LMFDB - Embedded newform 891.6.a.f.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [891,6,Mod(1,891)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(891, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("891.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 891 = 3^{4} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	891.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:	$142.901983453$
Analytic rank:	$1$
Dimension:	$23$
Twist minimal:	no (minimal twist has level 99)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Character	$\chi$	$=$	891.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.38380 q^{2} -26.3175 q^{4} +81.0633 q^{5} +5.57865 q^{7} +139.017 q^{8} +O(q^{10})$	\(q-2.38380 q^{2} -26.3175 q^{4} +81.0633 q^{5} +5.57865 q^{7} +139.017 q^{8} -193.239 q^{10} -121.000 q^{11} -265.146 q^{13} -13.2984 q^{14} +510.770 q^{16} +1713.72 q^{17} +487.657 q^{19} -2133.38 q^{20} +288.440 q^{22} -4673.54 q^{23} +3446.27 q^{25} +632.057 q^{26} -146.816 q^{28} -7834.50 q^{29} +872.241 q^{31} -5666.13 q^{32} -4085.16 q^{34} +452.224 q^{35} +8791.31 q^{37} -1162.48 q^{38} +11269.2 q^{40} +16877.0 q^{41} -10911.3 q^{43} +3184.42 q^{44} +11140.8 q^{46} -2434.74 q^{47} -16775.9 q^{49} -8215.21 q^{50} +6977.99 q^{52} -19335.5 q^{53} -9808.66 q^{55} +775.530 q^{56} +18675.9 q^{58} -36998.5 q^{59} -15036.4 q^{61} -2079.25 q^{62} -2837.71 q^{64} -21493.7 q^{65} +18289.2 q^{67} -45100.7 q^{68} -1078.01 q^{70} +51557.8 q^{71} -36311.6 q^{73} -20956.7 q^{74} -12833.9 q^{76} -675.017 q^{77} +68660.1 q^{79} +41404.7 q^{80} -40231.5 q^{82} -26254.7 q^{83} +138920. q^{85} +26010.4 q^{86} -16821.1 q^{88} -29798.6 q^{89} -1479.16 q^{91} +122996. q^{92} +5803.93 q^{94} +39531.1 q^{95} -74446.9 q^{97} +39990.4 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q + 320 q^{4} + 36 q^{5} - 167 q^{7} + 213 q^{8}+O(q^{10}) $ 23 * q + 320 * q^4 + 36 * q^5 - 167 * q^7 + 213 * q^8	$ 23 q + 320 q^{4} + 36 q^{5} - 167 q^{7} + 213 q^{8} - 600 q^{10} - 2783 q^{11} - 1871 q^{13} + 1329 q^{14} + 3584 q^{16} + 267 q^{17} - 3641 q^{19} + 1917 q^{20} + 8292 q^{23} + 10049 q^{25} + 9570 q^{26} + 3793 q^{28} + 5970 q^{29} - 9542 q^{31} + 3831 q^{32} - 2982 q^{34} + 3240 q^{35} - 16007 q^{37} - 1221 q^{38} - 40635 q^{40} - 12030 q^{41} - 25943 q^{43} - 38720 q^{44} - 77004 q^{46} - 9756 q^{47} + 6990 q^{49} - 101805 q^{50} - 144446 q^{52} + 53919 q^{53} - 4356 q^{55} + 16602 q^{56} - 95367 q^{58} + 20310 q^{59} - 100247 q^{61} - 15297 q^{62} - 84577 q^{64} - 20931 q^{65} - 84956 q^{67} - 168471 q^{68} - 212292 q^{70} + 36093 q^{71} - 173444 q^{73} - 86619 q^{74} - 340334 q^{76} + 20207 q^{77} - 123113 q^{79} - 15123 q^{80} - 199983 q^{82} - 30672 q^{83} - 268335 q^{85} + 211260 q^{86} - 25773 q^{88} + 32514 q^{89} - 328021 q^{91} + 196731 q^{92} - 230262 q^{94} + 325926 q^{95} - 357002 q^{97} - 214464 q^{98}+O(q^{100}) $ 23 * q + 320 * q^4 + 36 * q^5 - 167 * q^7 + 213 * q^8 - 600 * q^10 - 2783 * q^11 - 1871 * q^13 + 1329 * q^14 + 3584 * q^16 + 267 * q^17 - 3641 * q^19 + 1917 * q^20 + 8292 * q^23 + 10049 * q^25 + 9570 * q^26 + 3793 * q^28 + 5970 * q^29 - 9542 * q^31 + 3831 * q^32 - 2982 * q^34 + 3240 * q^35 - 16007 * q^37 - 1221 * q^38 - 40635 * q^40 - 12030 * q^41 - 25943 * q^43 - 38720 * q^44 - 77004 * q^46 - 9756 * q^47 + 6990 * q^49 - 101805 * q^50 - 144446 * q^52 + 53919 * q^53 - 4356 * q^55 + 16602 * q^56 - 95367 * q^58 + 20310 * q^59 - 100247 * q^61 - 15297 * q^62 - 84577 * q^64 - 20931 * q^65 - 84956 * q^67 - 168471 * q^68 - 212292 * q^70 + 36093 * q^71 - 173444 * q^73 - 86619 * q^74 - 340334 * q^76 + 20207 * q^77 - 123113 * q^79 - 15123 * q^80 - 199983 * q^82 - 30672 * q^83 - 268335 * q^85 + 211260 * q^86 - 25773 * q^88 + 32514 * q^89 - 328021 * q^91 + 196731 * q^92 - 230262 * q^94 + 325926 * q^95 - 357002 * q^97 - 214464 * q^98

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 $

$2$	−2.38380	−0.421401	−0.210700	−	0.977551i	$-0.567574\pi$
$2$	−2.38380	−0.421401	−0.210700	+	0.977551i	$0.567574\pi$
$3$	0	0
$3$	0	0
$4$	−26.3175	−0.822422
$5$	81.0633	1.45011	0.725053	−	0.688693i	$-0.241815\pi$
$5$	81.0633	1.45011	0.725053	+	0.688693i	$0.241815\pi$
$6$	0	0
$7$	5.57865	0.0430313	0.0215156	−	0.999769i	$-0.493151\pi$
$7$	5.57865	0.0430313	0.0215156	+	0.999769i	$0.493151\pi$
$8$	139.017	0.767970
$9$	0	0
$10$	−193.239	−0.611075
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	0	0
$13$	−265.146	−0.435138	−0.217569	−	0.976045i	$-0.569813\pi$
$13$	−265.146	−0.435138	−0.217569	+	0.976045i	$0.569813\pi$
$14$	−13.2984	−0.0181334
$15$	0	0
$16$	510.770	0.498799
$17$	1713.72	1.43819	0.719096	−	0.694911i	$-0.244556\pi$
$17$	1713.72	1.43819	0.719096	+	0.694911i	$0.244556\pi$
$18$	0	0
$19$	487.657	0.309907	0.154953	−	0.987922i	$-0.450477\pi$
$19$	487.657	0.309907	0.154953	+	0.987922i	$0.450477\pi$
$20$	−2133.38	−1.19260
$21$	0	0
$22$	288.440	0.127057
$23$	−4673.54	−1.84216	−0.921079	−	0.389377i	$-0.872690\pi$
$23$	−4673.54	−1.84216	−0.921079	+	0.389377i	$0.872690\pi$
$24$	0	0
$25$	3446.27	1.10281
$26$	632.057	0.183368
$27$	0	0
$28$	−146.816	−0.0353899
$29$	−7834.50	−1.72988	−0.864940	−	0.501875i	$-0.832644\pi$
$29$	−7834.50	−1.72988	−0.864940	+	0.501875i	$0.832644\pi$
$30$	0	0
$31$	872.241	0.163017	0.0815083	−	0.996673i	$-0.474026\pi$
$31$	872.241	0.163017	0.0815083	+	0.996673i	$0.474026\pi$
$32$	−5666.13	−0.978164
$33$	0	0
$34$	−4085.16	−0.606055
$35$	452.224	0.0623999
$36$	0	0
$37$	8791.31	1.05572	0.527860	−	0.849331i	$-0.322994\pi$
$37$	8791.31	1.05572	0.527860	+	0.849331i	$0.322994\pi$
$38$	−1162.48	−0.130595
$39$	0	0
$40$	11269.2	1.11364
$41$	16877.0	1.56797	0.783983	−	0.620782i	$-0.213185\pi$
$41$	16877.0	1.56797	0.783983	+	0.620782i	$0.213185\pi$
$42$	0	0
$43$	−10911.3	−0.899924	−0.449962	−	0.893048i	$-0.648562\pi$
$43$	−10911.3	−0.899924	−0.449962	+	0.893048i	$0.648562\pi$
$44$	3184.42	0.247969
$45$	0	0
$46$	11140.8	0.776286
$47$	−2434.74	−0.160771	−0.0803854	−	0.996764i	$-0.525615\pi$
$47$	−2434.74	−0.160771	−0.0803854	+	0.996764i	$0.525615\pi$
$48$	0	0
$49$	−16775.9	−0.998148
$50$	−8215.21	−0.464723
$51$	0	0
$52$	6977.99	0.357867
$53$	−19335.5	−0.945511	−0.472755	−	0.881194i	$-0.656741\pi$
$53$	−19335.5	−0.945511	−0.472755	+	0.881194i	$0.656741\pi$
$54$	0	0
$55$	−9808.66	−0.437223
$56$	775.530	0.0330467
$57$	0	0
$58$	18675.9	0.728973
$59$	−36998.5	−1.38374	−0.691869	−	0.722023i	$-0.743212\pi$
$59$	−36998.5	−1.38374	−0.691869	+	0.722023i	$0.743212\pi$
$60$	0	0
$61$	−15036.4	−0.517392	−0.258696	−	0.965959i	$-0.583293\pi$
$61$	−15036.4	−0.517392	−0.258696	+	0.965959i	$0.583293\pi$
$62$	−2079.25	−0.0686953
$63$	0	0
$64$	−2837.71	−0.0865999
$65$	−21493.7	−0.630996
$66$	0	0
$67$	18289.2	0.497747	0.248874	−	0.968536i	$-0.419940\pi$
$67$	18289.2	0.497747	0.248874	+	0.968536i	$0.419940\pi$
$68$	−45100.7	−1.18280
$69$	0	0
$70$	−1078.01	−0.0262954
$71$	51557.8	1.21380	0.606902	−	0.794776i	$-0.292412\pi$
$71$	51557.8	1.21380	0.606902	+	0.794776i	$0.292412\pi$
$72$	0	0
$73$	−36311.6	−0.797513	−0.398757	−	0.917057i	$-0.630558\pi$
$73$	−36311.6	−0.797513	−0.398757	+	0.917057i	$0.630558\pi$
$74$	−20956.7	−0.444881
$75$	0	0
$76$	−12833.9	−0.254874
$77$	−675.017	−0.0129744
$78$	0	0
$79$	68660.1	1.23776	0.618880	−	0.785486i	$-0.287587\pi$
$79$	68660.1	1.23776	0.618880	+	0.785486i	$0.287587\pi$
$80$	41404.7	0.723311
$81$	0	0
$82$	−40231.5	−0.660742
$83$	−26254.7	−0.418323	−0.209162	−	0.977881i	$-0.567073\pi$
$83$	−26254.7	−0.418323	−0.209162	+	0.977881i	$0.567073\pi$
$84$	0	0
$85$	138920.	2.08553
$86$	26010.4	0.379228
$87$	0	0
$88$	−16821.1	−0.231552
$89$	−29798.6	−0.398769	−0.199384	−	0.979921i	$-0.563894\pi$
$89$	−29798.6	−0.398769	−0.199384	+	0.979921i	$0.563894\pi$
$90$	0	0
$91$	−1479.16	−0.0187246
$92$	122996.	1.51503
$93$	0	0
$94$	5803.93	0.0677489
$95$	39531.1	0.449397
$96$	0	0
$97$	−74446.9	−0.803373	−0.401687	−	0.915777i	$-0.631576\pi$
$97$	−74446.9	−0.803373	−0.401687	+	0.915777i	$0.631576\pi$
$98$	39990.4	0.420620
$99$	0	0
$100$	−90697.1	−0.906971
$101$	−74757.4	−0.729207	−0.364603	−	0.931163i	$-0.618795\pi$
$101$	−74757.4	−0.729207	−0.364603	+	0.931163i	$0.618795\pi$
$102$	0	0
$103$	128768.	1.19595	0.597977	−	0.801513i	$-0.295971\pi$
$103$	128768.	1.19595	0.597977	+	0.801513i	$0.295971\pi$
$104$	−36860.0	−0.334173
$105$	0	0
$106$	46092.1	0.398439
$107$	162558.	1.37262	0.686308	−	0.727311i	$-0.259230\pi$
$107$	162558.	1.37262	0.686308	+	0.727311i	$0.259230\pi$
$108$	0	0
$109$	−7701.16	−0.0620855	−0.0310427	−	0.999518i	$-0.509883\pi$
$109$	−7701.16	−0.0620855	−0.0310427	+	0.999518i	$0.509883\pi$
$110$	23381.9	0.184246
$111$	0	0
$112$	2849.41	0.0214640
$113$	109518.	0.806843	0.403421	−	0.915014i	$-0.367821\pi$
$113$	109518.	0.806843	0.403421	+	0.915014i	$0.367821\pi$
$114$	0	0
$115$	−378853.	−2.67132
$116$	206184.	1.42269
$117$	0	0
$118$	88197.0	0.583108
$119$	9560.24	0.0618873
$120$	0	0
$121$	14641.0	0.0909091
$122$	35843.9	0.218029
$123$	0	0
$124$	−22955.2	−0.134068
$125$	26042.9	0.149078
$126$	0	0
$127$	168458.	0.926793	0.463396	−	0.886151i	$-0.346631\pi$
$127$	168458.	0.926793	0.463396	+	0.886151i	$0.346631\pi$
$128$	188081.	1.01466
$129$	0	0
$130$	51236.6	0.265902
$131$	−7059.29	−0.0359404	−0.0179702	−	0.999839i	$-0.505720\pi$
$131$	−7059.29	−0.0359404	−0.0179702	+	0.999839i	$0.505720\pi$
$132$	0	0
$133$	2720.47	0.0133357
$134$	−43597.9	−0.209751
$135$	0	0
$136$	238236.	1.10449
$137$	79290.8	0.360929	0.180464	−	0.983582i	$-0.442240\pi$
$137$	79290.8	0.360929	0.180464	+	0.983582i	$0.442240\pi$
$138$	0	0
$139$	−239967.	−1.05345	−0.526725	−	0.850036i	$-0.676580\pi$
$139$	−239967.	−1.05345	−0.526725	+	0.850036i	$0.676580\pi$
$140$	−11901.4	−0.0513190
$141$	0	0
$142$	−122904.	−0.511498
$143$	32082.7	0.131199
$144$	0	0
$145$	−635090.	−2.50851
$146$	86559.6	0.336073
$147$	0	0
$148$	−231365.	−0.868247
$149$	−442733.	−1.63372	−0.816858	−	0.576839i	$-0.804286\pi$
$149$	−442733.	−1.63372	−0.816858	+	0.576839i	$0.804286\pi$
$150$	0	0
$151$	388333.	1.38599	0.692997	−	0.720940i	$-0.256290\pi$
$151$	388333.	1.38599	0.692997	+	0.720940i	$0.256290\pi$
$152$	67792.8	0.237999
$153$	0	0
$154$	1609.11	0.00546743
$155$	70706.7	0.236391
$156$	0	0
$157$	−322126.	−1.04298	−0.521491	−	0.853257i	$-0.674624\pi$
$157$	−322126.	−1.04298	−0.521491	+	0.853257i	$0.674624\pi$
$158$	−163672.	−0.521593
$159$	0	0
$160$	−459315.	−1.41844
$161$	−26072.1	−0.0792704
$162$	0	0
$163$	−366095.	−1.07926	−0.539629	−	0.841903i	$-0.681435\pi$
$163$	−366095.	−1.07926	−0.539629	+	0.841903i	$0.681435\pi$
$164$	−444161.	−1.28953
$165$	0	0
$166$	62586.0	0.176282
$167$	597140.	1.65686	0.828429	−	0.560095i	$-0.189235\pi$
$167$	597140.	1.65686	0.828429	+	0.560095i	$0.189235\pi$
$168$	0	0
$169$	−300990.	−0.810655
$170$	−331157.	−0.878843
$171$	0	0
$172$	287158.	0.740117
$173$	−259604.	−0.659471	−0.329736	−	0.944073i	$-0.606960\pi$
$173$	−259604.	−0.659471	−0.329736	+	0.944073i	$0.606960\pi$
$174$	0	0
$175$	19225.5	0.0474551
$176$	−61803.2	−0.150393
$177$	0	0
$178$	71034.0	0.168041
$179$	248783.	0.580346	0.290173	−	0.956974i	$-0.406287\pi$
$179$	248783.	0.580346	0.290173	+	0.956974i	$0.406287\pi$
$180$	0	0
$181$	−513821.	−1.16578	−0.582888	−	0.812552i	$-0.698078\pi$
$181$	−513821.	−1.16578	−0.582888	+	0.812552i	$0.698078\pi$
$182$	3526.02	0.00789054
$183$	0	0
$184$	−649704.	−1.41472
$185$	712653.	1.53091
$186$	0	0
$187$	−207360.	−0.433631
$188$	64076.1	0.132221
$189$	0	0
$190$	−94234.4	−0.189376
$191$	656946.	1.30300	0.651502	−	0.758647i	$-0.274139\pi$
$191$	656946.	1.30300	0.651502	+	0.758647i	$0.274139\pi$
$192$	0	0
$193$	−685177.	−1.32406	−0.662032	−	0.749475i	$-0.730306\pi$
$193$	−685177.	−1.32406	−0.662032	+	0.749475i	$0.730306\pi$
$194$	177467.	0.338542
$195$	0	0
$196$	441499.	0.820899
$197$	−38460.1	−0.0706066	−0.0353033	−	0.999377i	$-0.511240\pi$
$197$	−38460.1	−0.0706066	−0.0353033	+	0.999377i	$0.511240\pi$
$198$	0	0
$199$	−160890.	−0.288002	−0.144001	−	0.989578i	$-0.545997\pi$
$199$	−160890.	−0.288002	−0.144001	+	0.989578i	$0.545997\pi$
$200$	479091.	0.846921
$201$	0	0
$202$	178207.	0.307288
$203$	−43705.9	−0.0744390
$204$	0	0
$205$	1.36811e6	2.27372
$206$	−306957.	−0.503976
$207$	0	0
$208$	−135429.	−0.217046
$209$	−59006.5	−0.0934403
$210$	0	0
$211$	−73487.3	−0.113633	−0.0568167	−	0.998385i	$-0.518095\pi$
$211$	−73487.3	−0.113633	−0.0568167	+	0.998385i	$0.518095\pi$
$212$	508863.	0.777608
$213$	0	0
$214$	−387506.	−0.578421
$215$	−884507.	−1.30498
$216$	0	0
$217$	4865.93	0.00701482
$218$	18358.0	0.0261629
$219$	0	0
$220$	258139.	0.359582
$221$	−454386.	−0.625813
$222$	0	0
$223$	−870358.	−1.17202	−0.586011	−	0.810303i	$-0.699302\pi$
$223$	−870358.	−1.17202	−0.586011	+	0.810303i	$0.699302\pi$
$224$	−31609.4	−0.0420916
$225$	0	0
$226$	−261069.	−0.340004
$227$	−657863.	−0.847366	−0.423683	−	0.905811i	$-0.639263\pi$
$227$	−657863.	−0.847366	−0.423683	+	0.905811i	$0.639263\pi$
$228$	0	0
$229$	−1.19516e6	−1.50604	−0.753019	−	0.657999i	$-0.771403\pi$
$229$	−1.19516e6	−1.50604	−0.753019	+	0.657999i	$0.771403\pi$
$230$	903111.	1.12570
$231$	0	0
$232$	−1.08913e6	−1.32850
$233$	−1.20879e6	−1.45868	−0.729340	−	0.684151i	$-0.760173\pi$
$233$	−1.20879e6	−1.45868	−0.729340	+	0.684151i	$0.760173\pi$
$234$	0	0
$235$	−197368.	−0.233135
$236$	973707.	1.13802
$237$	0	0
$238$	−22789.7	−0.0260793
$239$	−316787.	−0.358734	−0.179367	−	0.983782i	$-0.557405\pi$
$239$	−316787.	−0.358734	−0.179367	+	0.983782i	$0.557405\pi$
$240$	0	0
$241$	345641.	0.383338	0.191669	−	0.981460i	$-0.438610\pi$
$241$	345641.	0.383338	0.191669	+	0.981460i	$0.438610\pi$
$242$	−34901.2	−0.0383091
$243$	0	0
$244$	395721.	0.425515
$245$	−1.35991e6	−1.44742
$246$	0	0
$247$	−129301.	−0.134852
$248$	121257.	0.125192
$249$	0	0
$250$	−62081.1	−0.0628216
$251$	−1.05813e6	−1.06012	−0.530061	−	0.847960i	$-0.677831\pi$
$251$	−1.05813e6	−1.06012	−0.530061	+	0.847960i	$0.677831\pi$
$252$	0	0
$253$	565499.	0.555431
$254$	−401571.	−0.390551
$255$	0	0
$256$	−357540.	−0.340977
$257$	−1.69415e6	−1.60000	−0.800000	−	0.600001i	$-0.795167\pi$
$257$	−1.69415e6	−1.60000	−0.800000	+	0.600001i	$0.795167\pi$
$258$	0	0
$259$	49043.7	0.0454290
$260$	565659.	0.518945
$261$	0	0
$262$	16828.0	0.0151453
$263$	−829500.	−0.739481	−0.369740	−	0.929135i	$-0.620553\pi$
$263$	−829500.	−0.739481	−0.369740	+	0.929135i	$0.620553\pi$
$264$	0	0
$265$	−1.56740e6	−1.37109
$266$	−6485.07	−0.00561966
$267$	0	0
$268$	−481327.	−0.409358
$269$	−240172.	−0.202368	−0.101184	−	0.994868i	$-0.532263\pi$
$269$	−240172.	−0.202368	−0.101184	+	0.994868i	$0.532263\pi$
$270$	0	0
$271$	−686355.	−0.567708	−0.283854	−	0.958867i	$-0.591613\pi$
$271$	−686355.	−0.567708	−0.283854	+	0.958867i	$0.591613\pi$
$272$	875315.	0.717368
$273$	0	0
$274$	−189013.	−0.152096
$275$	−416998.	−0.332508
$276$	0	0
$277$	1.83135e6	1.43408	0.717039	−	0.697033i	$-0.245497\pi$
$277$	1.83135e6	1.43408	0.717039	+	0.697033i	$0.245497\pi$
$278$	572033.	0.443924
$279$	0	0
$280$	62867.0	0.0479212
$281$	−587912.	−0.444167	−0.222084	−	0.975028i	$-0.571286\pi$
$281$	−587912.	−0.444167	−0.222084	+	0.975028i	$0.571286\pi$
$282$	0	0
$283$	−1.23258e6	−0.914850	−0.457425	−	0.889248i	$-0.651228\pi$
$283$	−1.23258e6	−0.914850	−0.457425	+	0.889248i	$0.651228\pi$
$284$	−1.35687e6	−0.998259
$285$	0	0
$286$	−76478.8	−0.0552874
$287$	94151.2	0.0674716
$288$	0	0
$289$	1.51697e6	1.06840
$290$	1.51393e6	1.05709
$291$	0	0
$292$	955629.	0.655892
$293$	361948.	0.246307	0.123154	−	0.992388i	$-0.460699\pi$
$293$	361948.	0.246307	0.123154	+	0.992388i	$0.460699\pi$
$294$	0	0
$295$	−2.99922e6	−2.00657
$296$	1.22214e6	0.810761
$297$	0	0
$298$	1.05539e6	0.688449
$299$	1.23917e6	0.801593
$300$	0	0
$301$	−60870.4	−0.0387249
$302$	−925708.	−0.584059
$303$	0	0
$304$	249081.	0.154581
$305$	−1.21890e6	−0.750273
$306$	0	0
$307$	−1.77244e6	−1.07331	−0.536655	−	0.843802i	$-0.680312\pi$
$307$	−1.77244e6	−1.07331	−0.536655	+	0.843802i	$0.680312\pi$
$308$	17764.8	0.0106704
$309$	0	0
$310$	−168551.	−0.0996155
$311$	2.83932e6	1.66461	0.832305	−	0.554317i	$-0.187021\pi$
$311$	2.83932e6	1.66461	0.832305	+	0.554317i	$0.187021\pi$
$312$	0	0
$313$	660848.	0.381277	0.190639	−	0.981660i	$-0.438944\pi$
$313$	660848.	0.381277	0.190639	+	0.981660i	$0.438944\pi$
$314$	767885.	0.439513
$315$	0	0
$316$	−1.80696e6	−1.01796
$317$	−2.92561e6	−1.63519	−0.817596	−	0.575793i	$-0.804693\pi$
$317$	−2.92561e6	−1.63519	−0.817596	+	0.575793i	$0.804693\pi$
$318$	0	0
$319$	947974.	0.521579
$320$	−230034.	−0.125579
$321$	0	0
$322$	62150.7	0.0334046
$323$	835707.	0.445705
$324$	0	0
$325$	−913765.	−0.479873
$326$	872699.	0.454800
$327$	0	0
$328$	2.34620e6	1.20415
$329$	−13582.5	−0.00691817
$330$	0	0
$331$	1.20374e6	0.603897	0.301948	−	0.953324i	$-0.402363\pi$
$331$	1.20374e6	0.603897	0.301948	+	0.953324i	$0.402363\pi$
$332$	690958.	0.344038
$333$	0	0
$334$	−1.42346e6	−0.698201
$335$	1.48259e6	0.721786
$336$	0	0
$337$	−2.02796e6	−0.972713	−0.486356	−	0.873761i	$-0.661674\pi$
$337$	−2.02796e6	−0.972713	−0.486356	+	0.873761i	$0.661674\pi$
$338$	717501.	0.341610
$339$	0	0
$340$	−3.65602e6	−1.71518
$341$	−105541.	−0.0491514
$342$	0	0
$343$	−187347.	−0.0859829
$344$	−1.51686e6	−0.691114
$345$	0	0
$346$	618844.	0.277902
$347$	−1.52184e6	−0.678492	−0.339246	−	0.940698i	$-0.610172\pi$
$347$	−1.52184e6	−0.678492	−0.339246	+	0.940698i	$0.610172\pi$
$348$	0	0
$349$	−2.87478e6	−1.26340	−0.631701	−	0.775212i	$-0.717643\pi$
$349$	−2.87478e6	−1.26340	−0.631701	+	0.775212i	$0.717643\pi$
$350$	−45829.8	−0.0199976
$351$	0	0
$352$	685602.	0.294927
$353$	379865.	0.162253	0.0811265	−	0.996704i	$-0.474148\pi$
$353$	379865.	0.162253	0.0811265	+	0.996704i	$0.474148\pi$
$354$	0	0
$355$	4.17945e6	1.76014
$356$	784224.	0.327956
$357$	0	0
$358$	−593048.	−0.244558
$359$	−2.08326e6	−0.853115	−0.426558	−	0.904460i	$-0.640274\pi$
$359$	−2.08326e6	−0.853115	−0.426558	+	0.904460i	$0.640274\pi$
$360$	0	0
$361$	−2.23829e6	−0.903958
$362$	1.22485e6	0.491259
$363$	0	0
$364$	38927.8	0.0153995
$365$	−2.94354e6	−1.15648
$366$	0	0
$367$	862007.	0.334076	0.167038	−	0.985950i	$-0.446580\pi$
$367$	862007.	0.334076	0.167038	+	0.985950i	$0.446580\pi$
$368$	−2.38710e6	−0.918866
$369$	0	0
$370$	−1.69882e6	−0.645125
$371$	−107866.	−0.0406865
$372$	0	0
$373$	1.44298e6	0.537017	0.268508	−	0.963277i	$-0.413469\pi$
$373$	1.44298e6	0.537017	0.268508	+	0.963277i	$0.413469\pi$
$374$	494305.	0.182732
$375$	0	0
$376$	−338470.	−0.123467
$377$	2.07729e6	0.752737
$378$	0	0
$379$	−1.66143e6	−0.594133	−0.297067	−	0.954857i	$-0.596008\pi$
$379$	−1.66143e6	−0.594133	−0.297067	+	0.954857i	$0.596008\pi$
$380$	−1.04036e6	−0.369594
$381$	0	0
$382$	−1.56603e6	−0.549087
$383$	3.39686e6	1.18326	0.591630	−	0.806210i	$-0.298485\pi$
$383$	3.39686e6	1.18326	0.591630	+	0.806210i	$0.298485\pi$
$384$	0	0
$385$	−54719.1	−0.0188143
$386$	1.63333e6	0.557962
$387$	0	0
$388$	1.95926e6	0.660711
$389$	−3.57464e6	−1.19773	−0.598864	−	0.800850i	$-0.704381\pi$
$389$	−3.57464e6	−1.19773	−0.598864	+	0.800850i	$0.704381\pi$
$390$	0	0
$391$	−8.00913e6	−2.64938
$392$	−2.33214e6	−0.766547
$393$	0	0
$394$	91681.4	0.0297537
$395$	5.56581e6	1.79488
$396$	0	0
$397$	377066.	0.120072	0.0600360	−	0.998196i	$-0.480878\pi$
$397$	377066.	0.120072	0.0600360	+	0.998196i	$0.480878\pi$
$398$	383529.	0.121364
$399$	0	0
$400$	1.76025e6	0.550078
$401$	4.78006e6	1.48447	0.742236	−	0.670139i	$-0.233765\pi$
$401$	4.78006e6	1.48447	0.742236	+	0.670139i	$0.233765\pi$
$402$	0	0
$403$	−231271.	−0.0709348
$404$	1.96743e6	0.599715
$405$	0	0
$406$	104186.	0.0313686
$407$	−1.06375e6	−0.318312
$408$	0	0
$409$	3.13970e6	0.928067	0.464034	−	0.885818i	$-0.346402\pi$
$409$	3.13970e6	0.928067	0.464034	+	0.885818i	$0.346402\pi$
$410$	−3.26130e6	−0.958145
$411$	0	0
$412$	−3.38885e6	−0.983578
$413$	−206402.	−0.0595440
$414$	0	0
$415$	−2.12829e6	−0.606613
$416$	1.50235e6	0.425637
$417$	0	0
$418$	140660.	0.0393758
$419$	734774.	0.204465	0.102232	−	0.994761i	$-0.467401\pi$
$419$	734774.	0.204465	0.102232	+	0.994761i	$0.467401\pi$
$420$	0	0
$421$	2.90129e6	0.797785	0.398893	−	0.916998i	$-0.369395\pi$
$421$	2.90129e6	0.797785	0.398893	+	0.916998i	$0.369395\pi$
$422$	175179.	0.0478852
$423$	0	0
$424$	−2.68797e6	−0.726123
$425$	5.90593e6	1.58605
$426$	0	0
$427$	−83883.0	−0.0222641
$428$	−4.27812e6	−1.12887
$429$	0	0
$430$	2.10849e6	0.549921
$431$	3.50404e6	0.908606	0.454303	−	0.890847i	$-0.349888\pi$
$431$	3.50404e6	0.908606	0.454303	+	0.890847i	$0.349888\pi$
$432$	0	0
$433$	−3.33698e6	−0.855331	−0.427666	−	0.903937i	$-0.640664\pi$
$433$	−3.33698e6	−0.855331	−0.427666	+	0.903937i	$0.640664\pi$
$434$	−11599.4	−0.00295605
$435$	0	0
$436$	202675.	0.0510604
$437$	−2.27909e6	−0.570897
$438$	0	0
$439$	−6.41689e6	−1.58914	−0.794572	−	0.607170i	$-0.792305\pi$
$439$	−6.41689e6	−1.58914	−0.794572	+	0.607170i	$0.792305\pi$
$440$	−1.36357e6	−0.335774
$441$	0	0
$442$	1.08317e6	0.263718
$443$	−132603.	−0.0321030	−0.0160515	−	0.999871i	$-0.505110\pi$
$443$	−132603.	−0.0321030	−0.0160515	+	0.999871i	$0.505110\pi$
$444$	0	0
$445$	−2.41557e6	−0.578256
$446$	2.07476e6	0.493891
$447$	0	0
$448$	−15830.6	−0.00372651
$449$	2.76411e6	0.647053	0.323527	−	0.946219i	$-0.395131\pi$
$449$	2.76411e6	0.647053	0.323527	+	0.946219i	$0.395131\pi$
$450$	0	0
$451$	−2.04212e6	−0.472760
$452$	−2.88224e6	−0.663565
$453$	0	0
$454$	1.56822e6	0.357081
$455$	−119906.	−0.0271526
$456$	0	0
$457$	2.02476e6	0.453506	0.226753	−	0.973952i	$-0.427189\pi$
$457$	2.02476e6	0.453506	0.226753	+	0.973952i	$0.427189\pi$
$458$	2.84902e6	0.634645
$459$	0	0
$460$	9.97046e6	2.19695
$461$	−5.83370e6	−1.27847	−0.639237	−	0.769010i	$-0.720750\pi$
$461$	−5.83370e6	−1.27847	−0.639237	+	0.769010i	$0.720750\pi$
$462$	0	0
$463$	6.02936e6	1.30713	0.653565	−	0.756870i	$-0.273273\pi$
$463$	6.02936e6	1.30713	0.653565	+	0.756870i	$0.273273\pi$
$464$	−4.00162e6	−0.862862
$465$	0	0
$466$	2.88151e6	0.614689
$467$	−2.01833e6	−0.428253	−0.214127	−	0.976806i	$-0.568691\pi$
$467$	−2.01833e6	−0.428253	−0.214127	+	0.976806i	$0.568691\pi$
$468$	0	0
$469$	102029.	0.0214187
$470$	470486.	0.0982430
$471$	0	0
$472$	−5.14343e6	−1.06267
$473$	1.32027e6	0.271337
$474$	0	0
$475$	1.68060e6	0.341767
$476$	−251601.	−0.0508974
$477$	0	0
$478$	755158.	0.151171
$479$	−4.54025e6	−0.904151	−0.452075	−	0.891980i	$-0.649316\pi$
$479$	−4.54025e6	−0.904151	−0.452075	+	0.891980i	$0.649316\pi$
$480$	0	0
$481$	−2.33098e6	−0.459385
$482$	−823939.	−0.161539
$483$	0	0
$484$	−385314.	−0.0747656
$485$	−6.03491e6	−1.16498
$486$	0	0
$487$	6.46875e6	1.23594	0.617971	−	0.786201i	$-0.287955\pi$
$487$	6.46875e6	1.23594	0.617971	+	0.786201i	$0.287955\pi$
$488$	−2.09032e6	−0.397342
$489$	0	0
$490$	3.24175e6	0.609944
$491$	−4.79546e6	−0.897691	−0.448845	−	0.893609i	$-0.648165\pi$
$491$	−4.79546e6	−0.897691	−0.448845	+	0.893609i	$0.648165\pi$
$492$	0	0
$493$	−1.34261e7	−2.48790
$494$	308227.	0.0568268
$495$	0	0
$496$	445514.	0.0813125
$497$	287623.	0.0522316
$498$	0	0
$499$	−6.21706e6	−1.11772	−0.558861	−	0.829262i	$-0.688761\pi$
$499$	−6.21706e6	−1.11772	−0.558861	+	0.829262i	$0.688761\pi$
$500$	−685383.	−0.122605
$501$	0	0
$502$	2.52238e6	0.446736
$503$	−2.48047e6	−0.437133	−0.218567	−	0.975822i	$-0.570138\pi$
$503$	−2.48047e6	−0.437133	−0.218567	+	0.975822i	$0.570138\pi$
$504$	0	0
$505$	−6.06009e6	−1.05743
$506$	−1.34804e6	−0.234059
$507$	0	0
$508$	−4.43340e6	−0.762214
$509$	3.89804e6	0.666887	0.333443	−	0.942770i	$-0.391789\pi$
$509$	3.89804e6	0.666887	0.333443	+	0.942770i	$0.391789\pi$
$510$	0	0
$511$	−202570.	−0.0343180
$512$	−5.16628e6	−0.870969
$513$	0	0
$514$	4.03852e6	0.674241
$515$	1.04384e7	1.73426
$516$	0	0
$517$	294603.	0.0484742
$518$	−116910.	−0.0191438
$519$	0	0
$520$	−2.98799e6	−0.484586
$521$	−7.29862e6	−1.17800	−0.589002	−	0.808132i	$-0.700479\pi$
$521$	−7.29862e6	−1.17800	−0.589002	+	0.808132i	$0.700479\pi$
$522$	0	0
$523$	−1.36406e6	−0.218062	−0.109031	−	0.994038i	$-0.534775\pi$
$523$	−1.36406e6	−0.218062	−0.109031	+	0.994038i	$0.534775\pi$
$524$	185783.	0.0295582
$525$	0	0
$526$	1.97736e6	0.311618
$527$	1.49477e6	0.234449
$528$	0	0
$529$	1.54057e7	2.39354
$530$	3.73638e6	0.577778
$531$	0	0
$532$	−71596.0	−0.0109675
$533$	−4.47489e6	−0.682282
$534$	0	0
$535$	1.31775e7	1.99044
$536$	2.54252e6	0.382255
$537$	0	0
$538$	572523.	0.0852780
$539$	2.02988e6	0.300953
$540$	0	0
$541$	−1.94836e6	−0.286204	−0.143102	−	0.989708i	$-0.545708\pi$
$541$	−1.94836e6	−0.286204	−0.143102	+	0.989708i	$0.545708\pi$
$542$	1.63613e6	0.239233
$543$	0	0
$544$	−9.71014e6	−1.40679
$545$	−624282.	−0.0900305
$546$	0	0
$547$	−7.39187e6	−1.05630	−0.528148	−	0.849152i	$-0.677114\pi$
$547$	−7.39187e6	−1.05630	−0.528148	+	0.849152i	$0.677114\pi$
$548$	−2.08673e6	−0.296835
$549$	0	0
$550$	994041.	0.140119
$551$	−3.82055e6	−0.536101
$552$	0	0
$553$	383031.	0.0532624
$554$	−4.36558e6	−0.604321
$555$	0	0
$556$	6.31532e6	0.866380
$557$	−9.35650e6	−1.27784	−0.638919	−	0.769274i	$-0.720618\pi$
$557$	−9.35650e6	−1.27784	−0.638919	+	0.769274i	$0.720618\pi$
$558$	0	0
$559$	2.89309e6	0.391591
$560$	230983.	0.0311250
$561$	0	0
$562$	1.40147e6	0.187172
$563$	1.06865e7	1.42090	0.710452	−	0.703745i	$-0.248490\pi$
$563$	1.06865e7	1.42090	0.710452	+	0.703745i	$0.248490\pi$
$564$	0	0
$565$	8.87789e6	1.17001
$566$	2.93823e6	0.385518
$567$	0	0
$568$	7.16743e6	0.932165
$569$	1.26906e7	1.64324	0.821621	−	0.570034i	$-0.193070\pi$
$569$	1.26906e7	1.64324	0.821621	+	0.570034i	$0.193070\pi$
$570$	0	0
$571$	1.89539e6	0.243281	0.121640	−	0.992574i	$-0.461185\pi$
$571$	1.89539e6	0.243281	0.121640	+	0.992574i	$0.461185\pi$
$572$	−844337.	−0.107901
$573$	0	0
$574$	−224438.	−0.0284326
$575$	−1.61063e7	−2.03154
$576$	0	0
$577$	−2.22640e6	−0.278397	−0.139198	−	0.990265i	$-0.544453\pi$
$577$	−2.22640e6	−0.278397	−0.139198	+	0.990265i	$0.544453\pi$
$578$	−3.61616e6	−0.450223
$579$	0	0
$580$	1.67140e7	2.06305
$581$	−146466.	−0.0180010
$582$	0	0
$583$	2.33960e6	0.285082
$584$	−5.04794e6	−0.612466
$585$	0	0
$586$	−862812.	−0.103794
$587$	−8.98820e6	−1.07666	−0.538329	−	0.842735i	$-0.680944\pi$
$587$	−8.98820e6	−1.07666	−0.538329	+	0.842735i	$0.680944\pi$
$588$	0	0
$589$	425355.	0.0505199
$590$	7.14955e6	0.845568
$591$	0	0
$592$	4.49033e6	0.526592
$593$	1.53402e7	1.79140	0.895701	−	0.444656i	$-0.146674\pi$
$593$	1.53402e7	1.79140	0.895701	+	0.444656i	$0.146674\pi$
$594$	0	0
$595$	774985.	0.0897430
$596$	1.16516e7	1.34360
$597$	0	0
$598$	−2.95394e6	−0.337792
$599$	5.57851e6	0.635260	0.317630	−	0.948215i	$-0.397113\pi$
$599$	5.57851e6	0.635260	0.317630	+	0.948215i	$0.397113\pi$
$600$	0	0
$601$	9.68805e6	1.09408	0.547041	−	0.837106i	$-0.315754\pi$
$601$	9.68805e6	1.09408	0.547041	+	0.837106i	$0.315754\pi$
$602$	145103.	0.0163187
$603$	0	0
$604$	−1.02199e7	−1.13987
$605$	1.18685e6	0.131828
$606$	0	0
$607$	−4.05805e6	−0.447040	−0.223520	−	0.974699i	$-0.571755\pi$
$607$	−4.05805e6	−0.447040	−0.223520	+	0.974699i	$0.571755\pi$
$608$	−2.76313e6	−0.303139
$609$	0	0
$610$	2.90562e6	0.316166
$611$	645561.	0.0699575
$612$	0	0
$613$	−987590.	−0.106151	−0.0530757	−	0.998590i	$-0.516902\pi$
$613$	−987590.	−0.106151	−0.0530757	+	0.998590i	$0.516902\pi$
$614$	4.22514e6	0.452293
$615$	0	0
$616$	−93839.1	−0.00996396
$617$	661761.	0.0699824	0.0349912	−	0.999388i	$-0.488860\pi$
$617$	661761.	0.0699824	0.0349912	+	0.999388i	$0.488860\pi$
$618$	0	0
$619$	7.88034e6	0.826644	0.413322	−	0.910585i	$-0.364368\pi$
$619$	7.88034e6	0.826644	0.413322	+	0.910585i	$0.364368\pi$
$620$	−1.86082e6	−0.194413
$621$	0	0
$622$	−6.76837e6	−0.701468
$623$	−166236.	−0.0171595
$624$	0	0
$625$	−8.65846e6	−0.886626
$626$	−1.57533e6	−0.160671
$627$	0	0
$628$	8.47755e6	0.857771
$629$	1.50658e7	1.51833
$630$	0	0
$631$	8.08931e6	0.808794	0.404397	−	0.914584i	$-0.367481\pi$
$631$	8.08931e6	0.808794	0.404397	+	0.914584i	$0.367481\pi$
$632$	9.54494e6	0.950562
$633$	0	0
$634$	6.97408e6	0.689071
$635$	1.36558e7	1.34395
$636$	0	0
$637$	4.44806e6	0.434333
$638$	−2.25978e6	−0.219794
$639$	0	0
$640$	1.52464e7	1.47136
$641$	1.67048e7	1.60582	0.802910	−	0.596100i	$-0.203284\pi$
$641$	1.67048e7	1.60582	0.802910	+	0.596100i	$0.203284\pi$
$642$	0	0
$643$	−1.37829e7	−1.31466	−0.657331	−	0.753602i	$-0.728315\pi$
$643$	−1.37829e7	−1.31466	−0.657331	+	0.753602i	$0.728315\pi$
$644$	686152.	0.0651937
$645$	0	0
$646$	−1.99216e6	−0.187820
$647$	−1.48963e7	−1.39900	−0.699502	−	0.714631i	$-0.746595\pi$
$647$	−1.48963e7	−1.39900	−0.699502	+	0.714631i	$0.746595\pi$
$648$	0	0
$649$	4.47682e6	0.417213
$650$	2.17823e6	0.202219
$651$	0	0
$652$	9.63471e6	0.887605
$653$	101816.	0.00934403	0.00467202	−	0.999989i	$-0.498513\pi$
$653$	101816.	0.00934403	0.00467202	+	0.999989i	$0.498513\pi$
$654$	0	0
$655$	−572250.	−0.0521174
$656$	8.62029e6	0.782100
$657$	0	0
$658$	32378.1	0.00291532
$659$	1.05796e7	0.948977	0.474488	−	0.880262i	$-0.342633\pi$
$659$	1.05796e7	0.948977	0.474488	+	0.880262i	$0.342633\pi$
$660$	0	0
$661$	−6.90726e6	−0.614896	−0.307448	−	0.951565i	$-0.599475\pi$
$661$	−6.90726e6	−0.614896	−0.307448	+	0.951565i	$0.599475\pi$
$662$	−2.86948e6	−0.254483
$663$	0	0
$664$	−3.64986e6	−0.321260
$665$	220531.	0.0193381
$666$	0	0
$667$	3.66149e7	3.18671
$668$	−1.57152e7	−1.36264
$669$	0	0
$670$	−3.53419e6	−0.304161
$671$	1.81941e6	0.156000
$672$	0	0
$673$	8.32298e6	0.708339	0.354170	−	0.935181i	$-0.384764\pi$
$673$	8.32298e6	0.708339	0.354170	+	0.935181i	$0.384764\pi$
$674$	4.83425e6	0.409902
$675$	0	0
$676$	7.92131e6	0.666700
$677$	1.46773e7	1.23076	0.615381	−	0.788230i	$-0.289002\pi$
$677$	1.46773e7	1.23076	0.615381	+	0.788230i	$0.289002\pi$
$678$	0	0
$679$	−415314.	−0.0345702
$680$	1.93122e7	1.60162
$681$	0	0
$682$	251589.	0.0207124
$683$	2.19232e7	1.79826	0.899130	−	0.437683i	$-0.144201\pi$
$683$	2.19232e7	1.79826	0.899130	+	0.437683i	$0.144201\pi$
$684$	0	0
$685$	6.42758e6	0.523384
$686$	446599.	0.0362332
$687$	0	0
$688$	−5.57317e6	−0.448881
$689$	5.12675e6	0.411428
$690$	0	0
$691$	−1.41171e6	−0.112473	−0.0562366	−	0.998417i	$-0.517910\pi$
$691$	−1.41171e6	−0.112473	−0.0562366	+	0.998417i	$0.517910\pi$
$692$	6.83212e6	0.542363
$693$	0	0
$694$	3.62776e6	0.285917
$695$	−1.94525e7	−1.52761
$696$	0	0
$697$	2.89225e7	2.25504
$698$	6.85291e6	0.532398
$699$	0	0
$700$	−505968.	−0.0390281
$701$	−2.38560e7	−1.83359	−0.916794	−	0.399361i	$-0.869232\pi$
$701$	−2.38560e7	−1.83359	−0.916794	+	0.399361i	$0.869232\pi$
$702$	0	0
$703$	4.28714e6	0.327175
$704$	343363.	0.0261109
$705$	0	0
$706$	−905524.	−0.0683735
$707$	−417046.	−0.0313787
$708$	0	0
$709$	−6.31156e6	−0.471543	−0.235772	−	0.971809i	$-0.575762\pi$
$709$	−6.31156e6	−0.471543	−0.235772	+	0.971809i	$0.575762\pi$
$710$	−9.96298e6	−0.741726
$711$	0	0
$712$	−4.14252e6	−0.306242
$713$	−4.07645e6	−0.300302
$714$	0	0
$715$	2.60073e6	0.190253
$716$	−6.54733e6	−0.477289
$717$	0	0
$718$	4.96608e6	0.359503
$719$	1.74102e7	1.25597	0.627987	−	0.778223i	$-0.283879\pi$
$719$	1.74102e7	1.25597	0.627987	+	0.778223i	$0.283879\pi$
$720$	0	0
$721$	718351.	0.0514634
$722$	5.33564e6	0.380928
$723$	0	0
$724$	1.35225e7	0.958760
$725$	−2.69998e7	−1.90772
$726$	0	0
$727$	−231516.	−0.0162459	−0.00812297	−	0.999967i	$-0.502586\pi$
$727$	−231516.	−0.0162459	−0.00812297	+	0.999967i	$0.502586\pi$
$728$	−205629.	−0.0143799
$729$	0	0
$730$	7.01681e6	0.487341
$731$	−1.86989e7	−1.29426
$732$	0	0
$733$	1.76256e7	1.21167	0.605833	−	0.795592i	$-0.292840\pi$
$733$	1.76256e7	1.21167	0.605833	+	0.795592i	$0.292840\pi$
$734$	−2.05485e6	−0.140780
$735$	0	0
$736$	2.64809e7	1.80193
$737$	−2.21300e6	−0.150076
$738$	0	0
$739$	−2.69020e7	−1.81206	−0.906032	−	0.423208i	$-0.860904\pi$
$739$	−2.69020e7	−1.81206	−0.906032	+	0.423208i	$0.860904\pi$
$740$	−1.87552e7	−1.25905
$741$	0	0
$742$	257132.	0.0171453
$743$	−2.00353e7	−1.33145	−0.665724	−	0.746198i	$-0.731877\pi$
$743$	−2.00353e7	−1.33145	−0.665724	+	0.746198i	$0.731877\pi$
$744$	0	0
$745$	−3.58894e7	−2.36906
$746$	−3.43978e6	−0.226299
$747$	0	0
$748$	5.45719e6	0.356628
$749$	906855.	0.0590654
$750$	0	0
$751$	−1.15224e7	−0.745489	−0.372745	−	0.927934i	$-0.621583\pi$
$751$	−1.15224e7	−0.745489	−0.372745	+	0.927934i	$0.621583\pi$
$752$	−1.24359e6	−0.0801923
$753$	0	0
$754$	−4.95184e6	−0.317204
$755$	3.14795e7	2.00984
$756$	0	0
$757$	−1.44526e7	−0.916653	−0.458327	−	0.888784i	$-0.651551\pi$
$757$	−1.44526e7	−0.916653	−0.458327	+	0.888784i	$0.651551\pi$
$758$	3.96052e6	0.250368
$759$	0	0
$760$	5.49551e6	0.345123
$761$	8.34968e6	0.522647	0.261323	−	0.965251i	$-0.415841\pi$
$761$	8.34968e6	0.522647	0.261323	+	0.965251i	$0.415841\pi$
$762$	0	0
$763$	−42962.1	−0.00267162
$764$	−1.72892e7	−1.07162
$765$	0	0
$766$	−8.09743e6	−0.498627
$767$	9.81001e6	0.602118
$768$	0	0
$769$	−7.62509e6	−0.464975	−0.232487	−	0.972599i	$-0.574686\pi$
$769$	−7.62509e6	−0.464975	−0.232487	+	0.972599i	$0.574686\pi$
$770$	130440.	0.00792835
$771$	0	0
$772$	1.80321e7	1.08894
$773$	−5.66557e6	−0.341032	−0.170516	−	0.985355i	$-0.554543\pi$
$773$	−5.66557e6	−0.341032	−0.170516	+	0.985355i	$0.554543\pi$
$774$	0	0
$775$	3.00597e6	0.179776
$776$	−1.03494e7	−0.616966
$777$	0	0
$778$	8.52124e6	0.504724
$779$	8.23022e6	0.485923
$780$	0	0
$781$	−6.23850e6	−0.365976
$782$	1.90922e7	1.11645
$783$	0	0
$784$	−8.56861e6	−0.497875
$785$	−2.61126e7	−1.51243
$786$	0	0
$787$	−2.42213e7	−1.39399	−0.696996	−	0.717075i	$-0.745480\pi$
$787$	−2.42213e7	−1.39399	−0.696996	+	0.717075i	$0.745480\pi$
$788$	1.01217e6	0.0580684
$789$	0	0
$790$	−1.32678e7	−0.756364
$791$	610962.	0.0347195
$792$	0	0
$793$	3.98685e6	0.225137
$794$	−898852.	−0.0505984
$795$	0	0
$796$	4.23421e6	0.236859
$797$	−5.41937e6	−0.302206	−0.151103	−	0.988518i	$-0.548283\pi$
$797$	−5.41937e6	−0.302206	−0.151103	+	0.988518i	$0.548283\pi$
$798$	0	0
$799$	−4.17245e6	−0.231219
$800$	−1.95270e7	−1.07872
$801$	0	0
$802$	−1.13947e7	−0.625557
$803$	4.39370e6	0.240459
$804$	0	0
$805$	−2.11349e6	−0.114950
$806$	551305.	0.0298920
$807$	0	0
$808$	−1.03926e7	−0.560009
$809$	2.77364e7	1.48998	0.744988	−	0.667078i	$-0.232455\pi$
$809$	2.77364e7	1.48998	0.744988	+	0.667078i	$0.232455\pi$
$810$	0	0
$811$	2.37238e7	1.26658	0.633290	−	0.773914i	$-0.281704\pi$
$811$	2.37238e7	1.26658	0.633290	+	0.773914i	$0.281704\pi$
$812$	1.15023e6	0.0612202
$813$	0	0
$814$	2.53576e6	0.134137
$815$	−2.96769e7	−1.56504
$816$	0	0
$817$	−5.32098e6	−0.278892
$818$	−7.48441e6	−0.391088
$819$	0	0
$820$	−3.60052e7	−1.86995
$821$	−1.87064e7	−0.968572	−0.484286	−	0.874910i	$-0.660921\pi$
$821$	−1.87064e7	−0.968572	−0.484286	+	0.874910i	$0.660921\pi$
$822$	0	0
$823$	2.17854e7	1.12115	0.560577	−	0.828102i	$-0.310579\pi$
$823$	2.17854e7	1.12115	0.560577	+	0.828102i	$0.310579\pi$
$824$	1.79010e7	0.918456
$825$	0	0
$826$	492021.	0.0250919
$827$	2.98476e7	1.51756	0.758778	−	0.651349i	$-0.225797\pi$
$827$	2.98476e7	1.51756	0.758778	+	0.651349i	$0.225797\pi$
$828$	0	0
$829$	−2.03443e6	−0.102815	−0.0514074	−	0.998678i	$-0.516371\pi$
$829$	−2.03443e6	−0.102815	−0.0514074	+	0.998678i	$0.516371\pi$
$830$	5.07343e6	0.255627
$831$	0	0
$832$	752408.	0.0376830
$833$	−2.87491e7	−1.43553
$834$	0	0
$835$	4.84062e7	2.40262
$836$	1.55290e6	0.0768474
$837$	0	0
$838$	−1.75155e6	−0.0861616
$839$	2.75502e7	1.35120	0.675599	−	0.737269i	$-0.263885\pi$
$839$	2.75502e7	1.35120	0.675599	+	0.737269i	$0.263885\pi$
$840$	0	0
$841$	4.08682e7	1.99249
$842$	−6.91610e6	−0.336187
$843$	0	0
$844$	1.93400e6	0.0934546
$845$	−2.43993e7	−1.17553
$846$	0	0
$847$	81677.1	0.00391194
$848$	−9.87601e6	−0.471620
$849$	0	0
$850$	−1.40786e7	−0.668361
$851$	−4.10865e7	−1.94480
$852$	0	0
$853$	−2.88074e7	−1.35560	−0.677800	−	0.735246i	$-0.737067\pi$
$853$	−2.88074e7	−1.35560	−0.677800	+	0.735246i	$0.737067\pi$
$854$	199960.	0.00938209
$855$	0	0
$856$	2.25984e7	1.05413
$857$	−1.64709e7	−0.766064	−0.383032	−	0.923735i	$-0.625120\pi$
$857$	−1.64709e7	−0.766064	−0.383032	+	0.923735i	$0.625120\pi$
$858$	0	0
$859$	6.77453e6	0.313254	0.156627	−	0.987658i	$-0.449938\pi$
$859$	6.77453e6	0.313254	0.156627	+	0.987658i	$0.449938\pi$
$860$	2.32780e7	1.07325
$861$	0	0
$862$	−8.35293e6	−0.382887
$863$	−1.74900e7	−0.799396	−0.399698	−	0.916647i	$-0.630885\pi$
$863$	−1.74900e7	−0.799396	−0.399698	+	0.916647i	$0.630885\pi$
$864$	0	0
$865$	−2.10444e7	−0.956303
$866$	7.95471e6	0.360437
$867$	0	0
$868$	−128059.	−0.00576914
$869$	−8.30787e6	−0.373199
$870$	0	0
$871$	−4.84933e6	−0.216589
$872$	−1.07060e6	−0.0476798
$873$	0	0
$874$	5.43289e6	0.240576
$875$	145284.	0.00641503
$876$	0	0
$877$	−1.97539e7	−0.867267	−0.433634	−	0.901089i	$-0.642769\pi$
$877$	−1.97539e7	−0.867267	−0.433634	+	0.901089i	$0.642769\pi$
$878$	1.52966e7	0.669666
$879$	0	0
$880$	−5.00997e6	−0.218086
$881$	−4.07641e7	−1.76945	−0.884724	−	0.466115i	$-0.845653\pi$
$881$	−4.07641e7	−1.76945	−0.884724	+	0.466115i	$0.845653\pi$
$882$	0	0
$883$	−2.77034e7	−1.19572	−0.597861	−	0.801599i	$-0.703983\pi$
$883$	−2.77034e7	−1.19572	−0.597861	+	0.801599i	$0.703983\pi$
$884$	1.19583e7	0.514682
$885$	0	0
$886$	316100.	0.0135282
$887$	2.07061e7	0.883669	0.441835	−	0.897097i	$-0.354328\pi$
$887$	2.07061e7	0.883669	0.441835	+	0.897097i	$0.354328\pi$
$888$	0	0
$889$	939770.	0.0398811
$890$	5.75825e6	0.243678
$891$	0	0
$892$	2.29056e7	0.963896
$893$	−1.18732e6	−0.0498239
$894$	0	0
$895$	2.01671e7	0.841563
$896$	1.04924e6	0.0436620
$897$	0	0
$898$	−6.58910e6	−0.272669
$899$	−6.83357e6	−0.281999
$900$	0	0
$901$	−3.31356e7	−1.35983
$902$	4.86802e6	0.199221
$903$	0	0
$904$	1.52249e7	0.619631
$905$	−4.16521e7	−1.69050
$906$	0	0
$907$	−3.03635e7	−1.22556	−0.612779	−	0.790255i	$-0.709948\pi$
$907$	−3.03635e7	−1.22556	−0.612779	+	0.790255i	$0.709948\pi$
$908$	1.73133e7	0.696892
$909$	0	0
$910$	285831.	0.0114421
$911$	4.46860e7	1.78392	0.891960	−	0.452115i	$-0.149330\pi$
$911$	4.46860e7	1.78392	0.891960	+	0.452115i	$0.149330\pi$
$912$	0	0
$913$	3.17682e6	0.126129
$914$	−4.82663e6	−0.191108
$915$	0	0
$916$	3.14535e7	1.23860
$917$	−39381.4	−0.00154656
$918$	0	0
$919$	8.88611e6	0.347075	0.173537	−	0.984827i	$-0.444480\pi$
$919$	8.88611e6	0.347075	0.173537	+	0.984827i	$0.444480\pi$
$920$	−5.26671e7	−2.05149
$921$	0	0
$922$	1.39064e7	0.538750
$923$	−1.36704e7	−0.528173
$924$	0	0
$925$	3.02972e7	1.16425
$926$	−1.43728e7	−0.550826
$927$	0	0
$928$	4.43913e7	1.69211
$929$	−6.49409e6	−0.246876	−0.123438	−	0.992352i	$-0.539392\pi$
$929$	−6.49409e6	−0.246876	−0.123438	+	0.992352i	$0.539392\pi$
$930$	0	0
$931$	−8.18088e6	−0.309333
$932$	3.18123e7	1.19965
$933$	0	0
$934$	4.81131e6	0.180466
$935$	−1.68093e7	−0.628811
$936$	0	0
$937$	−2.75669e7	−1.02575	−0.512873	−	0.858465i	$-0.671419\pi$
$937$	−2.75669e7	−1.02575	−0.512873	+	0.858465i	$0.671419\pi$
$938$	−243218.	−0.00902585
$939$	0	0
$940$	5.19422e6	0.191735
$941$	−4.75077e7	−1.74900	−0.874500	−	0.485026i	$-0.838810\pi$
$941$	−4.75077e7	−1.74900	−0.874500	+	0.485026i	$0.838810\pi$
$942$	0	0
$943$	−7.88756e7	−2.88844
$944$	−1.88977e7	−0.690207
$945$	0	0
$946$	−3.14726e6	−0.114342
$947$	2.04194e7	0.739892	0.369946	−	0.929053i	$-0.379376\pi$
$947$	2.04194e7	0.739892	0.369946	+	0.929053i	$0.379376\pi$
$948$	0	0
$949$	9.62788e6	0.347029
$950$	−4.00621e6	−0.144021
$951$	0	0
$952$	1.32904e6	0.0475275
$953$	−3.93482e7	−1.40344	−0.701718	−	0.712455i	$-0.747583\pi$
$953$	−3.93482e7	−1.40344	−0.701718	+	0.712455i	$0.747583\pi$
$954$	0	0
$955$	5.32542e7	1.88949
$956$	8.33704e6	0.295031
$957$	0	0
$958$	1.08231e7	0.381010
$959$	442336.	0.0155312
$960$	0	0
$961$	−2.78683e7	−0.973426
$962$	5.55660e6	0.193585
$963$	0	0
$964$	−9.09640e6	−0.315266
$965$	−5.55427e7	−1.92003
$966$	0	0
$967$	−2.64246e7	−0.908745	−0.454372	−	0.890812i	$-0.650136\pi$
$967$	−2.64246e7	−0.908745	−0.454372	+	0.890812i	$0.650136\pi$
$968$	2.03535e6	0.0698154
$969$	0	0
$970$	1.43860e7	0.490921
$971$	−1.11743e6	−0.0380340	−0.0190170	−	0.999819i	$-0.506054\pi$
$971$	−1.11743e6	−0.0380340	−0.0190170	+	0.999819i	$0.506054\pi$
$972$	0	0
$973$	−1.33869e6	−0.0453313
$974$	−1.54202e7	−0.520826
$975$	0	0
$976$	−7.68015e6	−0.258075
$977$	4.37422e7	1.46610	0.733051	−	0.680174i	$-0.238096\pi$
$977$	4.37422e7	1.46610	0.733051	+	0.680174i	$0.238096\pi$
$978$	0	0
$979$	3.60563e6	0.120233
$980$	3.57894e7	1.19039
$981$	0	0
$982$	1.14314e7	0.378287
$983$	2.94701e6	0.0972742	0.0486371	−	0.998817i	$-0.484512\pi$
$983$	2.94701e6	0.0972742	0.0486371	+	0.998817i	$0.484512\pi$
$984$	0	0
$985$	−3.11771e6	−0.102387
$986$	3.20052e7	1.04840
$987$	0	0
$988$	3.40287e6	0.110905
$989$	5.09945e7	1.65780
$990$	0	0
$991$	−3.75745e7	−1.21537	−0.607686	−	0.794178i	$-0.707902\pi$
$991$	−3.75745e7	−1.21537	−0.607686	+	0.794178i	$0.707902\pi$
$992$	−4.94223e6	−0.159457
$993$	0	0
$994$	−685637.	−0.0220104
$995$	−1.30422e7	−0.417633
$996$	0	0
$997$	2.83475e6	0.0903184	0.0451592	−	0.998980i	$-0.485620\pi$
$997$	2.83475e6	0.0903184	0.0451592	+	0.998980i	$0.485620\pi$
$998$	1.48202e7	0.471009
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	891.6.a.f.1.11		23
3.2	odd	2		891.6.a.e.1.13		23
9.2	odd	6		297.6.e.a.199.11		46
9.4	even	3		99.6.e.a.34.13	✓	46
9.5	odd	6		297.6.e.a.100.11		46
9.7	even	3		99.6.e.a.67.13	yes	46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.6.e.a.34.13	✓	46	9.4	even	3
99.6.e.a.67.13	yes	46	9.7	even	3
297.6.e.a.100.11		46	9.5	odd	6
297.6.e.a.199.11		46	9.2	odd	6
891.6.a.e.1.13		23	3.2	odd	2
891.6.a.f.1.11		23	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:	\( N \)	\(=\)	\( 891 = 3^{4} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	891.a (trivial)

\(f(q)\)	\(=\)	\(q-2.38380 q^{2} -26.3175 q^{4} +81.0633 q^{5} +5.57865 q^{7} +139.017 q^{8} +O(q^{10})\)	\(q-2.38380 q^{2} -26.3175 q^{4} +81.0633 q^{5} +5.57865 q^{7} +139.017 q^{8} -193.239 q^{10} -121.000 q^{11} -265.146 q^{13} -13.2984 q^{14} +510.770 q^{16} +1713.72 q^{17} +487.657 q^{19} -2133.38 q^{20} +288.440 q^{22} -4673.54 q^{23} +3446.27 q^{25} +632.057 q^{26} -146.816 q^{28} -7834.50 q^{29} +872.241 q^{31} -5666.13 q^{32} -4085.16 q^{34} +452.224 q^{35} +8791.31 q^{37} -1162.48 q^{38} +11269.2 q^{40} +16877.0 q^{41} -10911.3 q^{43} +3184.42 q^{44} +11140.8 q^{46} -2434.74 q^{47} -16775.9 q^{49} -8215.21 q^{50} +6977.99 q^{52} -19335.5 q^{53} -9808.66 q^{55} +775.530 q^{56} +18675.9 q^{58} -36998.5 q^{59} -15036.4 q^{61} -2079.25 q^{62} -2837.71 q^{64} -21493.7 q^{65} +18289.2 q^{67} -45100.7 q^{68} -1078.01 q^{70} +51557.8 q^{71} -36311.6 q^{73} -20956.7 q^{74} -12833.9 q^{76} -675.017 q^{77} +68660.1 q^{79} +41404.7 q^{80} -40231.5 q^{82} -26254.7 q^{83} +138920. q^{85} +26010.4 q^{86} -16821.1 q^{88} -29798.6 q^{89} -1479.16 q^{91} +122996. q^{92} +5803.93 q^{94} +39531.1 q^{95} -74446.9 q^{97} +39990.4 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q + 320 q^{4} + 36 q^{5} - 167 q^{7} + 213 q^{8}+O(q^{10}) \) 23 * q + 320 * q^4 + 36 * q^5 - 167 * q^7 + 213 * q^8	\( 23 q + 320 q^{4} + 36 q^{5} - 167 q^{7} + 213 q^{8} - 600 q^{10} - 2783 q^{11} - 1871 q^{13} + 1329 q^{14} + 3584 q^{16} + 267 q^{17} - 3641 q^{19} + 1917 q^{20} + 8292 q^{23} + 10049 q^{25} + 9570 q^{26} + 3793 q^{28} + 5970 q^{29} - 9542 q^{31} + 3831 q^{32} - 2982 q^{34} + 3240 q^{35} - 16007 q^{37} - 1221 q^{38} - 40635 q^{40} - 12030 q^{41} - 25943 q^{43} - 38720 q^{44} - 77004 q^{46} - 9756 q^{47} + 6990 q^{49} - 101805 q^{50} - 144446 q^{52} + 53919 q^{53} - 4356 q^{55} + 16602 q^{56} - 95367 q^{58} + 20310 q^{59} - 100247 q^{61} - 15297 q^{62} - 84577 q^{64} - 20931 q^{65} - 84956 q^{67} - 168471 q^{68} - 212292 q^{70} + 36093 q^{71} - 173444 q^{73} - 86619 q^{74} - 340334 q^{76} + 20207 q^{77} - 123113 q^{79} - 15123 q^{80} - 199983 q^{82} - 30672 q^{83} - 268335 q^{85} + 211260 q^{86} - 25773 q^{88} + 32514 q^{89} - 328021 q^{91} + 196731 q^{92} - 230262 q^{94} + 325926 q^{95} - 357002 q^{97} - 214464 q^{98}+O(q^{100}) \) 23 * q + 320 * q^4 + 36 * q^5 - 167 * q^7 + 213 * q^8 - 600 * q^10 - 2783 * q^11 - 1871 * q^13 + 1329 * q^14 + 3584 * q^16 + 267 * q^17 - 3641 * q^19 + 1917 * q^20 + 8292 * q^23 + 10049 * q^25 + 9570 * q^26 + 3793 * q^28 + 5970 * q^29 - 9542 * q^31 + 3831 * q^32 - 2982 * q^34 + 3240 * q^35 - 16007 * q^37 - 1221 * q^38 - 40635 * q^40 - 12030 * q^41 - 25943 * q^43 - 38720 * q^44 - 77004 * q^46 - 9756 * q^47 + 6990 * q^49 - 101805 * q^50 - 144446 * q^52 + 53919 * q^53 - 4356 * q^55 + 16602 * q^56 - 95367 * q^58 + 20310 * q^59 - 100247 * q^61 - 15297 * q^62 - 84577 * q^64 - 20931 * q^65 - 84956 * q^67 - 168471 * q^68 - 212292 * q^70 + 36093 * q^71 - 173444 * q^73 - 86619 * q^74 - 340334 * q^76 + 20207 * q^77 - 123113 * q^79 - 15123 * q^80 - 199983 * q^82 - 30672 * q^83 - 268335 * q^85 + 211260 * q^86 - 25773 * q^88 + 32514 * q^89 - 328021 * q^91 + 196731 * q^92 - 230262 * q^94 + 325926 * q^95 - 357002 * q^97 - 214464 * q^98

Introduction

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