LMFDB - Embedded newform 1568.3.g.m.687.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1568,3,Mod(687,1568)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1568, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1568.687");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1568 = 2^{5} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1568.g (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:	$42.7249054517$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.292213762624.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 2x^{6} - 2x^{5} + 24x^{4} - 8x^{3} - 32x^{2} - 64x + 256 $ x^8 - x^7 - 2x^6 - 2x^5 + 24x^4 - 8x^3 - 32x^2 - 64x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			687.8
Root			$-1.67467 + 1.09337i$ of defining polynomial
Character	$\chi$	$=$	1568.687
Dual form			1568.3.g.m.687.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.56747 q^{3} +5.73252i q^{5} +11.8618 q^{9} +O(q^{10})$	$q+4.56747 q^{3} +5.73252i q^{5} +11.8618 q^{9} +1.40065 q^{11} -19.0821i q^{13} +26.1831i q^{15} +32.2699 q^{17} +12.5675 q^{19} +15.8893i q^{23} -7.86180 q^{25} +13.0712 q^{27} +3.29194i q^{29} -22.6705i q^{31} +6.39741 q^{33} +54.1537i q^{37} -87.1569i q^{39} +7.59607 q^{41} +20.8478 q^{43} +67.9980i q^{45} -21.6384i q^{47} +147.392 q^{51} -0.356667i q^{53} +8.02924i q^{55} +57.4016 q^{57} +26.8583 q^{59} +86.2287i q^{61} +109.389 q^{65} -114.523 q^{67} +72.5739i q^{69} -104.792i q^{71} +24.3974 q^{73} -35.9085 q^{75} +117.128i q^{79} -47.0539 q^{81} +79.2706 q^{83} +184.988i q^{85} +15.0359i q^{87} -2.66078 q^{89} -103.547i q^{93} +72.0433i q^{95} +52.0930 q^{97} +16.6142 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{3} + 48 q^{9}+O(q^{10}) $ 8 * q - 8 * q^3 + 48 * q^9	$ 8 q - 8 q^{3} + 48 q^{9} + 32 q^{11} + 80 q^{17} + 56 q^{19} - 16 q^{25} - 32 q^{27} - 32 q^{33} - 128 q^{41} + 368 q^{51} + 56 q^{57} + 104 q^{59} - 72 q^{65} - 304 q^{67} + 112 q^{73} + 72 q^{75} + 48 q^{81} + 72 q^{83} + 512 q^{89} - 64 q^{97} - 256 q^{99}+O(q^{100}) $ 8 * q - 8 * q^3 + 48 * q^9 + 32 * q^11 + 80 * q^17 + 56 * q^19 - 16 * q^25 - 32 * q^27 - 32 * q^33 - 128 * q^41 + 368 * q^51 + 56 * q^57 + 104 * q^59 - 72 * q^65 - 304 * q^67 + 112 * q^73 + 72 * q^75 + 48 * q^81 + 72 * q^83 + 512 * q^89 - 64 * q^97 - 256 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1568\mathbb{Z}\right)^\times$.

$n$	$197$	$1471$	$1473$
$\chi(n)$	$-1$	$-1$	$1$

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
$3$	4.56747	1.52249	0.761245	−	0.648464i	$-0.224588\pi$
$3$	4.56747	1.52249	0.761245	+	0.648464i	$0.224588\pi$
$4$	0	0
$5$	5.73252i	1.14650i	0.819379	+	0.573252i	$0.194318\pi$
$5$	5.73252i	1.14650i	−0.819379	+	0.573252i	$0.805682\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	11.8618	1.31798
$10$	0	0
$11$	1.40065	0.127332	0.0636658	−	0.997971i	$-0.479721\pi$
$11$	1.40065	0.127332	0.0636658	+	0.997971i	$0.479721\pi$
$12$	0	0
$13$	− 19.0821i	− 1.46785i	−0.679228	−	0.733927i	$-0.737685\pi$
$13$	− 19.0821i	− 1.46785i	0.679228	−	0.733927i	$-0.262315\pi$
$14$	0	0
$15$	26.1831i	1.74554i
$16$	0	0
$17$	32.2699	1.89823	0.949114	−	0.314932i	$-0.101982\pi$
$17$	32.2699	1.89823	0.949114	+	0.314932i	$0.101982\pi$
$18$	0	0
$19$	12.5675	0.661446	0.330723	−	0.943728i	$-0.392707\pi$
$19$	12.5675	0.661446	0.330723	+	0.943728i	$0.392707\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	15.8893i	0.690839i	0.938448	+	0.345419i	$0.112263\pi$
$23$	15.8893i	0.690839i	−0.938448	+	0.345419i	$0.887737\pi$
$24$	0	0
$25$	−7.86180	−0.314472
$26$	0	0
$27$	13.0712	0.484118
$28$	0	0
$29$	3.29194i	0.113515i	0.998388	+	0.0567576i	$0.0180762\pi$
$29$	3.29194i	0.113515i	−0.998388	+	0.0567576i	$0.981924\pi$
$30$	0	0
$31$	− 22.6705i	− 0.731306i	−0.930751	−	0.365653i	$-0.880846\pi$
$31$	− 22.6705i	− 0.731306i	0.930751	−	0.365653i	$-0.119154\pi$
$32$	0	0
$33$	6.39741	0.193861
$34$	0	0
$35$	0	0
$36$	0	0
$37$	54.1537i	1.46361i	0.681512	+	0.731807i	$0.261323\pi$
$37$	54.1537i	1.46361i	−0.681512	+	0.731807i	$0.738677\pi$
$38$	0	0
$39$	− 87.1569i	− 2.23479i
$40$	0	0
$41$	7.59607	0.185270	0.0926350	−	0.995700i	$-0.470471\pi$
$41$	7.59607	0.185270	0.0926350	+	0.995700i	$0.470471\pi$
$42$	0	0
$43$	20.8478	0.484833	0.242417	−	0.970172i	$-0.422060\pi$
$43$	20.8478	0.484833	0.242417	+	0.970172i	$0.422060\pi$
$44$	0	0
$45$	67.9980i	1.51107i
$46$	0	0
$47$	− 21.6384i	− 0.460392i	−0.973144	−	0.230196i	$-0.926063\pi$
$47$	− 21.6384i	− 0.460392i	0.973144	−	0.230196i	$-0.0739367\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	147.392	2.89004
$52$	0	0
$53$	− 0.356667i	− 0.00672957i	−0.999994	−	0.00336479i	$-0.998929\pi$
$53$	− 0.356667i	− 0.00672957i	0.999994	−	0.00336479i	$-0.00107105\pi$
$54$	0	0
$55$	8.02924i	0.145986i
$56$	0	0
$57$	57.4016	1.00705
$58$	0	0
$59$	26.8583	0.455226	0.227613	−	0.973752i	$-0.426908\pi$
$59$	26.8583	0.455226	0.227613	+	0.973752i	$0.426908\pi$
$60$	0	0
$61$	86.2287i	1.41359i	0.707420	+	0.706793i	$0.249859\pi$
$61$	86.2287i	1.41359i	−0.707420	+	0.706793i	$0.750141\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	109.389	1.68290
$66$	0	0
$67$	−114.523	−1.70929	−0.854646	−	0.519211i	$-0.826226\pi$
$67$	−114.523	−1.70929	−0.854646	+	0.519211i	$0.826226\pi$
$68$	0	0
$69$	72.5739i	1.05180i
$70$	0	0
$71$	− 104.792i	− 1.47594i	−0.674834	−	0.737969i	$-0.735785\pi$
$71$	− 104.792i	− 1.47594i	0.674834	−	0.737969i	$-0.264215\pi$
$72$	0	0
$73$	24.3974	0.334211	0.167106	−	0.985939i	$-0.446558\pi$
$73$	24.3974	0.334211	0.167106	+	0.985939i	$0.446558\pi$
$74$	0	0
$75$	−35.9085	−0.478781
$76$	0	0
$77$	0	0
$78$	0	0
$79$	117.128i	1.48263i	0.671157	+	0.741315i	$0.265798\pi$
$79$	117.128i	1.48263i	−0.671157	+	0.741315i	$0.734202\pi$
$80$	0	0
$81$	−47.0539	−0.580913
$82$	0	0
$83$	79.2706	0.955067	0.477534	−	0.878614i	$-0.341531\pi$
$83$	79.2706	0.955067	0.477534	+	0.878614i	$0.341531\pi$
$84$	0	0
$85$	184.988i	2.17633i
$86$	0	0
$87$	15.0359i	0.172826i
$88$	0	0
$89$	−2.66078	−0.0298964	−0.0149482	−	0.999888i	$-0.504758\pi$
$89$	−2.66078	−0.0298964	−0.0149482	+	0.999888i	$0.504758\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 103.547i	− 1.11341i
$94$	0	0
$95$	72.0433i	0.758350i
$96$	0	0
$97$	52.0930	0.537042	0.268521	−	0.963274i	$-0.413465\pi$
$97$	52.0930	0.537042	0.268521	+	0.963274i	$0.413465\pi$
$98$	0	0
$99$	16.6142	0.167820
$100$	0	0
$101$	91.4742i	0.905685i	0.891591	+	0.452842i	$0.149590\pi$
$101$	91.4742i	0.905685i	−0.891591	+	0.452842i	$0.850410\pi$
$102$	0	0
$103$	− 39.7891i	− 0.386302i	−0.981169	−	0.193151i	$-0.938129\pi$
$103$	− 39.7891i	− 0.386302i	0.981169	−	0.193151i	$-0.0618708\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−82.6631	−0.772552	−0.386276	−	0.922383i	$-0.626239\pi$
$107$	−82.6631	−0.772552	−0.386276	+	0.922383i	$0.626239\pi$
$108$	0	0
$109$	29.4719i	0.270384i	0.990819	+	0.135192i	$0.0431652\pi$
$109$	29.4719i	0.270384i	−0.990819	+	0.135192i	$0.956835\pi$
$110$	0	0
$111$	247.346i	2.22834i
$112$	0	0
$113$	159.133	1.40826	0.704130	−	0.710071i	$-0.251337\pi$
$113$	159.133	1.40826	0.704130	+	0.710071i	$0.251337\pi$
$114$	0	0
$115$	−91.0857	−0.792049
$116$	0	0
$117$	− 226.348i	− 1.93460i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−119.038	−0.983787
$122$	0	0
$123$	34.6948	0.282072
$124$	0	0
$125$	98.2451i	0.785961i
$126$	0	0
$127$	16.0834i	0.126641i	0.997993	+	0.0633205i	$0.0201690\pi$
$127$	16.0834i	0.126641i	−0.997993	+	0.0633205i	$0.979831\pi$
$128$	0	0
$129$	95.2219	0.738154
$130$	0	0
$131$	−118.136	−0.901799	−0.450899	−	0.892575i	$-0.648897\pi$
$131$	−118.136	−0.901799	−0.450899	+	0.892575i	$0.648897\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	74.9308i	0.555043i
$136$	0	0
$137$	−19.1708	−0.139933	−0.0699664	−	0.997549i	$-0.522289\pi$
$137$	−19.1708	−0.139933	−0.0699664	+	0.997549i	$0.522289\pi$
$138$	0	0
$139$	104.954	0.755062	0.377531	−	0.925997i	$-0.376773\pi$
$139$	104.954	0.755062	0.377531	+	0.925997i	$0.376773\pi$
$140$	0	0
$141$	− 98.8329i	− 0.700942i
$142$	0	0
$143$	− 26.7273i	− 0.186904i
$144$	0	0
$145$	−18.8711	−0.130146
$146$	0	0
$147$	0	0
$148$	0	0
$149$	82.3906i	0.552957i	0.961020	+	0.276478i	$0.0891674\pi$
$149$	82.3906i	0.552957i	−0.961020	+	0.276478i	$0.910833\pi$
$150$	0	0
$151$	− 57.7395i	− 0.382381i	−0.981553	−	0.191190i	$-0.938765\pi$
$151$	− 57.7395i	− 0.382381i	0.981553	−	0.191190i	$-0.0612347\pi$
$152$	0	0
$153$	382.779	2.50182
$154$	0	0
$155$	129.959	0.838445
$156$	0	0
$157$	3.72975i	0.0237564i	0.999929	+	0.0118782i	$0.00378104\pi$
$157$	3.72975i	0.0237564i	−0.999929	+	0.0118782i	$0.996219\pi$
$158$	0	0
$159$	− 1.62907i	− 0.0102457i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−77.7069	−0.476729	−0.238365	−	0.971176i	$-0.576611\pi$
$163$	−77.7069	−0.476729	−0.238365	+	0.971176i	$0.576611\pi$
$164$	0	0
$165$	36.6733i	0.222262i
$166$	0	0
$167$	− 62.0837i	− 0.371759i	−0.982573	−	0.185879i	$-0.940487\pi$
$167$	− 62.0837i	− 0.371759i	0.982573	−	0.185879i	$-0.0595133\pi$
$168$	0	0
$169$	−195.127	−1.15459
$170$	0	0
$171$	149.073	0.871771
$172$	0	0
$173$	− 195.614i	− 1.13072i	−0.824846	−	0.565358i	$-0.808738\pi$
$173$	− 195.614i	− 1.13072i	0.824846	−	0.565358i	$-0.191262\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	122.675	0.693077
$178$	0	0
$179$	−72.2099	−0.403407	−0.201704	−	0.979447i	$-0.564648\pi$
$179$	−72.2099	−0.403407	−0.201704	+	0.979447i	$0.564648\pi$
$180$	0	0
$181$	− 140.980i	− 0.778895i	−0.921049	−	0.389448i	$-0.872666\pi$
$181$	− 140.980i	− 0.778895i	0.921049	−	0.389448i	$-0.127334\pi$
$182$	0	0
$183$	393.847i	2.15217i
$184$	0	0
$185$	−310.437	−1.67804
$186$	0	0
$187$	45.1987	0.241704
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 284.473i	− 1.48939i	−0.667407	−	0.744693i	$-0.732596\pi$
$191$	− 284.473i	− 1.48939i	0.667407	−	0.744693i	$-0.267404\pi$
$192$	0	0
$193$	−123.850	−0.641710	−0.320855	−	0.947128i	$-0.603970\pi$
$193$	−123.850	−0.641710	−0.320855	+	0.947128i	$0.603970\pi$
$194$	0	0
$195$	499.629	2.56220
$196$	0	0
$197$	− 108.098i	− 0.548721i	−0.961627	−	0.274361i	$-0.911534\pi$
$197$	− 108.098i	− 0.548721i	0.961627	−	0.274361i	$-0.0884662\pi$
$198$	0	0
$199$	− 331.854i	− 1.66761i	−0.552060	−	0.833804i	$-0.686158\pi$
$199$	− 331.854i	− 1.66761i	0.552060	−	0.833804i	$-0.313842\pi$
$200$	0	0
$201$	−523.079	−2.60238
$202$	0	0
$203$	0	0
$204$	0	0
$205$	43.5446i	0.212413i
$206$	0	0
$207$	188.476i	0.910510i
$208$	0	0
$209$	17.6026	0.0842229
$210$	0	0
$211$	−26.3950	−0.125095	−0.0625475	−	0.998042i	$-0.519922\pi$
$211$	−26.3950	−0.125095	−0.0625475	+	0.998042i	$0.519922\pi$
$212$	0	0
$213$	− 478.633i	− 2.24710i
$214$	0	0
$215$	119.511i	0.555864i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	111.434	0.508833
$220$	0	0
$221$	− 615.777i	− 2.78632i
$222$	0	0
$223$	161.183i	0.722796i	0.932412	+	0.361398i	$0.117700\pi$
$223$	161.183i	0.722796i	−0.932412	+	0.361398i	$0.882300\pi$
$224$	0	0
$225$	−93.2551	−0.414467
$226$	0	0
$227$	−171.279	−0.754533	−0.377266	−	0.926105i	$-0.623136\pi$
$227$	−171.279	−0.754533	−0.377266	+	0.926105i	$0.623136\pi$
$228$	0	0
$229$	229.251i	1.00110i	0.865709	+	0.500548i	$0.166868\pi$
$229$	229.251i	1.00110i	−0.865709	+	0.500548i	$0.833132\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−270.154	−1.15946	−0.579730	−	0.814808i	$-0.696842\pi$
$233$	−270.154	−1.15946	−0.579730	+	0.814808i	$0.696842\pi$
$234$	0	0
$235$	124.043	0.527841
$236$	0	0
$237$	534.978i	2.25729i
$238$	0	0
$239$	− 157.155i	− 0.657551i	−0.944408	−	0.328776i	$-0.893364\pi$
$239$	− 157.155i	− 0.657551i	0.944408	−	0.328776i	$-0.106636\pi$
$240$	0	0
$241$	97.7124	0.405445	0.202723	−	0.979236i	$-0.435021\pi$
$241$	97.7124	0.405445	0.202723	+	0.979236i	$0.435021\pi$
$242$	0	0
$243$	−332.558	−1.36855
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 239.814i	− 0.970906i
$248$	0	0
$249$	362.066	1.45408
$250$	0	0
$251$	−313.145	−1.24759	−0.623796	−	0.781587i	$-0.714410\pi$
$251$	−313.145	−1.24759	−0.623796	+	0.781587i	$0.714410\pi$
$252$	0	0
$253$	22.2553i	0.0879655i
$254$	0	0
$255$	844.927i	3.31344i
$256$	0	0
$257$	348.855	1.35741	0.678707	−	0.734409i	$-0.262541\pi$
$257$	348.855	1.35741	0.678707	+	0.734409i	$0.262541\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	39.0484i	0.149611i
$262$	0	0
$263$	− 384.364i	− 1.46146i	−0.682667	−	0.730729i	$-0.739180\pi$
$263$	− 384.364i	− 1.46146i	0.682667	−	0.730729i	$-0.260820\pi$
$264$	0	0
$265$	2.04460	0.00771548
$266$	0	0
$267$	−12.1530	−0.0455170
$268$	0	0
$269$	− 37.7613i	− 0.140376i	−0.997534	−	0.0701882i	$-0.977640\pi$
$269$	− 37.7613i	− 0.140376i	0.997534	−	0.0701882i	$-0.0223600\pi$
$270$	0	0
$271$	− 308.730i	− 1.13922i	−0.821914	−	0.569612i	$-0.807093\pi$
$271$	− 308.730i	− 1.13922i	0.821914	−	0.569612i	$-0.192907\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−11.0116	−0.0400422
$276$	0	0
$277$	244.210i	0.881623i	0.897600	+	0.440812i	$0.145309\pi$
$277$	244.210i	0.881623i	−0.897600	+	0.440812i	$0.854691\pi$
$278$	0	0
$279$	− 268.913i	− 0.963845i
$280$	0	0
$281$	266.569	0.948646	0.474323	−	0.880351i	$-0.342693\pi$
$281$	266.569	0.948646	0.474323	+	0.880351i	$0.342693\pi$
$282$	0	0
$283$	165.605	0.585177	0.292589	−	0.956238i	$-0.405483\pi$
$283$	165.605	0.585177	0.292589	+	0.956238i	$0.405483\pi$
$284$	0	0
$285$	329.056i	1.15458i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	752.346	2.60327
$290$	0	0
$291$	237.933	0.817641
$292$	0	0
$293$	− 34.3652i	− 0.117288i	−0.998279	−	0.0586438i	$-0.981322\pi$
$293$	− 34.3652i	− 0.117288i	0.998279	−	0.0586438i	$-0.0186776\pi$
$294$	0	0
$295$	153.966i	0.521918i
$296$	0	0
$297$	18.3081	0.0616434
$298$	0	0
$299$	303.201	1.01405
$300$	0	0
$301$	0	0
$302$	0	0
$303$	417.806i	1.37890i
$304$	0	0
$305$	−494.308	−1.62068
$306$	0	0
$307$	−222.934	−0.726170	−0.363085	−	0.931756i	$-0.618276\pi$
$307$	−222.934	−0.726170	−0.363085	+	0.931756i	$0.618276\pi$
$308$	0	0
$309$	− 181.736i	− 0.588141i
$310$	0	0
$311$	419.934i	1.35027i	0.737694	+	0.675135i	$0.235915\pi$
$311$	419.934i	1.35027i	−0.737694	+	0.675135i	$0.764085\pi$
$312$	0	0
$313$	293.869	0.938878	0.469439	−	0.882965i	$-0.344456\pi$
$313$	293.869	0.938878	0.469439	+	0.882965i	$0.344456\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	423.461i	1.33584i	0.744234	+	0.667919i	$0.232815\pi$
$317$	423.461i	1.33584i	−0.744234	+	0.667919i	$0.767185\pi$
$318$	0	0
$319$	4.61085i	0.0144541i
$320$	0	0
$321$	−377.561	−1.17620
$322$	0	0
$323$	405.551	1.25558
$324$	0	0
$325$	150.020i	0.461599i
$326$	0	0
$327$	134.612i	0.411658i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	126.666	0.382678	0.191339	−	0.981524i	$-0.438717\pi$
$331$	126.666	0.382678	0.191339	+	0.981524i	$0.438717\pi$
$332$	0	0
$333$	642.361i	1.92901i
$334$	0	0
$335$	− 656.503i	− 1.95971i
$336$	0	0
$337$	302.404	0.897341	0.448671	−	0.893697i	$-0.351898\pi$
$337$	302.404	0.897341	0.448671	+	0.893697i	$0.351898\pi$
$338$	0	0
$339$	726.838	2.14406
$340$	0	0
$341$	− 31.7533i	− 0.0931183i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−416.031	−1.20589
$346$	0	0
$347$	−320.532	−0.923724	−0.461862	−	0.886952i	$-0.652818\pi$
$347$	−320.532	−0.923724	−0.461862	+	0.886952i	$0.652818\pi$
$348$	0	0
$349$	− 380.678i	− 1.09077i	−0.838186	−	0.545385i	$-0.816384\pi$
$349$	− 380.678i	− 1.09077i	0.838186	−	0.545385i	$-0.183616\pi$
$350$	0	0
$351$	− 249.426i	− 0.710614i
$352$	0	0
$353$	364.369	1.03221	0.516104	−	0.856526i	$-0.327382\pi$
$353$	364.369	1.03221	0.516104	+	0.856526i	$0.327382\pi$
$354$	0	0
$355$	600.720	1.69217
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 111.995i	− 0.311965i	−0.987760	−	0.155982i	$-0.950146\pi$
$359$	− 111.995i	− 0.311965i	0.987760	−	0.155982i	$-0.0498543\pi$
$360$	0	0
$361$	−203.059	−0.562489
$362$	0	0
$363$	−543.704	−1.49781
$364$	0	0
$365$	139.859i	0.383174i
$366$	0	0
$367$	− 439.042i	− 1.19630i	−0.801384	−	0.598150i	$-0.795903\pi$
$367$	− 439.042i	− 1.19630i	0.801384	−	0.598150i	$-0.204097\pi$
$368$	0	0
$369$	90.1030	0.244182
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 254.996i	− 0.683637i	−0.939766	−	0.341818i	$-0.888957\pi$
$373$	− 254.996i	− 0.683637i	0.939766	−	0.341818i	$-0.111043\pi$
$374$	0	0
$375$	448.732i	1.19662i
$376$	0	0
$377$	62.8172	0.166624
$378$	0	0
$379$	−603.048	−1.59116	−0.795578	−	0.605852i	$-0.792833\pi$
$379$	−603.048	−1.59116	−0.795578	+	0.605852i	$0.792833\pi$
$380$	0	0
$381$	73.4605i	0.192810i
$382$	0	0
$383$	− 73.3855i	− 0.191607i	−0.995400	−	0.0958035i	$-0.969458\pi$
$383$	− 73.3855i	− 0.191607i	0.995400	−	0.0958035i	$-0.0305420\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	247.293	0.639000
$388$	0	0
$389$	− 340.800i	− 0.876092i	−0.898953	−	0.438046i	$-0.855671\pi$
$389$	− 340.800i	− 0.876092i	0.898953	−	0.438046i	$-0.144329\pi$
$390$	0	0
$391$	512.745i	1.31137i
$392$	0	0
$393$	−539.581	−1.37298
$394$	0	0
$395$	−671.438	−1.69984
$396$	0	0
$397$	− 111.540i	− 0.280957i	−0.990084	−	0.140478i	$-0.955136\pi$
$397$	− 111.540i	− 0.280957i	0.990084	−	0.140478i	$-0.0448640\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	340.535	0.849215	0.424607	−	0.905378i	$-0.360412\pi$
$401$	340.535	0.849215	0.424607	+	0.905378i	$0.360412\pi$
$402$	0	0
$403$	−432.600	−1.07345
$404$	0	0
$405$	− 269.738i	− 0.666019i
$406$	0	0
$407$	75.8502i	0.186364i
$408$	0	0
$409$	−666.959	−1.63071	−0.815354	−	0.578963i	$-0.803457\pi$
$409$	−666.959	−1.63071	−0.815354	+	0.578963i	$0.803457\pi$
$410$	0	0
$411$	−87.5620	−0.213046
$412$	0	0
$413$	0	0
$414$	0	0
$415$	454.420i	1.09499i
$416$	0	0
$417$	479.373	1.14958
$418$	0	0
$419$	−200.191	−0.477783	−0.238891	−	0.971046i	$-0.576784\pi$
$419$	−200.191	−0.477783	−0.238891	+	0.971046i	$0.576784\pi$
$420$	0	0
$421$	15.9136i	0.0377996i	0.999821	+	0.0188998i	$0.00601636\pi$
$421$	15.9136i	0.0377996i	−0.999821	+	0.0188998i	$0.993984\pi$
$422$	0	0
$423$	− 256.671i	− 0.606786i
$424$	0	0
$425$	−253.699	−0.596940
$426$	0	0
$427$	0	0
$428$	0	0
$429$	− 122.076i	− 0.284560i
$430$	0	0
$431$	628.013i	1.45711i	0.684989	+	0.728553i	$0.259807\pi$
$431$	628.013i	1.45711i	−0.684989	+	0.728553i	$0.740193\pi$
$432$	0	0
$433$	−789.232	−1.82271	−0.911353	−	0.411625i	$-0.864961\pi$
$433$	−789.232	−1.82271	−0.911353	+	0.411625i	$0.864961\pi$
$434$	0	0
$435$	−86.1933	−0.198146
$436$	0	0
$437$	199.688i	0.456952i
$438$	0	0
$439$	665.570i	1.51610i	0.652194	+	0.758052i	$0.273849\pi$
$439$	665.570i	1.51610i	−0.652194	+	0.758052i	$0.726151\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−507.152	−1.14481	−0.572406	−	0.819970i	$-0.693990\pi$
$443$	−507.152	−1.14481	−0.572406	+	0.819970i	$0.693990\pi$
$444$	0	0
$445$	− 15.2530i	− 0.0342764i
$446$	0	0
$447$	376.317i	0.841872i
$448$	0	0
$449$	−279.029	−0.621446	−0.310723	−	0.950501i	$-0.600571\pi$
$449$	−279.029	−0.621446	−0.310723	+	0.950501i	$0.600571\pi$
$450$	0	0
$451$	10.6394	0.0235907
$452$	0	0
$453$	− 263.723i	− 0.582171i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−720.881	−1.57742	−0.788710	−	0.614765i	$-0.789251\pi$
$457$	−720.881	−1.57742	−0.788710	+	0.614765i	$0.789251\pi$
$458$	0	0
$459$	421.805	0.918966
$460$	0	0
$461$	− 483.262i	− 1.04829i	−0.851629	−	0.524145i	$-0.824385\pi$
$461$	− 483.262i	− 1.04829i	0.851629	−	0.524145i	$-0.175615\pi$
$462$	0	0
$463$	39.6326i	0.0855995i	0.999084	+	0.0427997i	$0.0136277\pi$
$463$	39.6326i	0.0855995i	−0.999084	+	0.0427997i	$0.986372\pi$
$464$	0	0
$465$	593.584	1.27652
$466$	0	0
$467$	17.7868	0.0380874	0.0190437	−	0.999819i	$-0.493938\pi$
$467$	17.7868	0.0380874	0.0190437	+	0.999819i	$0.493938\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	17.0355i	0.0361689i
$472$	0	0
$473$	29.2005	0.0617346
$474$	0	0
$475$	−98.8029	−0.208006
$476$	0	0
$477$	− 4.23072i	− 0.00886943i
$478$	0	0
$479$	− 668.616i	− 1.39586i	−0.716166	−	0.697930i	$-0.754105\pi$
$479$	− 668.616i	− 1.39586i	0.716166	−	0.697930i	$-0.245895\pi$
$480$	0	0
$481$	1033.37	2.14837
$482$	0	0
$483$	0	0
$484$	0	0
$485$	298.624i	0.615720i
$486$	0	0
$487$	− 418.484i	− 0.859311i	−0.902993	−	0.429656i	$-0.858635\pi$
$487$	− 418.484i	− 0.859311i	0.902993	−	0.429656i	$-0.141365\pi$
$488$	0	0
$489$	−354.924	−0.725816
$490$	0	0
$491$	−381.031	−0.776030	−0.388015	−	0.921653i	$-0.626839\pi$
$491$	−381.031	−0.776030	−0.388015	+	0.921653i	$0.626839\pi$
$492$	0	0
$493$	106.231i	0.215478i
$494$	0	0
$495$	95.2412i	0.192406i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	438.392	0.878541	0.439271	−	0.898355i	$-0.355237\pi$
$499$	438.392	0.878541	0.439271	+	0.898355i	$0.355237\pi$
$500$	0	0
$501$	− 283.565i	− 0.565999i
$502$	0	0
$503$	− 754.754i	− 1.50050i	−0.661151	−	0.750252i	$-0.729932\pi$
$503$	− 754.754i	− 1.50050i	0.661151	−	0.750252i	$-0.270068\pi$
$504$	0	0
$505$	−524.378	−1.03837
$506$	0	0
$507$	−891.235	−1.75786
$508$	0	0
$509$	− 494.029i	− 0.970588i	−0.874351	−	0.485294i	$-0.838713\pi$
$509$	− 494.029i	− 0.970588i	0.874351	−	0.485294i	$-0.161287\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	164.272	0.320218
$514$	0	0
$515$	228.092	0.442897
$516$	0	0
$517$	− 30.3078i	− 0.0586224i
$518$	0	0
$519$	− 893.460i	− 1.72150i
$520$	0	0
$521$	32.8747	0.0630993	0.0315496	−	0.999502i	$-0.489956\pi$
$521$	32.8747	0.0630993	0.0315496	+	0.999502i	$0.489956\pi$
$522$	0	0
$523$	−28.2755	−0.0540640	−0.0270320	−	0.999635i	$-0.508606\pi$
$523$	−28.2755	−0.0540640	−0.0270320	+	0.999635i	$0.508606\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 731.574i	− 1.38819i
$528$	0	0
$529$	276.531	0.522742
$530$	0	0
$531$	318.588	0.599977
$532$	0	0
$533$	− 144.949i	− 0.271949i
$534$	0	0
$535$	− 473.868i	− 0.885735i
$536$	0	0
$537$	−329.817	−0.614183
$538$	0	0
$539$	0	0
$540$	0	0
$541$	− 1071.59i	− 1.98077i	−0.138352	−	0.990383i	$-0.544180\pi$
$541$	− 1071.59i	− 1.98077i	0.138352	−	0.990383i	$-0.455820\pi$
$542$	0	0
$543$	− 643.922i	− 1.18586i
$544$	0	0
$545$	−168.948	−0.309997
$546$	0	0
$547$	986.888	1.80418	0.902091	−	0.431545i	$-0.142032\pi$
$547$	986.888	1.80418	0.902091	+	0.431545i	$0.142032\pi$
$548$	0	0
$549$	1022.83i	1.86307i
$550$	0	0
$551$	41.3714i	0.0750842i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−1417.91	−2.55480
$556$	0	0
$557$	483.550i	0.868133i	0.900881	+	0.434067i	$0.142922\pi$
$557$	483.550i	0.868133i	−0.900881	+	0.434067i	$0.857078\pi$
$558$	0	0
$559$	− 397.821i	− 0.711665i
$560$	0	0
$561$	206.444	0.367993
$562$	0	0
$563$	520.893	0.925210	0.462605	−	0.886564i	$-0.346915\pi$
$563$	520.893	0.925210	0.462605	+	0.886564i	$0.346915\pi$
$564$	0	0
$565$	912.236i	1.61458i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−732.959	−1.28815	−0.644077	−	0.764961i	$-0.722758\pi$
$569$	−732.959	−1.28815	−0.644077	+	0.764961i	$0.722758\pi$
$570$	0	0
$571$	999.584	1.75058	0.875292	−	0.483595i	$-0.160669\pi$
$571$	999.584	1.75058	0.875292	+	0.483595i	$0.160669\pi$
$572$	0	0
$573$	− 1299.32i	− 2.26758i
$574$	0	0
$575$	− 124.918i	− 0.217249i
$576$	0	0
$577$	−465.859	−0.807381	−0.403690	−	0.914896i	$-0.632273\pi$
$577$	−465.859	−0.807381	−0.403690	+	0.914896i	$0.632273\pi$
$578$	0	0
$579$	−565.682	−0.976998
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 0.499565i	0 0.000856887i
$584$	0	0
$585$	1297.54	2.21803
$586$	0	0
$587$	−574.851	−0.979303	−0.489651	−	0.871918i	$-0.662876\pi$
$587$	−574.851	−0.979303	−0.489651	+	0.871918i	$0.662876\pi$
$588$	0	0
$589$	− 284.911i	− 0.483719i
$590$	0	0
$591$	− 493.735i	− 0.835423i
$592$	0	0
$593$	−943.055	−1.59031	−0.795156	−	0.606405i	$-0.792611\pi$
$593$	−943.055	−1.59031	−0.795156	+	0.606405i	$0.792611\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 1515.73i	− 2.53892i
$598$	0	0
$599$	9.26699i	0.0154708i	0.999970	+	0.00773538i	$0.00246227\pi$
$599$	9.26699i	0.0154708i	−0.999970	+	0.00773538i	$0.997538\pi$
$600$	0	0
$601$	−57.7003	−0.0960072	−0.0480036	−	0.998847i	$-0.515286\pi$
$601$	−57.7003	−0.0960072	−0.0480036	+	0.998847i	$0.515286\pi$
$602$	0	0
$603$	−1358.44	−2.25281
$604$	0	0
$605$	− 682.389i	− 1.12792i
$606$	0	0
$607$	1024.68i	1.68810i	0.536264	+	0.844050i	$0.319835\pi$
$607$	1024.68i	1.68810i	−0.536264	+	0.844050i	$0.680165\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−412.906	−0.675788
$612$	0	0
$613$	404.818i	0.660389i	0.943913	+	0.330195i	$0.107114\pi$
$613$	404.818i	0.660389i	−0.943913	+	0.330195i	$0.892886\pi$
$614$	0	0
$615$	198.889i	0.323396i
$616$	0	0
$617$	894.209	1.44928	0.724642	−	0.689125i	$-0.242005\pi$
$617$	894.209	1.44928	0.724642	+	0.689125i	$0.242005\pi$
$618$	0	0
$619$	−779.388	−1.25911	−0.629554	−	0.776957i	$-0.716762\pi$
$619$	−779.388	−1.25911	−0.629554	+	0.776957i	$0.716762\pi$
$620$	0	0
$621$	207.692i	0.334447i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−759.737	−1.21558
$626$	0	0
$627$	80.3993	0.128229
$628$	0	0
$629$	1747.53i	2.77827i
$630$	0	0
$631$	− 780.191i	− 1.23644i	−0.786007	−	0.618218i	$-0.787855\pi$
$631$	− 780.191i	− 1.23644i	0.786007	−	0.618218i	$-0.212145\pi$
$632$	0	0
$633$	−120.559	−0.190456
$634$	0	0
$635$	−92.1985	−0.145194
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 1243.02i	− 1.94525i
$640$	0	0
$641$	−23.3139	−0.0363712	−0.0181856	−	0.999835i	$-0.505789\pi$
$641$	−23.3139	−0.0363712	−0.0181856	+	0.999835i	$0.505789\pi$
$642$	0	0
$643$	530.706	0.825360	0.412680	−	0.910876i	$-0.364593\pi$
$643$	530.706	0.825360	0.412680	+	0.910876i	$0.364593\pi$
$644$	0	0
$645$	545.862i	0.846297i
$646$	0	0
$647$	213.435i	0.329883i	0.986303	+	0.164942i	$0.0527436\pi$
$647$	213.435i	0.329883i	−0.986303	+	0.164942i	$0.947256\pi$
$648$	0	0
$649$	37.6190	0.0579646
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 274.874i	− 0.420941i	−0.977600	−	0.210470i	$-0.932500\pi$
$653$	− 274.874i	− 0.420941i	0.977600	−	0.210470i	$-0.0674995\pi$
$654$	0	0
$655$	− 677.215i	− 1.03392i
$656$	0	0
$657$	289.397	0.440483
$658$	0	0
$659$	−1234.48	−1.87327	−0.936633	−	0.350313i	$-0.886075\pi$
$659$	−1234.48	−1.87327	−0.936633	+	0.350313i	$0.886075\pi$
$660$	0	0
$661$	− 582.733i	− 0.881593i	−0.897607	−	0.440797i	$-0.854696\pi$
$661$	− 582.733i	− 0.881593i	0.897607	−	0.440797i	$-0.145304\pi$
$662$	0	0
$663$	− 2812.54i	− 4.24215i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−52.3066	−0.0784207
$668$	0	0
$669$	736.201i	1.10045i
$670$	0	0
$671$	120.776i	0.179994i
$672$	0	0
$673$	−399.145	−0.593083	−0.296542	−	0.955020i	$-0.595833\pi$
$673$	−399.145	−0.593083	−0.296542	+	0.955020i	$0.595833\pi$
$674$	0	0
$675$	−102.763	−0.152241
$676$	0	0
$677$	754.467i	1.11443i	0.830369	+	0.557214i	$0.188130\pi$
$677$	754.467i	1.11443i	−0.830369	+	0.557214i	$0.811870\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−782.312	−1.14877
$682$	0	0
$683$	−288.264	−0.422055	−0.211028	−	0.977480i	$-0.567681\pi$
$683$	−288.264	−0.422055	−0.211028	+	0.977480i	$0.567681\pi$
$684$	0	0
$685$	− 109.897i	− 0.160433i
$686$	0	0
$687$	1047.10i	1.52416i
$688$	0	0
$689$	−6.80596	−0.00987803
$690$	0	0
$691$	−156.692	−0.226761	−0.113380	−	0.993552i	$-0.536168\pi$
$691$	−156.692	−0.226761	−0.113380	+	0.993552i	$0.536168\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	601.649i	0.865682i
$696$	0	0
$697$	245.124	0.351685
$698$	0	0
$699$	−1233.92	−1.76527
$700$	0	0
$701$	1126.50i	1.60700i	0.595307	+	0.803498i	$0.297030\pi$
$701$	1126.50i	1.60700i	−0.595307	+	0.803498i	$0.702970\pi$
$702$	0	0
$703$	680.575i	0.968102i
$704$	0	0
$705$	566.562	0.803633
$706$	0	0
$707$	0	0
$708$	0	0
$709$	1096.17i	1.54608i	0.634356	+	0.773041i	$0.281266\pi$
$709$	1096.17i	1.54608i	−0.634356	+	0.773041i	$0.718734\pi$
$710$	0	0
$711$	1389.35i	1.95407i
$712$	0	0
$713$	360.218	0.505214
$714$	0	0
$715$	153.215	0.214286
$716$	0	0
$717$	− 717.800i	− 1.00112i
$718$	0	0
$719$	− 605.362i	− 0.841949i	−0.907072	−	0.420975i	$-0.861688\pi$
$719$	− 605.362i	− 0.841949i	0.907072	−	0.420975i	$-0.138312\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	446.298	0.617287
$724$	0	0
$725$	− 25.8806i	− 0.0356974i
$726$	0	0
$727$	443.659i	0.610260i	0.952311	+	0.305130i	$0.0986999\pi$
$727$	443.659i	0.610260i	−0.952311	+	0.305130i	$0.901300\pi$
$728$	0	0
$729$	−1095.46	−1.50269
$730$	0	0
$731$	672.757	0.920325
$732$	0	0
$733$	750.026i	1.02323i	0.859216	+	0.511614i	$0.170952\pi$
$733$	750.026i	1.02323i	−0.859216	+	0.511614i	$0.829048\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−160.406	−0.217647
$738$	0	0
$739$	−619.293	−0.838015	−0.419007	−	0.907983i	$-0.637622\pi$
$739$	−619.293	−0.838015	−0.419007	+	0.907983i	$0.637622\pi$
$740$	0	0
$741$	− 1095.34i	− 1.47819i
$742$	0	0
$743$	− 30.5255i	− 0.0410842i	−0.999789	−	0.0205421i	$-0.993461\pi$
$743$	− 30.5255i	− 0.0410842i	0.999789	−	0.0205421i	$-0.00653921\pi$
$744$	0	0
$745$	−472.306	−0.633967
$746$	0	0
$747$	940.291	1.25876
$748$	0	0
$749$	0	0
$750$	0	0
$751$	968.214i	1.28923i	0.764506	+	0.644616i	$0.222983\pi$
$751$	968.214i	1.28923i	−0.764506	+	0.644616i	$0.777017\pi$
$752$	0	0
$753$	−1430.28	−1.89945
$754$	0	0
$755$	330.993	0.438401
$756$	0	0
$757$	− 1171.15i	− 1.54710i	−0.633736	−	0.773550i	$-0.718479\pi$
$757$	− 1171.15i	− 1.54710i	0.633736	−	0.773550i	$-0.281521\pi$
$758$	0	0
$759$	101.650i	0.133927i
$760$	0	0
$761$	−235.996	−0.310113	−0.155057	−	0.987906i	$-0.549556\pi$
$761$	−235.996	−0.310113	−0.155057	+	0.987906i	$0.549556\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	2194.29i	2.86835i
$766$	0	0
$767$	− 512.513i	− 0.668205i
$768$	0	0
$769$	124.257	0.161582	0.0807912	−	0.996731i	$-0.474255\pi$
$769$	124.257	0.161582	0.0807912	+	0.996731i	$0.474255\pi$
$770$	0	0
$771$	1593.39	2.06665
$772$	0	0
$773$	− 178.223i	− 0.230560i	−0.993333	−	0.115280i	$-0.963224\pi$
$773$	− 178.223i	− 0.230560i	0.993333	−	0.115280i	$-0.0367765\pi$
$774$	0	0
$775$	178.231i	0.229975i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	95.4634	0.122546
$780$	0	0
$781$	− 146.776i	− 0.187933i
$782$	0	0
$783$	43.0296i	0.0549548i
$784$	0	0
$785$	−21.3809	−0.0272368
$786$	0	0
$787$	−1107.90	−1.40775	−0.703873	−	0.710326i	$-0.748547\pi$
$787$	−1107.90	−1.40775	−0.703873	+	0.710326i	$0.748547\pi$
$788$	0	0
$789$	− 1755.57i	− 2.22506i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	1645.43	2.07494
$794$	0	0
$795$	9.33867	0.0117468
$796$	0	0
$797$	1094.69i	1.37351i	0.726889	+	0.686755i	$0.240966\pi$
$797$	1094.69i	1.37351i	−0.726889	+	0.686755i	$0.759034\pi$
$798$	0	0
$799$	− 698.269i	− 0.873929i
$800$	0	0
$801$	−31.5617	−0.0394028
$802$	0	0
$803$	34.1722	0.0425556
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 172.473i	− 0.213722i
$808$	0	0
$809$	−1386.75	−1.71416	−0.857079	−	0.515185i	$-0.827723\pi$
$809$	−1386.75	−1.71416	−0.857079	+	0.515185i	$0.827723\pi$
$810$	0	0
$811$	−312.204	−0.384962	−0.192481	−	0.981301i	$-0.561653\pi$
$811$	−312.204	−0.384962	−0.192481	+	0.981301i	$0.561653\pi$
$812$	0	0
$813$	− 1410.11i	− 1.73446i
$814$	0	0
$815$	− 445.456i	− 0.546572i
$816$	0	0
$817$	262.005	0.320691
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 1092.89i	− 1.33117i	−0.746324	−	0.665583i	$-0.768183\pi$
$821$	− 1092.89i	− 1.33117i	0.746324	−	0.665583i	$-0.231817\pi$
$822$	0	0
$823$	907.162i	1.10226i	0.834419	+	0.551131i	$0.185804\pi$
$823$	907.162i	1.10226i	−0.834419	+	0.551131i	$0.814196\pi$
$824$	0	0
$825$	−50.2952	−0.0609638
$826$	0	0
$827$	607.144	0.734152	0.367076	−	0.930191i	$-0.380359\pi$
$827$	607.144	0.734152	0.367076	+	0.930191i	$0.380359\pi$
$828$	0	0
$829$	427.969i	0.516247i	0.966112	+	0.258124i	$0.0831042\pi$
$829$	427.969i	0.516247i	−0.966112	+	0.258124i	$0.916896\pi$
$830$	0	0
$831$	1115.42i	1.34226i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	355.896	0.426223
$836$	0	0
$837$	− 296.330i	− 0.354038i
$838$	0	0
$839$	− 1133.09i	− 1.35053i	−0.737575	−	0.675265i	$-0.764029\pi$
$839$	− 1133.09i	− 1.35053i	0.737575	−	0.675265i	$-0.235971\pi$
$840$	0	0
$841$	830.163	0.987114
$842$	0	0
$843$	1217.55	1.44430
$844$	0	0
$845$	− 1118.57i	− 1.32375i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	756.397	0.890927
$850$	0	0
$851$	−860.464	−1.01112
$852$	0	0
$853$	169.502i	0.198712i	0.995052	+	0.0993562i	$0.0316783\pi$
$853$	169.502i	0.198712i	−0.995052	+	0.0993562i	$0.968322\pi$
$854$	0	0
$855$	854.563i	0.999489i
$856$	0	0
$857$	234.079	0.273138	0.136569	−	0.990631i	$-0.456393\pi$
$857$	234.079	0.273138	0.136569	+	0.990631i	$0.456393\pi$
$858$	0	0
$859$	894.342	1.04114	0.520571	−	0.853818i	$-0.325719\pi$
$859$	894.342	1.04114	0.520571	+	0.853818i	$0.325719\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 778.580i	− 0.902178i	−0.892479	−	0.451089i	$-0.851036\pi$
$863$	− 778.580i	− 0.902178i	0.892479	−	0.451089i	$-0.148964\pi$
$864$	0	0
$865$	1121.36	1.29637
$866$	0	0
$867$	3436.32	3.96346
$868$	0	0
$869$	164.055i	0.188786i
$870$	0	0
$871$	2185.33i	2.50899i
$872$	0	0
$873$	617.917	0.707809
$874$	0	0
$875$	0	0
$876$	0	0
$877$	17.2780i	0.0197013i	0.999951	+	0.00985064i	$0.00313561\pi$
$877$	17.2780i	0.0197013i	−0.999951	+	0.00985064i	$0.996864\pi$
$878$	0	0
$879$	− 156.962i	− 0.178569i
$880$	0	0
$881$	−770.918	−0.875049	−0.437524	−	0.899207i	$-0.644145\pi$
$881$	−770.918	−0.875049	−0.437524	+	0.899207i	$0.644145\pi$
$882$	0	0
$883$	−776.362	−0.879232	−0.439616	−	0.898186i	$-0.644886\pi$
$883$	−776.362	−0.879232	−0.439616	+	0.898186i	$0.644886\pi$
$884$	0	0
$885$	703.235i	0.794616i
$886$	0	0
$887$	1630.80i	1.83856i	0.393603	+	0.919280i	$0.371228\pi$
$887$	1630.80i	1.83856i	−0.393603	+	0.919280i	$0.628772\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−65.9059	−0.0739685
$892$	0	0
$893$	− 271.940i	− 0.304524i
$894$	0	0
$895$	− 413.945i	− 0.462508i
$896$	0	0
$897$	1384.86	1.54388
$898$	0	0
$899$	74.6299	0.0830144
$900$	0	0
$901$	− 11.5096i	− 0.0127743i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	808.171	0.893007
$906$	0	0
$907$	953.863	1.05167	0.525834	−	0.850587i	$-0.323753\pi$
$907$	953.863	1.05167	0.525834	+	0.850587i	$0.323753\pi$
$908$	0	0
$909$	1085.05i	1.19367i
$910$	0	0
$911$	1681.15i	1.84539i	0.385534	+	0.922694i	$0.374017\pi$
$911$	1681.15i	1.84539i	−0.385534	+	0.922694i	$0.625983\pi$
$912$	0	0
$913$	111.030	0.121610
$914$	0	0
$915$	−2257.74	−2.46747
$916$	0	0
$917$	0	0
$918$	0	0
$919$	504.991i	0.549500i	0.961516	+	0.274750i	$0.0885952\pi$
$919$	504.991i	0.549500i	−0.961516	+	0.274750i	$0.911405\pi$
$920$	0	0
$921$	−1018.24	−1.10559
$922$	0	0
$923$	−1999.64	−2.16646
$924$	0	0
$925$	− 425.746i	− 0.460266i
$926$	0	0
$927$	− 471.971i	− 0.509138i
$928$	0	0
$929$	−983.851	−1.05904	−0.529521	−	0.848297i	$-0.677628\pi$
$929$	−983.851	−1.05904	−0.529521	+	0.848297i	$0.677628\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	1918.04i	2.05577i
$934$	0	0
$935$	259.103i	0.277115i
$936$	0	0
$937$	389.648	0.415846	0.207923	−	0.978145i	$-0.433330\pi$
$937$	389.648	0.415846	0.207923	+	0.978145i	$0.433330\pi$
$938$	0	0
$939$	1342.24	1.42943
$940$	0	0
$941$	875.465i	0.930356i	0.885217	+	0.465178i	$0.154010\pi$
$941$	875.465i	0.930356i	−0.885217	+	0.465178i	$0.845990\pi$
$942$	0	0
$943$	120.696i	0.127992i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1251.29	1.32132	0.660661	−	0.750685i	$-0.270276\pi$
$947$	1251.29	1.32132	0.660661	+	0.750685i	$0.270276\pi$
$948$	0	0
$949$	− 465.554i	− 0.490573i
$950$	0	0
$951$	1934.14i	2.03380i
$952$	0	0
$953$	−882.129	−0.925633	−0.462817	−	0.886454i	$-0.653161\pi$
$953$	−882.129	−0.925633	−0.462817	+	0.886454i	$0.653161\pi$
$954$	0	0
$955$	1630.75	1.70759
$956$	0	0
$957$	21.0599i	0.0220062i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	447.049	0.465192
$962$	0	0
$963$	−980.533	−1.01821
$964$	0	0
$965$	− 709.973i	− 0.735724i
$966$	0	0
$967$	1410.24i	1.45836i	0.684320	+	0.729182i	$0.260099\pi$
$967$	1410.24i	1.45836i	−0.684320	+	0.729182i	$0.739901\pi$
$968$	0	0
$969$	1852.34	1.91160
$970$	0	0
$971$	678.550	0.698815	0.349408	−	0.936971i	$-0.386383\pi$
$971$	678.550	0.698815	0.349408	+	0.936971i	$0.386383\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	685.210i	0.702780i
$976$	0	0
$977$	111.815	0.114447	0.0572235	−	0.998361i	$-0.481775\pi$
$977$	111.815	0.114447	0.0572235	+	0.998361i	$0.481775\pi$
$978$	0	0
$979$	−3.72682	−0.00380676
$980$	0	0
$981$	349.590i	0.356360i
$982$	0	0
$983$	202.226i	0.205723i	0.994696	+	0.102862i	$0.0327999\pi$
$983$	202.226i	0.205723i	−0.994696	+	0.102862i	$0.967200\pi$
$984$	0	0
$985$	619.675	0.629111
$986$	0	0
$987$	0	0
$988$	0	0
$989$	331.257i	0.334942i
$990$	0	0
$991$	189.064i	0.190781i	0.995440	+	0.0953907i	$0.0304100\pi$
$991$	189.064i	0.190781i	−0.995440	+	0.0953907i	$0.969590\pi$
$992$	0	0
$993$	578.546	0.582624
$994$	0	0
$995$	1902.36	1.91192
$996$	0	0
$997$	− 1632.91i	− 1.63783i	−0.573917	−	0.818914i	$-0.694577\pi$
$997$	− 1632.91i	− 1.63783i	0.573917	−	0.818914i	$-0.305423\pi$
$998$	0	0
$999$	707.853i	0.708562i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1568.3.g.m.687.8		8
4.3	odd	2		392.3.g.m.99.2		8
7.6	odd	2		224.3.g.b.15.1		8
8.3	odd	2	inner	1568.3.g.m.687.7		8
8.5	even	2		392.3.g.m.99.1		8
21.20	even	2		2016.3.g.b.1135.7		8
28.3	even	6		392.3.k.o.275.5		16
28.11	odd	6		392.3.k.n.275.5		16
28.19	even	6		392.3.k.o.67.7		16
28.23	odd	6		392.3.k.n.67.7		16
28.27	even	2		56.3.g.b.43.2	yes	8
56.5	odd	6		392.3.k.o.67.5		16
56.13	odd	2		56.3.g.b.43.1	✓	8
56.27	even	2		224.3.g.b.15.2		8
56.37	even	6		392.3.k.n.67.5		16
56.45	odd	6		392.3.k.o.275.7		16
56.53	even	6		392.3.k.n.275.7		16
84.83	odd	2		504.3.g.b.379.7		8
112.13	odd	4		1792.3.d.j.1023.3		16
112.27	even	4		1792.3.d.j.1023.4		16
112.69	odd	4		1792.3.d.j.1023.14		16
112.83	even	4		1792.3.d.j.1023.13		16
168.83	odd	2		2016.3.g.b.1135.2		8
168.125	even	2		504.3.g.b.379.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.3.g.b.43.1	✓	8	56.13	odd	2
56.3.g.b.43.2	yes	8	28.27	even	2
224.3.g.b.15.1		8	7.6	odd	2
224.3.g.b.15.2		8	56.27	even	2
392.3.g.m.99.1		8	8.5	even	2
392.3.g.m.99.2		8	4.3	odd	2
392.3.k.n.67.5		16	56.37	even	6
392.3.k.n.67.7		16	28.23	odd	6
392.3.k.n.275.5		16	28.11	odd	6
392.3.k.n.275.7		16	56.53	even	6
392.3.k.o.67.5		16	56.5	odd	6
392.3.k.o.67.7		16	28.19	even	6
392.3.k.o.275.5		16	28.3	even	6
392.3.k.o.275.7		16	56.45	odd	6
504.3.g.b.379.7		8	84.83	odd	2
504.3.g.b.379.8		8	168.125	even	2
1568.3.g.m.687.7		8	8.3	odd	2	inner
1568.3.g.m.687.8		8	1.1	even	1	trivial
1792.3.d.j.1023.3		16	112.13	odd	4
1792.3.d.j.1023.4		16	112.27	even	4
1792.3.d.j.1023.13		16	112.83	even	4
1792.3.d.j.1023.14		16	112.69	odd	4
2016.3.g.b.1135.2		8	168.83	odd	2
2016.3.g.b.1135.7		8	21.20	even	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:	\( N \)	\(=\)	\( 1568 = 2^{5} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1568.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.56747 q^{3} +5.73252i q^{5} +11.8618 q^{9} +O(q^{10})\)	\(q+4.56747 q^{3} +5.73252i q^{5} +11.8618 q^{9} +1.40065 q^{11} -19.0821i q^{13} +26.1831i q^{15} +32.2699 q^{17} +12.5675 q^{19} +15.8893i q^{23} -7.86180 q^{25} +13.0712 q^{27} +3.29194i q^{29} -22.6705i q^{31} +6.39741 q^{33} +54.1537i q^{37} -87.1569i q^{39} +7.59607 q^{41} +20.8478 q^{43} +67.9980i q^{45} -21.6384i q^{47} +147.392 q^{51} -0.356667i q^{53} +8.02924i q^{55} +57.4016 q^{57} +26.8583 q^{59} +86.2287i q^{61} +109.389 q^{65} -114.523 q^{67} +72.5739i q^{69} -104.792i q^{71} +24.3974 q^{73} -35.9085 q^{75} +117.128i q^{79} -47.0539 q^{81} +79.2706 q^{83} +184.988i q^{85} +15.0359i q^{87} -2.66078 q^{89} -103.547i q^{93} +72.0433i q^{95} +52.0930 q^{97} +16.6142 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{3} + 48 q^{9}+O(q^{10}) \) 8 * q - 8 * q^3 + 48 * q^9	\( 8 q - 8 q^{3} + 48 q^{9} + 32 q^{11} + 80 q^{17} + 56 q^{19} - 16 q^{25} - 32 q^{27} - 32 q^{33} - 128 q^{41} + 368 q^{51} + 56 q^{57} + 104 q^{59} - 72 q^{65} - 304 q^{67} + 112 q^{73} + 72 q^{75} + 48 q^{81} + 72 q^{83} + 512 q^{89} - 64 q^{97} - 256 q^{99}+O(q^{100}) \) 8 * q - 8 * q^3 + 48 * q^9 + 32 * q^11 + 80 * q^17 + 56 * q^19 - 16 * q^25 - 32 * q^27 - 32 * q^33 - 128 * q^41 + 368 * q^51 + 56 * q^57 + 104 * q^59 - 72 * q^65 - 304 * q^67 + 112 * q^73 + 72 * q^75 + 48 * q^81 + 72 * q^83 + 512 * q^89 - 64 * q^97 - 256 * q^99

Introduction

L-functions

Modular forms

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