LMFDB - Embedded newform 441.6.a.z.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,6,Mod(1,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, 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("441.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	441.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:	$70.7292645375$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{113})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 59x^{2} + 60x + 674 $ x^4 - 2x^3 - 59x^2 + 60*x + 674
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 7 $
Twist minimal:	no (minimal twist has level 49)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-6.22929$ of defining polynomial
Character	$\chi$	$=$	441.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+7.81507 q^{2} +29.0754 q^{4} +74.2753 q^{5} -22.8562 q^{8} +O(q^{10})$	$q+7.81507 q^{2} +29.0754 q^{4} +74.2753 q^{5} -22.8562 q^{8} +580.467 q^{10} +424.219 q^{11} -252.233 q^{13} -1109.03 q^{16} +1104.35 q^{17} -6.47100 q^{19} +2159.58 q^{20} +3315.30 q^{22} +3612.39 q^{23} +2391.82 q^{25} -1971.22 q^{26} +5005.02 q^{29} +2821.69 q^{31} -7935.79 q^{32} +8630.55 q^{34} -2046.88 q^{37} -50.5713 q^{38} -1697.65 q^{40} -9393.81 q^{41} +10320.8 q^{43} +12334.3 q^{44} +28231.1 q^{46} -17035.6 q^{47} +18692.3 q^{50} -7333.78 q^{52} +39506.7 q^{53} +31509.0 q^{55} +39114.6 q^{58} -33949.8 q^{59} +28295.2 q^{61} +22051.7 q^{62} -26529.7 q^{64} -18734.7 q^{65} +56100.9 q^{67} +32109.3 q^{68} +15537.4 q^{71} -78219.5 q^{73} -15996.5 q^{74} -188.147 q^{76} -45335.5 q^{79} -82373.9 q^{80} -73413.3 q^{82} -1381.82 q^{83} +82025.7 q^{85} +80657.6 q^{86} -9696.05 q^{88} +68879.4 q^{89} +105031. q^{92} -133134. q^{94} -480.635 q^{95} +108857. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 10 q^{2} + 10 q^{4} + 270 q^{8}+O(q^{10}) $ 4 * q + 10 * q^2 + 10 * q^4 + 270 * q^8	$ 4 q + 10 q^{2} + 10 q^{4} + 270 q^{8} + 1952 q^{11} - 1566 q^{16} + 3524 q^{22} + 7136 q^{23} + 2764 q^{25} + 3352 q^{29} - 27810 q^{32} - 9208 q^{37} + 20448 q^{43} - 1900 q^{44} + 56712 q^{46} + 43070 q^{50} + 102920 q^{53} + 96972 q^{58} - 40318 q^{64} + 63168 q^{65} - 22896 q^{67} + 153824 q^{71} - 17596 q^{74} - 90688 q^{79} + 272656 q^{85} + 161860 q^{86} + 154812 q^{88} + 212200 q^{92} - 108224 q^{95}+O(q^{100}) $ 4 * q + 10 * q^2 + 10 * q^4 + 270 * q^8 + 1952 * q^11 - 1566 * q^16 + 3524 * q^22 + 7136 * q^23 + 2764 * q^25 + 3352 * q^29 - 27810 * q^32 - 9208 * q^37 + 20448 * q^43 - 1900 * q^44 + 56712 * q^46 + 43070 * q^50 + 102920 * q^53 + 96972 * q^58 - 40318 * q^64 + 63168 * q^65 - 22896 * q^67 + 153824 * q^71 - 17596 * q^74 - 90688 * q^79 + 272656 * q^85 + 161860 * q^86 + 154812 * q^88 + 212200 * q^92 - 108224 * q^95

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 $

$2$	7.81507	1.38152	0.690761	−	0.723083i	$-0.257276\pi$
$2$	7.81507	1.38152	0.690761	+	0.723083i	$0.257276\pi$
$3$	0	0
$3$	0	0
$4$	29.0754	0.908605
$5$	74.2753	1.32868	0.664339	−	0.747432i	$-0.268713\pi$
$5$	74.2753	1.32868	0.664339	+	0.747432i	$0.268713\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−22.8562	−0.126264
$9$	0	0
$10$	580.467	1.83560
$11$	424.219	1.05708	0.528541	−	0.848908i	$-0.322739\pi$
$11$	424.219	1.05708	0.528541	+	0.848908i	$0.322739\pi$
$12$	0	0
$13$	−252.233	−0.413946	−0.206973	−	0.978347i	$-0.566361\pi$
$13$	−252.233	−0.413946	−0.206973	+	0.978347i	$0.566361\pi$
$14$	0	0
$15$	0	0
$16$	−1109.03	−1.08304
$17$	1104.35	0.926794	0.463397	−	0.886151i	$-0.346630\pi$
$17$	1104.35	0.926794	0.463397	+	0.886151i	$0.346630\pi$
$18$	0	0
$19$	−6.47100	−0.00411232	−0.00205616	−	0.999998i	$-0.500654\pi$
$19$	−6.47100	−0.00411232	−0.00205616	+	0.999998i	$0.500654\pi$
$20$	2159.58	1.20724
$21$	0	0
$22$	3315.30	1.46038
$23$	3612.39	1.42388	0.711942	−	0.702239i	$-0.247816\pi$
$23$	3612.39	1.42388	0.711942	+	0.702239i	$0.247816\pi$
$24$	0	0
$25$	2391.82	0.765383
$26$	−1971.22	−0.571876
$27$	0	0
$28$	0	0
$29$	5005.02	1.10512	0.552561	−	0.833472i	$-0.313650\pi$
$29$	5005.02	1.10512	0.552561	+	0.833472i	$0.313650\pi$
$30$	0	0
$31$	2821.69	0.527357	0.263679	−	0.964611i	$-0.415064\pi$
$31$	2821.69	0.527357	0.263679	+	0.964611i	$0.415064\pi$
$32$	−7935.79	−1.36998
$33$	0	0
$34$	8630.55	1.28039
$35$	0	0
$36$	0	0
$37$	−2046.88	−0.245803	−0.122902	−	0.992419i	$-0.539220\pi$
$37$	−2046.88	−0.245803	−0.122902	+	0.992419i	$0.539220\pi$
$38$	−50.5713	−0.00568127
$39$	0	0
$40$	−1697.65	−0.167764
$41$	−9393.81	−0.872734	−0.436367	−	0.899769i	$-0.643735\pi$
$41$	−9393.81	−0.872734	−0.436367	+	0.899769i	$0.643735\pi$
$42$	0	0
$43$	10320.8	0.851218	0.425609	−	0.904907i	$-0.360060\pi$
$43$	10320.8	0.851218	0.425609	+	0.904907i	$0.360060\pi$
$44$	12334.3	0.960470
$45$	0	0
$46$	28231.1	1.96713
$47$	−17035.6	−1.12490	−0.562448	−	0.826833i	$-0.690140\pi$
$47$	−17035.6	−1.12490	−0.562448	+	0.826833i	$0.690140\pi$
$48$	0	0
$49$	0	0
$50$	18692.3	1.05739
$51$	0	0
$52$	−7333.78	−0.376114
$53$	39506.7	1.93188	0.965941	−	0.258761i	$-0.0833143\pi$
$53$	39506.7	1.93188	0.965941	+	0.258761i	$0.0833143\pi$
$54$	0	0
$55$	31509.0	1.40452
$56$	0	0
$57$	0	0
$58$	39114.6	1.52675
$59$	−33949.8	−1.26972	−0.634859	−	0.772628i	$-0.718942\pi$
$59$	−33949.8	−1.26972	−0.634859	+	0.772628i	$0.718942\pi$
$60$	0	0
$61$	28295.2	0.973618	0.486809	−	0.873508i	$-0.338161\pi$
$61$	28295.2	0.973618	0.486809	+	0.873508i	$0.338161\pi$
$62$	22051.7	0.728556
$63$	0	0
$64$	−26529.7	−0.809621
$65$	−18734.7	−0.550001
$66$	0	0
$67$	56100.9	1.52680	0.763402	−	0.645924i	$-0.223528\pi$
$67$	56100.9	1.52680	0.763402	+	0.645924i	$0.223528\pi$
$68$	32109.3	0.842090
$69$	0	0
$70$	0	0
$71$	15537.4	0.365791	0.182895	−	0.983132i	$-0.441453\pi$
$71$	15537.4	0.365791	0.182895	+	0.983132i	$0.441453\pi$
$72$	0	0
$73$	−78219.5	−1.71794	−0.858970	−	0.512025i	$-0.828895\pi$
$73$	−78219.5	−1.71794	−0.858970	+	0.512025i	$0.828895\pi$
$74$	−15996.5	−0.339583
$75$	0	0
$76$	−188.147	−0.00373648
$77$	0	0
$78$	0	0
$79$	−45335.5	−0.817279	−0.408640	−	0.912696i	$-0.633997\pi$
$79$	−45335.5	−0.817279	−0.408640	+	0.912696i	$0.633997\pi$
$80$	−82373.9	−1.43901
$81$	0	0
$82$	−73413.3	−1.20570
$83$	−1381.82	−0.0220169	−0.0110085	−	0.999939i	$-0.503504\pi$
$83$	−1381.82	−0.0220169	−0.0110085	+	0.999939i	$0.503504\pi$
$84$	0	0
$85$	82025.7	1.23141
$86$	80657.6	1.17598
$87$	0	0
$88$	−9696.05	−0.133471
$89$	68879.4	0.921753	0.460876	−	0.887464i	$-0.347535\pi$
$89$	68879.4	0.921753	0.460876	+	0.887464i	$0.347535\pi$
$90$	0	0
$91$	0	0
$92$	105031.	1.29375
$93$	0	0
$94$	−133134.	−1.55407
$95$	−480.635	−0.00546395
$96$	0	0
$97$	108857.	1.17470	0.587351	−	0.809332i	$-0.300171\pi$
$97$	108857.	1.17470	0.587351	+	0.809332i	$0.300171\pi$
$98$	0	0
$99$	0	0
$100$	69543.1	0.695431
$101$	−17972.3	−0.175307	−0.0876535	−	0.996151i	$-0.527937\pi$
$101$	−17972.3	−0.175307	−0.0876535	+	0.996151i	$0.527937\pi$
$102$	0	0
$103$	31773.7	0.295103	0.147552	−	0.989054i	$-0.452861\pi$
$103$	31773.7	0.295103	0.147552	+	0.989054i	$0.452861\pi$
$104$	5765.11	0.0522666
$105$	0	0
$106$	308747.	2.66894
$107$	−8229.36	−0.0694875	−0.0347438	−	0.999396i	$-0.511062\pi$
$107$	−8229.36	−0.0694875	−0.0347438	+	0.999396i	$0.511062\pi$
$108$	0	0
$109$	−11068.7	−0.0892338	−0.0446169	−	0.999004i	$-0.514207\pi$
$109$	−11068.7	−0.0892338	−0.0446169	+	0.999004i	$0.514207\pi$
$110$	246245.	1.94038
$111$	0	0
$112$	0	0
$113$	−65184.3	−0.480228	−0.240114	−	0.970745i	$-0.577185\pi$
$113$	−65184.3	−0.480228	−0.240114	+	0.970745i	$0.577185\pi$
$114$	0	0
$115$	268311.	1.89188
$116$	145523.	1.00412
$117$	0	0
$118$	−265320.	−1.75414
$119$	0	0
$120$	0	0
$121$	18910.9	0.117422
$122$	221129.	1.34508
$123$	0	0
$124$	82041.7	0.479160
$125$	−54456.9	−0.311730
$126$	0	0
$127$	−194777.	−1.07159	−0.535796	−	0.844348i	$-0.679988\pi$
$127$	−194777.	−1.07159	−0.535796	+	0.844348i	$0.679988\pi$
$128$	46614.1	0.251473
$129$	0	0
$130$	−146413.	−0.759839
$131$	236503.	1.20409	0.602046	−	0.798462i	$-0.294353\pi$
$131$	236503.	1.20409	0.602046	+	0.798462i	$0.294353\pi$
$132$	0	0
$133$	0	0
$134$	438433.	2.10931
$135$	0	0
$136$	−25241.2	−0.117021
$137$	200903.	0.914503	0.457252	−	0.889337i	$-0.348834\pi$
$137$	200903.	0.914503	0.457252	+	0.889337i	$0.348834\pi$
$138$	0	0
$139$	52985.2	0.232604	0.116302	−	0.993214i	$-0.462896\pi$
$139$	52985.2	0.232604	0.116302	+	0.993214i	$0.462896\pi$
$140$	0	0
$141$	0	0
$142$	121426.	0.505348
$143$	−107002.	−0.437575
$144$	0	0
$145$	371749.	1.46835
$146$	−611291.	−2.37337
$147$	0	0
$148$	−59513.7	−0.223338
$149$	100770.	0.371849	0.185925	−	0.982564i	$-0.440472\pi$
$149$	100770.	0.371849	0.185925	+	0.982564i	$0.440472\pi$
$150$	0	0
$151$	−457904.	−1.63430	−0.817150	−	0.576425i	$-0.804447\pi$
$151$	−457904.	−1.63430	−0.817150	+	0.576425i	$0.804447\pi$
$152$	147.903	0.000519239 0
$153$	0	0
$154$	0	0
$155$	209582.	0.700688
$156$	0	0
$157$	−179037.	−0.579688	−0.289844	−	0.957074i	$-0.593603\pi$
$157$	−179037.	−0.579688	−0.289844	+	0.957074i	$0.593603\pi$
$158$	−354300.	−1.12909
$159$	0	0
$160$	−589433.	−1.82027
$161$	0	0
$162$	0	0
$163$	243610.	0.718168	0.359084	−	0.933305i	$-0.383089\pi$
$163$	243610.	0.718168	0.359084	+	0.933305i	$0.383089\pi$
$164$	−273128.	−0.792971
$165$	0	0
$166$	−10799.0	−0.0304169
$167$	−117033.	−0.324725	−0.162362	−	0.986731i	$-0.551911\pi$
$167$	−117033.	−0.324725	−0.162362	+	0.986731i	$0.551911\pi$
$168$	0	0
$169$	−307671.	−0.828648
$170$	641037.	1.70122
$171$	0	0
$172$	300080.	0.773421
$173$	−269733.	−0.685203	−0.342602	−	0.939481i	$-0.611308\pi$
$173$	−269733.	−0.685203	−0.342602	+	0.939481i	$0.611308\pi$
$174$	0	0
$175$	0	0
$176$	−470474.	−1.14486
$177$	0	0
$178$	538298.	1.27342
$179$	−376525.	−0.878336	−0.439168	−	0.898405i	$-0.644727\pi$
$179$	−376525.	−0.878336	−0.439168	+	0.898405i	$0.644727\pi$
$180$	0	0
$181$	−434641.	−0.986131	−0.493065	−	0.869992i	$-0.664124\pi$
$181$	−434641.	−0.986131	−0.493065	+	0.869992i	$0.664124\pi$
$182$	0	0
$183$	0	0
$184$	−82565.5	−0.179785
$185$	−152032.	−0.326593
$186$	0	0
$187$	468485.	0.979697
$188$	−495316.	−1.02209
$189$	0	0
$190$	−3756.20	−0.00754857
$191$	−565940.	−1.12250	−0.561250	−	0.827646i	$-0.689680\pi$
$191$	−565940.	−1.12250	−0.561250	+	0.827646i	$0.689680\pi$
$192$	0	0
$193$	514461.	0.994167	0.497084	−	0.867703i	$-0.334404\pi$
$193$	514461.	0.994167	0.497084	+	0.867703i	$0.334404\pi$
$194$	850727.	1.62288
$195$	0	0
$196$	0	0
$197$	298541.	0.548073	0.274037	−	0.961719i	$-0.411641\pi$
$197$	298541.	0.548073	0.274037	+	0.961719i	$0.411641\pi$
$198$	0	0
$199$	−591919.	−1.05957	−0.529785	−	0.848132i	$-0.677727\pi$
$199$	−591919.	−1.05957	−0.529785	+	0.848132i	$0.677727\pi$
$200$	−54668.1	−0.0966404
$201$	0	0
$202$	−140455.	−0.242191
$203$	0	0
$204$	0	0
$205$	−697728.	−1.15958
$206$	248313.	0.407692
$207$	0	0
$208$	279736.	0.448321
$209$	−2745.12	−0.00434706
$210$	0	0
$211$	−140535.	−0.217309	−0.108655	−	0.994080i	$-0.534654\pi$
$211$	−140535.	−0.217309	−0.108655	+	0.994080i	$0.534654\pi$
$212$	1.14867e6	1.75532
$213$	0	0
$214$	−64313.1	−0.0959986
$215$	766579.	1.13099
$216$	0	0
$217$	0	0
$218$	−86502.5	−0.123278
$219$	0	0
$220$	916136.	1.27615
$221$	−278553.	−0.383643
$222$	0	0
$223$	490.526	0.000660541 0	0.000330271	−	1.00000i	$-0.499895\pi$
$223$	490.526	0.000660541 0	0.000330271		1.00000i	$0.499895\pi$
$224$	0	0
$225$	0	0
$226$	−509420.	−0.663445
$227$	−593898.	−0.764975	−0.382488	−	0.923961i	$-0.624933\pi$
$227$	−593898.	−0.764975	−0.382488	+	0.923961i	$0.624933\pi$
$228$	0	0
$229$	−35880.3	−0.0452135	−0.0226067	−	0.999744i	$-0.507197\pi$
$229$	−35880.3	−0.0452135	−0.0226067	+	0.999744i	$0.507197\pi$
$230$	2.09687e6	2.61368
$231$	0	0
$232$	−114396.	−0.139537
$233$	−1.24822e6	−1.50626	−0.753131	−	0.657871i	$-0.771457\pi$
$233$	−1.24822e6	−1.50626	−0.753131	+	0.657871i	$0.771457\pi$
$234$	0	0
$235$	−1.26532e6	−1.49462
$236$	−987102.	−1.15367
$237$	0	0
$238$	0	0
$239$	576943.	0.653339	0.326669	−	0.945139i	$-0.394074\pi$
$239$	576943.	0.653339	0.326669	+	0.945139i	$0.394074\pi$
$240$	0	0
$241$	−1.38241e6	−1.53318	−0.766592	−	0.642135i	$-0.778049\pi$
$241$	−1.38241e6	−1.53318	−0.766592	+	0.642135i	$0.778049\pi$
$242$	147790.	0.162221
$243$	0	0
$244$	822694.	0.884634
$245$	0	0
$246$	0	0
$247$	1632.20	0.00170228
$248$	−64493.2	−0.0665863
$249$	0	0
$250$	−425585.	−0.430662
$251$	−323217.	−0.323824	−0.161912	−	0.986805i	$-0.551766\pi$
$251$	−323217.	−0.323824	−0.161912	+	0.986805i	$0.551766\pi$
$252$	0	0
$253$	1.53244e6	1.50516
$254$	−1.52220e6	−1.48043
$255$	0	0
$256$	1.21324e6	1.15704
$257$	−1.84601e6	−1.74342	−0.871711	−	0.490021i	$-0.836989\pi$
$257$	−1.84601e6	−1.74342	−0.871711	+	0.490021i	$0.836989\pi$
$258$	0	0
$259$	0	0
$260$	−544719.	−0.499734
$261$	0	0
$262$	1.84829e6	1.66348
$263$	458222.	0.408495	0.204248	−	0.978919i	$-0.434525\pi$
$263$	458222.	0.408495	0.204248	+	0.978919i	$0.434525\pi$
$264$	0	0
$265$	2.93437e6	2.56685
$266$	0	0
$267$	0	0
$268$	1.63115e6	1.38726
$269$	−416958.	−0.351327	−0.175663	−	0.984450i	$-0.556207\pi$
$269$	−416958.	−0.351327	−0.175663	+	0.984450i	$0.556207\pi$
$270$	0	0
$271$	−900379.	−0.744735	−0.372368	−	0.928085i	$-0.621454\pi$
$271$	−900379.	−0.744735	−0.372368	+	0.928085i	$0.621454\pi$
$272$	−1.22476e6	−1.00376
$273$	0	0
$274$	1.57007e6	1.26341
$275$	1.01466e6	0.809073
$276$	0	0
$277$	−447641.	−0.350535	−0.175267	−	0.984521i	$-0.556079\pi$
$277$	−447641.	−0.350535	−0.175267	+	0.984521i	$0.556079\pi$
$278$	414083.	0.321348
$279$	0	0
$280$	0	0
$281$	768521.	0.580617	0.290309	−	0.956933i	$-0.406242\pi$
$281$	768521.	0.580617	0.290309	+	0.956933i	$0.406242\pi$
$282$	0	0
$283$	−2.13220e6	−1.58256	−0.791282	−	0.611452i	$-0.790586\pi$
$283$	−2.13220e6	−1.58256	−0.791282	+	0.611452i	$0.790586\pi$
$284$	451756.	0.332359
$285$	0	0
$286$	−836230.	−0.604520
$287$	0	0
$288$	0	0
$289$	−200275.	−0.141053
$290$	2.90525e6	2.02856
$291$	0	0
$292$	−2.27426e6	−1.56093
$293$	2.42669e6	1.65138	0.825688	−	0.564128i	$-0.190787\pi$
$293$	2.42669e6	1.65138	0.825688	+	0.564128i	$0.190787\pi$
$294$	0	0
$295$	−2.52163e6	−1.68704
$296$	46783.9	0.0310361
$297$	0	0
$298$	787528.	0.513719
$299$	−911164.	−0.589411
$300$	0	0
$301$	0	0
$302$	−3.57855e6	−2.25782
$303$	0	0
$304$	7176.56	0.00445382
$305$	2.10164e6	1.29362
$306$	0	0
$307$	2.44328e6	1.47954	0.739772	−	0.672857i	$-0.234933\pi$
$307$	2.44328e6	1.47954	0.739772	+	0.672857i	$0.234933\pi$
$308$	0	0
$309$	0	0
$310$	1.63790e6	0.968016
$311$	1.15465e6	0.676938	0.338469	−	0.940978i	$-0.390091\pi$
$311$	1.15465e6	0.676938	0.338469	+	0.940978i	$0.390091\pi$
$312$	0	0
$313$	−1.65706e6	−0.956044	−0.478022	−	0.878348i	$-0.658646\pi$
$313$	−1.65706e6	−0.956044	−0.478022	+	0.878348i	$0.658646\pi$
$314$	−1.39919e6	−0.800852
$315$	0	0
$316$	−1.31815e6	−0.742584
$317$	−821361.	−0.459077	−0.229539	−	0.973300i	$-0.573722\pi$
$317$	−821361.	−0.459077	−0.229539	+	0.973300i	$0.573722\pi$
$318$	0	0
$319$	2.12322e6	1.16821
$320$	−1.97050e6	−1.07572
$321$	0	0
$322$	0	0
$323$	−7146.23	−0.00381128
$324$	0	0
$325$	−603298.	−0.316828
$326$	1.90383e6	0.992166
$327$	0	0
$328$	214707.	0.110195
$329$	0	0
$330$	0	0
$331$	−95670.7	−0.0479964	−0.0239982	−	0.999712i	$-0.507640\pi$
$331$	−95670.7	−0.0479964	−0.0239982	+	0.999712i	$0.507640\pi$
$332$	−40177.0	−0.0200047
$333$	0	0
$334$	−914618.	−0.448615
$335$	4.16691e6	2.02863
$336$	0	0
$337$	2.37020e6	1.13687	0.568435	−	0.822728i	$-0.307549\pi$
$337$	2.37020e6	1.13687	0.568435	+	0.822728i	$0.307549\pi$
$338$	−2.40447e6	−1.14480
$339$	0	0
$340$	2.38493e6	1.11887
$341$	1.19701e6	0.557460
$342$	0	0
$343$	0	0
$344$	−235894.	−0.107478
$345$	0	0
$346$	−2.10799e6	−0.946624
$347$	−490571.	−0.218715	−0.109358	−	0.994002i	$-0.534879\pi$
$347$	−490571.	−0.218715	−0.109358	+	0.994002i	$0.534879\pi$
$348$	0	0
$349$	−4.21208e6	−1.85111	−0.925557	−	0.378607i	$-0.876403\pi$
$349$	−4.21208e6	−1.85111	−0.925557	+	0.378607i	$0.876403\pi$
$350$	0	0
$351$	0	0
$352$	−3.36651e6	−1.44818
$353$	−3.17378e6	−1.35563	−0.677814	−	0.735234i	$-0.737072\pi$
$353$	−3.17378e6	−1.35563	−0.677814	+	0.735234i	$0.737072\pi$
$354$	0	0
$355$	1.15405e6	0.486018
$356$	2.00269e6	0.837509
$357$	0	0
$358$	−2.94257e6	−1.21344
$359$	3.40098e6	1.39273	0.696366	−	0.717687i	$-0.254799\pi$
$359$	3.40098e6	1.39273	0.696366	+	0.717687i	$0.254799\pi$
$360$	0	0
$361$	−2.47606e6	−0.999983
$362$	−3.39675e6	−1.36236
$363$	0	0
$364$	0	0
$365$	−5.80978e6	−2.28259
$366$	0	0
$367$	1.96872e6	0.762988	0.381494	−	0.924371i	$-0.375410\pi$
$367$	1.96872e6	0.762988	0.381494	+	0.924371i	$0.375410\pi$
$368$	−4.00626e6	−1.54213
$369$	0	0
$370$	−1.18814e6	−0.451196
$371$	0	0
$372$	0	0
$373$	−3.47889e6	−1.29470	−0.647349	−	0.762194i	$-0.724122\pi$
$373$	−3.47889e6	−1.29470	−0.647349	+	0.762194i	$0.724122\pi$
$374$	3.66124e6	1.35347
$375$	0	0
$376$	389369.	0.142034
$377$	−1.26243e6	−0.457462
$378$	0	0
$379$	−421294.	−0.150656	−0.0753281	−	0.997159i	$-0.524000\pi$
$379$	−421294.	−0.150656	−0.0753281	+	0.997159i	$0.524000\pi$
$380$	−13974.7	−0.00496457
$381$	0	0
$382$	−4.42286e6	−1.55076
$383$	−2.66910e6	−0.929754	−0.464877	−	0.885375i	$-0.653901\pi$
$383$	−2.66910e6	−0.929754	−0.464877	+	0.885375i	$0.653901\pi$
$384$	0	0
$385$	0	0
$386$	4.02055e6	1.37346
$387$	0	0
$388$	3.16506e6	1.06734
$389$	−3.10178e6	−1.03929	−0.519645	−	0.854382i	$-0.673936\pi$
$389$	−3.10178e6	−1.03929	−0.519645	+	0.854382i	$0.673936\pi$
$390$	0	0
$391$	3.98933e6	1.31965
$392$	0	0
$393$	0	0
$394$	2.33312e6	0.757176
$395$	−3.36731e6	−1.08590
$396$	0	0
$397$	613257.	0.195284	0.0976419	−	0.995222i	$-0.468870\pi$
$397$	613257.	0.195284	0.0976419	+	0.995222i	$0.468870\pi$
$398$	−4.62589e6	−1.46382
$399$	0	0
$400$	−2.65262e6	−0.828942
$401$	−2.82223e6	−0.876459	−0.438229	−	0.898863i	$-0.644394\pi$
$401$	−2.82223e6	−0.876459	−0.438229	+	0.898863i	$0.644394\pi$
$402$	0	0
$403$	−711724.	−0.218298
$404$	−522550.	−0.159285
$405$	0	0
$406$	0	0
$407$	−868324.	−0.259834
$408$	0	0
$409$	2.28350e6	0.674983	0.337492	−	0.941329i	$-0.390422\pi$
$409$	2.28350e6	0.674983	0.337492	+	0.941329i	$0.390422\pi$
$410$	−5.45279e6	−1.60199
$411$	0	0
$412$	923831.	0.268132
$413$	0	0
$414$	0	0
$415$	−102635.	−0.0292534
$416$	2.00167e6	0.567100
$417$	0	0
$418$	−21453.3	−0.00600556
$419$	2.65270e6	0.738163	0.369082	−	0.929397i	$-0.379672\pi$
$419$	2.65270e6	0.738163	0.369082	+	0.929397i	$0.379672\pi$
$420$	0	0
$421$	2.93674e6	0.807532	0.403766	−	0.914862i	$-0.367701\pi$
$421$	2.93674e6	0.807532	0.403766	+	0.914862i	$0.367701\pi$
$422$	−1.09829e6	−0.300218
$423$	0	0
$424$	−902974.	−0.243927
$425$	2.64140e6	0.709353
$426$	0	0
$427$	0	0
$428$	−239272.	−0.0631367
$429$	0	0
$430$	5.99087e6	1.56249
$431$	2.44565e6	0.634164	0.317082	−	0.948398i	$-0.397297\pi$
$431$	2.44565e6	0.634164	0.317082	+	0.948398i	$0.397297\pi$
$432$	0	0
$433$	−2.11718e6	−0.542673	−0.271336	−	0.962485i	$-0.587466\pi$
$433$	−2.11718e6	−0.542673	−0.271336	+	0.962485i	$0.587466\pi$
$434$	0	0
$435$	0	0
$436$	−321826.	−0.0810783
$437$	−23375.7	−0.00585547
$438$	0	0
$439$	4.64764e6	1.15099	0.575495	−	0.817806i	$-0.304810\pi$
$439$	4.64764e6	1.15099	0.575495	+	0.817806i	$0.304810\pi$
$440$	−720178.	−0.177340
$441$	0	0
$442$	−2.17691e6	−0.530012
$443$	−4.42925e6	−1.07231	−0.536155	−	0.844119i	$-0.680124\pi$
$443$	−4.42925e6	−1.07231	−0.536155	+	0.844119i	$0.680124\pi$
$444$	0	0
$445$	5.11604e6	1.22471
$446$	3833.50	0.000912553 0
$447$	0	0
$448$	0	0
$449$	6.70171e6	1.56881	0.784404	−	0.620250i	$-0.212969\pi$
$449$	6.70171e6	1.56881	0.784404	+	0.620250i	$0.212969\pi$
$450$	0	0
$451$	−3.98503e6	−0.922551
$452$	−1.89526e6	−0.436337
$453$	0	0
$454$	−4.64136e6	−1.05683
$455$	0	0
$456$	0	0
$457$	−5.88344e6	−1.31777	−0.658887	−	0.752242i	$-0.728972\pi$
$457$	−5.88344e6	−1.31777	−0.658887	+	0.752242i	$0.728972\pi$
$458$	−280407.	−0.0624634
$459$	0	0
$460$	7.80124e6	1.71897
$461$	−1.54764e6	−0.339171	−0.169585	−	0.985515i	$-0.554243\pi$
$461$	−1.54764e6	−0.339171	−0.169585	+	0.985515i	$0.554243\pi$
$462$	0	0
$463$	−3.93764e6	−0.853656	−0.426828	−	0.904333i	$-0.640369\pi$
$463$	−3.93764e6	−0.853656	−0.426828	+	0.904333i	$0.640369\pi$
$464$	−5.55074e6	−1.19689
$465$	0	0
$466$	−9.75491e6	−2.08093
$467$	6.81586e6	1.44620	0.723100	−	0.690743i	$-0.242716\pi$
$467$	6.81586e6	1.44620	0.723100	+	0.690743i	$0.242716\pi$
$468$	0	0
$469$	0	0
$470$	−9.88860e6	−2.06486
$471$	0	0
$472$	775964.	0.160320
$473$	4.37827e6	0.899807
$474$	0	0
$475$	−15477.5	−0.00314750
$476$	0	0
$477$	0	0
$478$	4.50885e6	0.902602
$479$	5.20406e6	1.03634	0.518171	−	0.855277i	$-0.326613\pi$
$479$	5.20406e6	1.03634	0.518171	+	0.855277i	$0.326613\pi$
$480$	0	0
$481$	516291.	0.101749
$482$	−1.08036e7	−2.11813
$483$	0	0
$484$	549840.	0.106690
$485$	8.08540e6	1.56080
$486$	0	0
$487$	−154998.	−0.0296145	−0.0148073	−	0.999890i	$-0.504713\pi$
$487$	−154998.	−0.0296145	−0.0148073	+	0.999890i	$0.504713\pi$
$488$	−646723.	−0.122933
$489$	0	0
$490$	0	0
$491$	1.61951e6	0.303165	0.151583	−	0.988445i	$-0.451563\pi$
$491$	1.61951e6	0.303165	0.151583	+	0.988445i	$0.451563\pi$
$492$	0	0
$493$	5.52727e6	1.02422
$494$	12755.8	0.00235174
$495$	0	0
$496$	−3.12935e6	−0.571150
$497$	0	0
$498$	0	0
$499$	4.10674e6	0.738322	0.369161	−	0.929366i	$-0.379645\pi$
$499$	4.10674e6	0.738322	0.369161	+	0.929366i	$0.379645\pi$
$500$	−1.58336e6	−0.283239
$501$	0	0
$502$	−2.52596e6	−0.447371
$503$	3.40748e6	0.600501	0.300250	−	0.953860i	$-0.402930\pi$
$503$	3.40748e6	0.600501	0.300250	+	0.953860i	$0.402930\pi$
$504$	0	0
$505$	−1.33490e6	−0.232927
$506$	1.19762e7	2.07941
$507$	0	0
$508$	−5.66322e6	−0.973654
$509$	−1.07091e7	−1.83213	−0.916067	−	0.401025i	$-0.868654\pi$
$509$	−1.07091e7	−1.83213	−0.916067	+	0.401025i	$0.868654\pi$
$510$	0	0
$511$	0	0
$512$	7.98992e6	1.34700
$513$	0	0
$514$	−1.44267e7	−2.40858
$515$	2.36000e6	0.392097
$516$	0	0
$517$	−7.22682e6	−1.18911
$518$	0	0
$519$	0	0
$520$	428205.	0.0694454
$521$	−7.92001e6	−1.27830	−0.639148	−	0.769084i	$-0.720713\pi$
$521$	−7.92001e6	−1.27830	−0.639148	+	0.769084i	$0.720713\pi$
$522$	0	0
$523$	−8.32746e6	−1.33125	−0.665623	−	0.746288i	$-0.731834\pi$
$523$	−8.32746e6	−1.33125	−0.665623	+	0.746288i	$0.731834\pi$
$524$	6.87643e6	1.09404
$525$	0	0
$526$	3.58104e6	0.564345
$527$	3.11612e6	0.488752
$528$	0	0
$529$	6.61298e6	1.02744
$530$	2.29323e7	3.54616
$531$	0	0
$532$	0	0
$533$	2.36943e6	0.361265
$534$	0	0
$535$	−611239.	−0.0923265
$536$	−1.28226e6	−0.192780
$537$	0	0
$538$	−3.25856e6	−0.485366
$539$	0	0
$540$	0	0
$541$	623261.	0.0915539	0.0457770	−	0.998952i	$-0.485424\pi$
$541$	623261.	0.0915539	0.0457770	+	0.998952i	$0.485424\pi$
$542$	−7.03653e6	−1.02887
$543$	0	0
$544$	−8.76386e6	−1.26969
$545$	−822129.	−0.118563
$546$	0	0
$547$	1.05691e7	1.51032	0.755159	−	0.655541i	$-0.227559\pi$
$547$	1.05691e7	1.51032	0.755159	+	0.655541i	$0.227559\pi$
$548$	5.84133e6	0.830922
$549$	0	0
$550$	7.92962e6	1.11775
$551$	−32387.5	−0.00454462
$552$	0	0
$553$	0	0
$554$	−3.49835e6	−0.484272
$555$	0	0
$556$	1.54056e6	0.211345
$557$	1.35398e7	1.84916	0.924579	−	0.380991i	$-0.124417\pi$
$557$	1.35398e7	1.84916	0.924579	+	0.380991i	$0.124417\pi$
$558$	0	0
$559$	−2.60324e6	−0.352359
$560$	0	0
$561$	0	0
$562$	6.00605e6	0.802136
$563$	1.39757e7	1.85824	0.929122	−	0.369774i	$-0.120565\pi$
$563$	1.39757e7	1.85824	0.929122	+	0.369774i	$0.120565\pi$
$564$	0	0
$565$	−4.84159e6	−0.638068
$566$	−1.66633e7	−2.18635
$567$	0	0
$568$	−355127.	−0.0461862
$569$	5.61993e6	0.727697	0.363848	−	0.931458i	$-0.381463\pi$
$569$	5.61993e6	0.727697	0.363848	+	0.931458i	$0.381463\pi$
$570$	0	0
$571$	8.70790e6	1.11769	0.558847	−	0.829271i	$-0.311244\pi$
$571$	8.70790e6	1.11769	0.558847	+	0.829271i	$0.311244\pi$
$572$	−3.11113e6	−0.397583
$573$	0	0
$574$	0	0
$575$	8.64019e6	1.08982
$576$	0	0
$577$	−6.63992e6	−0.830278	−0.415139	−	0.909758i	$-0.636267\pi$
$577$	−6.63992e6	−0.830278	−0.415139	+	0.909758i	$0.636267\pi$
$578$	−1.56517e6	−0.194868
$579$	0	0
$580$	1.08087e7	1.33415
$581$	0	0
$582$	0	0
$583$	1.67595e7	2.04216
$584$	1.78780e6	0.216914
$585$	0	0
$586$	1.89648e7	2.28141
$587$	1.36652e7	1.63689	0.818446	−	0.574583i	$-0.194836\pi$
$587$	1.36652e7	1.63689	0.818446	+	0.574583i	$0.194836\pi$
$588$	0	0
$589$	−18259.2	−0.00216866
$590$	−1.97067e7	−2.33069
$591$	0	0
$592$	2.27006e6	0.266215
$593$	7.02589e6	0.820474	0.410237	−	0.911979i	$-0.365446\pi$
$593$	7.02589e6	0.820474	0.410237	+	0.911979i	$0.365446\pi$
$594$	0	0
$595$	0	0
$596$	2.92994e6	0.337864
$597$	0	0
$598$	−7.12081e6	−0.814285
$599$	3.58663e6	0.408432	0.204216	−	0.978926i	$-0.434536\pi$
$599$	3.58663e6	0.408432	0.204216	+	0.978926i	$0.434536\pi$
$600$	0	0
$601$	1.58600e7	1.79108	0.895542	−	0.444977i	$-0.146788\pi$
$601$	1.58600e7	1.79108	0.895542	+	0.444977i	$0.146788\pi$
$602$	0	0
$603$	0	0
$604$	−1.33137e7	−1.48493
$605$	1.40461e6	0.156015
$606$	0	0
$607$	6.44170e6	0.709625	0.354812	−	0.934938i	$-0.384545\pi$
$607$	6.44170e6	0.709625	0.354812	+	0.934938i	$0.384545\pi$
$608$	51352.5	0.00563381
$609$	0	0
$610$	1.64244e7	1.78717
$611$	4.29694e6	0.465647
$612$	0	0
$613$	−4.58865e6	−0.493212	−0.246606	−	0.969116i	$-0.579315\pi$
$613$	−4.58865e6	−0.493212	−0.246606	+	0.969116i	$0.579315\pi$
$614$	1.90944e7	2.04402
$615$	0	0
$616$	0	0
$617$	−1.47104e6	−0.155565	−0.0777825	−	0.996970i	$-0.524784\pi$
$617$	−1.47104e6	−0.155565	−0.0777825	+	0.996970i	$0.524784\pi$
$618$	0	0
$619$	−3.13569e6	−0.328932	−0.164466	−	0.986383i	$-0.552590\pi$
$619$	−3.13569e6	−0.328932	−0.164466	+	0.986383i	$0.552590\pi$
$620$	6.09367e6	0.636649
$621$	0	0
$622$	9.02366e6	0.935205
$623$	0	0
$624$	0	0
$625$	−1.15193e7	−1.17957
$626$	−1.29501e7	−1.32080
$627$	0	0
$628$	−5.20557e6	−0.526707
$629$	−2.26046e6	−0.227809
$630$	0	0
$631$	484547.	0.0484465	0.0242233	−	0.999707i	$-0.492289\pi$
$631$	484547.	0.0484465	0.0242233	+	0.999707i	$0.492289\pi$
$632$	1.03620e6	0.103193
$633$	0	0
$634$	−6.41899e6	−0.634226
$635$	−1.44672e7	−1.42380
$636$	0	0
$637$	0	0
$638$	1.65932e7	1.61390
$639$	0	0
$640$	3.46228e6	0.334127
$641$	−3.04085e6	−0.292314	−0.146157	−	0.989261i	$-0.546690\pi$
$641$	−3.04085e6	−0.292314	−0.146157	+	0.989261i	$0.546690\pi$
$642$	0	0
$643$	5.25888e6	0.501609	0.250805	−	0.968038i	$-0.419305\pi$
$643$	5.25888e6	0.501609	0.250805	+	0.968038i	$0.419305\pi$
$644$	0	0
$645$	0	0
$646$	−55848.3	−0.00526536
$647$	−2.11970e7	−1.99074	−0.995368	−	0.0961386i	$-0.969351\pi$
$647$	−2.11970e7	−1.99074	−0.995368	+	0.0961386i	$0.969351\pi$
$648$	0	0
$649$	−1.44021e7	−1.34219
$650$	−4.71481e6	−0.437705
$651$	0	0
$652$	7.08305e6	0.652531
$653$	−1.30106e7	−1.19403	−0.597013	−	0.802232i	$-0.703646\pi$
$653$	−1.30106e7	−1.19403	−0.597013	+	0.802232i	$0.703646\pi$
$654$	0	0
$655$	1.75664e7	1.59985
$656$	1.04181e7	0.945208
$657$	0	0
$658$	0	0
$659$	1.59874e7	1.43405	0.717024	−	0.697049i	$-0.245504\pi$
$659$	1.59874e7	1.43405	0.717024	+	0.697049i	$0.245504\pi$
$660$	0	0
$661$	4.03142e6	0.358884	0.179442	−	0.983769i	$-0.442571\pi$
$661$	4.03142e6	0.358884	0.179442	+	0.983769i	$0.442571\pi$
$662$	−747673.	−0.0663082
$663$	0	0
$664$	31583.3	0.00277995
$665$	0	0
$666$	0	0
$667$	1.80800e7	1.57357
$668$	−3.40276e6	−0.295047
$669$	0	0
$670$	3.25647e7	2.80260
$671$	1.20034e7	1.02919
$672$	0	0
$673$	2.98234e6	0.253816	0.126908	−	0.991914i	$-0.459495\pi$
$673$	2.98234e6	0.253816	0.126908	+	0.991914i	$0.459495\pi$
$674$	1.85233e7	1.57061
$675$	0	0
$676$	−8.94566e6	−0.752914
$677$	−1.94696e6	−0.163262	−0.0816309	−	0.996663i	$-0.526013\pi$
$677$	−1.94696e6	−0.163262	−0.0816309	+	0.996663i	$0.526013\pi$
$678$	0	0
$679$	0	0
$680$	−1.87480e6	−0.155483
$681$	0	0
$682$	9.35476e6	0.770144
$683$	−1.19940e7	−0.983812	−0.491906	−	0.870648i	$-0.663700\pi$
$683$	−1.19940e7	−0.983812	−0.491906	+	0.870648i	$0.663700\pi$
$684$	0	0
$685$	1.49221e7	1.21508
$686$	0	0
$687$	0	0
$688$	−1.14461e7	−0.921905
$689$	−9.96490e6	−0.799696
$690$	0	0
$691$	−8.66304e6	−0.690201	−0.345100	−	0.938566i	$-0.612155\pi$
$691$	−8.66304e6	−0.690201	−0.345100	+	0.938566i	$0.612155\pi$
$692$	−7.84260e6	−0.622579
$693$	0	0
$694$	−3.83385e6	−0.302160
$695$	3.93549e6	0.309056
$696$	0	0
$697$	−1.03740e7	−0.808845
$698$	−3.29177e7	−2.55736
$699$	0	0
$700$	0	0
$701$	8.13382e6	0.625172	0.312586	−	0.949889i	$-0.398805\pi$
$701$	8.13382e6	0.625172	0.312586	+	0.949889i	$0.398805\pi$
$702$	0	0
$703$	13245.3	0.00101082
$704$	−1.12544e7	−0.855835
$705$	0	0
$706$	−2.48033e7	−1.87283
$707$	0	0
$708$	0	0
$709$	2.21326e7	1.65355	0.826773	−	0.562535i	$-0.190174\pi$
$709$	2.21326e7	1.65355	0.826773	+	0.562535i	$0.190174\pi$
$710$	9.01895e6	0.671445
$711$	0	0
$712$	−1.57432e6	−0.116384
$713$	1.01930e7	0.750896
$714$	0	0
$715$	−7.94762e6	−0.581396
$716$	−1.09476e7	−0.798061
$717$	0	0
$718$	2.65789e7	1.92409
$719$	7.23196e6	0.521716	0.260858	−	0.965377i	$-0.415995\pi$
$719$	7.23196e6	0.521716	0.260858	+	0.965377i	$0.415995\pi$
$720$	0	0
$721$	0	0
$722$	−1.93506e7	−1.38150
$723$	0	0
$724$	−1.26374e7	−0.896004
$725$	1.19711e7	0.845843
$726$	0	0
$727$	−1.70200e7	−1.19433	−0.597163	−	0.802120i	$-0.703705\pi$
$727$	−1.70200e7	−1.19433	−0.597163	+	0.802120i	$0.703705\pi$
$728$	0	0
$729$	0	0
$730$	−4.54039e7	−3.15345
$731$	1.13977e7	0.788904
$732$	0	0
$733$	1.00011e6	0.0687525	0.0343763	−	0.999409i	$-0.489056\pi$
$733$	1.00011e6	0.0687525	0.0343763	+	0.999409i	$0.489056\pi$
$734$	1.53857e7	1.05409
$735$	0	0
$736$	−2.86671e7	−1.95070
$737$	2.37991e7	1.61396
$738$	0	0
$739$	3.25979e6	0.219572	0.109786	−	0.993955i	$-0.464983\pi$
$739$	3.25979e6	0.219572	0.109786	+	0.993955i	$0.464983\pi$
$740$	−4.42040e6	−0.296744
$741$	0	0
$742$	0	0
$743$	−1.36125e7	−0.904617	−0.452309	−	0.891861i	$-0.649399\pi$
$743$	−1.36125e7	−0.904617	−0.452309	+	0.891861i	$0.649399\pi$
$744$	0	0
$745$	7.48475e6	0.494068
$746$	−2.71878e7	−1.78865
$747$	0	0
$748$	1.36214e7	0.890158
$749$	0	0
$750$	0	0
$751$	6.56544e6	0.424780	0.212390	−	0.977185i	$-0.431875\pi$
$751$	6.56544e6	0.424780	0.212390	+	0.977185i	$0.431875\pi$
$752$	1.88931e7	1.21831
$753$	0	0
$754$	−9.86600e6	−0.631994
$755$	−3.40110e7	−2.17146
$756$	0	0
$757$	2.62531e7	1.66510	0.832551	−	0.553948i	$-0.186879\pi$
$757$	2.62531e7	1.66510	0.832551	+	0.553948i	$0.186879\pi$
$758$	−3.29244e6	−0.208135
$759$	0	0
$760$	10985.5	0.000689901 0
$761$	5.25111e6	0.328692	0.164346	−	0.986403i	$-0.447449\pi$
$761$	5.25111e6	0.328692	0.164346	+	0.986403i	$0.447449\pi$
$762$	0	0
$763$	0	0
$764$	−1.64549e7	−1.01991
$765$	0	0
$766$	−2.08592e7	−1.28448
$767$	8.56327e6	0.525595
$768$	0	0
$769$	−1.77307e7	−1.08121	−0.540605	−	0.841277i	$-0.681805\pi$
$769$	−1.77307e7	−1.08121	−0.540605	+	0.841277i	$0.681805\pi$
$770$	0	0
$771$	0	0
$772$	1.49581e7	0.903305
$773$	−3.82592e6	−0.230296	−0.115148	−	0.993348i	$-0.536734\pi$
$773$	−3.82592e6	−0.230296	−0.115148	+	0.993348i	$0.536734\pi$
$774$	0	0
$775$	6.74898e6	0.403631
$776$	−2.48807e6	−0.148323
$777$	0	0
$778$	−2.42406e7	−1.43580
$779$	60787.3	0.00358896
$780$	0	0
$781$	6.59126e6	0.386671
$782$	3.11769e7	1.82312
$783$	0	0
$784$	0	0
$785$	−1.32980e7	−0.770218
$786$	0	0
$787$	2.94263e7	1.69355	0.846777	−	0.531948i	$-0.178540\pi$
$787$	2.94263e7	1.69355	0.846777	+	0.531948i	$0.178540\pi$
$788$	8.68019e6	0.497982
$789$	0	0
$790$	−2.63157e7	−1.50020
$791$	0	0
$792$	0	0
$793$	−7.13700e6	−0.403026
$794$	4.79265e6	0.269789
$795$	0	0
$796$	−1.72102e7	−0.962730
$797$	−1.35805e7	−0.757305	−0.378653	−	0.925539i	$-0.623613\pi$
$797$	−1.35805e7	−0.757305	−0.378653	+	0.925539i	$0.623613\pi$
$798$	0	0
$799$	−1.88132e7	−1.04255
$800$	−1.89810e7	−1.04856
$801$	0	0
$802$	−2.20559e7	−1.21085
$803$	−3.31822e7	−1.81600
$804$	0	0
$805$	0	0
$806$	−5.56218e6	−0.301583
$807$	0	0
$808$	410778.	0.0221350
$809$	1.15714e7	0.621604	0.310802	−	0.950475i	$-0.399402\pi$
$809$	1.15714e7	0.621604	0.310802	+	0.950475i	$0.399402\pi$
$810$	0	0
$811$	−3.52530e6	−0.188210	−0.0941052	−	0.995562i	$-0.529999\pi$
$811$	−3.52530e6	−0.188210	−0.0941052	+	0.995562i	$0.529999\pi$
$812$	0	0
$813$	0	0
$814$	−6.78602e6	−0.358966
$815$	1.80942e7	0.954214
$816$	0	0
$817$	−66785.7	−0.00350049
$818$	1.78457e7	0.932505
$819$	0	0
$820$	−2.02867e7	−1.05360
$821$	−1.78951e7	−0.926567	−0.463283	−	0.886210i	$-0.653329\pi$
$821$	−1.78951e7	−0.926567	−0.463283	+	0.886210i	$0.653329\pi$
$822$	0	0
$823$	−3.61421e7	−1.86000	−0.930001	−	0.367557i	$-0.880194\pi$
$823$	−3.61421e7	−1.86000	−0.930001	+	0.367557i	$0.880194\pi$
$824$	−726226.	−0.0372610
$825$	0	0
$826$	0	0
$827$	1.00605e7	0.511512	0.255756	−	0.966741i	$-0.417676\pi$
$827$	1.00605e7	0.511512	0.255756	+	0.966741i	$0.417676\pi$
$828$	0	0
$829$	2.03654e7	1.02921	0.514607	−	0.857426i	$-0.327938\pi$
$829$	2.03654e7	1.02921	0.514607	+	0.857426i	$0.327938\pi$
$830$	−802102.	−0.0404142
$831$	0	0
$832$	6.69166e6	0.335140
$833$	0	0
$834$	0	0
$835$	−8.69263e6	−0.431454
$836$	−79815.4	−0.00394976
$837$	0	0
$838$	2.07310e7	1.01979
$839$	5.95014e6	0.291825	0.145912	−	0.989298i	$-0.453388\pi$
$839$	5.95014e6	0.291825	0.145912	+	0.989298i	$0.453388\pi$
$840$	0	0
$841$	4.53905e6	0.221297
$842$	2.29508e7	1.11562
$843$	0	0
$844$	−4.08611e6	−0.197448
$845$	−2.28524e7	−1.10101
$846$	0	0
$847$	0	0
$848$	−4.38143e7	−2.09231
$849$	0	0
$850$	2.06428e7	0.979987
$851$	−7.39411e6	−0.349995
$852$	0	0
$853$	1.59836e7	0.752146	0.376073	−	0.926590i	$-0.377274\pi$
$853$	1.59836e7	0.752146	0.376073	+	0.926590i	$0.377274\pi$
$854$	0	0
$855$	0	0
$856$	188092.	0.00877378
$857$	2.34591e6	0.109109	0.0545544	−	0.998511i	$-0.482626\pi$
$857$	2.34591e6	0.109109	0.0545544	+	0.998511i	$0.482626\pi$
$858$	0	0
$859$	−1.30223e7	−0.602152	−0.301076	−	0.953600i	$-0.597346\pi$
$859$	−1.30223e7	−0.602152	−0.301076	+	0.953600i	$0.597346\pi$
$860$	2.22886e7	1.02763
$861$	0	0
$862$	1.91130e7	0.876112
$863$	−1.96688e7	−0.898983	−0.449492	−	0.893285i	$-0.648395\pi$
$863$	−1.96688e7	−0.898983	−0.449492	+	0.893285i	$0.648395\pi$
$864$	0	0
$865$	−2.00345e7	−0.910414
$866$	−1.65459e7	−0.749715
$867$	0	0
$868$	0	0
$869$	−1.92322e7	−0.863931
$870$	0	0
$871$	−1.41505e7	−0.632015
$872$	252988.	0.0112670
$873$	0	0
$874$	−182683.	−0.00808946
$875$	0	0
$876$	0	0
$877$	2.34581e7	1.02990	0.514950	−	0.857221i	$-0.327811\pi$
$877$	2.34581e7	1.02990	0.514950	+	0.857221i	$0.327811\pi$
$878$	3.63217e7	1.59012
$879$	0	0
$880$	−3.49446e7	−1.52115
$881$	−4.59257e6	−0.199350	−0.0996750	−	0.995020i	$-0.531780\pi$
$881$	−4.59257e6	−0.199350	−0.0996750	+	0.995020i	$0.531780\pi$
$882$	0	0
$883$	−1.23402e7	−0.532622	−0.266311	−	0.963887i	$-0.585805\pi$
$883$	−1.23402e7	−0.532622	−0.266311	+	0.963887i	$0.585805\pi$
$884$	−8.09903e6	−0.348580
$885$	0	0
$886$	−3.46149e7	−1.48142
$887$	−1.36554e7	−0.582769	−0.291384	−	0.956606i	$-0.594116\pi$
$887$	−1.36554e7	−0.582769	−0.291384	+	0.956606i	$0.594116\pi$
$888$	0	0
$889$	0	0
$890$	3.99822e7	1.69197
$891$	0	0
$892$	14262.2	0.000600171 0
$893$	110237.	0.00462594
$894$	0	0
$895$	−2.79665e7	−1.16703
$896$	0	0
$897$	0	0
$898$	5.23744e7	2.16735
$899$	1.41226e7	0.582795
$900$	0	0
$901$	4.36291e7	1.79046
$902$	−3.11433e7	−1.27453
$903$	0	0
$904$	1.48987e6	0.0606355
$905$	−3.22831e7	−1.31025
$906$	0	0
$907$	7.39599e6	0.298523	0.149262	−	0.988798i	$-0.452310\pi$
$907$	7.39599e6	0.298523	0.149262	+	0.988798i	$0.452310\pi$
$908$	−1.72678e7	−0.695060
$909$	0	0
$910$	0	0
$911$	3.51041e7	1.40140	0.700699	−	0.713457i	$-0.252871\pi$
$911$	3.51041e7	1.40140	0.700699	+	0.713457i	$0.252871\pi$
$912$	0	0
$913$	−586195.	−0.0232737
$914$	−4.59795e7	−1.82053
$915$	0	0
$916$	−1.04323e6	−0.0410812
$917$	0	0
$918$	0	0
$919$	−4.81337e7	−1.88001	−0.940005	−	0.341159i	$-0.889180\pi$
$919$	−4.81337e7	−1.88001	−0.940005	+	0.341159i	$0.889180\pi$
$920$	−6.13258e6	−0.238877
$921$	0	0
$922$	−1.20949e7	−0.468572
$923$	−3.91905e6	−0.151418
$924$	0	0
$925$	−4.89577e6	−0.188134
$926$	−3.07729e7	−1.17935
$927$	0	0
$928$	−3.97188e7	−1.51400
$929$	1.83602e7	0.697971	0.348986	−	0.937128i	$-0.386526\pi$
$929$	1.83602e7	0.697971	0.348986	+	0.937128i	$0.386526\pi$
$930$	0	0
$931$	0	0
$932$	−3.62924e7	−1.36860
$933$	0	0
$934$	5.32665e7	1.99796
$935$	3.47969e7	1.30170
$936$	0	0
$937$	2.27081e7	0.844951	0.422475	−	0.906374i	$-0.361161\pi$
$937$	2.27081e7	0.844951	0.422475	+	0.906374i	$0.361161\pi$
$938$	0	0
$939$	0	0
$940$	−3.67898e7	−1.35802
$941$	−4.15801e7	−1.53078	−0.765389	−	0.643568i	$-0.777453\pi$
$941$	−4.15801e7	−1.53078	−0.765389	+	0.643568i	$0.777453\pi$
$942$	0	0
$943$	−3.39340e7	−1.24267
$944$	3.76515e7	1.37516
$945$	0	0
$946$	3.42165e7	1.24310
$947$	−1.55341e7	−0.562874	−0.281437	−	0.959580i	$-0.590811\pi$
$947$	−1.55341e7	−0.562874	−0.281437	+	0.959580i	$0.590811\pi$
$948$	0	0
$949$	1.97296e7	0.711135
$950$	−120958.	−0.00434835
$951$	0	0
$952$	0	0
$953$	−3.94908e7	−1.40852	−0.704262	−	0.709940i	$-0.748722\pi$
$953$	−3.94908e7	−1.40852	−0.704262	+	0.709940i	$0.748722\pi$
$954$	0	0
$955$	−4.20354e7	−1.49144
$956$	1.67748e7	0.593627
$957$	0	0
$958$	4.06701e7	1.43173
$959$	0	0
$960$	0	0
$961$	−2.06672e7	−0.721894
$962$	4.03485e6	0.140569
$963$	0	0
$964$	−4.01941e7	−1.39306
$965$	3.82118e7	1.32093
$966$	0	0
$967$	−1.87472e7	−0.644718	−0.322359	−	0.946617i	$-0.604476\pi$
$967$	−1.87472e7	−0.644718	−0.322359	+	0.946617i	$0.604476\pi$
$968$	−432231.	−0.0148261
$969$	0	0
$970$	6.31880e7	2.15628
$971$	5.35423e7	1.82242	0.911211	−	0.411940i	$-0.135149\pi$
$971$	5.35423e7	1.82242	0.911211	+	0.411940i	$0.135149\pi$
$972$	0	0
$973$	0	0
$974$	−1.21132e6	−0.0409131
$975$	0	0
$976$	−3.13804e7	−1.05447
$977$	−4.46104e7	−1.49520	−0.747600	−	0.664149i	$-0.768794\pi$
$977$	−4.46104e7	−1.49520	−0.747600	+	0.664149i	$0.768794\pi$
$978$	0	0
$979$	2.92200e7	0.974368
$980$	0	0
$981$	0	0
$982$	1.26566e7	0.418829
$983$	2.00068e7	0.660380	0.330190	−	0.943914i	$-0.392887\pi$
$983$	2.00068e7	0.660380	0.330190	+	0.943914i	$0.392887\pi$
$984$	0	0
$985$	2.21742e7	0.728213
$986$	4.31961e7	1.41498
$987$	0	0
$988$	47456.9	0.00154670
$989$	3.72826e7	1.21204
$990$	0	0
$991$	3.24635e7	1.05005	0.525026	−	0.851086i	$-0.324056\pi$
$991$	3.24635e7	1.05005	0.525026	+	0.851086i	$0.324056\pi$
$992$	−2.23923e7	−0.722471
$993$	0	0
$994$	0	0
$995$	−4.39649e7	−1.40783
$996$	0	0
$997$	2.78940e7	0.888736	0.444368	−	0.895844i	$-0.353428\pi$
$997$	2.78940e7	0.888736	0.444368	+	0.895844i	$0.353428\pi$
$998$	3.20945e7	1.02001
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.6.a.z.1.4		4
3.2	odd	2		49.6.a.g.1.2	yes	4
7.6	odd	2	inner	441.6.a.z.1.3		4
12.11	even	2		784.6.a.bf.1.1		4
21.2	odd	6		49.6.c.h.18.3		8
21.5	even	6		49.6.c.h.18.4		8
21.11	odd	6		49.6.c.h.30.3		8
21.17	even	6		49.6.c.h.30.4		8
21.20	even	2		49.6.a.g.1.1	✓	4
84.83	odd	2		784.6.a.bf.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.6.a.g.1.1	✓	4	21.20	even	2
49.6.a.g.1.2	yes	4	3.2	odd	2
49.6.c.h.18.3		8	21.2	odd	6
49.6.c.h.18.4		8	21.5	even	6
49.6.c.h.30.3		8	21.11	odd	6
49.6.c.h.30.4		8	21.17	even	6
441.6.a.z.1.3		4	7.6	odd	2	inner
441.6.a.z.1.4		4	1.1	even	1	trivial
784.6.a.bf.1.1		4	12.11	even	2
784.6.a.bf.1.4		4	84.83	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	441.a (trivial)

\(f(q)\)	\(=\)	\(q+7.81507 q^{2} +29.0754 q^{4} +74.2753 q^{5} -22.8562 q^{8} +O(q^{10})\)	\(q+7.81507 q^{2} +29.0754 q^{4} +74.2753 q^{5} -22.8562 q^{8} +580.467 q^{10} +424.219 q^{11} -252.233 q^{13} -1109.03 q^{16} +1104.35 q^{17} -6.47100 q^{19} +2159.58 q^{20} +3315.30 q^{22} +3612.39 q^{23} +2391.82 q^{25} -1971.22 q^{26} +5005.02 q^{29} +2821.69 q^{31} -7935.79 q^{32} +8630.55 q^{34} -2046.88 q^{37} -50.5713 q^{38} -1697.65 q^{40} -9393.81 q^{41} +10320.8 q^{43} +12334.3 q^{44} +28231.1 q^{46} -17035.6 q^{47} +18692.3 q^{50} -7333.78 q^{52} +39506.7 q^{53} +31509.0 q^{55} +39114.6 q^{58} -33949.8 q^{59} +28295.2 q^{61} +22051.7 q^{62} -26529.7 q^{64} -18734.7 q^{65} +56100.9 q^{67} +32109.3 q^{68} +15537.4 q^{71} -78219.5 q^{73} -15996.5 q^{74} -188.147 q^{76} -45335.5 q^{79} -82373.9 q^{80} -73413.3 q^{82} -1381.82 q^{83} +82025.7 q^{85} +80657.6 q^{86} -9696.05 q^{88} +68879.4 q^{89} +105031. q^{92} -133134. q^{94} -480.635 q^{95} +108857. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 10 q^{2} + 10 q^{4} + 270 q^{8}+O(q^{10}) \) 4 * q + 10 * q^2 + 10 * q^4 + 270 * q^8	\( 4 q + 10 q^{2} + 10 q^{4} + 270 q^{8} + 1952 q^{11} - 1566 q^{16} + 3524 q^{22} + 7136 q^{23} + 2764 q^{25} + 3352 q^{29} - 27810 q^{32} - 9208 q^{37} + 20448 q^{43} - 1900 q^{44} + 56712 q^{46} + 43070 q^{50} + 102920 q^{53} + 96972 q^{58} - 40318 q^{64} + 63168 q^{65} - 22896 q^{67} + 153824 q^{71} - 17596 q^{74} - 90688 q^{79} + 272656 q^{85} + 161860 q^{86} + 154812 q^{88} + 212200 q^{92} - 108224 q^{95}+O(q^{100}) \) 4 * q + 10 * q^2 + 10 * q^4 + 270 * q^8 + 1952 * q^11 - 1566 * q^16 + 3524 * q^22 + 7136 * q^23 + 2764 * q^25 + 3352 * q^29 - 27810 * q^32 - 9208 * q^37 + 20448 * q^43 - 1900 * q^44 + 56712 * q^46 + 43070 * q^50 + 102920 * q^53 + 96972 * q^58 - 40318 * q^64 + 63168 * q^65 - 22896 * q^67 + 153824 * q^71 - 17596 * q^74 - 90688 * q^79 + 272656 * q^85 + 161860 * q^86 + 154812 * q^88 + 212200 * q^92 - 108224 * q^95

Introduction

L-functions

Modular forms

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