LMFDB - Embedded newform 4704.2.c.c.2353.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4704,2,Mod(2353,4704)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4704, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 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("4704.2353");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4704 = 2^{5} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4704.c (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:	$37.5616291108$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.386672896.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{6} - 2x^{5} + 2x^{4} - 4x^{3} - 4x^{2} + 16 $ x^8 - x^6 - 2x^5 + 2x^4 - 4x^3 - 4x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2353.8
Root			$0.621372 + 1.27039i$ of defining polynomial
Character	$\chi$	$=$	4704.2353
Dual form			4704.2.c.c.2353.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +3.69833i q^{5} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +3.69833i q^{5} -1.00000 q^{9} +3.21284i q^{11} +5.08157i q^{13} -3.69833 q^{15} -0.616762 q^{17} +4.48549i q^{19} -1.38324 q^{23} -8.67765 q^{25} -1.00000i q^{27} +5.67765i q^{29} +6.91117 q^{31} -3.21284 q^{33} +6.91117i q^{37} -5.08157 q^{39} +0.616762 q^{41} -7.99274i q^{43} -3.69833i q^{45} +4.97098 q^{47} -0.616762i q^{51} -4.48549i q^{53} -11.8822 q^{55} -4.48549 q^{57} -4.00000i q^{59} +12.4782i q^{61} -18.7933 q^{65} -9.56706i q^{67} -1.38324i q^{69} +15.2056 q^{71} -15.5598 q^{73} -8.67765i q^{75} +5.23352 q^{79} +1.00000 q^{81} +10.4257i q^{83} -2.28099i q^{85} -5.67765 q^{87} +14.1766 q^{89} +6.91117i q^{93} -16.5888 q^{95} -9.73746 q^{97} -3.21284i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^9	$ 8 q - 8 q^{9} + 4 q^{15} - 4 q^{17} - 12 q^{23} - 24 q^{25} + 8 q^{31} - 12 q^{33} - 8 q^{39} + 4 q^{41} - 8 q^{55} - 16 q^{57} - 16 q^{65} + 28 q^{71} + 8 q^{73} + 40 q^{79} + 8 q^{81} - 20 q^{89} - 40 q^{95} - 40 q^{97}+O(q^{100}) $ 8 * q - 8 * q^9 + 4 * q^15 - 4 * q^17 - 12 * q^23 - 24 * q^25 + 8 * q^31 - 12 * q^33 - 8 * q^39 + 4 * q^41 - 8 * q^55 - 16 * q^57 - 16 * q^65 + 28 * q^71 + 8 * q^73 + 40 * q^79 + 8 * q^81 - 20 * q^89 - 40 * q^95 - 40 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4704\mathbb{Z}\right)^\times$.

$n$	$1471$	$1765$	$3137$	$4609$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	0	0
$5$	3.69833i	1.65394i	0.562243	+	0.826972i	$0.309938\pi$
$5$	3.69833i	1.65394i	−0.562243	+	0.826972i	$0.690062\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	3.21284i	0.968708i	0.874872	+	0.484354i	$0.160945\pi$
$11$	3.21284i	0.968708i	−0.874872	+	0.484354i	$0.839055\pi$
$12$	0	0
$13$	5.08157i	1.40937i	0.709518	+	0.704687i	$0.248912\pi$
$13$	5.08157i	1.40937i	−0.709518	+	0.704687i	$0.751088\pi$
$14$	0	0
$15$	−3.69833	−0.954905
$16$	0	0
$17$	−0.616762	−0.149587	−0.0747933	−	0.997199i	$-0.523830\pi$
$17$	−0.616762	−0.149587	−0.0747933	+	0.997199i	$0.523830\pi$
$18$	0	0
$19$	4.48549i	1.02904i	0.857478	+	0.514521i	$0.172030\pi$
$19$	4.48549i	1.02904i	−0.857478	+	0.514521i	$0.827970\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.38324	−0.288425	−0.144213	−	0.989547i	$-0.546065\pi$
$23$	−1.38324	−0.288425	−0.144213	+	0.989547i	$0.546065\pi$
$24$	0	0
$25$	−8.67765	−1.73553
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	5.67765i	1.05431i	0.849768	+	0.527156i	$0.176742\pi$
$29$	5.67765i	1.05431i	−0.849768	+	0.527156i	$0.823258\pi$
$30$	0	0
$31$	6.91117	1.24128	0.620642	−	0.784094i	$-0.286872\pi$
$31$	6.91117	1.24128	0.620642	+	0.784094i	$0.286872\pi$
$32$	0	0
$33$	−3.21284	−0.559284
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.91117i	1.13619i	0.822963	+	0.568095i	$0.192319\pi$
$37$	6.91117i	1.13619i	−0.822963	+	0.568095i	$0.807681\pi$
$38$	0	0
$39$	−5.08157	−0.813702
$40$	0	0
$41$	0.616762	0.0963220	0.0481610	−	0.998840i	$-0.484664\pi$
$41$	0.616762	0.0963220	0.0481610	+	0.998840i	$0.484664\pi$
$42$	0	0
$43$	− 7.99274i	− 1.21888i	−0.792832	−	0.609441i	$-0.791394\pi$
$43$	− 7.99274i	− 1.21888i	0.792832	−	0.609441i	$-0.208606\pi$
$44$	0	0
$45$	− 3.69833i	− 0.551315i
$46$	0	0
$47$	4.97098	0.725092	0.362546	−	0.931966i	$-0.381908\pi$
$47$	4.97098	0.725092	0.362546	+	0.931966i	$0.381908\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	− 0.616762i	− 0.0863639i
$52$	0	0
$53$	− 4.48549i	− 0.616129i	−0.951365	−	0.308065i	$-0.900319\pi$
$53$	− 4.48549i	− 0.616129i	0.951365	−	0.308065i	$-0.0996813\pi$
$54$	0	0
$55$	−11.8822	−1.60219
$56$	0	0
$57$	−4.48549	−0.594118
$58$	0	0
$59$	− 4.00000i	− 0.520756i	−0.965507	−	0.260378i	$-0.916153\pi$
$59$	− 4.00000i	− 0.520756i	0.965507	−	0.260378i	$-0.0838471\pi$
$60$	0	0
$61$	12.4782i	1.59767i	0.601548	+	0.798837i	$0.294551\pi$
$61$	12.4782i	1.59767i	−0.601548	+	0.798837i	$0.705449\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−18.7933	−2.33102
$66$	0	0
$67$	− 9.56706i	− 1.16880i	−0.811465	−	0.584401i	$-0.801329\pi$
$67$	− 9.56706i	− 1.16880i	0.811465	−	0.584401i	$-0.198671\pi$
$68$	0	0
$69$	− 1.38324i	− 0.166522i
$70$	0	0
$71$	15.2056	1.80457	0.902285	−	0.431139i	$-0.141888\pi$
$71$	15.2056	1.80457	0.902285	+	0.431139i	$0.141888\pi$
$72$	0	0
$73$	−15.5598	−1.82114	−0.910568	−	0.413358i	$-0.864356\pi$
$73$	−15.5598	−1.82114	−0.910568	+	0.413358i	$0.864356\pi$
$74$	0	0
$75$	− 8.67765i	− 1.00201i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	5.23352	0.588817	0.294409	−	0.955680i	$-0.404877\pi$
$79$	5.23352	0.588817	0.294409	+	0.955680i	$0.404877\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	10.4257i	1.14437i	0.820125	+	0.572184i	$0.193904\pi$
$83$	10.4257i	1.14437i	−0.820125	+	0.572184i	$0.806096\pi$
$84$	0	0
$85$	− 2.28099i	− 0.247408i
$86$	0	0
$87$	−5.67765	−0.608708
$88$	0	0
$89$	14.1766	1.50271	0.751356	−	0.659897i	$-0.229400\pi$
$89$	14.1766	1.50271	0.751356	+	0.659897i	$0.229400\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	6.91117i	0.716655i
$94$	0	0
$95$	−16.5888	−1.70198
$96$	0	0
$97$	−9.73746	−0.988689	−0.494344	−	0.869266i	$-0.664592\pi$
$97$	−9.73746	−0.988689	−0.494344	+	0.869266i	$0.664592\pi$
$98$	0	0
$99$	− 3.21284i	− 0.322903i
$100$	0	0
$101$	− 5.27265i	− 0.524648i	−0.964980	−	0.262324i	$-0.915511\pi$
$101$	− 5.27265i	− 0.524648i	0.964980	−	0.262324i	$-0.0844889\pi$
$102$	0	0
$103$	1.94019	0.191173	0.0955865	−	0.995421i	$-0.469527\pi$
$103$	1.94019	0.191173	0.0955865	+	0.995421i	$0.469527\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.78716i	0.462792i	0.972860	+	0.231396i	$0.0743293\pi$
$107$	4.78716i	0.462792i	−0.972860	+	0.231396i	$0.925671\pi$
$108$	0	0
$109$	− 0.970978i	− 0.0930028i	−0.998918	−	0.0465014i	$-0.985193\pi$
$109$	− 0.970978i	− 0.0930028i	0.998918	−	0.0465014i	$-0.0148072\pi$
$110$	0	0
$111$	−6.91117	−0.655979
$112$	0	0
$113$	15.5598	1.46374	0.731871	−	0.681443i	$-0.238647\pi$
$113$	15.5598	1.46374	0.731871	+	0.681443i	$0.238647\pi$
$114$	0	0
$115$	− 5.11567i	− 0.477039i
$116$	0	0
$117$	− 5.08157i	− 0.469791i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	0.677649	0.0616045
$122$	0	0
$123$	0.616762i	0.0556115i
$124$	0	0
$125$	− 13.6012i	− 1.21652i
$126$	0	0
$127$	3.73746	0.331646	0.165823	−	0.986156i	$-0.446972\pi$
$127$	3.73746	0.331646	0.165823	+	0.986156i	$0.446972\pi$
$128$	0	0
$129$	7.99274	0.703722
$130$	0	0
$131$	− 3.39666i	− 0.296768i	−0.988930	−	0.148384i	$-0.952593\pi$
$131$	− 3.39666i	− 0.296768i	0.988930	−	0.148384i	$-0.0474071\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	3.69833	0.318302
$136$	0	0
$137$	13.0559	1.11544	0.557719	−	0.830030i	$-0.311677\pi$
$137$	13.0559	1.11544	0.557719	+	0.830030i	$0.311677\pi$
$138$	0	0
$139$	− 2.80784i	− 0.238158i	−0.992885	−	0.119079i	$-0.962006\pi$
$139$	− 2.80784i	− 0.238158i	0.992885	−	0.119079i	$-0.0379942\pi$
$140$	0	0
$141$	4.97098i	0.418632i
$142$	0	0
$143$	−16.3263	−1.36527
$144$	0	0
$145$	−20.9978	−1.74377
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 12.4855i	− 1.02285i	−0.859328	−	0.511426i	$-0.829118\pi$
$149$	− 12.4855i	− 1.02285i	0.859328	−	0.511426i	$-0.170882\pi$
$150$	0	0
$151$	−8.97098	−0.730048	−0.365024	−	0.930998i	$-0.618939\pi$
$151$	−8.97098	−0.730048	−0.365024	+	0.930998i	$0.618939\pi$
$152$	0	0
$153$	0.616762	0.0498622
$154$	0	0
$155$	25.5598i	2.05301i
$156$	0	0
$157$	15.1665i	1.21042i	0.796068	+	0.605208i	$0.206910\pi$
$157$	15.1665i	1.21042i	−0.796068	+	0.605208i	$0.793090\pi$
$158$	0	0
$159$	4.48549	0.355722
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 13.2263i	− 1.03596i	−0.855392	−	0.517980i	$-0.826684\pi$
$163$	− 13.2263i	− 1.03596i	0.855392	−	0.517980i	$-0.173316\pi$
$164$	0	0
$165$	− 11.8822i	− 0.925024i
$166$	0	0
$167$	3.05587	0.236470	0.118235	−	0.992986i	$-0.462276\pi$
$167$	3.05587	0.236470	0.118235	+	0.992986i	$0.462276\pi$
$168$	0	0
$169$	−12.8223	−0.986334
$170$	0	0
$171$	− 4.48549i	− 0.343014i
$172$	0	0
$173$	− 13.8615i	− 1.05387i	−0.849906	−	0.526934i	$-0.823342\pi$
$173$	− 13.8615i	− 1.05387i	0.849906	−	0.526934i	$-0.176658\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	4.00000	0.300658
$178$	0	0
$179$	− 18.3883i	− 1.37441i	−0.726465	−	0.687204i	$-0.758838\pi$
$179$	− 18.3883i	− 1.37441i	0.726465	−	0.687204i	$-0.241162\pi$
$180$	0	0
$181$	− 6.05255i	− 0.449882i	−0.974372	−	0.224941i	$-0.927781\pi$
$181$	− 6.05255i	− 0.449882i	0.974372	−	0.224941i	$-0.0722190\pi$
$182$	0	0
$183$	−12.4782	−0.922417
$184$	0	0
$185$	−25.5598	−1.87919
$186$	0	0
$187$	− 1.98156i	− 0.144906i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	15.2056	1.10024	0.550119	−	0.835086i	$-0.314582\pi$
$191$	15.2056	1.10024	0.550119	+	0.835086i	$0.314582\pi$
$192$	0	0
$193$	−19.4419	−1.39946	−0.699731	−	0.714406i	$-0.746697\pi$
$193$	−19.4419	−1.39946	−0.699731	+	0.714406i	$0.746697\pi$
$194$	0	0
$195$	− 18.7933i	− 1.34582i
$196$	0	0
$197$	15.2520i	1.08666i	0.839520	+	0.543329i	$0.182836\pi$
$197$	15.2520i	1.08666i	−0.839520	+	0.543329i	$0.817164\pi$
$198$	0	0
$199$	−3.02902	−0.214722	−0.107361	−	0.994220i	$-0.534240\pi$
$199$	−3.02902	−0.214722	−0.107361	+	0.994220i	$0.534240\pi$
$200$	0	0
$201$	9.56706	0.674808
$202$	0	0
$203$	0	0
$204$	0	0
$205$	2.28099i	0.159311i
$206$	0	0
$207$	1.38324	0.0961417
$208$	0	0
$209$	−14.4112	−0.996841
$210$	0	0
$211$	0.963719i	0.0663452i	0.999450	+	0.0331726i	$0.0105611\pi$
$211$	0.963719i	0.0663452i	−0.999450	+	0.0331726i	$0.989439\pi$
$212$	0	0
$213$	15.2056i	1.04187i
$214$	0	0
$215$	29.5598	2.01596
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 15.5598i	− 1.05143i
$220$	0	0
$221$	− 3.13412i	− 0.210823i
$222$	0	0
$223$	−22.1486	−1.48318	−0.741591	−	0.670853i	$-0.765928\pi$
$223$	−22.1486	−1.48318	−0.741591	+	0.670853i	$0.765928\pi$
$224$	0	0
$225$	8.67765	0.578510
$226$	0	0
$227$	− 0.929615i	− 0.0617007i	−0.999524	−	0.0308504i	$-0.990178\pi$
$227$	− 0.929615i	− 0.0617007i	0.999524	−	0.0308504i	$-0.00982153\pi$
$228$	0	0
$229$	1.42236i	0.0939924i	0.998895	+	0.0469962i	$0.0149649\pi$
$229$	1.42236i	0.0939924i	−0.998895	+	0.0469962i	$0.985035\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	10.9710	0.718733	0.359366	−	0.933197i	$-0.382993\pi$
$233$	10.9710	0.718733	0.359366	+	0.933197i	$0.382993\pi$
$234$	0	0
$235$	18.3843i	1.19926i
$236$	0	0
$237$	5.23352i	0.339954i
$238$	0	0
$239$	4.82126	0.311862	0.155931	−	0.987768i	$-0.450162\pi$
$239$	4.82126	0.311862	0.155931	+	0.987768i	$0.450162\pi$
$240$	0	0
$241$	0.204501	0.0131731	0.00658654	−	0.999978i	$-0.497903\pi$
$241$	0.204501	0.0131731	0.00658654	+	0.999978i	$0.497903\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−22.7933	−1.45030
$248$	0	0
$249$	−10.4257	−0.660701
$250$	0	0
$251$	− 3.10727i	− 0.196129i	−0.995180	−	0.0980646i	$-0.968735\pi$
$251$	− 3.10727i	− 0.196129i	0.995180	−	0.0980646i	$-0.0312652\pi$
$252$	0	0
$253$	− 4.44413i	− 0.279400i
$254$	0	0
$255$	2.28099	0.142841
$256$	0	0
$257$	14.1497	0.882635	0.441318	−	0.897351i	$-0.354511\pi$
$257$	14.1497	0.882635	0.441318	+	0.897351i	$0.354511\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	− 5.67765i	− 0.351438i
$262$	0	0
$263$	−2.94304	−0.181475	−0.0907377	−	0.995875i	$-0.528923\pi$
$263$	−2.94304	−0.181475	−0.0907377	+	0.995875i	$0.528923\pi$
$264$	0	0
$265$	16.5888	1.01904
$266$	0	0
$267$	14.1766i	0.867591i
$268$	0	0
$269$	0.642463i	0.0391717i	0.999808	+	0.0195858i	$0.00623476\pi$
$269$	0.642463i	0.0391717i	−0.999808	+	0.0195858i	$0.993765\pi$
$270$	0	0
$271$	−0.526852	−0.0320040	−0.0160020	−	0.999872i	$-0.505094\pi$
$271$	−0.526852	−0.0320040	−0.0160020	+	0.999872i	$0.505094\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 27.8799i	− 1.68122i
$276$	0	0
$277$	− 21.7190i	− 1.30497i	−0.757802	−	0.652484i	$-0.773727\pi$
$277$	− 21.7190i	− 1.30497i	0.757802	−	0.652484i	$-0.226273\pi$
$278$	0	0
$279$	−6.91117	−0.413761
$280$	0	0
$281$	−4.79332	−0.285946	−0.142973	−	0.989727i	$-0.545666\pi$
$281$	−4.79332	−0.285946	−0.142973	+	0.989727i	$0.545666\pi$
$282$	0	0
$283$	− 22.6486i	− 1.34632i	−0.739496	−	0.673161i	$-0.764936\pi$
$283$	− 22.6486i	− 1.34632i	0.739496	−	0.673161i	$-0.235064\pi$
$284$	0	0
$285$	− 16.5888i	− 0.982637i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−16.6196	−0.977624
$290$	0	0
$291$	− 9.73746i	− 0.570820i
$292$	0	0
$293$	− 24.3285i	− 1.42129i	−0.703552	−	0.710644i	$-0.748404\pi$
$293$	− 24.3285i	− 1.42129i	0.703552	−	0.710644i	$-0.251596\pi$
$294$	0	0
$295$	14.7933	0.861301
$296$	0	0
$297$	3.21284	0.186428
$298$	0	0
$299$	− 7.02902i	− 0.406499i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	5.27265	0.302906
$304$	0	0
$305$	−46.1486	−2.64246
$306$	0	0
$307$	3.51451i	0.200584i	0.994958	+	0.100292i	$0.0319777\pi$
$307$	3.51451i	0.200584i	−0.994958	+	0.100292i	$0.968022\pi$
$308$	0	0
$309$	1.94019i	0.110374i
$310$	0	0
$311$	−4.97098	−0.281878	−0.140939	−	0.990018i	$-0.545012\pi$
$311$	−4.97098	−0.281878	−0.140939	+	0.990018i	$0.545012\pi$
$312$	0	0
$313$	25.6447	1.44952	0.724762	−	0.689000i	$-0.241950\pi$
$313$	25.6447	1.44952	0.724762	+	0.689000i	$0.241950\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.51451i	0.197395i	0.995118	+	0.0986973i	$0.0314676\pi$
$317$	3.51451i	0.197395i	−0.995118	+	0.0986973i	$0.968532\pi$
$318$	0	0
$319$	−18.2414	−1.02132
$320$	0	0
$321$	−4.78716	−0.267193
$322$	0	0
$323$	− 2.76648i	− 0.153931i
$324$	0	0
$325$	− 44.0961i	− 2.44601i
$326$	0	0
$327$	0.970978	0.0536952
$328$	0	0
$329$	0	0
$330$	0	0
$331$	25.1849i	1.38429i	0.721760	+	0.692144i	$0.243334\pi$
$331$	25.1849i	1.38429i	−0.721760	+	0.692144i	$0.756666\pi$
$332$	0	0
$333$	− 6.91117i	− 0.378730i
$334$	0	0
$335$	35.3821	1.93313
$336$	0	0
$337$	21.7643	1.18558	0.592788	−	0.805358i	$-0.298027\pi$
$337$	21.7643	1.18558	0.592788	+	0.805358i	$0.298027\pi$
$338$	0	0
$339$	15.5598i	0.845092i
$340$	0	0
$341$	22.2045i	1.20244i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	5.11567	0.275419
$346$	0	0
$347$	− 31.9068i	− 1.71284i	−0.516276	−	0.856422i	$-0.672682\pi$
$347$	− 31.9068i	− 1.71284i	0.516276	−	0.856422i	$-0.327318\pi$
$348$	0	0
$349$	− 1.04021i	− 0.0556810i	−0.999612	−	0.0278405i	$-0.991137\pi$
$349$	− 1.04021i	− 0.0556810i	0.999612	−	0.0278405i	$-0.00886305\pi$
$350$	0	0
$351$	5.08157	0.271234
$352$	0	0
$353$	13.8503	0.737176	0.368588	−	0.929593i	$-0.379841\pi$
$353$	13.8503	0.737176	0.368588	+	0.929593i	$0.379841\pi$
$354$	0	0
$355$	56.2353i	2.98466i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−0.412260	−0.0217583	−0.0108791	−	0.999941i	$-0.503463\pi$
$359$	−0.412260	−0.0217583	−0.0108791	+	0.999941i	$0.503463\pi$
$360$	0	0
$361$	−1.11961	−0.0589269
$362$	0	0
$363$	0.677649i	0.0355674i
$364$	0	0
$365$	− 57.5453i	− 3.01206i
$366$	0	0
$367$	4.97098	0.259483	0.129741	−	0.991548i	$-0.458585\pi$
$367$	4.97098	0.259483	0.129741	+	0.991548i	$0.458585\pi$
$368$	0	0
$369$	−0.616762	−0.0321073
$370$	0	0
$371$	0	0
$372$	0	0
$373$	2.16314i	0.112003i	0.998431	+	0.0560015i	$0.0178352\pi$
$373$	2.16314i	0.112003i	−0.998431	+	0.0560015i	$0.982165\pi$
$374$	0	0
$375$	13.6012	0.702361
$376$	0	0
$377$	−28.8514	−1.48592
$378$	0	0
$379$	− 23.7570i	− 1.22032i	−0.792279	−	0.610159i	$-0.791106\pi$
$379$	− 23.7570i	− 1.22032i	0.792279	−	0.610159i	$-0.208894\pi$
$380$	0	0
$381$	3.73746i	0.191476i
$382$	0	0
$383$	−37.5598	−1.91922	−0.959608	−	0.281340i	$-0.909221\pi$
$383$	−37.5598	−1.91922	−0.959608	+	0.281340i	$0.909221\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	7.99274i	0.406294i
$388$	0	0
$389$	− 0.826283i	− 0.0418942i	−0.999781	−	0.0209471i	$-0.993332\pi$
$389$	− 0.826283i	− 0.0418942i	0.999781	−	0.0209471i	$-0.00666816\pi$
$390$	0	0
$391$	0.853128	0.0431446
$392$	0	0
$393$	3.39666	0.171339
$394$	0	0
$395$	19.3553i	0.973871i
$396$	0	0
$397$	8.81902i	0.442614i	0.975204	+	0.221307i	$0.0710323\pi$
$397$	8.81902i	0.442614i	−0.975204	+	0.221307i	$0.928968\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−8.64687	−0.431804	−0.215902	−	0.976415i	$-0.569269\pi$
$401$	−8.64687	−0.431804	−0.215902	+	0.976415i	$0.569269\pi$
$402$	0	0
$403$	35.1196i	1.74943i
$404$	0	0
$405$	3.69833i	0.183772i
$406$	0	0
$407$	−22.2045	−1.10064
$408$	0	0
$409$	18.7084	0.925072	0.462536	−	0.886600i	$-0.346940\pi$
$409$	18.7084	0.925072	0.462536	+	0.886600i	$0.346940\pi$
$410$	0	0
$411$	13.0559i	0.643998i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−38.5576	−1.89272
$416$	0	0
$417$	2.80784	0.137501
$418$	0	0
$419$	12.9710i	0.633674i	0.948480	+	0.316837i	$0.102621\pi$
$419$	12.9710i	0.633674i	−0.948480	+	0.316837i	$0.897379\pi$
$420$	0	0
$421$	− 3.54136i	− 0.172595i	−0.996269	−	0.0862976i	$-0.972496\pi$
$421$	− 3.54136i	− 0.172595i	0.996269	−	0.0862976i	$-0.0275036\pi$
$422$	0	0
$423$	−4.97098	−0.241697
$424$	0	0
$425$	5.35204	0.259612
$426$	0	0
$427$	0	0
$428$	0	0
$429$	− 16.3263i	− 0.788240i
$430$	0	0
$431$	−14.6168	−0.704065	−0.352032	−	0.935988i	$-0.614509\pi$
$431$	−14.6168	−0.704065	−0.352032	+	0.935988i	$0.614509\pi$
$432$	0	0
$433$	2.97098	0.142776	0.0713880	−	0.997449i	$-0.477257\pi$
$433$	2.97098	0.142776	0.0713880	+	0.997449i	$0.477257\pi$
$434$	0	0
$435$	− 20.9978i	− 1.00677i
$436$	0	0
$437$	− 6.20450i	− 0.296802i
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	27.1983i	1.29223i	0.763239	+	0.646116i	$0.223608\pi$
$443$	27.1983i	1.29223i	−0.763239	+	0.646116i	$0.776392\pi$
$444$	0	0
$445$	52.4296i	2.48540i
$446$	0	0
$447$	12.4855	0.590543
$448$	0	0
$449$	−31.5598	−1.48940	−0.744700	−	0.667400i	$-0.767407\pi$
$449$	−31.5598	−1.48940	−0.744700	+	0.667400i	$0.767407\pi$
$450$	0	0
$451$	1.98156i	0.0933079i
$452$	0	0
$453$	− 8.97098i	− 0.421493i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	28.1117	1.31501	0.657506	−	0.753450i	$-0.271612\pi$
$457$	28.1117	1.31501	0.657506	+	0.753450i	$0.271612\pi$
$458$	0	0
$459$	0.616762i	0.0287880i
$460$	0	0
$461$	5.19440i	0.241927i	0.992657	+	0.120964i	$0.0385984\pi$
$461$	5.19440i	0.241927i	−0.992657	+	0.120964i	$0.961402\pi$
$462$	0	0
$463$	10.7665	0.500361	0.250180	−	0.968199i	$-0.419510\pi$
$463$	10.7665	0.500361	0.250180	+	0.968199i	$0.419510\pi$
$464$	0	0
$465$	−25.5598	−1.18531
$466$	0	0
$467$	− 30.7520i	− 1.42303i	−0.702670	−	0.711515i	$-0.748009\pi$
$467$	− 30.7520i	− 1.42303i	0.702670	−	0.711515i	$-0.251991\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−15.1665	−0.698834
$472$	0	0
$473$	25.6794	1.18074
$474$	0	0
$475$	− 38.9235i	− 1.78593i
$476$	0	0
$477$	4.48549i	0.205376i
$478$	0	0
$479$	16.3263	0.745967	0.372983	−	0.927838i	$-0.378335\pi$
$479$	16.3263	0.745967	0.372983	+	0.927838i	$0.378335\pi$
$480$	0	0
$481$	−35.1196	−1.60132
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 36.0123i	− 1.63524i
$486$	0	0
$487$	−14.7933	−0.670349	−0.335175	−	0.942156i	$-0.608795\pi$
$487$	−14.7933	−0.670349	−0.335175	+	0.942156i	$0.608795\pi$
$488$	0	0
$489$	13.2263	0.598112
$490$	0	0
$491$	14.0475i	0.633956i	0.948433	+	0.316978i	$0.102668\pi$
$491$	14.0475i	0.633956i	−0.948433	+	0.316978i	$0.897332\pi$
$492$	0	0
$493$	− 3.50176i	− 0.157711i
$494$	0	0
$495$	11.8822	0.534063
$496$	0	0
$497$	0	0
$498$	0	0
$499$	34.1414i	1.52838i	0.644993	+	0.764189i	$0.276860\pi$
$499$	34.1414i	1.52838i	−0.644993	+	0.764189i	$0.723140\pi$
$500$	0	0
$501$	3.05587i	0.136526i
$502$	0	0
$503$	−27.3821	−1.22091	−0.610455	−	0.792051i	$-0.709013\pi$
$503$	−27.3821	−1.22091	−0.610455	+	0.792051i	$0.709013\pi$
$504$	0	0
$505$	19.5000	0.867738
$506$	0	0
$507$	− 12.8223i	− 0.569460i
$508$	0	0
$509$	2.49165i	0.110441i	0.998474	+	0.0552203i	$0.0175861\pi$
$509$	2.49165i	0.110441i	−0.998474	+	0.0552203i	$0.982414\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	4.48549	0.198039
$514$	0	0
$515$	7.17548i	0.316189i
$516$	0	0
$517$	15.9710i	0.702402i
$518$	0	0
$519$	13.8615	0.608451
$520$	0	0
$521$	3.00108	0.131480	0.0657399	−	0.997837i	$-0.479059\pi$
$521$	3.00108	0.131480	0.0657399	+	0.997837i	$0.479059\pi$
$522$	0	0
$523$	16.8514i	0.736859i	0.929656	+	0.368429i	$0.120104\pi$
$523$	16.8514i	0.736859i	−0.929656	+	0.368429i	$0.879896\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4.26254	−0.185679
$528$	0	0
$529$	−21.0867	−0.916811
$530$	0	0
$531$	4.00000i	0.173585i
$532$	0	0
$533$	3.13412i	0.135754i
$534$	0	0
$535$	−17.7045	−0.765432
$536$	0	0
$537$	18.3883	0.793515
$538$	0	0
$539$	0	0
$540$	0	0
$541$	24.6319i	1.05901i	0.848307	+	0.529505i	$0.177622\pi$
$541$	24.6319i	1.05901i	−0.848307	+	0.529505i	$0.822378\pi$
$542$	0	0
$543$	6.05255	0.259740
$544$	0	0
$545$	3.59100	0.153821
$546$	0	0
$547$	25.8564i	1.10554i	0.833333	+	0.552771i	$0.186430\pi$
$547$	25.8564i	1.10554i	−0.833333	+	0.552771i	$0.813570\pi$
$548$	0	0
$549$	− 12.4782i	− 0.532558i
$550$	0	0
$551$	−25.4670	−1.08493
$552$	0	0
$553$	0	0
$554$	0	0
$555$	− 25.5598i	− 1.08495i
$556$	0	0
$557$	27.2788i	1.15584i	0.816093	+	0.577920i	$0.196136\pi$
$557$	27.2788i	1.15584i	−0.816093	+	0.577920i	$0.803864\pi$
$558$	0	0
$559$	40.6157	1.71786
$560$	0	0
$561$	1.98156	0.0836614
$562$	0	0
$563$	13.1776i	0.555371i	0.960672	+	0.277686i	$0.0895674\pi$
$563$	13.1776i	0.555371i	−0.960672	+	0.277686i	$0.910433\pi$
$564$	0	0
$565$	57.5453i	2.42095i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	17.4380	0.731040	0.365520	−	0.930803i	$-0.380891\pi$
$569$	17.4380	0.731040	0.365520	+	0.930803i	$0.380891\pi$
$570$	0	0
$571$	4.71569i	0.197346i	0.995120	+	0.0986728i	$0.0314597\pi$
$571$	4.71569i	0.197346i	−0.995120	+	0.0986728i	$0.968540\pi$
$572$	0	0
$573$	15.2056i	0.635222i
$574$	0	0
$575$	12.0033	0.500570
$576$	0	0
$577$	11.8223	0.492171	0.246085	−	0.969248i	$-0.420856\pi$
$577$	11.8223	0.492171	0.246085	+	0.969248i	$0.420856\pi$
$578$	0	0
$579$	− 19.4419i	− 0.807980i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	14.4112	0.596849
$584$	0	0
$585$	18.7933	0.777008
$586$	0	0
$587$	− 13.7810i	− 0.568802i	−0.958705	−	0.284401i	$-0.908205\pi$
$587$	− 13.7810i	− 0.568802i	0.958705	−	0.284401i	$-0.0917947\pi$
$588$	0	0
$589$	31.0000i	1.27733i
$590$	0	0
$591$	−15.2520	−0.627382
$592$	0	0
$593$	−36.8872	−1.51477	−0.757387	−	0.652966i	$-0.773524\pi$
$593$	−36.8872	−1.51477	−0.757387	+	0.652966i	$0.773524\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 3.02902i	− 0.123970i
$598$	0	0
$599$	35.0895	1.43372	0.716859	−	0.697218i	$-0.245579\pi$
$599$	35.0895	1.43372	0.716859	+	0.697218i	$0.245579\pi$
$600$	0	0
$601$	−18.3263	−0.747544	−0.373772	−	0.927521i	$-0.621936\pi$
$601$	−18.3263	−0.747544	−0.373772	+	0.927521i	$0.621936\pi$
$602$	0	0
$603$	9.56706i	0.389601i
$604$	0	0
$605$	2.50617i	0.101890i
$606$	0	0
$607$	−12.4459	−0.505163	−0.252582	−	0.967576i	$-0.581280\pi$
$607$	−12.4459	−0.505163	−0.252582	+	0.967576i	$0.581280\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	25.2604i	1.02193i
$612$	0	0
$613$	14.2682i	0.576288i	0.957587	+	0.288144i	$0.0930383\pi$
$613$	14.2682i	0.576288i	−0.957587	+	0.288144i	$0.906962\pi$
$614$	0	0
$615$	−2.28099	−0.0919783
$616$	0	0
$617$	2.94413	0.118526	0.0592632	−	0.998242i	$-0.481125\pi$
$617$	2.94413	0.118526	0.0592632	+	0.998242i	$0.481125\pi$
$618$	0	0
$619$	13.2604i	0.532979i	0.963838	+	0.266490i	$0.0858638\pi$
$619$	13.2604i	0.532979i	−0.963838	+	0.266490i	$0.914136\pi$
$620$	0	0
$621$	1.38324i	0.0555074i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	6.91335	0.276534
$626$	0	0
$627$	− 14.4112i	− 0.575527i
$628$	0	0
$629$	− 4.26254i	− 0.169959i
$630$	0	0
$631$	−36.2335	−1.44243	−0.721217	−	0.692710i	$-0.756417\pi$
$631$	−36.2335	−1.44243	−0.721217	+	0.692710i	$0.756417\pi$
$632$	0	0
$633$	−0.963719	−0.0383044
$634$	0	0
$635$	13.8223i	0.548523i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−15.2056	−0.601524
$640$	0	0
$641$	25.9441	1.02473	0.512366	−	0.858767i	$-0.328769\pi$
$641$	25.9441	1.02473	0.512366	+	0.858767i	$0.328769\pi$
$642$	0	0
$643$	27.2643i	1.07520i	0.843200	+	0.537599i	$0.180669\pi$
$643$	27.2643i	1.07520i	−0.843200	+	0.537599i	$0.819331\pi$
$644$	0	0
$645$	29.5598i	1.16392i
$646$	0	0
$647$	−7.41118	−0.291364	−0.145682	−	0.989332i	$-0.546538\pi$
$647$	−7.41118	−0.291364	−0.145682	+	0.989332i	$0.546538\pi$
$648$	0	0
$649$	12.8514	0.504460
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 17.7045i	− 0.692830i	−0.938081	−	0.346415i	$-0.887399\pi$
$653$	− 17.7045i	− 0.692830i	0.938081	−	0.346415i	$-0.112601\pi$
$654$	0	0
$655$	12.5620	0.490837
$656$	0	0
$657$	15.5598	0.607046
$658$	0	0
$659$	− 39.2252i	− 1.52800i	−0.645219	−	0.763998i	$-0.723234\pi$
$659$	− 39.2252i	− 1.52800i	0.645219	−	0.763998i	$-0.276766\pi$
$660$	0	0
$661$	− 8.35862i	− 0.325113i	−0.986699	−	0.162556i	$-0.948026\pi$
$661$	− 8.35862i	− 0.325113i	0.986699	−	0.162556i	$-0.0519739\pi$
$662$	0	0
$663$	3.13412	0.121719
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 7.85354i	− 0.304090i
$668$	0	0
$669$	− 22.1486i	− 0.856315i
$670$	0	0
$671$	−40.0906	−1.54768
$672$	0	0
$673$	37.9090	1.46128	0.730642	−	0.682761i	$-0.239221\pi$
$673$	37.9090	1.46128	0.730642	+	0.682761i	$0.239221\pi$
$674$	0	0
$675$	8.67765i	0.334003i
$676$	0	0
$677$	− 41.5844i	− 1.59822i	−0.601186	−	0.799109i	$-0.705305\pi$
$677$	− 41.5844i	− 1.59822i	0.601186	−	0.799109i	$-0.294695\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0.929615	0.0356229
$682$	0	0
$683$	1.49559i	0.0572272i	0.999591	+	0.0286136i	$0.00910924\pi$
$683$	1.49559i	0.0572272i	−0.999591	+	0.0286136i	$0.990891\pi$
$684$	0	0
$685$	48.2849i	1.84487i
$686$	0	0
$687$	−1.42236	−0.0542665
$688$	0	0
$689$	22.7933	0.868356
$690$	0	0
$691$	19.9028i	0.757137i	0.925573	+	0.378568i	$0.123583\pi$
$691$	19.9028i	0.757137i	−0.925573	+	0.378568i	$0.876417\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	10.3843	0.393900
$696$	0	0
$697$	−0.380395	−0.0144085
$698$	0	0
$699$	10.9710i	0.414961i
$700$	0	0
$701$	44.5347i	1.68205i	0.540994	+	0.841026i	$0.318048\pi$
$701$	44.5347i	1.68205i	−0.540994	+	0.841026i	$0.681952\pi$
$702$	0	0
$703$	−31.0000	−1.16919
$704$	0	0
$705$	−18.3843	−0.692394
$706$	0	0
$707$	0	0
$708$	0	0
$709$	35.3553i	1.32780i	0.747823	+	0.663898i	$0.231099\pi$
$709$	35.3553i	1.32780i	−0.747823	+	0.663898i	$0.768901\pi$
$710$	0	0
$711$	−5.23352	−0.196272
$712$	0	0
$713$	−9.55980	−0.358017
$714$	0	0
$715$	− 60.3800i	− 2.25808i
$716$	0	0
$717$	4.82126i	0.180053i
$718$	0	0
$719$	11.0928	0.413690	0.206845	−	0.978374i	$-0.433680\pi$
$719$	11.0928	0.413690	0.206845	+	0.978374i	$0.433680\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0.204501i	0.00760548i
$724$	0	0
$725$	− 49.2686i	− 1.82979i
$726$	0	0
$727$	−38.1135	−1.41355	−0.706776	−	0.707438i	$-0.749851\pi$
$727$	−38.1135	−1.41355	−0.706776	+	0.707438i	$0.749851\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	4.92962i	0.182328i
$732$	0	0
$733$	− 27.9576i	− 1.03264i	−0.856396	−	0.516319i	$-0.827302\pi$
$733$	− 27.9576i	− 1.03264i	0.856396	−	0.516319i	$-0.172698\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	30.7374	1.13223
$738$	0	0
$739$	1.56706i	0.0576452i	0.999585	+	0.0288226i	$0.00917578\pi$
$739$	1.56706i	0.0576452i	−0.999585	+	0.0288226i	$0.990824\pi$
$740$	0	0
$741$	− 22.7933i	− 0.837334i
$742$	0	0
$743$	−44.0569	−1.61629	−0.808146	−	0.588982i	$-0.799529\pi$
$743$	−44.0569	−1.61629	−0.808146	+	0.588982i	$0.799529\pi$
$744$	0	0
$745$	46.1755	1.69174
$746$	0	0
$747$	− 10.4257i	− 0.381456i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	53.5039	1.95239	0.976193	−	0.216904i	$-0.0695960\pi$
$751$	53.5039	1.95239	0.976193	+	0.216904i	$0.0695960\pi$
$752$	0	0
$753$	3.10727	0.113235
$754$	0	0
$755$	− 33.1776i	− 1.20746i
$756$	0	0
$757$	− 34.1486i	− 1.24115i	−0.784146	−	0.620576i	$-0.786899\pi$
$757$	− 34.1486i	− 1.24115i	0.784146	−	0.620576i	$-0.213101\pi$
$758$	0	0
$759$	4.44413	0.161312
$760$	0	0
$761$	8.23460	0.298504	0.149252	−	0.988799i	$-0.452313\pi$
$761$	8.23460	0.298504	0.149252	+	0.988799i	$0.452313\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	2.28099i	0.0824693i
$766$	0	0
$767$	20.3263	0.733939
$768$	0	0
$769$	−17.4380	−0.628831	−0.314416	−	0.949285i	$-0.601809\pi$
$769$	−17.4380	−0.628831	−0.314416	+	0.949285i	$0.601809\pi$
$770$	0	0
$771$	14.1497i	0.509590i
$772$	0	0
$773$	13.8470i	0.498040i	0.968498	+	0.249020i	$0.0801085\pi$
$773$	13.8470i	0.498040i	−0.968498	+	0.249020i	$0.919891\pi$
$774$	0	0
$775$	−59.9727	−2.15428
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2.76648i	0.0991193i
$780$	0	0
$781$	48.8531i	1.74810i
$782$	0	0
$783$	5.67765	0.202903
$784$	0	0
$785$	−56.0906	−2.00196
$786$	0	0
$787$	− 48.8368i	− 1.74085i	−0.492305	−	0.870423i	$-0.663845\pi$
$787$	− 48.8368i	− 1.74085i	0.492305	−	0.870423i	$-0.336155\pi$
$788$	0	0
$789$	− 2.94304i	− 0.104775i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−63.4090	−2.25172
$794$	0	0
$795$	16.5888i	0.588345i
$796$	0	0
$797$	44.7330i	1.58453i	0.610180	+	0.792263i	$0.291097\pi$
$797$	44.7330i	1.58453i	−0.610180	+	0.792263i	$0.708903\pi$
$798$	0	0
$799$	−3.06591	−0.108464
$800$	0	0
$801$	−14.1766	−0.500904
$802$	0	0
$803$	− 49.9912i	− 1.76415i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−0.642463	−0.0226158
$808$	0	0
$809$	7.79550	0.274075	0.137038	−	0.990566i	$-0.456242\pi$
$809$	7.79550	0.274075	0.137038	+	0.990566i	$0.456242\pi$
$810$	0	0
$811$	9.53295i	0.334747i	0.985894	+	0.167374i	$0.0535286\pi$
$811$	9.53295i	0.334747i	−0.985894	+	0.167374i	$0.946471\pi$
$812$	0	0
$813$	− 0.526852i	− 0.0184775i
$814$	0	0
$815$	48.9151	1.71342
$816$	0	0
$817$	35.8514	1.25428
$818$	0	0
$819$	0	0
$820$	0	0
$821$	12.2498i	0.427521i	0.976886	+	0.213760i	$0.0685711\pi$
$821$	12.2498i	0.427521i	−0.976886	+	0.213760i	$0.931429\pi$
$822$	0	0
$823$	49.7665	1.73475	0.867375	−	0.497655i	$-0.165806\pi$
$823$	49.7665	1.73475	0.867375	+	0.497655i	$0.165806\pi$
$824$	0	0
$825$	27.8799	0.970654
$826$	0	0
$827$	13.1403i	0.456932i	0.973552	+	0.228466i	$0.0733710\pi$
$827$	13.1403i	0.456932i	−0.973552	+	0.228466i	$0.926629\pi$
$828$	0	0
$829$	− 0.961957i	− 0.0334102i	−0.999860	−	0.0167051i	$-0.994682\pi$
$829$	− 0.961957i	− 0.0334102i	0.999860	−	0.0167051i	$-0.00531764\pi$
$830$	0	0
$831$	21.7190	0.753424
$832$	0	0
$833$	0	0
$834$	0	0
$835$	11.3016i	0.391108i
$836$	0	0
$837$	− 6.91117i	− 0.238885i
$838$	0	0
$839$	51.4727	1.77704	0.888518	−	0.458842i	$-0.151736\pi$
$839$	51.4727	1.77704	0.888518	+	0.458842i	$0.151736\pi$
$840$	0	0
$841$	−3.23570	−0.111576
$842$	0	0
$843$	− 4.79332i	− 0.165091i
$844$	0	0
$845$	− 47.4213i	− 1.63134i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	22.6486	0.777299
$850$	0	0
$851$	− 9.55980i	− 0.327706i
$852$	0	0
$853$	− 3.72843i	− 0.127659i	−0.997961	−	0.0638296i	$-0.979669\pi$
$853$	− 3.72843i	− 0.127659i	0.997961	−	0.0638296i	$-0.0203314\pi$
$854$	0	0
$855$	16.5888	0.567326
$856$	0	0
$857$	0.473828	0.0161857	0.00809283	−	0.999967i	$-0.497424\pi$
$857$	0.473828	0.0161857	0.00809283	+	0.999967i	$0.497424\pi$
$858$	0	0
$859$	− 41.3078i	− 1.40940i	−0.709503	−	0.704702i	$-0.751081\pi$
$859$	− 41.3078i	− 1.40940i	0.709503	−	0.704702i	$-0.248919\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−29.9162	−1.01836	−0.509179	−	0.860660i	$-0.670051\pi$
$863$	−29.9162	−1.01836	−0.509179	+	0.860660i	$0.670051\pi$
$864$	0	0
$865$	51.2643	1.74304
$866$	0	0
$867$	− 16.6196i	− 0.564431i
$868$	0	0
$869$	16.8145i	0.570392i
$870$	0	0
$871$	48.6157	1.64728
$872$	0	0
$873$	9.73746	0.329563
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 18.4670i	− 0.623588i	−0.950150	−	0.311794i	$-0.899070\pi$
$877$	− 18.4670i	− 0.623588i	0.950150	−	0.311794i	$-0.100930\pi$
$878$	0	0
$879$	24.3285	0.820580
$880$	0	0
$881$	50.5297	1.70239	0.851194	−	0.524851i	$-0.175879\pi$
$881$	50.5297	1.70239	0.851194	+	0.524851i	$0.175879\pi$
$882$	0	0
$883$	53.8688i	1.81283i	0.422390	+	0.906414i	$0.361191\pi$
$883$	53.8688i	1.81283i	−0.422390	+	0.906414i	$0.638809\pi$
$884$	0	0
$885$	14.7933i	0.497272i
$886$	0	0
$887$	36.9978	1.24227	0.621133	−	0.783705i	$-0.286673\pi$
$887$	36.9978	1.24227	0.621133	+	0.783705i	$0.286673\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	3.21284i	0.107634i
$892$	0	0
$893$	22.2973i	0.746150i
$894$	0	0
$895$	68.0061	2.27319
$896$	0	0
$897$	7.02902	0.234692
$898$	0	0
$899$	39.2392i	1.30870i
$900$	0	0
$901$	2.76648i	0.0921647i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	22.3843	0.744080
$906$	0	0
$907$	− 53.5112i	− 1.77681i	−0.459061	−	0.888405i	$-0.651814\pi$
$907$	− 53.5112i	− 1.77681i	0.459061	−	0.888405i	$-0.348186\pi$
$908$	0	0
$909$	5.27265i	0.174883i
$910$	0	0
$911$	11.0860	0.367295	0.183647	−	0.982992i	$-0.441210\pi$
$911$	11.0860	0.367295	0.183647	+	0.982992i	$0.441210\pi$
$912$	0	0
$913$	−33.4961	−1.10856
$914$	0	0
$915$	− 46.1486i	− 1.52563i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−47.5286	−1.56782	−0.783912	−	0.620872i	$-0.786779\pi$
$919$	−47.5286	−1.56782	−0.783912	+	0.620872i	$0.786779\pi$
$920$	0	0
$921$	−3.51451	−0.115807
$922$	0	0
$923$	77.2682i	2.54331i
$924$	0	0
$925$	− 59.9727i	− 1.97189i
$926$	0	0
$927$	−1.94019	−0.0637243
$928$	0	0
$929$	33.2425	1.09065	0.545325	−	0.838225i	$-0.316406\pi$
$929$	33.2425	1.09065	0.545325	+	0.838225i	$0.316406\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	− 4.97098i	− 0.162743i
$934$	0	0
$935$	7.32845	0.239666
$936$	0	0
$937$	14.8514	0.485173	0.242586	−	0.970130i	$-0.422004\pi$
$937$	14.8514	0.485173	0.242586	+	0.970130i	$0.422004\pi$
$938$	0	0
$939$	25.6447i	0.836883i
$940$	0	0
$941$	53.6258i	1.74815i	0.485791	+	0.874075i	$0.338532\pi$
$941$	53.6258i	1.74815i	−0.485791	+	0.874075i	$0.661468\pi$
$942$	0	0
$943$	−0.853128	−0.0277817
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 24.1324i	− 0.784198i	−0.919923	−	0.392099i	$-0.871749\pi$
$947$	− 24.1324i	− 0.784198i	0.919923	−	0.392099i	$-0.128251\pi$
$948$	0	0
$949$	− 79.0682i	− 2.56666i
$950$	0	0
$951$	−3.51451	−0.113966
$952$	0	0
$953$	0.347434	0.0112545	0.00562725	−	0.999984i	$-0.498209\pi$
$953$	0.347434	0.0112545	0.00562725	+	0.999984i	$0.498209\pi$
$954$	0	0
$955$	56.2353i	1.81973i
$956$	0	0
$957$	− 18.2414i	− 0.589660i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	16.7643	0.540784
$962$	0	0
$963$	− 4.78716i	− 0.154264i
$964$	0	0
$965$	− 71.9028i	− 2.31463i
$966$	0	0
$967$	8.14646	0.261972	0.130986	−	0.991384i	$-0.458186\pi$
$967$	8.14646	0.261972	0.130986	+	0.991384i	$0.458186\pi$
$968$	0	0
$969$	2.76648	0.0888720
$970$	0	0
$971$	3.02902i	0.0972059i	0.998818	+	0.0486030i	$0.0154769\pi$
$971$	3.02902i	0.0972059i	−0.998818	+	0.0486030i	$0.984523\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	44.0961	1.41220
$976$	0	0
$977$	18.7353	0.599395	0.299697	−	0.954034i	$-0.403114\pi$
$977$	18.7353	0.599395	0.299697	+	0.954034i	$0.403114\pi$
$978$	0	0
$979$	45.5470i	1.45569i
$980$	0	0
$981$	0.970978i	0.0310009i
$982$	0	0
$983$	18.2313	0.581490	0.290745	−	0.956801i	$-0.406097\pi$
$983$	18.2313	0.581490	0.290745	+	0.956801i	$0.406097\pi$
$984$	0	0
$985$	−56.4068	−1.79727
$986$	0	0
$987$	0	0
$988$	0	0
$989$	11.0559i	0.351556i
$990$	0	0
$991$	21.2973	0.676530	0.338265	−	0.941051i	$-0.390160\pi$
$991$	21.2973	0.676530	0.338265	+	0.941051i	$0.390160\pi$
$992$	0	0
$993$	−25.1849	−0.799219
$994$	0	0
$995$	− 11.2023i	− 0.355138i
$996$	0	0
$997$	24.7900i	0.785107i	0.919729	+	0.392553i	$0.128408\pi$
$997$	24.7900i	0.785107i	−0.919729	+	0.392553i	$0.871592\pi$
$998$	0	0
$999$	6.91117	0.218660

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4704.2.c.c.2353.8		8
4.3	odd	2		1176.2.c.c.589.3		8
7.6	odd	2		672.2.c.b.337.1		8
8.3	odd	2		1176.2.c.c.589.4		8
8.5	even	2	inner	4704.2.c.c.2353.1		8
21.20	even	2		2016.2.c.e.1009.7		8
28.27	even	2		168.2.c.b.85.3	✓	8
56.13	odd	2		672.2.c.b.337.8		8
56.27	even	2		168.2.c.b.85.4	yes	8
84.83	odd	2		504.2.c.f.253.6		8
112.13	odd	4		5376.2.a.bq.1.1		4
112.27	even	4		5376.2.a.bp.1.4		4
112.69	odd	4		5376.2.a.bl.1.4		4
112.83	even	4		5376.2.a.bm.1.1		4
168.83	odd	2		504.2.c.f.253.5		8
168.125	even	2		2016.2.c.e.1009.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.c.b.85.3	✓	8	28.27	even	2
168.2.c.b.85.4	yes	8	56.27	even	2
504.2.c.f.253.5		8	168.83	odd	2
504.2.c.f.253.6		8	84.83	odd	2
672.2.c.b.337.1		8	7.6	odd	2
672.2.c.b.337.8		8	56.13	odd	2
1176.2.c.c.589.3		8	4.3	odd	2
1176.2.c.c.589.4		8	8.3	odd	2
2016.2.c.e.1009.2		8	168.125	even	2
2016.2.c.e.1009.7		8	21.20	even	2
4704.2.c.c.2353.1		8	8.5	even	2	inner
4704.2.c.c.2353.8		8	1.1	even	1	trivial
5376.2.a.bl.1.4		4	112.69	odd	4
5376.2.a.bm.1.1		4	112.83	even	4
5376.2.a.bp.1.4		4	112.27	even	4
5376.2.a.bq.1.1		4	112.13	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4704 = 2^{5} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4704.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +3.69833i q^{5} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +3.69833i q^{5} -1.00000 q^{9} +3.21284i q^{11} +5.08157i q^{13} -3.69833 q^{15} -0.616762 q^{17} +4.48549i q^{19} -1.38324 q^{23} -8.67765 q^{25} -1.00000i q^{27} +5.67765i q^{29} +6.91117 q^{31} -3.21284 q^{33} +6.91117i q^{37} -5.08157 q^{39} +0.616762 q^{41} -7.99274i q^{43} -3.69833i q^{45} +4.97098 q^{47} -0.616762i q^{51} -4.48549i q^{53} -11.8822 q^{55} -4.48549 q^{57} -4.00000i q^{59} +12.4782i q^{61} -18.7933 q^{65} -9.56706i q^{67} -1.38324i q^{69} +15.2056 q^{71} -15.5598 q^{73} -8.67765i q^{75} +5.23352 q^{79} +1.00000 q^{81} +10.4257i q^{83} -2.28099i q^{85} -5.67765 q^{87} +14.1766 q^{89} +6.91117i q^{93} -16.5888 q^{95} -9.73746 q^{97} -3.21284i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^9	\( 8 q - 8 q^{9} + 4 q^{15} - 4 q^{17} - 12 q^{23} - 24 q^{25} + 8 q^{31} - 12 q^{33} - 8 q^{39} + 4 q^{41} - 8 q^{55} - 16 q^{57} - 16 q^{65} + 28 q^{71} + 8 q^{73} + 40 q^{79} + 8 q^{81} - 20 q^{89} - 40 q^{95} - 40 q^{97}+O(q^{100}) \) 8 * q - 8 * q^9 + 4 * q^15 - 4 * q^17 - 12 * q^23 - 24 * q^25 + 8 * q^31 - 12 * q^33 - 8 * q^39 + 4 * q^41 - 8 * q^55 - 16 * q^57 - 16 * q^65 + 28 * q^71 + 8 * q^73 + 40 * q^79 + 8 * q^81 - 20 * q^89 - 40 * q^95 - 40 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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