LMFDB - Embedded newform 1104.6.a.i.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:	$3$
Coefficient field:	3.3.5333.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 11x + 8 $ x^3 - x^2 - 11*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 69)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.49331$ 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)

$f(q)$	$=$	$q-9.00000 q^{3} +59.1123 q^{5} +213.331 q^{7} +81.0000 q^{9} +O(q^{10})$	$q-9.00000 q^{3} +59.1123 q^{5} +213.331 q^{7} +81.0000 q^{9} -126.528 q^{11} -884.825 q^{13} -532.011 q^{15} -1179.98 q^{17} +1866.19 q^{19} -1919.98 q^{21} -529.000 q^{23} +369.263 q^{25} -729.000 q^{27} -6786.62 q^{29} +5146.34 q^{31} +1138.75 q^{33} +12610.5 q^{35} +5137.07 q^{37} +7963.42 q^{39} +12482.7 q^{41} -4198.66 q^{43} +4788.10 q^{45} -23006.9 q^{47} +28703.3 q^{49} +10619.8 q^{51} +25175.4 q^{53} -7479.38 q^{55} -16795.7 q^{57} +37118.1 q^{59} +26410.4 q^{61} +17279.8 q^{63} -52304.0 q^{65} +54398.5 q^{67} +4761.00 q^{69} -35684.7 q^{71} +33937.4 q^{73} -3323.36 q^{75} -26992.5 q^{77} +76625.8 q^{79} +6561.00 q^{81} +96627.2 q^{83} -69751.1 q^{85} +61079.6 q^{87} +30080.8 q^{89} -188761. q^{91} -46317.0 q^{93} +110315. q^{95} -14637.2 q^{97} -10248.8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 27 q^{3} - 56 q^{5} + 114 q^{7} + 243 q^{9}+O(q^{10}) $ 3 * q - 27 * q^3 - 56 * q^5 + 114 * q^7 + 243 * q^9	$ 3 q - 27 q^{3} - 56 q^{5} + 114 q^{7} + 243 q^{9} + 376 q^{11} - 858 q^{13} + 504 q^{15} - 2548 q^{17} + 2846 q^{19} - 1026 q^{21} - 1587 q^{23} + 753 q^{25} - 2187 q^{27} - 16370 q^{29} + 14756 q^{31} - 3384 q^{33} + 18520 q^{35} + 15874 q^{37} + 7722 q^{39} + 12606 q^{41} - 3154 q^{43} - 4536 q^{45} - 29928 q^{47} + 4471 q^{49} + 22932 q^{51} - 44084 q^{53} - 38360 q^{55} - 25614 q^{57} + 29300 q^{59} + 54010 q^{61} + 9234 q^{63} - 51216 q^{65} - 43390 q^{67} + 14283 q^{69} - 23424 q^{71} - 91402 q^{73} - 6777 q^{75} - 97208 q^{77} + 49398 q^{79} + 19683 q^{81} + 103936 q^{83} + 5888 q^{85} + 147330 q^{87} + 96112 q^{89} - 129228 q^{91} - 132804 q^{93} + 55928 q^{95} - 135318 q^{97} + 30456 q^{99}+O(q^{100}) $ 3 * q - 27 * q^3 - 56 * q^5 + 114 * q^7 + 243 * q^9 + 376 * q^11 - 858 * q^13 + 504 * q^15 - 2548 * q^17 + 2846 * q^19 - 1026 * q^21 - 1587 * q^23 + 753 * q^25 - 2187 * q^27 - 16370 * q^29 + 14756 * q^31 - 3384 * q^33 + 18520 * q^35 + 15874 * q^37 + 7722 * q^39 + 12606 * q^41 - 3154 * q^43 - 4536 * q^45 - 29928 * q^47 + 4471 * q^49 + 22932 * q^51 - 44084 * q^53 - 38360 * q^55 - 25614 * q^57 + 29300 * q^59 + 54010 * q^61 + 9234 * q^63 - 51216 * q^65 - 43390 * q^67 + 14283 * q^69 - 23424 * q^71 - 91402 * q^73 - 6777 * q^75 - 97208 * q^77 + 49398 * q^79 + 19683 * q^81 + 103936 * q^83 + 5888 * q^85 + 147330 * q^87 + 96112 * q^89 - 129228 * q^91 - 132804 * q^93 + 55928 * q^95 - 135318 * q^97 + 30456 * 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
$5$	59.1123	1.05743	0.528716	−	0.848799i	$-0.322674\pi$
$5$	59.1123	1.05743	0.528716	+	0.848799i	$0.322674\pi$
$6$	0	0
$7$	213.331	1.64554	0.822772	−	0.568371i	$-0.192426\pi$
$7$	213.331	1.64554	0.822772	+	0.568371i	$0.192426\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−126.528	−0.315287	−0.157643	−	0.987496i	$-0.550390\pi$
$11$	−126.528	−0.315287	−0.157643	+	0.987496i	$0.550390\pi$
$12$	0	0
$13$	−884.825	−1.45211	−0.726054	−	0.687638i	$-0.758648\pi$
$13$	−884.825	−1.45211	−0.726054	+	0.687638i	$0.758648\pi$
$14$	0	0
$15$	−532.011	−0.610509
$16$	0	0
$17$	−1179.98	−0.990264	−0.495132	−	0.868818i	$-0.664880\pi$
$17$	−1179.98	−0.990264	−0.495132	+	0.868818i	$0.664880\pi$
$18$	0	0
$19$	1866.19	1.18596	0.592981	−	0.805216i	$-0.297951\pi$
$19$	1866.19	1.18596	0.592981	+	0.805216i	$0.297951\pi$
$20$	0	0
$21$	−1919.98	−0.950056
$22$	0	0
$23$	−529.000	−0.208514
$23$	−529.000	−0.208514
$24$	0	0
$25$	369.263	0.118164
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	−6786.62	−1.49851	−0.749253	−	0.662284i	$-0.769587\pi$
$29$	−6786.62	−1.49851	−0.749253	+	0.662284i	$0.769587\pi$
$30$	0	0
$31$	5146.34	0.961820	0.480910	−	0.876770i	$-0.340306\pi$
$31$	5146.34	0.961820	0.480910	+	0.876770i	$0.340306\pi$
$32$	0	0
$33$	1138.75	0.182031
$34$	0	0
$35$	12610.5	1.74005
$36$	0	0
$37$	5137.07	0.616895	0.308448	−	0.951241i	$-0.400191\pi$
$37$	5137.07	0.616895	0.308448	+	0.951241i	$0.400191\pi$
$38$	0	0
$39$	7963.42	0.838375
$40$	0	0
$41$	12482.7	1.15971	0.579855	−	0.814720i	$-0.303109\pi$
$41$	12482.7	1.15971	0.579855	+	0.814720i	$0.303109\pi$
$42$	0	0
$43$	−4198.66	−0.346290	−0.173145	−	0.984896i	$-0.555393\pi$
$43$	−4198.66	−0.346290	−0.173145	+	0.984896i	$0.555393\pi$
$44$	0	0
$45$	4788.10	0.352478
$46$	0	0
$47$	−23006.9	−1.51919	−0.759596	−	0.650395i	$-0.774603\pi$
$47$	−23006.9	−1.51919	−0.759596	+	0.650395i	$0.774603\pi$
$48$	0	0
$49$	28703.3	1.70782
$50$	0	0
$51$	10619.8	0.571729
$52$	0	0
$53$	25175.4	1.23108	0.615541	−	0.788105i	$-0.288938\pi$
$53$	25175.4	1.23108	0.615541	+	0.788105i	$0.288938\pi$
$54$	0	0
$55$	−7479.38	−0.333395
$56$	0	0
$57$	−16795.7	−0.684716
$58$	0	0
$59$	37118.1	1.38821	0.694107	−	0.719872i	$-0.255800\pi$
$59$	37118.1	1.38821	0.694107	+	0.719872i	$0.255800\pi$
$60$	0	0
$61$	26410.4	0.908761	0.454381	−	0.890808i	$-0.349861\pi$
$61$	26410.4	0.908761	0.454381	+	0.890808i	$0.349861\pi$
$62$	0	0
$63$	17279.8	0.548515
$64$	0	0
$65$	−52304.0	−1.53551
$66$	0	0
$67$	54398.5	1.48047	0.740236	−	0.672347i	$-0.234714\pi$
$67$	54398.5	1.48047	0.740236	+	0.672347i	$0.234714\pi$
$68$	0	0
$69$	4761.00	0.120386
$70$	0	0
$71$	−35684.7	−0.840110	−0.420055	−	0.907499i	$-0.637989\pi$
$71$	−35684.7	−0.840110	−0.420055	+	0.907499i	$0.637989\pi$
$72$	0	0
$73$	33937.4	0.745369	0.372684	−	0.927958i	$-0.378437\pi$
$73$	33937.4	0.745369	0.372684	+	0.927958i	$0.378437\pi$
$74$	0	0
$75$	−3323.36	−0.0682220
$76$	0	0
$77$	−26992.5	−0.518819
$78$	0	0
$79$	76625.8	1.38136	0.690681	−	0.723160i	$-0.257311\pi$
$79$	76625.8	1.38136	0.690681	+	0.723160i	$0.257311\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	96627.2	1.53959	0.769794	−	0.638293i	$-0.220359\pi$
$83$	96627.2	1.53959	0.769794	+	0.638293i	$0.220359\pi$
$84$	0	0
$85$	−69751.1	−1.04714
$86$	0	0
$87$	61079.6	0.865163
$88$	0	0
$89$	30080.8	0.402545	0.201273	−	0.979535i	$-0.435492\pi$
$89$	30080.8	0.402545	0.201273	+	0.979535i	$0.435492\pi$
$90$	0	0
$91$	−188761.	−2.38951
$92$	0	0
$93$	−46317.0	−0.555307
$94$	0	0
$95$	110315.	1.25408
$96$	0	0
$97$	−14637.2	−0.157953	−0.0789766	−	0.996876i	$-0.525165\pi$
$97$	−14637.2	−0.157953	−0.0789766	+	0.996876i	$0.525165\pi$
$98$	0	0
$99$	−10248.8	−0.105096
$100$	0	0
$101$	−37693.7	−0.367676	−0.183838	−	0.982957i	$-0.558852\pi$
$101$	−37693.7	−0.367676	−0.183838	+	0.982957i	$0.558852\pi$
$102$	0	0
$103$	−125252.	−1.16330	−0.581649	−	0.813440i	$-0.697592\pi$
$103$	−125252.	−1.16330	−0.581649	+	0.813440i	$0.697592\pi$
$104$	0	0
$105$	−113495.	−1.00462
$106$	0	0
$107$	−28002.7	−0.236451	−0.118225	−	0.992987i	$-0.537721\pi$
$107$	−28002.7	−0.236451	−0.118225	+	0.992987i	$0.537721\pi$
$108$	0	0
$109$	−64151.4	−0.517178	−0.258589	−	0.965987i	$-0.583257\pi$
$109$	−64151.4	−0.517178	−0.258589	+	0.965987i	$0.583257\pi$
$110$	0	0
$111$	−46233.6	−0.356165
$112$	0	0
$113$	−25226.7	−0.185851	−0.0929253	−	0.995673i	$-0.529622\pi$
$113$	−25226.7	−0.185851	−0.0929253	+	0.995673i	$0.529622\pi$
$114$	0	0
$115$	−31270.4	−0.220490
$116$	0	0
$117$	−71670.8	−0.484036
$118$	0	0
$119$	−251726.	−1.62952
$120$	0	0
$121$	−145042.	−0.900594
$122$	0	0
$123$	−112344.	−0.669559
$124$	0	0
$125$	−162898.	−0.932482
$126$	0	0
$127$	−35783.9	−0.196870	−0.0984348	−	0.995144i	$-0.531384\pi$
$127$	−35783.9	−0.196870	−0.0984348	+	0.995144i	$0.531384\pi$
$128$	0	0
$129$	37787.9	0.199930
$130$	0	0
$131$	−78119.6	−0.397724	−0.198862	−	0.980028i	$-0.563725\pi$
$131$	−78119.6	−0.397724	−0.198862	+	0.980028i	$0.563725\pi$
$132$	0	0
$133$	398116.	1.95155
$134$	0	0
$135$	−43092.9	−0.203503
$136$	0	0
$137$	−92816.7	−0.422498	−0.211249	−	0.977432i	$-0.567753\pi$
$137$	−92816.7	−0.422498	−0.211249	+	0.977432i	$0.567753\pi$
$138$	0	0
$139$	417118.	1.83114	0.915570	−	0.402159i	$-0.131740\pi$
$139$	417118.	1.83114	0.915570	+	0.402159i	$0.131740\pi$
$140$	0	0
$141$	207062.	0.877106
$142$	0	0
$143$	111955.	0.457831
$144$	0	0
$145$	−401173.	−1.58457
$146$	0	0
$147$	−258330.	−0.986009
$148$	0	0
$149$	445936.	1.64553	0.822766	−	0.568380i	$-0.192430\pi$
$149$	445936.	1.64553	0.822766	+	0.568380i	$0.192430\pi$
$150$	0	0
$151$	260458.	0.929597	0.464799	−	0.885416i	$-0.346127\pi$
$151$	260458.	0.929597	0.464799	+	0.885416i	$0.346127\pi$
$152$	0	0
$153$	−95578.1	−0.330088
$154$	0	0
$155$	304212.	1.01706
$156$	0	0
$157$	46067.4	0.149157	0.0745787	−	0.997215i	$-0.476239\pi$
$157$	46067.4	0.149157	0.0745787	+	0.997215i	$0.476239\pi$
$158$	0	0
$159$	−226579.	−0.710765
$160$	0	0
$161$	−112852.	−0.343120
$162$	0	0
$163$	−188377.	−0.555341	−0.277671	−	0.960676i	$-0.589562\pi$
$163$	−188377.	−0.555341	−0.277671	+	0.960676i	$0.589562\pi$
$164$	0	0
$165$	67314.4	0.192485
$166$	0	0
$167$	−203214.	−0.563849	−0.281924	−	0.959437i	$-0.590973\pi$
$167$	−203214.	−0.563849	−0.281924	+	0.959437i	$0.590973\pi$
$168$	0	0
$169$	411622.	1.10862
$170$	0	0
$171$	151161.	0.395321
$172$	0	0
$173$	158051.	0.401497	0.200749	−	0.979643i	$-0.435663\pi$
$173$	158051.	0.401497	0.200749	+	0.979643i	$0.435663\pi$
$174$	0	0
$175$	78775.3	0.194444
$176$	0	0
$177$	−334063.	−0.801485
$178$	0	0
$179$	−166444.	−0.388271	−0.194136	−	0.980975i	$-0.562190\pi$
$179$	−166444.	−0.388271	−0.194136	+	0.980975i	$0.562190\pi$
$180$	0	0
$181$	185620.	0.421141	0.210571	−	0.977579i	$-0.432468\pi$
$181$	185620.	0.421141	0.210571	+	0.977579i	$0.432468\pi$
$182$	0	0
$183$	−237693.	−0.524673
$184$	0	0
$185$	303664.	0.652325
$186$	0	0
$187$	149300.	0.312217
$188$	0	0
$189$	−155519.	−0.316685
$190$	0	0
$191$	769733.	1.52671	0.763356	−	0.645978i	$-0.223550\pi$
$191$	769733.	1.52671	0.763356	+	0.645978i	$0.223550\pi$
$192$	0	0
$193$	−384145.	−0.742338	−0.371169	−	0.928565i	$-0.621043\pi$
$193$	−384145.	−0.742338	−0.371169	+	0.928565i	$0.621043\pi$
$194$	0	0
$195$	470736.	0.886525
$196$	0	0
$197$	266048.	0.488421	0.244211	−	0.969722i	$-0.421471\pi$
$197$	266048.	0.488421	0.244211	+	0.969722i	$0.421471\pi$
$198$	0	0
$199$	706070.	1.26391	0.631953	−	0.775007i	$-0.282254\pi$
$199$	706070.	1.26391	0.631953	+	0.775007i	$0.282254\pi$
$200$	0	0
$201$	−489587.	−0.854751
$202$	0	0
$203$	−1.44780e6	−2.46586
$204$	0	0
$205$	737882.	1.22631
$206$	0	0
$207$	−42849.0	−0.0695048
$208$	0	0
$209$	−236125.	−0.373918
$210$	0	0
$211$	290636.	0.449410	0.224705	−	0.974427i	$-0.427858\pi$
$211$	290636.	0.449410	0.224705	+	0.974427i	$0.427858\pi$
$212$	0	0
$213$	321162.	0.485038
$214$	0	0
$215$	−248192.	−0.366178
$216$	0	0
$217$	1.09788e6	1.58272
$218$	0	0
$219$	−305436.	−0.430339
$220$	0	0
$221$	1.04407e6	1.43797
$222$	0	0
$223$	574915.	0.774179	0.387089	−	0.922042i	$-0.373481\pi$
$223$	574915.	0.774179	0.387089	+	0.922042i	$0.373481\pi$
$224$	0	0
$225$	29910.3	0.0393880
$226$	0	0
$227$	1.37363e6	1.76931	0.884655	−	0.466246i	$-0.154394\pi$
$227$	1.37363e6	1.76931	0.884655	+	0.466246i	$0.154394\pi$
$228$	0	0
$229$	870725.	1.09722	0.548608	−	0.836080i	$-0.315158\pi$
$229$	870725.	1.09722	0.548608	+	0.836080i	$0.315158\pi$
$230$	0	0
$231$	242932.	0.299540
$232$	0	0
$233$	−1.61732e6	−1.95166	−0.975832	−	0.218522i	$-0.929876\pi$
$233$	−1.61732e6	−1.95166	−0.975832	+	0.218522i	$0.929876\pi$
$234$	0	0
$235$	−1.35999e6	−1.60644
$236$	0	0
$237$	−689633.	−0.797530
$238$	0	0
$239$	1.65281e6	1.87167	0.935833	−	0.352444i	$-0.114649\pi$
$239$	1.65281e6	1.87167	0.935833	+	0.352444i	$0.114649\pi$
$240$	0	0
$241$	928466.	1.02973	0.514865	−	0.857271i	$-0.327842\pi$
$241$	928466.	1.02973	0.514865	+	0.857271i	$0.327842\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	1.69672e6	1.80590
$246$	0	0
$247$	−1.65125e6	−1.72215
$248$	0	0
$249$	−869645.	−0.888881
$250$	0	0
$251$	909693.	0.911403	0.455702	−	0.890133i	$-0.349388\pi$
$251$	909693.	0.911403	0.455702	+	0.890133i	$0.349388\pi$
$252$	0	0
$253$	66933.5	0.0657419
$254$	0	0
$255$	627760.	0.604565
$256$	0	0
$257$	52666.7	0.0497397	0.0248699	−	0.999691i	$-0.492083\pi$
$257$	52666.7	0.0497397	0.0248699	+	0.999691i	$0.492083\pi$
$258$	0	0
$259$	1.09590e6	1.01513
$260$	0	0
$261$	−549716.	−0.499502
$262$	0	0
$263$	−464764.	−0.414327	−0.207164	−	0.978306i	$-0.566423\pi$
$263$	−464764.	−0.414327	−0.207164	+	0.978306i	$0.566423\pi$
$264$	0	0
$265$	1.48818e6	1.30179
$266$	0	0
$267$	−270727.	−0.232410
$268$	0	0
$269$	968420.	0.815987	0.407993	−	0.912985i	$-0.366229\pi$
$269$	968420.	0.815987	0.407993	+	0.912985i	$0.366229\pi$
$270$	0	0
$271$	−2.29262e6	−1.89630	−0.948152	−	0.317817i	$-0.897050\pi$
$271$	−2.29262e6	−1.89630	−0.948152	+	0.317817i	$0.897050\pi$
$272$	0	0
$273$	1.69885e6	1.37958
$274$	0	0
$275$	−46722.2	−0.0372556
$276$	0	0
$277$	290954.	0.227837	0.113919	−	0.993490i	$-0.463660\pi$
$277$	290954.	0.227837	0.113919	+	0.993490i	$0.463660\pi$
$278$	0	0
$279$	416853.	0.320607
$280$	0	0
$281$	−391978.	−0.296139	−0.148070	−	0.988977i	$-0.547306\pi$
$281$	−391978.	−0.296139	−0.148070	+	0.988977i	$0.547306\pi$
$282$	0	0
$283$	−732768.	−0.543877	−0.271938	−	0.962315i	$-0.587665\pi$
$283$	−732768.	−0.543877	−0.271938	+	0.962315i	$0.587665\pi$
$284$	0	0
$285$	−992831.	−0.724041
$286$	0	0
$287$	2.66295e6	1.90835
$288$	0	0
$289$	−27511.7	−0.0193764
$290$	0	0
$291$	131735.	0.0911944
$292$	0	0
$293$	1.62587e6	1.10641	0.553207	−	0.833044i	$-0.313404\pi$
$293$	1.62587e6	1.10641	0.553207	+	0.833044i	$0.313404\pi$
$294$	0	0
$295$	2.19414e6	1.46794
$296$	0	0
$297$	92239.1	0.0606770
$298$	0	0
$299$	468072.	0.302785
$300$	0	0
$301$	−895706.	−0.569835
$302$	0	0
$303$	339243.	0.212278
$304$	0	0
$305$	1.56118e6	0.960954
$306$	0	0
$307$	−1.24581e6	−0.754406	−0.377203	−	0.926131i	$-0.623114\pi$
$307$	−1.24581e6	−0.754406	−0.377203	+	0.926131i	$0.623114\pi$
$308$	0	0
$309$	1.12727e6	0.671631
$310$	0	0
$311$	3.26852e6	1.91624	0.958119	−	0.286369i	$-0.0924483\pi$
$311$	3.26852e6	1.91624	0.958119	+	0.286369i	$0.0924483\pi$
$312$	0	0
$313$	1.70161e6	0.981744	0.490872	−	0.871232i	$-0.336678\pi$
$313$	1.70161e6	0.981744	0.490872	+	0.871232i	$0.336678\pi$
$314$	0	0
$315$	1.02145e6	0.580018
$316$	0	0
$317$	−908317.	−0.507679	−0.253840	−	0.967246i	$-0.581694\pi$
$317$	−908317.	−0.507679	−0.253840	+	0.967246i	$0.581694\pi$
$318$	0	0
$319$	858699.	0.472459
$320$	0	0
$321$	252025.	0.136515
$322$	0	0
$323$	−2.20206e6	−1.17442
$324$	0	0
$325$	−326733.	−0.171587
$326$	0	0
$327$	577362.	0.298593
$328$	0	0
$329$	−4.90809e6	−2.49990
$330$	0	0
$331$	2.14345e6	1.07533	0.537666	−	0.843158i	$-0.319306\pi$
$331$	2.14345e6	1.07533	0.537666	+	0.843158i	$0.319306\pi$
$332$	0	0
$333$	416103.	0.205632
$334$	0	0
$335$	3.21562e6	1.56550
$336$	0	0
$337$	817498.	0.392114	0.196057	−	0.980593i	$-0.437186\pi$
$337$	817498.	0.392114	0.196057	+	0.980593i	$0.437186\pi$
$338$	0	0
$339$	227040.	0.107301
$340$	0	0
$341$	−651157.	−0.303249
$342$	0	0
$343$	2.53785e6	1.16475
$344$	0	0
$345$	281434.	0.127300
$346$	0	0
$347$	−3.20207e6	−1.42760	−0.713801	−	0.700349i	$-0.753028\pi$
$347$	−3.20207e6	−1.42760	−0.713801	+	0.700349i	$0.753028\pi$
$348$	0	0
$349$	−3.23916e6	−1.42354	−0.711768	−	0.702415i	$-0.752105\pi$
$349$	−3.23916e6	−1.42354	−0.711768	+	0.702415i	$0.752105\pi$
$350$	0	0
$351$	645037.	0.279458
$352$	0	0
$353$	−1.72676e6	−0.737555	−0.368777	−	0.929518i	$-0.620224\pi$
$353$	−1.72676e6	−0.737555	−0.368777	+	0.929518i	$0.620224\pi$
$354$	0	0
$355$	−2.10940e6	−0.888360
$356$	0	0
$357$	2.26554e6	0.940806
$358$	0	0
$359$	−3.12338e6	−1.27905	−0.639527	−	0.768769i	$-0.720870\pi$
$359$	−3.12338e6	−1.27905	−0.639527	+	0.768769i	$0.720870\pi$
$360$	0	0
$361$	1.00655e6	0.406507
$362$	0	0
$363$	1.30537e6	0.519958
$364$	0	0
$365$	2.00612e6	0.788177
$366$	0	0
$367$	−1.17031e6	−0.453560	−0.226780	−	0.973946i	$-0.572820\pi$
$367$	−1.17031e6	−0.453560	−0.226780	+	0.973946i	$0.572820\pi$
$368$	0	0
$369$	1.01110e6	0.386570
$370$	0	0
$371$	5.37070e6	2.02580
$372$	0	0
$373$	−692891.	−0.257865	−0.128933	−	0.991653i	$-0.541155\pi$
$373$	−692891.	−0.257865	−0.128933	+	0.991653i	$0.541155\pi$
$374$	0	0
$375$	1.46608e6	0.538369
$376$	0	0
$377$	6.00497e6	2.17599
$378$	0	0
$379$	−2.12163e6	−0.758702	−0.379351	−	0.925253i	$-0.623853\pi$
$379$	−2.12163e6	−0.758702	−0.379351	+	0.925253i	$0.623853\pi$
$380$	0	0
$381$	322055.	0.113663
$382$	0	0
$383$	287079.	0.100001	0.0500005	−	0.998749i	$-0.484078\pi$
$383$	287079.	0.100001	0.0500005	+	0.998749i	$0.484078\pi$
$384$	0	0
$385$	−1.59559e6	−0.548616
$386$	0	0
$387$	−340091.	−0.115430
$388$	0	0
$389$	679811.	0.227779	0.113890	−	0.993493i	$-0.463669\pi$
$389$	679811.	0.227779	0.113890	+	0.993493i	$0.463669\pi$
$390$	0	0
$391$	624208.	0.206484
$392$	0	0
$393$	703076.	0.229626
$394$	0	0
$395$	4.52953e6	1.46070
$396$	0	0
$397$	−5.39917e6	−1.71930	−0.859648	−	0.510886i	$-0.829317\pi$
$397$	−5.39917e6	−1.71930	−0.859648	+	0.510886i	$0.829317\pi$
$398$	0	0
$399$	−3.58304e6	−1.12673
$400$	0	0
$401$	−2.17808e6	−0.676416	−0.338208	−	0.941071i	$-0.609821\pi$
$401$	−2.17808e6	−0.676416	−0.338208	+	0.941071i	$0.609821\pi$
$402$	0	0
$403$	−4.55361e6	−1.39667
$404$	0	0
$405$	387836.	0.117493
$406$	0	0
$407$	−649985.	−0.194499
$408$	0	0
$409$	−356567.	−0.105398	−0.0526991	−	0.998610i	$-0.516782\pi$
$409$	−356567.	−0.105398	−0.0526991	+	0.998610i	$0.516782\pi$
$410$	0	0
$411$	835350.	0.243929
$412$	0	0
$413$	7.91847e6	2.28437
$414$	0	0
$415$	5.71186e6	1.62801
$416$	0	0
$417$	−3.75406e6	−1.05721
$418$	0	0
$419$	−3.82634e6	−1.06475	−0.532376	−	0.846508i	$-0.678701\pi$
$419$	−3.82634e6	−1.06475	−0.532376	+	0.846508i	$0.678701\pi$
$420$	0	0
$421$	−3.00814e6	−0.827167	−0.413584	−	0.910466i	$-0.635723\pi$
$421$	−3.00814e6	−0.827167	−0.413584	+	0.910466i	$0.635723\pi$
$422$	0	0
$423$	−1.86356e6	−0.506397
$424$	0	0
$425$	−435721.	−0.117014
$426$	0	0
$427$	5.63416e6	1.49541
$428$	0	0
$429$	−1.00760e6	−0.264329
$430$	0	0
$431$	−577809.	−0.149827	−0.0749136	−	0.997190i	$-0.523868\pi$
$431$	−577809.	−0.149827	−0.0749136	+	0.997190i	$0.523868\pi$
$432$	0	0
$433$	7.21909e6	1.85039	0.925194	−	0.379494i	$-0.123902\pi$
$433$	7.21909e6	1.85039	0.925194	+	0.379494i	$0.123902\pi$
$434$	0	0
$435$	3.61055e6	0.914852
$436$	0	0
$437$	−987212.	−0.247290
$438$	0	0
$439$	760935.	0.188446	0.0942229	−	0.995551i	$-0.469963\pi$
$439$	760935.	0.188446	0.0942229	+	0.995551i	$0.469963\pi$
$440$	0	0
$441$	2.32497e6	0.569273
$442$	0	0
$443$	6.96892e6	1.68716	0.843580	−	0.537003i	$-0.180444\pi$
$443$	6.96892e6	1.68716	0.843580	+	0.537003i	$0.180444\pi$
$444$	0	0
$445$	1.77815e6	0.425665
$446$	0	0
$447$	−4.01342e6	−0.950049
$448$	0	0
$449$	5.48897e6	1.28492	0.642458	−	0.766321i	$-0.277915\pi$
$449$	5.48897e6	1.28492	0.642458	+	0.766321i	$0.277915\pi$
$450$	0	0
$451$	−1.57942e6	−0.365641
$452$	0	0
$453$	−2.34412e6	−0.536703
$454$	0	0
$455$	−1.11581e7	−2.52675
$456$	0	0
$457$	−2.49968e6	−0.559879	−0.279940	−	0.960018i	$-0.590314\pi$
$457$	−2.49968e6	−0.559879	−0.279940	+	0.960018i	$0.590314\pi$
$458$	0	0
$459$	860203.	0.190576
$460$	0	0
$461$	−8.10339e6	−1.77588	−0.887942	−	0.459956i	$-0.847865\pi$
$461$	−8.10339e6	−1.77588	−0.887942	+	0.459956i	$0.847865\pi$
$462$	0	0
$463$	4.71421e6	1.02201	0.511007	−	0.859577i	$-0.329273\pi$
$463$	4.71421e6	1.02201	0.511007	+	0.859577i	$0.329273\pi$
$464$	0	0
$465$	−2.73791e6	−0.587200
$466$	0	0
$467$	1.03346e6	0.219281	0.109641	−	0.993971i	$-0.465030\pi$
$467$	1.03346e6	0.219281	0.109641	+	0.993971i	$0.465030\pi$
$468$	0	0
$469$	1.16049e7	2.43618
$470$	0	0
$471$	−414607.	−0.0861161
$472$	0	0
$473$	531249.	0.109181
$474$	0	0
$475$	689113.	0.140138
$476$	0	0
$477$	2.03921e6	0.410360
$478$	0	0
$479$	−1.13020e6	−0.225069	−0.112534	−	0.993648i	$-0.535897\pi$
$479$	−1.13020e6	−0.225069	−0.112534	+	0.993648i	$0.535897\pi$
$480$	0	0
$481$	−4.54541e6	−0.895798
$482$	0	0
$483$	1.01567e6	0.198100
$484$	0	0
$485$	−865238.	−0.167025
$486$	0	0
$487$	793366.	0.151583	0.0757916	−	0.997124i	$-0.475852\pi$
$487$	793366.	0.151583	0.0757916	+	0.997124i	$0.475852\pi$
$488$	0	0
$489$	1.69540e6	0.320626
$490$	0	0
$491$	3.66618e6	0.686294	0.343147	−	0.939282i	$-0.388507\pi$
$491$	3.66618e6	0.686294	0.343147	+	0.939282i	$0.388507\pi$
$492$	0	0
$493$	8.00805e6	1.48392
$494$	0	0
$495$	−605829.	−0.111132
$496$	0	0
$497$	−7.61267e6	−1.38244
$498$	0	0
$499$	5.08658e6	0.914480	0.457240	−	0.889343i	$-0.348838\pi$
$499$	5.08658e6	0.914480	0.457240	+	0.889343i	$0.348838\pi$
$500$	0	0
$501$	1.82893e6	0.325538
$502$	0	0
$503$	−6.49233e6	−1.14414	−0.572072	−	0.820203i	$-0.693860\pi$
$503$	−6.49233e6	−1.14414	−0.572072	+	0.820203i	$0.693860\pi$
$504$	0	0
$505$	−2.22816e6	−0.388793
$506$	0	0
$507$	−3.70460e6	−0.640061
$508$	0	0
$509$	−5.58035e6	−0.954700	−0.477350	−	0.878713i	$-0.658403\pi$
$509$	−5.58035e6	−0.954700	−0.477350	+	0.878713i	$0.658403\pi$
$510$	0	0
$511$	7.23991e6	1.22654
$512$	0	0
$513$	−1.36045e6	−0.228239
$514$	0	0
$515$	−7.40393e6	−1.23011
$516$	0	0
$517$	2.91102e6	0.478981
$518$	0	0
$519$	−1.42246e6	−0.231804
$520$	0	0
$521$	−4.02579e6	−0.649766	−0.324883	−	0.945754i	$-0.605325\pi$
$521$	−4.02579e6	−0.649766	−0.324883	+	0.945754i	$0.605325\pi$
$522$	0	0
$523$	9.41059e6	1.50440	0.752198	−	0.658937i	$-0.228993\pi$
$523$	9.41059e6	1.50440	0.752198	+	0.658937i	$0.228993\pi$
$524$	0	0
$525$	−708978.	−0.112262
$526$	0	0
$527$	−6.07256e6	−0.952457
$528$	0	0
$529$	279841.	0.0434783
$530$	0	0
$531$	3.00657e6	0.462738
$532$	0	0
$533$	−1.10450e7	−1.68402
$534$	0	0
$535$	−1.65531e6	−0.250031
$536$	0	0
$537$	1.49800e6	0.224169
$538$	0	0
$539$	−3.63178e6	−0.538453
$540$	0	0
$541$	−2.68147e6	−0.393894	−0.196947	−	0.980414i	$-0.563103\pi$
$541$	−2.68147e6	−0.393894	−0.196947	+	0.980414i	$0.563103\pi$
$542$	0	0
$543$	−1.67058e6	−0.243146
$544$	0	0
$545$	−3.79214e6	−0.546881
$546$	0	0
$547$	−8.74676e6	−1.24991	−0.624955	−	0.780660i	$-0.714883\pi$
$547$	−8.74676e6	−1.24991	−0.624955	+	0.780660i	$0.714883\pi$
$548$	0	0
$549$	2.13924e6	0.302920
$550$	0	0
$551$	−1.26651e7	−1.77717
$552$	0	0
$553$	1.63467e7	2.27309
$554$	0	0
$555$	−2.73298e6	−0.376620
$556$	0	0
$557$	2.07905e6	0.283940	0.141970	−	0.989871i	$-0.454656\pi$
$557$	2.07905e6	0.283940	0.141970	+	0.989871i	$0.454656\pi$
$558$	0	0
$559$	3.71508e6	0.502850
$560$	0	0
$561$	−1.34370e6	−0.180259
$562$	0	0
$563$	−8.52157e6	−1.13305	−0.566524	−	0.824045i	$-0.691712\pi$
$563$	−8.52157e6	−1.13305	−0.566524	+	0.824045i	$0.691712\pi$
$564$	0	0
$565$	−1.49121e6	−0.196524
$566$	0	0
$567$	1.39967e6	0.182838
$568$	0	0
$569$	6.58526e6	0.852692	0.426346	−	0.904560i	$-0.359801\pi$
$569$	6.58526e6	0.852692	0.426346	+	0.904560i	$0.359801\pi$
$570$	0	0
$571$	6.31614e6	0.810702	0.405351	−	0.914161i	$-0.367149\pi$
$571$	6.31614e6	0.810702	0.405351	+	0.914161i	$0.367149\pi$
$572$	0	0
$573$	−6.92760e6	−0.881447
$574$	0	0
$575$	−195340.	−0.0246389
$576$	0	0
$577$	4.48677e6	0.561040	0.280520	−	0.959848i	$-0.409493\pi$
$577$	4.48677e6	0.561040	0.280520	+	0.959848i	$0.409493\pi$
$578$	0	0
$579$	3.45730e6	0.428589
$580$	0	0
$581$	2.06136e7	2.53346
$582$	0	0
$583$	−3.18540e6	−0.388144
$584$	0	0
$585$	−4.23663e6	−0.511836
$586$	0	0
$587$	7.42328e6	0.889203	0.444601	−	0.895729i	$-0.353345\pi$
$587$	7.42328e6	0.889203	0.444601	+	0.895729i	$0.353345\pi$
$588$	0	0
$589$	9.60402e6	1.14068
$590$	0	0
$591$	−2.39443e6	−0.281990
$592$	0	0
$593$	7.02182e6	0.819998	0.409999	−	0.912086i	$-0.365529\pi$
$593$	7.02182e6	0.819998	0.409999	+	0.912086i	$0.365529\pi$
$594$	0	0
$595$	−1.48801e7	−1.72311
$596$	0	0
$597$	−6.35463e6	−0.729717
$598$	0	0
$599$	−5.02165e6	−0.571846	−0.285923	−	0.958253i	$-0.592300\pi$
$599$	−5.02165e6	−0.571846	−0.285923	+	0.958253i	$0.592300\pi$
$600$	0	0
$601$	4.66921e6	0.527300	0.263650	−	0.964618i	$-0.415074\pi$
$601$	4.66921e6	0.527300	0.263650	+	0.964618i	$0.415074\pi$
$602$	0	0
$603$	4.40628e6	0.493491
$604$	0	0
$605$	−8.57374e6	−0.952318
$606$	0	0
$607$	1.48942e7	1.64076	0.820380	−	0.571819i	$-0.193762\pi$
$607$	1.48942e7	1.64076	0.820380	+	0.571819i	$0.193762\pi$
$608$	0	0
$609$	1.30302e7	1.42366
$610$	0	0
$611$	2.03570e7	2.20603
$612$	0	0
$613$	1.09408e7	1.17598	0.587989	−	0.808869i	$-0.299920\pi$
$613$	1.09408e7	1.17598	0.587989	+	0.808869i	$0.299920\pi$
$614$	0	0
$615$	−6.64093e6	−0.708013
$616$	0	0
$617$	−6.42199e6	−0.679136	−0.339568	−	0.940582i	$-0.610281\pi$
$617$	−6.42199e6	−0.679136	−0.339568	+	0.940582i	$0.610281\pi$
$618$	0	0
$619$	3.05345e6	0.320305	0.160152	−	0.987092i	$-0.448801\pi$
$619$	3.05345e6	0.320305	0.160152	+	0.987092i	$0.448801\pi$
$620$	0	0
$621$	385641.	0.0401286
$622$	0	0
$623$	6.41719e6	0.662406
$624$	0	0
$625$	−1.07832e7	−1.10420
$626$	0	0
$627$	2.12513e6	0.215882
$628$	0	0
$629$	−6.06162e6	−0.610889
$630$	0	0
$631$	−1.05012e7	−1.04994	−0.524971	−	0.851120i	$-0.675924\pi$
$631$	−1.05012e7	−1.04994	−0.524971	+	0.851120i	$0.675924\pi$
$632$	0	0
$633$	−2.61572e6	−0.259467
$634$	0	0
$635$	−2.11527e6	−0.208176
$636$	0	0
$637$	−2.53974e7	−2.47994
$638$	0	0
$639$	−2.89046e6	−0.280037
$640$	0	0
$641$	−1.57928e7	−1.51815	−0.759073	−	0.651006i	$-0.774347\pi$
$641$	−1.57928e7	−1.51815	−0.759073	+	0.651006i	$0.774347\pi$
$642$	0	0
$643$	−551159.	−0.0525714	−0.0262857	−	0.999654i	$-0.508368\pi$
$643$	−551159.	−0.0525714	−0.0262857	+	0.999654i	$0.508368\pi$
$644$	0	0
$645$	2.23373e6	0.211413
$646$	0	0
$647$	−1.12498e7	−1.05653	−0.528267	−	0.849078i	$-0.677158\pi$
$647$	−1.12498e7	−1.05653	−0.528267	+	0.849078i	$0.677158\pi$
$648$	0	0
$649$	−4.69649e6	−0.437685
$650$	0	0
$651$	−9.88088e6	−0.913783
$652$	0	0
$653$	−2.56046e6	−0.234982	−0.117491	−	0.993074i	$-0.537485\pi$
$653$	−2.56046e6	−0.234982	−0.117491	+	0.993074i	$0.537485\pi$
$654$	0	0
$655$	−4.61783e6	−0.420566
$656$	0	0
$657$	2.74893e6	0.248456
$658$	0	0
$659$	734143.	0.0658518	0.0329259	−	0.999458i	$-0.489517\pi$
$659$	734143.	0.0658518	0.0329259	+	0.999458i	$0.489517\pi$
$660$	0	0
$661$	1.92354e7	1.71237	0.856185	−	0.516670i	$-0.172828\pi$
$661$	1.92354e7	1.71237	0.856185	+	0.516670i	$0.172828\pi$
$662$	0	0
$663$	−9.39666e6	−0.830213
$664$	0	0
$665$	2.35336e7	2.06364
$666$	0	0
$667$	3.59012e6	0.312460
$668$	0	0
$669$	−5.17423e6	−0.446972
$670$	0	0
$671$	−3.34166e6	−0.286520
$672$	0	0
$673$	1.20397e7	1.02465	0.512326	−	0.858791i	$-0.328784\pi$
$673$	1.20397e7	1.02465	0.512326	+	0.858791i	$0.328784\pi$
$674$	0	0
$675$	−269192.	−0.0227407
$676$	0	0
$677$	99259.2	0.00832337	0.00416168	−	0.999991i	$-0.498675\pi$
$677$	99259.2	0.00832337	0.00416168	+	0.999991i	$0.498675\pi$
$678$	0	0
$679$	−3.12257e6	−0.259919
$680$	0	0
$681$	−1.23626e7	−1.02151
$682$	0	0
$683$	−9.22059e6	−0.756323	−0.378161	−	0.925740i	$-0.623444\pi$
$683$	−9.22059e6	−0.756323	−0.378161	+	0.925740i	$0.623444\pi$
$684$	0	0
$685$	−5.48661e6	−0.446763
$686$	0	0
$687$	−7.83652e6	−0.633478
$688$	0	0
$689$	−2.22758e7	−1.78766
$690$	0	0
$691$	1.77887e7	1.41726	0.708629	−	0.705581i	$-0.249314\pi$
$691$	1.77887e7	1.41726	0.708629	+	0.705581i	$0.249314\pi$
$692$	0	0
$693$	−2.18639e6	−0.172940
$694$	0	0
$695$	2.46568e7	1.93631
$696$	0	0
$697$	−1.47293e7	−1.14842
$698$	0	0
$699$	1.45558e7	1.12679
$700$	0	0
$701$	1.60390e7	1.23277	0.616384	−	0.787446i	$-0.288597\pi$
$701$	1.60390e7	1.23277	0.616384	+	0.787446i	$0.288597\pi$
$702$	0	0
$703$	9.58673e6	0.731614
$704$	0	0
$705$	1.22399e7	0.927480
$706$	0	0
$707$	−8.04125e6	−0.605028
$708$	0	0
$709$	−1.64673e7	−1.23029	−0.615145	−	0.788414i	$-0.710903\pi$
$709$	−1.64673e7	−1.23029	−0.615145	+	0.788414i	$0.710903\pi$
$710$	0	0
$711$	6.20669e6	0.460454
$712$	0	0
$713$	−2.72241e6	−0.200553
$714$	0	0
$715$	6.61794e6	0.484125
$716$	0	0
$717$	−1.48753e7	−1.08061
$718$	0	0
$719$	−5.48538e6	−0.395717	−0.197859	−	0.980231i	$-0.563399\pi$
$719$	−5.48538e6	−0.395717	−0.197859	+	0.980231i	$0.563399\pi$
$720$	0	0
$721$	−2.67202e7	−1.91426
$722$	0	0
$723$	−8.35619e6	−0.594515
$724$	0	0
$725$	−2.50604e6	−0.177070
$726$	0	0
$727$	2.75484e7	1.93313	0.966563	−	0.256431i	$-0.0825465\pi$
$727$	2.75484e7	1.93313	0.966563	+	0.256431i	$0.0825465\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	4.95432e6	0.342918
$732$	0	0
$733$	1.51233e7	1.03965	0.519825	−	0.854273i	$-0.325997\pi$
$733$	1.51233e7	1.03965	0.519825	+	0.854273i	$0.325997\pi$
$734$	0	0
$735$	−1.52705e7	−1.04264
$736$	0	0
$737$	−6.88295e6	−0.466773
$738$	0	0
$739$	1.56004e7	1.05081	0.525405	−	0.850852i	$-0.323914\pi$
$739$	1.56004e7	1.05081	0.525405	+	0.850852i	$0.323914\pi$
$740$	0	0
$741$	1.48612e7	0.994281
$742$	0	0
$743$	−1.77279e6	−0.117811	−0.0589053	−	0.998264i	$-0.518761\pi$
$743$	−1.77279e6	−0.117811	−0.0589053	+	0.998264i	$0.518761\pi$
$744$	0	0
$745$	2.63603e7	1.74004
$746$	0	0
$747$	7.82681e6	0.513196
$748$	0	0
$749$	−5.97386e6	−0.389090
$750$	0	0
$751$	−5.07089e6	−0.328083	−0.164042	−	0.986453i	$-0.552453\pi$
$751$	−5.07089e6	−0.328083	−0.164042	+	0.986453i	$0.552453\pi$
$752$	0	0
$753$	−8.18724e6	−0.526199
$754$	0	0
$755$	1.53962e7	0.982987
$756$	0	0
$757$	−733561.	−0.0465261	−0.0232631	−	0.999729i	$-0.507406\pi$
$757$	−733561.	−0.0465261	−0.0232631	+	0.999729i	$0.507406\pi$
$758$	0	0
$759$	−602401.	−0.0379561
$760$	0	0
$761$	2.90077e6	0.181573	0.0907866	−	0.995870i	$-0.471062\pi$
$761$	2.90077e6	0.181573	0.0907866	+	0.995870i	$0.471062\pi$
$762$	0	0
$763$	−1.36855e7	−0.851039
$764$	0	0
$765$	−5.64984e6	−0.349046
$766$	0	0
$767$	−3.28431e7	−2.01584
$768$	0	0
$769$	−6.02882e6	−0.367635	−0.183817	−	0.982960i	$-0.558846\pi$
$769$	−6.02882e6	−0.367635	−0.183817	+	0.982960i	$0.558846\pi$
$770$	0	0
$771$	−474001.	−0.0287173
$772$	0	0
$773$	1.48827e7	0.895844	0.447922	−	0.894073i	$-0.352164\pi$
$773$	1.48827e7	0.895844	0.447922	+	0.894073i	$0.352164\pi$
$774$	0	0
$775$	1.90035e6	0.113653
$776$	0	0
$777$	−9.86309e6	−0.586085
$778$	0	0
$779$	2.32951e7	1.37537
$780$	0	0
$781$	4.51512e6	0.264876
$782$	0	0
$783$	4.94745e6	0.288388
$784$	0	0
$785$	2.72315e6	0.157724
$786$	0	0
$787$	−2.19846e7	−1.26526	−0.632631	−	0.774453i	$-0.718025\pi$
$787$	−2.19846e7	−1.26526	−0.632631	+	0.774453i	$0.718025\pi$
$788$	0	0
$789$	4.18288e6	0.239212
$790$	0	0
$791$	−5.38164e6	−0.305825
$792$	0	0
$793$	−2.33685e7	−1.31962
$794$	0	0
$795$	−1.33936e7	−0.751586
$796$	0	0
$797$	−2.92578e6	−0.163153	−0.0815765	−	0.996667i	$-0.525995\pi$
$797$	−2.92578e6	−0.163153	−0.0815765	+	0.996667i	$0.525995\pi$
$798$	0	0
$799$	2.71476e7	1.50440
$800$	0	0
$801$	2.43655e6	0.134182
$802$	0	0
$803$	−4.29404e6	−0.235005
$804$	0	0
$805$	−6.67096e6	−0.362826
$806$	0	0
$807$	−8.71578e6	−0.471110
$808$	0	0
$809$	−2.36363e6	−0.126972	−0.0634860	−	0.997983i	$-0.520222\pi$
$809$	−2.36363e6	−0.126972	−0.0634860	+	0.997983i	$0.520222\pi$
$810$	0	0
$811$	1.40226e7	0.748648	0.374324	−	0.927298i	$-0.377875\pi$
$811$	1.40226e7	0.748648	0.374324	+	0.927298i	$0.377875\pi$
$812$	0	0
$813$	2.06335e7	1.09483
$814$	0	0
$815$	−1.11354e7	−0.587236
$816$	0	0
$817$	−7.83548e6	−0.410686
$818$	0	0
$819$	−1.52896e7	−0.796503
$820$	0	0
$821$	3.57170e7	1.84934	0.924670	−	0.380769i	$-0.124341\pi$
$821$	3.57170e7	1.84934	0.924670	+	0.380769i	$0.124341\pi$
$822$	0	0
$823$	1.15100e7	0.592344	0.296172	−	0.955135i	$-0.404290\pi$
$823$	1.15100e7	0.592344	0.296172	+	0.955135i	$0.404290\pi$
$824$	0	0
$825$	420499.	0.0215095
$826$	0	0
$827$	1.24864e7	0.634854	0.317427	−	0.948283i	$-0.397181\pi$
$827$	1.24864e7	0.634854	0.317427	+	0.948283i	$0.397181\pi$
$828$	0	0
$829$	−1.13398e7	−0.573087	−0.286544	−	0.958067i	$-0.592506\pi$
$829$	−1.13398e7	−0.573087	−0.286544	+	0.958067i	$0.592506\pi$
$830$	0	0
$831$	−2.61858e6	−0.131542
$832$	0	0
$833$	−3.38692e7	−1.69119
$834$	0	0
$835$	−1.20125e7	−0.596232
$836$	0	0
$837$	−3.75168e6	−0.185102
$838$	0	0
$839$	−2.91430e7	−1.42932	−0.714659	−	0.699473i	$-0.753418\pi$
$839$	−2.91430e7	−1.42932	−0.714659	+	0.699473i	$0.753418\pi$
$840$	0	0
$841$	2.55471e7	1.24552
$842$	0	0
$843$	3.52780e6	0.170976
$844$	0	0
$845$	2.43319e7	1.17229
$846$	0	0
$847$	−3.09419e7	−1.48197
$848$	0	0
$849$	6.59491e6	0.314007
$850$	0	0
$851$	−2.71751e6	−0.128632
$852$	0	0
$853$	−9.77420e6	−0.459948	−0.229974	−	0.973197i	$-0.573864\pi$
$853$	−9.77420e6	−0.459948	−0.229974	+	0.973197i	$0.573864\pi$
$854$	0	0
$855$	8.93548e6	0.418025
$856$	0	0
$857$	−2.33262e7	−1.08491	−0.542453	−	0.840086i	$-0.682504\pi$
$857$	−2.33262e7	−1.08491	−0.542453	+	0.840086i	$0.682504\pi$
$858$	0	0
$859$	1.14340e7	0.528708	0.264354	−	0.964426i	$-0.414841\pi$
$859$	1.14340e7	0.528708	0.264354	+	0.964426i	$0.414841\pi$
$860$	0	0
$861$	−2.39666e7	−1.10179
$862$	0	0
$863$	7.70458e6	0.352145	0.176073	−	0.984377i	$-0.443661\pi$
$863$	7.70458e6	0.352145	0.176073	+	0.984377i	$0.443661\pi$
$864$	0	0
$865$	9.34277e6	0.424556
$866$	0	0
$867$	247605.	0.0111870
$868$	0	0
$869$	−9.69534e6	−0.435525
$870$	0	0
$871$	−4.81332e7	−2.14981
$872$	0	0
$873$	−1.18561e6	−0.0526511
$874$	0	0
$875$	−3.47513e7	−1.53444
$876$	0	0
$877$	1.42476e7	0.625520	0.312760	−	0.949832i	$-0.398746\pi$
$877$	1.42476e7	0.625520	0.312760	+	0.949832i	$0.398746\pi$
$878$	0	0
$879$	−1.46328e7	−0.638788
$880$	0	0
$881$	−1.76114e7	−0.764459	−0.382230	−	0.924067i	$-0.624844\pi$
$881$	−1.76114e7	−0.764459	−0.382230	+	0.924067i	$0.624844\pi$
$882$	0	0
$883$	9.40461e6	0.405919	0.202959	−	0.979187i	$-0.434944\pi$
$883$	9.40461e6	0.405919	0.202959	+	0.979187i	$0.434944\pi$
$884$	0	0
$885$	−1.97472e7	−0.847517
$886$	0	0
$887$	−1.38034e7	−0.589086	−0.294543	−	0.955638i	$-0.595167\pi$
$887$	−1.38034e7	−0.589086	−0.294543	+	0.955638i	$0.595167\pi$
$888$	0	0
$889$	−7.63383e6	−0.323958
$890$	0	0
$891$	−830152.	−0.0350319
$892$	0	0
$893$	−4.29351e7	−1.80170
$894$	0	0
$895$	−9.83888e6	−0.410571
$896$	0	0
$897$	−4.21265e6	−0.174813
$898$	0	0
$899$	−3.49262e7	−1.44129
$900$	0	0
$901$	−2.97064e7	−1.21910
$902$	0	0
$903$	8.06135e6	0.328994
$904$	0	0
$905$	1.09724e7	0.445328
$906$	0	0
$907$	−802157.	−0.0323773	−0.0161887	−	0.999869i	$-0.505153\pi$
$907$	−802157.	−0.0323773	−0.0161887	+	0.999869i	$0.505153\pi$
$908$	0	0
$909$	−3.05319e6	−0.122559
$910$	0	0
$911$	−4.80457e6	−0.191804	−0.0959022	−	0.995391i	$-0.530574\pi$
$911$	−4.80457e6	−0.191804	−0.0959022	+	0.995391i	$0.530574\pi$
$912$	0	0
$913$	−1.22261e7	−0.485412
$914$	0	0
$915$	−1.40506e7	−0.554807
$916$	0	0
$917$	−1.66654e7	−0.654473
$918$	0	0
$919$	−2.84366e7	−1.11068	−0.555339	−	0.831624i	$-0.687412\pi$
$919$	−2.84366e7	−1.11068	−0.555339	+	0.831624i	$0.687412\pi$
$920$	0	0
$921$	1.12123e7	0.435556
$922$	0	0
$923$	3.15747e7	1.21993
$924$	0	0
$925$	1.89693e6	0.0728948
$926$	0	0
$927$	−1.01454e7	−0.387766
$928$	0	0
$929$	5.06040e6	0.192374	0.0961868	−	0.995363i	$-0.469335\pi$
$929$	5.06040e6	0.192374	0.0961868	+	0.995363i	$0.469335\pi$
$930$	0	0
$931$	5.35657e7	2.02541
$932$	0	0
$933$	−2.94166e7	−1.10634
$934$	0	0
$935$	8.82549e6	0.330149
$936$	0	0
$937$	1.27197e7	0.473291	0.236646	−	0.971596i	$-0.423952\pi$
$937$	1.27197e7	0.473291	0.236646	+	0.971596i	$0.423952\pi$
$938$	0	0
$939$	−1.53145e7	−0.566810
$940$	0	0
$941$	−1.66122e7	−0.611581	−0.305790	−	0.952099i	$-0.598921\pi$
$941$	−1.66122e7	−0.611581	−0.305790	+	0.952099i	$0.598921\pi$
$942$	0	0
$943$	−6.60335e6	−0.241816
$944$	0	0
$945$	−9.19306e6	−0.334873
$946$	0	0
$947$	−4.77751e7	−1.73112	−0.865558	−	0.500808i	$-0.833036\pi$
$947$	−4.77751e7	−1.73112	−0.865558	+	0.500808i	$0.833036\pi$
$948$	0	0
$949$	−3.00286e7	−1.08236
$950$	0	0
$951$	8.17485e6	0.293109
$952$	0	0
$953$	−3.77139e7	−1.34514	−0.672572	−	0.740032i	$-0.734810\pi$
$953$	−3.77139e7	−1.34514	−0.672572	+	0.740032i	$0.734810\pi$
$954$	0	0
$955$	4.55007e7	1.61439
$956$	0	0
$957$	−7.72829e6	−0.272775
$958$	0	0
$959$	−1.98007e7	−0.695239
$960$	0	0
$961$	−2.14436e6	−0.0749014
$962$	0	0
$963$	−2.26822e6	−0.0788169
$964$	0	0
$965$	−2.27077e7	−0.784973
$966$	0	0
$967$	1.36407e7	0.469105	0.234553	−	0.972103i	$-0.424638\pi$
$967$	1.36407e7	0.469105	0.234553	+	0.972103i	$0.424638\pi$
$968$	0	0
$969$	1.98185e7	0.678050
$970$	0	0
$971$	3.96602e7	1.34992	0.674958	−	0.737856i	$-0.264162\pi$
$971$	3.96602e7	1.34992	0.674958	+	0.737856i	$0.264162\pi$
$972$	0	0
$973$	8.89843e7	3.01322
$974$	0	0
$975$	2.94059e6	0.0990658
$976$	0	0
$977$	−9.70600e6	−0.325315	−0.162657	−	0.986683i	$-0.552007\pi$
$977$	−9.70600e6	−0.325315	−0.162657	+	0.986683i	$0.552007\pi$
$978$	0	0
$979$	−3.80608e6	−0.126917
$980$	0	0
$981$	−5.19626e6	−0.172393
$982$	0	0
$983$	2.88398e7	0.951937	0.475969	−	0.879462i	$-0.342098\pi$
$983$	2.88398e7	0.951937	0.475969	+	0.879462i	$0.342098\pi$
$984$	0	0
$985$	1.57267e7	0.516473
$986$	0	0
$987$	4.41728e7	1.44332
$988$	0	0
$989$	2.22109e6	0.0722064
$990$	0	0
$991$	−3.94595e7	−1.27634	−0.638172	−	0.769894i	$-0.720309\pi$
$991$	−3.94595e7	−1.27634	−0.638172	+	0.769894i	$0.720309\pi$
$992$	0	0
$993$	−1.92910e7	−0.620844
$994$	0	0
$995$	4.17374e7	1.33650
$996$	0	0
$997$	3.61342e7	1.15128	0.575639	−	0.817704i	$-0.304753\pi$
$997$	3.61342e7	1.15128	0.575639	+	0.817704i	$0.304753\pi$
$998$	0	0
$999$	−3.74492e6	−0.118722

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1104.6.a.i.1.3		3
4.3	odd	2		69.6.a.b.1.2	✓	3
12.11	even	2		207.6.a.c.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.6.a.b.1.2	✓	3	4.3	odd	2
207.6.a.c.1.2		3	12.11	even	2
1104.6.a.i.1.3		3	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)

\(f(q)\)	\(=\)	\(q-9.00000 q^{3} +59.1123 q^{5} +213.331 q^{7} +81.0000 q^{9} +O(q^{10})\)	\(q-9.00000 q^{3} +59.1123 q^{5} +213.331 q^{7} +81.0000 q^{9} -126.528 q^{11} -884.825 q^{13} -532.011 q^{15} -1179.98 q^{17} +1866.19 q^{19} -1919.98 q^{21} -529.000 q^{23} +369.263 q^{25} -729.000 q^{27} -6786.62 q^{29} +5146.34 q^{31} +1138.75 q^{33} +12610.5 q^{35} +5137.07 q^{37} +7963.42 q^{39} +12482.7 q^{41} -4198.66 q^{43} +4788.10 q^{45} -23006.9 q^{47} +28703.3 q^{49} +10619.8 q^{51} +25175.4 q^{53} -7479.38 q^{55} -16795.7 q^{57} +37118.1 q^{59} +26410.4 q^{61} +17279.8 q^{63} -52304.0 q^{65} +54398.5 q^{67} +4761.00 q^{69} -35684.7 q^{71} +33937.4 q^{73} -3323.36 q^{75} -26992.5 q^{77} +76625.8 q^{79} +6561.00 q^{81} +96627.2 q^{83} -69751.1 q^{85} +61079.6 q^{87} +30080.8 q^{89} -188761. q^{91} -46317.0 q^{93} +110315. q^{95} -14637.2 q^{97} -10248.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 27 q^{3} - 56 q^{5} + 114 q^{7} + 243 q^{9}+O(q^{10}) \) 3 * q - 27 * q^3 - 56 * q^5 + 114 * q^7 + 243 * q^9	\( 3 q - 27 q^{3} - 56 q^{5} + 114 q^{7} + 243 q^{9} + 376 q^{11} - 858 q^{13} + 504 q^{15} - 2548 q^{17} + 2846 q^{19} - 1026 q^{21} - 1587 q^{23} + 753 q^{25} - 2187 q^{27} - 16370 q^{29} + 14756 q^{31} - 3384 q^{33} + 18520 q^{35} + 15874 q^{37} + 7722 q^{39} + 12606 q^{41} - 3154 q^{43} - 4536 q^{45} - 29928 q^{47} + 4471 q^{49} + 22932 q^{51} - 44084 q^{53} - 38360 q^{55} - 25614 q^{57} + 29300 q^{59} + 54010 q^{61} + 9234 q^{63} - 51216 q^{65} - 43390 q^{67} + 14283 q^{69} - 23424 q^{71} - 91402 q^{73} - 6777 q^{75} - 97208 q^{77} + 49398 q^{79} + 19683 q^{81} + 103936 q^{83} + 5888 q^{85} + 147330 q^{87} + 96112 q^{89} - 129228 q^{91} - 132804 q^{93} + 55928 q^{95} - 135318 q^{97} + 30456 q^{99}+O(q^{100}) \) 3 * q - 27 * q^3 - 56 * q^5 + 114 * q^7 + 243 * q^9 + 376 * q^11 - 858 * q^13 + 504 * q^15 - 2548 * q^17 + 2846 * q^19 - 1026 * q^21 - 1587 * q^23 + 753 * q^25 - 2187 * q^27 - 16370 * q^29 + 14756 * q^31 - 3384 * q^33 + 18520 * q^35 + 15874 * q^37 + 7722 * q^39 + 12606 * q^41 - 3154 * q^43 - 4536 * q^45 - 29928 * q^47 + 4471 * q^49 + 22932 * q^51 - 44084 * q^53 - 38360 * q^55 - 25614 * q^57 + 29300 * q^59 + 54010 * q^61 + 9234 * q^63 - 51216 * q^65 - 43390 * q^67 + 14283 * q^69 - 23424 * q^71 - 91402 * q^73 - 6777 * q^75 - 97208 * q^77 + 49398 * q^79 + 19683 * q^81 + 103936 * q^83 + 5888 * q^85 + 147330 * q^87 + 96112 * q^89 - 129228 * q^91 - 132804 * q^93 + 55928 * q^95 - 135318 * q^97 + 30456 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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