LMFDB - Embedded newform 368.6.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [368,6,Mod(1,368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(368, 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("368.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 368 = 2^{4} \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	368.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:	$59.0212456912$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.7925.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 13x + 12 $ x^3 - x^2 - 13*x + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 23)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.57511$ of defining polynomial
Character	$\chi$	$=$	368.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.09413 q^{3} +3.86330 q^{5} +226.379 q^{7} -160.297 q^{9} +O(q^{10})$	$q-9.09413 q^{3} +3.86330 q^{5} +226.379 q^{7} -160.297 q^{9} -18.3261 q^{11} -711.185 q^{13} -35.1334 q^{15} +165.441 q^{17} -213.254 q^{19} -2058.72 q^{21} +529.000 q^{23} -3110.07 q^{25} +3667.63 q^{27} +5475.82 q^{29} -6077.13 q^{31} +166.660 q^{33} +874.570 q^{35} +11499.7 q^{37} +6467.61 q^{39} -13183.0 q^{41} +21455.2 q^{43} -619.274 q^{45} -12960.8 q^{47} +34440.5 q^{49} -1504.54 q^{51} -29103.4 q^{53} -70.7993 q^{55} +1939.36 q^{57} -10506.2 q^{59} -34103.3 q^{61} -36287.8 q^{63} -2747.52 q^{65} -12988.2 q^{67} -4810.80 q^{69} -45243.0 q^{71} -52589.2 q^{73} +28283.4 q^{75} -4148.65 q^{77} -57603.3 q^{79} +5598.14 q^{81} +40816.8 q^{83} +639.148 q^{85} -49797.8 q^{87} +54699.9 q^{89} -160997. q^{91} +55266.2 q^{93} -823.864 q^{95} -117497. q^{97} +2937.62 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 20 q^{3} - 58 q^{5} + 282 q^{7} + 121 q^{9}+O(q^{10}) $ 3 * q + 20 * q^3 - 58 * q^5 + 282 * q^7 + 121 * q^9	$ 3 q + 20 q^{3} - 58 q^{5} + 282 q^{7} + 121 q^{9} - 136 q^{11} - 1116 q^{13} - 750 q^{15} - 896 q^{17} - 1654 q^{19} - 1670 q^{21} + 1587 q^{23} - 7347 q^{25} + 10700 q^{27} - 844 q^{29} + 3020 q^{31} - 7370 q^{33} - 1072 q^{35} + 8938 q^{37} - 16020 q^{39} - 12792 q^{41} + 16730 q^{43} - 3936 q^{45} - 22500 q^{47} + 2887 q^{49} - 50290 q^{51} + 17108 q^{53} + 436 q^{55} - 61960 q^{57} - 54176 q^{59} - 71324 q^{61} - 40696 q^{63} + 846 q^{65} + 62960 q^{67} + 10580 q^{69} - 98400 q^{71} - 81772 q^{73} - 44800 q^{75} - 304 q^{77} - 58224 q^{79} + 149947 q^{81} - 9892 q^{83} + 15536 q^{85} - 90500 q^{87} + 27542 q^{89} - 151974 q^{91} + 157330 q^{93} + 20644 q^{95} - 273672 q^{97} - 183082 q^{99}+O(q^{100}) $ 3 * q + 20 * q^3 - 58 * q^5 + 282 * q^7 + 121 * q^9 - 136 * q^11 - 1116 * q^13 - 750 * q^15 - 896 * q^17 - 1654 * q^19 - 1670 * q^21 + 1587 * q^23 - 7347 * q^25 + 10700 * q^27 - 844 * q^29 + 3020 * q^31 - 7370 * q^33 - 1072 * q^35 + 8938 * q^37 - 16020 * q^39 - 12792 * q^41 + 16730 * q^43 - 3936 * q^45 - 22500 * q^47 + 2887 * q^49 - 50290 * q^51 + 17108 * q^53 + 436 * q^55 - 61960 * q^57 - 54176 * q^59 - 71324 * q^61 - 40696 * q^63 + 846 * q^65 + 62960 * q^67 + 10580 * q^69 - 98400 * q^71 - 81772 * q^73 - 44800 * q^75 - 304 * q^77 - 58224 * q^79 + 149947 * q^81 - 9892 * q^83 + 15536 * q^85 - 90500 * q^87 + 27542 * q^89 - 151974 * q^91 + 157330 * q^93 + 20644 * q^95 - 273672 * q^97 - 183082 * 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
$3$	−9.09413	−0.583389	−0.291694	−	0.956512i	$-0.594219\pi$
$3$	−9.09413	−0.583389	−0.291694	+	0.956512i	$0.594219\pi$
$4$	0	0
$5$	3.86330	0.0691088	0.0345544	−	0.999403i	$-0.488999\pi$
$5$	3.86330	0.0691088	0.0345544	+	0.999403i	$0.488999\pi$
$6$	0	0
$7$	226.379	1.74619	0.873094	−	0.487551i	$-0.162110\pi$
$7$	226.379	1.74619	0.873094	+	0.487551i	$0.162110\pi$
$8$	0	0
$9$	−160.297	−0.659657
$10$	0	0
$11$	−18.3261	−0.0456656	−0.0228328	−	0.999739i	$-0.507269\pi$
$11$	−18.3261	−0.0456656	−0.0228328	+	0.999739i	$0.507269\pi$
$12$	0	0
$13$	−711.185	−1.16714	−0.583572	−	0.812062i	$-0.698345\pi$
$13$	−711.185	−1.16714	−0.583572	+	0.812062i	$0.698345\pi$
$14$	0	0
$15$	−35.1334	−0.0403173
$16$	0	0
$17$	165.441	0.138842	0.0694210	−	0.997587i	$-0.477885\pi$
$17$	165.441	0.138842	0.0694210	+	0.997587i	$0.477885\pi$
$18$	0	0
$19$	−213.254	−0.135523	−0.0677615	−	0.997702i	$-0.521586\pi$
$19$	−213.254	−0.135523	−0.0677615	+	0.997702i	$0.521586\pi$
$20$	0	0
$21$	−2058.72	−1.01871
$22$	0	0
$23$	529.000	0.208514
$23$	529.000	0.208514
$24$	0	0
$25$	−3110.07	−0.995224
$26$	0	0
$27$	3667.63	0.968226
$28$	0	0
$29$	5475.82	1.20908	0.604539	−	0.796576i	$-0.293357\pi$
$29$	5475.82	1.20908	0.604539	+	0.796576i	$0.293357\pi$
$30$	0	0
$31$	−6077.13	−1.13578	−0.567890	−	0.823104i	$-0.692240\pi$
$31$	−6077.13	−1.13578	−0.567890	+	0.823104i	$0.692240\pi$
$32$	0	0
$33$	166.660	0.0266408
$34$	0	0
$35$	874.570	0.120677
$36$	0	0
$37$	11499.7	1.38096	0.690481	−	0.723350i	$-0.257399\pi$
$37$	11499.7	1.38096	0.690481	+	0.723350i	$0.257399\pi$
$38$	0	0
$39$	6467.61	0.680899
$40$	0	0
$41$	−13183.0	−1.22477	−0.612384	−	0.790561i	$-0.709789\pi$
$41$	−13183.0	−1.22477	−0.612384	+	0.790561i	$0.709789\pi$
$42$	0	0
$43$	21455.2	1.76955	0.884773	−	0.466022i	$-0.154313\pi$
$43$	21455.2	1.76955	0.884773	+	0.466022i	$0.154313\pi$
$44$	0	0
$45$	−619.274	−0.0455881
$46$	0	0
$47$	−12960.8	−0.855830	−0.427915	−	0.903819i	$-0.640752\pi$
$47$	−12960.8	−0.855830	−0.427915	+	0.903819i	$0.640752\pi$
$48$	0	0
$49$	34440.5	2.04917
$50$	0	0
$51$	−1504.54	−0.0809988
$52$	0	0
$53$	−29103.4	−1.42316	−0.711581	−	0.702604i	$-0.752021\pi$
$53$	−29103.4	−1.42316	−0.711581	+	0.702604i	$0.752021\pi$
$54$	0	0
$55$	−70.7993	−0.00315589
$56$	0	0
$57$	1939.36	0.0790626
$58$	0	0
$59$	−10506.2	−0.392930	−0.196465	−	0.980511i	$-0.562946\pi$
$59$	−10506.2	−0.392930	−0.196465	+	0.980511i	$0.562946\pi$
$60$	0	0
$61$	−34103.3	−1.17347	−0.586734	−	0.809780i	$-0.699587\pi$
$61$	−34103.3	−1.17347	−0.586734	+	0.809780i	$0.699587\pi$
$62$	0	0
$63$	−36287.8	−1.15189
$64$	0	0
$65$	−2747.52	−0.0806599
$66$	0	0
$67$	−12988.2	−0.353476	−0.176738	−	0.984258i	$-0.556555\pi$
$67$	−12988.2	−0.353476	−0.176738	+	0.984258i	$0.556555\pi$
$68$	0	0
$69$	−4810.80	−0.121645
$70$	0	0
$71$	−45243.0	−1.06514	−0.532569	−	0.846387i	$-0.678773\pi$
$71$	−45243.0	−1.06514	−0.532569	+	0.846387i	$0.678773\pi$
$72$	0	0
$73$	−52589.2	−1.15502	−0.577510	−	0.816384i	$-0.695976\pi$
$73$	−52589.2	−1.15502	−0.577510	+	0.816384i	$0.695976\pi$
$74$	0	0
$75$	28283.4	0.580603
$76$	0	0
$77$	−4148.65	−0.0797407
$78$	0	0
$79$	−57603.3	−1.03844	−0.519218	−	0.854642i	$-0.673777\pi$
$79$	−57603.3	−1.03844	−0.519218	+	0.854642i	$0.673777\pi$
$80$	0	0
$81$	5598.14	0.0948051
$82$	0	0
$83$	40816.8	0.650345	0.325173	−	0.945655i	$-0.394578\pi$
$83$	40816.8	0.650345	0.325173	+	0.945655i	$0.394578\pi$
$84$	0	0
$85$	639.148	0.00959520
$86$	0	0
$87$	−49797.8	−0.705362
$88$	0	0
$89$	54699.9	0.732000	0.366000	−	0.930615i	$-0.380727\pi$
$89$	54699.9	0.732000	0.366000	+	0.930615i	$0.380727\pi$
$90$	0	0
$91$	−160997.	−2.03805
$92$	0	0
$93$	55266.2	0.662602
$94$	0	0
$95$	−823.864	−0.00936583
$96$	0	0
$97$	−117497.	−1.26794	−0.633968	−	0.773360i	$-0.718575\pi$
$97$	−117497.	−1.26794	−0.633968	+	0.773360i	$0.718575\pi$
$98$	0	0
$99$	2937.62	0.0301236
$100$	0	0
$101$	−141707.	−1.38226	−0.691128	−	0.722733i	$-0.742886\pi$
$101$	−141707.	−1.38226	−0.691128	+	0.722733i	$0.742886\pi$
$102$	0	0
$103$	−60185.0	−0.558979	−0.279489	−	0.960149i	$-0.590165\pi$
$103$	−60185.0	−0.558979	−0.279489	+	0.960149i	$0.590165\pi$
$104$	0	0
$105$	−7953.46	−0.0704017
$106$	0	0
$107$	186999.	1.57899	0.789495	−	0.613757i	$-0.210342\pi$
$107$	186999.	1.57899	0.789495	+	0.613757i	$0.210342\pi$
$108$	0	0
$109$	−63597.2	−0.512710	−0.256355	−	0.966583i	$-0.582522\pi$
$109$	−63597.2	−0.512710	−0.256355	+	0.966583i	$0.582522\pi$
$110$	0	0
$111$	−104580.	−0.805638
$112$	0	0
$113$	42281.3	0.311496	0.155748	−	0.987797i	$-0.450221\pi$
$113$	42281.3	0.311496	0.155748	+	0.987797i	$0.450221\pi$
$114$	0	0
$115$	2043.69	0.0144102
$116$	0	0
$117$	114001.	0.769915
$118$	0	0
$119$	37452.4	0.242444
$120$	0	0
$121$	−160715.	−0.997915
$122$	0	0
$123$	119888.	0.714516
$124$	0	0
$125$	−24088.0	−0.137888
$126$	0	0
$127$	119942.	0.659877	0.329938	−	0.944002i	$-0.392972\pi$
$127$	119942.	0.659877	0.329938	+	0.944002i	$0.392972\pi$
$128$	0	0
$129$	−195117.	−1.03233
$130$	0	0
$131$	−188708.	−0.960753	−0.480376	−	0.877062i	$-0.659500\pi$
$131$	−188708.	−0.960753	−0.480376	+	0.877062i	$0.659500\pi$
$132$	0	0
$133$	−48276.2	−0.236649
$134$	0	0
$135$	14169.2	0.0669129
$136$	0	0
$137$	−320875.	−1.46061	−0.730305	−	0.683121i	$-0.760622\pi$
$137$	−320875.	−1.46061	−0.730305	+	0.683121i	$0.760622\pi$
$138$	0	0
$139$	−174260.	−0.765000	−0.382500	−	0.923955i	$-0.624937\pi$
$139$	−174260.	−0.765000	−0.382500	+	0.923955i	$0.624937\pi$
$140$	0	0
$141$	117867.	0.499282
$142$	0	0
$143$	13033.3	0.0532983
$144$	0	0
$145$	21154.7	0.0835579
$146$	0	0
$147$	−313206.	−1.19547
$148$	0	0
$149$	−144356.	−0.532682	−0.266341	−	0.963879i	$-0.585815\pi$
$149$	−144356.	−0.532682	−0.266341	+	0.963879i	$0.585815\pi$
$150$	0	0
$151$	122240.	0.436286	0.218143	−	0.975917i	$-0.430000\pi$
$151$	122240.	0.436286	0.218143	+	0.975917i	$0.430000\pi$
$152$	0	0
$153$	−26519.6	−0.0915881
$154$	0	0
$155$	−23477.8	−0.0784924
$156$	0	0
$157$	−115272.	−0.373227	−0.186614	−	0.982433i	$-0.559751\pi$
$157$	−115272.	−0.373227	−0.186614	+	0.982433i	$0.559751\pi$
$158$	0	0
$159$	264670.	0.830257
$160$	0	0
$161$	119755.	0.364106
$162$	0	0
$163$	−120609.	−0.355557	−0.177778	−	0.984071i	$-0.556891\pi$
$163$	−120609.	−0.355557	−0.177778	+	0.984071i	$0.556891\pi$
$164$	0	0
$165$	643.858	0.00184111
$166$	0	0
$167$	−476969.	−1.32342	−0.661711	−	0.749759i	$-0.730170\pi$
$167$	−476969.	−1.32342	−0.661711	+	0.749759i	$0.730170\pi$
$168$	0	0
$169$	134491.	0.362224
$170$	0	0
$171$	34183.9	0.0893987
$172$	0	0
$173$	341220.	0.866801	0.433400	−	0.901202i	$-0.357314\pi$
$173$	341220.	0.866801	0.433400	+	0.901202i	$0.357314\pi$
$174$	0	0
$175$	−704056.	−1.73785
$176$	0	0
$177$	95544.7	0.229231
$178$	0	0
$179$	−157948.	−0.368452	−0.184226	−	0.982884i	$-0.558978\pi$
$179$	−157948.	−0.368452	−0.184226	+	0.982884i	$0.558978\pi$
$180$	0	0
$181$	−133395.	−0.302652	−0.151326	−	0.988484i	$-0.548354\pi$
$181$	−133395.	−0.302652	−0.151326	+	0.988484i	$0.548354\pi$
$182$	0	0
$183$	310140.	0.684588
$184$	0	0
$185$	44426.8	0.0954367
$186$	0	0
$187$	−3031.89	−0.00634030
$188$	0	0
$189$	830276.	1.69070
$190$	0	0
$191$	819880.	1.62617	0.813087	−	0.582142i	$-0.197785\pi$
$191$	819880.	1.62617	0.813087	+	0.582142i	$0.197785\pi$
$192$	0	0
$193$	389602.	0.752884	0.376442	−	0.926440i	$-0.377147\pi$
$193$	389602.	0.752884	0.376442	+	0.926440i	$0.377147\pi$
$194$	0	0
$195$	24986.3	0.0470561
$196$	0	0
$197$	251073.	0.460930	0.230465	−	0.973081i	$-0.425975\pi$
$197$	251073.	0.460930	0.230465	+	0.973081i	$0.425975\pi$
$198$	0	0
$199$	−487551.	−0.872745	−0.436373	−	0.899766i	$-0.643737\pi$
$199$	−487551.	−0.872745	−0.436373	+	0.899766i	$0.643737\pi$
$200$	0	0
$201$	118116.	0.206214
$202$	0	0
$203$	1.23961e6	2.11128
$204$	0	0
$205$	−50929.8	−0.0846423
$206$	0	0
$207$	−84797.0	−0.137548
$208$	0	0
$209$	3908.12	0.00618873
$210$	0	0
$211$	274245.	0.424064	0.212032	−	0.977263i	$-0.431992\pi$
$211$	274245.	0.424064	0.212032	+	0.977263i	$0.431992\pi$
$212$	0	0
$213$	411446.	0.621390
$214$	0	0
$215$	82888.0	0.122291
$216$	0	0
$217$	−1.37573e6	−1.98329
$218$	0	0
$219$	478253.	0.673826
$220$	0	0
$221$	−117659.	−0.162048
$222$	0	0
$223$	215015.	0.289539	0.144769	−	0.989465i	$-0.453756\pi$
$223$	215015.	0.289539	0.144769	+	0.989465i	$0.453756\pi$
$224$	0	0
$225$	498535.	0.656507
$226$	0	0
$227$	−837285.	−1.07847	−0.539236	−	0.842155i	$-0.681287\pi$
$227$	−837285.	−1.07847	−0.539236	+	0.842155i	$0.681287\pi$
$228$	0	0
$229$	−1.33914e6	−1.68748	−0.843740	−	0.536753i	$-0.819651\pi$
$229$	−1.33914e6	−1.68748	−0.843740	+	0.536753i	$0.819651\pi$
$230$	0	0
$231$	37728.4	0.0465198
$232$	0	0
$233$	−320747.	−0.387055	−0.193528	−	0.981095i	$-0.561993\pi$
$233$	−320747.	−0.387055	−0.193528	+	0.981095i	$0.561993\pi$
$234$	0	0
$235$	−50071.5	−0.0591454
$236$	0	0
$237$	523852.	0.605812
$238$	0	0
$239$	278679.	0.315580	0.157790	−	0.987473i	$-0.449563\pi$
$239$	278679.	0.315580	0.157790	+	0.987473i	$0.449563\pi$
$240$	0	0
$241$	−680862.	−0.755120	−0.377560	−	0.925985i	$-0.623237\pi$
$241$	−680862.	−0.755120	−0.377560	+	0.925985i	$0.623237\pi$
$242$	0	0
$243$	−942145.	−1.02353
$244$	0	0
$245$	133054.	0.141616
$246$	0	0
$247$	151663.	0.158175
$248$	0	0
$249$	−371194.	−0.379404
$250$	0	0
$251$	−410308.	−0.411079	−0.205540	−	0.978649i	$-0.565895\pi$
$251$	−410308.	−0.411079	−0.205540	+	0.978649i	$0.565895\pi$
$252$	0	0
$253$	−9694.52	−0.00952193
$254$	0	0
$255$	−5812.50	−0.00559773
$256$	0	0
$257$	1.69725e6	1.60292	0.801460	−	0.598048i	$-0.204057\pi$
$257$	1.69725e6	1.60292	0.801460	+	0.598048i	$0.204057\pi$
$258$	0	0
$259$	2.60329e6	2.41142
$260$	0	0
$261$	−877756.	−0.797577
$262$	0	0
$263$	−1.11082e6	−0.990269	−0.495135	−	0.868816i	$-0.664881\pi$
$263$	−1.11082e6	−0.990269	−0.495135	+	0.868816i	$0.664881\pi$
$264$	0	0
$265$	−112435.	−0.0983531
$266$	0	0
$267$	−497448.	−0.427041
$268$	0	0
$269$	133948.	0.112864	0.0564322	−	0.998406i	$-0.482028\pi$
$269$	133948.	0.112864	0.0564322	+	0.998406i	$0.482028\pi$
$270$	0	0
$271$	−93482.4	−0.0773226	−0.0386613	−	0.999252i	$-0.512309\pi$
$271$	−93482.4	−0.0773226	−0.0386613	+	0.999252i	$0.512309\pi$
$272$	0	0
$273$	1.46413e6	1.18898
$274$	0	0
$275$	56995.6	0.0454475
$276$	0	0
$277$	576551.	0.451479	0.225740	−	0.974188i	$-0.427520\pi$
$277$	576551.	0.451479	0.225740	+	0.974188i	$0.427520\pi$
$278$	0	0
$279$	974144.	0.749226
$280$	0	0
$281$	−434678.	−0.328399	−0.164200	−	0.986427i	$-0.552504\pi$
$281$	−434678.	−0.328399	−0.164200	+	0.986427i	$0.552504\pi$
$282$	0	0
$283$	1.52198e6	1.12965	0.564823	−	0.825212i	$-0.308944\pi$
$283$	1.52198e6	1.12965	0.564823	+	0.825212i	$0.308944\pi$
$284$	0	0
$285$	7492.33	0.00546392
$286$	0	0
$287$	−2.98435e6	−2.13868
$288$	0	0
$289$	−1.39249e6	−0.980723
$290$	0	0
$291$	1.06853e6	0.739699
$292$	0	0
$293$	1.36342e6	0.927813	0.463907	−	0.885884i	$-0.346447\pi$
$293$	1.36342e6	0.927813	0.463907	+	0.885884i	$0.346447\pi$
$294$	0	0
$295$	−40588.6	−0.0271549
$296$	0	0
$297$	−67213.5	−0.0442146
$298$	0	0
$299$	−376217.	−0.243366
$300$	0	0
$301$	4.85701e6	3.08996
$302$	0	0
$303$	1.28870e6	0.806392
$304$	0	0
$305$	−131751.	−0.0810970
$306$	0	0
$307$	3.03847e6	1.83996	0.919980	−	0.391966i	$-0.128205\pi$
$307$	3.03847e6	1.83996	0.919980	+	0.391966i	$0.128205\pi$
$308$	0	0
$309$	547331.	0.326102
$310$	0	0
$311$	3.32689e6	1.95046	0.975232	−	0.221186i	$-0.0709928\pi$
$311$	3.32689e6	1.95046	0.975232	+	0.221186i	$0.0709928\pi$
$312$	0	0
$313$	−134964.	−0.0778677	−0.0389339	−	0.999242i	$-0.512396\pi$
$313$	−134964.	−0.0778677	−0.0389339	+	0.999242i	$0.512396\pi$
$314$	0	0
$315$	−140191.	−0.0796055
$316$	0	0
$317$	2.83907e6	1.58682	0.793411	−	0.608687i	$-0.208303\pi$
$317$	2.83907e6	1.58682	0.793411	+	0.608687i	$0.208303\pi$
$318$	0	0
$319$	−100351.	−0.0552132
$320$	0	0
$321$	−1.70059e6	−0.921166
$322$	0	0
$323$	−35280.9	−0.0188163
$324$	0	0
$325$	2.21184e6	1.16157
$326$	0	0
$327$	578361.	0.299109
$328$	0	0
$329$	−2.93406e6	−1.49444
$330$	0	0
$331$	−2.26789e6	−1.13777	−0.568883	−	0.822419i	$-0.692624\pi$
$331$	−2.26789e6	−1.13777	−0.568883	+	0.822419i	$0.692624\pi$
$332$	0	0
$333$	−1.84336e6	−0.910962
$334$	0	0
$335$	−50177.1	−0.0244283
$336$	0	0
$337$	−1.16585e6	−0.559199	−0.279599	−	0.960117i	$-0.590202\pi$
$337$	−1.16585e6	−0.559199	−0.279599	+	0.960117i	$0.590202\pi$
$338$	0	0
$339$	−384512.	−0.181723
$340$	0	0
$341$	111370.	0.0518660
$342$	0	0
$343$	3.99185e6	1.83206
$344$	0	0
$345$	−18585.6	−0.00840674
$346$	0	0
$347$	−247150.	−0.110189	−0.0550944	−	0.998481i	$-0.517546\pi$
$347$	−247150.	−0.110189	−0.0550944	+	0.998481i	$0.517546\pi$
$348$	0	0
$349$	869941.	0.382319	0.191160	−	0.981559i	$-0.438775\pi$
$349$	869941.	0.382319	0.191160	+	0.981559i	$0.438775\pi$
$350$	0	0
$351$	−2.60837e6	−1.13006
$352$	0	0
$353$	3.16976e6	1.35391	0.676954	−	0.736025i	$-0.263299\pi$
$353$	3.16976e6	1.35391	0.676954	+	0.736025i	$0.263299\pi$
$354$	0	0
$355$	−174787.	−0.0736104
$356$	0	0
$357$	−340597.	−0.141439
$358$	0	0
$359$	−1.48523e6	−0.608215	−0.304108	−	0.952638i	$-0.598358\pi$
$359$	−1.48523e6	−0.608215	−0.304108	+	0.952638i	$0.598358\pi$
$360$	0	0
$361$	−2.43062e6	−0.981634
$362$	0	0
$363$	1.46157e6	0.582172
$364$	0	0
$365$	−203168.	−0.0798221
$366$	0	0
$367$	−2.27000e6	−0.879752	−0.439876	−	0.898059i	$-0.644978\pi$
$367$	−2.27000e6	−0.879752	−0.439876	+	0.898059i	$0.644978\pi$
$368$	0	0
$369$	2.11319e6	0.807927
$370$	0	0
$371$	−6.58841e6	−2.48511
$372$	0	0
$373$	2.90005e6	1.07928	0.539640	−	0.841896i	$-0.318560\pi$
$373$	2.90005e6	1.07928	0.539640	+	0.841896i	$0.318560\pi$
$374$	0	0
$375$	219059.	0.0804421
$376$	0	0
$377$	−3.89432e6	−1.41117
$378$	0	0
$379$	−1.01778e6	−0.363960	−0.181980	−	0.983302i	$-0.558251\pi$
$379$	−1.01778e6	−0.363960	−0.181980	+	0.983302i	$0.558251\pi$
$380$	0	0
$381$	−1.09077e6	−0.384965
$382$	0	0
$383$	4.19874e6	1.46259	0.731293	−	0.682063i	$-0.238917\pi$
$383$	4.19874e6	1.46259	0.731293	+	0.682063i	$0.238917\pi$
$384$	0	0
$385$	−16027.5	−0.00551079
$386$	0	0
$387$	−3.43920e6	−1.16729
$388$	0	0
$389$	1.67191e6	0.560193	0.280097	−	0.959972i	$-0.409634\pi$
$389$	1.67191e6	0.560193	0.280097	+	0.959972i	$0.409634\pi$
$390$	0	0
$391$	87518.2	0.0289505
$392$	0	0
$393$	1.71613e6	0.560493
$394$	0	0
$395$	−222539.	−0.0717651
$396$	0	0
$397$	−1.36652e6	−0.435152	−0.217576	−	0.976043i	$-0.569815\pi$
$397$	−1.36652e6	−0.435152	−0.217576	+	0.976043i	$0.569815\pi$
$398$	0	0
$399$	439030.	0.138058
$400$	0	0
$401$	−1.30417e6	−0.405017	−0.202509	−	0.979280i	$-0.564909\pi$
$401$	−1.30417e6	−0.405017	−0.202509	+	0.979280i	$0.564909\pi$
$402$	0	0
$403$	4.32196e6	1.32562
$404$	0	0
$405$	21627.3	0.00655187
$406$	0	0
$407$	−210745.	−0.0630625
$408$	0	0
$409$	441139.	0.130397	0.0651984	−	0.997872i	$-0.479232\pi$
$409$	441139.	0.130397	0.0651984	+	0.997872i	$0.479232\pi$
$410$	0	0
$411$	2.91808e6	0.852104
$412$	0	0
$413$	−2.37838e6	−0.686130
$414$	0	0
$415$	157688.	0.0449446
$416$	0	0
$417$	1.58475e6	0.446293
$418$	0	0
$419$	−1.89676e6	−0.527809	−0.263904	−	0.964549i	$-0.585010\pi$
$419$	−1.89676e6	−0.527809	−0.263904	+	0.964549i	$0.585010\pi$
$420$	0	0
$421$	−1.39879e6	−0.384634	−0.192317	−	0.981333i	$-0.561600\pi$
$421$	−1.39879e6	−0.384634	−0.192317	+	0.981333i	$0.561600\pi$
$422$	0	0
$423$	2.07757e6	0.564554
$424$	0	0
$425$	−514534.	−0.138179
$426$	0	0
$427$	−7.72026e6	−2.04910
$428$	0	0
$429$	−118526.	−0.0310936
$430$	0	0
$431$	2.45338e6	0.636168	0.318084	−	0.948063i	$-0.396961\pi$
$431$	2.45338e6	0.636168	0.318084	+	0.948063i	$0.396961\pi$
$432$	0	0
$433$	−7.33998e6	−1.88137	−0.940686	−	0.339277i	$-0.889818\pi$
$433$	−7.33998e6	−1.88137	−0.940686	+	0.339277i	$0.889818\pi$
$434$	0	0
$435$	−192384.	−0.0487468
$436$	0	0
$437$	−112811.	−0.0282585
$438$	0	0
$439$	−4.56023e6	−1.12934	−0.564671	−	0.825316i	$-0.690997\pi$
$439$	−4.56023e6	−1.12934	−0.564671	+	0.825316i	$0.690997\pi$
$440$	0	0
$441$	−5.52070e6	−1.35175
$442$	0	0
$443$	−4.31520e6	−1.04470	−0.522350	−	0.852731i	$-0.674945\pi$
$443$	−4.31520e6	−1.04470	−0.522350	+	0.852731i	$0.674945\pi$
$444$	0	0
$445$	211322.	0.0505877
$446$	0	0
$447$	1.31279e6	0.310761
$448$	0	0
$449$	2.53058e6	0.592386	0.296193	−	0.955128i	$-0.404283\pi$
$449$	2.53058e6	0.592386	0.296193	+	0.955128i	$0.404283\pi$
$450$	0	0
$451$	241593.	0.0559297
$452$	0	0
$453$	−1.11167e6	−0.254524
$454$	0	0
$455$	−621981.	−0.140847
$456$	0	0
$457$	1.61209e6	0.361077	0.180538	−	0.983568i	$-0.442216\pi$
$457$	1.61209e6	0.361077	0.180538	+	0.983568i	$0.442216\pi$
$458$	0	0
$459$	606777.	0.134430
$460$	0	0
$461$	360859.	0.0790833	0.0395417	−	0.999218i	$-0.487410\pi$
$461$	360859.	0.0790833	0.0395417	+	0.999218i	$0.487410\pi$
$462$	0	0
$463$	4.98842e6	1.08146	0.540730	−	0.841196i	$-0.318148\pi$
$463$	4.98842e6	1.08146	0.540730	+	0.841196i	$0.318148\pi$
$464$	0	0
$465$	213510.	0.0457916
$466$	0	0
$467$	3.38295e6	0.717800	0.358900	−	0.933376i	$-0.383152\pi$
$467$	3.38295e6	0.717800	0.358900	+	0.933376i	$0.383152\pi$
$468$	0	0
$469$	−2.94025e6	−0.617236
$470$	0	0
$471$	1.04830e6	0.217737
$472$	0	0
$473$	−393191.	−0.0808073
$474$	0	0
$475$	663235.	0.134876
$476$	0	0
$477$	4.66518e6	0.938799
$478$	0	0
$479$	8.41871e6	1.67651	0.838256	−	0.545276i	$-0.183575\pi$
$479$	8.41871e6	1.67651	0.838256	+	0.545276i	$0.183575\pi$
$480$	0	0
$481$	−8.17841e6	−1.61178
$482$	0	0
$483$	−1.08906e6	−0.212415
$484$	0	0
$485$	−453926.	−0.0876255
$486$	0	0
$487$	−5.35500e6	−1.02314	−0.511572	−	0.859240i	$-0.670937\pi$
$487$	−5.35500e6	−1.02314	−0.511572	+	0.859240i	$0.670937\pi$
$488$	0	0
$489$	1.09683e6	0.207428
$490$	0	0
$491$	2.09557e6	0.392282	0.196141	−	0.980576i	$-0.437159\pi$
$491$	2.09557e6	0.392282	0.196141	+	0.980576i	$0.437159\pi$
$492$	0	0
$493$	905925.	0.167871
$494$	0	0
$495$	11348.9	0.00208181
$496$	0	0
$497$	−1.02421e7	−1.85993
$498$	0	0
$499$	−7.41055e6	−1.33229	−0.666146	−	0.745821i	$-0.732057\pi$
$499$	−7.41055e6	−1.33229	−0.666146	+	0.745821i	$0.732057\pi$
$500$	0	0
$501$	4.33762e6	0.772070
$502$	0	0
$503$	−8.19468e6	−1.44415	−0.722075	−	0.691815i	$-0.756811\pi$
$503$	−8.19468e6	−1.44415	−0.722075	+	0.691815i	$0.756811\pi$
$504$	0	0
$505$	−547457.	−0.0955260
$506$	0	0
$507$	−1.22308e6	−0.211318
$508$	0	0
$509$	−6.72046e6	−1.14975	−0.574876	−	0.818240i	$-0.694950\pi$
$509$	−6.72046e6	−1.14975	−0.574876	+	0.818240i	$0.694950\pi$
$510$	0	0
$511$	−1.19051e7	−2.01688
$512$	0	0
$513$	−782137.	−0.131217
$514$	0	0
$515$	−232513.	−0.0386304
$516$	0	0
$517$	237521.	0.0390820
$518$	0	0
$519$	−3.10310e6	−0.505682
$520$	0	0
$521$	−4.57081e6	−0.737732	−0.368866	−	0.929483i	$-0.620254\pi$
$521$	−4.57081e6	−0.737732	−0.368866	+	0.929483i	$0.620254\pi$
$522$	0	0
$523$	35425.6	0.00566321	0.00283161	−	0.999996i	$-0.499099\pi$
$523$	35425.6	0.00566321	0.00283161	+	0.999996i	$0.499099\pi$
$524$	0	0
$525$	6.40278e6	1.01384
$526$	0	0
$527$	−1.00541e6	−0.157694
$528$	0	0
$529$	279841.	0.0434783
$530$	0	0
$531$	1.68411e6	0.259199
$532$	0	0
$533$	9.37554e6	1.42948
$534$	0	0
$535$	722433.	0.109122
$536$	0	0
$537$	1.43640e6	0.214951
$538$	0	0
$539$	−631160.	−0.0935767
$540$	0	0
$541$	2.95315e6	0.433803	0.216902	−	0.976193i	$-0.430405\pi$
$541$	2.95315e6	0.433803	0.216902	+	0.976193i	$0.430405\pi$
$542$	0	0
$543$	1.21312e6	0.176564
$544$	0	0
$545$	−245695.	−0.0354328
$546$	0	0
$547$	3.04020e6	0.434444	0.217222	−	0.976122i	$-0.430300\pi$
$547$	3.04020e6	0.434444	0.217222	+	0.976122i	$0.430300\pi$
$548$	0	0
$549$	5.46664e6	0.774087
$550$	0	0
$551$	−1.16774e6	−0.163858
$552$	0	0
$553$	−1.30402e7	−1.81330
$554$	0	0
$555$	−404023.	−0.0556767
$556$	0	0
$557$	−3.03791e6	−0.414894	−0.207447	−	0.978246i	$-0.566516\pi$
$557$	−3.03791e6	−0.414894	−0.207447	+	0.978246i	$0.566516\pi$
$558$	0	0
$559$	−1.52586e7	−2.06531
$560$	0	0
$561$	27572.4	0.00369886
$562$	0	0
$563$	1.05140e6	0.139796	0.0698982	−	0.997554i	$-0.477733\pi$
$563$	1.05140e6	0.139796	0.0698982	+	0.997554i	$0.477733\pi$
$564$	0	0
$565$	163345.	0.0215271
$566$	0	0
$567$	1.26730e6	0.165548
$568$	0	0
$569$	1.67008e6	0.216250	0.108125	−	0.994137i	$-0.465515\pi$
$569$	1.67008e6	0.216250	0.108125	+	0.994137i	$0.465515\pi$
$570$	0	0
$571$	−2.71320e6	−0.348250	−0.174125	−	0.984724i	$-0.555710\pi$
$571$	−2.71320e6	−0.348250	−0.174125	+	0.984724i	$0.555710\pi$
$572$	0	0
$573$	−7.45610e6	−0.948692
$574$	0	0
$575$	−1.64523e6	−0.207519
$576$	0	0
$577$	8.83363e6	1.10459	0.552293	−	0.833650i	$-0.313753\pi$
$577$	8.83363e6	1.10459	0.552293	+	0.833650i	$0.313753\pi$
$578$	0	0
$579$	−3.54310e6	−0.439224
$580$	0	0
$581$	9.24008e6	1.13563
$582$	0	0
$583$	533353.	0.0649895
$584$	0	0
$585$	440419.	0.0532079
$586$	0	0
$587$	1.35169e7	1.61913	0.809566	−	0.587029i	$-0.199702\pi$
$587$	1.35169e7	1.61913	0.809566	+	0.587029i	$0.199702\pi$
$588$	0	0
$589$	1.29597e6	0.153924
$590$	0	0
$591$	−2.28329e6	−0.268901
$592$	0	0
$593$	848223.	0.0990543	0.0495272	−	0.998773i	$-0.484229\pi$
$593$	848223.	0.0990543	0.0495272	+	0.998773i	$0.484229\pi$
$594$	0	0
$595$	144690.	0.0167550
$596$	0	0
$597$	4.43386e6	0.509150
$598$	0	0
$599$	1.05945e7	1.20646	0.603229	−	0.797568i	$-0.293880\pi$
$599$	1.05945e7	1.20646	0.603229	+	0.797568i	$0.293880\pi$
$600$	0	0
$601$	−209181.	−0.0236231	−0.0118115	−	0.999930i	$-0.503760\pi$
$601$	−209181.	−0.0236231	−0.0118115	+	0.999930i	$0.503760\pi$
$602$	0	0
$603$	2.08196e6	0.233173
$604$	0	0
$605$	−620891.	−0.0689647
$606$	0	0
$607$	−1.02964e7	−1.13427	−0.567133	−	0.823627i	$-0.691947\pi$
$607$	−1.02964e7	−1.13427	−0.567133	+	0.823627i	$0.691947\pi$
$608$	0	0
$609$	−1.12732e7	−1.23170
$610$	0	0
$611$	9.21753e6	0.998876
$612$	0	0
$613$	1.62158e7	1.74296	0.871479	−	0.490432i	$-0.163161\pi$
$613$	1.62158e7	1.74296	0.871479	+	0.490432i	$0.163161\pi$
$614$	0	0
$615$	463162.	0.0493794
$616$	0	0
$617$	6.25365e6	0.661334	0.330667	−	0.943747i	$-0.392726\pi$
$617$	6.25365e6	0.661334	0.330667	+	0.943747i	$0.392726\pi$
$618$	0	0
$619$	1.06232e7	1.11437	0.557183	−	0.830390i	$-0.311882\pi$
$619$	1.06232e7	1.11437	0.557183	+	0.830390i	$0.311882\pi$
$620$	0	0
$621$	1.94018e6	0.201889
$622$	0	0
$623$	1.23829e7	1.27821
$624$	0	0
$625$	9.62593e6	0.985695
$626$	0	0
$627$	−35540.9	−0.00361044
$628$	0	0
$629$	1.90252e6	0.191736
$630$	0	0
$631$	−8.45928e6	−0.845785	−0.422893	−	0.906180i	$-0.638985\pi$
$631$	−8.45928e6	−0.845785	−0.422893	+	0.906180i	$0.638985\pi$
$632$	0	0
$633$	−2.49402e6	−0.247395
$634$	0	0
$635$	463373.	0.0456033
$636$	0	0
$637$	−2.44936e7	−2.39168
$638$	0	0
$639$	7.25231e6	0.702626
$640$	0	0
$641$	−2.03496e7	−1.95619	−0.978095	−	0.208157i	$-0.933253\pi$
$641$	−2.03496e7	−1.95619	−0.978095	+	0.208157i	$0.933253\pi$
$642$	0	0
$643$	6.43818e6	0.614095	0.307047	−	0.951694i	$-0.400659\pi$
$643$	6.43818e6	0.614095	0.307047	+	0.951694i	$0.400659\pi$
$644$	0	0
$645$	−753794.	−0.0713434
$646$	0	0
$647$	9.20281e6	0.864290	0.432145	−	0.901804i	$-0.357757\pi$
$647$	9.20281e6	0.864290	0.432145	+	0.901804i	$0.357757\pi$
$648$	0	0
$649$	192538.	0.0179434
$650$	0	0
$651$	1.25111e7	1.15703
$652$	0	0
$653$	8.14140e6	0.747164	0.373582	−	0.927597i	$-0.378129\pi$
$653$	8.14140e6	0.747164	0.373582	+	0.927597i	$0.378129\pi$
$654$	0	0
$655$	−729035.	−0.0663965
$656$	0	0
$657$	8.42988e6	0.761917
$658$	0	0
$659$	8.11833e6	0.728204	0.364102	−	0.931359i	$-0.381376\pi$
$659$	8.11833e6	0.728204	0.364102	+	0.931359i	$0.381376\pi$
$660$	0	0
$661$	7.02602e6	0.625469	0.312735	−	0.949841i	$-0.398755\pi$
$661$	7.02602e6	0.625469	0.312735	+	0.949841i	$0.398755\pi$
$662$	0	0
$663$	1.07001e6	0.0945373
$664$	0	0
$665$	−186505.	−0.0163545
$666$	0	0
$667$	2.89671e6	0.252110
$668$	0	0
$669$	−1.95537e6	−0.168914
$670$	0	0
$671$	624981.	0.0535871
$672$	0	0
$673$	−1.13585e7	−0.966683	−0.483341	−	0.875432i	$-0.660577\pi$
$673$	−1.13585e7	−0.966683	−0.483341	+	0.875432i	$0.660577\pi$
$674$	0	0
$675$	−1.14066e7	−0.963601
$676$	0	0
$677$	7.60093e6	0.637375	0.318687	−	0.947860i	$-0.396758\pi$
$677$	7.60093e6	0.637375	0.318687	+	0.947860i	$0.396758\pi$
$678$	0	0
$679$	−2.65988e7	−2.21405
$680$	0	0
$681$	7.61439e6	0.629169
$682$	0	0
$683$	7.85446e6	0.644265	0.322133	−	0.946695i	$-0.395600\pi$
$683$	7.85446e6	0.644265	0.322133	+	0.946695i	$0.395600\pi$
$684$	0	0
$685$	−1.23964e6	−0.100941
$686$	0	0
$687$	1.21784e7	0.984457
$688$	0	0
$689$	2.06979e7	1.66103
$690$	0	0
$691$	2.73211e6	0.217672	0.108836	−	0.994060i	$-0.465288\pi$
$691$	2.73211e6	0.217672	0.108836	+	0.994060i	$0.465288\pi$
$692$	0	0
$693$	665015.	0.0526015
$694$	0	0
$695$	−673220.	−0.0528683
$696$	0	0
$697$	−2.18100e6	−0.170049
$698$	0	0
$699$	2.91692e6	0.225804
$700$	0	0
$701$	−3.62295e6	−0.278463	−0.139231	−	0.990260i	$-0.544463\pi$
$701$	−3.62295e6	−0.278463	−0.139231	+	0.990260i	$0.544463\pi$
$702$	0	0
$703$	−2.45235e6	−0.187152
$704$	0	0
$705$	455357.	0.0345048
$706$	0	0
$707$	−3.20795e7	−2.41368
$708$	0	0
$709$	−4.19896e6	−0.313708	−0.156854	−	0.987622i	$-0.550135\pi$
$709$	−4.19896e6	−0.313708	−0.156854	+	0.987622i	$0.550135\pi$
$710$	0	0
$711$	9.23362e6	0.685012
$712$	0	0
$713$	−3.21480e6	−0.236827
$714$	0	0
$715$	50351.4	0.00368338
$716$	0	0
$717$	−2.53434e6	−0.184106
$718$	0	0
$719$	−1.31323e7	−0.947369	−0.473684	−	0.880695i	$-0.657076\pi$
$719$	−1.31323e7	−0.947369	−0.473684	+	0.880695i	$0.657076\pi$
$720$	0	0
$721$	−1.36246e7	−0.976083
$722$	0	0
$723$	6.19185e6	0.440529
$724$	0	0
$725$	−1.70302e7	−1.20330
$726$	0	0
$727$	1.90858e7	1.33929	0.669644	−	0.742683i	$-0.266447\pi$
$727$	1.90858e7	1.33929	0.669644	+	0.742683i	$0.266447\pi$
$728$	0	0
$729$	7.20765e6	0.502313
$730$	0	0
$731$	3.54957e6	0.245687
$732$	0	0
$733$	−1.58603e7	−1.09031	−0.545157	−	0.838334i	$-0.683530\pi$
$733$	−1.58603e7	−1.09031	−0.545157	+	0.838334i	$0.683530\pi$
$734$	0	0
$735$	−1.21001e6	−0.0826172
$736$	0	0
$737$	238022.	0.0161417
$738$	0	0
$739$	−5.61015e6	−0.377888	−0.188944	−	0.981988i	$-0.560506\pi$
$739$	−5.61015e6	−0.377888	−0.188944	+	0.981988i	$0.560506\pi$
$740$	0	0
$741$	−1.37924e6	−0.0922774
$742$	0	0
$743$	−4.05128e6	−0.269228	−0.134614	−	0.990898i	$-0.542979\pi$
$743$	−4.05128e6	−0.269228	−0.134614	+	0.990898i	$0.542979\pi$
$744$	0	0
$745$	−557689.	−0.0368130
$746$	0	0
$747$	−6.54280e6	−0.429005
$748$	0	0
$749$	4.23326e7	2.75722
$750$	0	0
$751$	500386.	0.0323747	0.0161873	−	0.999869i	$-0.494847\pi$
$751$	500386.	0.0323747	0.0161873	+	0.999869i	$0.494847\pi$
$752$	0	0
$753$	3.73140e6	0.239819
$754$	0	0
$755$	472250.	0.0301512
$756$	0	0
$757$	2.77142e7	1.75777	0.878886	−	0.477032i	$-0.158288\pi$
$757$	2.77142e7	1.75777	0.878886	+	0.477032i	$0.158288\pi$
$758$	0	0
$759$	88163.3	0.00555499
$760$	0	0
$761$	2.10372e7	1.31682	0.658411	−	0.752659i	$-0.271229\pi$
$761$	2.10372e7	1.31682	0.658411	+	0.752659i	$0.271229\pi$
$762$	0	0
$763$	−1.43971e7	−0.895288
$764$	0	0
$765$	−102453.	−0.00632954
$766$	0	0
$767$	7.47185e6	0.458606
$768$	0	0
$769$	1.52722e7	0.931295	0.465647	−	0.884970i	$-0.345822\pi$
$769$	1.52722e7	0.931295	0.465647	+	0.884970i	$0.345822\pi$
$770$	0	0
$771$	−1.54350e7	−0.935126
$772$	0	0
$773$	−5.03712e6	−0.303203	−0.151602	−	0.988442i	$-0.548443\pi$
$773$	−5.03712e6	−0.303203	−0.151602	+	0.988442i	$0.548443\pi$
$774$	0	0
$775$	1.89003e7	1.13036
$776$	0	0
$777$	−2.36747e7	−1.40680
$778$	0	0
$779$	2.81132e6	0.165984
$780$	0	0
$781$	829129.	0.0486401
$782$	0	0
$783$	2.00833e7	1.17066
$784$	0	0
$785$	−445329.	−0.0257933
$786$	0	0
$787$	4.02266e6	0.231514	0.115757	−	0.993278i	$-0.463071\pi$
$787$	4.02266e6	0.231514	0.115757	+	0.993278i	$0.463071\pi$
$788$	0	0
$789$	1.01019e7	0.577712
$790$	0	0
$791$	9.57161e6	0.543931
$792$	0	0
$793$	2.42537e7	1.36961
$794$	0	0
$795$	1.02250e6	0.0573781
$796$	0	0
$797$	−3.18310e7	−1.77503	−0.887514	−	0.460781i	$-0.847569\pi$
$797$	−3.18310e7	−1.77503	−0.887514	+	0.460781i	$0.847569\pi$
$798$	0	0
$799$	−2.14425e6	−0.118825
$800$	0	0
$801$	−8.76821e6	−0.482869
$802$	0	0
$803$	963756.	0.0527446
$804$	0	0
$805$	462648.	0.0251629
$806$	0	0
$807$	−1.21814e6	−0.0658438
$808$	0	0
$809$	1.26438e7	0.679213	0.339606	−	0.940568i	$-0.389706\pi$
$809$	1.26438e7	0.679213	0.339606	+	0.940568i	$0.389706\pi$
$810$	0	0
$811$	1.06282e7	0.567426	0.283713	−	0.958909i	$-0.408434\pi$
$811$	1.06282e7	0.567426	0.283713	+	0.958909i	$0.408434\pi$
$812$	0	0
$813$	850142.	0.0451092
$814$	0	0
$815$	−465947.	−0.0245721
$816$	0	0
$817$	−4.57541e6	−0.239814
$818$	0	0
$819$	2.58074e7	1.34442
$820$	0	0
$821$	3.63064e7	1.87986	0.939930	−	0.341366i	$-0.110890\pi$
$821$	3.63064e7	1.87986	0.939930	+	0.341366i	$0.110890\pi$
$822$	0	0
$823$	6.98474e6	0.359460	0.179730	−	0.983716i	$-0.442478\pi$
$823$	6.98474e6	0.359460	0.179730	+	0.983716i	$0.442478\pi$
$824$	0	0
$825$	−518326.	−0.0265136
$826$	0	0
$827$	−2.85277e7	−1.45045	−0.725227	−	0.688510i	$-0.758265\pi$
$827$	−2.85277e7	−1.45045	−0.725227	+	0.688510i	$0.758265\pi$
$828$	0	0
$829$	−2.16227e7	−1.09276	−0.546378	−	0.837538i	$-0.683994\pi$
$829$	−2.16227e7	−1.09276	−0.546378	+	0.837538i	$0.683994\pi$
$830$	0	0
$831$	−5.24323e6	−0.263388
$832$	0	0
$833$	5.69786e6	0.284511
$834$	0	0
$835$	−1.84267e6	−0.0914602
$836$	0	0
$837$	−2.22887e7	−1.09969
$838$	0	0
$839$	3.42589e7	1.68023	0.840115	−	0.542408i	$-0.182487\pi$
$839$	3.42589e7	1.68023	0.840115	+	0.542408i	$0.182487\pi$
$840$	0	0
$841$	9.47345e6	0.461868
$842$	0	0
$843$	3.95302e6	0.191584
$844$	0	0
$845$	519581.	0.0250329
$846$	0	0
$847$	−3.63825e7	−1.74255
$848$	0	0
$849$	−1.38411e7	−0.659023
$850$	0	0
$851$	6.08334e6	0.287951
$852$	0	0
$853$	−3.62802e7	−1.70725	−0.853624	−	0.520890i	$-0.825600\pi$
$853$	−3.62802e7	−1.70725	−0.853624	+	0.520890i	$0.825600\pi$
$854$	0	0
$855$	132063.	0.00617824
$856$	0	0
$857$	−1.47179e7	−0.684531	−0.342265	−	0.939603i	$-0.611194\pi$
$857$	−1.47179e7	−0.684531	−0.342265	+	0.939603i	$0.611194\pi$
$858$	0	0
$859$	1.22343e7	0.565713	0.282856	−	0.959162i	$-0.408718\pi$
$859$	1.22343e7	0.565713	0.282856	+	0.959162i	$0.408718\pi$
$860$	0	0
$861$	2.71401e7	1.24768
$862$	0	0
$863$	−2.54861e7	−1.16487	−0.582434	−	0.812878i	$-0.697900\pi$
$863$	−2.54861e7	−1.16487	−0.582434	+	0.812878i	$0.697900\pi$
$864$	0	0
$865$	1.31824e6	0.0599036
$866$	0	0
$867$	1.26635e7	0.572143
$868$	0	0
$869$	1.05565e6	0.0474208
$870$	0	0
$871$	9.23698e6	0.412558
$872$	0	0
$873$	1.88344e7	0.836403
$874$	0	0
$875$	−5.45301e6	−0.240778
$876$	0	0
$877$	1.28458e7	0.563978	0.281989	−	0.959418i	$-0.409006\pi$
$877$	1.28458e7	0.563978	0.281989	+	0.959418i	$0.409006\pi$
$878$	0	0
$879$	−1.23991e7	−0.541276
$880$	0	0
$881$	2.97345e7	1.29069	0.645344	−	0.763892i	$-0.276714\pi$
$881$	2.97345e7	1.29069	0.645344	+	0.763892i	$0.276714\pi$
$882$	0	0
$883$	−7.58412e6	−0.327343	−0.163672	−	0.986515i	$-0.552334\pi$
$883$	−7.58412e6	−0.327343	−0.163672	+	0.986515i	$0.552334\pi$
$884$	0	0
$885$	369118.	0.0158419
$886$	0	0
$887$	4.35352e7	1.85794	0.928970	−	0.370155i	$-0.120695\pi$
$887$	4.35352e7	1.85794	0.928970	+	0.370155i	$0.120695\pi$
$888$	0	0
$889$	2.71524e7	1.15227
$890$	0	0
$891$	−102592.	−0.00432933
$892$	0	0
$893$	2.76394e6	0.115985
$894$	0	0
$895$	−610199.	−0.0254633
$896$	0	0
$897$	3.42137e6	0.141977
$898$	0	0
$899$	−3.32773e7	−1.37325
$900$	0	0
$901$	−4.81490e6	−0.197595
$902$	0	0
$903$	−4.41703e7	−1.80265
$904$	0	0
$905$	−515346.	−0.0209160
$906$	0	0
$907$	−3.27804e7	−1.32311	−0.661555	−	0.749897i	$-0.730103\pi$
$907$	−3.27804e7	−1.32311	−0.661555	+	0.749897i	$0.730103\pi$
$908$	0	0
$909$	2.27152e7	0.911815
$910$	0	0
$911$	3.93141e7	1.56947	0.784733	−	0.619834i	$-0.212800\pi$
$911$	3.93141e7	1.56947	0.784733	+	0.619834i	$0.212800\pi$
$912$	0	0
$913$	−748014.	−0.0296984
$914$	0	0
$915$	1.19816e6	0.0473111
$916$	0	0
$917$	−4.27195e7	−1.67766
$918$	0	0
$919$	2.97680e7	1.16268	0.581342	−	0.813660i	$-0.302528\pi$
$919$	2.97680e7	1.16268	0.581342	+	0.813660i	$0.302528\pi$
$920$	0	0
$921$	−2.76322e7	−1.07341
$922$	0	0
$923$	3.21762e7	1.24317
$924$	0	0
$925$	−3.57649e7	−1.37437
$926$	0	0
$927$	9.64746e6	0.368735
$928$	0	0
$929$	3.60951e7	1.37217	0.686086	−	0.727521i	$-0.259327\pi$
$929$	3.60951e7	1.37217	0.686086	+	0.727521i	$0.259327\pi$
$930$	0	0
$931$	−7.34457e6	−0.277710
$932$	0	0
$933$	−3.02552e7	−1.13788
$934$	0	0
$935$	−11713.1	−0.000438170 0
$936$	0	0
$937$	−4.14725e7	−1.54316	−0.771581	−	0.636131i	$-0.780534\pi$
$937$	−4.14725e7	−1.54316	−0.771581	+	0.636131i	$0.780534\pi$
$938$	0	0
$939$	1.22738e6	0.0454272
$940$	0	0
$941$	3.05025e6	0.112295	0.0561476	−	0.998422i	$-0.482118\pi$
$941$	3.05025e6	0.112295	0.0561476	+	0.998422i	$0.482118\pi$
$942$	0	0
$943$	−6.97379e6	−0.255382
$944$	0	0
$945$	3.20760e6	0.116843
$946$	0	0
$947$	2.14155e7	0.775983	0.387992	−	0.921663i	$-0.373169\pi$
$947$	2.14155e7	0.775983	0.387992	+	0.921663i	$0.373169\pi$
$948$	0	0
$949$	3.74007e7	1.34807
$950$	0	0
$951$	−2.58189e7	−0.925734
$952$	0	0
$953$	−4.74956e7	−1.69403	−0.847015	−	0.531569i	$-0.821602\pi$
$953$	−4.74956e7	−1.69403	−0.847015	+	0.531569i	$0.821602\pi$
$954$	0	0
$955$	3.16744e6	0.112383
$956$	0	0
$957$	912601.	0.0322108
$958$	0	0
$959$	−7.26394e7	−2.55050
$960$	0	0
$961$	8.30235e6	0.289996
$962$	0	0
$963$	−2.99753e7	−1.04159
$964$	0	0
$965$	1.50515e6	0.0520309
$966$	0	0
$967$	−761545.	−0.0261896	−0.0130948	−	0.999914i	$-0.504168\pi$
$967$	−761545.	−0.0261896	−0.0130948	+	0.999914i	$0.504168\pi$
$968$	0	0
$969$	320849.	0.0109772
$970$	0	0
$971$	−1.28371e7	−0.436936	−0.218468	−	0.975844i	$-0.570106\pi$
$971$	−1.28371e7	−0.436936	−0.218468	+	0.975844i	$0.570106\pi$
$972$	0	0
$973$	−3.94489e7	−1.33583
$974$	0	0
$975$	−2.01148e7	−0.677647
$976$	0	0
$977$	−1.89960e7	−0.636688	−0.318344	−	0.947975i	$-0.603127\pi$
$977$	−1.89960e7	−0.636688	−0.318344	+	0.947975i	$0.603127\pi$
$978$	0	0
$979$	−1.00244e6	−0.0334272
$980$	0	0
$981$	1.01944e7	0.338213
$982$	0	0
$983$	−3.27541e7	−1.08114	−0.540569	−	0.841300i	$-0.681791\pi$
$983$	−3.27541e7	−1.08114	−0.540569	+	0.841300i	$0.681791\pi$
$984$	0	0
$985$	969972.	0.0318543
$986$	0	0
$987$	2.66827e7	0.871840
$988$	0	0
$989$	1.13498e7	0.368976
$990$	0	0
$991$	7.38784e6	0.238965	0.119482	−	0.992836i	$-0.461877\pi$
$991$	7.38784e6	0.238965	0.119482	+	0.992836i	$0.461877\pi$
$992$	0	0
$993$	2.06245e7	0.663760
$994$	0	0
$995$	−1.88356e6	−0.0603144
$996$	0	0
$997$	−3.41271e7	−1.08733	−0.543666	−	0.839302i	$-0.682964\pi$
$997$	−3.41271e7	−1.08733	−0.543666	+	0.839302i	$0.682964\pi$
$998$	0	0
$999$	4.21767e7	1.33708

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	368.6.a.e.1.1		3
4.3	odd	2		23.6.a.a.1.1	✓	3
12.11	even	2		207.6.a.b.1.3		3
20.19	odd	2		575.6.a.b.1.3		3
92.91	even	2		529.6.a.a.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.6.a.a.1.1	✓	3	4.3	odd	2
207.6.a.b.1.3		3	12.11	even	2
368.6.a.e.1.1		3	1.1	even	1	trivial
529.6.a.a.1.1		3	92.91	even	2
575.6.a.b.1.3		3	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 368 = 2^{4} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	368.a (trivial)

\(f(q)\)	\(=\)	\(q-9.09413 q^{3} +3.86330 q^{5} +226.379 q^{7} -160.297 q^{9} +O(q^{10})\)	\(q-9.09413 q^{3} +3.86330 q^{5} +226.379 q^{7} -160.297 q^{9} -18.3261 q^{11} -711.185 q^{13} -35.1334 q^{15} +165.441 q^{17} -213.254 q^{19} -2058.72 q^{21} +529.000 q^{23} -3110.07 q^{25} +3667.63 q^{27} +5475.82 q^{29} -6077.13 q^{31} +166.660 q^{33} +874.570 q^{35} +11499.7 q^{37} +6467.61 q^{39} -13183.0 q^{41} +21455.2 q^{43} -619.274 q^{45} -12960.8 q^{47} +34440.5 q^{49} -1504.54 q^{51} -29103.4 q^{53} -70.7993 q^{55} +1939.36 q^{57} -10506.2 q^{59} -34103.3 q^{61} -36287.8 q^{63} -2747.52 q^{65} -12988.2 q^{67} -4810.80 q^{69} -45243.0 q^{71} -52589.2 q^{73} +28283.4 q^{75} -4148.65 q^{77} -57603.3 q^{79} +5598.14 q^{81} +40816.8 q^{83} +639.148 q^{85} -49797.8 q^{87} +54699.9 q^{89} -160997. q^{91} +55266.2 q^{93} -823.864 q^{95} -117497. q^{97} +2937.62 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 20 q^{3} - 58 q^{5} + 282 q^{7} + 121 q^{9}+O(q^{10}) \) 3 * q + 20 * q^3 - 58 * q^5 + 282 * q^7 + 121 * q^9	\( 3 q + 20 q^{3} - 58 q^{5} + 282 q^{7} + 121 q^{9} - 136 q^{11} - 1116 q^{13} - 750 q^{15} - 896 q^{17} - 1654 q^{19} - 1670 q^{21} + 1587 q^{23} - 7347 q^{25} + 10700 q^{27} - 844 q^{29} + 3020 q^{31} - 7370 q^{33} - 1072 q^{35} + 8938 q^{37} - 16020 q^{39} - 12792 q^{41} + 16730 q^{43} - 3936 q^{45} - 22500 q^{47} + 2887 q^{49} - 50290 q^{51} + 17108 q^{53} + 436 q^{55} - 61960 q^{57} - 54176 q^{59} - 71324 q^{61} - 40696 q^{63} + 846 q^{65} + 62960 q^{67} + 10580 q^{69} - 98400 q^{71} - 81772 q^{73} - 44800 q^{75} - 304 q^{77} - 58224 q^{79} + 149947 q^{81} - 9892 q^{83} + 15536 q^{85} - 90500 q^{87} + 27542 q^{89} - 151974 q^{91} + 157330 q^{93} + 20644 q^{95} - 273672 q^{97} - 183082 q^{99}+O(q^{100}) \) 3 * q + 20 * q^3 - 58 * q^5 + 282 * q^7 + 121 * q^9 - 136 * q^11 - 1116 * q^13 - 750 * q^15 - 896 * q^17 - 1654 * q^19 - 1670 * q^21 + 1587 * q^23 - 7347 * q^25 + 10700 * q^27 - 844 * q^29 + 3020 * q^31 - 7370 * q^33 - 1072 * q^35 + 8938 * q^37 - 16020 * q^39 - 12792 * q^41 + 16730 * q^43 - 3936 * q^45 - 22500 * q^47 + 2887 * q^49 - 50290 * q^51 + 17108 * q^53 + 436 * q^55 - 61960 * q^57 - 54176 * q^59 - 71324 * q^61 - 40696 * q^63 + 846 * q^65 + 62960 * q^67 + 10580 * q^69 - 98400 * q^71 - 81772 * q^73 - 44800 * q^75 - 304 * q^77 - 58224 * q^79 + 149947 * q^81 - 9892 * q^83 + 15536 * q^85 - 90500 * q^87 + 27542 * q^89 - 151974 * q^91 + 157330 * q^93 + 20644 * q^95 - 273672 * q^97 - 183082 * q^99

Introduction

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