LMFDB - Embedded newform 700.6.a.p.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [700,6,Mod(1,700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(700, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("700.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 700 = 2^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	700.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$112.268673869$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 1359x^{6} + 2771x^{5} + 533790x^{4} + 30024x^{3} - 57192500x^{2} - 308465700x - 415017000 $ x^8 - 3x^7 - 1359x^6 + 2771x^5 + 533790x^4 + 30024x^3 - 57192500x^2 - 308465700*x - 415017000
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{7}\cdot 3\cdot 5^{5} $
Twist minimal:	no (minimal twist has level 140)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$27.3503$ of defining polynomial
Character	$\chi$	$=$	700.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-25.3503 q^{3} +49.0000 q^{7} +399.638 q^{9} -141.450 q^{11} +297.641 q^{13} +1226.74 q^{17} +1777.30 q^{19} -1242.16 q^{21} +3158.14 q^{23} -3970.81 q^{27} -7419.42 q^{29} +970.605 q^{31} +3585.80 q^{33} +2337.19 q^{37} -7545.29 q^{39} -561.020 q^{41} +1974.49 q^{43} -10976.5 q^{47} +2401.00 q^{49} -31098.1 q^{51} +33500.0 q^{53} -45055.2 q^{57} +32227.0 q^{59} -34441.8 q^{61} +19582.2 q^{63} -12670.3 q^{67} -80059.9 q^{69} -83271.0 q^{71} -89632.0 q^{73} -6931.05 q^{77} +4871.34 q^{79} +3549.22 q^{81} +67688.7 q^{83} +188085. q^{87} -86637.4 q^{89} +14584.4 q^{91} -24605.1 q^{93} +6607.75 q^{97} -56528.7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 13 q^{3} + 392 q^{7} + 803 q^{9} + 147 q^{11} + 173 q^{13} + 1755 q^{17} + 1756 q^{19} + 637 q^{21} + 4422 q^{23} + 6127 q^{27} + 10435 q^{29} + 2014 q^{31} - 195 q^{33} + 5986 q^{37} + 12679 q^{39}+ \cdots - 146982 q^{99}+O(q^{100}) $ 8 * q + 13 * q^3 + 392 * q^7 + 803 * q^9 + 147 * q^11 + 173 * q^13 + 1755 * q^17 + 1756 * q^19 + 637 * q^21 + 4422 * q^23 + 6127 * q^27 + 10435 * q^29 + 2014 * q^31 - 195 * q^33 + 5986 * q^37 + 12679 * q^39 + 12476 * q^41 + 11596 * q^43 - 17655 * q^47 + 19208 * q^49 - 6151 * q^51 + 3526 * q^53 - 30712 * q^57 - 18078 * q^59 - 7278 * q^61 + 39347 * q^63 + 29410 * q^67 - 49670 * q^69 - 23996 * q^71 + 51730 * q^73 + 7203 * q^77 - 215 * q^79 + 78104 * q^81 + 203292 * q^83 + 179271 * q^87 - 152090 * q^89 + 8477 * q^91 + 255098 * q^93 + 142427 * q^97 - 146982 * q^99

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
$3$	−25.3503	−1.62622	−0.813111	−	0.582108i	$-0.802228\pi$
$3$	−25.3503	−1.62622	−0.813111	+	0.582108i	$0.802228\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	0	0
$9$	399.638	1.64460
$10$	0	0
$11$	−141.450	−0.352469	−0.176235	−	0.984348i	$-0.556392\pi$
$11$	−141.450	−0.352469	−0.176235	+	0.984348i	$0.556392\pi$
$12$	0	0
$13$	297.641	0.488466	0.244233	−	0.969717i	$-0.421464\pi$
$13$	297.641	0.488466	0.244233	+	0.969717i	$0.421464\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1226.74	1.02951	0.514753	−	0.857339i	$-0.327884\pi$
$17$	1226.74	1.02951	0.514753	+	0.857339i	$0.327884\pi$
$18$	0	0
$19$	1777.30	1.12948	0.564739	−	0.825270i	$-0.308977\pi$
$19$	1777.30	1.12948	0.564739	+	0.825270i	$0.308977\pi$
$20$	0	0
$21$	−1242.16	−0.614654
$22$	0	0
$23$	3158.14	1.24484	0.622418	−	0.782685i	$-0.286150\pi$
$23$	3158.14	1.24484	0.622418	+	0.782685i	$0.286150\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−3970.81	−1.04826
$28$	0	0
$29$	−7419.42	−1.63823	−0.819115	−	0.573629i	$-0.805535\pi$
$29$	−7419.42	−1.63823	−0.819115	+	0.573629i	$0.805535\pi$
$30$	0	0
$31$	970.605	0.181400	0.0907002	−	0.995878i	$-0.471090\pi$
$31$	970.605	0.181400	0.0907002	+	0.995878i	$0.471090\pi$
$32$	0	0
$33$	3585.80	0.573193
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2337.19	0.280666	0.140333	−	0.990104i	$-0.455183\pi$
$37$	2337.19	0.280666	0.140333	+	0.990104i	$0.455183\pi$
$38$	0	0
$39$	−7545.29	−0.794354
$40$	0	0
$41$	−561.020	−0.0521217	−0.0260609	−	0.999660i	$-0.508296\pi$
$41$	−561.020	−0.0521217	−0.0260609	+	0.999660i	$0.508296\pi$
$42$	0	0
$43$	1974.49	0.162848	0.0814242	−	0.996680i	$-0.474053\pi$
$43$	1974.49	0.162848	0.0814242	+	0.996680i	$0.474053\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−10976.5	−0.724802	−0.362401	−	0.932022i	$-0.618043\pi$
$47$	−10976.5	−0.724802	−0.362401	+	0.932022i	$0.618043\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	−31098.1	−1.67421
$52$	0	0
$53$	33500.0	1.63816	0.819079	−	0.573681i	$-0.194485\pi$
$53$	33500.0	1.63816	0.819079	+	0.573681i	$0.194485\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−45055.2	−1.83678
$58$	0	0
$59$	32227.0	1.20529	0.602643	−	0.798011i	$-0.294114\pi$
$59$	32227.0	1.20529	0.602643	+	0.798011i	$0.294114\pi$
$60$	0	0
$61$	−34441.8	−1.18512	−0.592558	−	0.805528i	$-0.701882\pi$
$61$	−34441.8	−1.18512	−0.592558	+	0.805528i	$0.701882\pi$
$62$	0	0
$63$	19582.2	0.621600
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−12670.3	−0.344827	−0.172413	−	0.985025i	$-0.555156\pi$
$67$	−12670.3	−0.344827	−0.172413	+	0.985025i	$0.555156\pi$
$68$	0	0
$69$	−80059.9	−2.02438
$70$	0	0
$71$	−83271.0	−1.96042	−0.980208	−	0.197972i	$-0.936565\pi$
$71$	−83271.0	−1.96042	−0.980208	+	0.197972i	$0.936565\pi$
$72$	0	0
$73$	−89632.0	−1.96859	−0.984297	−	0.176521i	$-0.943516\pi$
$73$	−89632.0	−1.96859	−0.984297	+	0.176521i	$0.943516\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−6931.05	−0.133221
$78$	0	0
$79$	4871.34	0.0878174	0.0439087	−	0.999036i	$-0.486019\pi$
$79$	4871.34	0.0878174	0.0439087	+	0.999036i	$0.486019\pi$
$80$	0	0
$81$	3549.22	0.0601064
$82$	0	0
$83$	67688.7	1.07850	0.539251	−	0.842145i	$-0.318707\pi$
$83$	67688.7	1.07850	0.539251	+	0.842145i	$0.318707\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	188085.	2.66413
$88$	0	0
$89$	−86637.4	−1.15939	−0.579696	−	0.814833i	$-0.696829\pi$
$89$	−86637.4	−1.15939	−0.579696	+	0.814833i	$0.696829\pi$
$90$	0	0
$91$	14584.4	0.184623
$92$	0	0
$93$	−24605.1	−0.294997
$94$	0	0
$95$	0	0
$96$	0	0
$97$	6607.75	0.0713057	0.0356528	−	0.999364i	$-0.488649\pi$
$97$	6607.75	0.0713057	0.0356528	+	0.999364i	$0.488649\pi$
$98$	0	0
$99$	−56528.7	−0.579671
$100$	0	0
$101$	−38774.9	−0.378223	−0.189111	−	0.981956i	$-0.560561\pi$
$101$	−38774.9	−0.378223	−0.189111	+	0.981956i	$0.560561\pi$
$102$	0	0
$103$	−1658.22	−0.0154010	−0.00770049	−	0.999970i	$-0.502451\pi$
$103$	−1658.22	−0.0154010	−0.00770049	+	0.999970i	$0.502451\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	192852.	1.62842	0.814208	−	0.580573i	$-0.197171\pi$
$107$	192852.	1.62842	0.814208	+	0.580573i	$0.197171\pi$
$108$	0	0
$109$	125324.	1.01034	0.505171	−	0.863019i	$-0.331429\pi$
$109$	125324.	1.01034	0.505171	+	0.863019i	$0.331429\pi$
$110$	0	0
$111$	−59248.4	−0.456425
$112$	0	0
$113$	118360.	0.871984	0.435992	−	0.899951i	$-0.356398\pi$
$113$	118360.	0.871984	0.435992	+	0.899951i	$0.356398\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	118948.	0.803330
$118$	0	0
$119$	60110.1	0.389117
$120$	0	0
$121$	−141043.	−0.875765
$122$	0	0
$123$	14222.0	0.0847615
$124$	0	0
$125$	0	0
$126$	0	0
$127$	78506.1	0.431911	0.215955	−	0.976403i	$-0.430713\pi$
$127$	78506.1	0.431911	0.215955	+	0.976403i	$0.430713\pi$
$128$	0	0
$129$	−50053.9	−0.264828
$130$	0	0
$131$	−21998.0	−0.111996	−0.0559982	−	0.998431i	$-0.517834\pi$
$131$	−21998.0	−0.111996	−0.0559982	+	0.998431i	$0.517834\pi$
$132$	0	0
$133$	87087.9	0.426902
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−330457.	−1.50423	−0.752113	−	0.659034i	$-0.770965\pi$
$137$	−330457.	−1.50423	−0.752113	+	0.659034i	$0.770965\pi$
$138$	0	0
$139$	403251.	1.77027	0.885133	−	0.465337i	$-0.154067\pi$
$139$	403251.	1.77027	0.885133	+	0.465337i	$0.154067\pi$
$140$	0	0
$141$	278258.	1.17869
$142$	0	0
$143$	−42101.3	−0.172169
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−60866.1	−0.232317
$148$	0	0
$149$	441499.	1.62916	0.814581	−	0.580050i	$-0.196967\pi$
$149$	441499.	1.62916	0.814581	+	0.580050i	$0.196967\pi$
$150$	0	0
$151$	424395.	1.51471	0.757353	−	0.653006i	$-0.226492\pi$
$151$	424395.	1.51471	0.757353	+	0.653006i	$0.226492\pi$
$152$	0	0
$153$	490250.	1.69312
$154$	0	0
$155$	0	0
$156$	0	0
$157$	154250.	0.499431	0.249715	−	0.968319i	$-0.419663\pi$
$157$	154250.	0.499431	0.249715	+	0.968319i	$0.419663\pi$
$158$	0	0
$159$	−849236.	−2.66401
$160$	0	0
$161$	154749.	0.470504
$162$	0	0
$163$	156114.	0.460227	0.230113	−	0.973164i	$-0.426090\pi$
$163$	156114.	0.460227	0.230113	+	0.973164i	$0.426090\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−84036.3	−0.233172	−0.116586	−	0.993181i	$-0.537195\pi$
$167$	−84036.3	−0.233172	−0.116586	+	0.993181i	$0.537195\pi$
$168$	0	0
$169$	−282703.	−0.761401
$170$	0	0
$171$	710277.	1.85754
$172$	0	0
$173$	155939.	0.396133	0.198066	−	0.980189i	$-0.436534\pi$
$173$	155939.	0.396133	0.198066	+	0.980189i	$0.436534\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−816965.	−1.96006
$178$	0	0
$179$	485375.	1.13226	0.566129	−	0.824317i	$-0.308440\pi$
$179$	485375.	1.13226	0.566129	+	0.824317i	$0.308440\pi$
$180$	0	0
$181$	9996.05	0.0226794	0.0113397	−	0.999936i	$-0.496390\pi$
$181$	9996.05	0.0226794	0.0113397	+	0.999936i	$0.496390\pi$
$182$	0	0
$183$	873109.	1.92726
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−173522.	−0.362869
$188$	0	0
$189$	−194570.	−0.396205
$190$	0	0
$191$	398572.	0.790540	0.395270	−	0.918565i	$-0.370651\pi$
$191$	398572.	0.790540	0.395270	+	0.918565i	$0.370651\pi$
$192$	0	0
$193$	669914.	1.29457	0.647286	−	0.762248i	$-0.275904\pi$
$193$	669914.	1.29457	0.647286	+	0.762248i	$0.275904\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	937695.	1.72146	0.860728	−	0.509065i	$-0.170009\pi$
$197$	937695.	1.72146	0.860728	+	0.509065i	$0.170009\pi$
$198$	0	0
$199$	104138.	0.186413	0.0932066	−	0.995647i	$-0.470288\pi$
$199$	104138.	0.186413	0.0932066	+	0.995647i	$0.470288\pi$
$200$	0	0
$201$	321197.	0.560765
$202$	0	0
$203$	−363552.	−0.619193
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.26211e6	2.04726
$208$	0	0
$209$	−251400.	−0.398106
$210$	0	0
$211$	−1.11877e6	−1.72995	−0.864975	−	0.501815i	$-0.832666\pi$
$211$	−1.11877e6	−1.72995	−0.864975	+	0.501815i	$0.832666\pi$
$212$	0	0
$213$	2.11095e6	3.18807
$214$	0	0
$215$	0	0
$216$	0	0
$217$	47559.6	0.0685629
$218$	0	0
$219$	2.27220e6	3.20137
$220$	0	0
$221$	365127.	0.502879
$222$	0	0
$223$	−972345.	−1.30936	−0.654679	−	0.755907i	$-0.727196\pi$
$223$	−972345.	−1.30936	−0.654679	+	0.755907i	$0.727196\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−378096.	−0.487010	−0.243505	−	0.969900i	$-0.578297\pi$
$227$	−378096.	−0.487010	−0.243505	+	0.969900i	$0.578297\pi$
$228$	0	0
$229$	−1.18847e6	−1.49761	−0.748807	−	0.662788i	$-0.769373\pi$
$229$	−1.18847e6	−1.49761	−0.748807	+	0.662788i	$0.769373\pi$
$230$	0	0
$231$	175704.	0.216647
$232$	0	0
$233$	1.18457e6	1.42946	0.714729	−	0.699402i	$-0.246550\pi$
$233$	1.18457e6	1.42946	0.714729	+	0.699402i	$0.246550\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−123490.	−0.142811
$238$	0	0
$239$	803487.	0.909880	0.454940	−	0.890522i	$-0.349661\pi$
$239$	803487.	0.909880	0.454940	+	0.890522i	$0.349661\pi$
$240$	0	0
$241$	344716.	0.382313	0.191156	−	0.981560i	$-0.438776\pi$
$241$	344716.	0.382313	0.191156	+	0.981560i	$0.438776\pi$
$242$	0	0
$243$	874932.	0.950515
$244$	0	0
$245$	0	0
$246$	0	0
$247$	528998.	0.551711
$248$	0	0
$249$	−1.71593e6	−1.75388
$250$	0	0
$251$	−76632.3	−0.0767764	−0.0383882	−	0.999263i	$-0.512222\pi$
$251$	−76632.3	−0.0767764	−0.0383882	+	0.999263i	$0.512222\pi$
$252$	0	0
$253$	−446719.	−0.438766
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−734767.	−0.693931	−0.346966	−	0.937878i	$-0.612788\pi$
$257$	−734767.	−0.693931	−0.346966	+	0.937878i	$0.612788\pi$
$258$	0	0
$259$	114522.	0.106082
$260$	0	0
$261$	−2.96508e6	−2.69423
$262$	0	0
$263$	−448950.	−0.400229	−0.200115	−	0.979772i	$-0.564131\pi$
$263$	−448950.	−0.400229	−0.200115	+	0.979772i	$0.564131\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2.19628e6	1.88543
$268$	0	0
$269$	−1.66975e6	−1.40692	−0.703462	−	0.710733i	$-0.748363\pi$
$269$	−1.66975e6	−1.40692	−0.703462	+	0.710733i	$0.748363\pi$
$270$	0	0
$271$	466892.	0.386183	0.193092	−	0.981181i	$-0.438149\pi$
$271$	466892.	0.386183	0.193092	+	0.981181i	$0.438149\pi$
$272$	0	0
$273$	−369719.	−0.300238
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.86028e6	1.45673	0.728364	−	0.685190i	$-0.240281\pi$
$277$	1.86028e6	1.45673	0.728364	+	0.685190i	$0.240281\pi$
$278$	0	0
$279$	387890.	0.298331
$280$	0	0
$281$	384958.	0.290836	0.145418	−	0.989370i	$-0.453547\pi$
$281$	384958.	0.290836	0.145418	+	0.989370i	$0.453547\pi$
$282$	0	0
$283$	−255926.	−0.189954	−0.0949769	−	0.995479i	$-0.530278\pi$
$283$	−255926.	−0.189954	−0.0949769	+	0.995479i	$0.530278\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−27490.0	−0.0197002
$288$	0	0
$289$	85024.4	0.0598824
$290$	0	0
$291$	−167508.	−0.115959
$292$	0	0
$293$	−345511.	−0.235122	−0.117561	−	0.993066i	$-0.537508\pi$
$293$	−345511.	−0.235122	−0.117561	+	0.993066i	$0.537508\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	561671.	0.369480
$298$	0	0
$299$	939992.	0.608060
$300$	0	0
$301$	96750.0	0.0615509
$302$	0	0
$303$	982956.	0.615074
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−1.38493e6	−0.838651	−0.419326	−	0.907836i	$-0.637733\pi$
$307$	−1.38493e6	−0.838651	−0.419326	+	0.907836i	$0.637733\pi$
$308$	0	0
$309$	42036.3	0.0250454
$310$	0	0
$311$	900004.	0.527647	0.263824	−	0.964571i	$-0.415016\pi$
$311$	900004.	0.527647	0.263824	+	0.964571i	$0.415016\pi$
$312$	0	0
$313$	−2.80649e6	−1.61921	−0.809603	−	0.586978i	$-0.800318\pi$
$313$	−2.80649e6	−1.61921	−0.809603	+	0.586978i	$0.800318\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2.45481e6	−1.37205	−0.686024	−	0.727579i	$-0.740645\pi$
$317$	−2.45481e6	−1.37205	−0.686024	+	0.727579i	$0.740645\pi$
$318$	0	0
$319$	1.04948e6	0.577426
$320$	0	0
$321$	−4.88886e6	−2.64817
$322$	0	0
$323$	2.18028e6	1.16280
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−3.17700e6	−1.64304
$328$	0	0
$329$	−537849.	−0.273949
$330$	0	0
$331$	−3.87397e6	−1.94351	−0.971755	−	0.235993i	$-0.924166\pi$
$331$	−3.87397e6	−1.94351	−0.971755	+	0.235993i	$0.924166\pi$
$332$	0	0
$333$	934029.	0.461583
$334$	0	0
$335$	0	0
$336$	0	0
$337$	3.04334e6	1.45974	0.729871	−	0.683584i	$-0.239580\pi$
$337$	3.04334e6	1.45974	0.729871	+	0.683584i	$0.239580\pi$
$338$	0	0
$339$	−3.00046e6	−1.41804
$340$	0	0
$341$	−137292.	−0.0639380
$342$	0	0
$343$	117649.	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.78331e6	0.795068	0.397534	−	0.917587i	$-0.369866\pi$
$347$	1.78331e6	0.795068	0.397534	+	0.917587i	$0.369866\pi$
$348$	0	0
$349$	1.76269e6	0.774661	0.387330	−	0.921941i	$-0.373397\pi$
$349$	1.76269e6	0.774661	0.387330	+	0.921941i	$0.373397\pi$
$350$	0	0
$351$	−1.18187e6	−0.512040
$352$	0	0
$353$	675787.	0.288651	0.144325	−	0.989530i	$-0.453899\pi$
$353$	675787.	0.288651	0.144325	+	0.989530i	$0.453899\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−1.52381e6	−0.632790
$358$	0	0
$359$	−1.77828e6	−0.728223	−0.364111	−	0.931355i	$-0.618627\pi$
$359$	−1.77828e6	−0.728223	−0.364111	+	0.931355i	$0.618627\pi$
$360$	0	0
$361$	682710.	0.275720
$362$	0	0
$363$	3.57548e6	1.42419
$364$	0	0
$365$	0	0
$366$	0	0
$367$	4.35734e6	1.68871	0.844357	−	0.535781i	$-0.179983\pi$
$367$	4.35734e6	1.68871	0.844357	+	0.535781i	$0.179983\pi$
$368$	0	0
$369$	−224205.	−0.0857193
$370$	0	0
$371$	1.64150e6	0.619166
$372$	0	0
$373$	−2.87221e6	−1.06892	−0.534458	−	0.845195i	$-0.679484\pi$
$373$	−2.87221e6	−1.06892	−0.534458	+	0.845195i	$0.679484\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.20832e6	−0.800220
$378$	0	0
$379$	2.11884e6	0.757705	0.378853	−	0.925457i	$-0.376319\pi$
$379$	2.11884e6	0.757705	0.378853	+	0.925457i	$0.376319\pi$
$380$	0	0
$381$	−1.99015e6	−0.702383
$382$	0	0
$383$	3.78103e6	1.31708	0.658542	−	0.752544i	$-0.271174\pi$
$383$	3.78103e6	1.31708	0.658542	+	0.752544i	$0.271174\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	789080.	0.267820
$388$	0	0
$389$	−3.66654e6	−1.22852	−0.614261	−	0.789103i	$-0.710546\pi$
$389$	−3.66654e6	−1.22852	−0.614261	+	0.789103i	$0.710546\pi$
$390$	0	0
$391$	3.87421e6	1.28157
$392$	0	0
$393$	557655.	0.182131
$394$	0	0
$395$	0	0
$396$	0	0
$397$	2.32579e6	0.740618	0.370309	−	0.928909i	$-0.379252\pi$
$397$	2.32579e6	0.740618	0.370309	+	0.928909i	$0.379252\pi$
$398$	0	0
$399$	−2.20770e6	−0.694238
$400$	0	0
$401$	1.74464e6	0.541807	0.270904	−	0.962607i	$-0.412678\pi$
$401$	1.74464e6	0.541807	0.270904	+	0.962607i	$0.412678\pi$
$402$	0	0
$403$	288892.	0.0886079
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−330595.	−0.0989261
$408$	0	0
$409$	−669116.	−0.197785	−0.0988924	−	0.995098i	$-0.531530\pi$
$409$	−669116.	−0.197785	−0.0988924	+	0.995098i	$0.531530\pi$
$410$	0	0
$411$	8.37718e6	2.44621
$412$	0	0
$413$	1.57912e6	0.455555
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−1.02225e7	−2.87885
$418$	0	0
$419$	4.67106e6	1.29981	0.649906	−	0.760014i	$-0.274808\pi$
$419$	4.67106e6	1.29981	0.649906	+	0.760014i	$0.274808\pi$
$420$	0	0
$421$	4.55778e6	1.25328	0.626640	−	0.779309i	$-0.284430\pi$
$421$	4.55778e6	1.25328	0.626640	+	0.779309i	$0.284430\pi$
$422$	0	0
$423$	−4.38662e6	−1.19201
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−1.68765e6	−0.447932
$428$	0	0
$429$	1.06728e6	0.279985
$430$	0	0
$431$	426321.	0.110546	0.0552731	−	0.998471i	$-0.482397\pi$
$431$	426321.	0.110546	0.0552731	+	0.998471i	$0.482397\pi$
$432$	0	0
$433$	−3.32358e6	−0.851896	−0.425948	−	0.904748i	$-0.640059\pi$
$433$	−3.32358e6	−0.851896	−0.425948	+	0.904748i	$0.640059\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.61298e6	1.40601
$438$	0	0
$439$	4.11543e6	1.01919	0.509594	−	0.860415i	$-0.329796\pi$
$439$	4.11543e6	1.01919	0.509594	+	0.860415i	$0.329796\pi$
$440$	0	0
$441$	959530.	0.234943
$442$	0	0
$443$	−2.88152e6	−0.697610	−0.348805	−	0.937195i	$-0.613412\pi$
$443$	−2.88152e6	−0.697610	−0.348805	+	0.937195i	$0.613412\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−1.11921e7	−2.64938
$448$	0	0
$449$	4.35526e6	1.01952	0.509762	−	0.860315i	$-0.329733\pi$
$449$	4.35526e6	1.01952	0.509762	+	0.860315i	$0.329733\pi$
$450$	0	0
$451$	79356.3	0.0183713
$452$	0	0
$453$	−1.07585e7	−2.46325
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−7.79301e6	−1.74548	−0.872739	−	0.488186i	$-0.837659\pi$
$457$	−7.79301e6	−1.74548	−0.872739	+	0.488186i	$0.837659\pi$
$458$	0	0
$459$	−4.87113e6	−1.07919
$460$	0	0
$461$	−2.48971e6	−0.545627	−0.272814	−	0.962067i	$-0.587954\pi$
$461$	−2.48971e6	−0.545627	−0.272814	+	0.962067i	$0.587954\pi$
$462$	0	0
$463$	6.12486e6	1.32783	0.663917	−	0.747807i	$-0.268893\pi$
$463$	6.12486e6	1.32783	0.663917	+	0.747807i	$0.268893\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−6.48662e6	−1.37634	−0.688171	−	0.725549i	$-0.741586\pi$
$467$	−6.48662e6	−1.37634	−0.688171	+	0.725549i	$0.741586\pi$
$468$	0	0
$469$	−620846.	−0.130332
$470$	0	0
$471$	−3.91028e6	−0.812186
$472$	0	0
$473$	−279291.	−0.0573991
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1.33879e7	2.69411
$478$	0	0
$479$	4.61538e6	0.919112	0.459556	−	0.888149i	$-0.348009\pi$
$479$	4.61538e6	0.919112	0.459556	+	0.888149i	$0.348009\pi$
$480$	0	0
$481$	695643.	0.137096
$482$	0	0
$483$	−3.92293e6	−0.765144
$484$	0	0
$485$	0	0
$486$	0	0
$487$	8.66187e6	1.65497	0.827484	−	0.561490i	$-0.189772\pi$
$487$	8.66187e6	1.65497	0.827484	+	0.561490i	$0.189772\pi$
$488$	0	0
$489$	−3.95753e6	−0.748431
$490$	0	0
$491$	1.87139e6	0.350316	0.175158	−	0.984540i	$-0.443956\pi$
$491$	1.87139e6	0.350316	0.175158	+	0.984540i	$0.443956\pi$
$492$	0	0
$493$	−9.10167e6	−1.68657
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−4.08028e6	−0.740967
$498$	0	0
$499$	6.42247e6	1.15465	0.577325	−	0.816514i	$-0.304096\pi$
$499$	6.42247e6	1.15465	0.577325	+	0.816514i	$0.304096\pi$
$500$	0	0
$501$	2.13034e6	0.379189
$502$	0	0
$503$	−6.83277e6	−1.20414	−0.602070	−	0.798443i	$-0.705657\pi$
$503$	−6.83277e6	−1.20414	−0.602070	+	0.798443i	$0.705657\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	7.16660e6	1.23821
$508$	0	0
$509$	9.16537e6	1.56803	0.784017	−	0.620740i	$-0.213168\pi$
$509$	9.16537e6	1.56803	0.784017	+	0.620740i	$0.213168\pi$
$510$	0	0
$511$	−4.39197e6	−0.744058
$512$	0	0
$513$	−7.05733e6	−1.18399
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1.55263e6	0.255470
$518$	0	0
$519$	−3.95311e6	−0.644200
$520$	0	0
$521$	6.71054e6	1.08309	0.541543	−	0.840673i	$-0.317840\pi$
$521$	6.71054e6	1.08309	0.541543	+	0.840673i	$0.317840\pi$
$522$	0	0
$523$	−3.31400e6	−0.529783	−0.264892	−	0.964278i	$-0.585336\pi$
$523$	−3.31400e6	−0.529783	−0.264892	+	0.964278i	$0.585336\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.19068e6	0.186753
$528$	0	0
$529$	3.53752e6	0.549617
$530$	0	0
$531$	1.28791e7	1.98221
$532$	0	0
$533$	−166982.	−0.0254597
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−1.23044e7	−1.84130
$538$	0	0
$539$	−339621.	−0.0503528
$540$	0	0
$541$	4.39605e6	0.645757	0.322879	−	0.946440i	$-0.395349\pi$
$541$	4.39605e6	0.645757	0.322879	+	0.946440i	$0.395349\pi$
$542$	0	0
$543$	−253403.	−0.0368818
$544$	0	0
$545$	0	0
$546$	0	0
$547$	1.03221e7	1.47503	0.737515	−	0.675331i	$-0.235999\pi$
$547$	1.03221e7	1.47503	0.737515	+	0.675331i	$0.235999\pi$
$548$	0	0
$549$	−1.37642e7	−1.94904
$550$	0	0
$551$	−1.31866e7	−1.85035
$552$	0	0
$553$	238695.	0.0331918
$554$	0	0
$555$	0	0
$556$	0	0
$557$	4.02433e6	0.549611	0.274805	−	0.961500i	$-0.411387\pi$
$557$	4.02433e6	0.549611	0.274805	+	0.961500i	$0.411387\pi$
$558$	0	0
$559$	587689.	0.0795459
$560$	0	0
$561$	4.39883e6	0.590106
$562$	0	0
$563$	7.76463e6	1.03240	0.516202	−	0.856467i	$-0.327345\pi$
$563$	7.76463e6	1.03240	0.516202	+	0.856467i	$0.327345\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	173912.	0.0227181
$568$	0	0
$569$	962147.	0.124584	0.0622918	−	0.998058i	$-0.480159\pi$
$569$	962147.	0.124584	0.0622918	+	0.998058i	$0.480159\pi$
$570$	0	0
$571$	7.97240e6	1.02329	0.511645	−	0.859197i	$-0.329036\pi$
$571$	7.97240e6	1.02329	0.511645	+	0.859197i	$0.329036\pi$
$572$	0	0
$573$	−1.01039e7	−1.28559
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−3.13108e6	−0.391521	−0.195761	−	0.980652i	$-0.562718\pi$
$577$	−3.13108e6	−0.391521	−0.195761	+	0.980652i	$0.562718\pi$
$578$	0	0
$579$	−1.69825e7	−2.10526
$580$	0	0
$581$	3.31675e6	0.407635
$582$	0	0
$583$	−4.73858e6	−0.577400
$584$	0	0
$585$	0	0
$586$	0	0
$587$	5.27415e6	0.631767	0.315884	−	0.948798i	$-0.397699\pi$
$587$	5.27415e6	0.631767	0.315884	+	0.948798i	$0.397699\pi$
$588$	0	0
$589$	1.72506e6	0.204888
$590$	0	0
$591$	−2.37708e7	−2.79947
$592$	0	0
$593$	−4.94938e6	−0.577981	−0.288991	−	0.957332i	$-0.593320\pi$
$593$	−4.94938e6	−0.577981	−0.288991	+	0.957332i	$0.593320\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−2.63993e6	−0.303149
$598$	0	0
$599$	2.70972e6	0.308573	0.154286	−	0.988026i	$-0.450692\pi$
$599$	2.70972e6	0.308573	0.154286	+	0.988026i	$0.450692\pi$
$600$	0	0
$601$	−992330.	−0.112065	−0.0560325	−	0.998429i	$-0.517845\pi$
$601$	−992330.	−0.112065	−0.0560325	+	0.998429i	$0.517845\pi$
$602$	0	0
$603$	−5.06354e6	−0.567102
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−1.35542e7	−1.49315	−0.746574	−	0.665302i	$-0.768303\pi$
$607$	−1.35542e7	−1.49315	−0.746574	+	0.665302i	$0.768303\pi$
$608$	0	0
$609$	9.21614e6	1.00695
$610$	0	0
$611$	−3.26706e6	−0.354041
$612$	0	0
$613$	−8.37239e6	−0.899908	−0.449954	−	0.893052i	$-0.648560\pi$
$613$	−8.37239e6	−0.899908	−0.449954	+	0.893052i	$0.648560\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6.49138e6	0.686474	0.343237	−	0.939249i	$-0.388477\pi$
$617$	6.49138e6	0.686474	0.343237	+	0.939249i	$0.388477\pi$
$618$	0	0
$619$	8.98202e6	0.942210	0.471105	−	0.882077i	$-0.343855\pi$
$619$	8.98202e6	0.942210	0.471105	+	0.882077i	$0.343855\pi$
$620$	0	0
$621$	−1.25404e7	−1.30491
$622$	0	0
$623$	−4.24523e6	−0.438209
$624$	0	0
$625$	0	0
$626$	0	0
$627$	6.37306e6	0.647409
$628$	0	0
$629$	2.86711e6	0.288947
$630$	0	0
$631$	−1.68639e7	−1.68611	−0.843053	−	0.537831i	$-0.819244\pi$
$631$	−1.68639e7	−1.68611	−0.843053	+	0.537831i	$0.819244\pi$
$632$	0	0
$633$	2.83611e7	2.81328
$634$	0	0
$635$	0	0
$636$	0	0
$637$	714636.	0.0697808
$638$	0	0
$639$	−3.32782e7	−3.22410
$640$	0	0
$641$	−1.35136e7	−1.29905	−0.649524	−	0.760341i	$-0.725032\pi$
$641$	−1.35136e7	−1.29905	−0.649524	+	0.760341i	$0.725032\pi$
$642$	0	0
$643$	−9.15022e6	−0.872779	−0.436390	−	0.899758i	$-0.643743\pi$
$643$	−9.15022e6	−0.872779	−0.436390	+	0.899758i	$0.643743\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−3.76700e6	−0.353781	−0.176891	−	0.984231i	$-0.556604\pi$
$647$	−3.76700e6	−0.353781	−0.176891	+	0.984231i	$0.556604\pi$
$648$	0	0
$649$	−4.55851e6	−0.424826
$650$	0	0
$651$	−1.20565e6	−0.111498
$652$	0	0
$653$	1.36182e7	1.24979	0.624897	−	0.780708i	$-0.285141\pi$
$653$	1.36182e7	1.24979	0.624897	+	0.780708i	$0.285141\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−3.58203e7	−3.23755
$658$	0	0
$659$	1.63668e6	0.146808	0.0734039	−	0.997302i	$-0.476614\pi$
$659$	1.63668e6	0.146808	0.0734039	+	0.997302i	$0.476614\pi$
$660$	0	0
$661$	−1.43398e7	−1.27656	−0.638278	−	0.769806i	$-0.720353\pi$
$661$	−1.43398e7	−1.27656	−0.638278	+	0.769806i	$0.720353\pi$
$662$	0	0
$663$	−9.25607e6	−0.817792
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−2.34316e7	−2.03933
$668$	0	0
$669$	2.46492e7	2.12931
$670$	0	0
$671$	4.87179e6	0.417717
$672$	0	0
$673$	9.54325e6	0.812192	0.406096	−	0.913830i	$-0.366890\pi$
$673$	9.54325e6	0.812192	0.406096	+	0.913830i	$0.366890\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−1.76523e7	−1.48023	−0.740116	−	0.672479i	$-0.765229\pi$
$677$	−1.76523e7	−1.48023	−0.740116	+	0.672479i	$0.765229\pi$
$678$	0	0
$679$	323780.	0.0269510
$680$	0	0
$681$	9.58485e6	0.791986
$682$	0	0
$683$	−3.15865e6	−0.259089	−0.129545	−	0.991574i	$-0.541352\pi$
$683$	−3.15865e6	−0.259089	−0.129545	+	0.991574i	$0.541352\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	3.01281e7	2.43545
$688$	0	0
$689$	9.97099e6	0.800184
$690$	0	0
$691$	624449.	0.0497510	0.0248755	−	0.999691i	$-0.492081\pi$
$691$	624449.	0.0497510	0.0248755	+	0.999691i	$0.492081\pi$
$692$	0	0
$693$	−2.76991e6	−0.219095
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−688223.	−0.0536596
$698$	0	0
$699$	−3.00292e7	−2.32462
$700$	0	0
$701$	1.57395e7	1.20975	0.604875	−	0.796320i	$-0.293223\pi$
$701$	1.57395e7	1.20975	0.604875	+	0.796320i	$0.293223\pi$
$702$	0	0
$703$	4.15390e6	0.317006
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.89997e6	−0.142955
$708$	0	0
$709$	−5.74101e6	−0.428917	−0.214458	−	0.976733i	$-0.568799\pi$
$709$	−5.74101e6	−0.428917	−0.214458	+	0.976733i	$0.568799\pi$
$710$	0	0
$711$	1.94677e6	0.144424
$712$	0	0
$713$	3.06531e6	0.225814
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−2.03686e7	−1.47967
$718$	0	0
$719$	−3.79854e6	−0.274028	−0.137014	−	0.990569i	$-0.543750\pi$
$719$	−3.79854e6	−0.274028	−0.137014	+	0.990569i	$0.543750\pi$
$720$	0	0
$721$	−81252.6	−0.00582102
$722$	0	0
$723$	−8.73865e6	−0.621725
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1.68558e7	1.18280	0.591402	−	0.806377i	$-0.298575\pi$
$727$	1.68558e7	1.18280	0.591402	+	0.806377i	$0.298575\pi$
$728$	0	0
$729$	−2.30423e7	−1.60585
$730$	0	0
$731$	2.42218e6	0.167653
$732$	0	0
$733$	2.16490e7	1.48826	0.744130	−	0.668035i	$-0.232864\pi$
$733$	2.16490e7	1.48826	0.744130	+	0.668035i	$0.232864\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.79222e6	0.121541
$738$	0	0
$739$	1.64235e7	1.10625	0.553125	−	0.833098i	$-0.313435\pi$
$739$	1.64235e7	1.10625	0.553125	+	0.833098i	$0.313435\pi$
$740$	0	0
$741$	−1.34103e7	−0.897205
$742$	0	0
$743$	−5.84269e6	−0.388276	−0.194138	−	0.980974i	$-0.562191\pi$
$743$	−5.84269e6	−0.388276	−0.194138	+	0.980974i	$0.562191\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	2.70509e7	1.77370
$748$	0	0
$749$	9.44977e6	0.615484
$750$	0	0
$751$	1.02447e7	0.662823	0.331412	−	0.943486i	$-0.392475\pi$
$751$	1.02447e7	0.662823	0.331412	+	0.943486i	$0.392475\pi$
$752$	0	0
$753$	1.94265e6	0.124855
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−1.30935e7	−0.830452	−0.415226	−	0.909718i	$-0.636297\pi$
$757$	−1.30935e7	−0.830452	−0.415226	+	0.909718i	$0.636297\pi$
$758$	0	0
$759$	1.13245e7	0.713532
$760$	0	0
$761$	−6.99157e6	−0.437636	−0.218818	−	0.975766i	$-0.570220\pi$
$761$	−6.99157e6	−0.437636	−0.218818	+	0.975766i	$0.570220\pi$
$762$	0	0
$763$	6.14088e6	0.381874
$764$	0	0
$765$	0	0
$766$	0	0
$767$	9.59208e6	0.588741
$768$	0	0
$769$	1.46387e7	0.892660	0.446330	−	0.894868i	$-0.352731\pi$
$769$	1.46387e7	0.892660	0.446330	+	0.894868i	$0.352731\pi$
$770$	0	0
$771$	1.86265e7	1.12849
$772$	0	0
$773$	−2.11451e7	−1.27280	−0.636402	−	0.771357i	$-0.719578\pi$
$773$	−2.11451e7	−1.27280	−0.636402	+	0.771357i	$0.719578\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−2.90317e6	−0.172512
$778$	0	0
$779$	−997103.	−0.0588703
$780$	0	0
$781$	1.17787e7	0.690986
$782$	0	0
$783$	2.94611e7	1.71729
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1.80434e7	1.03844	0.519219	−	0.854641i	$-0.326223\pi$
$787$	1.80434e7	1.03844	0.519219	+	0.854641i	$0.326223\pi$
$788$	0	0
$789$	1.13810e7	0.650862
$790$	0	0
$791$	5.79964e6	0.329579
$792$	0	0
$793$	−1.02513e7	−0.578889
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2.14077e7	1.19378	0.596890	−	0.802323i	$-0.296403\pi$
$797$	2.14077e7	1.19378	0.596890	+	0.802323i	$0.296403\pi$
$798$	0	0
$799$	−1.34653e7	−0.746188
$800$	0	0
$801$	−3.46236e7	−1.90674
$802$	0	0
$803$	1.26785e7	0.693869
$804$	0	0
$805$	0	0
$806$	0	0
$807$	4.23286e7	2.28797
$808$	0	0
$809$	−3.85342e6	−0.207002	−0.103501	−	0.994629i	$-0.533005\pi$
$809$	−3.85342e6	−0.207002	−0.103501	+	0.994629i	$0.533005\pi$
$810$	0	0
$811$	1.82776e7	0.975814	0.487907	−	0.872896i	$-0.337761\pi$
$811$	1.82776e7	0.975814	0.487907	+	0.872896i	$0.337761\pi$
$812$	0	0
$813$	−1.18359e7	−0.628020
$814$	0	0
$815$	0	0
$816$	0	0
$817$	3.50927e6	0.183934
$818$	0	0
$819$	5.82848e6	0.303630
$820$	0	0
$821$	2.63108e7	1.36231	0.681156	−	0.732138i	$-0.261477\pi$
$821$	2.63108e7	1.36231	0.681156	+	0.732138i	$0.261477\pi$
$822$	0	0
$823$	3.23546e7	1.66509	0.832543	−	0.553960i	$-0.186884\pi$
$823$	3.23546e7	1.66509	0.832543	+	0.553960i	$0.186884\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	5.30031e6	0.269487	0.134743	−	0.990881i	$-0.456979\pi$
$827$	5.30031e6	0.269487	0.134743	+	0.990881i	$0.456979\pi$
$828$	0	0
$829$	2.12766e7	1.07526	0.537632	−	0.843180i	$-0.319319\pi$
$829$	2.12766e7	1.07526	0.537632	+	0.843180i	$0.319319\pi$
$830$	0	0
$831$	−4.71586e7	−2.36896
$832$	0	0
$833$	2.94539e6	0.147072
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−3.85408e6	−0.190155
$838$	0	0
$839$	−2.68060e6	−0.131470	−0.0657350	−	0.997837i	$-0.520939\pi$
$839$	−2.68060e6	−0.131470	−0.0657350	+	0.997837i	$0.520939\pi$
$840$	0	0
$841$	3.45367e7	1.68380
$842$	0	0
$843$	−9.75881e6	−0.472964
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−6.91110e6	−0.331008
$848$	0	0
$849$	6.48780e6	0.308907
$850$	0	0
$851$	7.38118e6	0.349383
$852$	0	0
$853$	−5.65172e6	−0.265955	−0.132977	−	0.991119i	$-0.542454\pi$
$853$	−5.65172e6	−0.265955	−0.132977	+	0.991119i	$0.542454\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2.76367e7	1.28539	0.642694	−	0.766123i	$-0.277817\pi$
$857$	2.76367e7	1.28539	0.642694	+	0.766123i	$0.277817\pi$
$858$	0	0
$859$	−2.44636e6	−0.113119	−0.0565597	−	0.998399i	$-0.518013\pi$
$859$	−2.44636e6	−0.113119	−0.0565597	+	0.998399i	$0.518013\pi$
$860$	0	0
$861$	696879.	0.0320368
$862$	0	0
$863$	3.06078e7	1.39896	0.699481	−	0.714652i	$-0.253415\pi$
$863$	3.06078e7	1.39896	0.699481	+	0.714652i	$0.253415\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−2.15539e6	−0.0973821
$868$	0	0
$869$	−689051.	−0.0309529
$870$	0	0
$871$	−3.77121e6	−0.168436
$872$	0	0
$873$	2.64070e6	0.117269
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−2.14914e7	−0.943552	−0.471776	−	0.881718i	$-0.656387\pi$
$877$	−2.14914e7	−0.943552	−0.471776	+	0.881718i	$0.656387\pi$
$878$	0	0
$879$	8.75881e6	0.382360
$880$	0	0
$881$	2.79915e7	1.21503	0.607514	−	0.794309i	$-0.292167\pi$
$881$	2.79915e7	1.21503	0.607514	+	0.794309i	$0.292167\pi$
$882$	0	0
$883$	2.40091e7	1.03627	0.518136	−	0.855298i	$-0.326626\pi$
$883$	2.40091e7	1.03627	0.518136	+	0.855298i	$0.326626\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	4.01622e7	1.71399	0.856995	−	0.515326i	$-0.172329\pi$
$887$	4.01622e7	1.71399	0.856995	+	0.515326i	$0.172329\pi$
$888$	0	0
$889$	3.84680e6	0.163247
$890$	0	0
$891$	−502038.	−0.0211857
$892$	0	0
$893$	−1.95086e7	−0.818648
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−2.38291e7	−0.988841
$898$	0	0
$899$	−7.20132e6	−0.297176
$900$	0	0
$901$	4.10957e7	1.68649
$902$	0	0
$903$	−2.45264e6	−0.100095
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−2.19331e7	−0.885283	−0.442641	−	0.896699i	$-0.645959\pi$
$907$	−2.19331e7	−0.885283	−0.442641	+	0.896699i	$0.645959\pi$
$908$	0	0
$909$	−1.54959e7	−0.622025
$910$	0	0
$911$	−2.21245e7	−0.883236	−0.441618	−	0.897203i	$-0.645595\pi$
$911$	−2.21245e7	−0.883236	−0.441618	+	0.897203i	$0.645595\pi$
$912$	0	0
$913$	−9.57457e6	−0.380139
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1.07790e6	−0.0423307
$918$	0	0
$919$	−1.82158e7	−0.711476	−0.355738	−	0.934586i	$-0.615770\pi$
$919$	−1.82158e7	−0.711476	−0.355738	+	0.934586i	$0.615770\pi$
$920$	0	0
$921$	3.51084e7	1.36383
$922$	0	0
$923$	−2.47849e7	−0.957596
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−662686.	−0.0253284
$928$	0	0
$929$	1.48086e7	0.562958	0.281479	−	0.959567i	$-0.409175\pi$
$929$	1.48086e7	0.562958	0.281479	+	0.959567i	$0.409175\pi$
$930$	0	0
$931$	4.26731e6	0.161354
$932$	0	0
$933$	−2.28154e7	−0.858071
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−3.96760e7	−1.47631	−0.738157	−	0.674629i	$-0.764304\pi$
$937$	−3.96760e7	−1.47631	−0.738157	+	0.674629i	$0.764304\pi$
$938$	0	0
$939$	7.11453e7	2.63319
$940$	0	0
$941$	−2.74007e7	−1.00876	−0.504380	−	0.863482i	$-0.668279\pi$
$941$	−2.74007e7	−1.00876	−0.504380	+	0.863482i	$0.668279\pi$
$942$	0	0
$943$	−1.77178e6	−0.0648830
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.70369e7	1.34202	0.671011	−	0.741448i	$-0.265860\pi$
$947$	3.70369e7	1.34202	0.671011	+	0.741448i	$0.265860\pi$
$948$	0	0
$949$	−2.66782e7	−0.961591
$950$	0	0
$951$	6.22301e7	2.23125
$952$	0	0
$953$	3.01349e7	1.07482	0.537412	−	0.843320i	$-0.319402\pi$
$953$	3.01349e7	1.07482	0.537412	+	0.843320i	$0.319402\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−2.66046e7	−0.939023
$958$	0	0
$959$	−1.61924e7	−0.568544
$960$	0	0
$961$	−2.76871e7	−0.967094
$962$	0	0
$963$	7.70710e7	2.67809
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−5.27724e7	−1.81485	−0.907425	−	0.420213i	$-0.861955\pi$
$967$	−5.27724e7	−1.81485	−0.907425	+	0.420213i	$0.861955\pi$
$968$	0	0
$969$	−5.52708e7	−1.89098
$970$	0	0
$971$	2.09055e7	0.711560	0.355780	−	0.934570i	$-0.384215\pi$
$971$	2.09055e7	0.711560	0.355780	+	0.934570i	$0.384215\pi$
$972$	0	0
$973$	1.97593e7	0.669098
$974$	0	0
$975$	0	0
$976$	0	0
$977$	609167.	0.0204174	0.0102087	−	0.999948i	$-0.496750\pi$
$977$	609167.	0.0204174	0.0102087	+	0.999948i	$0.496750\pi$
$978$	0	0
$979$	1.22549e7	0.408650
$980$	0	0
$981$	5.00842e7	1.66161
$982$	0	0
$983$	4.01952e6	0.132676	0.0663378	−	0.997797i	$-0.478869\pi$
$983$	4.01952e6	0.132676	0.0663378	+	0.997797i	$0.478869\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1.36346e7	0.445503
$988$	0	0
$989$	6.23572e6	0.202720
$990$	0	0
$991$	2.08792e6	0.0675351	0.0337676	−	0.999430i	$-0.489249\pi$
$991$	2.08792e6	0.0675351	0.0337676	+	0.999430i	$0.489249\pi$
$992$	0	0
$993$	9.82064e7	3.16058
$994$	0	0
$995$	0	0
$996$	0	0
$997$	6.09104e6	0.194068	0.0970340	−	0.995281i	$-0.469064\pi$
$997$	6.09104e6	0.194068	0.0970340	+	0.995281i	$0.469064\pi$
$998$	0	0
$999$	−9.28053e6	−0.294211

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	700.6.a.p.1.1		8
5.2	odd	4		140.6.e.a.29.15	yes	16
5.3	odd	4		140.6.e.a.29.2	✓	16
5.4	even	2		700.6.a.o.1.8		8
20.3	even	4		560.6.g.f.449.15		16
20.7	even	4		560.6.g.f.449.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.6.e.a.29.2	✓	16	5.3	odd	4
140.6.e.a.29.15	yes	16	5.2	odd	4
560.6.g.f.449.2		16	20.7	even	4
560.6.g.f.449.15		16	20.3	even	4
700.6.a.o.1.8		8	5.4	even	2
700.6.a.p.1.1		8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	700.a (trivial)

\(f(q)\)	\(=\)	\(q-25.3503 q^{3} +49.0000 q^{7} +399.638 q^{9} -141.450 q^{11} +297.641 q^{13} +1226.74 q^{17} +1777.30 q^{19} -1242.16 q^{21} +3158.14 q^{23} -3970.81 q^{27} -7419.42 q^{29} +970.605 q^{31} +3585.80 q^{33} +2337.19 q^{37} -7545.29 q^{39} -561.020 q^{41} +1974.49 q^{43} -10976.5 q^{47} +2401.00 q^{49} -31098.1 q^{51} +33500.0 q^{53} -45055.2 q^{57} +32227.0 q^{59} -34441.8 q^{61} +19582.2 q^{63} -12670.3 q^{67} -80059.9 q^{69} -83271.0 q^{71} -89632.0 q^{73} -6931.05 q^{77} +4871.34 q^{79} +3549.22 q^{81} +67688.7 q^{83} +188085. q^{87} -86637.4 q^{89} +14584.4 q^{91} -24605.1 q^{93} +6607.75 q^{97} -56528.7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 13 q^{3} + 392 q^{7} + 803 q^{9} + 147 q^{11} + 173 q^{13} + 1755 q^{17} + 1756 q^{19} + 637 q^{21} + 4422 q^{23} + 6127 q^{27} + 10435 q^{29} + 2014 q^{31} - 195 q^{33} + 5986 q^{37} + 12679 q^{39}+ \cdots - 146982 q^{99}+O(q^{100}) \) 8 * q + 13 * q^3 + 392 * q^7 + 803 * q^9 + 147 * q^11 + 173 * q^13 + 1755 * q^17 + 1756 * q^19 + 637 * q^21 + 4422 * q^23 + 6127 * q^27 + 10435 * q^29 + 2014 * q^31 - 195 * q^33 + 5986 * q^37 + 12679 * q^39 + 12476 * q^41 + 11596 * q^43 - 17655 * q^47 + 19208 * q^49 - 6151 * q^51 + 3526 * q^53 - 30712 * q^57 - 18078 * q^59 - 7278 * q^61 + 39347 * q^63 + 29410 * q^67 - 49670 * q^69 - 23996 * q^71 + 51730 * q^73 + 7203 * q^77 - 215 * q^79 + 78104 * q^81 + 203292 * q^83 + 179271 * q^87 - 152090 * q^89 + 8477 * q^91 + 255098 * q^93 + 142427 * q^97 - 146982 * q^99

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