LMFDB - Embedded newform 1104.6.a.ba.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1104,6,Mod(1,1104)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1104, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1104.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1104 = 2^{4} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1104.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:	$177.063737074$
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} - 4608x^{6} - 3161x^{5} + 6284039x^{4} - 8279002x^{3} - 2677454576x^{2} + 13573192447x + 77901299860 $ x^8 - 4608x^6 - 3161x^5 + 6284039x^4 - 8279002x^3 - 2677454576x^2 + 13573192447x + 77901299860
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{18}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 552)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-31.3446$ of defining polynomial
Character	$\chi$	$=$	1104.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.00000 q^{3} -60.6892 q^{5} -174.569 q^{7} +81.0000 q^{9} +O(q^{10})$	$q-9.00000 q^{3} -60.6892 q^{5} -174.569 q^{7} +81.0000 q^{9} +481.705 q^{11} +161.616 q^{13} +546.203 q^{15} -1846.65 q^{17} -1656.53 q^{19} +1571.12 q^{21} +529.000 q^{23} +558.182 q^{25} -729.000 q^{27} +7289.65 q^{29} -10639.5 q^{31} -4335.35 q^{33} +10594.5 q^{35} -36.0322 q^{37} -1454.54 q^{39} -17644.8 q^{41} -5122.84 q^{43} -4915.83 q^{45} -7297.42 q^{47} +13667.4 q^{49} +16619.8 q^{51} -12588.2 q^{53} -29234.3 q^{55} +14908.7 q^{57} +25462.7 q^{59} -22070.4 q^{61} -14140.1 q^{63} -9808.33 q^{65} -39060.4 q^{67} -4761.00 q^{69} -64854.3 q^{71} -45246.3 q^{73} -5023.64 q^{75} -84091.0 q^{77} -64271.1 q^{79} +6561.00 q^{81} -115471. q^{83} +112072. q^{85} -65606.8 q^{87} +94339.3 q^{89} -28213.1 q^{91} +95755.3 q^{93} +100533. q^{95} -13195.4 q^{97} +39018.1 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 72 q^{3} + 16 q^{5} + 36 q^{7} + 648 q^{9}+O(q^{10}) $ 8 * q - 72 * q^3 + 16 * q^5 + 36 * q^7 + 648 * q^9	$ 8 q - 72 q^{3} + 16 q^{5} + 36 q^{7} + 648 q^{9} + 148 q^{11} + 696 q^{13} - 144 q^{15} + 572 q^{17} - 2456 q^{19} - 324 q^{21} + 4232 q^{23} + 11896 q^{25} - 5832 q^{27} + 6504 q^{29} - 3128 q^{31} - 1332 q^{33} - 6960 q^{35} + 3844 q^{37} - 6264 q^{39} + 6440 q^{41} - 7048 q^{43} + 1296 q^{45} - 30464 q^{47} + 46984 q^{49} - 5148 q^{51} + 63696 q^{53} - 32688 q^{55} + 22104 q^{57} - 54872 q^{59} + 12108 q^{61} + 2916 q^{63} + 124088 q^{65} - 139216 q^{67} - 38088 q^{69} - 76216 q^{71} + 13632 q^{73} - 107064 q^{75} - 78248 q^{77} - 126380 q^{79} + 52488 q^{81} - 238196 q^{83} + 117536 q^{85} - 58536 q^{87} + 123668 q^{89} - 248176 q^{91} + 28152 q^{93} - 183408 q^{95} + 18576 q^{97} + 11988 q^{99}+O(q^{100}) $ 8 * q - 72 * q^3 + 16 * q^5 + 36 * q^7 + 648 * q^9 + 148 * q^11 + 696 * q^13 - 144 * q^15 + 572 * q^17 - 2456 * q^19 - 324 * q^21 + 4232 * q^23 + 11896 * q^25 - 5832 * q^27 + 6504 * q^29 - 3128 * q^31 - 1332 * q^33 - 6960 * q^35 + 3844 * q^37 - 6264 * q^39 + 6440 * q^41 - 7048 * q^43 + 1296 * q^45 - 30464 * q^47 + 46984 * q^49 - 5148 * q^51 + 63696 * q^53 - 32688 * q^55 + 22104 * q^57 - 54872 * q^59 + 12108 * q^61 + 2916 * q^63 + 124088 * q^65 - 139216 * q^67 - 38088 * q^69 - 76216 * q^71 + 13632 * q^73 - 107064 * q^75 - 78248 * q^77 - 126380 * q^79 + 52488 * q^81 - 238196 * q^83 + 117536 * q^85 - 58536 * q^87 + 123668 * q^89 - 248176 * q^91 + 28152 * q^93 - 183408 * q^95 + 18576 * q^97 + 11988 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	0	0
$5$	−60.6892	−1.08564	−0.542821	−	0.839848i	$-0.682644\pi$
$5$	−60.6892	−1.08564	−0.542821	+	0.839848i	$0.682644\pi$
$6$	0	0
$7$	−174.569	−1.34655	−0.673275	−	0.739392i	$-0.735113\pi$
$7$	−174.569	−1.34655	−0.673275	+	0.739392i	$0.735113\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	481.705	1.20033	0.600164	−	0.799877i	$-0.295102\pi$
$11$	481.705	1.20033	0.600164	+	0.799877i	$0.295102\pi$
$12$	0	0
$13$	161.616	0.265231	0.132616	−	0.991168i	$-0.457662\pi$
$13$	161.616	0.265231	0.132616	+	0.991168i	$0.457662\pi$
$14$	0	0
$15$	546.203	0.626796
$16$	0	0
$17$	−1846.65	−1.54975	−0.774875	−	0.632114i	$-0.782187\pi$
$17$	−1846.65	−1.54975	−0.774875	+	0.632114i	$0.782187\pi$
$18$	0	0
$19$	−1656.53	−1.05272	−0.526362	−	0.850261i	$-0.676444\pi$
$19$	−1656.53	−1.05272	−0.526362	+	0.850261i	$0.676444\pi$
$20$	0	0
$21$	1571.12	0.777432
$22$	0	0
$23$	529.000	0.208514
$23$	529.000	0.208514
$24$	0	0
$25$	558.182	0.178618
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	7289.65	1.60958	0.804788	−	0.593563i	$-0.202279\pi$
$29$	7289.65	1.60958	0.804788	+	0.593563i	$0.202279\pi$
$30$	0	0
$31$	−10639.5	−1.98846	−0.994229	−	0.107282i	$-0.965785\pi$
$31$	−10639.5	−1.98846	−0.994229	+	0.107282i	$0.965785\pi$
$32$	0	0
$33$	−4335.35	−0.693009
$34$	0	0
$35$	10594.5	1.46187
$36$	0	0
$37$	−36.0322	−0.00432700	−0.00216350	−	0.999998i	$-0.500689\pi$
$37$	−36.0322	−0.00432700	−0.00216350	+	0.999998i	$0.500689\pi$
$38$	0	0
$39$	−1454.54	−0.153131
$40$	0	0
$41$	−17644.8	−1.63930	−0.819650	−	0.572865i	$-0.805832\pi$
$41$	−17644.8	−1.63930	−0.819650	+	0.572865i	$0.805832\pi$
$42$	0	0
$43$	−5122.84	−0.422512	−0.211256	−	0.977431i	$-0.567755\pi$
$43$	−5122.84	−0.422512	−0.211256	+	0.977431i	$0.567755\pi$
$44$	0	0
$45$	−4915.83	−0.361881
$46$	0	0
$47$	−7297.42	−0.481864	−0.240932	−	0.970542i	$-0.577453\pi$
$47$	−7297.42	−0.481864	−0.240932	+	0.970542i	$0.577453\pi$
$48$	0	0
$49$	13667.4	0.813199
$50$	0	0
$51$	16619.8	0.894749
$52$	0	0
$53$	−12588.2	−0.615564	−0.307782	−	0.951457i	$-0.599587\pi$
$53$	−12588.2	−0.615564	−0.307782	+	0.951457i	$0.599587\pi$
$54$	0	0
$55$	−29234.3	−1.30313
$56$	0	0
$57$	14908.7	0.607790
$58$	0	0
$59$	25462.7	0.952303	0.476151	−	0.879363i	$-0.342031\pi$
$59$	25462.7	0.952303	0.476151	+	0.879363i	$0.342031\pi$
$60$	0	0
$61$	−22070.4	−0.759427	−0.379714	−	0.925104i	$-0.623977\pi$
$61$	−22070.4	−0.759427	−0.379714	+	0.925104i	$0.623977\pi$
$62$	0	0
$63$	−14140.1	−0.448850
$64$	0	0
$65$	−9808.33	−0.287946
$66$	0	0
$67$	−39060.4	−1.06304	−0.531521	−	0.847045i	$-0.678379\pi$
$67$	−39060.4	−1.06304	−0.531521	+	0.847045i	$0.678379\pi$
$68$	0	0
$69$	−4761.00	−0.120386
$70$	0	0
$71$	−64854.3	−1.52684	−0.763419	−	0.645903i	$-0.776481\pi$
$71$	−64854.3	−1.52684	−0.763419	+	0.645903i	$0.776481\pi$
$72$	0	0
$73$	−45246.3	−0.993748	−0.496874	−	0.867823i	$-0.665519\pi$
$73$	−45246.3	−0.993748	−0.496874	+	0.867823i	$0.665519\pi$
$74$	0	0
$75$	−5023.64	−0.103125
$76$	0	0
$77$	−84091.0	−1.61630
$78$	0	0
$79$	−64271.1	−1.15864	−0.579320	−	0.815100i	$-0.696682\pi$
$79$	−64271.1	−1.15864	−0.579320	+	0.815100i	$0.696682\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−115471.	−1.83982	−0.919912	−	0.392126i	$-0.871740\pi$
$83$	−115471.	−1.83982	−0.919912	+	0.392126i	$0.871740\pi$
$84$	0	0
$85$	112072.	1.68247
$86$	0	0
$87$	−65606.8	−0.929289
$88$	0	0
$89$	94339.3	1.26246	0.631230	−	0.775596i	$-0.282550\pi$
$89$	94339.3	1.26246	0.631230	+	0.775596i	$0.282550\pi$
$90$	0	0
$91$	−28213.1	−0.357148
$92$	0	0
$93$	95755.3	1.14804
$94$	0	0
$95$	100533.	1.14288
$96$	0	0
$97$	−13195.4	−0.142394	−0.0711970	−	0.997462i	$-0.522682\pi$
$97$	−13195.4	−0.142394	−0.0711970	+	0.997462i	$0.522682\pi$
$98$	0	0
$99$	39018.1	0.400109
$100$	0	0
$101$	−140482.	−1.37030	−0.685152	−	0.728400i	$-0.740264\pi$
$101$	−140482.	−1.37030	−0.685152	+	0.728400i	$0.740264\pi$
$102$	0	0
$103$	−20534.2	−0.190715	−0.0953573	−	0.995443i	$-0.530399\pi$
$103$	−20534.2	−0.190715	−0.0953573	+	0.995443i	$0.530399\pi$
$104$	0	0
$105$	−95350.3	−0.844012
$106$	0	0
$107$	−208609.	−1.76146	−0.880730	−	0.473618i	$-0.842948\pi$
$107$	−208609.	−1.76146	−0.880730	+	0.473618i	$0.842948\pi$
$108$	0	0
$109$	162723.	1.31185	0.655924	−	0.754827i	$-0.272279\pi$
$109$	162723.	1.31185	0.655924	+	0.754827i	$0.272279\pi$
$110$	0	0
$111$	324.290	0.00249819
$112$	0	0
$113$	192061.	1.41496	0.707478	−	0.706736i	$-0.249833\pi$
$113$	192061.	1.41496	0.707478	+	0.706736i	$0.249833\pi$
$114$	0	0
$115$	−32104.6	−0.226372
$116$	0	0
$117$	13090.9	0.0884105
$118$	0	0
$119$	322368.	2.08682
$120$	0	0
$121$	70989.0	0.440786
$122$	0	0
$123$	158804.	0.946450
$124$	0	0
$125$	155778.	0.891726
$126$	0	0
$127$	−136779.	−0.752507	−0.376253	−	0.926517i	$-0.622788\pi$
$127$	−136779.	−0.752507	−0.376253	+	0.926517i	$0.622788\pi$
$128$	0	0
$129$	46105.5	0.243938
$130$	0	0
$131$	245626.	1.25054	0.625268	−	0.780410i	$-0.284990\pi$
$131$	245626.	1.25054	0.625268	+	0.780410i	$0.284990\pi$
$132$	0	0
$133$	289179.	1.41755
$134$	0	0
$135$	44242.4	0.208932
$136$	0	0
$137$	124080.	0.564808	0.282404	−	0.959296i	$-0.408868\pi$
$137$	124080.	0.564808	0.282404	+	0.959296i	$0.408868\pi$
$138$	0	0
$139$	−294003.	−1.29067	−0.645333	−	0.763901i	$-0.723282\pi$
$139$	−294003.	−1.29067	−0.645333	+	0.763901i	$0.723282\pi$
$140$	0	0
$141$	65676.8	0.278204
$142$	0	0
$143$	77851.1	0.318365
$144$	0	0
$145$	−442403.	−1.74742
$146$	0	0
$147$	−123007.	−0.469501
$148$	0	0
$149$	144843.	0.534479	0.267240	−	0.963630i	$-0.413888\pi$
$149$	144843.	0.534479	0.267240	+	0.963630i	$0.413888\pi$
$150$	0	0
$151$	445043.	1.58840	0.794200	−	0.607656i	$-0.207890\pi$
$151$	445043.	1.58840	0.794200	+	0.607656i	$0.207890\pi$
$152$	0	0
$153$	−149578.	−0.516584
$154$	0	0
$155$	645702.	2.15875
$156$	0	0
$157$	−221350.	−0.716690	−0.358345	−	0.933589i	$-0.616659\pi$
$157$	−221350.	−0.716690	−0.358345	+	0.933589i	$0.616659\pi$
$158$	0	0
$159$	113294.	0.355396
$160$	0	0
$161$	−92347.2	−0.280775
$162$	0	0
$163$	162717.	0.479694	0.239847	−	0.970811i	$-0.422903\pi$
$163$	162717.	0.479694	0.239847	+	0.970811i	$0.422903\pi$
$164$	0	0
$165$	263109.	0.752360
$166$	0	0
$167$	−252133.	−0.699583	−0.349792	−	0.936828i	$-0.613748\pi$
$167$	−252133.	−0.699583	−0.349792	+	0.936828i	$0.613748\pi$
$168$	0	0
$169$	−345173.	−0.929652
$170$	0	0
$171$	−134179.	−0.350908
$172$	0	0
$173$	369881.	0.939607	0.469804	−	0.882771i	$-0.344325\pi$
$173$	369881.	0.939607	0.469804	+	0.882771i	$0.344325\pi$
$174$	0	0
$175$	−97441.4	−0.240519
$176$	0	0
$177$	−229165.	−0.549812
$178$	0	0
$179$	−766256.	−1.78748	−0.893740	−	0.448585i	$-0.851928\pi$
$179$	−766256.	−1.78748	−0.893740	+	0.448585i	$0.851928\pi$
$180$	0	0
$181$	419759.	0.952365	0.476183	−	0.879346i	$-0.342020\pi$
$181$	419759.	0.952365	0.476183	+	0.879346i	$0.342020\pi$
$182$	0	0
$183$	198634.	0.438456
$184$	0	0
$185$	2186.77	0.00469757
$186$	0	0
$187$	−889540.	−1.86021
$188$	0	0
$189$	127261.	0.259144
$190$	0	0
$191$	−921353.	−1.82744	−0.913719	−	0.406347i	$-0.866802\pi$
$191$	−921353.	−1.82744	−0.913719	+	0.406347i	$0.866802\pi$
$192$	0	0
$193$	136558.	0.263891	0.131945	−	0.991257i	$-0.457878\pi$
$193$	136558.	0.263891	0.131945	+	0.991257i	$0.457878\pi$
$194$	0	0
$195$	88275.0	0.166246
$196$	0	0
$197$	693025.	1.27228	0.636140	−	0.771573i	$-0.280530\pi$
$197$	693025.	1.27228	0.636140	+	0.771573i	$0.280530\pi$
$198$	0	0
$199$	308782.	0.552738	0.276369	−	0.961052i	$-0.410869\pi$
$199$	308782.	0.552738	0.276369	+	0.961052i	$0.410869\pi$
$200$	0	0
$201$	351544.	0.613747
$202$	0	0
$203$	−1.27255e6	−2.16738
$204$	0	0
$205$	1.07085e6	1.77969
$206$	0	0
$207$	42849.0	0.0695048
$208$	0	0
$209$	−797957.	−1.26361
$210$	0	0
$211$	164387.	0.254191	0.127096	−	0.991890i	$-0.459435\pi$
$211$	164387.	0.254191	0.127096	+	0.991890i	$0.459435\pi$
$212$	0	0
$213$	583689.	0.881521
$214$	0	0
$215$	310901.	0.458697
$216$	0	0
$217$	1.85733e6	2.67756
$218$	0	0
$219$	407217.	0.573740
$220$	0	0
$221$	−298447.	−0.411043
$222$	0	0
$223$	163364.	0.219985	0.109992	−	0.993932i	$-0.464917\pi$
$223$	163364.	0.219985	0.109992	+	0.993932i	$0.464917\pi$
$224$	0	0
$225$	45212.7	0.0595394
$226$	0	0
$227$	−853879.	−1.09985	−0.549923	−	0.835215i	$-0.685343\pi$
$227$	−853879.	−1.09985	−0.549923	+	0.835215i	$0.685343\pi$
$228$	0	0
$229$	220633.	0.278024	0.139012	−	0.990291i	$-0.455607\pi$
$229$	220633.	0.278024	0.139012	+	0.990291i	$0.455607\pi$
$230$	0	0
$231$	756819.	0.933172
$232$	0	0
$233$	17761.9	0.0214338	0.0107169	−	0.999943i	$-0.496589\pi$
$233$	17761.9	0.0214338	0.0107169	+	0.999943i	$0.496589\pi$
$234$	0	0
$235$	442875.	0.523132
$236$	0	0
$237$	578440.	0.668941
$238$	0	0
$239$	1.35996e6	1.54004	0.770021	−	0.638018i	$-0.220246\pi$
$239$	1.35996e6	1.54004	0.770021	+	0.638018i	$0.220246\pi$
$240$	0	0
$241$	−946071.	−1.04925	−0.524627	−	0.851332i	$-0.675795\pi$
$241$	−946071.	−1.04925	−0.524627	+	0.851332i	$0.675795\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	−829466.	−0.882843
$246$	0	0
$247$	−267721.	−0.279215
$248$	0	0
$249$	1.03924e6	1.06222
$250$	0	0
$251$	−570536.	−0.571609	−0.285804	−	0.958288i	$-0.592261\pi$
$251$	−570536.	−0.571609	−0.285804	+	0.958288i	$0.592261\pi$
$252$	0	0
$253$	254822.	0.250286
$254$	0	0
$255$	−1.00864e6	−0.971377
$256$	0	0
$257$	854156.	0.806686	0.403343	−	0.915049i	$-0.367848\pi$
$257$	854156.	0.806686	0.403343	+	0.915049i	$0.367848\pi$
$258$	0	0
$259$	6290.12	0.00582652
$260$	0	0
$261$	590461.	0.536525
$262$	0	0
$263$	447169.	0.398641	0.199321	−	0.979934i	$-0.436126\pi$
$263$	447169.	0.398641	0.199321	+	0.979934i	$0.436126\pi$
$264$	0	0
$265$	763967.	0.668282
$266$	0	0
$267$	−849053.	−0.728881
$268$	0	0
$269$	354739.	0.298902	0.149451	−	0.988769i	$-0.452249\pi$
$269$	354739.	0.298902	0.149451	+	0.988769i	$0.452249\pi$
$270$	0	0
$271$	1.47922e6	1.22352	0.611760	−	0.791044i	$-0.290462\pi$
$271$	1.47922e6	1.22352	0.611760	+	0.791044i	$0.290462\pi$
$272$	0	0
$273$	253918.	0.206199
$274$	0	0
$275$	268879.	0.214400
$276$	0	0
$277$	−2.05668e6	−1.61053	−0.805263	−	0.592917i	$-0.797976\pi$
$277$	−2.05668e6	−1.61053	−0.805263	+	0.592917i	$0.797976\pi$
$278$	0	0
$279$	−861798.	−0.662819
$280$	0	0
$281$	−264152.	−0.199567	−0.0997834	−	0.995009i	$-0.531815\pi$
$281$	−264152.	−0.199567	−0.0997834	+	0.995009i	$0.531815\pi$
$282$	0	0
$283$	288785.	0.214343	0.107171	−	0.994241i	$-0.465821\pi$
$283$	288785.	0.214343	0.107171	+	0.994241i	$0.465821\pi$
$284$	0	0
$285$	−904800.	−0.659842
$286$	0	0
$287$	3.08025e6	2.20740
$288$	0	0
$289$	1.99025e6	1.40173
$290$	0	0
$291$	118758.	0.0822112
$292$	0	0
$293$	−2.15332e6	−1.46535	−0.732673	−	0.680581i	$-0.761728\pi$
$293$	−2.15332e6	−1.46535	−0.732673	+	0.680581i	$0.761728\pi$
$294$	0	0
$295$	−1.54531e6	−1.03386
$296$	0	0
$297$	−351163.	−0.231003
$298$	0	0
$299$	85494.7	0.0553046
$300$	0	0
$301$	894290.	0.568935
$302$	0	0
$303$	1.26434e6	0.791146
$304$	0	0
$305$	1.33944e6	0.824466
$306$	0	0
$307$	2.51491e6	1.52292	0.761459	−	0.648213i	$-0.224483\pi$
$307$	2.51491e6	1.52292	0.761459	+	0.648213i	$0.224483\pi$
$308$	0	0
$309$	184808.	0.110109
$310$	0	0
$311$	−2.37167e6	−1.39044	−0.695220	−	0.718797i	$-0.744693\pi$
$311$	−2.37167e6	−1.39044	−0.695220	+	0.718797i	$0.744693\pi$
$312$	0	0
$313$	1.76518e6	1.01842	0.509211	−	0.860642i	$-0.329937\pi$
$313$	1.76518e6	1.01842	0.509211	+	0.860642i	$0.329937\pi$
$314$	0	0
$315$	858153.	0.487291
$316$	0	0
$317$	−52032.6	−0.0290822	−0.0145411	−	0.999894i	$-0.504629\pi$
$317$	−52032.6	−0.0290822	−0.0145411	+	0.999894i	$0.504629\pi$
$318$	0	0
$319$	3.51146e6	1.93202
$320$	0	0
$321$	1.87748e6	1.01698
$322$	0	0
$323$	3.05902e6	1.63146
$324$	0	0
$325$	90210.9	0.0473752
$326$	0	0
$327$	−1.46451e6	−0.757395
$328$	0	0
$329$	1.27391e6	0.648855
$330$	0	0
$331$	508028.	0.254869	0.127435	−	0.991847i	$-0.459326\pi$
$331$	508028.	0.254869	0.127435	+	0.991847i	$0.459326\pi$
$332$	0	0
$333$	−2918.61	−0.00144233
$334$	0	0
$335$	2.37055e6	1.15408
$336$	0	0
$337$	−1.80066e6	−0.863688	−0.431844	−	0.901948i	$-0.642137\pi$
$337$	−1.80066e6	−0.863688	−0.431844	+	0.901948i	$0.642137\pi$
$338$	0	0
$339$	−1.72855e6	−0.816925
$340$	0	0
$341$	−5.12510e6	−2.38680
$342$	0	0
$343$	548071.	0.251537
$344$	0	0
$345$	288941.	0.130696
$346$	0	0
$347$	788587.	0.351581	0.175791	−	0.984428i	$-0.443752\pi$
$347$	788587.	0.351581	0.175791	+	0.984428i	$0.443752\pi$
$348$	0	0
$349$	2.25294e6	0.990117	0.495058	−	0.868860i	$-0.335147\pi$
$349$	2.25294e6	0.990117	0.495058	+	0.868860i	$0.335147\pi$
$350$	0	0
$351$	−117818.	−0.0510438
$352$	0	0
$353$	−4.45702e6	−1.90374	−0.951871	−	0.306499i	$-0.900842\pi$
$353$	−4.45702e6	−1.90374	−0.951871	+	0.306499i	$0.900842\pi$
$354$	0	0
$355$	3.93596e6	1.65760
$356$	0	0
$357$	−2.90131e6	−1.20482
$358$	0	0
$359$	2.23009e6	0.913242	0.456621	−	0.889661i	$-0.349059\pi$
$359$	2.23009e6	0.913242	0.456621	+	0.889661i	$0.349059\pi$
$360$	0	0
$361$	267979.	0.108226
$362$	0	0
$363$	−638901.	−0.254488
$364$	0	0
$365$	2.74596e6	1.07885
$366$	0	0
$367$	946480.	0.366814	0.183407	−	0.983037i	$-0.441287\pi$
$367$	946480.	0.366814	0.183407	+	0.983037i	$0.441287\pi$
$368$	0	0
$369$	−1.42923e6	−0.546433
$370$	0	0
$371$	2.19751e6	0.828888
$372$	0	0
$373$	−170002.	−0.0632678	−0.0316339	−	0.999500i	$-0.510071\pi$
$373$	−170002.	−0.0632678	−0.0316339	+	0.999500i	$0.510071\pi$
$374$	0	0
$375$	−1.40200e6	−0.514838
$376$	0	0
$377$	1.17812e6	0.426910
$378$	0	0
$379$	−3.84137e6	−1.37369	−0.686843	−	0.726805i	$-0.741004\pi$
$379$	−3.84137e6	−1.37369	−0.686843	+	0.726805i	$0.741004\pi$
$380$	0	0
$381$	1.23101e6	0.434460
$382$	0	0
$383$	3.91015e6	1.36206	0.681031	−	0.732255i	$-0.261532\pi$
$383$	3.91015e6	1.36206	0.681031	+	0.732255i	$0.261532\pi$
$384$	0	0
$385$	5.10342e6	1.75473
$386$	0	0
$387$	−414950.	−0.140837
$388$	0	0
$389$	5.43733e6	1.82185	0.910923	−	0.412576i	$-0.135371\pi$
$389$	5.43733e6	1.82185	0.910923	+	0.412576i	$0.135371\pi$
$390$	0	0
$391$	−976877.	−0.323145
$392$	0	0
$393$	−2.21063e6	−0.721997
$394$	0	0
$395$	3.90057e6	1.25787
$396$	0	0
$397$	1.16849e6	0.372089	0.186045	−	0.982541i	$-0.440433\pi$
$397$	1.16849e6	0.372089	0.186045	+	0.982541i	$0.440433\pi$
$398$	0	0
$399$	−2.60261e6	−0.818420
$400$	0	0
$401$	−2.31140e6	−0.717818	−0.358909	−	0.933372i	$-0.616851\pi$
$401$	−2.31140e6	−0.717818	−0.358909	+	0.933372i	$0.616851\pi$
$402$	0	0
$403$	−1.71951e6	−0.527401
$404$	0	0
$405$	−398182.	−0.120627
$406$	0	0
$407$	−17356.9	−0.00519381
$408$	0	0
$409$	6.38521e6	1.88741	0.943707	−	0.330784i	$-0.107313\pi$
$409$	6.38521e6	1.88741	0.943707	+	0.330784i	$0.107313\pi$
$410$	0	0
$411$	−1.11672e6	−0.326092
$412$	0	0
$413$	−4.44501e6	−1.28232
$414$	0	0
$415$	7.00782e6	1.99739
$416$	0	0
$417$	2.64602e6	0.745167
$418$	0	0
$419$	−3.33189e6	−0.927162	−0.463581	−	0.886055i	$-0.653436\pi$
$419$	−3.33189e6	−0.927162	−0.463581	+	0.886055i	$0.653436\pi$
$420$	0	0
$421$	−68350.6	−0.0187948	−0.00939739	−	0.999956i	$-0.502991\pi$
$421$	−68350.6	−0.0187948	−0.00939739	+	0.999956i	$0.502991\pi$
$422$	0	0
$423$	−591091.	−0.160621
$424$	0	0
$425$	−1.03077e6	−0.276814
$426$	0	0
$427$	3.85282e6	1.02261
$428$	0	0
$429$	−700660.	−0.183808
$430$	0	0
$431$	4.08680e6	1.05972	0.529859	−	0.848086i	$-0.322245\pi$
$431$	4.08680e6	1.05972	0.529859	+	0.848086i	$0.322245\pi$
$432$	0	0
$433$	−128511.	−0.0329397	−0.0164699	−	0.999864i	$-0.505243\pi$
$433$	−128511.	−0.0329397	−0.0164699	+	0.999864i	$0.505243\pi$
$434$	0	0
$435$	3.98163e6	1.00888
$436$	0	0
$437$	−876302.	−0.219508
$438$	0	0
$439$	−3.96252e6	−0.981320	−0.490660	−	0.871351i	$-0.663244\pi$
$439$	−3.96252e6	−0.981320	−0.490660	+	0.871351i	$0.663244\pi$
$440$	0	0
$441$	1.10706e6	0.271066
$442$	0	0
$443$	2.17034e6	0.525434	0.262717	−	0.964873i	$-0.415382\pi$
$443$	2.17034e6	0.525434	0.262717	+	0.964873i	$0.415382\pi$
$444$	0	0
$445$	−5.72538e6	−1.37058
$446$	0	0
$447$	−1.30358e6	−0.308582
$448$	0	0
$449$	−1.60700e6	−0.376184	−0.188092	−	0.982151i	$-0.560230\pi$
$449$	−1.60700e6	−0.376184	−0.188092	+	0.982151i	$0.560230\pi$
$450$	0	0
$451$	−8.49962e6	−1.96770
$452$	0	0
$453$	−4.00539e6	−0.917063
$454$	0	0
$455$	1.71223e6	0.387734
$456$	0	0
$457$	−3.02643e6	−0.677860	−0.338930	−	0.940812i	$-0.610065\pi$
$457$	−3.02643e6	−0.677860	−0.338930	+	0.940812i	$0.610065\pi$
$458$	0	0
$459$	1.34621e6	0.298250
$460$	0	0
$461$	−3.05908e6	−0.670408	−0.335204	−	0.942146i	$-0.608805\pi$
$461$	−3.05908e6	−0.670408	−0.335204	+	0.942146i	$0.608805\pi$
$462$	0	0
$463$	−6.65702e6	−1.44320	−0.721601	−	0.692309i	$-0.756594\pi$
$463$	−6.65702e6	−1.44320	−0.721601	+	0.692309i	$0.756594\pi$
$464$	0	0
$465$	−5.81132e6	−1.24636
$466$	0	0
$467$	−1.61730e6	−0.343162	−0.171581	−	0.985170i	$-0.554887\pi$
$467$	−1.61730e6	−0.343162	−0.171581	+	0.985170i	$0.554887\pi$
$468$	0	0
$469$	6.81875e6	1.43144
$470$	0	0
$471$	1.99215e6	0.413781
$472$	0	0
$473$	−2.46770e6	−0.507153
$474$	0	0
$475$	−924643.	−0.188036
$476$	0	0
$477$	−1.01964e6	−0.205188
$478$	0	0
$479$	6.81345e6	1.35684	0.678420	−	0.734675i	$-0.262665\pi$
$479$	6.81345e6	1.35684	0.678420	+	0.734675i	$0.262665\pi$
$480$	0	0
$481$	−5823.37	−0.00114766
$482$	0	0
$483$	831124.	0.162106
$484$	0	0
$485$	800816.	0.154589
$486$	0	0
$487$	2.87219e6	0.548771	0.274386	−	0.961620i	$-0.411526\pi$
$487$	2.87219e6	0.548771	0.274386	+	0.961620i	$0.411526\pi$
$488$	0	0
$489$	−1.46445e6	−0.276951
$490$	0	0
$491$	5.91552e6	1.10736	0.553681	−	0.832729i	$-0.313223\pi$
$491$	5.91552e6	1.10736	0.553681	+	0.832729i	$0.313223\pi$
$492$	0	0
$493$	−1.34614e7	−2.49444
$494$	0	0
$495$	−2.36798e6	−0.434375
$496$	0	0
$497$	1.13216e7	2.05597
$498$	0	0
$499$	968049.	0.174039	0.0870193	−	0.996207i	$-0.472266\pi$
$499$	968049.	0.174039	0.0870193	+	0.996207i	$0.472266\pi$
$500$	0	0
$501$	2.26920e6	0.403905
$502$	0	0
$503$	−7.74071e6	−1.36415	−0.682073	−	0.731284i	$-0.738921\pi$
$503$	−7.74071e6	−1.36415	−0.682073	+	0.731284i	$0.738921\pi$
$504$	0	0
$505$	8.52574e6	1.48766
$506$	0	0
$507$	3.10656e6	0.536735
$508$	0	0
$509$	154667.	0.0264608	0.0132304	−	0.999912i	$-0.495789\pi$
$509$	154667.	0.0264608	0.0132304	+	0.999912i	$0.495789\pi$
$510$	0	0
$511$	7.89862e6	1.33813
$512$	0	0
$513$	1.20761e6	0.202597
$514$	0	0
$515$	1.24620e6	0.207048
$516$	0	0
$517$	−3.51521e6	−0.578395
$518$	0	0
$519$	−3.32893e6	−0.542482
$520$	0	0
$521$	77107.6	0.0124452	0.00622261	−	0.999981i	$-0.498019\pi$
$521$	77107.6	0.0124452	0.00622261	+	0.999981i	$0.498019\pi$
$522$	0	0
$523$	−7.72482e6	−1.23491	−0.617453	−	0.786608i	$-0.711836\pi$
$523$	−7.72482e6	−1.23491	−0.617453	+	0.786608i	$0.711836\pi$
$524$	0	0
$525$	876973.	0.138863
$526$	0	0
$527$	1.96474e7	3.08161
$528$	0	0
$529$	279841.	0.0434783
$530$	0	0
$531$	2.06248e6	0.317434
$532$	0	0
$533$	−2.85168e6	−0.434794
$534$	0	0
$535$	1.26603e7	1.91231
$536$	0	0
$537$	6.89630e6	1.03200
$538$	0	0
$539$	6.58368e6	0.976105
$540$	0	0
$541$	7.00760e6	1.02938	0.514691	−	0.857376i	$-0.327907\pi$
$541$	7.00760e6	1.02938	0.514691	+	0.857376i	$0.327907\pi$
$542$	0	0
$543$	−3.77783e6	−0.549848
$544$	0	0
$545$	−9.87555e6	−1.42420
$546$	0	0
$547$	83596.3	0.0119459	0.00597295	−	0.999982i	$-0.498099\pi$
$547$	83596.3	0.0119459	0.00597295	+	0.999982i	$0.498099\pi$
$548$	0	0
$549$	−1.78771e6	−0.253142
$550$	0	0
$551$	−1.20755e7	−1.69444
$552$	0	0
$553$	1.12198e7	1.56017
$554$	0	0
$555$	−19680.9	−0.00271214
$556$	0	0
$557$	−7.18713e6	−0.981561	−0.490781	−	0.871283i	$-0.663288\pi$
$557$	−7.18713e6	−0.981561	−0.490781	+	0.871283i	$0.663288\pi$
$558$	0	0
$559$	−827931.	−0.112064
$560$	0	0
$561$	8.00586e6	1.07399
$562$	0	0
$563$	−5.69572e6	−0.757317	−0.378659	−	0.925536i	$-0.623615\pi$
$563$	−5.69572e6	−0.757317	−0.378659	+	0.925536i	$0.623615\pi$
$564$	0	0
$565$	−1.16560e7	−1.53613
$566$	0	0
$567$	−1.14535e6	−0.149617
$568$	0	0
$569$	−6.92076e6	−0.896135	−0.448067	−	0.894000i	$-0.647888\pi$
$569$	−6.92076e6	−0.896135	−0.448067	+	0.894000i	$0.647888\pi$
$570$	0	0
$571$	7.60133e6	0.975662	0.487831	−	0.872938i	$-0.337788\pi$
$571$	7.60133e6	0.975662	0.487831	+	0.872938i	$0.337788\pi$
$572$	0	0
$573$	8.29218e6	1.05507
$574$	0	0
$575$	295278.	0.0372445
$576$	0	0
$577$	−4.30535e6	−0.538356	−0.269178	−	0.963091i	$-0.586752\pi$
$577$	−4.30535e6	−0.538356	−0.269178	+	0.963091i	$0.586752\pi$
$578$	0	0
$579$	−1.22902e6	−0.152357
$580$	0	0
$581$	2.01576e7	2.47742
$582$	0	0
$583$	−6.06379e6	−0.738878
$584$	0	0
$585$	−794475.	−0.0959821
$586$	0	0
$587$	−1.26135e7	−1.51092	−0.755458	−	0.655197i	$-0.772586\pi$
$587$	−1.26135e7	−1.51092	−0.755458	+	0.655197i	$0.772586\pi$
$588$	0	0
$589$	1.76246e7	2.09330
$590$	0	0
$591$	−6.23722e6	−0.734552
$592$	0	0
$593$	−1.52475e6	−0.178058	−0.0890289	−	0.996029i	$-0.528376\pi$
$593$	−1.52475e6	−0.178058	−0.0890289	+	0.996029i	$0.528376\pi$
$594$	0	0
$595$	−1.95643e7	−2.26554
$596$	0	0
$597$	−2.77904e6	−0.319123
$598$	0	0
$599$	−3.07078e6	−0.349689	−0.174844	−	0.984596i	$-0.555942\pi$
$599$	−3.07078e6	−0.349689	−0.174844	+	0.984596i	$0.555942\pi$
$600$	0	0
$601$	−1.67106e7	−1.88714	−0.943571	−	0.331169i	$-0.892557\pi$
$601$	−1.67106e7	−1.88714	−0.943571	+	0.331169i	$0.892557\pi$
$602$	0	0
$603$	−3.16390e6	−0.354347
$604$	0	0
$605$	−4.30827e6	−0.478536
$606$	0	0
$607$	1.51462e7	1.66853	0.834264	−	0.551366i	$-0.185893\pi$
$607$	1.51462e7	1.66853	0.834264	+	0.551366i	$0.185893\pi$
$608$	0	0
$609$	1.14529e7	1.25133
$610$	0	0
$611$	−1.17938e6	−0.127806
$612$	0	0
$613$	−348260.	−0.0374328	−0.0187164	−	0.999825i	$-0.505958\pi$
$613$	−348260.	−0.0374328	−0.0187164	+	0.999825i	$0.505958\pi$
$614$	0	0
$615$	−9.63767e6	−1.02751
$616$	0	0
$617$	−7.67910e6	−0.812077	−0.406038	−	0.913856i	$-0.633090\pi$
$617$	−7.67910e6	−0.812077	−0.406038	+	0.913856i	$0.633090\pi$
$618$	0	0
$619$	−4.87853e6	−0.511755	−0.255878	−	0.966709i	$-0.582364\pi$
$619$	−4.87853e6	−0.511755	−0.255878	+	0.966709i	$0.582364\pi$
$620$	0	0
$621$	−385641.	−0.0401286
$622$	0	0
$623$	−1.64687e7	−1.69997
$624$	0	0
$625$	−1.11984e7	−1.14671
$626$	0	0
$627$	7.18162e6	0.729547
$628$	0	0
$629$	66538.8	0.00670577
$630$	0	0
$631$	7.45376e6	0.745250	0.372625	−	0.927982i	$-0.378458\pi$
$631$	7.45376e6	0.745250	0.372625	+	0.927982i	$0.378458\pi$
$632$	0	0
$633$	−1.47948e6	−0.146757
$634$	0	0
$635$	8.30102e6	0.816953
$636$	0	0
$637$	2.20887e6	0.215686
$638$	0	0
$639$	−5.25320e6	−0.508946
$640$	0	0
$641$	−1.65227e6	−0.158831	−0.0794157	−	0.996842i	$-0.525305\pi$
$641$	−1.65227e6	−0.158831	−0.0794157	+	0.996842i	$0.525305\pi$
$642$	0	0
$643$	−1.43037e7	−1.36433	−0.682167	−	0.731196i	$-0.738962\pi$
$643$	−1.43037e7	−1.36433	−0.682167	+	0.731196i	$0.738962\pi$
$644$	0	0
$645$	−2.79811e6	−0.264829
$646$	0	0
$647$	4.51167e6	0.423718	0.211859	−	0.977300i	$-0.432048\pi$
$647$	4.51167e6	0.423718	0.211859	+	0.977300i	$0.432048\pi$
$648$	0	0
$649$	1.22655e7	1.14307
$650$	0	0
$651$	−1.67159e7	−1.54589
$652$	0	0
$653$	−1.03974e7	−0.954207	−0.477103	−	0.878847i	$-0.658313\pi$
$653$	−1.03974e7	−0.954207	−0.477103	+	0.878847i	$0.658313\pi$
$654$	0	0
$655$	−1.49069e7	−1.35763
$656$	0	0
$657$	−3.66495e6	−0.331249
$658$	0	0
$659$	2.79900e6	0.251067	0.125534	−	0.992089i	$-0.459936\pi$
$659$	2.79900e6	0.251067	0.125534	+	0.992089i	$0.459936\pi$
$660$	0	0
$661$	1.15087e7	1.02452	0.512262	−	0.858829i	$-0.328808\pi$
$661$	1.15087e7	1.02452	0.512262	+	0.858829i	$0.328808\pi$
$662$	0	0
$663$	2.68602e6	0.237316
$664$	0	0
$665$	−1.75500e7	−1.53895
$666$	0	0
$667$	3.85622e6	0.335620
$668$	0	0
$669$	−1.47027e6	−0.127008
$670$	0	0
$671$	−1.06314e7	−0.911562
$672$	0	0
$673$	1.60729e7	1.36791	0.683955	−	0.729525i	$-0.260259\pi$
$673$	1.60729e7	1.36791	0.683955	+	0.729525i	$0.260259\pi$
$674$	0	0
$675$	−406915.	−0.0343751
$676$	0	0
$677$	194320.	0.0162947	0.00814735	−	0.999967i	$-0.497407\pi$
$677$	194320.	0.0162947	0.00814735	+	0.999967i	$0.497407\pi$
$678$	0	0
$679$	2.30350e6	0.191741
$680$	0	0
$681$	7.68491e6	0.634996
$682$	0	0
$683$	6.80001e6	0.557773	0.278887	−	0.960324i	$-0.410035\pi$
$683$	6.80001e6	0.557773	0.278887	+	0.960324i	$0.410035\pi$
$684$	0	0
$685$	−7.53033e6	−0.613180
$686$	0	0
$687$	−1.98570e6	−0.160517
$688$	0	0
$689$	−2.03445e6	−0.163267
$690$	0	0
$691$	1.80151e7	1.43530	0.717648	−	0.696406i	$-0.245218\pi$
$691$	1.80151e7	1.43530	0.717648	+	0.696406i	$0.245218\pi$
$692$	0	0
$693$	−6.81137e6	−0.538767
$694$	0	0
$695$	1.78428e7	1.40120
$696$	0	0
$697$	3.25838e7	2.54050
$698$	0	0
$699$	−159857.	−0.0123748
$700$	0	0
$701$	−1.61747e7	−1.24320	−0.621598	−	0.783336i	$-0.713516\pi$
$701$	−1.61747e7	−1.24320	−0.621598	+	0.783336i	$0.713516\pi$
$702$	0	0
$703$	59688.3	0.00455513
$704$	0	0
$705$	−3.98587e6	−0.302030
$706$	0	0
$707$	2.45238e7	1.84518
$708$	0	0
$709$	1.12963e7	0.843961	0.421980	−	0.906605i	$-0.361335\pi$
$709$	1.12963e7	0.843961	0.421980	+	0.906605i	$0.361335\pi$
$710$	0	0
$711$	−5.20596e6	−0.386213
$712$	0	0
$713$	−5.62829e6	−0.414622
$714$	0	0
$715$	−4.72472e6	−0.345630
$716$	0	0
$717$	−1.22397e7	−0.889144
$718$	0	0
$719$	−1.34650e7	−0.971371	−0.485686	−	0.874134i	$-0.661430\pi$
$719$	−1.34650e7	−0.971371	−0.485686	+	0.874134i	$0.661430\pi$
$720$	0	0
$721$	3.58464e6	0.256807
$722$	0	0
$723$	8.51464e6	0.605788
$724$	0	0
$725$	4.06895e6	0.287500
$726$	0	0
$727$	−1.46483e7	−1.02790	−0.513951	−	0.857820i	$-0.671819\pi$
$727$	−1.46483e7	−1.02790	−0.513951	+	0.857820i	$0.671819\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	9.46008e6	0.654789
$732$	0	0
$733$	−2.05894e7	−1.41541	−0.707706	−	0.706507i	$-0.750270\pi$
$733$	−2.05894e7	−1.41541	−0.707706	+	0.706507i	$0.750270\pi$
$734$	0	0
$735$	7.46520e6	0.509710
$736$	0	0
$737$	−1.88156e7	−1.27600
$738$	0	0
$739$	−5.80976e6	−0.391333	−0.195667	−	0.980670i	$-0.562687\pi$
$739$	−5.80976e6	−0.391333	−0.195667	+	0.980670i	$0.562687\pi$
$740$	0	0
$741$	2.40948e6	0.161205
$742$	0	0
$743$	−1.81349e7	−1.20515	−0.602577	−	0.798061i	$-0.705859\pi$
$743$	−1.81349e7	−1.20515	−0.602577	+	0.798061i	$0.705859\pi$
$744$	0	0
$745$	−8.79039e6	−0.580253
$746$	0	0
$747$	−9.35312e6	−0.613274
$748$	0	0
$749$	3.64167e7	2.37190
$750$	0	0
$751$	1.65497e7	1.07076	0.535378	−	0.844613i	$-0.320169\pi$
$751$	1.65497e7	1.07076	0.535378	+	0.844613i	$0.320169\pi$
$752$	0	0
$753$	5.13483e6	0.330018
$754$	0	0
$755$	−2.70093e7	−1.72443
$756$	0	0
$757$	1.41218e7	0.895677	0.447839	−	0.894114i	$-0.352194\pi$
$757$	1.41218e7	0.895677	0.447839	+	0.894114i	$0.352194\pi$
$758$	0	0
$759$	−2.29340e6	−0.144502
$760$	0	0
$761$	1.53991e7	0.963904	0.481952	−	0.876198i	$-0.339928\pi$
$761$	1.53991e7	0.963904	0.481952	+	0.876198i	$0.339928\pi$
$762$	0	0
$763$	−2.84065e7	−1.76647
$764$	0	0
$765$	9.07780e6	0.560825
$766$	0	0
$767$	4.11518e6	0.252581
$768$	0	0
$769$	−2.08839e7	−1.27349	−0.636745	−	0.771075i	$-0.719720\pi$
$769$	−2.08839e7	−1.27349	−0.636745	+	0.771075i	$0.719720\pi$
$770$	0	0
$771$	−7.68740e6	−0.465740
$772$	0	0
$773$	1.91859e7	1.15487	0.577436	−	0.816436i	$-0.304053\pi$
$773$	1.91859e7	1.15487	0.577436	+	0.816436i	$0.304053\pi$
$774$	0	0
$775$	−5.93877e6	−0.355175
$776$	0	0
$777$	−56611.1	−0.00336394
$778$	0	0
$779$	2.92291e7	1.72573
$780$	0	0
$781$	−3.12407e7	−1.83271
$782$	0	0
$783$	−5.31415e6	−0.309763
$784$	0	0
$785$	1.34336e7	0.778068
$786$	0	0
$787$	−1.52457e7	−0.877427	−0.438713	−	0.898627i	$-0.644566\pi$
$787$	−1.52457e7	−0.877427	−0.438713	+	0.898627i	$0.644566\pi$
$788$	0	0
$789$	−4.02452e6	−0.230156
$790$	0	0
$791$	−3.35279e7	−1.90531
$792$	0	0
$793$	−3.56693e6	−0.201424
$794$	0	0
$795$	−6.87570e6	−0.385833
$796$	0	0
$797$	−3.03991e7	−1.69518	−0.847588	−	0.530654i	$-0.821946\pi$
$797$	−3.03991e7	−1.69518	−0.847588	+	0.530654i	$0.821946\pi$
$798$	0	0
$799$	1.34758e7	0.746769
$800$	0	0
$801$	7.64148e6	0.420820
$802$	0	0
$803$	−2.17954e7	−1.19282
$804$	0	0
$805$	5.60448e6	0.304821
$806$	0	0
$807$	−3.19266e6	−0.172571
$808$	0	0
$809$	−417317.	−0.0224179	−0.0112089	−	0.999937i	$-0.503568\pi$
$809$	−417317.	−0.0224179	−0.0112089	+	0.999937i	$0.503568\pi$
$810$	0	0
$811$	−1.46642e7	−0.782901	−0.391451	−	0.920199i	$-0.628027\pi$
$811$	−1.46642e7	−0.782901	−0.391451	+	0.920199i	$0.628027\pi$
$812$	0	0
$813$	−1.33130e7	−0.706399
$814$	0	0
$815$	−9.87517e6	−0.520776
$816$	0	0
$817$	8.48612e6	0.444789
$818$	0	0
$819$	−2.28526e6	−0.119049
$820$	0	0
$821$	8.11523e6	0.420187	0.210094	−	0.977681i	$-0.432623\pi$
$821$	8.11523e6	0.420187	0.210094	+	0.977681i	$0.432623\pi$
$822$	0	0
$823$	2.07025e7	1.06543	0.532713	−	0.846296i	$-0.321173\pi$
$823$	2.07025e7	1.06543	0.532713	+	0.846296i	$0.321173\pi$
$824$	0	0
$825$	−2.41991e6	−0.123784
$826$	0	0
$827$	−6.99590e6	−0.355697	−0.177848	−	0.984058i	$-0.556914\pi$
$827$	−6.99590e6	−0.355697	−0.177848	+	0.984058i	$0.556914\pi$
$828$	0	0
$829$	3.02334e7	1.52792	0.763960	−	0.645263i	$-0.223252\pi$
$829$	3.02334e7	1.52792	0.763960	+	0.645263i	$0.223252\pi$
$830$	0	0
$831$	1.85101e7	0.929838
$832$	0	0
$833$	−2.52389e7	−1.26026
$834$	0	0
$835$	1.53018e7	0.759497
$836$	0	0
$837$	7.75618e6	0.382679
$838$	0	0
$839$	−2.00359e6	−0.0982664	−0.0491332	−	0.998792i	$-0.515646\pi$
$839$	−2.00359e6	−0.0982664	−0.0491332	+	0.998792i	$0.515646\pi$
$840$	0	0
$841$	3.26278e7	1.59073
$842$	0	0
$843$	2.37737e6	0.115220
$844$	0	0
$845$	2.09483e7	1.00927
$846$	0	0
$847$	−1.23925e7	−0.593541
$848$	0	0
$849$	−2.59906e6	−0.123751
$850$	0	0
$851$	−19061.0	−0.000902241 0
$852$	0	0
$853$	1.25692e7	0.591474	0.295737	−	0.955269i	$-0.404435\pi$
$853$	1.25692e7	0.591474	0.295737	+	0.955269i	$0.404435\pi$
$854$	0	0
$855$	8.14320e6	0.380960
$856$	0	0
$857$	3.26615e7	1.51909	0.759545	−	0.650455i	$-0.225422\pi$
$857$	3.26615e7	1.51909	0.759545	+	0.650455i	$0.225422\pi$
$858$	0	0
$859$	3.22021e7	1.48902	0.744510	−	0.667611i	$-0.232683\pi$
$859$	3.22021e7	1.48902	0.744510	+	0.667611i	$0.232683\pi$
$860$	0	0
$861$	−2.77222e7	−1.27444
$862$	0	0
$863$	−2.43670e7	−1.11372	−0.556860	−	0.830607i	$-0.687994\pi$
$863$	−2.43670e7	−1.11372	−0.556860	+	0.830607i	$0.687994\pi$
$864$	0	0
$865$	−2.24478e7	−1.02008
$866$	0	0
$867$	−1.79123e7	−0.809287
$868$	0	0
$869$	−3.09598e7	−1.39075
$870$	0	0
$871$	−6.31278e6	−0.281952
$872$	0	0
$873$	−1.06882e6	−0.0474647
$874$	0	0
$875$	−2.71941e7	−1.20075
$876$	0	0
$877$	−1.43643e6	−0.0630647	−0.0315323	−	0.999503i	$-0.510039\pi$
$877$	−1.43643e6	−0.0630647	−0.0315323	+	0.999503i	$0.510039\pi$
$878$	0	0
$879$	1.93799e7	0.846018
$880$	0	0
$881$	−3.68679e7	−1.60033	−0.800164	−	0.599782i	$-0.795254\pi$
$881$	−3.68679e7	−1.60033	−0.800164	+	0.599782i	$0.795254\pi$
$882$	0	0
$883$	3.55152e7	1.53290	0.766448	−	0.642307i	$-0.222022\pi$
$883$	3.55152e7	1.53290	0.766448	+	0.642307i	$0.222022\pi$
$884$	0	0
$885$	1.39078e7	0.596899
$886$	0	0
$887$	3.92660e7	1.67574	0.837872	−	0.545866i	$-0.183799\pi$
$887$	3.92660e7	1.67574	0.837872	+	0.545866i	$0.183799\pi$
$888$	0	0
$889$	2.38774e7	1.01329
$890$	0	0
$891$	3.16047e6	0.133370
$892$	0	0
$893$	1.20884e7	0.507270
$894$	0	0
$895$	4.65035e7	1.94056
$896$	0	0
$897$	−769452.	−0.0319301
$898$	0	0
$899$	−7.75581e7	−3.20057
$900$	0	0
$901$	2.32459e7	0.953970
$902$	0	0
$903$	−8.04861e6	−0.328475
$904$	0	0
$905$	−2.54748e7	−1.03393
$906$	0	0
$907$	−1.69117e7	−0.682603	−0.341301	−	0.939954i	$-0.610868\pi$
$907$	−1.69117e7	−0.682603	−0.341301	+	0.939954i	$0.610868\pi$
$908$	0	0
$909$	−1.13790e7	−0.456768
$910$	0	0
$911$	−4.40872e6	−0.176002	−0.0880008	−	0.996120i	$-0.528048\pi$
$911$	−4.40872e6	−0.176002	−0.0880008	+	0.996120i	$0.528048\pi$
$912$	0	0
$913$	−5.56228e7	−2.20839
$914$	0	0
$915$	−1.20549e7	−0.476006
$916$	0	0
$917$	−4.28788e7	−1.68391
$918$	0	0
$919$	−1.99354e7	−0.778638	−0.389319	−	0.921103i	$-0.627290\pi$
$919$	−1.99354e7	−0.778638	−0.389319	+	0.921103i	$0.627290\pi$
$920$	0	0
$921$	−2.26342e7	−0.879257
$922$	0	0
$923$	−1.04815e7	−0.404966
$924$	0	0
$925$	−20112.5	−0.000772880 0
$926$	0	0
$927$	−1.66327e6	−0.0635716
$928$	0	0
$929$	−4.20242e7	−1.59757	−0.798786	−	0.601615i	$-0.794524\pi$
$929$	−4.20242e7	−1.59757	−0.798786	+	0.601615i	$0.794524\pi$
$930$	0	0
$931$	−2.26405e7	−0.856074
$932$	0	0
$933$	2.13450e7	0.802771
$934$	0	0
$935$	5.39855e7	2.01952
$936$	0	0
$937$	2.02341e7	0.752896	0.376448	−	0.926438i	$-0.377145\pi$
$937$	2.02341e7	0.752896	0.376448	+	0.926438i	$0.377145\pi$
$938$	0	0
$939$	−1.58866e7	−0.587986
$940$	0	0
$941$	3.98731e7	1.46793	0.733966	−	0.679186i	$-0.237667\pi$
$941$	3.98731e7	1.46793	0.733966	+	0.679186i	$0.237667\pi$
$942$	0	0
$943$	−9.33412e6	−0.341817
$944$	0	0
$945$	−7.72337e6	−0.281337
$946$	0	0
$947$	−1.46849e7	−0.532104	−0.266052	−	0.963959i	$-0.585719\pi$
$947$	−1.46849e7	−0.532104	−0.266052	+	0.963959i	$0.585719\pi$
$948$	0	0
$949$	−7.31251e6	−0.263573
$950$	0	0
$951$	468293.	0.0167906
$952$	0	0
$953$	3.62875e6	0.129427	0.0647134	−	0.997904i	$-0.479387\pi$
$953$	3.62875e6	0.129427	0.0647134	+	0.997904i	$0.479387\pi$
$954$	0	0
$955$	5.59162e7	1.98394
$956$	0	0
$957$	−3.16032e7	−1.11545
$958$	0	0
$959$	−2.16606e7	−0.760543
$960$	0	0
$961$	8.45694e7	2.95396
$962$	0	0
$963$	−1.68973e7	−0.587153
$964$	0	0
$965$	−8.28761e6	−0.286491
$966$	0	0
$967$	−4.11518e7	−1.41522	−0.707608	−	0.706605i	$-0.750226\pi$
$967$	−4.11518e7	−1.41522	−0.707608	+	0.706605i	$0.750226\pi$
$968$	0	0
$969$	−2.75312e7	−0.941923
$970$	0	0
$971$	−1.85469e7	−0.631283	−0.315641	−	0.948879i	$-0.602220\pi$
$971$	−1.85469e7	−0.631283	−0.315641	+	0.948879i	$0.602220\pi$
$972$	0	0
$973$	5.13238e7	1.73795
$974$	0	0
$975$	−811898.	−0.0273521
$976$	0	0
$977$	1.25402e7	0.420308	0.210154	−	0.977668i	$-0.432604\pi$
$977$	1.25402e7	0.420308	0.210154	+	0.977668i	$0.432604\pi$
$978$	0	0
$979$	4.54437e7	1.51536
$980$	0	0
$981$	1.31806e7	0.437282
$982$	0	0
$983$	2.97225e6	0.0981074	0.0490537	−	0.998796i	$-0.484379\pi$
$983$	2.97225e6	0.0981074	0.0490537	+	0.998796i	$0.484379\pi$
$984$	0	0
$985$	−4.20591e7	−1.38124
$986$	0	0
$987$	−1.14651e7	−0.374616
$988$	0	0
$989$	−2.70998e6	−0.0880999
$990$	0	0
$991$	4.95892e7	1.60399	0.801997	−	0.597328i	$-0.203771\pi$
$991$	4.95892e7	1.60399	0.801997	+	0.597328i	$0.203771\pi$
$992$	0	0
$993$	−4.57225e6	−0.147149
$994$	0	0
$995$	−1.87397e7	−0.600075
$996$	0	0
$997$	1.25319e7	0.399281	0.199640	−	0.979869i	$-0.436023\pi$
$997$	1.25319e7	0.399281	0.199640	+	0.979869i	$0.436023\pi$
$998$	0	0
$999$	26267.5	0.000832731 0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1104.6.a.ba.1.3		8
4.3	odd	2		552.6.a.i.1.3	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.6.a.i.1.3	✓	8	4.3	odd	2
1104.6.a.ba.1.3		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:	\( N \)	\(=\)	\( 1104 = 2^{4} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1104.a (trivial)

\(f(q)\)	\(=\)	\(q-9.00000 q^{3} -60.6892 q^{5} -174.569 q^{7} +81.0000 q^{9} +O(q^{10})\)	\(q-9.00000 q^{3} -60.6892 q^{5} -174.569 q^{7} +81.0000 q^{9} +481.705 q^{11} +161.616 q^{13} +546.203 q^{15} -1846.65 q^{17} -1656.53 q^{19} +1571.12 q^{21} +529.000 q^{23} +558.182 q^{25} -729.000 q^{27} +7289.65 q^{29} -10639.5 q^{31} -4335.35 q^{33} +10594.5 q^{35} -36.0322 q^{37} -1454.54 q^{39} -17644.8 q^{41} -5122.84 q^{43} -4915.83 q^{45} -7297.42 q^{47} +13667.4 q^{49} +16619.8 q^{51} -12588.2 q^{53} -29234.3 q^{55} +14908.7 q^{57} +25462.7 q^{59} -22070.4 q^{61} -14140.1 q^{63} -9808.33 q^{65} -39060.4 q^{67} -4761.00 q^{69} -64854.3 q^{71} -45246.3 q^{73} -5023.64 q^{75} -84091.0 q^{77} -64271.1 q^{79} +6561.00 q^{81} -115471. q^{83} +112072. q^{85} -65606.8 q^{87} +94339.3 q^{89} -28213.1 q^{91} +95755.3 q^{93} +100533. q^{95} -13195.4 q^{97} +39018.1 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 72 q^{3} + 16 q^{5} + 36 q^{7} + 648 q^{9}+O(q^{10}) \) 8 * q - 72 * q^3 + 16 * q^5 + 36 * q^7 + 648 * q^9	\( 8 q - 72 q^{3} + 16 q^{5} + 36 q^{7} + 648 q^{9} + 148 q^{11} + 696 q^{13} - 144 q^{15} + 572 q^{17} - 2456 q^{19} - 324 q^{21} + 4232 q^{23} + 11896 q^{25} - 5832 q^{27} + 6504 q^{29} - 3128 q^{31} - 1332 q^{33} - 6960 q^{35} + 3844 q^{37} - 6264 q^{39} + 6440 q^{41} - 7048 q^{43} + 1296 q^{45} - 30464 q^{47} + 46984 q^{49} - 5148 q^{51} + 63696 q^{53} - 32688 q^{55} + 22104 q^{57} - 54872 q^{59} + 12108 q^{61} + 2916 q^{63} + 124088 q^{65} - 139216 q^{67} - 38088 q^{69} - 76216 q^{71} + 13632 q^{73} - 107064 q^{75} - 78248 q^{77} - 126380 q^{79} + 52488 q^{81} - 238196 q^{83} + 117536 q^{85} - 58536 q^{87} + 123668 q^{89} - 248176 q^{91} + 28152 q^{93} - 183408 q^{95} + 18576 q^{97} + 11988 q^{99}+O(q^{100}) \) 8 * q - 72 * q^3 + 16 * q^5 + 36 * q^7 + 648 * q^9 + 148 * q^11 + 696 * q^13 - 144 * q^15 + 572 * q^17 - 2456 * q^19 - 324 * q^21 + 4232 * q^23 + 11896 * q^25 - 5832 * q^27 + 6504 * q^29 - 3128 * q^31 - 1332 * q^33 - 6960 * q^35 + 3844 * q^37 - 6264 * q^39 + 6440 * q^41 - 7048 * q^43 + 1296 * q^45 - 30464 * q^47 + 46984 * q^49 - 5148 * q^51 + 63696 * q^53 - 32688 * q^55 + 22104 * q^57 - 54872 * q^59 + 12108 * q^61 + 2916 * q^63 + 124088 * q^65 - 139216 * q^67 - 38088 * q^69 - 76216 * q^71 + 13632 * q^73 - 107064 * q^75 - 78248 * q^77 - 126380 * q^79 + 52488 * q^81 - 238196 * q^83 + 117536 * q^85 - 58536 * q^87 + 123668 * q^89 - 248176 * q^91 + 28152 * q^93 - 183408 * q^95 + 18576 * q^97 + 11988 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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