LMFDB - Embedded newform 3200.2.c.ba.2049.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3200,2,Mod(2049,3200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3200, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3200.2049");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3200 = 2^{7} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3200.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:	$25.5521286468$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2049.2
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	3200.2049
Dual form			3200.2.c.ba.2049.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.414214i q^{3} -4.82843i q^{7} +2.82843 q^{9} +O(q^{10})$	$q-0.414214i q^{3} -4.82843i q^{7} +2.82843 q^{9} -3.24264 q^{11} +5.65685i q^{13} +5.82843i q^{17} -4.41421 q^{19} -2.00000 q^{21} -0.828427i q^{23} -2.41421i q^{27} -8.00000 q^{29} -4.82843 q^{31} +1.34315i q^{33} -3.65685i q^{37} +2.34315 q^{39} -0.656854 q^{41} +10.0000i q^{43} +1.65685i q^{47} -16.3137 q^{49} +2.41421 q^{51} -3.65685i q^{53} +1.82843i q^{57} -7.65685 q^{59} +6.00000 q^{61} -13.6569i q^{63} +11.2426i q^{67} -0.343146 q^{69} -1.65685 q^{71} +0.171573i q^{73} +15.6569i q^{77} +7.17157 q^{79} +7.48528 q^{81} -7.24264i q^{83} +3.31371i q^{87} -11.8284 q^{89} +27.3137 q^{91} +2.00000i q^{93} -5.31371i q^{97} -9.17157 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 4 q^{11} - 12 q^{19} - 8 q^{21} - 32 q^{29} - 8 q^{31} + 32 q^{39} + 20 q^{41} - 20 q^{49} + 4 q^{51} - 8 q^{59} + 24 q^{61} - 24 q^{69} + 16 q^{71} + 40 q^{79} - 4 q^{81} - 36 q^{89} + 64 q^{91} - 48 q^{99}+O(q^{100}) $ 4 * q + 4 * q^11 - 12 * q^19 - 8 * q^21 - 32 * q^29 - 8 * q^31 + 32 * q^39 + 20 * q^41 - 20 * q^49 + 4 * q^51 - 8 * q^59 + 24 * q^61 - 24 * q^69 + 16 * q^71 + 40 * q^79 - 4 * q^81 - 36 * q^89 + 64 * q^91 - 48 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3200\mathbb{Z}\right)^\times$.

$n$	$901$	$1151$	$2177$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	− 0.414214i	− 0.239146i	−0.992825	−	0.119573i	$-0.961847\pi$
$3$	− 0.414214i	− 0.239146i	0.992825	−	0.119573i	$-0.0381526\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 4.82843i	− 1.82497i	−0.409106	−	0.912487i	$-0.634159\pi$
$7$	− 4.82843i	− 1.82497i	0.409106	−	0.912487i	$-0.365841\pi$
$8$	0	0
$9$	2.82843	0.942809
$10$	0	0
$11$	−3.24264	−0.977693	−0.488846	−	0.872370i	$-0.662582\pi$
$11$	−3.24264	−0.977693	−0.488846	+	0.872370i	$0.662582\pi$
$12$	0	0
$13$	5.65685i	1.56893i	0.620174	+	0.784465i	$0.287062\pi$
$13$	5.65685i	1.56893i	−0.620174	+	0.784465i	$0.712938\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.82843i	1.41360i	0.707413	+	0.706801i	$0.249862\pi$
$17$	5.82843i	1.41360i	−0.707413	+	0.706801i	$0.750138\pi$
$18$	0	0
$19$	−4.41421	−1.01269	−0.506345	−	0.862331i	$-0.669004\pi$
$19$	−4.41421	−1.01269	−0.506345	+	0.862331i	$0.669004\pi$
$20$	0	0
$21$	−2.00000	−0.436436
$22$	0	0
$23$	− 0.828427i	− 0.172739i	−0.996263	−	0.0863695i	$-0.972473\pi$
$23$	− 0.828427i	− 0.172739i	0.996263	−	0.0863695i	$-0.0275266\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 2.41421i	− 0.464616i
$28$	0	0
$29$	−8.00000	−1.48556	−0.742781	−	0.669534i	$-0.766494\pi$
$29$	−8.00000	−1.48556	−0.742781	+	0.669534i	$0.766494\pi$
$30$	0	0
$31$	−4.82843	−0.867211	−0.433606	−	0.901103i	$-0.642759\pi$
$31$	−4.82843	−0.867211	−0.433606	+	0.901103i	$0.642759\pi$
$32$	0	0
$33$	1.34315i	0.233812i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 3.65685i	− 0.601183i	−0.953753	−	0.300592i	$-0.902816\pi$
$37$	− 3.65685i	− 0.601183i	0.953753	−	0.300592i	$-0.0971841\pi$
$38$	0	0
$39$	2.34315	0.375204
$40$	0	0
$41$	−0.656854	−0.102583	−0.0512917	−	0.998684i	$-0.516334\pi$
$41$	−0.656854	−0.102583	−0.0512917	+	0.998684i	$0.516334\pi$
$42$	0	0
$43$	10.0000i	1.52499i	0.646997	+	0.762493i	$0.276025\pi$
$43$	10.0000i	1.52499i	−0.646997	+	0.762493i	$0.723975\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.65685i	0.241677i	0.992672	+	0.120839i	$0.0385583\pi$
$47$	1.65685i	0.241677i	−0.992672	+	0.120839i	$0.961442\pi$
$48$	0	0
$49$	−16.3137	−2.33053
$50$	0	0
$51$	2.41421	0.338058
$52$	0	0
$53$	− 3.65685i	− 0.502308i	−0.967947	−	0.251154i	$-0.919190\pi$
$53$	− 3.65685i	− 0.502308i	0.967947	−	0.251154i	$-0.0808100\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.82843i	0.242181i
$58$	0	0
$59$	−7.65685	−0.996838	−0.498419	−	0.866936i	$-0.666086\pi$
$59$	−7.65685	−0.996838	−0.498419	+	0.866936i	$0.666086\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	− 13.6569i	− 1.72060i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	11.2426i	1.37351i	0.726890	+	0.686754i	$0.240965\pi$
$67$	11.2426i	1.37351i	−0.726890	+	0.686754i	$0.759035\pi$
$68$	0	0
$69$	−0.343146	−0.0413099
$70$	0	0
$71$	−1.65685	−0.196632	−0.0983162	−	0.995155i	$-0.531346\pi$
$71$	−1.65685	−0.196632	−0.0983162	+	0.995155i	$0.531346\pi$
$72$	0	0
$73$	0.171573i	0.0200811i	0.999950	+	0.0100405i	$0.00319606\pi$
$73$	0.171573i	0.0200811i	−0.999950	+	0.0100405i	$0.996804\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	15.6569i	1.78426i
$78$	0	0
$79$	7.17157	0.806865	0.403432	−	0.915009i	$-0.367817\pi$
$79$	7.17157	0.806865	0.403432	+	0.915009i	$0.367817\pi$
$80$	0	0
$81$	7.48528	0.831698
$82$	0	0
$83$	− 7.24264i	− 0.794983i	−0.917606	−	0.397492i	$-0.869881\pi$
$83$	− 7.24264i	− 0.794983i	0.917606	−	0.397492i	$-0.130119\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	3.31371i	0.355267i
$88$	0	0
$89$	−11.8284	−1.25381	−0.626905	−	0.779095i	$-0.715679\pi$
$89$	−11.8284	−1.25381	−0.626905	+	0.779095i	$0.715679\pi$
$90$	0	0
$91$	27.3137	2.86325
$92$	0	0
$93$	2.00000i	0.207390i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 5.31371i	− 0.539525i	−0.962927	−	0.269763i	$-0.913055\pi$
$97$	− 5.31371i	− 0.539525i	0.962927	−	0.269763i	$-0.0869453\pi$
$98$	0	0
$99$	−9.17157	−0.921778
$100$	0	0
$101$	14.9706	1.48963	0.744813	−	0.667273i	$-0.232539\pi$
$101$	14.9706	1.48963	0.744813	+	0.667273i	$0.232539\pi$
$102$	0	0
$103$	9.65685i	0.951518i	0.879576	+	0.475759i	$0.157827\pi$
$103$	9.65685i	0.951518i	−0.879576	+	0.475759i	$0.842173\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16.0711i	1.55365i	0.629717	+	0.776824i	$0.283171\pi$
$107$	16.0711i	1.55365i	−0.629717	+	0.776824i	$0.716829\pi$
$108$	0	0
$109$	−0.343146	−0.0328674	−0.0164337	−	0.999865i	$-0.505231\pi$
$109$	−0.343146	−0.0328674	−0.0164337	+	0.999865i	$0.505231\pi$
$110$	0	0
$111$	−1.51472	−0.143771
$112$	0	0
$113$	14.6569i	1.37880i	0.724380	+	0.689400i	$0.242126\pi$
$113$	14.6569i	1.37880i	−0.724380	+	0.689400i	$0.757874\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	16.0000i	1.47920i
$118$	0	0
$119$	28.1421	2.57979
$120$	0	0
$121$	−0.485281	−0.0441165
$122$	0	0
$123$	0.272078i	0.0245324i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 8.14214i	− 0.722498i	−0.932469	−	0.361249i	$-0.882350\pi$
$127$	− 8.14214i	− 0.722498i	0.932469	−	0.361249i	$-0.117650\pi$
$128$	0	0
$129$	4.14214	0.364695
$130$	0	0
$131$	−17.3137	−1.51271	−0.756353	−	0.654164i	$-0.773021\pi$
$131$	−17.3137	−1.51271	−0.756353	+	0.654164i	$0.773021\pi$
$132$	0	0
$133$	21.3137i	1.84813i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.00000i	0.427179i	0.976924	+	0.213589i	$0.0685155\pi$
$137$	5.00000i	0.427179i	−0.976924	+	0.213589i	$0.931485\pi$
$138$	0	0
$139$	8.89949	0.754845	0.377423	−	0.926041i	$-0.376810\pi$
$139$	8.89949	0.754845	0.377423	+	0.926041i	$0.376810\pi$
$140$	0	0
$141$	0.686292	0.0577962
$142$	0	0
$143$	− 18.3431i	− 1.53393i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	6.75736i	0.557338i
$148$	0	0
$149$	−19.3137	−1.58224	−0.791120	−	0.611661i	$-0.790502\pi$
$149$	−19.3137	−1.58224	−0.791120	+	0.611661i	$0.790502\pi$
$150$	0	0
$151$	10.4853	0.853280	0.426640	−	0.904422i	$-0.359697\pi$
$151$	10.4853	0.853280	0.426640	+	0.904422i	$0.359697\pi$
$152$	0	0
$153$	16.4853i	1.33276i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 8.34315i	− 0.665856i	−0.942952	−	0.332928i	$-0.891963\pi$
$157$	− 8.34315i	− 0.665856i	0.942952	−	0.332928i	$-0.108037\pi$
$158$	0	0
$159$	−1.51472	−0.120125
$160$	0	0
$161$	−4.00000	−0.315244
$162$	0	0
$163$	5.72792i	0.448645i	0.974515	+	0.224323i	$0.0720170\pi$
$163$	5.72792i	0.448645i	−0.974515	+	0.224323i	$0.927983\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 13.6569i	− 1.05680i	−0.848996	−	0.528400i	$-0.822792\pi$
$167$	− 13.6569i	− 1.05680i	0.848996	−	0.528400i	$-0.177208\pi$
$168$	0	0
$169$	−19.0000	−1.46154
$170$	0	0
$171$	−12.4853	−0.954773
$172$	0	0
$173$	− 1.65685i	− 0.125968i	−0.998015	−	0.0629841i	$-0.979938\pi$
$173$	− 1.65685i	− 0.125968i	0.998015	−	0.0629841i	$-0.0200618\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	3.17157i	0.238390i
$178$	0	0
$179$	−24.4142	−1.82480	−0.912402	−	0.409295i	$-0.865775\pi$
$179$	−24.4142	−1.82480	−0.912402	+	0.409295i	$0.865775\pi$
$180$	0	0
$181$	4.34315	0.322823	0.161412	−	0.986887i	$-0.448395\pi$
$181$	4.34315	0.322823	0.161412	+	0.986887i	$0.448395\pi$
$182$	0	0
$183$	− 2.48528i	− 0.183717i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 18.8995i	− 1.38207i
$188$	0	0
$189$	−11.6569	−0.847911
$190$	0	0
$191$	−18.4853	−1.33755	−0.668774	−	0.743466i	$-0.733181\pi$
$191$	−18.4853	−1.33755	−0.668774	+	0.743466i	$0.733181\pi$
$192$	0	0
$193$	− 13.8284i	− 0.995392i	−0.867352	−	0.497696i	$-0.834180\pi$
$193$	− 13.8284i	− 0.995392i	0.867352	−	0.497696i	$-0.165820\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3.65685i	0.260540i	0.991479	+	0.130270i	$0.0415844\pi$
$197$	3.65685i	0.260540i	−0.991479	+	0.130270i	$0.958416\pi$
$198$	0	0
$199$	14.3431	1.01676	0.508379	−	0.861133i	$-0.330245\pi$
$199$	14.3431	1.01676	0.508379	+	0.861133i	$0.330245\pi$
$200$	0	0
$201$	4.65685	0.328469
$202$	0	0
$203$	38.6274i	2.71111i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 2.34315i	− 0.162860i
$208$	0	0
$209$	14.3137	0.990100
$210$	0	0
$211$	8.89949	0.612666	0.306333	−	0.951924i	$-0.400898\pi$
$211$	8.89949	0.612666	0.306333	+	0.951924i	$0.400898\pi$
$212$	0	0
$213$	0.686292i	0.0470239i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	23.3137i	1.58264i
$218$	0	0
$219$	0.0710678	0.00480232
$220$	0	0
$221$	−32.9706	−2.21784
$222$	0	0
$223$	− 2.34315i	− 0.156909i	−0.996918	−	0.0784543i	$-0.975002\pi$
$223$	− 2.34315i	− 0.156909i	0.996918	−	0.0784543i	$-0.0249985\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 1.31371i	− 0.0871939i	−0.999049	−	0.0435969i	$-0.986118\pi$
$227$	− 1.31371i	− 0.0871939i	0.999049	−	0.0435969i	$-0.0138817\pi$
$228$	0	0
$229$	−23.3137	−1.54061	−0.770307	−	0.637674i	$-0.779897\pi$
$229$	−23.3137	−1.54061	−0.770307	+	0.637674i	$0.779897\pi$
$230$	0	0
$231$	6.48528	0.426700
$232$	0	0
$233$	9.31371i	0.610161i	0.952327	+	0.305081i	$0.0986834\pi$
$233$	9.31371i	0.610161i	−0.952327	+	0.305081i	$0.901317\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 2.97056i	− 0.192959i
$238$	0	0
$239$	−26.6274	−1.72238	−0.861192	−	0.508279i	$-0.830282\pi$
$239$	−26.6274	−1.72238	−0.861192	+	0.508279i	$0.830282\pi$
$240$	0	0
$241$	4.17157	0.268715	0.134357	−	0.990933i	$-0.457103\pi$
$241$	4.17157	0.268715	0.134357	+	0.990933i	$0.457103\pi$
$242$	0	0
$243$	− 10.3431i	− 0.663513i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 24.9706i	− 1.58884i
$248$	0	0
$249$	−3.00000	−0.190117
$250$	0	0
$251$	17.2426	1.08835	0.544173	−	0.838973i	$-0.316844\pi$
$251$	17.2426	1.08835	0.544173	+	0.838973i	$0.316844\pi$
$252$	0	0
$253$	2.68629i	0.168886i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	17.3137i	1.08000i	0.841665	+	0.540000i	$0.181576\pi$
$257$	17.3137i	1.08000i	−0.841665	+	0.540000i	$0.818424\pi$
$258$	0	0
$259$	−17.6569	−1.09714
$260$	0	0
$261$	−22.6274	−1.40060
$262$	0	0
$263$	6.48528i	0.399900i	0.979806	+	0.199950i	$0.0640779\pi$
$263$	6.48528i	0.399900i	−0.979806	+	0.199950i	$0.935922\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	4.89949i	0.299844i
$268$	0	0
$269$	17.6569	1.07656	0.538279	−	0.842767i	$-0.319075\pi$
$269$	17.6569	1.07656	0.538279	+	0.842767i	$0.319075\pi$
$270$	0	0
$271$	−24.8284	−1.50822	−0.754110	−	0.656748i	$-0.771931\pi$
$271$	−24.8284	−1.50822	−0.754110	+	0.656748i	$0.771931\pi$
$272$	0	0
$273$	− 11.3137i	− 0.684737i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.65685i	0.0995507i	0.998760	+	0.0497754i	$0.0158505\pi$
$277$	1.65685i	0.0995507i	−0.998760	+	0.0497754i	$0.984149\pi$
$278$	0	0
$279$	−13.6569	−0.817614
$280$	0	0
$281$	24.6274	1.46915	0.734574	−	0.678528i	$-0.237382\pi$
$281$	24.6274	1.46915	0.734574	+	0.678528i	$0.237382\pi$
$282$	0	0
$283$	15.7279i	0.934928i	0.884012	+	0.467464i	$0.154832\pi$
$283$	15.7279i	0.934928i	−0.884012	+	0.467464i	$0.845168\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.17157i	0.187212i
$288$	0	0
$289$	−16.9706	−0.998268
$290$	0	0
$291$	−2.20101	−0.129025
$292$	0	0
$293$	− 5.31371i	− 0.310430i	−0.987881	−	0.155215i	$-0.950393\pi$
$293$	− 5.31371i	− 0.310430i	0.987881	−	0.155215i	$-0.0496071\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	7.82843i	0.454251i
$298$	0	0
$299$	4.68629	0.271015
$300$	0	0
$301$	48.2843	2.78306
$302$	0	0
$303$	− 6.20101i	− 0.356239i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 5.24264i	− 0.299213i	−0.988746	−	0.149607i	$-0.952199\pi$
$307$	− 5.24264i	− 0.299213i	0.988746	−	0.149607i	$-0.0478007\pi$
$308$	0	0
$309$	4.00000	0.227552
$310$	0	0
$311$	5.51472	0.312711	0.156356	−	0.987701i	$-0.450025\pi$
$311$	5.51472	0.312711	0.156356	+	0.987701i	$0.450025\pi$
$312$	0	0
$313$	− 10.0000i	− 0.565233i	−0.959233	−	0.282617i	$-0.908798\pi$
$313$	− 10.0000i	− 0.565233i	0.959233	−	0.282617i	$-0.0912024\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 13.6569i	− 0.767045i	−0.923531	−	0.383523i	$-0.874711\pi$
$317$	− 13.6569i	− 0.767045i	0.923531	−	0.383523i	$-0.125289\pi$
$318$	0	0
$319$	25.9411	1.45242
$320$	0	0
$321$	6.65685	0.371549
$322$	0	0
$323$	− 25.7279i	− 1.43154i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0.142136i	0.00786012i
$328$	0	0
$329$	8.00000	0.441054
$330$	0	0
$331$	3.10051	0.170419	0.0852096	−	0.996363i	$-0.472844\pi$
$331$	3.10051	0.170419	0.0852096	+	0.996363i	$0.472844\pi$
$332$	0	0
$333$	− 10.3431i	− 0.566801i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 18.6569i	− 1.01630i	−0.861268	−	0.508152i	$-0.830329\pi$
$337$	− 18.6569i	− 1.01630i	0.861268	−	0.508152i	$-0.169671\pi$
$338$	0	0
$339$	6.07107	0.329735
$340$	0	0
$341$	15.6569	0.847866
$342$	0	0
$343$	44.9706i	2.42818i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 0.899495i	− 0.0482874i	−0.999708	−	0.0241437i	$-0.992314\pi$
$347$	− 0.899495i	− 0.0482874i	0.999708	−	0.0241437i	$-0.00768593\pi$
$348$	0	0
$349$	−6.68629	−0.357909	−0.178954	−	0.983857i	$-0.557271\pi$
$349$	−6.68629	−0.357909	−0.178954	+	0.983857i	$0.557271\pi$
$350$	0	0
$351$	13.6569	0.728949
$352$	0	0
$353$	32.6274i	1.73658i	0.496055	+	0.868291i	$0.334781\pi$
$353$	32.6274i	1.73658i	−0.496055	+	0.868291i	$0.665219\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 11.6569i	− 0.616946i
$358$	0	0
$359$	3.17157	0.167389	0.0836946	−	0.996491i	$-0.473328\pi$
$359$	3.17157	0.167389	0.0836946	+	0.996491i	$0.473328\pi$
$360$	0	0
$361$	0.485281	0.0255411
$362$	0	0
$363$	0.201010i	0.0105503i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	32.9706i	1.72105i	0.509409	+	0.860525i	$0.329864\pi$
$367$	32.9706i	1.72105i	−0.509409	+	0.860525i	$0.670136\pi$
$368$	0	0
$369$	−1.85786	−0.0967166
$370$	0	0
$371$	−17.6569	−0.916698
$372$	0	0
$373$	− 10.6863i	− 0.553315i	−0.960969	−	0.276658i	$-0.910773\pi$
$373$	− 10.6863i	− 0.553315i	0.960969	−	0.276658i	$-0.0892268\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 45.2548i	− 2.33074i
$378$	0	0
$379$	−2.89949	−0.148937	−0.0744685	−	0.997223i	$-0.523726\pi$
$379$	−2.89949	−0.148937	−0.0744685	+	0.997223i	$0.523726\pi$
$380$	0	0
$381$	−3.37258	−0.172783
$382$	0	0
$383$	− 23.4558i	− 1.19854i	−0.800548	−	0.599269i	$-0.795458\pi$
$383$	− 23.4558i	− 1.19854i	0.800548	−	0.599269i	$-0.204542\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	28.2843i	1.43777i
$388$	0	0
$389$	17.3137	0.877840	0.438920	−	0.898526i	$-0.355361\pi$
$389$	17.3137	0.877840	0.438920	+	0.898526i	$0.355361\pi$
$390$	0	0
$391$	4.82843	0.244184
$392$	0	0
$393$	7.17157i	0.361758i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	31.3137i	1.57159i	0.618487	+	0.785795i	$0.287746\pi$
$397$	31.3137i	1.57159i	−0.618487	+	0.785795i	$0.712254\pi$
$398$	0	0
$399$	8.82843	0.441974
$400$	0	0
$401$	−30.4558	−1.52089	−0.760446	−	0.649401i	$-0.775020\pi$
$401$	−30.4558	−1.52089	−0.760446	+	0.649401i	$0.775020\pi$
$402$	0	0
$403$	− 27.3137i	− 1.36059i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	11.8579i	0.587773i
$408$	0	0
$409$	27.9706	1.38306	0.691528	−	0.722350i	$-0.256938\pi$
$409$	27.9706	1.38306	0.691528	+	0.722350i	$0.256938\pi$
$410$	0	0
$411$	2.07107	0.102158
$412$	0	0
$413$	36.9706i	1.81920i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 3.68629i	− 0.180518i
$418$	0	0
$419$	24.5563	1.19966	0.599828	−	0.800129i	$-0.295236\pi$
$419$	24.5563	1.19966	0.599828	+	0.800129i	$0.295236\pi$
$420$	0	0
$421$	−4.00000	−0.194948	−0.0974740	−	0.995238i	$-0.531076\pi$
$421$	−4.00000	−0.194948	−0.0974740	+	0.995238i	$0.531076\pi$
$422$	0	0
$423$	4.68629i	0.227855i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 28.9706i	− 1.40198i
$428$	0	0
$429$	−7.59798	−0.366834
$430$	0	0
$431$	16.1421	0.777539	0.388770	−	0.921335i	$-0.372900\pi$
$431$	16.1421	0.777539	0.388770	+	0.921335i	$0.372900\pi$
$432$	0	0
$433$	− 16.1716i	− 0.777156i	−0.921416	−	0.388578i	$-0.872966\pi$
$433$	− 16.1716i	− 0.777156i	0.921416	−	0.388578i	$-0.127034\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.65685i	0.174931i
$438$	0	0
$439$	23.3137	1.11270	0.556351	−	0.830947i	$-0.312201\pi$
$439$	23.3137	1.11270	0.556351	+	0.830947i	$0.312201\pi$
$440$	0	0
$441$	−46.1421	−2.19724
$442$	0	0
$443$	− 8.55635i	− 0.406524i	−0.979124	−	0.203262i	$-0.934846\pi$
$443$	− 8.55635i	− 0.406524i	0.979124	−	0.203262i	$-0.0651544\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	8.00000i	0.378387i
$448$	0	0
$449$	−16.4558	−0.776599	−0.388300	−	0.921533i	$-0.626937\pi$
$449$	−16.4558	−0.776599	−0.388300	+	0.921533i	$0.626937\pi$
$450$	0	0
$451$	2.12994	0.100295
$452$	0	0
$453$	− 4.34315i	− 0.204059i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 28.6569i	− 1.34051i	−0.742130	−	0.670256i	$-0.766184\pi$
$457$	− 28.6569i	− 1.34051i	0.742130	−	0.670256i	$-0.233816\pi$
$458$	0	0
$459$	14.0711	0.656781
$460$	0	0
$461$	−5.65685	−0.263466	−0.131733	−	0.991285i	$-0.542054\pi$
$461$	−5.65685	−0.263466	−0.131733	+	0.991285i	$0.542054\pi$
$462$	0	0
$463$	− 20.9706i	− 0.974585i	−0.873239	−	0.487292i	$-0.837985\pi$
$463$	− 20.9706i	− 0.974585i	0.873239	−	0.487292i	$-0.162015\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	13.3137i	0.616085i	0.951373	+	0.308042i	$0.0996739\pi$
$467$	13.3137i	0.616085i	−0.951373	+	0.308042i	$0.900326\pi$
$468$	0	0
$469$	54.2843	2.50661
$470$	0	0
$471$	−3.45584	−0.159237
$472$	0	0
$473$	− 32.4264i	− 1.49097i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 10.3431i	− 0.473580i
$478$	0	0
$479$	−40.8284	−1.86550	−0.932749	−	0.360526i	$-0.882597\pi$
$479$	−40.8284	−1.86550	−0.932749	+	0.360526i	$0.882597\pi$
$480$	0	0
$481$	20.6863	0.943214
$482$	0	0
$483$	1.65685i	0.0753895i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	37.1127i	1.68174i	0.541240	+	0.840868i	$0.317955\pi$
$487$	37.1127i	1.68174i	−0.541240	+	0.840868i	$0.682045\pi$
$488$	0	0
$489$	2.37258	0.107292
$490$	0	0
$491$	28.6274	1.29194	0.645969	−	0.763364i	$-0.276454\pi$
$491$	28.6274	1.29194	0.645969	+	0.763364i	$0.276454\pi$
$492$	0	0
$493$	− 46.6274i	− 2.09999i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	8.00000i	0.358849i
$498$	0	0
$499$	14.0000	0.626726	0.313363	−	0.949633i	$-0.398544\pi$
$499$	14.0000	0.626726	0.313363	+	0.949633i	$0.398544\pi$
$500$	0	0
$501$	−5.65685	−0.252730
$502$	0	0
$503$	− 16.9706i	− 0.756680i	−0.925667	−	0.378340i	$-0.876495\pi$
$503$	− 16.9706i	− 0.756680i	0.925667	−	0.378340i	$-0.123505\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	7.87006i	0.349522i
$508$	0	0
$509$	−14.0000	−0.620539	−0.310270	−	0.950649i	$-0.600419\pi$
$509$	−14.0000	−0.620539	−0.310270	+	0.950649i	$0.600419\pi$
$510$	0	0
$511$	0.828427	0.0366475
$512$	0	0
$513$	10.6569i	0.470512i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 5.37258i	− 0.236286i
$518$	0	0
$519$	−0.686292	−0.0301249
$520$	0	0
$521$	−6.31371	−0.276609	−0.138304	−	0.990390i	$-0.544165\pi$
$521$	−6.31371	−0.276609	−0.138304	+	0.990390i	$0.544165\pi$
$522$	0	0
$523$	− 22.4142i	− 0.980105i	−0.871693	−	0.490053i	$-0.836978\pi$
$523$	− 22.4142i	− 0.980105i	0.871693	−	0.490053i	$-0.163022\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 28.1421i	− 1.22589i
$528$	0	0
$529$	22.3137	0.970161
$530$	0	0
$531$	−21.6569	−0.939827
$532$	0	0
$533$	− 3.71573i	− 0.160946i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	10.1127i	0.436395i
$538$	0	0
$539$	52.8995	2.27854
$540$	0	0
$541$	−6.34315	−0.272713	−0.136357	−	0.990660i	$-0.543539\pi$
$541$	−6.34315	−0.272713	−0.136357	+	0.990660i	$0.543539\pi$
$542$	0	0
$543$	− 1.79899i	− 0.0772020i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 16.5563i	− 0.707898i	−0.935265	−	0.353949i	$-0.884839\pi$
$547$	− 16.5563i	− 0.707898i	0.935265	−	0.353949i	$-0.115161\pi$
$548$	0	0
$549$	16.9706	0.724286
$550$	0	0
$551$	35.3137	1.50441
$552$	0	0
$553$	− 34.6274i	− 1.47251i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	37.6569i	1.59557i	0.602941	+	0.797786i	$0.293996\pi$
$557$	37.6569i	1.59557i	−0.602941	+	0.797786i	$0.706004\pi$
$558$	0	0
$559$	−56.5685	−2.39259
$560$	0	0
$561$	−7.82843	−0.330516
$562$	0	0
$563$	26.9706i	1.13667i	0.822796	+	0.568337i	$0.192413\pi$
$563$	26.9706i	1.13667i	−0.822796	+	0.568337i	$0.807587\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 36.1421i	− 1.51783i
$568$	0	0
$569$	3.68629	0.154537	0.0772687	−	0.997010i	$-0.475380\pi$
$569$	3.68629	0.154537	0.0772687	+	0.997010i	$0.475380\pi$
$570$	0	0
$571$	−35.9411	−1.50409	−0.752045	−	0.659112i	$-0.770932\pi$
$571$	−35.9411	−1.50409	−0.752045	+	0.659112i	$0.770932\pi$
$572$	0	0
$573$	7.65685i	0.319870i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	15.2843i	0.636293i	0.948042	+	0.318146i	$0.103060\pi$
$577$	15.2843i	0.636293i	−0.948042	+	0.318146i	$0.896940\pi$
$578$	0	0
$579$	−5.72792	−0.238044
$580$	0	0
$581$	−34.9706	−1.45082
$582$	0	0
$583$	11.8579i	0.491103i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 14.0711i	− 0.580775i	−0.956909	−	0.290388i	$-0.906216\pi$
$587$	− 14.0711i	− 0.580775i	0.956909	−	0.290388i	$-0.0937842\pi$
$588$	0	0
$589$	21.3137	0.878216
$590$	0	0
$591$	1.51472	0.0623072
$592$	0	0
$593$	− 7.00000i	− 0.287456i	−0.989617	−	0.143728i	$-0.954091\pi$
$593$	− 7.00000i	− 0.287456i	0.989617	−	0.143728i	$-0.0459090\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 5.94113i	− 0.243154i
$598$	0	0
$599$	−46.7696	−1.91095	−0.955476	−	0.295069i	$-0.904657\pi$
$599$	−46.7696	−1.91095	−0.955476	+	0.295069i	$0.904657\pi$
$600$	0	0
$601$	6.85786	0.279738	0.139869	−	0.990170i	$-0.455332\pi$
$601$	6.85786	0.279738	0.139869	+	0.990170i	$0.455332\pi$
$602$	0	0
$603$	31.7990i	1.29495i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	36.0000i	1.46119i	0.682808	+	0.730597i	$0.260758\pi$
$607$	36.0000i	1.46119i	−0.682808	+	0.730597i	$0.739242\pi$
$608$	0	0
$609$	16.0000	0.648353
$610$	0	0
$611$	−9.37258	−0.379174
$612$	0	0
$613$	− 29.3137i	− 1.18397i	−0.805949	−	0.591985i	$-0.798345\pi$
$613$	− 29.3137i	− 1.18397i	0.805949	−	0.591985i	$-0.201655\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	18.0000i	0.724653i	0.932051	+	0.362326i	$0.118017\pi$
$617$	18.0000i	0.724653i	−0.932051	+	0.362326i	$0.881983\pi$
$618$	0	0
$619$	−9.31371	−0.374350	−0.187175	−	0.982327i	$-0.559933\pi$
$619$	−9.31371	−0.374350	−0.187175	+	0.982327i	$0.559933\pi$
$620$	0	0
$621$	−2.00000	−0.0802572
$622$	0	0
$623$	57.1127i	2.28817i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 5.92893i	− 0.236779i
$628$	0	0
$629$	21.3137	0.849833
$630$	0	0
$631$	−44.1421	−1.75727	−0.878635	−	0.477493i	$-0.841545\pi$
$631$	−44.1421	−1.75727	−0.878635	+	0.477493i	$0.841545\pi$
$632$	0	0
$633$	− 3.68629i	− 0.146517i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 92.2843i	− 3.65644i
$638$	0	0
$639$	−4.68629	−0.185387
$640$	0	0
$641$	−22.6863	−0.896055	−0.448027	−	0.894020i	$-0.647873\pi$
$641$	−22.6863	−0.896055	−0.448027	+	0.894020i	$0.647873\pi$
$642$	0	0
$643$	− 37.3137i	− 1.47151i	−0.677248	−	0.735755i	$-0.736828\pi$
$643$	− 37.3137i	− 1.47151i	0.677248	−	0.735755i	$-0.263172\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	5.65685i	0.222394i	0.993798	+	0.111197i	$0.0354684\pi$
$647$	5.65685i	0.222394i	−0.993798	+	0.111197i	$0.964532\pi$
$648$	0	0
$649$	24.8284	0.974601
$650$	0	0
$651$	9.65685	0.378482
$652$	0	0
$653$	6.00000i	0.234798i	0.993085	+	0.117399i	$0.0374557\pi$
$653$	6.00000i	0.234798i	−0.993085	+	0.117399i	$0.962544\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0.485281i	0.0189326i
$658$	0	0
$659$	−25.1838	−0.981020	−0.490510	−	0.871435i	$-0.663190\pi$
$659$	−25.1838	−0.981020	−0.490510	+	0.871435i	$0.663190\pi$
$660$	0	0
$661$	17.6569	0.686772	0.343386	−	0.939194i	$-0.388426\pi$
$661$	17.6569	0.686772	0.343386	+	0.939194i	$0.388426\pi$
$662$	0	0
$663$	13.6569i	0.530388i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	6.62742i	0.256615i
$668$	0	0
$669$	−0.970563	−0.0375241
$670$	0	0
$671$	−19.4558	−0.751085
$672$	0	0
$673$	− 2.68629i	− 0.103549i	−0.998659	−	0.0517745i	$-0.983512\pi$
$673$	− 2.68629i	− 0.103549i	0.998659	−	0.0517745i	$-0.0164877\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 51.3137i	− 1.97215i	−0.166313	−	0.986073i	$-0.553186\pi$
$677$	− 51.3137i	− 1.97215i	0.166313	−	0.986073i	$-0.446814\pi$
$678$	0	0
$679$	−25.6569	−0.984620
$680$	0	0
$681$	−0.544156	−0.0208521
$682$	0	0
$683$	− 42.6985i	− 1.63381i	−0.576771	−	0.816906i	$-0.695687\pi$
$683$	− 42.6985i	− 1.63381i	0.576771	−	0.816906i	$-0.304313\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	9.65685i	0.368432i
$688$	0	0
$689$	20.6863	0.788085
$690$	0	0
$691$	−6.21320	−0.236361	−0.118181	−	0.992992i	$-0.537706\pi$
$691$	−6.21320	−0.236361	−0.118181	+	0.992992i	$0.537706\pi$
$692$	0	0
$693$	44.2843i	1.68222i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 3.82843i	− 0.145012i
$698$	0	0
$699$	3.85786	0.145918
$700$	0	0
$701$	−24.6863	−0.932388	−0.466194	−	0.884682i	$-0.654375\pi$
$701$	−24.6863	−0.932388	−0.466194	+	0.884682i	$0.654375\pi$
$702$	0	0
$703$	16.1421i	0.608812i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 72.2843i	− 2.71853i
$708$	0	0
$709$	−45.9411	−1.72536	−0.862678	−	0.505754i	$-0.831214\pi$
$709$	−45.9411	−1.72536	−0.862678	+	0.505754i	$0.831214\pi$
$710$	0	0
$711$	20.2843	0.760720
$712$	0	0
$713$	4.00000i	0.149801i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	11.0294i	0.411902i
$718$	0	0
$719$	27.4558	1.02393	0.511965	−	0.859006i	$-0.328918\pi$
$719$	27.4558	1.02393	0.511965	+	0.859006i	$0.328918\pi$
$720$	0	0
$721$	46.6274	1.73650
$722$	0	0
$723$	− 1.72792i	− 0.0642621i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	32.0000i	1.18681i	0.804902	+	0.593407i	$0.202218\pi$
$727$	32.0000i	1.18681i	−0.804902	+	0.593407i	$0.797782\pi$
$728$	0	0
$729$	18.1716	0.673021
$730$	0	0
$731$	−58.2843	−2.15572
$732$	0	0
$733$	− 37.6569i	− 1.39089i	−0.718580	−	0.695444i	$-0.755208\pi$
$733$	− 37.6569i	− 1.39089i	0.718580	−	0.695444i	$-0.244792\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 36.4558i	− 1.34287i
$738$	0	0
$739$	−16.6274	−0.611649	−0.305825	−	0.952088i	$-0.598932\pi$
$739$	−16.6274	−0.611649	−0.305825	+	0.952088i	$0.598932\pi$
$740$	0	0
$741$	−10.3431	−0.379965
$742$	0	0
$743$	13.7990i	0.506236i	0.967435	+	0.253118i	$0.0814561\pi$
$743$	13.7990i	0.506236i	−0.967435	+	0.253118i	$0.918544\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 20.4853i	− 0.749517i
$748$	0	0
$749$	77.5980	2.83537
$750$	0	0
$751$	−10.6274	−0.387800	−0.193900	−	0.981021i	$-0.562114\pi$
$751$	−10.6274	−0.387800	−0.193900	+	0.981021i	$0.562114\pi$
$752$	0	0
$753$	− 7.14214i	− 0.260274i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 14.3431i	− 0.521310i	−0.965432	−	0.260655i	$-0.916061\pi$
$757$	− 14.3431i	− 0.521310i	0.965432	−	0.260655i	$-0.0839386\pi$
$758$	0	0
$759$	1.11270	0.0403884
$760$	0	0
$761$	17.2843	0.626554	0.313277	−	0.949662i	$-0.398573\pi$
$761$	17.2843	0.626554	0.313277	+	0.949662i	$0.398573\pi$
$762$	0	0
$763$	1.65685i	0.0599822i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 43.3137i	− 1.56397i
$768$	0	0
$769$	28.1127	1.01377	0.506885	−	0.862014i	$-0.330797\pi$
$769$	28.1127	1.01377	0.506885	+	0.862014i	$0.330797\pi$
$770$	0	0
$771$	7.17157	0.258278
$772$	0	0
$773$	− 3.31371i	− 0.119186i	−0.998223	−	0.0595929i	$-0.981020\pi$
$773$	− 3.31371i	− 0.119186i	0.998223	−	0.0595929i	$-0.0189803\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	7.31371i	0.262378i
$778$	0	0
$779$	2.89949	0.103885
$780$	0	0
$781$	5.37258	0.192246
$782$	0	0
$783$	19.3137i	0.690216i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 28.6274i	− 1.02046i	−0.860039	−	0.510229i	$-0.829561\pi$
$787$	− 28.6274i	− 1.02046i	0.860039	−	0.510229i	$-0.170439\pi$
$788$	0	0
$789$	2.68629	0.0956345
$790$	0	0
$791$	70.7696	2.51628
$792$	0	0
$793$	33.9411i	1.20528i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 42.0000i	− 1.48772i	−0.668338	−	0.743858i	$-0.732994\pi$
$797$	− 42.0000i	− 1.48772i	0.668338	−	0.743858i	$-0.267006\pi$
$798$	0	0
$799$	−9.65685	−0.341635
$800$	0	0
$801$	−33.4558	−1.18210
$802$	0	0
$803$	− 0.556349i	− 0.0196331i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 7.31371i	− 0.257455i
$808$	0	0
$809$	4.62742	0.162691	0.0813457	−	0.996686i	$-0.474078\pi$
$809$	4.62742	0.162691	0.0813457	+	0.996686i	$0.474078\pi$
$810$	0	0
$811$	31.9411	1.12160	0.560802	−	0.827950i	$-0.310493\pi$
$811$	31.9411	1.12160	0.560802	+	0.827950i	$0.310493\pi$
$812$	0	0
$813$	10.2843i	0.360685i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 44.1421i	− 1.54434i
$818$	0	0
$819$	77.2548	2.69950
$820$	0	0
$821$	4.62742	0.161498	0.0807490	−	0.996734i	$-0.474269\pi$
$821$	4.62742	0.161498	0.0807490	+	0.996734i	$0.474269\pi$
$822$	0	0
$823$	− 20.9706i	− 0.730988i	−0.930814	−	0.365494i	$-0.880900\pi$
$823$	− 20.9706i	− 0.730988i	0.930814	−	0.365494i	$-0.119100\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	33.9289i	1.17982i	0.807467	+	0.589912i	$0.200838\pi$
$827$	33.9289i	1.17982i	−0.807467	+	0.589912i	$0.799162\pi$
$828$	0	0
$829$	−20.0000	−0.694629	−0.347314	−	0.937749i	$-0.612906\pi$
$829$	−20.0000	−0.694629	−0.347314	+	0.937749i	$0.612906\pi$
$830$	0	0
$831$	0.686292	0.0238072
$832$	0	0
$833$	− 95.0833i	− 3.29444i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	11.6569i	0.402920i
$838$	0	0
$839$	4.00000	0.138095	0.0690477	−	0.997613i	$-0.478004\pi$
$839$	4.00000	0.138095	0.0690477	+	0.997613i	$0.478004\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	0	0
$843$	− 10.2010i	− 0.351341i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2.34315i	0.0805114i
$848$	0	0
$849$	6.51472	0.223585
$850$	0	0
$851$	−3.02944	−0.103848
$852$	0	0
$853$	− 28.6274i	− 0.980184i	−0.871671	−	0.490092i	$-0.836963\pi$
$853$	− 28.6274i	− 0.980184i	0.871671	−	0.490092i	$-0.163037\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 11.0000i	− 0.375753i	−0.982193	−	0.187876i	$-0.939840\pi$
$857$	− 11.0000i	− 0.375753i	0.982193	−	0.187876i	$-0.0601604\pi$
$858$	0	0
$859$	−11.7868	−0.402160	−0.201080	−	0.979575i	$-0.564445\pi$
$859$	−11.7868	−0.402160	−0.201080	+	0.979575i	$0.564445\pi$
$860$	0	0
$861$	1.31371	0.0447711
$862$	0	0
$863$	4.00000i	0.136162i	0.997680	+	0.0680808i	$0.0216876\pi$
$863$	4.00000i	0.136162i	−0.997680	+	0.0680808i	$0.978312\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	7.02944i	0.238732i
$868$	0	0
$869$	−23.2548	−0.788866
$870$	0	0
$871$	−63.5980	−2.15494
$872$	0	0
$873$	− 15.0294i	− 0.508669i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	37.6569i	1.27158i	0.771861	+	0.635791i	$0.219326\pi$
$877$	37.6569i	1.27158i	−0.771861	+	0.635791i	$0.780674\pi$
$878$	0	0
$879$	−2.20101	−0.0742382
$880$	0	0
$881$	−37.3137	−1.25713	−0.628565	−	0.777757i	$-0.716358\pi$
$881$	−37.3137	−1.25713	−0.628565	+	0.777757i	$0.716358\pi$
$882$	0	0
$883$	− 34.7574i	− 1.16968i	−0.811149	−	0.584839i	$-0.801158\pi$
$883$	− 34.7574i	− 1.16968i	0.811149	−	0.584839i	$-0.198842\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	33.6569i	1.13009i	0.825061	+	0.565043i	$0.191141\pi$
$887$	33.6569i	1.13009i	−0.825061	+	0.565043i	$0.808859\pi$
$888$	0	0
$889$	−39.3137	−1.31854
$890$	0	0
$891$	−24.2721	−0.813145
$892$	0	0
$893$	− 7.31371i	− 0.244744i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 1.94113i	− 0.0648123i
$898$	0	0
$899$	38.6274	1.28830
$900$	0	0
$901$	21.3137	0.710063
$902$	0	0
$903$	− 20.0000i	− 0.665558i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 22.6863i	− 0.753286i	−0.926359	−	0.376643i	$-0.877078\pi$
$907$	− 22.6863i	− 0.753286i	0.926359	−	0.376643i	$-0.122922\pi$
$908$	0	0
$909$	42.3431	1.40443
$910$	0	0
$911$	20.2843	0.672048	0.336024	−	0.941853i	$-0.390918\pi$
$911$	20.2843	0.672048	0.336024	+	0.941853i	$0.390918\pi$
$912$	0	0
$913$	23.4853i	0.777249i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	83.5980i	2.76065i
$918$	0	0
$919$	−44.8284	−1.47875	−0.739377	−	0.673292i	$-0.764880\pi$
$919$	−44.8284	−1.47875	−0.739377	+	0.673292i	$0.764880\pi$
$920$	0	0
$921$	−2.17157	−0.0715558
$922$	0	0
$923$	− 9.37258i	− 0.308502i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	27.3137i	0.897100i
$928$	0	0
$929$	44.6274	1.46418	0.732089	−	0.681209i	$-0.238545\pi$
$929$	44.6274	1.46418	0.732089	+	0.681209i	$0.238545\pi$
$930$	0	0
$931$	72.0122	2.36010
$932$	0	0
$933$	− 2.28427i	− 0.0747837i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 42.1127i	− 1.37576i	−0.725824	−	0.687881i	$-0.758541\pi$
$937$	− 42.1127i	− 1.37576i	0.725824	−	0.687881i	$-0.241459\pi$
$938$	0	0
$939$	−4.14214	−0.135173
$940$	0	0
$941$	7.02944	0.229153	0.114577	−	0.993414i	$-0.463449\pi$
$941$	7.02944	0.229153	0.114577	+	0.993414i	$0.463449\pi$
$942$	0	0
$943$	0.544156i	0.0177202i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	45.5980i	1.48174i	0.671651	+	0.740868i	$0.265585\pi$
$947$	45.5980i	1.48174i	−0.671651	+	0.740868i	$0.734415\pi$
$948$	0	0
$949$	−0.970563	−0.0315058
$950$	0	0
$951$	−5.65685	−0.183436
$952$	0	0
$953$	10.3137i	0.334094i	0.985949	+	0.167047i	$0.0534231\pi$
$953$	10.3137i	0.334094i	−0.985949	+	0.167047i	$0.946577\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 10.7452i	− 0.347342i
$958$	0	0
$959$	24.1421	0.779590
$960$	0	0
$961$	−7.68629	−0.247945
$962$	0	0
$963$	45.4558i	1.46479i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 2.62742i	− 0.0844920i	−0.999107	−	0.0422460i	$-0.986549\pi$
$967$	− 2.62742i	− 0.0844920i	0.999107	−	0.0422460i	$-0.0134513\pi$
$968$	0	0
$969$	−10.6569	−0.342347
$970$	0	0
$971$	23.1005	0.741330	0.370665	−	0.928767i	$-0.379130\pi$
$971$	23.1005	0.741330	0.370665	+	0.928767i	$0.379130\pi$
$972$	0	0
$973$	− 42.9706i	− 1.37757i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	38.1127i	1.21933i	0.792658	+	0.609667i	$0.208697\pi$
$977$	38.1127i	1.21933i	−0.792658	+	0.609667i	$0.791303\pi$
$978$	0	0
$979$	38.3553	1.22584
$980$	0	0
$981$	−0.970563	−0.0309877
$982$	0	0
$983$	5.79899i	0.184959i	0.995715	+	0.0924795i	$0.0294793\pi$
$983$	5.79899i	0.184959i	−0.995715	+	0.0924795i	$0.970521\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 3.31371i	− 0.105477i
$988$	0	0
$989$	8.28427	0.263425
$990$	0	0
$991$	34.0833	1.08269	0.541345	−	0.840800i	$-0.317915\pi$
$991$	34.0833	1.08269	0.541345	+	0.840800i	$0.317915\pi$
$992$	0	0
$993$	− 1.28427i	− 0.0407551i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	16.9706i	0.537463i	0.963215	+	0.268732i	$0.0866045\pi$
$997$	16.9706i	0.537463i	−0.963215	+	0.268732i	$0.913396\pi$
$998$	0	0
$999$	−8.82843	−0.279319

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3200.2.c.ba.2049.2		4
4.3	odd	2		3200.2.c.y.2049.3		4
5.2	odd	4		3200.2.a.bn.1.1	yes	2
5.3	odd	4		3200.2.a.bd.1.2	yes	2
5.4	even	2	inner	3200.2.c.ba.2049.3		4
8.3	odd	2		3200.2.c.bb.2049.2		4
8.5	even	2		3200.2.c.z.2049.3		4
20.3	even	4		3200.2.a.bm.1.1	yes	2
20.7	even	4		3200.2.a.bc.1.2	✓	2
20.19	odd	2		3200.2.c.y.2049.2		4
40.3	even	4		3200.2.a.bh.1.2	yes	2
40.13	odd	4		3200.2.a.bi.1.1	yes	2
40.19	odd	2		3200.2.c.bb.2049.3		4
40.27	even	4		3200.2.a.bj.1.1	yes	2
40.29	even	2		3200.2.c.z.2049.2		4
40.37	odd	4		3200.2.a.bg.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3200.2.a.bc.1.2	✓	2	20.7	even	4
3200.2.a.bd.1.2	yes	2	5.3	odd	4
3200.2.a.bg.1.2	yes	2	40.37	odd	4
3200.2.a.bh.1.2	yes	2	40.3	even	4
3200.2.a.bi.1.1	yes	2	40.13	odd	4
3200.2.a.bj.1.1	yes	2	40.27	even	4
3200.2.a.bm.1.1	yes	2	20.3	even	4
3200.2.a.bn.1.1	yes	2	5.2	odd	4
3200.2.c.y.2049.2		4	20.19	odd	2
3200.2.c.y.2049.3		4	4.3	odd	2
3200.2.c.z.2049.2		4	40.29	even	2
3200.2.c.z.2049.3		4	8.5	even	2
3200.2.c.ba.2049.2		4	1.1	even	1	trivial
3200.2.c.ba.2049.3		4	5.4	even	2	inner
3200.2.c.bb.2049.2		4	8.3	odd	2
3200.2.c.bb.2049.3		4	40.19	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:	\( N \)	\(=\)	\( 3200 = 2^{7} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3200.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.414214i q^{3} -4.82843i q^{7} +2.82843 q^{9} +O(q^{10})\)	\(q-0.414214i q^{3} -4.82843i q^{7} +2.82843 q^{9} -3.24264 q^{11} +5.65685i q^{13} +5.82843i q^{17} -4.41421 q^{19} -2.00000 q^{21} -0.828427i q^{23} -2.41421i q^{27} -8.00000 q^{29} -4.82843 q^{31} +1.34315i q^{33} -3.65685i q^{37} +2.34315 q^{39} -0.656854 q^{41} +10.0000i q^{43} +1.65685i q^{47} -16.3137 q^{49} +2.41421 q^{51} -3.65685i q^{53} +1.82843i q^{57} -7.65685 q^{59} +6.00000 q^{61} -13.6569i q^{63} +11.2426i q^{67} -0.343146 q^{69} -1.65685 q^{71} +0.171573i q^{73} +15.6569i q^{77} +7.17157 q^{79} +7.48528 q^{81} -7.24264i q^{83} +3.31371i q^{87} -11.8284 q^{89} +27.3137 q^{91} +2.00000i q^{93} -5.31371i q^{97} -9.17157 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 4 q^{11} - 12 q^{19} - 8 q^{21} - 32 q^{29} - 8 q^{31} + 32 q^{39} + 20 q^{41} - 20 q^{49} + 4 q^{51} - 8 q^{59} + 24 q^{61} - 24 q^{69} + 16 q^{71} + 40 q^{79} - 4 q^{81} - 36 q^{89} + 64 q^{91} - 48 q^{99}+O(q^{100}) \) 4 * q + 4 * q^11 - 12 * q^19 - 8 * q^21 - 32 * q^29 - 8 * q^31 + 32 * q^39 + 20 * q^41 - 20 * q^49 + 4 * q^51 - 8 * q^59 + 24 * q^61 - 24 * q^69 + 16 * q^71 + 40 * q^79 - 4 * q^81 - 36 * q^89 + 64 * q^91 - 48 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more