LMFDB - Embedded newform 448.6.a.z.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [448,6,Mod(1,448)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("448.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(448, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	448.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,26,0,62] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$71.8519512762$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{177}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 44 $ x^2 - x - 44
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 56)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$7.15207$ of defining polynomial
Character	$\chi$	$=$	448.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.304135 q^{3} -35.5207 q^{5} +49.0000 q^{7} -242.908 q^{9} -565.825 q^{11} -983.594 q^{13} +10.8031 q^{15} +200.175 q^{17} -828.747 q^{19} -14.9026 q^{21} +4435.43 q^{23} -1863.28 q^{25} +147.781 q^{27} +3717.74 q^{29} +992.462 q^{31} +172.087 q^{33} -1740.51 q^{35} +8359.69 q^{37} +299.145 q^{39} -13473.0 q^{41} -298.798 q^{43} +8628.24 q^{45} -18736.5 q^{47} +2401.00 q^{49} -60.8802 q^{51} -16036.4 q^{53} +20098.5 q^{55} +252.051 q^{57} -12749.2 q^{59} +34975.9 q^{61} -11902.5 q^{63} +34937.9 q^{65} -11978.9 q^{67} -1348.97 q^{69} -12924.9 q^{71} +81177.2 q^{73} +566.689 q^{75} -27725.4 q^{77} +46998.8 q^{79} +58981.6 q^{81} +111544. q^{83} -7110.36 q^{85} -1130.69 q^{87} -34726.8 q^{89} -48196.1 q^{91} -301.842 q^{93} +29437.7 q^{95} +92655.6 q^{97} +137443. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 26 q^{3} + 62 q^{5} + 98 q^{7} + 206 q^{9} - 972 q^{11} - 78 q^{13} + 2576 q^{15} + 560 q^{17} - 2642 q^{19} + 1274 q^{21} + 2272 q^{23} + 4522 q^{25} + 5564 q^{27} + 7808 q^{29} + 5444 q^{31} - 10512 q^{33}+ \cdots - 44892 q^{99}+O(q^{100}) $ 2 * q + 26 * q^3 + 62 * q^5 + 98 * q^7 + 206 * q^9 - 972 * q^11 - 78 * q^13 + 2576 * q^15 + 560 * q^17 - 2642 * q^19 + 1274 * q^21 + 2272 * q^23 + 4522 * q^25 + 5564 * q^27 + 7808 * q^29 + 5444 * q^31 - 10512 * q^33 + 3038 * q^35 - 576 * q^37 + 24120 * q^39 - 16888 * q^41 + 8396 * q^43 + 52406 * q^45 - 4532 * q^47 + 4802 * q^49 + 9404 * q^51 - 1420 * q^53 - 19512 * q^55 - 47444 * q^57 - 34146 * q^59 - 19106 * q^61 + 10094 * q^63 + 123252 * q^65 - 56952 * q^67 - 58256 * q^69 - 7224 * q^71 + 128828 * q^73 + 168526 * q^75 - 47628 * q^77 + 52808 * q^79 + 92366 * q^81 + 84486 * q^83 + 27980 * q^85 + 106460 * q^87 - 130972 * q^89 - 3822 * q^91 + 116792 * q^93 - 147392 * q^95 + 194624 * q^97 - 44892 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−0.304135	−0.0195102	−0.00975512	−	0.999952i	$-0.503105\pi$
$3$	−0.304135	−0.0195102	−0.00975512	+	0.999952i	$0.503105\pi$
$4$	0	0
$5$	−35.5207	−0.635413	−0.317707	−	0.948189i	$-0.602913\pi$
$5$	−35.5207	−0.635413	−0.317707	+	0.948189i	$0.602913\pi$
$6$	0	0
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	0	0
$9$	−242.908	−0.999619
$10$	0	0
$11$	−565.825	−1.40994	−0.704969	−	0.709238i	$-0.749039\pi$
$11$	−565.825	−1.40994	−0.704969	+	0.709238i	$0.749039\pi$
$12$	0	0
$13$	−983.594	−1.61420	−0.807100	−	0.590415i	$-0.798964\pi$
$13$	−983.594	−1.61420	−0.807100	+	0.590415i	$0.798964\pi$
$14$	0	0
$15$	10.8031	0.0123971
$16$	0	0
$17$	200.175	0.167992	0.0839959	−	0.996466i	$-0.473232\pi$
$17$	200.175	0.167992	0.0839959	+	0.996466i	$0.473232\pi$
$18$	0	0
$19$	−828.747	−0.526669	−0.263335	−	0.964705i	$-0.584822\pi$
$19$	−828.747	−0.526669	−0.263335	+	0.964705i	$0.584822\pi$
$20$	0	0
$21$	−14.9026	−0.00737418
$22$	0	0
$23$	4435.43	1.74830	0.874149	−	0.485657i	$-0.161420\pi$
$23$	4435.43	1.74830	0.874149	+	0.485657i	$0.161420\pi$
$24$	0	0
$25$	−1863.28	−0.596250
$26$	0	0
$27$	147.781	0.0390131
$28$	0	0
$29$	3717.74	0.820889	0.410444	−	0.911886i	$-0.365374\pi$
$29$	3717.74	0.820889	0.410444	+	0.911886i	$0.365374\pi$
$30$	0	0
$31$	992.462	0.185485	0.0927427	−	0.995690i	$-0.470437\pi$
$31$	992.462	0.185485	0.0927427	+	0.995690i	$0.470437\pi$
$32$	0	0
$33$	172.087	0.0275083
$34$	0	0
$35$	−1740.51	−0.240164
$36$	0	0
$37$	8359.69	1.00389	0.501945	−	0.864900i	$-0.332618\pi$
$37$	8359.69	1.00389	0.501945	+	0.864900i	$0.332618\pi$
$38$	0	0
$39$	299.145	0.0314934
$40$	0	0
$41$	−13473.0	−1.25171	−0.625855	−	0.779940i	$-0.715250\pi$
$41$	−13473.0	−1.25171	−0.625855	+	0.779940i	$0.715250\pi$
$42$	0	0
$43$	−298.798	−0.0246437	−0.0123218	−	0.999924i	$-0.503922\pi$
$43$	−298.798	−0.0246437	−0.0123218	+	0.999924i	$0.503922\pi$
$44$	0	0
$45$	8628.24	0.635171
$46$	0	0
$47$	−18736.5	−1.23721	−0.618606	−	0.785701i	$-0.712302\pi$
$47$	−18736.5	−1.23721	−0.618606	+	0.785701i	$0.712302\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	−60.8802	−0.00327756
$52$	0	0
$53$	−16036.4	−0.784181	−0.392090	−	0.919927i	$-0.628248\pi$
$53$	−16036.4	−0.784181	−0.392090	+	0.919927i	$0.628248\pi$
$54$	0	0
$55$	20098.5	0.895894
$56$	0	0
$57$	252.051	0.0102754
$58$	0	0
$59$	−12749.2	−0.476817	−0.238408	−	0.971165i	$-0.576626\pi$
$59$	−12749.2	−0.476817	−0.238408	+	0.971165i	$0.576626\pi$
$60$	0	0
$61$	34975.9	1.20350	0.601748	−	0.798686i	$-0.294471\pi$
$61$	34975.9	1.20350	0.601748	+	0.798686i	$0.294471\pi$
$62$	0	0
$63$	−11902.5	−0.377821
$64$	0	0
$65$	34937.9	1.02568
$66$	0	0
$67$	−11978.9	−0.326009	−0.163004	−	0.986625i	$-0.552118\pi$
$67$	−11978.9	−0.326009	−0.163004	+	0.986625i	$0.552118\pi$
$68$	0	0
$69$	−1348.97	−0.0341097
$70$	0	0
$71$	−12924.9	−0.304285	−0.152143	−	0.988359i	$-0.548617\pi$
$71$	−12924.9	−0.304285	−0.152143	+	0.988359i	$0.548617\pi$
$72$	0	0
$73$	81177.2	1.78290	0.891450	−	0.453119i	$-0.149689\pi$
$73$	81177.2	1.78290	0.891450	+	0.453119i	$0.149689\pi$
$74$	0	0
$75$	566.689	0.0116330
$76$	0	0
$77$	−27725.4	−0.532907
$78$	0	0
$79$	46998.8	0.847265	0.423632	−	0.905834i	$-0.360755\pi$
$79$	46998.8	0.847265	0.423632	+	0.905834i	$0.360755\pi$
$80$	0	0
$81$	58981.6	0.998858
$82$	0	0
$83$	111544.	1.77726	0.888632	−	0.458621i	$-0.151656\pi$
$83$	111544.	1.77726	0.888632	+	0.458621i	$0.151656\pi$
$84$	0	0
$85$	−7110.36	−0.106744
$86$	0	0
$87$	−1130.69	−0.0160157
$88$	0	0
$89$	−34726.8	−0.464719	−0.232359	−	0.972630i	$-0.574645\pi$
$89$	−34726.8	−0.464719	−0.232359	+	0.972630i	$0.574645\pi$
$90$	0	0
$91$	−48196.1	−0.610110
$92$	0	0
$93$	−301.842	−0.00361887
$94$	0	0
$95$	29437.7	0.334653
$96$	0	0
$97$	92655.6	0.999867	0.499933	−	0.866064i	$-0.333358\pi$
$97$	92655.6	0.999867	0.499933	+	0.866064i	$0.333358\pi$
$98$	0	0
$99$	137443.	1.40940
$100$	0	0
$101$	154765.	1.50963	0.754814	−	0.655939i	$-0.227727\pi$
$101$	154765.	1.50963	0.754814	+	0.655939i	$0.227727\pi$
$102$	0	0
$103$	−197146.	−1.83102	−0.915512	−	0.402290i	$-0.868214\pi$
$103$	−197146.	−1.83102	−0.915512	+	0.402290i	$0.868214\pi$
$104$	0	0
$105$	529.350	0.00468565
$106$	0	0
$107$	−63901.3	−0.539573	−0.269786	−	0.962920i	$-0.586953\pi$
$107$	−63901.3	−0.539573	−0.269786	+	0.962920i	$0.586953\pi$
$108$	0	0
$109$	−66910.4	−0.539421	−0.269710	−	0.962941i	$-0.586928\pi$
$109$	−66910.4	−0.539421	−0.269710	+	0.962941i	$0.586928\pi$
$110$	0	0
$111$	−2542.47	−0.0195861
$112$	0	0
$113$	−100524.	−0.740580	−0.370290	−	0.928916i	$-0.620742\pi$
$113$	−100524.	−0.740580	−0.370290	+	0.928916i	$0.620742\pi$
$114$	0	0
$115$	−157549.	−1.11089
$116$	0	0
$117$	238922.	1.61359
$118$	0	0
$119$	9808.58	0.0634949
$120$	0	0
$121$	159107.	0.987928
$122$	0	0
$123$	4097.60	0.0244212
$124$	0	0
$125$	177187.	1.01428
$126$	0	0
$127$	−155009.	−0.852800	−0.426400	−	0.904535i	$-0.640218\pi$
$127$	−155009.	−0.852800	−0.426400	+	0.904535i	$0.640218\pi$
$128$	0	0
$129$	90.8747	0.000480805 0
$130$	0	0
$131$	70396.8	0.358405	0.179203	−	0.983812i	$-0.442648\pi$
$131$	70396.8	0.358405	0.179203	+	0.983812i	$0.442648\pi$
$132$	0	0
$133$	−40608.6	−0.199062
$134$	0	0
$135$	−5249.29	−0.0247894
$136$	0	0
$137$	175732.	0.799925	0.399962	−	0.916532i	$-0.369023\pi$
$137$	175732.	0.799925	0.399962	+	0.916532i	$0.369023\pi$
$138$	0	0
$139$	−44017.9	−0.193238	−0.0966189	−	0.995321i	$-0.530803\pi$
$139$	−44017.9	−0.193238	−0.0966189	+	0.995321i	$0.530803\pi$
$140$	0	0
$141$	5698.43	0.0241383
$142$	0	0
$143$	556542.	2.27592
$144$	0	0
$145$	−132057.	−0.521603
$146$	0	0
$147$	−730.227	−0.00278718
$148$	0	0
$149$	121136.	0.447001	0.223500	−	0.974704i	$-0.428252\pi$
$149$	121136.	0.447001	0.223500	+	0.974704i	$0.428252\pi$
$150$	0	0
$151$	398463.	1.42215	0.711075	−	0.703116i	$-0.248209\pi$
$151$	398463.	1.42215	0.711075	+	0.703116i	$0.248209\pi$
$152$	0	0
$153$	−48624.1	−0.167928
$154$	0	0
$155$	−35252.9	−0.117860
$156$	0	0
$157$	136739.	0.442733	0.221367	−	0.975191i	$-0.428948\pi$
$157$	136739.	0.442733	0.221367	+	0.975191i	$0.428948\pi$
$158$	0	0
$159$	4877.21	0.0152996
$160$	0	0
$161$	217336.	0.660795
$162$	0	0
$163$	−642409.	−1.89384	−0.946919	−	0.321473i	$-0.895822\pi$
$163$	−642409.	−1.89384	−0.946919	+	0.321473i	$0.895822\pi$
$164$	0	0
$165$	−6112.64	−0.0174791
$166$	0	0
$167$	−290841.	−0.806984	−0.403492	−	0.914983i	$-0.632204\pi$
$167$	−290841.	−0.806984	−0.403492	+	0.914983i	$0.632204\pi$
$168$	0	0
$169$	596163.	1.60564
$170$	0	0
$171$	201309.	0.526469
$172$	0	0
$173$	562234.	1.42824	0.714121	−	0.700022i	$-0.246827\pi$
$173$	562234.	1.42824	0.714121	+	0.700022i	$0.246827\pi$
$174$	0	0
$175$	−91300.8	−0.225361
$176$	0	0
$177$	3877.46	0.00930281
$178$	0	0
$179$	312167.	0.728207	0.364103	−	0.931359i	$-0.381375\pi$
$179$	312167.	0.728207	0.364103	+	0.931359i	$0.381375\pi$
$180$	0	0
$181$	−254232.	−0.576811	−0.288405	−	0.957508i	$-0.593125\pi$
$181$	−254232.	−0.576811	−0.288405	+	0.957508i	$0.593125\pi$
$182$	0	0
$183$	−10637.4	−0.0234805
$184$	0	0
$185$	−296942.	−0.637884
$186$	0	0
$187$	−113264.	−0.236858
$188$	0	0
$189$	7241.29	0.0147456
$190$	0	0
$191$	−189060.	−0.374986	−0.187493	−	0.982266i	$-0.560036\pi$
$191$	−189060.	−0.374986	−0.187493	+	0.982266i	$0.560036\pi$
$192$	0	0
$193$	660426.	1.27624	0.638118	−	0.769939i	$-0.279713\pi$
$193$	660426.	1.27624	0.638118	+	0.769939i	$0.279713\pi$
$194$	0	0
$195$	−10625.8	−0.0200113
$196$	0	0
$197$	786619.	1.44410	0.722052	−	0.691838i	$-0.243199\pi$
$197$	786619.	1.44410	0.722052	+	0.691838i	$0.243199\pi$
$198$	0	0
$199$	360530.	0.645370	0.322685	−	0.946506i	$-0.395414\pi$
$199$	360530.	0.645370	0.322685	+	0.946506i	$0.395414\pi$
$200$	0	0
$201$	3643.19	0.00636051
$202$	0	0
$203$	182169.	0.310267
$204$	0	0
$205$	478569.	0.795353
$206$	0	0
$207$	−1.07740e6	−1.74763
$208$	0	0
$209$	468926.	0.742571
$210$	0	0
$211$	1.13491e6	1.75491	0.877455	−	0.479658i	$-0.159239\pi$
$211$	1.13491e6	1.75491	0.877455	+	0.479658i	$0.159239\pi$
$212$	0	0
$213$	3930.91	0.00593668
$214$	0	0
$215$	10613.5	0.0156589
$216$	0	0
$217$	48630.7	0.0701069
$218$	0	0
$219$	−24688.8	−0.0347848
$220$	0	0
$221$	−196891.	−0.271172
$222$	0	0
$223$	−783934.	−1.05564	−0.527822	−	0.849355i	$-0.676991\pi$
$223$	−783934.	−1.05564	−0.527822	+	0.849355i	$0.676991\pi$
$224$	0	0
$225$	452605.	0.596023
$226$	0	0
$227$	−600754.	−0.773806	−0.386903	−	0.922120i	$-0.626455\pi$
$227$	−600754.	−0.773806	−0.386903	+	0.922120i	$0.626455\pi$
$228$	0	0
$229$	−1.47992e6	−1.86487	−0.932435	−	0.361337i	$-0.882320\pi$
$229$	−1.47992e6	−1.86487	−0.932435	+	0.361337i	$0.882320\pi$
$230$	0	0
$231$	8432.26	0.0103971
$232$	0	0
$233$	−1.04350e6	−1.25922	−0.629610	−	0.776911i	$-0.716785\pi$
$233$	−1.04350e6	−1.25922	−0.629610	+	0.776911i	$0.716785\pi$
$234$	0	0
$235$	665534.	0.786141
$236$	0	0
$237$	−14294.0	−0.0165303
$238$	0	0
$239$	262948.	0.297767	0.148883	−	0.988855i	$-0.452432\pi$
$239$	262948.	0.297767	0.148883	+	0.988855i	$0.452432\pi$
$240$	0	0
$241$	−1.79946e6	−1.99571	−0.997857	−	0.0654284i	$-0.979159\pi$
$241$	−1.79946e6	−1.99571	−0.997857	+	0.0654284i	$0.979159\pi$
$242$	0	0
$243$	−53849.2	−0.0585010
$244$	0	0
$245$	−85285.1	−0.0907733
$246$	0	0
$247$	815150.	0.850149
$248$	0	0
$249$	−33924.5	−0.0346749
$250$	0	0
$251$	−1.06631e6	−1.06831	−0.534156	−	0.845386i	$-0.679370\pi$
$251$	−1.06631e6	−1.06831	−0.534156	+	0.845386i	$0.679370\pi$
$252$	0	0
$253$	−2.50967e6	−2.46499
$254$	0	0
$255$	2162.51	0.00208261
$256$	0	0
$257$	675178.	0.637654	0.318827	−	0.947813i	$-0.396711\pi$
$257$	675178.	0.637654	0.318827	+	0.947813i	$0.396711\pi$
$258$	0	0
$259$	409625.	0.379434
$260$	0	0
$261$	−903067.	−0.820576
$262$	0	0
$263$	301937.	0.269170	0.134585	−	0.990902i	$-0.457030\pi$
$263$	301937.	0.269170	0.134585	+	0.990902i	$0.457030\pi$
$264$	0	0
$265$	569622.	0.498279
$266$	0	0
$267$	10561.6	0.00906678
$268$	0	0
$269$	817428.	0.688762	0.344381	−	0.938830i	$-0.388089\pi$
$269$	817428.	0.688762	0.344381	+	0.938830i	$0.388089\pi$
$270$	0	0
$271$	896619.	0.741626	0.370813	−	0.928708i	$-0.379079\pi$
$271$	896619.	0.741626	0.370813	+	0.928708i	$0.379079\pi$
$272$	0	0
$273$	14658.1	0.0119034
$274$	0	0
$275$	1.05429e6	0.840676
$276$	0	0
$277$	909620.	0.712296	0.356148	−	0.934430i	$-0.384090\pi$
$277$	909620.	0.712296	0.356148	+	0.934430i	$0.384090\pi$
$278$	0	0
$279$	−241077.	−0.185415
$280$	0	0
$281$	762090.	0.575758	0.287879	−	0.957667i	$-0.407050\pi$
$281$	762090.	0.575758	0.287879	+	0.957667i	$0.407050\pi$
$282$	0	0
$283$	−931060.	−0.691054	−0.345527	−	0.938409i	$-0.612300\pi$
$283$	−931060.	−0.691054	−0.345527	+	0.938409i	$0.612300\pi$
$284$	0	0
$285$	−8953.01	−0.00652916
$286$	0	0
$287$	−660175.	−0.473102
$288$	0	0
$289$	−1.37979e6	−0.971779
$290$	0	0
$291$	−28179.8	−0.0195076
$292$	0	0
$293$	−204188.	−0.138951	−0.0694755	−	0.997584i	$-0.522133\pi$
$293$	−204188.	−0.138951	−0.0694755	+	0.997584i	$0.522133\pi$
$294$	0	0
$295$	452859.	0.302976
$296$	0	0
$297$	−83618.3	−0.0550060
$298$	0	0
$299$	−4.36266e6	−2.82210
$300$	0	0
$301$	−14641.1	−0.00931444
$302$	0	0
$303$	−47069.5	−0.0294532
$304$	0	0
$305$	−1.24237e6	−0.764717
$306$	0	0
$307$	1.00786e6	0.610314	0.305157	−	0.952302i	$-0.401291\pi$
$307$	1.00786e6	0.610314	0.305157	+	0.952302i	$0.401291\pi$
$308$	0	0
$309$	59958.8	0.0357238
$310$	0	0
$311$	1.77736e6	1.04202	0.521009	−	0.853551i	$-0.325556\pi$
$311$	1.77736e6	1.04202	0.521009	+	0.853551i	$0.325556\pi$
$312$	0	0
$313$	−1.73100e6	−0.998705	−0.499352	−	0.866399i	$-0.666429\pi$
$313$	−1.73100e6	−0.998705	−0.499352	+	0.866399i	$0.666429\pi$
$314$	0	0
$315$	422784.	0.240072
$316$	0	0
$317$	2.22072e6	1.24121	0.620605	−	0.784123i	$-0.286887\pi$
$317$	2.22072e6	1.24121	0.620605	+	0.784123i	$0.286887\pi$
$318$	0	0
$319$	−2.10359e6	−1.15740
$320$	0	0
$321$	19434.6	0.0105272
$322$	0	0
$323$	−165895.	−0.0884761
$324$	0	0
$325$	1.83271e6	0.962467
$326$	0	0
$327$	20349.8	0.0105242
$328$	0	0
$329$	−918089.	−0.467622
$330$	0	0
$331$	2.45218e6	1.23022	0.615109	−	0.788442i	$-0.289112\pi$
$331$	2.45218e6	1.23022	0.615109	+	0.788442i	$0.289112\pi$
$332$	0	0
$333$	−2.03063e6	−1.00351
$334$	0	0
$335$	425498.	0.207150
$336$	0	0
$337$	−2.06932e6	−0.992549	−0.496275	−	0.868166i	$-0.665299\pi$
$337$	−2.06932e6	−0.992549	−0.496275	+	0.868166i	$0.665299\pi$
$338$	0	0
$339$	30572.7	0.0144489
$340$	0	0
$341$	−561560.	−0.261523
$342$	0	0
$343$	117649.	0.0539949
$344$	0	0
$345$	47916.2	0.0216738
$346$	0	0
$347$	2.21939e6	0.989487	0.494743	−	0.869039i	$-0.335262\pi$
$347$	2.21939e6	0.989487	0.494743	+	0.869039i	$0.335262\pi$
$348$	0	0
$349$	−2.92580e6	−1.28582	−0.642911	−	0.765941i	$-0.722273\pi$
$349$	−2.92580e6	−1.28582	−0.642911	+	0.765941i	$0.722273\pi$
$350$	0	0
$351$	−145357.	−0.0629749
$352$	0	0
$353$	−3.03319e6	−1.29558	−0.647788	−	0.761821i	$-0.724306\pi$
$353$	−3.03319e6	−1.29558	−0.647788	+	0.761821i	$0.724306\pi$
$354$	0	0
$355$	459101.	0.193347
$356$	0	0
$357$	−2983.13	−0.00123880
$358$	0	0
$359$	3.66596e6	1.50125	0.750623	−	0.660731i	$-0.229754\pi$
$359$	3.66596e6	1.50125	0.750623	+	0.660731i	$0.229754\pi$
$360$	0	0
$361$	−1.78928e6	−0.722619
$362$	0	0
$363$	−48389.9	−0.0192747
$364$	0	0
$365$	−2.88347e6	−1.13288
$366$	0	0
$367$	−3.47355e6	−1.34620	−0.673098	−	0.739553i	$-0.735037\pi$
$367$	−3.47355e6	−1.34620	−0.673098	+	0.739553i	$0.735037\pi$
$368$	0	0
$369$	3.27268e6	1.25123
$370$	0	0
$371$	−785782.	−0.296392
$372$	0	0
$373$	−581430.	−0.216384	−0.108192	−	0.994130i	$-0.534506\pi$
$373$	−581430.	−0.216384	−0.108192	+	0.994130i	$0.534506\pi$
$374$	0	0
$375$	−53888.8	−0.0197888
$376$	0	0
$377$	−3.65675e6	−1.32508
$378$	0	0
$379$	−1.83859e6	−0.657486	−0.328743	−	0.944419i	$-0.606625\pi$
$379$	−1.83859e6	−0.657486	−0.328743	+	0.944419i	$0.606625\pi$
$380$	0	0
$381$	47143.6	0.0166383
$382$	0	0
$383$	139133.	0.0484655	0.0242328	−	0.999706i	$-0.492286\pi$
$383$	139133.	0.0484655	0.0242328	+	0.999706i	$0.492286\pi$
$384$	0	0
$385$	984825.	0.338616
$386$	0	0
$387$	72580.2	0.0246343
$388$	0	0
$389$	−2.21985e6	−0.743788	−0.371894	−	0.928275i	$-0.621292\pi$
$389$	−2.21985e6	−0.743788	−0.371894	+	0.928275i	$0.621292\pi$
$390$	0	0
$391$	887862.	0.293700
$392$	0	0
$393$	−21410.1	−0.00699258
$394$	0	0
$395$	−1.66943e6	−0.538363
$396$	0	0
$397$	−505998.	−0.161129	−0.0805643	−	0.996749i	$-0.525672\pi$
$397$	−505998.	−0.161129	−0.0805643	+	0.996749i	$0.525672\pi$
$398$	0	0
$399$	12350.5	0.00388375
$400$	0	0
$401$	2.06663e6	0.641802	0.320901	−	0.947113i	$-0.396014\pi$
$401$	2.06663e6	0.641802	0.320901	+	0.947113i	$0.396014\pi$
$402$	0	0
$403$	−976180.	−0.299411
$404$	0	0
$405$	−2.09507e6	−0.634688
$406$	0	0
$407$	−4.73012e6	−1.41542
$408$	0	0
$409$	5.96711e6	1.76383	0.881913	−	0.471411i	$-0.156255\pi$
$409$	5.96711e6	1.76383	0.881913	+	0.471411i	$0.156255\pi$
$410$	0	0
$411$	−53446.2	−0.0156067
$412$	0	0
$413$	−624709.	−0.180220
$414$	0	0
$415$	−3.96213e6	−1.12930
$416$	0	0
$417$	13387.4	0.00377012
$418$	0	0
$419$	569081.	0.158358	0.0791788	−	0.996860i	$-0.474770\pi$
$419$	569081.	0.158358	0.0791788	+	0.996860i	$0.474770\pi$
$420$	0	0
$421$	171377.	0.0471247	0.0235623	−	0.999722i	$-0.492499\pi$
$421$	171377.	0.0471247	0.0235623	+	0.999722i	$0.492499\pi$
$422$	0	0
$423$	4.55124e6	1.23674
$424$	0	0
$425$	−372983.	−0.100165
$426$	0	0
$427$	1.71382e6	0.454879
$428$	0	0
$429$	−169264.	−0.0444038
$430$	0	0
$431$	4.75930e6	1.23410	0.617049	−	0.786924i	$-0.288328\pi$
$431$	4.75930e6	1.23410	0.617049	+	0.786924i	$0.288328\pi$
$432$	0	0
$433$	623369.	0.159781	0.0798906	−	0.996804i	$-0.474543\pi$
$433$	623369.	0.159781	0.0798906	+	0.996804i	$0.474543\pi$
$434$	0	0
$435$	40163.0	0.0101766
$436$	0	0
$437$	−3.67585e6	−0.920775
$438$	0	0
$439$	−5.24578e6	−1.29912	−0.649559	−	0.760311i	$-0.725046\pi$
$439$	−5.24578e6	−1.29912	−0.649559	+	0.760311i	$0.725046\pi$
$440$	0	0
$441$	−583221.	−0.142803
$442$	0	0
$443$	−5.64926e6	−1.36767	−0.683837	−	0.729635i	$-0.739690\pi$
$443$	−5.64926e6	−1.36767	−0.683837	+	0.729635i	$0.739690\pi$
$444$	0	0
$445$	1.23352e6	0.295288
$446$	0	0
$447$	−36841.7	−0.00872110
$448$	0	0
$449$	812156.	0.190118	0.0950591	−	0.995472i	$-0.469696\pi$
$449$	812156.	0.190118	0.0950591	+	0.995472i	$0.469696\pi$
$450$	0	0
$451$	7.62334e6	1.76483
$452$	0	0
$453$	−121186.	−0.0277465
$454$	0	0
$455$	1.71196e6	0.387672
$456$	0	0
$457$	−926920.	−0.207612	−0.103806	−	0.994598i	$-0.533102\pi$
$457$	−926920.	−0.207612	−0.103806	+	0.994598i	$0.533102\pi$
$458$	0	0
$459$	29582.2	0.00655387
$460$	0	0
$461$	3.67043e6	0.804386	0.402193	−	0.915555i	$-0.368248\pi$
$461$	3.67043e6	0.804386	0.402193	+	0.915555i	$0.368248\pi$
$462$	0	0
$463$	−2.70462e6	−0.586345	−0.293173	−	0.956060i	$-0.594711\pi$
$463$	−2.70462e6	−0.586345	−0.293173	+	0.956060i	$0.594711\pi$
$464$	0	0
$465$	10721.6	0.00229948
$466$	0	0
$467$	8.24431e6	1.74929	0.874645	−	0.484763i	$-0.161094\pi$
$467$	8.24431e6	1.74929	0.874645	+	0.484763i	$0.161094\pi$
$468$	0	0
$469$	−586965.	−0.123220
$470$	0	0
$471$	−41587.0	−0.00863784
$472$	0	0
$473$	169067.	0.0347461
$474$	0	0
$475$	1.54419e6	0.314027
$476$	0	0
$477$	3.89535e6	0.783882
$478$	0	0
$479$	−2.55003e6	−0.507816	−0.253908	−	0.967228i	$-0.581716\pi$
$479$	−2.55003e6	−0.507816	−0.253908	+	0.967228i	$0.581716\pi$
$480$	0	0
$481$	−8.22253e6	−1.62048
$482$	0	0
$483$	−66099.4	−0.0128923
$484$	0	0
$485$	−3.29119e6	−0.635328
$486$	0	0
$487$	7.66815e6	1.46510	0.732552	−	0.680711i	$-0.238329\pi$
$487$	7.66815e6	1.46510	0.732552	+	0.680711i	$0.238329\pi$
$488$	0	0
$489$	195379.	0.0369492
$490$	0	0
$491$	−9.33268e6	−1.74704	−0.873519	−	0.486790i	$-0.838168\pi$
$491$	−9.33268e6	−1.74704	−0.873519	+	0.486790i	$0.838168\pi$
$492$	0	0
$493$	744200.	0.137903
$494$	0	0
$495$	−4.88207e6	−0.895553
$496$	0	0
$497$	−633320.	−0.115009
$498$	0	0
$499$	1.19065e6	0.214059	0.107030	−	0.994256i	$-0.465866\pi$
$499$	1.19065e6	0.214059	0.107030	+	0.994256i	$0.465866\pi$
$500$	0	0
$501$	88454.9	0.0157445
$502$	0	0
$503$	2.34213e6	0.412753	0.206377	−	0.978473i	$-0.433833\pi$
$503$	2.34213e6	0.412753	0.206377	+	0.978473i	$0.433833\pi$
$504$	0	0
$505$	−5.49736e6	−0.959237
$506$	0	0
$507$	−181314.	−0.0313265
$508$	0	0
$509$	6.69767e6	1.14585	0.572927	−	0.819606i	$-0.305808\pi$
$509$	6.69767e6	1.14585	0.572927	+	0.819606i	$0.305808\pi$
$510$	0	0
$511$	3.97768e6	0.673873
$512$	0	0
$513$	−122473.	−0.0205470
$514$	0	0
$515$	7.00275e6	1.16346
$516$	0	0
$517$	1.06016e7	1.74439
$518$	0	0
$519$	−170995.	−0.0278654
$520$	0	0
$521$	5.81727e6	0.938911	0.469456	−	0.882956i	$-0.344450\pi$
$521$	5.81727e6	0.938911	0.469456	+	0.882956i	$0.344450\pi$
$522$	0	0
$523$	−1.02571e7	−1.63972	−0.819858	−	0.572568i	$-0.805947\pi$
$523$	−1.02571e7	−1.63972	−0.819858	+	0.572568i	$0.805947\pi$
$524$	0	0
$525$	27767.7	0.00439686
$526$	0	0
$527$	198666.	0.0311600
$528$	0	0
$529$	1.32367e7	2.05655
$530$	0	0
$531$	3.09687e6	0.476635
$532$	0	0
$533$	1.32519e7	2.02051
$534$	0	0
$535$	2.26982e6	0.342852
$536$	0	0
$537$	−94940.9	−0.0142075
$538$	0	0
$539$	−1.35855e6	−0.201420
$540$	0	0
$541$	−5.26601e6	−0.773550	−0.386775	−	0.922174i	$-0.626411\pi$
$541$	−5.26601e6	−0.773550	−0.386775	+	0.922174i	$0.626411\pi$
$542$	0	0
$543$	77320.7	0.0112537
$544$	0	0
$545$	2.37670e6	0.342755
$546$	0	0
$547$	−4.89900e6	−0.700066	−0.350033	−	0.936737i	$-0.613830\pi$
$547$	−4.89900e6	−0.700066	−0.350033	+	0.936737i	$0.613830\pi$
$548$	0	0
$549$	−8.49592e6	−1.20304
$550$	0	0
$551$	−3.08107e6	−0.432337
$552$	0	0
$553$	2.30294e6	0.320236
$554$	0	0
$555$	90310.3	0.0124453
$556$	0	0
$557$	9.17946e6	1.25366	0.626829	−	0.779157i	$-0.284352\pi$
$557$	9.17946e6	1.25366	0.626829	+	0.779157i	$0.284352\pi$
$558$	0	0
$559$	293895.	0.0397799
$560$	0	0
$561$	34447.5	0.00462116
$562$	0	0
$563$	249472.	0.0331704	0.0165852	−	0.999862i	$-0.494721\pi$
$563$	249472.	0.0331704	0.0165852	+	0.999862i	$0.494721\pi$
$564$	0	0
$565$	3.57066e6	0.470574
$566$	0	0
$567$	2.89010e6	0.377533
$568$	0	0
$569$	1.31498e7	1.70270	0.851351	−	0.524596i	$-0.175784\pi$
$569$	1.31498e7	1.70270	0.851351	+	0.524596i	$0.175784\pi$
$570$	0	0
$571$	−1.51038e7	−1.93863	−0.969316	−	0.245817i	$-0.920944\pi$
$571$	−1.51038e7	−1.93863	−0.969316	+	0.245817i	$0.920944\pi$
$572$	0	0
$573$	57499.6	0.00731608
$574$	0	0
$575$	−8.26445e6	−1.04242
$576$	0	0
$577$	−727822.	−0.0910093	−0.0455046	−	0.998964i	$-0.514490\pi$
$577$	−727822.	−0.0910093	−0.0455046	+	0.998964i	$0.514490\pi$
$578$	0	0
$579$	−200858.	−0.0248997
$580$	0	0
$581$	5.46567e6	0.671743
$582$	0	0
$583$	9.07377e6	1.10565
$584$	0	0
$585$	−8.48668e6	−1.02529
$586$	0	0
$587$	−1.24256e7	−1.48841	−0.744207	−	0.667949i	$-0.767172\pi$
$587$	−1.24256e7	−1.48841	−0.744207	+	0.667949i	$0.767172\pi$
$588$	0	0
$589$	−822500.	−0.0976895
$590$	0	0
$591$	−239238.	−0.0281748
$592$	0	0
$593$	6.17981e6	0.721669	0.360835	−	0.932630i	$-0.382492\pi$
$593$	6.17981e6	0.721669	0.360835	+	0.932630i	$0.382492\pi$
$594$	0	0
$595$	−348408.	−0.0403455
$596$	0	0
$597$	−109650.	−0.0125913
$598$	0	0
$599$	1.30603e7	1.48726	0.743630	−	0.668592i	$-0.233103\pi$
$599$	1.30603e7	1.48726	0.743630	+	0.668592i	$0.233103\pi$
$600$	0	0
$601$	9.01833e6	1.01845	0.509225	−	0.860633i	$-0.329932\pi$
$601$	9.01833e6	1.01845	0.509225	+	0.860633i	$0.329932\pi$
$602$	0	0
$603$	2.90976e6	0.325884
$604$	0	0
$605$	−5.65158e6	−0.627742
$606$	0	0
$607$	−1.36835e7	−1.50739	−0.753694	−	0.657226i	$-0.771730\pi$
$607$	−1.36835e7	−1.50739	−0.753694	+	0.657226i	$0.771730\pi$
$608$	0	0
$609$	−55404.0	−0.00605338
$610$	0	0
$611$	1.84291e7	1.99711
$612$	0	0
$613$	4.23266e6	0.454949	0.227474	−	0.973784i	$-0.426953\pi$
$613$	4.23266e6	0.454949	0.227474	+	0.973784i	$0.426953\pi$
$614$	0	0
$615$	−145549.	−0.0155175
$616$	0	0
$617$	3.32584e6	0.351713	0.175856	−	0.984416i	$-0.443731\pi$
$617$	3.32584e6	0.351713	0.175856	+	0.984416i	$0.443731\pi$
$618$	0	0
$619$	1.19987e7	1.25866	0.629328	−	0.777140i	$-0.283330\pi$
$619$	1.19987e7	1.25866	0.629328	+	0.777140i	$0.283330\pi$
$620$	0	0
$621$	655473.	0.0682065
$622$	0	0
$623$	−1.70162e6	−0.175647
$624$	0	0
$625$	−471051.	−0.0482356
$626$	0	0
$627$	−142617.	−0.0144878
$628$	0	0
$629$	1.67340e6	0.168645
$630$	0	0
$631$	9.99962e6	0.999793	0.499896	−	0.866085i	$-0.333371\pi$
$631$	9.99962e6	0.999793	0.499896	+	0.866085i	$0.333371\pi$
$632$	0	0
$633$	−345165.	−0.0342387
$634$	0	0
$635$	5.50602e6	0.541880
$636$	0	0
$637$	−2.36161e6	−0.230600
$638$	0	0
$639$	3.13955e6	0.304170
$640$	0	0
$641$	−3.16624e6	−0.304368	−0.152184	−	0.988352i	$-0.548631\pi$
$641$	−3.16624e6	−0.304368	−0.152184	+	0.988352i	$0.548631\pi$
$642$	0	0
$643$	−950578.	−0.0906693	−0.0453346	−	0.998972i	$-0.514435\pi$
$643$	−950578.	−0.0906693	−0.0453346	+	0.998972i	$0.514435\pi$
$644$	0	0
$645$	−3227.93	−0.000305510 0
$646$	0	0
$647$	−1.14671e7	−1.07694	−0.538470	−	0.842645i	$-0.680997\pi$
$647$	−1.14671e7	−1.07694	−0.538470	+	0.842645i	$0.680997\pi$
$648$	0	0
$649$	7.21379e6	0.672282
$650$	0	0
$651$	−14790.3	−0.00136780
$652$	0	0
$653$	2.80936e6	0.257824	0.128912	−	0.991656i	$-0.458851\pi$
$653$	2.80936e6	0.257824	0.128912	+	0.991656i	$0.458851\pi$
$654$	0	0
$655$	−2.50054e6	−0.227735
$656$	0	0
$657$	−1.97186e7	−1.78222
$658$	0	0
$659$	4.23794e6	0.380138	0.190069	−	0.981771i	$-0.439129\pi$
$659$	4.23794e6	0.380138	0.190069	+	0.981771i	$0.439129\pi$
$660$	0	0
$661$	−1.70903e6	−0.152141	−0.0760703	−	0.997102i	$-0.524237\pi$
$661$	−1.70903e6	−0.152141	−0.0760703	+	0.997102i	$0.524237\pi$
$662$	0	0
$663$	59881.4	0.00529064
$664$	0	0
$665$	1.44244e6	0.126487
$666$	0	0
$667$	1.64898e7	1.43516
$668$	0	0
$669$	238421.	0.0205959
$670$	0	0
$671$	−1.97903e7	−1.69686
$672$	0	0
$673$	1.11371e7	0.947836	0.473918	−	0.880569i	$-0.342839\pi$
$673$	1.11371e7	0.947836	0.473918	+	0.880569i	$0.342839\pi$
$674$	0	0
$675$	−275358.	−0.0232616
$676$	0	0
$677$	−8.09873e6	−0.679118	−0.339559	−	0.940585i	$-0.610278\pi$
$677$	−8.09873e6	−0.679118	−0.339559	+	0.940585i	$0.610278\pi$
$678$	0	0
$679$	4.54012e6	0.377914
$680$	0	0
$681$	182710.	0.0150971
$682$	0	0
$683$	1.49864e7	1.22926	0.614632	−	0.788814i	$-0.289305\pi$
$683$	1.49864e7	1.22926	0.614632	+	0.788814i	$0.289305\pi$
$684$	0	0
$685$	−6.24211e6	−0.508283
$686$	0	0
$687$	450094.	0.0363841
$688$	0	0
$689$	1.57733e7	1.26582
$690$	0	0
$691$	−2.18569e7	−1.74138	−0.870691	−	0.491831i	$-0.836328\pi$
$691$	−2.18569e7	−1.74138	−0.870691	+	0.491831i	$0.836328\pi$
$692$	0	0
$693$	6.73471e6	0.532704
$694$	0	0
$695$	1.56354e6	0.122786
$696$	0	0
$697$	−2.69695e6	−0.210277
$698$	0	0
$699$	317364.	0.0245677
$700$	0	0
$701$	1.21819e7	0.936307	0.468154	−	0.883647i	$-0.344919\pi$
$701$	1.21819e7	0.936307	0.468154	+	0.883647i	$0.344919\pi$
$702$	0	0
$703$	−6.92807e6	−0.528718
$704$	0	0
$705$	−202412.	−0.0153378
$706$	0	0
$707$	7.58349e6	0.570585
$708$	0	0
$709$	8.03040e6	0.599959	0.299980	−	0.953946i	$-0.403020\pi$
$709$	8.03040e6	0.599959	0.299980	+	0.953946i	$0.403020\pi$
$710$	0	0
$711$	−1.14164e7	−0.846942
$712$	0	0
$713$	4.40199e6	0.324284
$714$	0	0
$715$	−1.97687e7	−1.44615
$716$	0	0
$717$	−79971.8	−0.00580950
$718$	0	0
$719$	−1.76681e7	−1.27458	−0.637290	−	0.770624i	$-0.719945\pi$
$719$	−1.76681e7	−1.27458	−0.637290	+	0.770624i	$0.719945\pi$
$720$	0	0
$721$	−9.66014e6	−0.692062
$722$	0	0
$723$	547277.	0.0389369
$724$	0	0
$725$	−6.92720e6	−0.489455
$726$	0	0
$727$	1.71905e7	1.20629	0.603146	−	0.797631i	$-0.293914\pi$
$727$	1.71905e7	1.20629	0.603146	+	0.797631i	$0.293914\pi$
$728$	0	0
$729$	−1.43161e7	−0.997717
$730$	0	0
$731$	−59811.9	−0.00413994
$732$	0	0
$733$	−9.30219e6	−0.639478	−0.319739	−	0.947506i	$-0.603595\pi$
$733$	−9.30219e6	−0.639478	−0.319739	+	0.947506i	$0.603595\pi$
$734$	0	0
$735$	25938.2	0.00177101
$736$	0	0
$737$	6.77794e6	0.459652
$738$	0	0
$739$	1.29625e7	0.873129	0.436564	−	0.899673i	$-0.356195\pi$
$739$	1.29625e7	0.873129	0.436564	+	0.899673i	$0.356195\pi$
$740$	0	0
$741$	−247915.	−0.0165866
$742$	0	0
$743$	1.45466e7	0.966693	0.483347	−	0.875429i	$-0.339421\pi$
$743$	1.45466e7	0.966693	0.483347	+	0.875429i	$0.339421\pi$
$744$	0	0
$745$	−4.30284e6	−0.284030
$746$	0	0
$747$	−2.70949e7	−1.77659
$748$	0	0
$749$	−3.13116e6	−0.203939
$750$	0	0
$751$	1.44950e7	0.937816	0.468908	−	0.883247i	$-0.344648\pi$
$751$	1.44950e7	0.937816	0.468908	+	0.883247i	$0.344648\pi$
$752$	0	0
$753$	324301.	0.0208430
$754$	0	0
$755$	−1.41537e7	−0.903653
$756$	0	0
$757$	−2.49809e7	−1.58441	−0.792207	−	0.610252i	$-0.791068\pi$
$757$	−2.49809e7	−1.58441	−0.792207	+	0.610252i	$0.791068\pi$
$758$	0	0
$759$	763279.	0.0480927
$760$	0	0
$761$	1.56594e7	0.980196	0.490098	−	0.871667i	$-0.336961\pi$
$761$	1.56594e7	0.980196	0.490098	+	0.871667i	$0.336961\pi$
$762$	0	0
$763$	−3.27861e6	−0.203882
$764$	0	0
$765$	1.72716e6	0.106704
$766$	0	0
$767$	1.25400e7	0.769677
$768$	0	0
$769$	1.30420e7	0.795292	0.397646	−	0.917539i	$-0.369827\pi$
$769$	1.30420e7	0.795292	0.397646	+	0.917539i	$0.369827\pi$
$770$	0	0
$771$	−205345.	−0.0124408
$772$	0	0
$773$	1.46084e7	0.879336	0.439668	−	0.898160i	$-0.355096\pi$
$773$	1.46084e7	0.879336	0.439668	+	0.898160i	$0.355096\pi$
$774$	0	0
$775$	−1.84924e6	−0.110596
$776$	0	0
$777$	−124581.	−0.00740286
$778$	0	0
$779$	1.11657e7	0.659237
$780$	0	0
$781$	7.31323e6	0.429024
$782$	0	0
$783$	549413.	0.0320254
$784$	0	0
$785$	−4.85705e6	−0.281319
$786$	0	0
$787$	2.62167e7	1.50883	0.754416	−	0.656396i	$-0.227920\pi$
$787$	2.62167e7	1.50883	0.754416	+	0.656396i	$0.227920\pi$
$788$	0	0
$789$	−91829.4	−0.00525157
$790$	0	0
$791$	−4.92565e6	−0.279913
$792$	0	0
$793$	−3.44021e7	−1.94268
$794$	0	0
$795$	−173242.	−0.00972154
$796$	0	0
$797$	−3.41107e6	−0.190215	−0.0951075	−	0.995467i	$-0.530319\pi$
$797$	−3.41107e6	−0.190215	−0.0951075	+	0.995467i	$0.530319\pi$
$798$	0	0
$799$	−3.75059e6	−0.207841
$800$	0	0
$801$	8.43541e6	0.464542
$802$	0	0
$803$	−4.59321e7	−2.51378
$804$	0	0
$805$	−7.71992e6	−0.419878
$806$	0	0
$807$	−248608.	−0.0134379
$808$	0	0
$809$	−1.72606e7	−0.927225	−0.463612	−	0.886038i	$-0.653447\pi$
$809$	−1.72606e7	−0.927225	−0.463612	+	0.886038i	$0.653447\pi$
$810$	0	0
$811$	−2.42616e7	−1.29529	−0.647647	−	0.761941i	$-0.724247\pi$
$811$	−2.42616e7	−1.29529	−0.647647	+	0.761941i	$0.724247\pi$
$812$	0	0
$813$	−272693.	−0.0144693
$814$	0	0
$815$	2.28188e7	1.20337
$816$	0	0
$817$	247628.	0.0129791
$818$	0	0
$819$	1.17072e7	0.609878
$820$	0	0
$821$	3.29927e6	0.170829	0.0854143	−	0.996346i	$-0.472779\pi$
$821$	3.29927e6	0.170829	0.0854143	+	0.996346i	$0.472779\pi$
$822$	0	0
$823$	2.15209e7	1.10754	0.553772	−	0.832669i	$-0.313188\pi$
$823$	2.15209e7	1.10754	0.553772	+	0.832669i	$0.313188\pi$
$824$	0	0
$825$	−320646.	−0.0164018
$826$	0	0
$827$	7.31588e6	0.371966	0.185983	−	0.982553i	$-0.440453\pi$
$827$	7.31588e6	0.371966	0.185983	+	0.982553i	$0.440453\pi$
$828$	0	0
$829$	2.33238e7	1.17873	0.589364	−	0.807867i	$-0.299378\pi$
$829$	2.33238e7	1.17873	0.589364	+	0.807867i	$0.299378\pi$
$830$	0	0
$831$	−276647.	−0.0138971
$832$	0	0
$833$	480621.	0.0239988
$834$	0	0
$835$	1.03309e7	0.512768
$836$	0	0
$837$	146667.	0.00723636
$838$	0	0
$839$	−2.69417e7	−1.32136	−0.660678	−	0.750669i	$-0.729731\pi$
$839$	−2.69417e7	−1.32136	−0.660678	+	0.750669i	$0.729731\pi$
$840$	0	0
$841$	−6.68954e6	−0.326142
$842$	0	0
$843$	−231778.	−0.0112332
$844$	0	0
$845$	−2.11761e7	−1.02025
$846$	0	0
$847$	7.79623e6	0.373402
$848$	0	0
$849$	283168.	0.0134826
$850$	0	0
$851$	3.70788e7	1.75510
$852$	0	0
$853$	1.20124e7	0.565270	0.282635	−	0.959228i	$-0.408792\pi$
$853$	1.20124e7	0.565270	0.282635	+	0.959228i	$0.408792\pi$
$854$	0	0
$855$	−7.15063e6	−0.334525
$856$	0	0
$857$	−2.71627e6	−0.126334	−0.0631672	−	0.998003i	$-0.520120\pi$
$857$	−2.71627e6	−0.126334	−0.0631672	+	0.998003i	$0.520120\pi$
$858$	0	0
$859$	7.59580e6	0.351229	0.175615	−	0.984459i	$-0.443809\pi$
$859$	7.59580e6	0.351229	0.175615	+	0.984459i	$0.443809\pi$
$860$	0	0
$861$	200782.	0.00923033
$862$	0	0
$863$	−1.72463e6	−0.0788260	−0.0394130	−	0.999223i	$-0.512549\pi$
$863$	−1.72463e6	−0.0788260	−0.0394130	+	0.999223i	$0.512549\pi$
$864$	0	0
$865$	−1.99709e7	−0.907524
$866$	0	0
$867$	419641.	0.0189596
$868$	0	0
$869$	−2.65931e7	−1.19459
$870$	0	0
$871$	1.17823e7	0.526243
$872$	0	0
$873$	−2.25067e7	−0.999486
$874$	0	0
$875$	8.68217e6	0.383361
$876$	0	0
$877$	−7.20923e6	−0.316512	−0.158256	−	0.987398i	$-0.550587\pi$
$877$	−7.20923e6	−0.316512	−0.158256	+	0.987398i	$0.550587\pi$
$878$	0	0
$879$	62100.7	0.00271097
$880$	0	0
$881$	−3.88596e7	−1.68678	−0.843391	−	0.537301i	$-0.819444\pi$
$881$	−3.88596e7	−1.68678	−0.843391	+	0.537301i	$0.819444\pi$
$882$	0	0
$883$	−1.52433e7	−0.657927	−0.328964	−	0.944343i	$-0.606699\pi$
$883$	−1.52433e7	−0.657927	−0.328964	+	0.944343i	$0.606699\pi$
$884$	0	0
$885$	−137730.	−0.00591113
$886$	0	0
$887$	−2.66999e6	−0.113947	−0.0569733	−	0.998376i	$-0.518145\pi$
$887$	−2.66999e6	−0.113947	−0.0569733	+	0.998376i	$0.518145\pi$
$888$	0	0
$889$	−7.59543e6	−0.322328
$890$	0	0
$891$	−3.33732e7	−1.40833
$892$	0	0
$893$	1.55278e7	0.651602
$894$	0	0
$895$	−1.10884e7	−0.462712
$896$	0	0
$897$	1.32684e6	0.0550600
$898$	0	0
$899$	3.68972e6	0.152263
$900$	0	0
$901$	−3.21008e6	−0.131736
$902$	0	0
$903$	4452.86	0.000181727 0
$904$	0	0
$905$	9.03048e6	0.366513
$906$	0	0
$907$	8.48123e6	0.342327	0.171163	−	0.985243i	$-0.445247\pi$
$907$	8.48123e6	0.342327	0.171163	+	0.985243i	$0.445247\pi$
$908$	0	0
$909$	−3.75936e7	−1.50905
$910$	0	0
$911$	−3.89967e7	−1.55680	−0.778399	−	0.627770i	$-0.783968\pi$
$911$	−3.89967e7	−1.55680	−0.778399	+	0.627770i	$0.783968\pi$
$912$	0	0
$913$	−6.31145e7	−2.50583
$914$	0	0
$915$	377847.	0.0149198
$916$	0	0
$917$	3.44944e6	0.135464
$918$	0	0
$919$	−2.65313e7	−1.03626	−0.518130	−	0.855302i	$-0.673372\pi$
$919$	−2.65313e7	−1.03626	−0.518130	+	0.855302i	$0.673372\pi$
$920$	0	0
$921$	−306525.	−0.0119074
$922$	0	0
$923$	1.27128e7	0.491177
$924$	0	0
$925$	−1.55765e7	−0.598569
$926$	0	0
$927$	4.78882e7	1.83033
$928$	0	0
$929$	9.74701e6	0.370537	0.185269	−	0.982688i	$-0.440684\pi$
$929$	9.74701e6	0.370537	0.185269	+	0.982688i	$0.440684\pi$
$930$	0	0
$931$	−1.98982e6	−0.0752385
$932$	0	0
$933$	−540557.	−0.0203300
$934$	0	0
$935$	4.02322e6	0.150503
$936$	0	0
$937$	3.21792e7	1.19736	0.598682	−	0.800987i	$-0.295691\pi$
$937$	3.21792e7	1.19736	0.598682	+	0.800987i	$0.295691\pi$
$938$	0	0
$939$	526458.	0.0194850
$940$	0	0
$941$	−2.89802e7	−1.06691	−0.533454	−	0.845829i	$-0.679106\pi$
$941$	−2.89802e7	−1.06691	−0.533454	+	0.845829i	$0.679106\pi$
$942$	0	0
$943$	−5.97583e7	−2.18836
$944$	0	0
$945$	−257215.	−0.00936952
$946$	0	0
$947$	3.99329e6	0.144696	0.0723479	−	0.997379i	$-0.476951\pi$
$947$	3.99329e6	0.144696	0.0723479	+	0.997379i	$0.476951\pi$
$948$	0	0
$949$	−7.98454e7	−2.87796
$950$	0	0
$951$	−675398.	−0.0242163
$952$	0	0
$953$	3.26669e7	1.16513	0.582566	−	0.812783i	$-0.302049\pi$
$953$	3.26669e7	1.16513	0.582566	+	0.812783i	$0.302049\pi$
$954$	0	0
$955$	6.71553e6	0.238271
$956$	0	0
$957$	639775.	0.0225812
$958$	0	0
$959$	8.61086e6	0.302343
$960$	0	0
$961$	−2.76442e7	−0.965595
$962$	0	0
$963$	1.55221e7	0.539367
$964$	0	0
$965$	−2.34588e7	−0.810937
$966$	0	0
$967$	−9.51237e6	−0.327132	−0.163566	−	0.986532i	$-0.552300\pi$
$967$	−9.51237e6	−0.327132	−0.163566	+	0.986532i	$0.552300\pi$
$968$	0	0
$969$	50454.3	0.00172619
$970$	0	0
$971$	−2.99431e7	−1.01917	−0.509587	−	0.860419i	$-0.670202\pi$
$971$	−2.99431e7	−1.01917	−0.509587	+	0.860419i	$0.670202\pi$
$972$	0	0
$973$	−2.15688e6	−0.0730370
$974$	0	0
$975$	−557391.	−0.0187780
$976$	0	0
$977$	4.93182e7	1.65299	0.826496	−	0.562943i	$-0.190331\pi$
$977$	4.93182e7	1.65299	0.826496	+	0.562943i	$0.190331\pi$
$978$	0	0
$979$	1.96493e7	0.655225
$980$	0	0
$981$	1.62530e7	0.539215
$982$	0	0
$983$	4.39536e7	1.45081	0.725405	−	0.688322i	$-0.241653\pi$
$983$	4.39536e7	1.45081	0.725405	+	0.688322i	$0.241653\pi$
$984$	0	0
$985$	−2.79412e7	−0.917603
$986$	0	0
$987$	279223.	0.00912343
$988$	0	0
$989$	−1.32529e6	−0.0430845
$990$	0	0
$991$	2.45885e7	0.795331	0.397665	−	0.917531i	$-0.369820\pi$
$991$	2.45885e7	0.795331	0.397665	+	0.917531i	$0.369820\pi$
$992$	0	0
$993$	−745793.	−0.0240019
$994$	0	0
$995$	−1.28063e7	−0.410077
$996$	0	0
$997$	2.71682e7	0.865612	0.432806	−	0.901487i	$-0.357524\pi$
$997$	2.71682e7	0.865612	0.432806	+	0.901487i	$0.357524\pi$
$998$	0	0
$999$	1.23541e6	0.0391648

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.6.a.z.1.1		2
4.3	odd	2		448.6.a.q.1.2		2
8.3	odd	2		112.6.a.k.1.1		2
8.5	even	2		56.6.a.c.1.2	✓	2
24.5	odd	2		504.6.a.s.1.1		2
24.11	even	2		1008.6.a.bt.1.1		2
56.5	odd	6		392.6.i.g.361.2		4
56.13	odd	2		392.6.a.f.1.1		2
56.27	even	2		784.6.a.p.1.2		2
56.37	even	6		392.6.i.l.361.1		4
56.45	odd	6		392.6.i.g.177.2		4
56.53	even	6		392.6.i.l.177.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.6.a.c.1.2	✓	2	8.5	even	2
112.6.a.k.1.1		2	8.3	odd	2
392.6.a.f.1.1		2	56.13	odd	2
392.6.i.g.177.2		4	56.45	odd	6
392.6.i.g.361.2		4	56.5	odd	6
392.6.i.l.177.1		4	56.53	even	6
392.6.i.l.361.1		4	56.37	even	6
448.6.a.q.1.2		2	4.3	odd	2
448.6.a.z.1.1		2	1.1	even	1	trivial
504.6.a.s.1.1		2	24.5	odd	2
784.6.a.p.1.2		2	56.27	even	2
1008.6.a.bt.1.1		2	24.11	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	448.a (trivial)

\(f(q)\)	\(=\)	\(q-0.304135 q^{3} -35.5207 q^{5} +49.0000 q^{7} -242.908 q^{9} -565.825 q^{11} -983.594 q^{13} +10.8031 q^{15} +200.175 q^{17} -828.747 q^{19} -14.9026 q^{21} +4435.43 q^{23} -1863.28 q^{25} +147.781 q^{27} +3717.74 q^{29} +992.462 q^{31} +172.087 q^{33} -1740.51 q^{35} +8359.69 q^{37} +299.145 q^{39} -13473.0 q^{41} -298.798 q^{43} +8628.24 q^{45} -18736.5 q^{47} +2401.00 q^{49} -60.8802 q^{51} -16036.4 q^{53} +20098.5 q^{55} +252.051 q^{57} -12749.2 q^{59} +34975.9 q^{61} -11902.5 q^{63} +34937.9 q^{65} -11978.9 q^{67} -1348.97 q^{69} -12924.9 q^{71} +81177.2 q^{73} +566.689 q^{75} -27725.4 q^{77} +46998.8 q^{79} +58981.6 q^{81} +111544. q^{83} -7110.36 q^{85} -1130.69 q^{87} -34726.8 q^{89} -48196.1 q^{91} -301.842 q^{93} +29437.7 q^{95} +92655.6 q^{97} +137443. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 26 q^{3} + 62 q^{5} + 98 q^{7} + 206 q^{9} - 972 q^{11} - 78 q^{13} + 2576 q^{15} + 560 q^{17} - 2642 q^{19} + 1274 q^{21} + 2272 q^{23} + 4522 q^{25} + 5564 q^{27} + 7808 q^{29} + 5444 q^{31} - 10512 q^{33}+ \cdots - 44892 q^{99}+O(q^{100}) \) 2 * q + 26 * q^3 + 62 * q^5 + 98 * q^7 + 206 * q^9 - 972 * q^11 - 78 * q^13 + 2576 * q^15 + 560 * q^17 - 2642 * q^19 + 1274 * q^21 + 2272 * q^23 + 4522 * q^25 + 5564 * q^27 + 7808 * q^29 + 5444 * q^31 - 10512 * q^33 + 3038 * q^35 - 576 * q^37 + 24120 * q^39 - 16888 * q^41 + 8396 * q^43 + 52406 * q^45 - 4532 * q^47 + 4802 * q^49 + 9404 * q^51 - 1420 * q^53 - 19512 * q^55 - 47444 * q^57 - 34146 * q^59 - 19106 * q^61 + 10094 * q^63 + 123252 * q^65 - 56952 * q^67 - 58256 * q^69 - 7224 * q^71 + 128828 * q^73 + 168526 * q^75 - 47628 * q^77 + 52808 * q^79 + 92366 * q^81 + 84486 * q^83 + 27980 * q^85 + 106460 * q^87 - 130972 * q^89 - 3822 * q^91 + 116792 * q^93 - 147392 * q^95 + 194624 * q^97 - 44892 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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