LMFDB - Embedded newform 704.6.a.m.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [704,6,Mod(1,704)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(704, 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("704.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 704 = 2^{6} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	704.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:	$112.910209148$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{31}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 31 $ x^2 - 31
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 44)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$5.56776$ of defining polynomial
Character	$\chi$	$=$	704.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+19.2711 q^{3} -33.5421 q^{5} -67.1868 q^{7} +128.374 q^{9} -121.000 q^{11} -504.645 q^{13} -646.392 q^{15} +984.740 q^{17} +281.597 q^{19} -1294.76 q^{21} -359.008 q^{23} -1999.93 q^{25} -2208.97 q^{27} +5025.14 q^{29} +7013.56 q^{31} -2331.80 q^{33} +2253.59 q^{35} +5243.52 q^{37} -9725.04 q^{39} -13875.3 q^{41} +20156.6 q^{43} -4305.92 q^{45} -6782.51 q^{47} -12292.9 q^{49} +18977.0 q^{51} +27257.7 q^{53} +4058.60 q^{55} +5426.68 q^{57} +19091.6 q^{59} -24047.7 q^{61} -8625.02 q^{63} +16926.9 q^{65} +53262.7 q^{67} -6918.46 q^{69} +44240.8 q^{71} +21914.6 q^{73} -38540.7 q^{75} +8129.61 q^{77} +26364.9 q^{79} -73764.0 q^{81} -19420.6 q^{83} -33030.3 q^{85} +96839.8 q^{87} -31325.2 q^{89} +33905.5 q^{91} +135159. q^{93} -9445.37 q^{95} +134081. q^{97} -15533.2 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} + 22 q^{5} - 268 q^{7} + 524 q^{9} - 242 q^{11} - 1232 q^{13} - 2050 q^{15} - 124 q^{17} + 1944 q^{19} + 3780 q^{21} - 3346 q^{23} - 2040 q^{25} - 6066 q^{27} + 6576 q^{29} - 2498 q^{31} + 726 q^{33}+ \cdots - 63404 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 + 22 * q^5 - 268 * q^7 + 524 * q^9 - 242 * q^11 - 1232 * q^13 - 2050 * q^15 - 124 * q^17 + 1944 * q^19 + 3780 * q^21 - 3346 * q^23 - 2040 * q^25 - 6066 * q^27 + 6576 * q^29 - 2498 * q^31 + 726 * q^33 - 8900 * q^35 + 14674 * q^37 + 8656 * q^39 - 6504 * q^41 + 11628 * q^43 + 17668 * q^45 - 36816 * q^47 + 11226 * q^49 + 46996 * q^51 + 3292 * q^53 - 2662 * q^55 - 36584 * q^57 + 12126 * q^59 - 73128 * q^61 - 88072 * q^63 - 23472 * q^65 + 29334 * q^67 + 68566 * q^69 + 46122 * q^71 + 8240 * q^73 - 37528 * q^75 + 32428 * q^77 + 14780 * q^79 - 72430 * q^81 - 74564 * q^83 - 94612 * q^85 + 57648 * q^87 - 33698 * q^89 + 179968 * q^91 + 375526 * q^93 + 82888 * q^95 + 126162 * q^97 - 63404 * 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
$3$	19.2711	1.23624	0.618119	−	0.786084i	$-0.287895\pi$
$3$	19.2711	1.23624	0.618119	+	0.786084i	$0.287895\pi$
$4$	0	0
$5$	−33.5421	−0.600020	−0.300010	−	0.953936i	$-0.596990\pi$
$5$	−33.5421	−0.600020	−0.300010	+	0.953936i	$0.596990\pi$
$6$	0	0
$7$	−67.1868	−0.518250	−0.259125	−	0.965844i	$-0.583434\pi$
$7$	−67.1868	−0.518250	−0.259125	+	0.965844i	$0.583434\pi$
$8$	0	0
$9$	128.374	0.528287
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	0	0
$13$	−504.645	−0.828185	−0.414092	−	0.910235i	$-0.635901\pi$
$13$	−504.645	−0.828185	−0.414092	+	0.910235i	$0.635901\pi$
$14$	0	0
$15$	−646.392	−0.741768
$16$	0	0
$17$	984.740	0.826417	0.413208	−	0.910636i	$-0.364408\pi$
$17$	984.740	0.826417	0.413208	+	0.910636i	$0.364408\pi$
$18$	0	0
$19$	281.597	0.178955	0.0894776	−	0.995989i	$-0.471480\pi$
$19$	281.597	0.178955	0.0894776	+	0.995989i	$0.471480\pi$
$20$	0	0
$21$	−1294.76	−0.640680
$22$	0	0
$23$	−359.008	−0.141509	−0.0707545	−	0.997494i	$-0.522541\pi$
$23$	−359.008	−0.141509	−0.0707545	+	0.997494i	$0.522541\pi$
$24$	0	0
$25$	−1999.93	−0.639976
$26$	0	0
$27$	−2208.97	−0.583150
$28$	0	0
$29$	5025.14	1.10957	0.554783	−	0.831995i	$-0.312801\pi$
$29$	5025.14	1.10957	0.554783	+	0.831995i	$0.312801\pi$
$30$	0	0
$31$	7013.56	1.31079	0.655397	−	0.755285i	$-0.272501\pi$
$31$	7013.56	1.31079	0.655397	+	0.755285i	$0.272501\pi$
$32$	0	0
$33$	−2331.80	−0.372740
$34$	0	0
$35$	2253.59	0.310960
$36$	0	0
$37$	5243.52	0.629678	0.314839	−	0.949145i	$-0.398049\pi$
$37$	5243.52	0.629678	0.314839	+	0.949145i	$0.398049\pi$
$38$	0	0
$39$	−9725.04	−1.02383
$40$	0	0
$41$	−13875.3	−1.28909	−0.644544	−	0.764567i	$-0.722953\pi$
$41$	−13875.3	−1.28909	−0.644544	+	0.764567i	$0.722953\pi$
$42$	0	0
$43$	20156.6	1.66244	0.831219	−	0.555946i	$-0.187644\pi$
$43$	20156.6	1.66244	0.831219	+	0.555946i	$0.187644\pi$
$44$	0	0
$45$	−4305.92	−0.316982
$46$	0	0
$47$	−6782.51	−0.447864	−0.223932	−	0.974605i	$-0.571889\pi$
$47$	−6782.51	−0.447864	−0.223932	+	0.974605i	$0.571889\pi$
$48$	0	0
$49$	−12292.9	−0.731417
$50$	0	0
$51$	18977.0	1.02165
$52$	0	0
$53$	27257.7	1.33291	0.666453	−	0.745547i	$-0.267812\pi$
$53$	27257.7	1.33291	0.666453	+	0.745547i	$0.267812\pi$
$54$	0	0
$55$	4058.60	0.180913
$56$	0	0
$57$	5426.68	0.221231
$58$	0	0
$59$	19091.6	0.714022	0.357011	−	0.934100i	$-0.383796\pi$
$59$	19091.6	0.714022	0.357011	+	0.934100i	$0.383796\pi$
$60$	0	0
$61$	−24047.7	−0.827463	−0.413731	−	0.910399i	$-0.635775\pi$
$61$	−24047.7	−0.827463	−0.413731	+	0.910399i	$0.635775\pi$
$62$	0	0
$63$	−8625.02	−0.273784
$64$	0	0
$65$	16926.9	0.496927
$66$	0	0
$67$	53262.7	1.44956	0.724781	−	0.688980i	$-0.241941\pi$
$67$	53262.7	1.44956	0.724781	+	0.688980i	$0.241941\pi$
$68$	0	0
$69$	−6918.46	−0.174939
$70$	0	0
$71$	44240.8	1.04154	0.520771	−	0.853696i	$-0.325645\pi$
$71$	44240.8	1.04154	0.520771	+	0.853696i	$0.325645\pi$
$72$	0	0
$73$	21914.6	0.481311	0.240656	−	0.970611i	$-0.422638\pi$
$73$	21914.6	0.481311	0.240656	+	0.970611i	$0.422638\pi$
$74$	0	0
$75$	−38540.7	−0.791164
$76$	0	0
$77$	8129.61	0.156258
$78$	0	0
$79$	26364.9	0.475290	0.237645	−	0.971352i	$-0.423624\pi$
$79$	26364.9	0.475290	0.237645	+	0.971352i	$0.423624\pi$
$80$	0	0
$81$	−73764.0	−1.24920
$82$	0	0
$83$	−19420.6	−0.309434	−0.154717	−	0.987959i	$-0.549447\pi$
$83$	−19420.6	−0.309434	−0.154717	+	0.987959i	$0.549447\pi$
$84$	0	0
$85$	−33030.3	−0.495866
$86$	0	0
$87$	96839.8	1.37169
$88$	0	0
$89$	−31325.2	−0.419197	−0.209599	−	0.977787i	$-0.567216\pi$
$89$	−31325.2	−0.419197	−0.209599	+	0.977787i	$0.567216\pi$
$90$	0	0
$91$	33905.5	0.429207
$92$	0	0
$93$	135159.	1.62045
$94$	0	0
$95$	−9445.37	−0.107377
$96$	0	0
$97$	134081.	1.44690	0.723450	−	0.690377i	$-0.242555\pi$
$97$	134081.	1.44690	0.723450	+	0.690377i	$0.242555\pi$
$98$	0	0
$99$	−15533.2	−0.159284
$100$	0	0
$101$	160519.	1.56575	0.782875	−	0.622179i	$-0.213752\pi$
$101$	160519.	1.56575	0.782875	+	0.622179i	$0.213752\pi$
$102$	0	0
$103$	−26257.2	−0.243868	−0.121934	−	0.992538i	$-0.538910\pi$
$103$	−26257.2	−0.243868	−0.121934	+	0.992538i	$0.538910\pi$
$104$	0	0
$105$	43429.0	0.384421
$106$	0	0
$107$	225739.	1.90611	0.953053	−	0.302803i	$-0.0979226\pi$
$107$	225739.	1.90611	0.953053	+	0.302803i	$0.0979226\pi$
$108$	0	0
$109$	−65776.2	−0.530277	−0.265138	−	0.964210i	$-0.585418\pi$
$109$	−65776.2	−0.530277	−0.265138	+	0.964210i	$0.585418\pi$
$110$	0	0
$111$	101048.	0.778433
$112$	0	0
$113$	219652.	1.61823	0.809115	−	0.587651i	$-0.199947\pi$
$113$	219652.	1.61823	0.809115	+	0.587651i	$0.199947\pi$
$114$	0	0
$115$	12041.9	0.0849082
$116$	0	0
$117$	−64783.1	−0.437519
$118$	0	0
$119$	−66161.5	−0.428290
$120$	0	0
$121$	14641.0	0.0909091
$122$	0	0
$123$	−267392.	−1.59362
$124$	0	0
$125$	171901.	0.984018
$126$	0	0
$127$	−220271.	−1.21185	−0.605924	−	0.795522i	$-0.707196\pi$
$127$	−220271.	−1.21185	−0.605924	+	0.795522i	$0.707196\pi$
$128$	0	0
$129$	388438.	2.05517
$130$	0	0
$131$	−279273.	−1.42184	−0.710921	−	0.703272i	$-0.751722\pi$
$131$	−279273.	−1.42184	−0.710921	+	0.703272i	$0.751722\pi$
$132$	0	0
$133$	−18919.6	−0.0927435
$134$	0	0
$135$	74093.6	0.349902
$136$	0	0
$137$	−100347.	−0.456774	−0.228387	−	0.973570i	$-0.573345\pi$
$137$	−100347.	−0.456774	−0.228387	+	0.973570i	$0.573345\pi$
$138$	0	0
$139$	242456.	1.06438	0.532189	−	0.846625i	$-0.321369\pi$
$139$	242456.	1.06438	0.532189	+	0.846625i	$0.321369\pi$
$140$	0	0
$141$	−130706.	−0.553666
$142$	0	0
$143$	61062.0	0.249707
$144$	0	0
$145$	−168554.	−0.665762
$146$	0	0
$147$	−236898.	−0.904206
$148$	0	0
$149$	−8611.73	−0.0317779	−0.0158889	−	0.999874i	$-0.505058\pi$
$149$	−8611.73	−0.0317779	−0.0158889	+	0.999874i	$0.505058\pi$
$150$	0	0
$151$	−433376.	−1.54676	−0.773380	−	0.633943i	$-0.781435\pi$
$151$	−433376.	−1.54676	−0.773380	+	0.633943i	$0.781435\pi$
$152$	0	0
$153$	126415.	0.436585
$154$	0	0
$155$	−235250.	−0.786502
$156$	0	0
$157$	−335878.	−1.08751	−0.543754	−	0.839245i	$-0.682997\pi$
$157$	−335878.	−1.08751	−0.543754	+	0.839245i	$0.682997\pi$
$158$	0	0
$159$	525285.	1.64779
$160$	0	0
$161$	24120.6	0.0733370
$162$	0	0
$163$	−375816.	−1.10792	−0.553958	−	0.832545i	$-0.686883\pi$
$163$	−375816.	−1.10792	−0.553958	+	0.832545i	$0.686883\pi$
$164$	0	0
$165$	78213.4	0.223651
$166$	0	0
$167$	379939.	1.05420	0.527100	−	0.849803i	$-0.323279\pi$
$167$	379939.	1.05420	0.527100	+	0.849803i	$0.323279\pi$
$168$	0	0
$169$	−116627.	−0.314110
$170$	0	0
$171$	36149.7	0.0945397
$172$	0	0
$173$	405780.	1.03080	0.515401	−	0.856949i	$-0.327643\pi$
$173$	405780.	1.03080	0.515401	+	0.856949i	$0.327643\pi$
$174$	0	0
$175$	134369.	0.331668
$176$	0	0
$177$	367915.	0.882702
$178$	0	0
$179$	−131144.	−0.305926	−0.152963	−	0.988232i	$-0.548881\pi$
$179$	−131144.	−0.305926	−0.152963	+	0.988232i	$0.548881\pi$
$180$	0	0
$181$	726589.	1.64851	0.824256	−	0.566217i	$-0.191594\pi$
$181$	726589.	1.64851	0.824256	+	0.566217i	$0.191594\pi$
$182$	0	0
$183$	−463424.	−1.02294
$184$	0	0
$185$	−175879.	−0.377819
$186$	0	0
$187$	−119154.	−0.249174
$188$	0	0
$189$	148414.	0.302218
$190$	0	0
$191$	−524638.	−1.04058	−0.520291	−	0.853989i	$-0.674177\pi$
$191$	−524638.	−1.04058	−0.520291	+	0.853989i	$0.674177\pi$
$192$	0	0
$193$	524652.	1.01386	0.506930	−	0.861987i	$-0.330780\pi$
$193$	524652.	1.01386	0.506930	+	0.861987i	$0.330780\pi$
$194$	0	0
$195$	326198.	0.614321
$196$	0	0
$197$	−485331.	−0.890988	−0.445494	−	0.895285i	$-0.646972\pi$
$197$	−485331.	−0.890988	−0.445494	+	0.895285i	$0.646972\pi$
$198$	0	0
$199$	−867363.	−1.55263	−0.776315	−	0.630345i	$-0.782913\pi$
$199$	−867363.	−1.55263	−0.776315	+	0.630345i	$0.782913\pi$
$200$	0	0
$201$	1.02643e6	1.79200
$202$	0	0
$203$	−337623.	−0.575033
$204$	0	0
$205$	465407.	0.773478
$206$	0	0
$207$	−46087.1	−0.0747573
$208$	0	0
$209$	−34073.3	−0.0539570
$210$	0	0
$211$	752734.	1.16395	0.581976	−	0.813206i	$-0.302280\pi$
$211$	752734.	1.16395	0.581976	+	0.813206i	$0.302280\pi$
$212$	0	0
$213$	852567.	1.28759
$214$	0	0
$215$	−676094.	−0.997495
$216$	0	0
$217$	−471219.	−0.679319
$218$	0	0
$219$	422317.	0.595016
$220$	0	0
$221$	−496944.	−0.684426
$222$	0	0
$223$	533342.	0.718197	0.359099	−	0.933300i	$-0.383084\pi$
$223$	533342.	0.718197	0.359099	+	0.933300i	$0.383084\pi$
$224$	0	0
$225$	−256738.	−0.338091
$226$	0	0
$227$	792585.	1.02089	0.510447	−	0.859909i	$-0.329480\pi$
$227$	792585.	1.02089	0.510447	+	0.859909i	$0.329480\pi$
$228$	0	0
$229$	1.53926e6	1.93964	0.969822	−	0.243814i	$-0.0783985\pi$
$229$	1.53926e6	1.93964	0.969822	+	0.243814i	$0.0783985\pi$
$230$	0	0
$231$	156666.	0.193172
$232$	0	0
$233$	−397433.	−0.479594	−0.239797	−	0.970823i	$-0.577081\pi$
$233$	−397433.	−0.479594	−0.239797	+	0.970823i	$0.577081\pi$
$234$	0	0
$235$	227500.	0.268727
$236$	0	0
$237$	508080.	0.587573
$238$	0	0
$239$	−360080.	−0.407760	−0.203880	−	0.978996i	$-0.565355\pi$
$239$	−360080.	−0.407760	−0.203880	+	0.978996i	$0.565355\pi$
$240$	0	0
$241$	915010.	1.01481	0.507403	−	0.861709i	$-0.330605\pi$
$241$	915010.	1.01481	0.507403	+	0.861709i	$0.330605\pi$
$242$	0	0
$243$	−884730.	−0.961159
$244$	0	0
$245$	412331.	0.438865
$246$	0	0
$247$	−142107.	−0.148208
$248$	0	0
$249$	−374256.	−0.382534
$250$	0	0
$251$	377210.	0.377919	0.188960	−	0.981985i	$-0.439488\pi$
$251$	377210.	0.377919	0.188960	+	0.981985i	$0.439488\pi$
$252$	0	0
$253$	43439.9	0.0426666
$254$	0	0
$255$	−636528.	−0.613009
$256$	0	0
$257$	618791.	0.584402	0.292201	−	0.956357i	$-0.405612\pi$
$257$	618791.	0.584402	0.292201	+	0.956357i	$0.405612\pi$
$258$	0	0
$259$	−352296.	−0.326331
$260$	0	0
$261$	645096.	0.586169
$262$	0	0
$263$	1.48074e6	1.32004	0.660022	−	0.751246i	$-0.270547\pi$
$263$	1.48074e6	1.32004	0.660022	+	0.751246i	$0.270547\pi$
$264$	0	0
$265$	−914281.	−0.799770
$266$	0	0
$267$	−603669.	−0.518228
$268$	0	0
$269$	−620662.	−0.522967	−0.261484	−	0.965208i	$-0.584212\pi$
$269$	−620662.	−0.522967	−0.261484	+	0.965208i	$0.584212\pi$
$270$	0	0
$271$	210316.	0.173960	0.0869798	−	0.996210i	$-0.472278\pi$
$271$	210316.	0.173960	0.0869798	+	0.996210i	$0.472278\pi$
$272$	0	0
$273$	653394.	0.530602
$274$	0	0
$275$	241991.	0.192960
$276$	0	0
$277$	1.24683e6	0.976354	0.488177	−	0.872745i	$-0.337662\pi$
$277$	1.24683e6	0.976354	0.488177	+	0.872745i	$0.337662\pi$
$278$	0	0
$279$	900357.	0.692475
$280$	0	0
$281$	1.50249e6	1.13513	0.567564	−	0.823329i	$-0.307886\pi$
$281$	1.50249e6	1.13513	0.567564	+	0.823329i	$0.307886\pi$
$282$	0	0
$283$	−539947.	−0.400761	−0.200380	−	0.979718i	$-0.564218\pi$
$283$	−539947.	−0.400761	−0.200380	+	0.979718i	$0.564218\pi$
$284$	0	0
$285$	−182022.	−0.132743
$286$	0	0
$287$	932237.	0.668069
$288$	0	0
$289$	−450145.	−0.317035
$290$	0	0
$291$	2.58389e6	1.78871
$292$	0	0
$293$	1.43256e6	0.974861	0.487431	−	0.873162i	$-0.337934\pi$
$293$	1.43256e6	0.974861	0.487431	+	0.873162i	$0.337934\pi$
$294$	0	0
$295$	−640372.	−0.428427
$296$	0	0
$297$	267285.	0.175826
$298$	0	0
$299$	181171.	0.117196
$300$	0	0
$301$	−1.35426e6	−0.861558
$302$	0	0
$303$	3.09337e6	1.93564
$304$	0	0
$305$	806610.	0.496494
$306$	0	0
$307$	−1.41065e6	−0.854224	−0.427112	−	0.904199i	$-0.640469\pi$
$307$	−1.41065e6	−0.854224	−0.427112	+	0.904199i	$0.640469\pi$
$308$	0	0
$309$	−506003.	−0.301479
$310$	0	0
$311$	−875210.	−0.513111	−0.256555	−	0.966530i	$-0.582588\pi$
$311$	−875210.	−0.513111	−0.256555	+	0.966530i	$0.582588\pi$
$312$	0	0
$313$	996513.	0.574939	0.287470	−	0.957790i	$-0.407186\pi$
$313$	996513.	0.574939	0.287470	+	0.957790i	$0.407186\pi$
$314$	0	0
$315$	289301.	0.164276
$316$	0	0
$317$	−2.25829e6	−1.26221	−0.631104	−	0.775698i	$-0.717398\pi$
$317$	−2.25829e6	−1.26221	−0.631104	+	0.775698i	$0.717398\pi$
$318$	0	0
$319$	−608042.	−0.334547
$320$	0	0
$321$	4.35023e6	2.35640
$322$	0	0
$323$	277300.	0.147892
$324$	0	0
$325$	1.00925e6	0.530019
$326$	0	0
$327$	−1.26758e6	−0.655549
$328$	0	0
$329$	455695.	0.232105
$330$	0	0
$331$	188468.	0.0945512	0.0472756	−	0.998882i	$-0.484946\pi$
$331$	188468.	0.0945512	0.0472756	+	0.998882i	$0.484946\pi$
$332$	0	0
$333$	673130.	0.332651
$334$	0	0
$335$	−1.78655e6	−0.869765
$336$	0	0
$337$	−1.43240e6	−0.687050	−0.343525	−	0.939143i	$-0.611621\pi$
$337$	−1.43240e6	−0.687050	−0.343525	+	0.939143i	$0.611621\pi$
$338$	0	0
$339$	4.23294e6	2.00052
$340$	0	0
$341$	−848641.	−0.395219
$342$	0	0
$343$	1.95513e6	0.897306
$344$	0	0
$345$	232060.	0.104967
$346$	0	0
$347$	−2.64009e6	−1.17705	−0.588526	−	0.808478i	$-0.700292\pi$
$347$	−2.64009e6	−1.17705	−0.588526	+	0.808478i	$0.700292\pi$
$348$	0	0
$349$	3.38705e6	1.48853	0.744267	−	0.667883i	$-0.232799\pi$
$349$	3.38705e6	1.48853	0.744267	+	0.667883i	$0.232799\pi$
$350$	0	0
$351$	1.11475e6	0.482956
$352$	0	0
$353$	−54279.5	−0.0231846	−0.0115923	−	0.999933i	$-0.503690\pi$
$353$	−54279.5	−0.0231846	−0.0115923	+	0.999933i	$0.503690\pi$
$354$	0	0
$355$	−1.48393e6	−0.624946
$356$	0	0
$357$	−1.27500e6	−0.529469
$358$	0	0
$359$	2.12836e6	0.871584	0.435792	−	0.900047i	$-0.356468\pi$
$359$	2.12836e6	0.871584	0.435792	+	0.900047i	$0.356468\pi$
$360$	0	0
$361$	−2.39680e6	−0.967975
$362$	0	0
$363$	282148.	0.112385
$364$	0	0
$365$	−735061.	−0.288796
$366$	0	0
$367$	−1.17965e6	−0.457183	−0.228591	−	0.973522i	$-0.573412\pi$
$367$	−1.17965e6	−0.457183	−0.228591	+	0.973522i	$0.573412\pi$
$368$	0	0
$369$	−1.78122e6	−0.681008
$370$	0	0
$371$	−1.83136e6	−0.690779
$372$	0	0
$373$	2.30580e6	0.858125	0.429062	−	0.903275i	$-0.358844\pi$
$373$	2.30580e6	0.858125	0.429062	+	0.903275i	$0.358844\pi$
$374$	0	0
$375$	3.31271e6	1.21648
$376$	0	0
$377$	−2.53591e6	−0.918926
$378$	0	0
$379$	1.17613e6	0.420588	0.210294	−	0.977638i	$-0.432558\pi$
$379$	1.17613e6	0.420588	0.210294	+	0.977638i	$0.432558\pi$
$380$	0	0
$381$	−4.24486e6	−1.49813
$382$	0	0
$383$	2.05506e6	0.715858	0.357929	−	0.933749i	$-0.383483\pi$
$383$	2.05506e6	0.715858	0.357929	+	0.933749i	$0.383483\pi$
$384$	0	0
$385$	−272684.	−0.0937580
$386$	0	0
$387$	2.58757e6	0.878243
$388$	0	0
$389$	−2.93102e6	−0.982074	−0.491037	−	0.871139i	$-0.663382\pi$
$389$	−2.93102e6	−0.982074	−0.491037	+	0.871139i	$0.663382\pi$
$390$	0	0
$391$	−353529.	−0.116945
$392$	0	0
$393$	−5.38189e6	−1.75774
$394$	0	0
$395$	−884336.	−0.285184
$396$	0	0
$397$	−4.06241e6	−1.29362	−0.646810	−	0.762651i	$-0.723897\pi$
$397$	−4.06241e6	−1.29362	−0.646810	+	0.762651i	$0.723897\pi$
$398$	0	0
$399$	−364601.	−0.114653
$400$	0	0
$401$	3.50063e6	1.08714	0.543570	−	0.839364i	$-0.317072\pi$
$401$	3.50063e6	1.08714	0.543570	+	0.839364i	$0.317072\pi$
$402$	0	0
$403$	−3.53936e6	−1.08558
$404$	0	0
$405$	2.47420e6	0.749544
$406$	0	0
$407$	−634466.	−0.189855
$408$	0	0
$409$	3.14103e6	0.928462	0.464231	−	0.885714i	$-0.346331\pi$
$409$	3.14103e6	0.928462	0.464231	+	0.885714i	$0.346331\pi$
$410$	0	0
$411$	−1.93379e6	−0.564682
$412$	0	0
$413$	−1.28270e6	−0.370042
$414$	0	0
$415$	651408.	0.185666
$416$	0	0
$417$	4.67239e6	1.31583
$418$	0	0
$419$	−435980.	−0.121320	−0.0606598	−	0.998158i	$-0.519320\pi$
$419$	−435980.	−0.121320	−0.0606598	+	0.998158i	$0.519320\pi$
$420$	0	0
$421$	2.17633e6	0.598439	0.299220	−	0.954184i	$-0.403274\pi$
$421$	2.17633e6	0.598439	0.299220	+	0.954184i	$0.403274\pi$
$422$	0	0
$423$	−870695.	−0.236600
$424$	0	0
$425$	−1.96941e6	−0.528887
$426$	0	0
$427$	1.61569e6	0.428832
$428$	0	0
$429$	1.17673e6	0.308698
$430$	0	0
$431$	787788.	0.204275	0.102138	−	0.994770i	$-0.467432\pi$
$431$	787788.	0.204275	0.102138	+	0.994770i	$0.467432\pi$
$432$	0	0
$433$	956782.	0.245241	0.122620	−	0.992454i	$-0.460870\pi$
$433$	956782.	0.245241	0.122620	+	0.992454i	$0.460870\pi$
$434$	0	0
$435$	−3.24821e6	−0.823041
$436$	0	0
$437$	−101096.	−0.0253238
$438$	0	0
$439$	−6.51277e6	−1.61289	−0.806445	−	0.591309i	$-0.798611\pi$
$439$	−6.51277e6	−1.61289	−0.806445	+	0.591309i	$0.798611\pi$
$440$	0	0
$441$	−1.57809e6	−0.386398
$442$	0	0
$443$	124827.	0.0302203	0.0151102	−	0.999886i	$-0.495190\pi$
$443$	124827.	0.0302203	0.0151102	+	0.999886i	$0.495190\pi$
$444$	0	0
$445$	1.05071e6	0.251527
$446$	0	0
$447$	−165957.	−0.0392850
$448$	0	0
$449$	−8.06847e6	−1.88875	−0.944377	−	0.328865i	$-0.893334\pi$
$449$	−8.06847e6	−1.88875	−0.944377	+	0.328865i	$0.893334\pi$
$450$	0	0
$451$	1.67891e6	0.388675
$452$	0	0
$453$	−8.35162e6	−1.91216
$454$	0	0
$455$	−1.13726e6	−0.257532
$456$	0	0
$457$	6.85566e6	1.53553	0.767766	−	0.640731i	$-0.221368\pi$
$457$	6.85566e6	1.53553	0.767766	+	0.640731i	$0.221368\pi$
$458$	0	0
$459$	−2.17526e6	−0.481925
$460$	0	0
$461$	−4.18486e6	−0.917125	−0.458562	−	0.888662i	$-0.651635\pi$
$461$	−4.18486e6	−0.917125	−0.458562	+	0.888662i	$0.651635\pi$
$462$	0	0
$463$	−2.44902e6	−0.530934	−0.265467	−	0.964120i	$-0.585526\pi$
$463$	−2.44902e6	−0.530934	−0.265467	+	0.964120i	$0.585526\pi$
$464$	0	0
$465$	−4.53351e6	−0.972304
$466$	0	0
$467$	2.99629e6	0.635758	0.317879	−	0.948131i	$-0.397029\pi$
$467$	2.99629e6	0.635758	0.317879	+	0.948131i	$0.397029\pi$
$468$	0	0
$469$	−3.57855e6	−0.751235
$470$	0	0
$471$	−6.47272e6	−1.34442
$472$	0	0
$473$	−2.43894e6	−0.501244
$474$	0	0
$475$	−563174.	−0.114527
$476$	0	0
$477$	3.49917e6	0.704157
$478$	0	0
$479$	−6.25520e6	−1.24567	−0.622834	−	0.782354i	$-0.714019\pi$
$479$	−6.25520e6	−1.24567	−0.622834	+	0.782354i	$0.714019\pi$
$480$	0	0
$481$	−2.64611e6	−0.521490
$482$	0	0
$483$	464829.	0.0906620
$484$	0	0
$485$	−4.49736e6	−0.868168
$486$	0	0
$487$	−8.04889e6	−1.53785	−0.768924	−	0.639340i	$-0.779208\pi$
$487$	−8.04889e6	−1.53785	−0.768924	+	0.639340i	$0.779208\pi$
$488$	0	0
$489$	−7.24238e6	−1.36965
$490$	0	0
$491$	−1.00629e7	−1.88374	−0.941869	−	0.335981i	$-0.890932\pi$
$491$	−1.00629e7	−1.88374	−0.941869	+	0.335981i	$0.890932\pi$
$492$	0	0
$493$	4.94846e6	0.916965
$494$	0	0
$495$	521017.	0.0955738
$496$	0	0
$497$	−2.97240e6	−0.539779
$498$	0	0
$499$	6.54572e6	1.17681	0.588405	−	0.808566i	$-0.299756\pi$
$499$	6.54572e6	1.17681	0.588405	+	0.808566i	$0.299756\pi$
$500$	0	0
$501$	7.32183e6	1.30324
$502$	0	0
$503$	−1.64809e6	−0.290443	−0.145222	−	0.989399i	$-0.546389\pi$
$503$	−1.64809e6	−0.290443	−0.145222	+	0.989399i	$0.546389\pi$
$504$	0	0
$505$	−5.38414e6	−0.939481
$506$	0	0
$507$	−2.24752e6	−0.388315
$508$	0	0
$509$	2.66813e6	0.456471	0.228235	−	0.973606i	$-0.426704\pi$
$509$	2.66813e6	0.456471	0.228235	+	0.973606i	$0.426704\pi$
$510$	0	0
$511$	−1.47237e6	−0.249439
$512$	0	0
$513$	−622040.	−0.104358
$514$	0	0
$515$	880721.	0.146326
$516$	0	0
$517$	820683.	0.135036
$518$	0	0
$519$	7.81981e6	1.27432
$520$	0	0
$521$	667289.	0.107701	0.0538505	−	0.998549i	$-0.482851\pi$
$521$	667289.	0.107701	0.0538505	+	0.998549i	$0.482851\pi$
$522$	0	0
$523$	1.34478e6	0.214979	0.107490	−	0.994206i	$-0.465719\pi$
$523$	1.34478e6	0.214979	0.107490	+	0.994206i	$0.465719\pi$
$524$	0	0
$525$	2.58943e6	0.410020
$526$	0	0
$527$	6.90653e6	1.08326
$528$	0	0
$529$	−6.30746e6	−0.979975
$530$	0	0
$531$	2.45085e6	0.377208
$532$	0	0
$533$	7.00209e6	1.06760
$534$	0	0
$535$	−7.57176e6	−1.14370
$536$	0	0
$537$	−2.52728e6	−0.378197
$538$	0	0
$539$	1.48744e6	0.220531
$540$	0	0
$541$	−1.29223e6	−0.189822	−0.0949108	−	0.995486i	$-0.530257\pi$
$541$	−1.29223e6	−0.189822	−0.0949108	+	0.995486i	$0.530257\pi$
$542$	0	0
$543$	1.40021e7	2.03795
$544$	0	0
$545$	2.20627e6	0.318176
$546$	0	0
$547$	1.03623e7	1.48077	0.740385	−	0.672183i	$-0.234643\pi$
$547$	1.03623e7	1.48077	0.740385	+	0.672183i	$0.234643\pi$
$548$	0	0
$549$	−3.08709e6	−0.437137
$550$	0	0
$551$	1.41507e6	0.198563
$552$	0	0
$553$	−1.77138e6	−0.246319
$554$	0	0
$555$	−3.38937e6	−0.467075
$556$	0	0
$557$	−7.49816e6	−1.02404	−0.512020	−	0.858974i	$-0.671102\pi$
$557$	−7.49816e6	−1.02404	−0.512020	+	0.858974i	$0.671102\pi$
$558$	0	0
$559$	−1.01719e7	−1.37681
$560$	0	0
$561$	−2.29621e6	−0.308039
$562$	0	0
$563$	−8.55298e6	−1.13723	−0.568613	−	0.822605i	$-0.692520\pi$
$563$	−8.55298e6	−1.13723	−0.568613	+	0.822605i	$0.692520\pi$
$564$	0	0
$565$	−7.36761e6	−0.970969
$566$	0	0
$567$	4.95597e6	0.647397
$568$	0	0
$569$	1.24374e7	1.61045	0.805227	−	0.592967i	$-0.202044\pi$
$569$	1.24374e7	1.61045	0.805227	+	0.592967i	$0.202044\pi$
$570$	0	0
$571$	−9.99711e6	−1.28317	−0.641585	−	0.767052i	$-0.721723\pi$
$571$	−9.99711e6	−1.28317	−0.641585	+	0.767052i	$0.721723\pi$
$572$	0	0
$573$	−1.01103e7	−1.28641
$574$	0	0
$575$	717989.	0.0905624
$576$	0	0
$577$	−1.59907e6	−0.199954	−0.0999768	−	0.994990i	$-0.531877\pi$
$577$	−1.59907e6	−0.199954	−0.0999768	+	0.994990i	$0.531877\pi$
$578$	0	0
$579$	1.01106e7	1.25337
$580$	0	0
$581$	1.30481e6	0.160364
$582$	0	0
$583$	−3.29818e6	−0.401887
$584$	0	0
$585$	2.17296e6	0.262520
$586$	0	0
$587$	−5.78685e6	−0.693181	−0.346591	−	0.938016i	$-0.612661\pi$
$587$	−5.78685e6	−0.693181	−0.346591	+	0.938016i	$0.612661\pi$
$588$	0	0
$589$	1.97500e6	0.234573
$590$	0	0
$591$	−9.35283e6	−1.10147
$592$	0	0
$593$	7.00622e6	0.818177	0.409088	−	0.912495i	$-0.365847\pi$
$593$	7.00622e6	0.818177	0.409088	+	0.912495i	$0.365847\pi$
$594$	0	0
$595$	2.21920e6	0.256983
$596$	0	0
$597$	−1.67150e7	−1.91942
$598$	0	0
$599$	5.64793e6	0.643165	0.321582	−	0.946882i	$-0.395785\pi$
$599$	5.64793e6	0.643165	0.321582	+	0.946882i	$0.395785\pi$
$600$	0	0
$601$	−1.25824e6	−0.142094	−0.0710472	−	0.997473i	$-0.522634\pi$
$601$	−1.25824e6	−0.142094	−0.0710472	+	0.997473i	$0.522634\pi$
$602$	0	0
$603$	6.83753e6	0.765784
$604$	0	0
$605$	−491090.	−0.0545472
$606$	0	0
$607$	1.75185e7	1.92986	0.964928	−	0.262514i	$-0.0845515\pi$
$607$	1.75185e7	1.92986	0.964928	+	0.262514i	$0.0845515\pi$
$608$	0	0
$609$	−6.50636e6	−0.710878
$610$	0	0
$611$	3.42276e6	0.370914
$612$	0	0
$613$	−6.33766e6	−0.681205	−0.340603	−	0.940207i	$-0.610631\pi$
$613$	−6.33766e6	−0.681205	−0.340603	+	0.940207i	$0.610631\pi$
$614$	0	0
$615$	8.96888e6	0.956204
$616$	0	0
$617$	−9.63352e6	−1.01876	−0.509380	−	0.860542i	$-0.670125\pi$
$617$	−9.63352e6	−1.01876	−0.509380	+	0.860542i	$0.670125\pi$
$618$	0	0
$619$	1.06496e7	1.11714	0.558569	−	0.829458i	$-0.311350\pi$
$619$	1.06496e7	1.11714	0.558569	+	0.829458i	$0.311350\pi$
$620$	0	0
$621$	793037.	0.0825210
$622$	0	0
$623$	2.10464e6	0.217249
$624$	0	0
$625$	483852.	0.0495464
$626$	0	0
$627$	−656628.	−0.0667038
$628$	0	0
$629$	5.16350e6	0.520377
$630$	0	0
$631$	4.49516e6	0.449440	0.224720	−	0.974423i	$-0.427853\pi$
$631$	4.49516e6	0.449440	0.224720	+	0.974423i	$0.427853\pi$
$632$	0	0
$633$	1.45060e7	1.43892
$634$	0	0
$635$	7.38836e6	0.727133
$636$	0	0
$637$	6.20356e6	0.605749
$638$	0	0
$639$	5.67935e6	0.550233
$640$	0	0
$641$	−4.93215e6	−0.474123	−0.237062	−	0.971495i	$-0.576184\pi$
$641$	−4.93215e6	−0.474123	−0.237062	+	0.971495i	$0.576184\pi$
$642$	0	0
$643$	122478.	0.0116823	0.00584117	−	0.999983i	$-0.498141\pi$
$643$	122478.	0.0116823	0.00584117	+	0.999983i	$0.498141\pi$
$644$	0	0
$645$	−1.30290e7	−1.23314
$646$	0	0
$647$	−7.69881e6	−0.723041	−0.361520	−	0.932364i	$-0.617742\pi$
$647$	−7.69881e6	−0.723041	−0.361520	+	0.932364i	$0.617742\pi$
$648$	0	0
$649$	−2.31008e6	−0.215286
$650$	0	0
$651$	−9.08089e6	−0.839800
$652$	0	0
$653$	−2.51171e6	−0.230509	−0.115254	−	0.993336i	$-0.536768\pi$
$653$	−2.51171e6	−0.230509	−0.115254	+	0.993336i	$0.536768\pi$
$654$	0	0
$655$	9.36742e6	0.853133
$656$	0	0
$657$	2.81325e6	0.254270
$658$	0	0
$659$	−4.10385e6	−0.368110	−0.184055	−	0.982916i	$-0.558922\pi$
$659$	−4.10385e6	−0.368110	−0.184055	+	0.982916i	$0.558922\pi$
$660$	0	0
$661$	−1.22236e7	−1.08817	−0.544085	−	0.839030i	$-0.683123\pi$
$661$	−1.22236e7	−1.08817	−0.544085	+	0.839030i	$0.683123\pi$
$662$	0	0
$663$	−9.57663e6	−0.846114
$664$	0	0
$665$	634604.	0.0556479
$666$	0	0
$667$	−1.80406e6	−0.157014
$668$	0	0
$669$	1.02781e7	0.887863
$670$	0	0
$671$	2.90977e6	0.249489
$672$	0	0
$673$	−1.27239e7	−1.08288	−0.541442	−	0.840738i	$-0.682121\pi$
$673$	−1.27239e7	−1.08288	−0.541442	+	0.840738i	$0.682121\pi$
$674$	0	0
$675$	4.41778e6	0.373203
$676$	0	0
$677$	−1.85518e7	−1.55566	−0.777830	−	0.628474i	$-0.783680\pi$
$677$	−1.85518e7	−1.55566	−0.777830	+	0.628474i	$0.783680\pi$
$678$	0	0
$679$	−9.00849e6	−0.749855
$680$	0	0
$681$	1.52739e7	1.26207
$682$	0	0
$683$	−1.62954e7	−1.33664	−0.668318	−	0.743876i	$-0.732985\pi$
$683$	−1.62954e7	−1.33664	−0.668318	+	0.743876i	$0.732985\pi$
$684$	0	0
$685$	3.36584e6	0.274074
$686$	0	0
$687$	2.96631e7	2.39786
$688$	0	0
$689$	−1.37555e7	−1.10389
$690$	0	0
$691$	1.06584e6	0.0849176	0.0424588	−	0.999098i	$-0.486481\pi$
$691$	1.06584e6	0.0849176	0.0424588	+	0.999098i	$0.486481\pi$
$692$	0	0
$693$	1.04363e6	0.0825491
$694$	0	0
$695$	−8.13249e6	−0.638648
$696$	0	0
$697$	−1.36636e7	−1.06532
$698$	0	0
$699$	−7.65896e6	−0.592893
$700$	0	0
$701$	1.97109e7	1.51499	0.757496	−	0.652839i	$-0.226422\pi$
$701$	1.97109e7	1.51499	0.757496	+	0.652839i	$0.226422\pi$
$702$	0	0
$703$	1.47656e6	0.112684
$704$	0	0
$705$	4.38416e6	0.332211
$706$	0	0
$707$	−1.07848e7	−0.811450
$708$	0	0
$709$	2.47096e7	1.84608	0.923039	−	0.384705i	$-0.125697\pi$
$709$	2.47096e7	1.84608	0.923039	+	0.384705i	$0.125697\pi$
$710$	0	0
$711$	3.38456e6	0.251090
$712$	0	0
$713$	−2.51792e6	−0.185489
$714$	0	0
$715$	−2.04815e6	−0.149829
$716$	0	0
$717$	−6.93913e6	−0.504089
$718$	0	0
$719$	2.89197e6	0.208627	0.104314	−	0.994544i	$-0.466735\pi$
$719$	2.89197e6	0.208627	0.104314	+	0.994544i	$0.466735\pi$
$720$	0	0
$721$	1.76414e6	0.126384
$722$	0	0
$723$	1.76332e7	1.25454
$724$	0	0
$725$	−1.00499e7	−0.710097
$726$	0	0
$727$	79481.6	0.00557739	0.00278869	−	0.999996i	$-0.499112\pi$
$727$	79481.6	0.00557739	0.00278869	+	0.999996i	$0.499112\pi$
$728$	0	0
$729$	874962.	0.0609776
$730$	0	0
$731$	1.98490e7	1.37387
$732$	0	0
$733$	−4.91934e6	−0.338179	−0.169090	−	0.985601i	$-0.554083\pi$
$733$	−4.91934e6	−0.338179	−0.169090	+	0.985601i	$0.554083\pi$
$734$	0	0
$735$	7.94605e6	0.542542
$736$	0	0
$737$	−6.44479e6	−0.437059
$738$	0	0
$739$	3.51508e6	0.236768	0.118384	−	0.992968i	$-0.462229\pi$
$739$	3.51508e6	0.236768	0.118384	+	0.992968i	$0.462229\pi$
$740$	0	0
$741$	−2.73854e6	−0.183221
$742$	0	0
$743$	−1.67752e7	−1.11480	−0.557399	−	0.830244i	$-0.688201\pi$
$743$	−1.67752e7	−1.11480	−0.557399	+	0.830244i	$0.688201\pi$
$744$	0	0
$745$	288856.	0.0190673
$746$	0	0
$747$	−2.49309e6	−0.163470
$748$	0	0
$749$	−1.51667e7	−0.987839
$750$	0	0
$751$	8.77489e6	0.567730	0.283865	−	0.958864i	$-0.408383\pi$
$751$	8.77489e6	0.567730	0.283865	+	0.958864i	$0.408383\pi$
$752$	0	0
$753$	7.26924e6	0.467198
$754$	0	0
$755$	1.45364e7	0.928086
$756$	0	0
$757$	2.36081e7	1.49734	0.748672	−	0.662940i	$-0.230692\pi$
$757$	2.36081e7	1.49734	0.748672	+	0.662940i	$0.230692\pi$
$758$	0	0
$759$	837133.	0.0527461
$760$	0	0
$761$	9.08185e6	0.568477	0.284238	−	0.958754i	$-0.408259\pi$
$761$	9.08185e6	0.568477	0.284238	+	0.958754i	$0.408259\pi$
$762$	0	0
$763$	4.41929e6	0.274816
$764$	0	0
$765$	−4.24021e6	−0.261960
$766$	0	0
$767$	−9.63446e6	−0.591342
$768$	0	0
$769$	1.30617e7	0.796495	0.398248	−	0.917278i	$-0.369619\pi$
$769$	1.30617e7	0.796495	0.398248	+	0.917278i	$0.369619\pi$
$770$	0	0
$771$	1.19248e7	0.722460
$772$	0	0
$773$	−2.62713e7	−1.58137	−0.790683	−	0.612226i	$-0.790274\pi$
$773$	−2.62713e7	−1.58137	−0.790683	+	0.612226i	$0.790274\pi$
$774$	0	0
$775$	−1.40266e7	−0.838877
$776$	0	0
$777$	−6.78911e6	−0.403422
$778$	0	0
$779$	−3.90724e6	−0.230689
$780$	0	0
$781$	−5.35313e6	−0.314037
$782$	0	0
$783$	−1.11004e7	−0.647044
$784$	0	0
$785$	1.12661e7	0.652526
$786$	0	0
$787$	−1.64871e6	−0.0948870	−0.0474435	−	0.998874i	$-0.515107\pi$
$787$	−1.64871e6	−0.0948870	−0.0474435	+	0.998874i	$0.515107\pi$
$788$	0	0
$789$	2.85354e7	1.63189
$790$	0	0
$791$	−1.47578e7	−0.838647
$792$	0	0
$793$	1.21355e7	0.685292
$794$	0	0
$795$	−1.76192e7	−0.988707
$796$	0	0
$797$	2.84759e7	1.58793	0.793965	−	0.607964i	$-0.208013\pi$
$797$	2.84759e7	1.58793	0.793965	+	0.607964i	$0.208013\pi$
$798$	0	0
$799$	−6.67900e6	−0.370122
$800$	0	0
$801$	−4.02133e6	−0.221456
$802$	0	0
$803$	−2.65166e6	−0.145121
$804$	0	0
$805$	−809055.	−0.0440036
$806$	0	0
$807$	−1.19608e7	−0.646513
$808$	0	0
$809$	2.52803e7	1.35804	0.679018	−	0.734121i	$-0.262406\pi$
$809$	2.52803e7	1.35804	0.679018	+	0.734121i	$0.262406\pi$
$810$	0	0
$811$	−3.23455e7	−1.72688	−0.863439	−	0.504453i	$-0.831694\pi$
$811$	−3.23455e7	−1.72688	−0.863439	+	0.504453i	$0.831694\pi$
$812$	0	0
$813$	4.05300e6	0.215056
$814$	0	0
$815$	1.26057e7	0.664771
$816$	0	0
$817$	5.67603e6	0.297502
$818$	0	0
$819$	4.35257e6	0.226744
$820$	0	0
$821$	1.65807e7	0.858511	0.429256	−	0.903183i	$-0.358776\pi$
$821$	1.65807e7	0.858511	0.429256	+	0.903183i	$0.358776\pi$
$822$	0	0
$823$	−8.09835e6	−0.416770	−0.208385	−	0.978047i	$-0.566821\pi$
$823$	−8.09835e6	−0.416770	−0.208385	+	0.978047i	$0.566821\pi$
$824$	0	0
$825$	4.66342e6	0.238545
$826$	0	0
$827$	−3.87720e6	−0.197131	−0.0985655	−	0.995131i	$-0.531425\pi$
$827$	−3.87720e6	−0.197131	−0.0985655	+	0.995131i	$0.531425\pi$
$828$	0	0
$829$	1.50786e7	0.762033	0.381016	−	0.924568i	$-0.375574\pi$
$829$	1.50786e7	0.762033	0.381016	+	0.924568i	$0.375574\pi$
$830$	0	0
$831$	2.40277e7	1.20701
$832$	0	0
$833$	−1.21053e7	−0.604456
$834$	0	0
$835$	−1.27440e7	−0.632540
$836$	0	0
$837$	−1.54928e7	−0.764390
$838$	0	0
$839$	−4.85698e6	−0.238211	−0.119105	−	0.992882i	$-0.538003\pi$
$839$	−4.85698e6	−0.238211	−0.119105	+	0.992882i	$0.538003\pi$
$840$	0	0
$841$	4.74091e6	0.231138
$842$	0	0
$843$	2.89545e7	1.40329
$844$	0	0
$845$	3.91191e6	0.188472
$846$	0	0
$847$	−983682.	−0.0471136
$848$	0	0
$849$	−1.04054e7	−0.495436
$850$	0	0
$851$	−1.88246e6	−0.0891051
$852$	0	0
$853$	4.18760e7	1.97057	0.985287	−	0.170910i	$-0.0546707\pi$
$853$	4.18760e7	1.97057	0.985287	+	0.170910i	$0.0546707\pi$
$854$	0	0
$855$	−1.21254e6	−0.0567256
$856$	0	0
$857$	3.67316e7	1.70840	0.854198	−	0.519949i	$-0.174049\pi$
$857$	3.67316e7	1.70840	0.854198	+	0.519949i	$0.174049\pi$
$858$	0	0
$859$	−1.14765e7	−0.530673	−0.265336	−	0.964156i	$-0.585483\pi$
$859$	−1.14765e7	−0.530673	−0.265336	+	0.964156i	$0.585483\pi$
$860$	0	0
$861$	1.79652e7	0.825893
$862$	0	0
$863$	−3.52102e7	−1.60932	−0.804659	−	0.593738i	$-0.797652\pi$
$863$	−3.52102e7	−1.60932	−0.804659	+	0.593738i	$0.797652\pi$
$864$	0	0
$865$	−1.36107e7	−0.618501
$866$	0	0
$867$	−8.67476e6	−0.391931
$868$	0	0
$869$	−3.19016e6	−0.143305
$870$	0	0
$871$	−2.68788e7	−1.20050
$872$	0	0
$873$	1.72125e7	0.764378
$874$	0	0
$875$	−1.15495e7	−0.509967
$876$	0	0
$877$	1.81618e7	0.797369	0.398684	−	0.917088i	$-0.369467\pi$
$877$	1.81618e7	0.797369	0.398684	+	0.917088i	$0.369467\pi$
$878$	0	0
$879$	2.76069e7	1.20516
$880$	0	0
$881$	−4.48001e7	−1.94464	−0.972321	−	0.233650i	$-0.924933\pi$
$881$	−4.48001e7	−1.94464	−0.972321	+	0.233650i	$0.924933\pi$
$882$	0	0
$883$	−3.84155e6	−0.165808	−0.0829039	−	0.996558i	$-0.526419\pi$
$883$	−3.84155e6	−0.165808	−0.0829039	+	0.996558i	$0.526419\pi$
$884$	0	0
$885$	−1.23406e7	−0.529638
$886$	0	0
$887$	1.89020e7	0.806675	0.403337	−	0.915051i	$-0.367850\pi$
$887$	1.89020e7	0.806675	0.403337	+	0.915051i	$0.367850\pi$
$888$	0	0
$889$	1.47993e7	0.628040
$890$	0	0
$891$	8.92544e6	0.376648
$892$	0	0
$893$	−1.90994e6	−0.0801475
$894$	0	0
$895$	4.39885e6	0.183561
$896$	0	0
$897$	3.49136e6	0.144882
$898$	0	0
$899$	3.52441e7	1.45441
$900$	0	0
$901$	2.68418e7	1.10154
$902$	0	0
$903$	−2.60979e7	−1.06509
$904$	0	0
$905$	−2.43713e7	−0.989139
$906$	0	0
$907$	−1.77294e7	−0.715609	−0.357805	−	0.933796i	$-0.616475\pi$
$907$	−1.77294e7	−0.715609	−0.357805	+	0.933796i	$0.616475\pi$
$908$	0	0
$909$	2.06064e7	0.827165
$910$	0	0
$911$	−2.41161e7	−0.962743	−0.481372	−	0.876517i	$-0.659861\pi$
$911$	−2.41161e7	−0.962743	−0.481372	+	0.876517i	$0.659861\pi$
$912$	0	0
$913$	2.34989e6	0.0932978
$914$	0	0
$915$	1.55442e7	0.613785
$916$	0	0
$917$	1.87635e7	0.736869
$918$	0	0
$919$	1.25701e7	0.490962	0.245481	−	0.969401i	$-0.421054\pi$
$919$	1.25701e7	0.490962	0.245481	+	0.969401i	$0.421054\pi$
$920$	0	0
$921$	−2.71846e7	−1.05603
$922$	0	0
$923$	−2.23259e7	−0.862590
$924$	0	0
$925$	−1.04867e7	−0.402979
$926$	0	0
$927$	−3.37073e6	−0.128832
$928$	0	0
$929$	4.61354e7	1.75386	0.876930	−	0.480618i	$-0.159588\pi$
$929$	4.61354e7	1.75386	0.876930	+	0.480618i	$0.159588\pi$
$930$	0	0
$931$	−3.46165e6	−0.130891
$932$	0	0
$933$	−1.68662e7	−0.634328
$934$	0	0
$935$	3.99666e6	0.149509
$936$	0	0
$937$	1.53113e7	0.569721	0.284860	−	0.958569i	$-0.408053\pi$
$937$	1.53113e7	0.569721	0.284860	+	0.958569i	$0.408053\pi$
$938$	0	0
$939$	1.92039e7	0.710762
$940$	0	0
$941$	4.22371e7	1.55496	0.777482	−	0.628905i	$-0.216497\pi$
$941$	4.22371e7	1.55496	0.777482	+	0.628905i	$0.216497\pi$
$942$	0	0
$943$	4.98134e6	0.182418
$944$	0	0
$945$	−4.97811e6	−0.181336
$946$	0	0
$947$	−2.32087e7	−0.840961	−0.420480	−	0.907302i	$-0.638138\pi$
$947$	−2.32087e7	−0.840961	−0.420480	+	0.907302i	$0.638138\pi$
$948$	0	0
$949$	−1.10591e7	−0.398615
$950$	0	0
$951$	−4.35196e7	−1.56039
$952$	0	0
$953$	−2.43736e7	−0.869335	−0.434668	−	0.900591i	$-0.643134\pi$
$953$	−2.43736e7	−0.869335	−0.434668	+	0.900591i	$0.643134\pi$
$954$	0	0
$955$	1.75975e7	0.624370
$956$	0	0
$957$	−1.17176e7	−0.413580
$958$	0	0
$959$	6.74198e6	0.236723
$960$	0	0
$961$	2.05609e7	0.718181
$962$	0	0
$963$	2.89789e7	1.00697
$964$	0	0
$965$	−1.75979e7	−0.608335
$966$	0	0
$967$	1.64611e6	0.0566099	0.0283049	−	0.999599i	$-0.490989\pi$
$967$	1.64611e6	0.0566099	0.0283049	+	0.999599i	$0.490989\pi$
$968$	0	0
$969$	5.34386e6	0.182829
$970$	0	0
$971$	−5.05232e7	−1.71966	−0.859830	−	0.510581i	$-0.829431\pi$
$971$	−5.05232e7	−1.71966	−0.859830	+	0.510581i	$0.829431\pi$
$972$	0	0
$973$	−1.62899e7	−0.551614
$974$	0	0
$975$	1.94494e7	0.655230
$976$	0	0
$977$	3.46101e7	1.16002	0.580012	−	0.814608i	$-0.303048\pi$
$977$	3.46101e7	1.16002	0.580012	+	0.814608i	$0.303048\pi$
$978$	0	0
$979$	3.79035e6	0.126393
$980$	0	0
$981$	−8.44393e6	−0.280138
$982$	0	0
$983$	−3.01142e7	−0.994003	−0.497001	−	0.867750i	$-0.665566\pi$
$983$	−3.01142e7	−0.994003	−0.497001	+	0.867750i	$0.665566\pi$
$984$	0	0
$985$	1.62790e7	0.534610
$986$	0	0
$987$	8.78173e6	0.286937
$988$	0	0
$989$	−7.23636e6	−0.235250
$990$	0	0
$991$	2.58640e7	0.836589	0.418294	−	0.908312i	$-0.362628\pi$
$991$	2.58640e7	0.836589	0.418294	+	0.908312i	$0.362628\pi$
$992$	0	0
$993$	3.63197e6	0.116888
$994$	0	0
$995$	2.90932e7	0.931608
$996$	0	0
$997$	−5.44951e7	−1.73628	−0.868139	−	0.496321i	$-0.834684\pi$
$997$	−5.44951e7	−1.73628	−0.868139	+	0.496321i	$0.834684\pi$
$998$	0	0
$999$	−1.15828e7	−0.367197

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	704.6.a.m.1.2		2
4.3	odd	2		704.6.a.n.1.1		2
8.3	odd	2		44.6.a.b.1.2	✓	2
8.5	even	2		176.6.a.g.1.1		2
24.11	even	2		396.6.a.f.1.1		2
40.3	even	4		1100.6.b.c.749.3		4
40.19	odd	2		1100.6.a.b.1.1		2
40.27	even	4		1100.6.b.c.749.2		4
88.43	even	2		484.6.a.d.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
44.6.a.b.1.2	✓	2	8.3	odd	2
176.6.a.g.1.1		2	8.5	even	2
396.6.a.f.1.1		2	24.11	even	2
484.6.a.d.1.2		2	88.43	even	2
704.6.a.m.1.2		2	1.1	even	1	trivial
704.6.a.n.1.1		2	4.3	odd	2
1100.6.a.b.1.1		2	40.19	odd	2
1100.6.b.c.749.2		4	40.27	even	4
1100.6.b.c.749.3		4	40.3	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 704 = 2^{6} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	704.a (trivial)

\(f(q)\)	\(=\)	\(q+19.2711 q^{3} -33.5421 q^{5} -67.1868 q^{7} +128.374 q^{9} -121.000 q^{11} -504.645 q^{13} -646.392 q^{15} +984.740 q^{17} +281.597 q^{19} -1294.76 q^{21} -359.008 q^{23} -1999.93 q^{25} -2208.97 q^{27} +5025.14 q^{29} +7013.56 q^{31} -2331.80 q^{33} +2253.59 q^{35} +5243.52 q^{37} -9725.04 q^{39} -13875.3 q^{41} +20156.6 q^{43} -4305.92 q^{45} -6782.51 q^{47} -12292.9 q^{49} +18977.0 q^{51} +27257.7 q^{53} +4058.60 q^{55} +5426.68 q^{57} +19091.6 q^{59} -24047.7 q^{61} -8625.02 q^{63} +16926.9 q^{65} +53262.7 q^{67} -6918.46 q^{69} +44240.8 q^{71} +21914.6 q^{73} -38540.7 q^{75} +8129.61 q^{77} +26364.9 q^{79} -73764.0 q^{81} -19420.6 q^{83} -33030.3 q^{85} +96839.8 q^{87} -31325.2 q^{89} +33905.5 q^{91} +135159. q^{93} -9445.37 q^{95} +134081. q^{97} -15533.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} + 22 q^{5} - 268 q^{7} + 524 q^{9} - 242 q^{11} - 1232 q^{13} - 2050 q^{15} - 124 q^{17} + 1944 q^{19} + 3780 q^{21} - 3346 q^{23} - 2040 q^{25} - 6066 q^{27} + 6576 q^{29} - 2498 q^{31} + 726 q^{33}+ \cdots - 63404 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 + 22 * q^5 - 268 * q^7 + 524 * q^9 - 242 * q^11 - 1232 * q^13 - 2050 * q^15 - 124 * q^17 + 1944 * q^19 + 3780 * q^21 - 3346 * q^23 - 2040 * q^25 - 6066 * q^27 + 6576 * q^29 - 2498 * q^31 + 726 * q^33 - 8900 * q^35 + 14674 * q^37 + 8656 * q^39 - 6504 * q^41 + 11628 * q^43 + 17668 * q^45 - 36816 * q^47 + 11226 * q^49 + 46996 * q^51 + 3292 * q^53 - 2662 * q^55 - 36584 * q^57 + 12126 * q^59 - 73128 * q^61 - 88072 * q^63 - 23472 * q^65 + 29334 * q^67 + 68566 * q^69 + 46122 * q^71 + 8240 * q^73 - 37528 * q^75 + 32428 * q^77 + 14780 * q^79 - 72430 * q^81 - 74564 * q^83 - 94612 * q^85 + 57648 * q^87 - 33698 * q^89 + 179968 * q^91 + 375526 * q^93 + 82888 * q^95 + 126162 * q^97 - 63404 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more