 32 * q^9	\( 64 q - 32 q^{9} - 32 q^{25} - 320 q^{49} + 448 q^{81}+O(q^{100}) \) 64 * q - 32 * q^9 - 32 * q^25 - 320 * q^49 + 448 * q^81

Introduction

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