LMFDB - Embedded newform 9680.2.a.by.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9680,2,Mod(1,9680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9680, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("9680.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9680 = 2^{4} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9680.a (trivial)

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:	yes
Analytic conductor:	$77.2951891566$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.788.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x - 3 $ x^3 - x^2 - 7*x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4840)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.476452$ of defining polynomial
Character	$\chi$	$=$	9680.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.476452 q^{3} -1.00000 q^{5} -3.34364 q^{7} -2.77299 q^{9} +O(q^{10})$	$q+0.476452 q^{3} -1.00000 q^{5} -3.34364 q^{7} -2.77299 q^{9} -3.82009 q^{13} -0.476452 q^{15} -6.59308 q^{17} -1.82009 q^{19} -1.59308 q^{21} -2.77299 q^{23} +1.00000 q^{25} -2.75055 q^{27} -0.952904 q^{29} +4.17991 q^{31} +3.34364 q^{35} -2.95290 q^{37} -1.82009 q^{39} +4.82009 q^{41} -9.93672 q^{43} +2.77299 q^{45} -3.16373 q^{47} +4.17991 q^{49} -3.14129 q^{51} +7.46027 q^{53} -0.867185 q^{57} -2.77299 q^{59} -11.7730 q^{61} +9.27188 q^{63} +3.82009 q^{65} -7.42936 q^{67} -1.32120 q^{69} -14.0534 q^{71} -7.64018 q^{73} +0.476452 q^{75} -8.59308 q^{79} +7.00847 q^{81} -4.86719 q^{83} +6.59308 q^{85} -0.454013 q^{87} -1.59308 q^{89} +12.7730 q^{91} +1.99153 q^{93} +1.82009 q^{95} -3.82009 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{3} - 3 q^{5} - 3 q^{7} + 6 q^{9}+O(q^{10}) $ 3 * q - q^3 - 3 * q^5 - 3 * q^7 + 6 * q^9	$ 3 q - q^{3} - 3 q^{5} - 3 q^{7} + 6 q^{9} - 2 q^{13} + q^{15} + 4 q^{17} + 4 q^{19} + 19 q^{21} + 6 q^{23} + 3 q^{25} - 25 q^{27} + 2 q^{29} + 22 q^{31} + 3 q^{35} - 4 q^{37} + 4 q^{39} + 5 q^{41} + q^{43} - 6 q^{45} + 7 q^{47} + 22 q^{49} - 24 q^{51} - 6 q^{53} + 2 q^{57} + 6 q^{59} - 21 q^{61} - 20 q^{63} + 2 q^{65} - 15 q^{67} - 28 q^{69} + 10 q^{71} - 4 q^{73} - q^{75} - 2 q^{79} + 31 q^{81} - 10 q^{83} - 4 q^{85} - 30 q^{87} + 19 q^{89} + 24 q^{91} - 4 q^{93} - 4 q^{95} - 2 q^{97}+O(q^{100}) $ 3 * q - q^3 - 3 * q^5 - 3 * q^7 + 6 * q^9 - 2 * q^13 + q^15 + 4 * q^17 + 4 * q^19 + 19 * q^21 + 6 * q^23 + 3 * q^25 - 25 * q^27 + 2 * q^29 + 22 * q^31 + 3 * q^35 - 4 * q^37 + 4 * q^39 + 5 * q^41 + q^43 - 6 * q^45 + 7 * q^47 + 22 * q^49 - 24 * q^51 - 6 * q^53 + 2 * q^57 + 6 * q^59 - 21 * q^61 - 20 * q^63 + 2 * q^65 - 15 * q^67 - 28 * q^69 + 10 * q^71 - 4 * q^73 - q^75 - 2 * q^79 + 31 * q^81 - 10 * q^83 - 4 * q^85 - 30 * q^87 + 19 * q^89 + 24 * q^91 - 4 * q^93 - 4 * q^95 - 2 * q^97

Display 100 coefficients 10 coefficients

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
$3$	0.476452	0.275080	0.137540	−	0.990496i	$-0.456080\pi$
$3$	0.476452	0.275080	0.137540	+	0.990496i	$0.456080\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−3.34364	−1.26378	−0.631888	−	0.775060i	$-0.717720\pi$
$7$	−3.34364	−1.26378	−0.631888	+	0.775060i	$0.717720\pi$
$8$	0	0
$9$	−2.77299	−0.924331
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−3.82009	−1.05950	−0.529751	−	0.848153i	$-0.677715\pi$
$13$	−3.82009	−1.05950	−0.529751	+	0.848153i	$0.677715\pi$
$14$	0	0
$15$	−0.476452	−0.123019
$16$	0	0
$17$	−6.59308	−1.59906	−0.799529	−	0.600628i	$-0.794917\pi$
$17$	−6.59308	−1.59906	−0.799529	+	0.600628i	$0.794917\pi$
$18$	0	0
$19$	−1.82009	−0.417557	−0.208779	−	0.977963i	$-0.566949\pi$
$19$	−1.82009	−0.417557	−0.208779	+	0.977963i	$0.566949\pi$
$20$	0	0
$21$	−1.59308	−0.347639
$22$	0	0
$23$	−2.77299	−0.578209	−0.289105	−	0.957298i	$-0.593358\pi$
$23$	−2.77299	−0.578209	−0.289105	+	0.957298i	$0.593358\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−2.75055	−0.529344
$28$	0	0
$29$	−0.952904	−0.176950	−0.0884749	−	0.996078i	$-0.528199\pi$
$29$	−0.952904	−0.176950	−0.0884749	+	0.996078i	$0.528199\pi$
$30$	0	0
$31$	4.17991	0.750734	0.375367	−	0.926876i	$-0.377517\pi$
$31$	4.17991	0.750734	0.375367	+	0.926876i	$0.377517\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.34364	0.565178
$36$	0	0
$37$	−2.95290	−0.485454	−0.242727	−	0.970095i	$-0.578042\pi$
$37$	−2.95290	−0.485454	−0.242727	+	0.970095i	$0.578042\pi$
$38$	0	0
$39$	−1.82009	−0.291448
$40$	0	0
$41$	4.82009	0.752771	0.376386	−	0.926463i	$-0.377167\pi$
$41$	4.82009	0.752771	0.376386	+	0.926463i	$0.377167\pi$
$42$	0	0
$43$	−9.93672	−1.51534	−0.757668	−	0.652640i	$-0.773661\pi$
$43$	−9.93672	−1.51534	−0.757668	+	0.652640i	$0.773661\pi$
$44$	0	0
$45$	2.77299	0.413373
$46$	0	0
$47$	−3.16373	−0.461477	−0.230738	−	0.973016i	$-0.574114\pi$
$47$	−3.16373	−0.461477	−0.230738	+	0.973016i	$0.574114\pi$
$48$	0	0
$49$	4.17991	0.597130
$50$	0	0
$51$	−3.14129	−0.439868
$52$	0	0
$53$	7.46027	1.02475	0.512373	−	0.858763i	$-0.328766\pi$
$53$	7.46027	1.02475	0.512373	+	0.858763i	$0.328766\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−0.867185	−0.114861
$58$	0	0
$59$	−2.77299	−0.361013	−0.180506	−	0.983574i	$-0.557774\pi$
$59$	−2.77299	−0.361013	−0.180506	+	0.983574i	$0.557774\pi$
$60$	0	0
$61$	−11.7730	−1.50738	−0.753689	−	0.657232i	$-0.771727\pi$
$61$	−11.7730	−1.50738	−0.753689	+	0.657232i	$0.771727\pi$
$62$	0	0
$63$	9.27188	1.16815
$64$	0	0
$65$	3.82009	0.473824
$66$	0	0
$67$	−7.42936	−0.907640	−0.453820	−	0.891093i	$-0.649939\pi$
$67$	−7.42936	−0.907640	−0.453820	+	0.891093i	$0.649939\pi$
$68$	0	0
$69$	−1.32120	−0.159054
$70$	0	0
$71$	−14.0534	−1.66783	−0.833913	−	0.551896i	$-0.813905\pi$
$71$	−14.0534	−1.66783	−0.833913	+	0.551896i	$0.813905\pi$
$72$	0	0
$73$	−7.64018	−0.894215	−0.447108	−	0.894480i	$-0.647546\pi$
$73$	−7.64018	−0.894215	−0.447108	+	0.894480i	$0.647546\pi$
$74$	0	0
$75$	0.476452	0.0550159
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−8.59308	−0.966797	−0.483399	−	0.875400i	$-0.660598\pi$
$79$	−8.59308	−0.966797	−0.483399	+	0.875400i	$0.660598\pi$
$80$	0	0
$81$	7.00847	0.778719
$82$	0	0
$83$	−4.86719	−0.534243	−0.267121	−	0.963663i	$-0.586073\pi$
$83$	−4.86719	−0.534243	−0.267121	+	0.963663i	$0.586073\pi$
$84$	0	0
$85$	6.59308	0.715120
$86$	0	0
$87$	−0.454013	−0.0486753
$88$	0	0
$89$	−1.59308	−0.168866	−0.0844332	−	0.996429i	$-0.526908\pi$
$89$	−1.59308	−0.168866	−0.0844332	+	0.996429i	$0.526908\pi$
$90$	0	0
$91$	12.7730	1.33897
$92$	0	0
$93$	1.99153	0.206512
$94$	0	0
$95$	1.82009	0.186737
$96$	0	0
$97$	−3.82009	−0.387871	−0.193936	−	0.981014i	$-0.562125\pi$
$97$	−3.82009	−0.387871	−0.193936	+	0.981014i	$0.562125\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	15.3275	1.52514	0.762569	−	0.646907i	$-0.223938\pi$
$101$	15.3275	1.52514	0.762569	+	0.646907i	$0.223938\pi$
$102$	0	0
$103$	18.0534	1.77885	0.889425	−	0.457082i	$-0.151105\pi$
$103$	18.0534	1.77885	0.889425	+	0.457082i	$0.151105\pi$
$104$	0	0
$105$	1.59308	0.155469
$106$	0	0
$107$	−10.1166	−0.978012	−0.489006	−	0.872281i	$-0.662640\pi$
$107$	−10.1166	−0.978012	−0.489006	+	0.872281i	$0.662640\pi$
$108$	0	0
$109$	17.6788	1.69332	0.846661	−	0.532133i	$-0.178609\pi$
$109$	17.6788	1.69332	0.846661	+	0.532133i	$0.178609\pi$
$110$	0	0
$111$	−1.40692	−0.133539
$112$	0	0
$113$	−15.3745	−1.44632	−0.723158	−	0.690683i	$-0.757310\pi$
$113$	−15.3745	−1.44632	−0.723158	+	0.690683i	$0.757310\pi$
$114$	0	0
$115$	2.77299	0.258583
$116$	0	0
$117$	10.5931	0.979331
$118$	0	0
$119$	22.0449	2.02085
$120$	0	0
$121$	0	0
$122$	0	0
$123$	2.29654	0.207072
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−15.0695	−1.33720	−0.668602	−	0.743620i	$-0.733107\pi$
$127$	−15.0695	−1.33720	−0.668602	+	0.743620i	$0.733107\pi$
$128$	0	0
$129$	−4.73437	−0.416838
$130$	0	0
$131$	11.5545	1.00952	0.504759	−	0.863260i	$-0.331581\pi$
$131$	11.5545	1.00952	0.504759	+	0.863260i	$0.331581\pi$
$132$	0	0
$133$	6.08572	0.527699
$134$	0	0
$135$	2.75055	0.236730
$136$	0	0
$137$	20.1475	1.72132	0.860660	−	0.509179i	$-0.170051\pi$
$137$	20.1475	1.72132	0.860660	+	0.509179i	$0.170051\pi$
$138$	0	0
$139$	−2.27410	−0.192887	−0.0964434	−	0.995338i	$-0.530747\pi$
$139$	−2.27410	−0.192887	−0.0964434	+	0.995338i	$0.530747\pi$
$140$	0	0
$141$	−1.50736	−0.126943
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0.952904	0.0791344
$146$	0	0
$147$	1.99153	0.164258
$148$	0	0
$149$	8.46027	0.693092	0.346546	−	0.938033i	$-0.387354\pi$
$149$	8.46027	0.693092	0.346546	+	0.938033i	$0.387354\pi$
$150$	0	0
$151$	16.1475	1.31407	0.657034	−	0.753861i	$-0.271811\pi$
$151$	16.1475	1.31407	0.657034	+	0.753861i	$0.271811\pi$
$152$	0	0
$153$	18.2826	1.47806
$154$	0	0
$155$	−4.17991	−0.335739
$156$	0	0
$157$	−1.73437	−0.138418	−0.0692089	−	0.997602i	$-0.522048\pi$
$157$	−1.73437	−0.138418	−0.0692089	+	0.997602i	$0.522048\pi$
$158$	0	0
$159$	3.55446	0.281887
$160$	0	0
$161$	9.27188	0.730727
$162$	0	0
$163$	10.6564	0.834671	0.417335	−	0.908753i	$-0.362964\pi$
$163$	10.6564	0.834671	0.417335	+	0.908753i	$0.362964\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−21.7483	−1.68294	−0.841468	−	0.540306i	$-0.818308\pi$
$167$	−21.7483	−1.68294	−0.841468	+	0.540306i	$0.818308\pi$
$168$	0	0
$169$	1.59308	0.122545
$170$	0	0
$171$	5.04710	0.385961
$172$	0	0
$173$	5.91428	0.449654	0.224827	−	0.974399i	$-0.427818\pi$
$173$	5.91428	0.449654	0.224827	+	0.974399i	$0.427818\pi$
$174$	0	0
$175$	−3.34364	−0.252755
$176$	0	0
$177$	−1.32120	−0.0993074
$178$	0	0
$179$	13.6402	1.01951	0.509757	−	0.860318i	$-0.329735\pi$
$179$	13.6402	1.01951	0.509757	+	0.860318i	$0.329735\pi$
$180$	0	0
$181$	−12.8734	−0.956875	−0.478438	−	0.878122i	$-0.658797\pi$
$181$	−12.8734	−0.956875	−0.478438	+	0.878122i	$0.658797\pi$
$182$	0	0
$183$	−5.60927	−0.414649
$184$	0	0
$185$	2.95290	0.217102
$186$	0	0
$187$	0	0
$188$	0	0
$189$	9.19686	0.668973
$190$	0	0
$191$	−12.4132	−0.898186	−0.449093	−	0.893485i	$-0.648253\pi$
$191$	−12.4132	−0.898186	−0.449093	+	0.893485i	$0.648253\pi$
$192$	0	0
$193$	−12.9529	−0.932370	−0.466185	−	0.884687i	$-0.654372\pi$
$193$	−12.9529	−0.932370	−0.466185	+	0.884687i	$0.654372\pi$
$194$	0	0
$195$	1.82009	0.130339
$196$	0	0
$197$	10.8587	0.773651	0.386826	−	0.922153i	$-0.373572\pi$
$197$	10.8587	0.773651	0.386826	+	0.922153i	$0.373572\pi$
$198$	0	0
$199$	−3.73437	−0.264723	−0.132361	−	0.991202i	$-0.542256\pi$
$199$	−3.73437	−0.264723	−0.132361	+	0.991202i	$0.542256\pi$
$200$	0	0
$201$	−3.53973	−0.249673
$202$	0	0
$203$	3.18617	0.223625
$204$	0	0
$205$	−4.82009	−0.336650
$206$	0	0
$207$	7.68949	0.534457
$208$	0	0
$209$	0	0
$210$	0	0
$211$	13.7259	0.944930	0.472465	−	0.881349i	$-0.343364\pi$
$211$	13.7259	0.944930	0.472465	+	0.881349i	$0.343364\pi$
$212$	0	0
$213$	−6.69575	−0.458785
$214$	0	0
$215$	9.93672	0.677679
$216$	0	0
$217$	−13.9761	−0.948760
$218$	0	0
$219$	−3.64018	−0.245980
$220$	0	0
$221$	25.1862	1.69420
$222$	0	0
$223$	23.2494	1.55690	0.778449	−	0.627708i	$-0.216007\pi$
$223$	23.2494	1.55690	0.778449	+	0.627708i	$0.216007\pi$
$224$	0	0
$225$	−2.77299	−0.184866
$226$	0	0
$227$	−9.01618	−0.598425	−0.299213	−	0.954186i	$-0.596724\pi$
$227$	−9.01618	−0.598425	−0.299213	+	0.954186i	$0.596724\pi$
$228$	0	0
$229$	7.68727	0.507989	0.253995	−	0.967206i	$-0.418255\pi$
$229$	7.68727	0.507989	0.253995	+	0.967206i	$0.418255\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.94665	0.127529	0.0637646	−	0.997965i	$-0.479689\pi$
$233$	1.94665	0.127529	0.0637646	+	0.997965i	$0.479689\pi$
$234$	0	0
$235$	3.16373	0.206379
$236$	0	0
$237$	−4.09419	−0.265946
$238$	0	0
$239$	−24.7406	−1.60034	−0.800169	−	0.599775i	$-0.795257\pi$
$239$	−24.7406	−1.60034	−0.800169	+	0.599775i	$0.795257\pi$
$240$	0	0
$241$	18.0063	1.15988	0.579942	−	0.814657i	$-0.303075\pi$
$241$	18.0063	1.15988	0.579942	+	0.814657i	$0.303075\pi$
$242$	0	0
$243$	11.5909	0.743554
$244$	0	0
$245$	−4.17991	−0.267045
$246$	0	0
$247$	6.95290	0.442403
$248$	0	0
$249$	−2.31898	−0.146959
$250$	0	0
$251$	−4.86719	−0.307214	−0.153607	−	0.988132i	$-0.549089\pi$
$251$	−4.86719	−0.307214	−0.153607	+	0.988132i	$0.549089\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	3.14129	0.196715
$256$	0	0
$257$	−13.8201	−0.862073	−0.431037	−	0.902334i	$-0.641852\pi$
$257$	−13.8201	−0.862073	−0.431037	+	0.902334i	$0.641852\pi$
$258$	0	0
$259$	9.87344	0.613506
$260$	0	0
$261$	2.64240	0.163560
$262$	0	0
$263$	−0.539732	−0.0332813	−0.0166406	−	0.999862i	$-0.505297\pi$
$263$	−0.539732	−0.0332813	−0.0166406	+	0.999862i	$0.505297\pi$
$264$	0	0
$265$	−7.46027	−0.458281
$266$	0	0
$267$	−0.759028	−0.0464517
$268$	0	0
$269$	19.1475	1.16745	0.583723	−	0.811953i	$-0.301595\pi$
$269$	19.1475	1.16745	0.583723	+	0.811953i	$0.301595\pi$
$270$	0	0
$271$	−8.32745	−0.505857	−0.252928	−	0.967485i	$-0.581394\pi$
$271$	−8.32745	−0.505857	−0.252928	+	0.967485i	$0.581394\pi$
$272$	0	0
$273$	6.08572	0.368324
$274$	0	0
$275$	0	0
$276$	0	0
$277$	24.9529	1.49927	0.749637	−	0.661849i	$-0.230228\pi$
$277$	24.9529	1.49927	0.749637	+	0.661849i	$0.230228\pi$
$278$	0	0
$279$	−11.5909	−0.693927
$280$	0	0
$281$	−13.8116	−0.823932	−0.411966	−	0.911199i	$-0.635158\pi$
$281$	−13.8116	−0.823932	−0.411966	+	0.911199i	$0.635158\pi$
$282$	0	0
$283$	−30.3499	−1.80411	−0.902057	−	0.431617i	$-0.857943\pi$
$283$	−30.3499	−1.80411	−0.902057	+	0.431617i	$0.857943\pi$
$284$	0	0
$285$	0.867185	0.0513676
$286$	0	0
$287$	−16.1166	−0.951335
$288$	0	0
$289$	26.4687	1.55698
$290$	0	0
$291$	−1.82009	−0.106696
$292$	0	0
$293$	1.49264	0.0872007	0.0436004	−	0.999049i	$-0.486117\pi$
$293$	1.49264	0.0872007	0.0436004	+	0.999049i	$0.486117\pi$
$294$	0	0
$295$	2.77299	0.161450
$296$	0	0
$297$	0	0
$298$	0	0
$299$	10.5931	0.612614
$300$	0	0
$301$	33.2248	1.91504
$302$	0	0
$303$	7.30280	0.419535
$304$	0	0
$305$	11.7730	0.674120
$306$	0	0
$307$	−16.8672	−0.962661	−0.481331	−	0.876539i	$-0.659846\pi$
$307$	−16.8672	−0.962661	−0.481331	+	0.876539i	$0.659846\pi$
$308$	0	0
$309$	8.60156	0.489325
$310$	0	0
$311$	31.3661	1.77861	0.889304	−	0.457317i	$-0.151190\pi$
$311$	31.3661	1.77861	0.889304	+	0.457317i	$0.151190\pi$
$312$	0	0
$313$	33.5051	1.89382	0.946911	−	0.321495i	$-0.104185\pi$
$313$	33.5051	1.89382	0.946911	+	0.321495i	$0.104185\pi$
$314$	0	0
$315$	−9.27188	−0.522412
$316$	0	0
$317$	27.1862	1.52693	0.763464	−	0.645851i	$-0.223497\pi$
$317$	27.1862	1.52693	0.763464	+	0.645851i	$0.223497\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−4.82009	−0.269031
$322$	0	0
$323$	12.0000	0.667698
$324$	0	0
$325$	−3.82009	−0.211900
$326$	0	0
$327$	8.42310	0.465799
$328$	0	0
$329$	10.5784	0.583204
$330$	0	0
$331$	−33.7877	−1.85714	−0.928571	−	0.371156i	$-0.878962\pi$
$331$	−33.7877	−1.85714	−0.928571	+	0.371156i	$0.878962\pi$
$332$	0	0
$333$	8.18838	0.448721
$334$	0	0
$335$	7.42936	0.405909
$336$	0	0
$337$	7.63171	0.415725	0.207863	−	0.978158i	$-0.433349\pi$
$337$	7.63171	0.415725	0.207863	+	0.978158i	$0.433349\pi$
$338$	0	0
$339$	−7.32524	−0.397852
$340$	0	0
$341$	0	0
$342$	0	0
$343$	9.42936	0.509137
$344$	0	0
$345$	1.32120	0.0711309
$346$	0	0
$347$	−35.7483	−1.91907	−0.959536	−	0.281587i	$-0.909139\pi$
$347$	−35.7483	−1.91907	−0.959536	+	0.281587i	$0.909139\pi$
$348$	0	0
$349$	−13.3127	−0.712614	−0.356307	−	0.934369i	$-0.615964\pi$
$349$	−13.3127	−0.712614	−0.356307	+	0.934369i	$0.615964\pi$
$350$	0	0
$351$	10.5074	0.560842
$352$	0	0
$353$	0.601556	0.0320176	0.0160088	−	0.999872i	$-0.494904\pi$
$353$	0.601556	0.0320176	0.0160088	+	0.999872i	$0.494904\pi$
$354$	0	0
$355$	14.0534	0.745874
$356$	0	0
$357$	10.5033	0.555895
$358$	0	0
$359$	−16.5846	−0.875302	−0.437651	−	0.899145i	$-0.644189\pi$
$359$	−16.5846	−0.875302	−0.437651	+	0.899145i	$0.644189\pi$
$360$	0	0
$361$	−15.6873	−0.825646
$362$	0	0
$363$	0	0
$364$	0	0
$365$	7.64018	0.399905
$366$	0	0
$367$	−1.39699	−0.0729222	−0.0364611	−	0.999335i	$-0.511609\pi$
$367$	−1.39699	−0.0729222	−0.0364611	+	0.999335i	$0.511609\pi$
$368$	0	0
$369$	−13.3661	−0.695810
$370$	0	0
$371$	−24.9444	−1.29505
$372$	0	0
$373$	−4.60156	−0.238260	−0.119130	−	0.992879i	$-0.538010\pi$
$373$	−4.60156	−0.238260	−0.119130	+	0.992879i	$0.538010\pi$
$374$	0	0
$375$	−0.476452	−0.0246039
$376$	0	0
$377$	3.64018	0.187479
$378$	0	0
$379$	−33.9507	−1.74393	−0.871965	−	0.489569i	$-0.837154\pi$
$379$	−33.9507	−1.74393	−0.871965	+	0.489569i	$0.837154\pi$
$380$	0	0
$381$	−7.17991	−0.367838
$382$	0	0
$383$	18.9121	0.966361	0.483181	−	0.875521i	$-0.339481\pi$
$383$	18.9121	0.966361	0.483181	+	0.875521i	$0.339481\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	27.5545	1.40067
$388$	0	0
$389$	−21.2866	−1.07927	−0.539637	−	0.841898i	$-0.681439\pi$
$389$	−21.2866	−1.07927	−0.539637	+	0.841898i	$0.681439\pi$
$390$	0	0
$391$	18.2826	0.924590
$392$	0	0
$393$	5.50515	0.277698
$394$	0	0
$395$	8.59308	0.432365
$396$	0	0
$397$	−17.7344	−0.890063	−0.445031	−	0.895515i	$-0.646807\pi$
$397$	−17.7344	−0.890063	−0.445031	+	0.895515i	$0.646807\pi$
$398$	0	0
$399$	2.89955	0.145159
$400$	0	0
$401$	−12.8116	−0.639782	−0.319891	−	0.947454i	$-0.603646\pi$
$401$	−12.8116	−0.639782	−0.319891	+	0.947454i	$0.603646\pi$
$402$	0	0
$403$	−15.9676	−0.795404
$404$	0	0
$405$	−7.00847	−0.348254
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−16.4603	−0.813908	−0.406954	−	0.913449i	$-0.633409\pi$
$409$	−16.4603	−0.813908	−0.406954	+	0.913449i	$0.633409\pi$
$410$	0	0
$411$	9.59934	0.473500
$412$	0	0
$413$	9.27188	0.456240
$414$	0	0
$415$	4.86719	0.238921
$416$	0	0
$417$	−1.08350	−0.0530593
$418$	0	0
$419$	−1.32120	−0.0645448	−0.0322724	−	0.999479i	$-0.510274\pi$
$419$	−1.32120	−0.0645448	−0.0322724	+	0.999479i	$0.510274\pi$
$420$	0	0
$421$	15.7406	0.767151	0.383576	−	0.923509i	$-0.374693\pi$
$421$	15.7406	0.767151	0.383576	+	0.923509i	$0.374693\pi$
$422$	0	0
$423$	8.77299	0.426558
$424$	0	0
$425$	−6.59308	−0.319811
$426$	0	0
$427$	39.3646	1.90499
$428$	0	0
$429$	0	0
$430$	0	0
$431$	21.7792	1.04907	0.524535	−	0.851389i	$-0.324239\pi$
$431$	21.7792	1.04907	0.524535	+	0.851389i	$0.324239\pi$
$432$	0	0
$433$	17.6317	0.847326	0.423663	−	0.905820i	$-0.360744\pi$
$433$	17.6317	0.847326	0.423663	+	0.905820i	$0.360744\pi$
$434$	0	0
$435$	0.454013	0.0217683
$436$	0	0
$437$	5.04710	0.241435
$438$	0	0
$439$	10.4047	0.496589	0.248295	−	0.968685i	$-0.420130\pi$
$439$	10.4047	0.496589	0.248295	+	0.968685i	$0.420130\pi$
$440$	0	0
$441$	−11.5909	−0.551946
$442$	0	0
$443$	−25.4827	−1.21072	−0.605360	−	0.795952i	$-0.706971\pi$
$443$	−25.4827	−1.21072	−0.605360	+	0.795952i	$0.706971\pi$
$444$	0	0
$445$	1.59308	0.0755194
$446$	0	0
$447$	4.03091	0.190656
$448$	0	0
$449$	19.9614	0.942036	0.471018	−	0.882124i	$-0.343887\pi$
$449$	19.9614	0.942036	0.471018	+	0.882124i	$0.343887\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	7.69353	0.361474
$454$	0	0
$455$	−12.7730	−0.598807
$456$	0	0
$457$	12.9121	0.604001	0.302000	−	0.953308i	$-0.402346\pi$
$457$	12.9121	0.604001	0.302000	+	0.953308i	$0.402346\pi$
$458$	0	0
$459$	18.1346	0.846452
$460$	0	0
$461$	13.5931	0.633093	0.316546	−	0.948577i	$-0.397477\pi$
$461$	13.5931	0.633093	0.316546	+	0.948577i	$0.397477\pi$
$462$	0	0
$463$	7.96909	0.370355	0.185177	−	0.982705i	$-0.440714\pi$
$463$	7.96909	0.370355	0.185177	+	0.982705i	$0.440714\pi$
$464$	0	0
$465$	−1.99153	−0.0923549
$466$	0	0
$467$	−12.8039	−0.592494	−0.296247	−	0.955111i	$-0.595735\pi$
$467$	−12.8039	−0.592494	−0.296247	+	0.955111i	$0.595735\pi$
$468$	0	0
$469$	24.8411	1.14705
$470$	0	0
$471$	−0.826344	−0.0380759
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−1.82009	−0.0835114
$476$	0	0
$477$	−20.6873	−0.947205
$478$	0	0
$479$	−33.4194	−1.52697	−0.763486	−	0.645824i	$-0.776514\pi$
$479$	−33.4194	−1.52697	−0.763486	+	0.645824i	$0.776514\pi$
$480$	0	0
$481$	11.2804	0.514340
$482$	0	0
$483$	4.41761	0.201008
$484$	0	0
$485$	3.82009	0.173461
$486$	0	0
$487$	8.69575	0.394042	0.197021	−	0.980399i	$-0.436873\pi$
$487$	8.69575	0.394042	0.197021	+	0.980399i	$0.436873\pi$
$488$	0	0
$489$	5.07725	0.229601
$490$	0	0
$491$	11.5136	0.519602	0.259801	−	0.965662i	$-0.416343\pi$
$491$	11.5136	0.519602	0.259801	+	0.965662i	$0.416343\pi$
$492$	0	0
$493$	6.28258	0.282953
$494$	0	0
$495$	0	0
$496$	0	0
$497$	46.9893	2.10776
$498$	0	0
$499$	38.6549	1.73043	0.865216	−	0.501400i	$-0.167181\pi$
$499$	38.6549	1.73043	0.865216	+	0.501400i	$0.167181\pi$
$500$	0	0
$501$	−10.3620	−0.462942
$502$	0	0
$503$	8.99229	0.400946	0.200473	−	0.979699i	$-0.435752\pi$
$503$	8.99229	0.400946	0.200473	+	0.979699i	$0.435752\pi$
$504$	0	0
$505$	−15.3275	−0.682063
$506$	0	0
$507$	0.759028	0.0337096
$508$	0	0
$509$	−22.1538	−0.981950	−0.490975	−	0.871174i	$-0.663359\pi$
$509$	−22.1538	−0.981950	−0.490975	+	0.871174i	$0.663359\pi$
$510$	0	0
$511$	25.5460	1.13009
$512$	0	0
$513$	5.00626	0.221032
$514$	0	0
$515$	−18.0534	−0.795526
$516$	0	0
$517$	0	0
$518$	0	0
$519$	2.81787	0.123691
$520$	0	0
$521$	31.1391	1.36423	0.682114	−	0.731246i	$-0.261061\pi$
$521$	31.1391	1.36423	0.682114	+	0.731246i	$0.261061\pi$
$522$	0	0
$523$	−1.36608	−0.0597343	−0.0298672	−	0.999554i	$-0.509508\pi$
$523$	−1.36608	−0.0597343	−0.0298672	+	0.999554i	$0.509508\pi$
$524$	0	0
$525$	−1.59308	−0.0695278
$526$	0	0
$527$	−27.5585	−1.20047
$528$	0	0
$529$	−15.3105	−0.665674
$530$	0	0
$531$	7.68949	0.333696
$532$	0	0
$533$	−18.4132	−0.797563
$534$	0	0
$535$	10.1166	0.437380
$536$	0	0
$537$	6.49889	0.280448
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−22.3745	−0.961957	−0.480979	−	0.876732i	$-0.659718\pi$
$541$	−22.3745	−0.961957	−0.480979	+	0.876732i	$0.659718\pi$
$542$	0	0
$543$	−6.13358	−0.263217
$544$	0	0
$545$	−17.6788	−0.757277
$546$	0	0
$547$	16.3359	0.698474	0.349237	−	0.937034i	$-0.386441\pi$
$547$	16.3359	0.698474	0.349237	+	0.937034i	$0.386441\pi$
$548$	0	0
$549$	32.6464	1.39332
$550$	0	0
$551$	1.73437	0.0738867
$552$	0	0
$553$	28.7322	1.22182
$554$	0	0
$555$	1.40692	0.0597203
$556$	0	0
$557$	−12.3275	−0.522331	−0.261165	−	0.965294i	$-0.584107\pi$
$557$	−12.3275	−0.522331	−0.261165	+	0.965294i	$0.584107\pi$
$558$	0	0
$559$	37.9592	1.60550
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−7.85947	−0.331237	−0.165619	−	0.986190i	$-0.552962\pi$
$563$	−7.85947	−0.331237	−0.165619	+	0.986190i	$0.552962\pi$
$564$	0	0
$565$	15.3745	0.646812
$566$	0	0
$567$	−23.4338	−0.984127
$568$	0	0
$569$	−21.1475	−0.886551	−0.443276	−	0.896385i	$-0.646184\pi$
$569$	−21.1475	−0.886551	−0.443276	+	0.896385i	$0.646184\pi$
$570$	0	0
$571$	−4.78147	−0.200098	−0.100049	−	0.994983i	$-0.531900\pi$
$571$	−4.78147	−0.200098	−0.100049	+	0.994983i	$0.531900\pi$
$572$	0	0
$573$	−5.91428	−0.247073
$574$	0	0
$575$	−2.77299	−0.115642
$576$	0	0
$577$	4.30647	0.179281	0.0896404	−	0.995974i	$-0.471428\pi$
$577$	4.30647	0.179281	0.0896404	+	0.995974i	$0.471428\pi$
$578$	0	0
$579$	−6.17144	−0.256476
$580$	0	0
$581$	16.2741	0.675164
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−10.5931	−0.437970
$586$	0	0
$587$	16.2965	0.672630	0.336315	−	0.941750i	$-0.390819\pi$
$587$	16.2965	0.672630	0.336315	+	0.941750i	$0.390819\pi$
$588$	0	0
$589$	−7.60781	−0.313474
$590$	0	0
$591$	5.17366	0.212816
$592$	0	0
$593$	3.10045	0.127320	0.0636600	−	0.997972i	$-0.479723\pi$
$593$	3.10045	0.127320	0.0636600	+	0.997972i	$0.479723\pi$
$594$	0	0
$595$	−22.0449	−0.903752
$596$	0	0
$597$	−1.77925	−0.0728198
$598$	0	0
$599$	13.2888	0.542967	0.271483	−	0.962443i	$-0.412486\pi$
$599$	13.2888	0.542967	0.271483	+	0.962443i	$0.412486\pi$
$600$	0	0
$601$	−31.1538	−1.27079	−0.635395	−	0.772187i	$-0.719163\pi$
$601$	−31.1538	−1.27079	−0.635395	+	0.772187i	$0.719163\pi$
$602$	0	0
$603$	20.6016	0.838960
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−16.8348	−0.683304	−0.341652	−	0.939827i	$-0.610986\pi$
$607$	−16.8348	−0.683304	−0.341652	+	0.939827i	$0.610986\pi$
$608$	0	0
$609$	1.51806	0.0615147
$610$	0	0
$611$	12.0857	0.488936
$612$	0	0
$613$	−31.6851	−1.27975	−0.639874	−	0.768480i	$-0.721013\pi$
$613$	−31.6851	−1.27975	−0.639874	+	0.768480i	$0.721013\pi$
$614$	0	0
$615$	−2.29654	−0.0926055
$616$	0	0
$617$	−35.8650	−1.44387	−0.721935	−	0.691961i	$-0.756747\pi$
$617$	−35.8650	−1.44387	−0.721935	+	0.691961i	$0.756747\pi$
$618$	0	0
$619$	−48.6380	−1.95492	−0.977462	−	0.211110i	$-0.932292\pi$
$619$	−48.6380	−1.95492	−0.977462	+	0.211110i	$0.932292\pi$
$620$	0	0
$621$	7.62727	0.306072
$622$	0	0
$623$	5.32669	0.213409
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	19.4687	0.776270
$630$	0	0
$631$	3.10892	0.123764	0.0618821	−	0.998083i	$-0.480290\pi$
$631$	3.10892	0.123764	0.0618821	+	0.998083i	$0.480290\pi$
$632$	0	0
$633$	6.53973	0.259931
$634$	0	0
$635$	15.0695	0.598016
$636$	0	0
$637$	−15.9676	−0.632661
$638$	0	0
$639$	38.9699	1.54162
$640$	0	0
$641$	−38.2951	−1.51256	−0.756282	−	0.654245i	$-0.772986\pi$
$641$	−38.2951	−1.51256	−0.756282	+	0.654245i	$0.772986\pi$
$642$	0	0
$643$	−24.1251	−0.951401	−0.475701	−	0.879607i	$-0.657805\pi$
$643$	−24.1251	−0.951401	−0.475701	+	0.879607i	$0.657805\pi$
$644$	0	0
$645$	4.73437	0.186416
$646$	0	0
$647$	−35.7160	−1.40414	−0.702070	−	0.712108i	$-0.747741\pi$
$647$	−35.7160	−1.40414	−0.702070	+	0.712108i	$0.747741\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−6.65894	−0.260985
$652$	0	0
$653$	33.0596	1.29372	0.646861	−	0.762608i	$-0.276081\pi$
$653$	33.0596	1.29372	0.646861	+	0.762608i	$0.276081\pi$
$654$	0	0
$655$	−11.5545	−0.451470
$656$	0	0
$657$	21.1862	0.826551
$658$	0	0
$659$	35.6851	1.39009	0.695046	−	0.718965i	$-0.255384\pi$
$659$	35.6851	1.39009	0.695046	+	0.718965i	$0.255384\pi$
$660$	0	0
$661$	−33.2094	−1.29169	−0.645847	−	0.763467i	$-0.723496\pi$
$661$	−33.2094	−1.29169	−0.645847	+	0.763467i	$0.723496\pi$
$662$	0	0
$663$	12.0000	0.466041
$664$	0	0
$665$	−6.08572	−0.235994
$666$	0	0
$667$	2.64240	0.102314
$668$	0	0
$669$	11.0772	0.428271
$670$	0	0
$671$	0	0
$672$	0	0
$673$	34.4581	1.32826	0.664130	−	0.747617i	$-0.268802\pi$
$673$	34.4581	1.32826	0.664130	+	0.747617i	$0.268802\pi$
$674$	0	0
$675$	−2.75055	−0.105869
$676$	0	0
$677$	−12.4665	−0.479127	−0.239564	−	0.970881i	$-0.577004\pi$
$677$	−12.4665	−0.479127	−0.239564	+	0.970881i	$0.577004\pi$
$678$	0	0
$679$	12.7730	0.490183
$680$	0	0
$681$	−4.29578	−0.164615
$682$	0	0
$683$	−16.4680	−0.630130	−0.315065	−	0.949070i	$-0.602026\pi$
$683$	−16.4680	−0.630130	−0.315065	+	0.949070i	$0.602026\pi$
$684$	0	0
$685$	−20.1475	−0.769798
$686$	0	0
$687$	3.66262	0.139738
$688$	0	0
$689$	−28.4989	−1.08572
$690$	0	0
$691$	−12.9614	−0.493074	−0.246537	−	0.969133i	$-0.579293\pi$
$691$	−12.9614	−0.493074	−0.246537	+	0.969133i	$0.579293\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2.27410	0.0862616
$696$	0	0
$697$	−31.7792	−1.20372
$698$	0	0
$699$	0.927485	0.0350807
$700$	0	0
$701$	25.0596	0.946488	0.473244	−	0.880931i	$-0.343083\pi$
$701$	25.0596	0.946488	0.473244	+	0.880931i	$0.343083\pi$
$702$	0	0
$703$	5.37455	0.202705
$704$	0	0
$705$	1.50736	0.0567706
$706$	0	0
$707$	−51.2494	−1.92743
$708$	0	0
$709$	−7.07725	−0.265792	−0.132896	−	0.991130i	$-0.542428\pi$
$709$	−7.07725	−0.265792	−0.132896	+	0.991130i	$0.542428\pi$
$710$	0	0
$711$	23.8286	0.893641
$712$	0	0
$713$	−11.5909	−0.434081
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−11.7877	−0.440221
$718$	0	0
$719$	21.9592	0.818938	0.409469	−	0.912324i	$-0.365714\pi$
$719$	21.9592	0.818938	0.409469	+	0.912324i	$0.365714\pi$
$720$	0	0
$721$	−60.3639	−2.24807
$722$	0	0
$723$	8.57912	0.319061
$724$	0	0
$725$	−0.952904	−0.0353900
$726$	0	0
$727$	25.9044	0.960739	0.480370	−	0.877066i	$-0.340503\pi$
$727$	25.9044	0.960739	0.480370	+	0.877066i	$0.340503\pi$
$728$	0	0
$729$	−15.5029	−0.574183
$730$	0	0
$731$	65.5136	2.42311
$732$	0	0
$733$	−23.5909	−0.871348	−0.435674	−	0.900104i	$-0.643490\pi$
$733$	−23.5909	−0.871348	−0.435674	+	0.900104i	$0.643490\pi$
$734$	0	0
$735$	−1.99153	−0.0734586
$736$	0	0
$737$	0	0
$738$	0	0
$739$	9.50111	0.349504	0.174752	−	0.984612i	$-0.444088\pi$
$739$	9.50111	0.349504	0.174752	+	0.984612i	$0.444088\pi$
$740$	0	0
$741$	3.31273	0.121696
$742$	0	0
$743$	−35.9901	−1.32035	−0.660174	−	0.751113i	$-0.729517\pi$
$743$	−35.9901	−1.32035	−0.660174	+	0.751113i	$0.729517\pi$
$744$	0	0
$745$	−8.46027	−0.309960
$746$	0	0
$747$	13.4967	0.493817
$748$	0	0
$749$	33.8263	1.23599
$750$	0	0
$751$	50.4665	1.84155	0.920775	−	0.390095i	$-0.127558\pi$
$751$	50.4665	1.84155	0.920775	+	0.390095i	$0.127558\pi$
$752$	0	0
$753$	−2.31898	−0.0845083
$754$	0	0
$755$	−16.1475	−0.587669
$756$	0	0
$757$	−26.5846	−0.966234	−0.483117	−	0.875556i	$-0.660495\pi$
$757$	−26.5846	−0.966234	−0.483117	+	0.875556i	$0.660495\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−15.3745	−0.557327	−0.278663	−	0.960389i	$-0.589891\pi$
$761$	−15.3745	−0.557327	−0.278663	+	0.960389i	$0.589891\pi$
$762$	0	0
$763$	−59.1115	−2.13998
$764$	0	0
$765$	−18.2826	−0.661008
$766$	0	0
$767$	10.5931	0.382494
$768$	0	0
$769$	−37.4364	−1.34999	−0.674995	−	0.737822i	$-0.735854\pi$
$769$	−37.4364	−1.34999	−0.674995	+	0.737822i	$0.735854\pi$
$770$	0	0
$771$	−6.58461	−0.237139
$772$	0	0
$773$	37.0511	1.33264	0.666318	−	0.745667i	$-0.267869\pi$
$773$	37.0511	1.33264	0.666318	+	0.745667i	$0.267869\pi$
$774$	0	0
$775$	4.17991	0.150147
$776$	0	0
$777$	4.70422	0.168763
$778$	0	0
$779$	−8.77299	−0.314325
$780$	0	0
$781$	0	0
$782$	0	0
$783$	2.62101	0.0936674
$784$	0	0
$785$	1.73437	0.0619023
$786$	0	0
$787$	−43.6626	−1.55640	−0.778202	−	0.628014i	$-0.783868\pi$
$787$	−43.6626	−1.55640	−0.778202	+	0.628014i	$0.783868\pi$
$788$	0	0
$789$	−0.257156	−0.00915501
$790$	0	0
$791$	51.4069	1.82782
$792$	0	0
$793$	44.9739	1.59707
$794$	0	0
$795$	−3.55446	−0.126064
$796$	0	0
$797$	−23.0386	−0.816070	−0.408035	−	0.912966i	$-0.633786\pi$
$797$	−23.0386	−0.816070	−0.408035	+	0.912966i	$0.633786\pi$
$798$	0	0
$799$	20.8587	0.737928
$800$	0	0
$801$	4.41761	0.156089
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−9.27188	−0.326791
$806$	0	0
$807$	9.12289	0.321141
$808$	0	0
$809$	50.3275	1.76942	0.884710	−	0.466143i	$-0.154357\pi$
$809$	50.3275	1.76942	0.884710	+	0.466143i	$0.154357\pi$
$810$	0	0
$811$	−55.9268	−1.96386	−0.981928	−	0.189257i	$-0.939392\pi$
$811$	−55.9268	−1.96386	−0.981928	+	0.189257i	$0.939392\pi$
$812$	0	0
$813$	−3.96763	−0.139151
$814$	0	0
$815$	−10.6564	−0.373276
$816$	0	0
$817$	18.0857	0.632739
$818$	0	0
$819$	−35.4194	−1.23765
$820$	0	0
$821$	−34.3661	−1.19938	−0.599692	−	0.800231i	$-0.704710\pi$
$821$	−34.3661	−1.19938	−0.599692	+	0.800231i	$0.704710\pi$
$822$	0	0
$823$	−16.5832	−0.578052	−0.289026	−	0.957321i	$-0.593331\pi$
$823$	−16.5832	−0.578052	−0.289026	+	0.957321i	$0.593331\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	37.7693	1.31337	0.656684	−	0.754166i	$-0.271959\pi$
$827$	37.7693	1.31337	0.656684	+	0.754166i	$0.271959\pi$
$828$	0	0
$829$	27.8348	0.966743	0.483372	−	0.875415i	$-0.339412\pi$
$829$	27.8348	0.966743	0.483372	+	0.875415i	$0.339412\pi$
$830$	0	0
$831$	11.8889	0.412420
$832$	0	0
$833$	−27.5585	−0.954845
$834$	0	0
$835$	21.7483	0.752632
$836$	0	0
$837$	−11.4971	−0.397397
$838$	0	0
$839$	22.2951	0.769712	0.384856	−	0.922977i	$-0.374251\pi$
$839$	22.2951	0.769712	0.384856	+	0.922977i	$0.374251\pi$
$840$	0	0
$841$	−28.0920	−0.968689
$842$	0	0
$843$	−6.58057	−0.226647
$844$	0	0
$845$	−1.59308	−0.0548037
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−14.4603	−0.496275
$850$	0	0
$851$	8.18838	0.280694
$852$	0	0
$853$	17.3745	0.594893	0.297447	−	0.954738i	$-0.403865\pi$
$853$	17.3745	0.594893	0.297447	+	0.954738i	$0.403865\pi$
$854$	0	0
$855$	−5.04710	−0.172607
$856$	0	0
$857$	−2.39219	−0.0817156	−0.0408578	−	0.999165i	$-0.513009\pi$
$857$	−2.39219	−0.0817156	−0.0408578	+	0.999165i	$0.513009\pi$
$858$	0	0
$859$	46.1475	1.57453	0.787267	−	0.616612i	$-0.211495\pi$
$859$	46.1475	1.57453	0.787267	+	0.616612i	$0.211495\pi$
$860$	0	0
$861$	−7.67880	−0.261693
$862$	0	0
$863$	19.1961	0.653443	0.326721	−	0.945121i	$-0.394056\pi$
$863$	19.1961	0.653443	0.326721	+	0.945121i	$0.394056\pi$
$864$	0	0
$865$	−5.91428	−0.201092
$866$	0	0
$867$	12.6111	0.428295
$868$	0	0
$869$	0	0
$870$	0	0
$871$	28.3808	0.961647
$872$	0	0
$873$	10.5931	0.358522
$874$	0	0
$875$	3.34364	0.113036
$876$	0	0
$877$	−5.30425	−0.179112	−0.0895559	−	0.995982i	$-0.528545\pi$
$877$	−5.30425	−0.179112	−0.0895559	+	0.995982i	$0.528545\pi$
$878$	0	0
$879$	0.711169	0.0239872
$880$	0	0
$881$	38.0681	1.28255	0.641273	−	0.767313i	$-0.278407\pi$
$881$	38.0681	1.28255	0.641273	+	0.767313i	$0.278407\pi$
$882$	0	0
$883$	−24.6788	−0.830508	−0.415254	−	0.909706i	$-0.636307\pi$
$883$	−24.6788	−0.830508	−0.415254	+	0.909706i	$0.636307\pi$
$884$	0	0
$885$	1.32120	0.0444116
$886$	0	0
$887$	−10.8124	−0.363044	−0.181522	−	0.983387i	$-0.558102\pi$
$887$	−10.8124	−0.363044	−0.181522	+	0.983387i	$0.558102\pi$
$888$	0	0
$889$	50.3871	1.68993
$890$	0	0
$891$	0	0
$892$	0	0
$893$	5.75827	0.192693
$894$	0	0
$895$	−13.6402	−0.455941
$896$	0	0
$897$	5.04710	0.168518
$898$	0	0
$899$	−3.98305	−0.132842
$900$	0	0
$901$	−49.1862	−1.63863
$902$	0	0
$903$	15.8300	0.526790
$904$	0	0
$905$	12.8734	0.427928
$906$	0	0
$907$	−8.56217	−0.284302	−0.142151	−	0.989845i	$-0.545402\pi$
$907$	−8.56217	−0.284302	−0.142151	+	0.989845i	$0.545402\pi$
$908$	0	0
$909$	−42.5029	−1.40973
$910$	0	0
$911$	15.8734	0.525911	0.262955	−	0.964808i	$-0.415303\pi$
$911$	15.8734	0.525911	0.262955	+	0.964808i	$0.415303\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	5.60927	0.185437
$916$	0	0
$917$	−38.6339	−1.27580
$918$	0	0
$919$	−11.0386	−0.364131	−0.182065	−	0.983286i	$-0.558278\pi$
$919$	−11.0386	−0.364131	−0.182065	+	0.983286i	$0.558278\pi$
$920$	0	0
$921$	−8.03640	−0.264809
$922$	0	0
$923$	53.6851	1.76707
$924$	0	0
$925$	−2.95290	−0.0970909
$926$	0	0
$927$	−50.0618	−1.64425
$928$	0	0
$929$	−56.2009	−1.84389	−0.921946	−	0.387319i	$-0.873401\pi$
$929$	−56.2009	−1.84389	−0.921946	+	0.387319i	$0.873401\pi$
$930$	0	0
$931$	−7.60781	−0.249336
$932$	0	0
$933$	14.9444	0.489259
$934$	0	0
$935$	0	0
$936$	0	0
$937$	25.0596	0.818662	0.409331	−	0.912386i	$-0.365762\pi$
$937$	25.0596	0.818662	0.409331	+	0.912386i	$0.365762\pi$
$938$	0	0
$939$	15.9636	0.520952
$940$	0	0
$941$	−5.01695	−0.163548	−0.0817739	−	0.996651i	$-0.526059\pi$
$941$	−5.01695	−0.163548	−0.0817739	+	0.996651i	$0.526059\pi$
$942$	0	0
$943$	−13.3661	−0.435259
$944$	0	0
$945$	−9.19686	−0.299174
$946$	0	0
$947$	−35.4603	−1.15230	−0.576152	−	0.817343i	$-0.695446\pi$
$947$	−35.4603	−1.15230	−0.576152	+	0.817343i	$0.695446\pi$
$948$	0	0
$949$	29.1862	0.947423
$950$	0	0
$951$	12.9529	0.420027
$952$	0	0
$953$	42.5283	1.37763	0.688814	−	0.724938i	$-0.258132\pi$
$953$	42.5283	1.37763	0.688814	+	0.724938i	$0.258132\pi$
$954$	0	0
$955$	12.4132	0.401681
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−67.3661	−2.17536
$960$	0	0
$961$	−13.5283	−0.436398
$962$	0	0
$963$	28.0534	0.904007
$964$	0	0
$965$	12.9529	0.416969
$966$	0	0
$967$	−24.6788	−0.793617	−0.396808	−	0.917901i	$-0.629882\pi$
$967$	−24.6788	−0.793617	−0.396808	+	0.917901i	$0.629882\pi$
$968$	0	0
$969$	5.71742	0.183670
$970$	0	0
$971$	−47.2480	−1.51626	−0.758130	−	0.652103i	$-0.773887\pi$
$971$	−47.2480	−1.51626	−0.758130	+	0.652103i	$0.773887\pi$
$972$	0	0
$973$	7.60377	0.243766
$974$	0	0
$975$	−1.82009	−0.0582895
$976$	0	0
$977$	25.8734	0.827765	0.413882	−	0.910330i	$-0.364172\pi$
$977$	25.8734	0.827765	0.413882	+	0.910330i	$0.364172\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−49.0232	−1.56519
$982$	0	0
$983$	42.3948	1.35218	0.676092	−	0.736818i	$-0.263672\pi$
$983$	42.3948	1.35218	0.676092	+	0.736818i	$0.263672\pi$
$984$	0	0
$985$	−10.8587	−0.345987
$986$	0	0
$987$	5.04008	0.160427
$988$	0	0
$989$	27.5545	0.876181
$990$	0	0
$991$	30.0982	0.956102	0.478051	−	0.878332i	$-0.341343\pi$
$991$	30.0982	0.956102	0.478051	+	0.878332i	$0.341343\pi$
$992$	0	0
$993$	−16.0982	−0.510862
$994$	0	0
$995$	3.73437	0.118388
$996$	0	0
$997$	3.66407	0.116042	0.0580212	−	0.998315i	$-0.481521\pi$
$997$	3.66407	0.116042	0.0580212	+	0.998315i	$0.481521\pi$
$998$	0	0
$999$	8.12212	0.256973

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9680.2.a.by.1.2		3
4.3	odd	2		4840.2.a.v.1.2	yes	3
11.10	odd	2		9680.2.a.cd.1.2		3
44.43	even	2		4840.2.a.s.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4840.2.a.s.1.2	✓	3	44.43	even	2
4840.2.a.v.1.2	yes	3	4.3	odd	2
9680.2.a.by.1.2		3	1.1	even	1	trivial
9680.2.a.cd.1.2		3	11.10	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:	\( N \)	\(=\)	\( 9680 = 2^{4} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9680.a (trivial)

\(f(q)\)	\(=\)	\(q+0.476452 q^{3} -1.00000 q^{5} -3.34364 q^{7} -2.77299 q^{9} +O(q^{10})\)	\(q+0.476452 q^{3} -1.00000 q^{5} -3.34364 q^{7} -2.77299 q^{9} -3.82009 q^{13} -0.476452 q^{15} -6.59308 q^{17} -1.82009 q^{19} -1.59308 q^{21} -2.77299 q^{23} +1.00000 q^{25} -2.75055 q^{27} -0.952904 q^{29} +4.17991 q^{31} +3.34364 q^{35} -2.95290 q^{37} -1.82009 q^{39} +4.82009 q^{41} -9.93672 q^{43} +2.77299 q^{45} -3.16373 q^{47} +4.17991 q^{49} -3.14129 q^{51} +7.46027 q^{53} -0.867185 q^{57} -2.77299 q^{59} -11.7730 q^{61} +9.27188 q^{63} +3.82009 q^{65} -7.42936 q^{67} -1.32120 q^{69} -14.0534 q^{71} -7.64018 q^{73} +0.476452 q^{75} -8.59308 q^{79} +7.00847 q^{81} -4.86719 q^{83} +6.59308 q^{85} -0.454013 q^{87} -1.59308 q^{89} +12.7730 q^{91} +1.99153 q^{93} +1.82009 q^{95} -3.82009 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{3} - 3 q^{5} - 3 q^{7} + 6 q^{9}+O(q^{10}) \) 3 * q - q^3 - 3 * q^5 - 3 * q^7 + 6 * q^9	\( 3 q - q^{3} - 3 q^{5} - 3 q^{7} + 6 q^{9} - 2 q^{13} + q^{15} + 4 q^{17} + 4 q^{19} + 19 q^{21} + 6 q^{23} + 3 q^{25} - 25 q^{27} + 2 q^{29} + 22 q^{31} + 3 q^{35} - 4 q^{37} + 4 q^{39} + 5 q^{41} + q^{43} - 6 q^{45} + 7 q^{47} + 22 q^{49} - 24 q^{51} - 6 q^{53} + 2 q^{57} + 6 q^{59} - 21 q^{61} - 20 q^{63} + 2 q^{65} - 15 q^{67} - 28 q^{69} + 10 q^{71} - 4 q^{73} - q^{75} - 2 q^{79} + 31 q^{81} - 10 q^{83} - 4 q^{85} - 30 q^{87} + 19 q^{89} + 24 q^{91} - 4 q^{93} - 4 q^{95} - 2 q^{97}+O(q^{100}) \) 3 * q - q^3 - 3 * q^5 - 3 * q^7 + 6 * q^9 - 2 * q^13 + q^15 + 4 * q^17 + 4 * q^19 + 19 * q^21 + 6 * q^23 + 3 * q^25 - 25 * q^27 + 2 * q^29 + 22 * q^31 + 3 * q^35 - 4 * q^37 + 4 * q^39 + 5 * q^41 + q^43 - 6 * q^45 + 7 * q^47 + 22 * q^49 - 24 * q^51 - 6 * q^53 + 2 * q^57 + 6 * q^59 - 21 * q^61 - 20 * q^63 + 2 * q^65 - 15 * q^67 - 28 * q^69 + 10 * q^71 - 4 * q^73 - q^75 - 2 * q^79 + 31 * q^81 - 10 * q^83 - 4 * q^85 - 30 * q^87 + 19 * q^89 + 24 * q^91 - 4 * q^93 - 4 * q^95 - 2 * q^97

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