LMFDB - Embedded newform 225.8.a.v.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,8,Mod(1,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 8, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("225.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	225.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:	$70.2866307339$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{649}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 162 $ x^2 - x - 162
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 25)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$13.2377$ of defining polynomial
Character	$\chi$	$=$	225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.23774 q^{2} -100.566 q^{4} -769.970 q^{7} +1197.17 q^{8} +O(q^{10})$	$q-5.23774 q^{2} -100.566 q^{4} -769.970 q^{7} +1197.17 q^{8} -5356.43 q^{11} -14037.0 q^{13} +4032.90 q^{14} +6602.00 q^{16} +12402.4 q^{17} -5038.89 q^{19} +28055.6 q^{22} -75190.5 q^{23} +73522.1 q^{26} +77432.9 q^{28} -195529. q^{29} -93568.2 q^{31} -187817. q^{32} -64960.4 q^{34} -161554. q^{37} +26392.4 q^{38} +28767.5 q^{41} -739076. q^{43} +538676. q^{44} +393828. q^{46} +1.06799e6 q^{47} -230689. q^{49} +1.41165e6 q^{52} +626442. q^{53} -921785. q^{56} +1.02413e6 q^{58} -2.14861e6 q^{59} -2.57497e6 q^{61} +490086. q^{62} +138682. q^{64} +807676. q^{67} -1.24726e6 q^{68} +1.72722e6 q^{71} +1.74519e6 q^{73} +846178. q^{74} +506742. q^{76} +4.12429e6 q^{77} -2.46887e6 q^{79} -150677. q^{82} +6.90850e6 q^{83} +3.87109e6 q^{86} -6.41256e6 q^{88} -2.63567e6 q^{89} +1.08081e7 q^{91} +7.56161e6 q^{92} -5.59385e6 q^{94} +1.01234e7 q^{97} +1.20829e6 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 15 q^{2} + 181 q^{4} + 600 q^{7} + 4305 q^{8}+O(q^{10}) $ 2 * q + 15 * q^2 + 181 * q^4 + 600 * q^7 + 4305 * q^8	$ 2 q + 15 q^{2} + 181 q^{4} + 600 q^{7} + 4305 q^{8} - 4344 q^{11} - 17680 q^{13} + 31758 q^{14} + 33457 q^{16} + 6870 q^{17} + 18200 q^{19} + 48545 q^{22} + 21120 q^{23} - 204 q^{26} + 463170 q^{28} - 55800 q^{29} - 301776 q^{31} - 42135 q^{32} - 176923 q^{34} - 609860 q^{37} + 496695 q^{38} + 108486 q^{41} - 966400 q^{43} + 823743 q^{44} + 2342934 q^{46} + 1787880 q^{47} + 822586 q^{49} + 385900 q^{52} + 130740 q^{53} + 3335850 q^{56} + 3851920 q^{58} - 2067600 q^{59} + 582044 q^{61} - 3723570 q^{62} - 350479 q^{64} - 255720 q^{67} - 2804985 q^{68} + 4728216 q^{71} + 1339430 q^{73} - 8226522 q^{74} + 7050025 q^{76} + 5511300 q^{77} - 7186200 q^{79} + 1462645 q^{82} + 12049560 q^{83} - 729444 q^{86} - 3266085 q^{88} + 5990850 q^{89} + 5817264 q^{91} + 34679370 q^{92} + 8975132 q^{94} + 17120020 q^{97} + 22524195 q^{98}+O(q^{100}) $ 2 * q + 15 * q^2 + 181 * q^4 + 600 * q^7 + 4305 * q^8 - 4344 * q^11 - 17680 * q^13 + 31758 * q^14 + 33457 * q^16 + 6870 * q^17 + 18200 * q^19 + 48545 * q^22 + 21120 * q^23 - 204 * q^26 + 463170 * q^28 - 55800 * q^29 - 301776 * q^31 - 42135 * q^32 - 176923 * q^34 - 609860 * q^37 + 496695 * q^38 + 108486 * q^41 - 966400 * q^43 + 823743 * q^44 + 2342934 * q^46 + 1787880 * q^47 + 822586 * q^49 + 385900 * q^52 + 130740 * q^53 + 3335850 * q^56 + 3851920 * q^58 - 2067600 * q^59 + 582044 * q^61 - 3723570 * q^62 - 350479 * q^64 - 255720 * q^67 - 2804985 * q^68 + 4728216 * q^71 + 1339430 * q^73 - 8226522 * q^74 + 7050025 * q^76 + 5511300 * q^77 - 7186200 * q^79 + 1462645 * q^82 + 12049560 * q^83 - 729444 * q^86 - 3266085 * q^88 + 5990850 * q^89 + 5817264 * q^91 + 34679370 * q^92 + 8975132 * q^94 + 17120020 * q^97 + 22524195 * 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 $

$2$	−5.23774	−0.462955	−0.231478	−	0.972840i	$-0.574356\pi$
$2$	−5.23774	−0.462955	−0.231478	+	0.972840i	$0.574356\pi$
$3$	0	0
$3$	0	0
$4$	−100.566	−0.785673
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−769.970	−0.848459	−0.424229	−	0.905555i	$-0.639455\pi$
$7$	−769.970	−0.848459	−0.424229	+	0.905555i	$0.639455\pi$
$8$	1197.17	0.826686
$9$	0	0
$10$	0	0
$11$	−5356.43	−1.21339	−0.606696	−	0.794934i	$-0.707506\pi$
$11$	−5356.43	−1.21339	−0.606696	+	0.794934i	$0.707506\pi$
$12$	0	0
$13$	−14037.0	−1.77204	−0.886018	−	0.463651i	$-0.846539\pi$
$13$	−14037.0	−1.77204	−0.886018	+	0.463651i	$0.846539\pi$
$14$	4032.90	0.392798
$15$	0	0
$16$	6602.00	0.402954
$17$	12402.4	0.612256	0.306128	−	0.951990i	$-0.400966\pi$
$17$	12402.4	0.612256	0.306128	+	0.951990i	$0.400966\pi$
$18$	0	0
$19$	−5038.89	−0.168538	−0.0842689	−	0.996443i	$-0.526855\pi$
$19$	−5038.89	−0.168538	−0.0842689	+	0.996443i	$0.526855\pi$
$20$	0	0
$21$	0	0
$22$	28055.6	0.561746
$23$	−75190.5	−1.28859	−0.644296	−	0.764776i	$-0.722849\pi$
$23$	−75190.5	−1.28859	−0.644296	+	0.764776i	$0.722849\pi$
$24$	0	0
$25$	0	0
$26$	73522.1	0.820373
$27$	0	0
$28$	77432.9	0.666611
$29$	−195529.	−1.48874	−0.744368	−	0.667770i	$-0.767249\pi$
$29$	−195529.	−1.48874	−0.744368	+	0.667770i	$0.767249\pi$
$30$	0	0
$31$	−93568.2	−0.564108	−0.282054	−	0.959399i	$-0.591016\pi$
$31$	−93568.2	−0.564108	−0.282054	+	0.959399i	$0.591016\pi$
$32$	−187817.	−1.01324
$33$	0	0
$34$	−64960.4	−0.283447
$35$	0	0
$36$	0	0
$37$	−161554.	−0.524338	−0.262169	−	0.965022i	$-0.584438\pi$
$37$	−161554.	−0.524338	−0.262169	+	0.965022i	$0.584438\pi$
$38$	26392.4	0.0780254
$39$	0	0
$40$	0	0
$41$	28767.5	0.0651867	0.0325933	−	0.999469i	$-0.489623\pi$
$41$	28767.5	0.0651867	0.0325933	+	0.999469i	$0.489623\pi$
$42$	0	0
$43$	−739076.	−1.41759	−0.708793	−	0.705417i	$-0.750760\pi$
$43$	−739076.	−1.41759	−0.708793	+	0.705417i	$0.750760\pi$
$44$	538676.	0.953329
$45$	0	0
$46$	393828.	0.596560
$47$	1.06799e6	1.50046	0.750229	−	0.661178i	$-0.229943\pi$
$47$	1.06799e6	1.50046	0.750229	+	0.661178i	$0.229943\pi$
$48$	0	0
$49$	−230689.	−0.280118
$50$	0	0
$51$	0	0
$52$	1.41165e6	1.39224
$53$	626442.	0.577983	0.288992	−	0.957332i	$-0.406680\pi$
$53$	626442.	0.577983	0.288992	+	0.957332i	$0.406680\pi$
$54$	0	0
$55$	0	0
$56$	−921785.	−0.701409
$57$	0	0
$58$	1.02413e6	0.689217
$59$	−2.14861e6	−1.36199	−0.680997	−	0.732287i	$-0.738453\pi$
$59$	−2.14861e6	−1.36199	−0.680997	+	0.732287i	$0.738453\pi$
$60$	0	0
$61$	−2.57497e6	−1.45251	−0.726253	−	0.687428i	$-0.758740\pi$
$61$	−2.57497e6	−1.45251	−0.726253	+	0.687428i	$0.758740\pi$
$62$	490086.	0.261157
$63$	0	0
$64$	138682.	0.0661288
$65$	0	0
$66$	0	0
$67$	807676.	0.328077	0.164038	−	0.986454i	$-0.447548\pi$
$67$	807676.	0.328077	0.164038	+	0.986454i	$0.447548\pi$
$68$	−1.24726e6	−0.481033
$69$	0	0
$70$	0	0
$71$	1.72722e6	0.572722	0.286361	−	0.958122i	$-0.407554\pi$
$71$	1.72722e6	0.572722	0.286361	+	0.958122i	$0.407554\pi$
$72$	0	0
$73$	1.74519e6	0.525064	0.262532	−	0.964923i	$-0.415442\pi$
$73$	1.74519e6	0.525064	0.262532	+	0.964923i	$0.415442\pi$
$74$	846178.	0.242745
$75$	0	0
$76$	506742.	0.132416
$77$	4.12429e6	1.02951
$78$	0	0
$79$	−2.46887e6	−0.563382	−0.281691	−	0.959505i	$-0.590895\pi$
$79$	−2.46887e6	−0.563382	−0.281691	+	0.959505i	$0.590895\pi$
$80$	0	0
$81$	0	0
$82$	−150677.	−0.0301785
$83$	6.90850e6	1.32620	0.663102	−	0.748529i	$-0.269240\pi$
$83$	6.90850e6	1.32620	0.663102	+	0.748529i	$0.269240\pi$
$84$	0	0
$85$	0	0
$86$	3.87109e6	0.656279
$87$	0	0
$88$	−6.41256e6	−1.00310
$89$	−2.63567e6	−0.396302	−0.198151	−	0.980171i	$-0.563494\pi$
$89$	−2.63567e6	−0.396302	−0.198151	+	0.980171i	$0.563494\pi$
$90$	0	0
$91$	1.08081e7	1.50350
$92$	7.56161e6	1.01241
$93$	0	0
$94$	−5.59385e6	−0.694645
$95$	0	0
$96$	0	0
$97$	1.01234e7	1.12622	0.563112	−	0.826381i	$-0.309604\pi$
$97$	1.01234e7	1.12622	0.563112	+	0.826381i	$0.309604\pi$
$98$	1.20829e6	0.129682
$99$	0	0
$100$	0	0
$101$	8.33196e6	0.804679	0.402339	−	0.915491i	$-0.368197\pi$
$101$	8.33196e6	0.804679	0.402339	+	0.915491i	$0.368197\pi$
$102$	0	0
$103$	7.75782e6	0.699535	0.349767	−	0.936837i	$-0.386261\pi$
$103$	7.75782e6	0.699535	0.349767	+	0.936837i	$0.386261\pi$
$104$	−1.68047e7	−1.46492
$105$	0	0
$106$	−3.28114e6	−0.267580
$107$	8.86598e6	0.699654	0.349827	−	0.936814i	$-0.386240\pi$
$107$	8.86598e6	0.699654	0.349827	+	0.936814i	$0.386240\pi$
$108$	0	0
$109$	1.55125e6	0.114734	0.0573668	−	0.998353i	$-0.481730\pi$
$109$	1.55125e6	0.114734	0.0573668	+	0.998353i	$0.481730\pi$
$110$	0	0
$111$	0	0
$112$	−5.08334e6	−0.341890
$113$	−1.80311e7	−1.17557	−0.587783	−	0.809019i	$-0.699999\pi$
$113$	−1.80311e7	−1.17557	−0.587783	+	0.809019i	$0.699999\pi$
$114$	0	0
$115$	0	0
$116$	1.96636e7	1.16966
$117$	0	0
$118$	1.12538e7	0.630542
$119$	−9.54945e6	−0.519474
$120$	0	0
$121$	9.20422e6	0.472322
$122$	1.34870e7	0.672445
$123$	0	0
$124$	9.40979e6	0.443204
$125$	0	0
$126$	0	0
$127$	1.98814e7	0.861258	0.430629	−	0.902529i	$-0.358292\pi$
$127$	1.98814e7	0.861258	0.430629	+	0.902529i	$0.358292\pi$
$128$	2.33142e7	0.982621
$129$	0	0
$130$	0	0
$131$	5.81990e6	0.226186	0.113093	−	0.993584i	$-0.463924\pi$
$131$	5.81990e6	0.226186	0.113093	+	0.993584i	$0.463924\pi$
$132$	0	0
$133$	3.87979e6	0.142997
$134$	−4.23040e6	−0.151885
$135$	0	0
$136$	1.48477e7	0.506144
$137$	3.83328e7	1.27365	0.636823	−	0.771010i	$-0.280248\pi$
$137$	3.83328e7	1.27365	0.636823	+	0.771010i	$0.280248\pi$
$138$	0	0
$139$	−3.60826e7	−1.13958	−0.569791	−	0.821789i	$-0.692976\pi$
$139$	−3.60826e7	−1.13958	−0.569791	+	0.821789i	$0.692976\pi$
$140$	0	0
$141$	0	0
$142$	−9.04673e6	−0.265144
$143$	7.51883e7	2.15018
$144$	0	0
$145$	0	0
$146$	−9.14084e6	−0.243081
$147$	0	0
$148$	1.62469e7	0.411958
$149$	4.91341e6	0.121683	0.0608416	−	0.998147i	$-0.480622\pi$
$149$	4.91341e6	0.121683	0.0608416	+	0.998147i	$0.480622\pi$
$150$	0	0
$151$	4.95186e7	1.17044	0.585220	−	0.810874i	$-0.301008\pi$
$151$	4.95186e7	1.17044	0.585220	+	0.810874i	$0.301008\pi$
$152$	−6.03241e6	−0.139328
$153$	0	0
$154$	−2.16020e7	−0.476619
$155$	0	0
$156$	0	0
$157$	5.84517e6	0.120545	0.0602724	−	0.998182i	$-0.480803\pi$
$157$	5.84517e6	0.120545	0.0602724	+	0.998182i	$0.480803\pi$
$158$	1.29313e7	0.260821
$159$	0	0
$160$	0	0
$161$	5.78944e7	1.09332
$162$	0	0
$163$	−1.79731e7	−0.325062	−0.162531	−	0.986703i	$-0.551966\pi$
$163$	−1.79731e7	−0.325062	−0.162531	+	0.986703i	$0.551966\pi$
$164$	−2.89304e6	−0.0512154
$165$	0	0
$166$	−3.61849e7	−0.613973
$167$	5.72460e7	0.951124	0.475562	−	0.879682i	$-0.342245\pi$
$167$	5.72460e7	0.951124	0.475562	+	0.879682i	$0.342245\pi$
$168$	0	0
$169$	1.34289e8	2.14011
$170$	0	0
$171$	0	0
$172$	7.43260e7	1.11376
$173$	−1.09265e8	−1.60442	−0.802212	−	0.597039i	$-0.796344\pi$
$173$	−1.09265e8	−1.60442	−0.802212	+	0.597039i	$0.796344\pi$
$174$	0	0
$175$	0	0
$176$	−3.53632e7	−0.488941
$177$	0	0
$178$	1.38050e7	0.183470
$179$	−1.98072e7	−0.258130	−0.129065	−	0.991636i	$-0.541198\pi$
$179$	−1.98072e7	−0.258130	−0.129065	+	0.991636i	$0.541198\pi$
$180$	0	0
$181$	−1.00547e8	−1.26036	−0.630178	−	0.776451i	$-0.717018\pi$
$181$	−1.00547e8	−1.26036	−0.630178	+	0.776451i	$0.717018\pi$
$182$	−5.66098e7	−0.696053
$183$	0	0
$184$	−9.00157e7	−1.06526
$185$	0	0
$186$	0	0
$187$	−6.64325e7	−0.742908
$188$	−1.07403e8	−1.17887
$189$	0	0
$190$	0	0
$191$	−2.28106e7	−0.236875	−0.118437	−	0.992962i	$-0.537789\pi$
$191$	−2.28106e7	−0.236875	−0.118437	+	0.992962i	$0.537789\pi$
$192$	0	0
$193$	1.92250e6	0.0192493	0.00962465	−	0.999954i	$-0.496936\pi$
$193$	1.92250e6	0.0192493	0.00962465	+	0.999954i	$0.496936\pi$
$194$	−5.30237e7	−0.521391
$195$	0	0
$196$	2.31995e7	0.220081
$197$	−1.98118e7	−0.184626	−0.0923129	−	0.995730i	$-0.529426\pi$
$197$	−1.98118e7	−0.184626	−0.0923129	+	0.995730i	$0.529426\pi$
$198$	0	0
$199$	−1.14071e8	−1.02610	−0.513048	−	0.858360i	$-0.671484\pi$
$199$	−1.14071e8	−1.02610	−0.513048	+	0.858360i	$0.671484\pi$
$200$	0	0
$201$	0	0
$202$	−4.36407e7	−0.372530
$203$	1.50551e8	1.26313
$204$	0	0
$205$	0	0
$206$	−4.06334e7	−0.323853
$207$	0	0
$208$	−9.26722e7	−0.714049
$209$	2.69905e7	0.204503
$210$	0	0
$211$	9.50285e6	0.0696410	0.0348205	−	0.999394i	$-0.488914\pi$
$211$	9.50285e6	0.0696410	0.0348205	+	0.999394i	$0.488914\pi$
$212$	−6.29988e7	−0.454106
$213$	0	0
$214$	−4.64377e7	−0.323909
$215$	0	0
$216$	0	0
$217$	7.20447e7	0.478622
$218$	−8.12507e6	−0.0531165
$219$	0	0
$220$	0	0
$221$	−1.74092e8	−1.08494
$222$	0	0
$223$	−1.92299e8	−1.16121	−0.580603	−	0.814187i	$-0.697183\pi$
$223$	−1.92299e8	−1.16121	−0.580603	+	0.814187i	$0.697183\pi$
$224$	1.44614e8	0.859689
$225$	0	0
$226$	9.44420e7	0.544234
$227$	−1.60189e8	−0.908955	−0.454478	−	0.890758i	$-0.650174\pi$
$227$	−1.60189e8	−0.908955	−0.454478	+	0.890758i	$0.650174\pi$
$228$	0	0
$229$	1.97660e8	1.08766	0.543832	−	0.839194i	$-0.316973\pi$
$229$	1.97660e8	1.08766	0.543832	+	0.839194i	$0.316973\pi$
$230$	0	0
$231$	0	0
$232$	−2.34081e8	−1.23072
$233$	−1.99180e8	−1.03157	−0.515786	−	0.856717i	$-0.672500\pi$
$233$	−1.99180e8	−1.03157	−0.515786	+	0.856717i	$0.672500\pi$
$234$	0	0
$235$	0	0
$236$	2.16077e8	1.07008
$237$	0	0
$238$	5.00175e7	0.240493
$239$	3.31111e8	1.56885	0.784424	−	0.620225i	$-0.212959\pi$
$239$	3.31111e8	1.56885	0.784424	+	0.620225i	$0.212959\pi$
$240$	0	0
$241$	−3.68270e8	−1.69475	−0.847377	−	0.530991i	$-0.821820\pi$
$241$	−3.68270e8	−1.69475	−0.847377	+	0.530991i	$0.821820\pi$
$242$	−4.82093e7	−0.218664
$243$	0	0
$244$	2.58955e8	1.14119
$245$	0	0
$246$	0	0
$247$	7.07309e7	0.298655
$248$	−1.12017e8	−0.466340
$249$	0	0
$250$	0	0
$251$	−3.74255e8	−1.49386	−0.746929	−	0.664903i	$-0.768473\pi$
$251$	−3.74255e8	−1.49386	−0.746929	+	0.664903i	$0.768473\pi$
$252$	0	0
$253$	4.02753e8	1.56357
$254$	−1.04133e8	−0.398724
$255$	0	0
$256$	−1.39865e8	−0.521038
$257$	−4.57062e8	−1.67961	−0.839807	−	0.542885i	$-0.817332\pi$
$257$	−4.57062e8	−1.67961	−0.839807	+	0.542885i	$0.817332\pi$
$258$	0	0
$259$	1.24392e8	0.444880
$260$	0	0
$261$	0	0
$262$	−3.04831e7	−0.104714
$263$	6.77338e7	0.229594	0.114797	−	0.993389i	$-0.463378\pi$
$263$	6.77338e7	0.229594	0.114797	+	0.993389i	$0.463378\pi$
$264$	0	0
$265$	0	0
$266$	−2.03214e7	−0.0662014
$267$	0	0
$268$	−8.12248e7	−0.257761
$269$	−2.21000e7	−0.0692244	−0.0346122	−	0.999401i	$-0.511020\pi$
$269$	−2.21000e7	−0.0692244	−0.0346122	+	0.999401i	$0.511020\pi$
$270$	0	0
$271$	4.22269e8	1.28883	0.644416	−	0.764675i	$-0.277100\pi$
$271$	4.22269e8	1.28883	0.644416	+	0.764675i	$0.277100\pi$
$272$	8.18804e7	0.246711
$273$	0	0
$274$	−2.00777e8	−0.589641
$275$	0	0
$276$	0	0
$277$	−1.98868e8	−0.562195	−0.281097	−	0.959679i	$-0.590698\pi$
$277$	−1.98868e8	−0.562195	−0.281097	+	0.959679i	$0.590698\pi$
$278$	1.88991e8	0.527576
$279$	0	0
$280$	0	0
$281$	−3.88135e8	−1.04354	−0.521771	−	0.853085i	$-0.674729\pi$
$281$	−3.88135e8	−1.04354	−0.521771	+	0.853085i	$0.674729\pi$
$282$	0	0
$283$	−2.98951e8	−0.784056	−0.392028	−	0.919953i	$-0.628226\pi$
$283$	−2.98951e8	−0.784056	−0.392028	+	0.919953i	$0.628226\pi$
$284$	−1.73700e8	−0.449972
$285$	0	0
$286$	−3.93817e8	−0.995435
$287$	−2.21501e7	−0.0553082
$288$	0	0
$289$	−2.56520e8	−0.625142
$290$	0	0
$291$	0	0
$292$	−1.75507e8	−0.412528
$293$	1.77029e8	0.411158	0.205579	−	0.978641i	$-0.434092\pi$
$293$	1.77029e8	0.411158	0.205579	+	0.978641i	$0.434092\pi$
$294$	0	0
$295$	0	0
$296$	−1.93408e8	−0.433463
$297$	0	0
$298$	−2.57351e7	−0.0563339
$299$	1.05545e9	2.28343
$300$	0	0
$301$	5.69066e8	1.20276
$302$	−2.59366e8	−0.541862
$303$	0	0
$304$	−3.32667e7	−0.0679130
$305$	0	0
$306$	0	0
$307$	−2.25726e8	−0.445243	−0.222622	−	0.974905i	$-0.571461\pi$
$307$	−2.25726e8	−0.445243	−0.222622	+	0.974905i	$0.571461\pi$
$308$	−4.14764e8	−0.808861
$309$	0	0
$310$	0	0
$311$	8.14288e8	1.53503	0.767515	−	0.641031i	$-0.221493\pi$
$311$	8.14288e8	1.53503	0.767515	+	0.641031i	$0.221493\pi$
$312$	0	0
$313$	−6.60615e7	−0.121771	−0.0608854	−	0.998145i	$-0.519392\pi$
$313$	−6.60615e7	−0.121771	−0.0608854	+	0.998145i	$0.519392\pi$
$314$	−3.06155e7	−0.0558068
$315$	0	0
$316$	2.48284e8	0.442634
$317$	6.63770e7	0.117033	0.0585167	−	0.998286i	$-0.481363\pi$
$317$	6.63770e7	0.117033	0.0585167	+	0.998286i	$0.481363\pi$
$318$	0	0
$319$	1.04734e9	1.80642
$320$	0	0
$321$	0	0
$322$	−3.03236e8	−0.506157
$323$	−6.24942e7	−0.103188
$324$	0	0
$325$	0	0
$326$	9.41385e7	0.150489
$327$	0	0
$328$	3.44396e7	0.0538889
$329$	−8.22319e8	−1.27308
$330$	0	0
$331$	−5.59199e8	−0.847557	−0.423778	−	0.905766i	$-0.639296\pi$
$331$	−5.59199e8	−0.847557	−0.423778	+	0.905766i	$0.639296\pi$
$332$	−6.94761e8	−1.04196
$333$	0	0
$334$	−2.99839e8	−0.440328
$335$	0	0
$336$	0	0
$337$	4.77074e8	0.679018	0.339509	−	0.940603i	$-0.389739\pi$
$337$	4.77074e8	0.679018	0.339509	+	0.940603i	$0.389739\pi$
$338$	−7.03370e8	−0.990775
$339$	0	0
$340$	0	0
$341$	5.01192e8	0.684485
$342$	0	0
$343$	8.11727e8	1.08613
$344$	−8.84799e8	−1.17190
$345$	0	0
$346$	5.72301e8	0.742777
$347$	5.49825e8	0.706433	0.353217	−	0.935542i	$-0.385088\pi$
$347$	5.49825e8	0.706433	0.353217	+	0.935542i	$0.385088\pi$
$348$	0	0
$349$	−2.51578e8	−0.316799	−0.158400	−	0.987375i	$-0.550633\pi$
$349$	−2.51578e8	−0.316799	−0.158400	+	0.987375i	$0.550633\pi$
$350$	0	0
$351$	0	0
$352$	1.00603e9	1.22945
$353$	8.04432e8	0.973370	0.486685	−	0.873577i	$-0.338206\pi$
$353$	8.04432e8	0.973370	0.486685	+	0.873577i	$0.338206\pi$
$354$	0	0
$355$	0	0
$356$	2.65059e8	0.311364
$357$	0	0
$358$	1.03745e8	0.119503
$359$	−1.13834e8	−0.129850	−0.0649250	−	0.997890i	$-0.520681\pi$
$359$	−1.13834e8	−0.129850	−0.0649250	+	0.997890i	$0.520681\pi$
$360$	0	0
$361$	−8.68481e8	−0.971595
$362$	5.26638e8	0.583488
$363$	0	0
$364$	−1.08693e9	−1.18126
$365$	0	0
$366$	0	0
$367$	−2.25920e8	−0.238574	−0.119287	−	0.992860i	$-0.538061\pi$
$367$	−2.25920e8	−0.238574	−0.119287	+	0.992860i	$0.538061\pi$
$368$	−4.96407e8	−0.519243
$369$	0	0
$370$	0	0
$371$	−4.82342e8	−0.490395
$372$	0	0
$373$	−8.19128e8	−0.817280	−0.408640	−	0.912696i	$-0.633997\pi$
$373$	−8.19128e8	−0.817280	−0.408640	+	0.912696i	$0.633997\pi$
$374$	3.47956e8	0.343933
$375$	0	0
$376$	1.27856e9	1.24041
$377$	2.74464e9	2.63809
$378$	0	0
$379$	−1.86356e8	−0.175836	−0.0879178	−	0.996128i	$-0.528021\pi$
$379$	−1.86356e8	−0.175836	−0.0879178	+	0.996128i	$0.528021\pi$
$380$	0	0
$381$	0	0
$382$	1.19476e8	0.109662
$383$	1.43839e9	1.30822	0.654109	−	0.756400i	$-0.273044\pi$
$383$	1.43839e9	1.30822	0.654109	+	0.756400i	$0.273044\pi$
$384$	0	0
$385$	0	0
$386$	−1.00695e7	−0.00891156
$387$	0	0
$388$	−1.01807e9	−0.884843
$389$	1.00373e9	0.864558	0.432279	−	0.901740i	$-0.357710\pi$
$389$	1.00373e9	0.864558	0.432279	+	0.901740i	$0.357710\pi$
$390$	0	0
$391$	−9.32540e8	−0.788949
$392$	−2.76174e8	−0.231570
$393$	0	0
$394$	1.03769e8	0.0854735
$395$	0	0
$396$	0	0
$397$	9.05187e8	0.726058	0.363029	−	0.931778i	$-0.381743\pi$
$397$	9.05187e8	0.726058	0.363029	+	0.931778i	$0.381743\pi$
$398$	5.97472e8	0.475036
$399$	0	0
$400$	0	0
$401$	−2.05917e9	−1.59473	−0.797363	−	0.603500i	$-0.793772\pi$
$401$	−2.05917e9	−1.59473	−0.797363	+	0.603500i	$0.793772\pi$
$402$	0	0
$403$	1.31342e9	0.999619
$404$	−8.37913e8	−0.632214
$405$	0	0
$406$	−7.88548e8	−0.584773
$407$	8.65354e8	0.636229
$408$	0	0
$409$	−6.09853e8	−0.440751	−0.220376	−	0.975415i	$-0.570728\pi$
$409$	−6.09853e8	−0.440751	−0.220376	+	0.975415i	$0.570728\pi$
$410$	0	0
$411$	0	0
$412$	−7.80173e8	−0.549605
$413$	1.65436e9	1.15559
$414$	0	0
$415$	0	0
$416$	2.63639e9	1.79549
$417$	0	0
$418$	−1.41369e8	−0.0946755
$419$	−2.76626e9	−1.83715	−0.918574	−	0.395248i	$-0.870659\pi$
$419$	−2.76626e9	−1.83715	−0.918574	+	0.395248i	$0.870659\pi$
$420$	0	0
$421$	1.11034e9	0.725219	0.362609	−	0.931941i	$-0.381886\pi$
$421$	1.11034e9	0.725219	0.362609	+	0.931941i	$0.381886\pi$
$422$	−4.97734e7	−0.0322407
$423$	0	0
$424$	7.49957e8	0.477811
$425$	0	0
$426$	0	0
$427$	1.98265e9	1.23239
$428$	−8.91617e8	−0.549699
$429$	0	0
$430$	0	0
$431$	1.55134e9	0.933331	0.466666	−	0.884434i	$-0.345455\pi$
$431$	1.55134e9	0.933331	0.466666	+	0.884434i	$0.345455\pi$
$432$	0	0
$433$	1.19534e9	0.707593	0.353797	−	0.935322i	$-0.384891\pi$
$433$	1.19534e9	0.707593	0.353797	+	0.935322i	$0.384891\pi$
$434$	−3.77351e8	−0.221581
$435$	0	0
$436$	−1.56004e8	−0.0901430
$437$	3.78876e8	0.217176
$438$	0	0
$439$	−2.81223e8	−0.158644	−0.0793221	−	0.996849i	$-0.525276\pi$
$439$	−2.81223e8	−0.158644	−0.0793221	+	0.996849i	$0.525276\pi$
$440$	0	0
$441$	0	0
$442$	9.11849e8	0.502279
$443$	−2.07780e9	−1.13551	−0.567756	−	0.823197i	$-0.692188\pi$
$443$	−2.07780e9	−1.13551	−0.567756	+	0.823197i	$0.692188\pi$
$444$	0	0
$445$	0	0
$446$	1.00721e9	0.537586
$447$	0	0
$448$	−1.06781e8	−0.0561075
$449$	1.73277e9	0.903397	0.451698	−	0.892171i	$-0.350818\pi$
$449$	1.73277e9	0.903397	0.451698	+	0.892171i	$0.350818\pi$
$450$	0	0
$451$	−1.54091e8	−0.0790971
$452$	1.81331e9	0.923609
$453$	0	0
$454$	8.39029e8	0.420805
$455$	0	0
$456$	0	0
$457$	1.78846e9	0.876541	0.438270	−	0.898843i	$-0.355591\pi$
$457$	1.78846e9	0.876541	0.438270	+	0.898843i	$0.355591\pi$
$458$	−1.03529e9	−0.503539
$459$	0	0
$460$	0	0
$461$	3.91667e8	0.186193	0.0930965	−	0.995657i	$-0.470323\pi$
$461$	3.91667e8	0.186193	0.0930965	+	0.995657i	$0.470323\pi$
$462$	0	0
$463$	1.05509e9	0.494034	0.247017	−	0.969011i	$-0.420550\pi$
$463$	1.05509e9	0.494034	0.247017	+	0.969011i	$0.420550\pi$
$464$	−1.29088e9	−0.599892
$465$	0	0
$466$	1.04325e9	0.477572
$467$	−1.73541e9	−0.788485	−0.394243	−	0.919006i	$-0.628993\pi$
$467$	−1.73541e9	−0.788485	−0.394243	+	0.919006i	$0.628993\pi$
$468$	0	0
$469$	−6.21886e8	−0.278360
$470$	0	0
$471$	0	0
$472$	−2.57225e9	−1.12594
$473$	3.95881e9	1.72009
$474$	0	0
$475$	0	0
$476$	9.60351e8	0.408137
$477$	0	0
$478$	−1.73427e9	−0.726306
$479$	−2.46088e8	−0.102310	−0.0511548	−	0.998691i	$-0.516290\pi$
$479$	−2.46088e8	−0.102310	−0.0511548	+	0.998691i	$0.516290\pi$
$480$	0	0
$481$	2.26773e9	0.929146
$482$	1.92890e9	0.784595
$483$	0	0
$484$	−9.25633e8	−0.371091
$485$	0	0
$486$	0	0
$487$	−2.60089e9	−1.02040	−0.510201	−	0.860055i	$-0.670429\pi$
$487$	−2.60089e9	−1.02040	−0.510201	+	0.860055i	$0.670429\pi$
$488$	−3.08267e9	−1.20077
$489$	0	0
$490$	0	0
$491$	−3.18729e9	−1.21517	−0.607584	−	0.794256i	$-0.707861\pi$
$491$	−3.18729e9	−1.21517	−0.607584	+	0.794256i	$0.707861\pi$
$492$	0	0
$493$	−2.42502e9	−0.911488
$494$	−3.70470e8	−0.138264
$495$	0	0
$496$	−6.17737e8	−0.227310
$497$	−1.32991e9	−0.485931
$498$	0	0
$499$	−3.08098e9	−1.11004	−0.555018	−	0.831838i	$-0.687289\pi$
$499$	−3.08098e9	−1.11004	−0.555018	+	0.831838i	$0.687289\pi$
$500$	0	0
$501$	0	0
$502$	1.96025e9	0.691590
$503$	2.17354e9	0.761517	0.380759	−	0.924674i	$-0.375663\pi$
$503$	2.17354e9	0.761517	0.380759	+	0.924674i	$0.375663\pi$
$504$	0	0
$505$	0	0
$506$	−2.10951e9	−0.723862
$507$	0	0
$508$	−1.99939e9	−0.676667
$509$	1.91522e9	0.643733	0.321866	−	0.946785i	$-0.395690\pi$
$509$	1.91522e9	0.643733	0.321866	+	0.946785i	$0.395690\pi$
$510$	0	0
$511$	−1.34374e9	−0.445495
$512$	−2.25164e9	−0.741404
$513$	0	0
$514$	2.39397e9	0.777586
$515$	0	0
$516$	0	0
$517$	−5.72061e9	−1.82065
$518$	−6.51532e8	−0.205959
$519$	0	0
$520$	0	0
$521$	6.78513e8	0.210197	0.105098	−	0.994462i	$-0.466484\pi$
$521$	6.78513e8	0.210197	0.105098	+	0.994462i	$0.466484\pi$
$522$	0	0
$523$	5.85186e9	1.78870	0.894351	−	0.447366i	$-0.147638\pi$
$523$	5.85186e9	1.78870	0.894351	+	0.447366i	$0.147638\pi$
$524$	−5.85285e8	−0.177708
$525$	0	0
$526$	−3.54772e8	−0.106292
$527$	−1.16047e9	−0.345379
$528$	0	0
$529$	2.24878e9	0.660468
$530$	0	0
$531$	0	0
$532$	−3.90176e8	−0.112349
$533$	−4.03810e8	−0.115513
$534$	0	0
$535$	0	0
$536$	9.66925e8	0.271216
$537$	0	0
$538$	1.15754e8	0.0320478
$539$	1.23567e9	0.339893
$540$	0	0
$541$	−7.71811e8	−0.209566	−0.104783	−	0.994495i	$-0.533415\pi$
$541$	−7.71811e8	−0.209566	−0.104783	+	0.994495i	$0.533415\pi$
$542$	−2.21173e9	−0.596672
$543$	0	0
$544$	−2.32938e9	−0.620360
$545$	0	0
$546$	0	0
$547$	5.26686e9	1.37593	0.687965	−	0.725744i	$-0.258504\pi$
$547$	5.26686e9	1.37593	0.687965	+	0.725744i	$0.258504\pi$
$548$	−3.85498e9	−1.00067
$549$	0	0
$550$	0	0
$551$	9.85247e8	0.250908
$552$	0	0
$553$	1.90095e9	0.478006
$554$	1.04162e9	0.260271
$555$	0	0
$556$	3.62869e9	0.895339
$557$	−8.78323e7	−0.0215358	−0.0107679	−	0.999942i	$-0.503428\pi$
$557$	−8.78323e7	−0.0215358	−0.0107679	+	0.999942i	$0.503428\pi$
$558$	0	0
$559$	1.03744e10	2.51201
$560$	0	0
$561$	0	0
$562$	2.03295e9	0.483114
$563$	5.32226e8	0.125695	0.0628473	−	0.998023i	$-0.479982\pi$
$563$	5.32226e8	0.125695	0.0628473	+	0.998023i	$0.479982\pi$
$564$	0	0
$565$	0	0
$566$	1.56583e9	0.362983
$567$	0	0
$568$	2.06778e9	0.473461
$569$	7.55216e9	1.71861	0.859306	−	0.511461i	$-0.170896\pi$
$569$	7.55216e9	1.71861	0.859306	+	0.511461i	$0.170896\pi$
$570$	0	0
$571$	−2.55147e9	−0.573541	−0.286771	−	0.957999i	$-0.592582\pi$
$571$	−2.55147e9	−0.573541	−0.286771	+	0.957999i	$0.592582\pi$
$572$	−7.56139e9	−1.68933
$573$	0	0
$574$	1.16017e8	0.0256052
$575$	0	0
$576$	0	0
$577$	−5.53852e9	−1.20027	−0.600135	−	0.799899i	$-0.704886\pi$
$577$	−5.53852e9	−1.20027	−0.600135	+	0.799899i	$0.704886\pi$
$578$	1.34358e9	0.289413
$579$	0	0
$580$	0	0
$581$	−5.31934e9	−1.12523
$582$	0	0
$583$	−3.35550e9	−0.701321
$584$	2.08929e9	0.434063
$585$	0	0
$586$	−9.27233e8	−0.190348
$587$	4.73642e9	0.966533	0.483266	−	0.875473i	$-0.339450\pi$
$587$	4.73642e9	0.966533	0.483266	+	0.875473i	$0.339450\pi$
$588$	0	0
$589$	4.71480e8	0.0950735
$590$	0	0
$591$	0	0
$592$	−1.06658e9	−0.211284
$593$	1.44860e9	0.285272	0.142636	−	0.989775i	$-0.454442\pi$
$593$	1.44860e9	0.285272	0.142636	+	0.989775i	$0.454442\pi$
$594$	0	0
$595$	0	0
$596$	−4.94122e8	−0.0956032
$597$	0	0
$598$	−5.52816e9	−1.05713
$599$	−1.04150e9	−0.198001	−0.0990005	−	0.995087i	$-0.531565\pi$
$599$	−1.04150e9	−0.198001	−0.0990005	+	0.995087i	$0.531565\pi$
$600$	0	0
$601$	−7.31995e9	−1.37546	−0.687729	−	0.725967i	$-0.741392\pi$
$601$	−7.31995e9	−1.37546	−0.687729	+	0.725967i	$0.741392\pi$
$602$	−2.98062e9	−0.556825
$603$	0	0
$604$	−4.97990e9	−0.919583
$605$	0	0
$606$	0	0
$607$	−6.16291e9	−1.11847	−0.559237	−	0.829008i	$-0.688906\pi$
$607$	−6.16291e9	−1.11847	−0.559237	+	0.829008i	$0.688906\pi$
$608$	9.46391e8	0.170769
$609$	0	0
$610$	0	0
$611$	−1.49914e10	−2.65887
$612$	0	0
$613$	−3.18382e8	−0.0558260	−0.0279130	−	0.999610i	$-0.508886\pi$
$613$	−3.18382e8	−0.0558260	−0.0279130	+	0.999610i	$0.508886\pi$
$614$	1.18229e9	0.206128
$615$	0	0
$616$	4.93748e9	0.851085
$617$	−7.13555e9	−1.22301	−0.611505	−	0.791241i	$-0.709435\pi$
$617$	−7.13555e9	−1.22301	−0.611505	+	0.791241i	$0.709435\pi$
$618$	0	0
$619$	−9.96303e9	−1.68840	−0.844198	−	0.536031i	$-0.819923\pi$
$619$	−9.96303e9	−1.68840	−0.844198	+	0.536031i	$0.819923\pi$
$620$	0	0
$621$	0	0
$622$	−4.26503e9	−0.710650
$623$	2.02939e9	0.336246
$624$	0	0
$625$	0	0
$626$	3.46013e8	0.0563744
$627$	0	0
$628$	−5.87826e8	−0.0947087
$629$	−2.00365e9	−0.321030
$630$	0	0
$631$	−4.94300e9	−0.783227	−0.391614	−	0.920130i	$-0.628083\pi$
$631$	−4.94300e9	−0.783227	−0.391614	+	0.920130i	$0.628083\pi$
$632$	−2.95565e9	−0.465740
$633$	0	0
$634$	−3.47665e8	−0.0541813
$635$	0	0
$636$	0	0
$637$	3.23818e9	0.496379
$638$	−5.48568e9	−0.836292
$639$	0	0
$640$	0	0
$641$	2.18024e9	0.326965	0.163482	−	0.986546i	$-0.447727\pi$
$641$	2.18024e9	0.326965	0.163482	+	0.986546i	$0.447727\pi$
$642$	0	0
$643$	−8.70849e9	−1.29183	−0.645913	−	0.763411i	$-0.723523\pi$
$643$	−8.70849e9	−1.29183	−0.645913	+	0.763411i	$0.723523\pi$
$644$	−5.82221e9	−0.858989
$645$	0	0
$646$	3.27328e8	0.0477716
$647$	4.57610e9	0.664249	0.332124	−	0.943236i	$-0.392235\pi$
$647$	4.57610e9	0.664249	0.332124	+	0.943236i	$0.392235\pi$
$648$	0	0
$649$	1.15089e10	1.65263
$650$	0	0
$651$	0	0
$652$	1.80749e9	0.255393
$653$	−9.23098e9	−1.29733	−0.648667	−	0.761072i	$-0.724673\pi$
$653$	−9.23098e9	−1.29733	−0.648667	+	0.761072i	$0.724673\pi$
$654$	0	0
$655$	0	0
$656$	1.89923e8	0.0262672
$657$	0	0
$658$	4.30709e9	0.589378
$659$	8.30221e9	1.13004	0.565021	−	0.825077i	$-0.308868\pi$
$659$	8.30221e9	1.13004	0.565021	+	0.825077i	$0.308868\pi$
$660$	0	0
$661$	2.67892e9	0.360791	0.180395	−	0.983594i	$-0.442262\pi$
$661$	2.67892e9	0.360791	0.180395	+	0.983594i	$0.442262\pi$
$662$	2.92894e9	0.392381
$663$	0	0
$664$	8.27064e9	1.09635
$665$	0	0
$666$	0	0
$667$	1.47019e10	1.91837
$668$	−5.75700e9	−0.747272
$669$	0	0
$670$	0	0
$671$	1.37927e10	1.76246
$672$	0	0
$673$	8.63768e9	1.09231	0.546153	−	0.837685i	$-0.316092\pi$
$673$	8.63768e9	1.09231	0.546153	+	0.837685i	$0.316092\pi$
$674$	−2.49879e9	−0.314355
$675$	0	0
$676$	−1.35049e10	−1.68143
$677$	−8.17701e9	−1.01282	−0.506412	−	0.862291i	$-0.669029\pi$
$677$	−8.17701e9	−1.01282	−0.506412	+	0.862291i	$0.669029\pi$
$678$	0	0
$679$	−7.79471e9	−0.955555
$680$	0	0
$681$	0	0
$682$	−2.62511e9	−0.316886
$683$	−3.81811e9	−0.458539	−0.229269	−	0.973363i	$-0.573634\pi$
$683$	−3.81811e9	−0.458539	−0.229269	+	0.973363i	$0.573634\pi$
$684$	0	0
$685$	0	0
$686$	−4.25162e9	−0.502828
$687$	0	0
$688$	−4.87938e9	−0.571222
$689$	−8.79336e9	−1.02421
$690$	0	0
$691$	1.38590e10	1.59794	0.798969	−	0.601372i	$-0.205379\pi$
$691$	1.38590e10	1.59794	0.798969	+	0.601372i	$0.205379\pi$
$692$	1.09883e10	1.26055
$693$	0	0
$694$	−2.87984e9	−0.327047
$695$	0	0
$696$	0	0
$697$	3.56785e8	0.0399110
$698$	1.31770e9	0.146664
$699$	0	0
$700$	0	0
$701$	4.38274e9	0.480543	0.240271	−	0.970706i	$-0.422764\pi$
$701$	4.38274e9	0.480543	0.240271	+	0.970706i	$0.422764\pi$
$702$	0	0
$703$	8.14053e8	0.0883708
$704$	−7.42842e8	−0.0802402
$705$	0	0
$706$	−4.21341e9	−0.450627
$707$	−6.41536e9	−0.682737
$708$	0	0
$709$	−5.46537e9	−0.575914	−0.287957	−	0.957643i	$-0.592976\pi$
$709$	−5.46537e9	−0.575914	−0.287957	+	0.957643i	$0.592976\pi$
$710$	0	0
$711$	0	0
$712$	−3.15535e9	−0.327618
$713$	7.03543e9	0.726905
$714$	0	0
$715$	0	0
$716$	1.99194e9	0.202806
$717$	0	0
$718$	5.96233e8	0.0601147
$719$	7.24738e9	0.727160	0.363580	−	0.931563i	$-0.381554\pi$
$719$	7.24738e9	0.727160	0.363580	+	0.931563i	$0.381554\pi$
$720$	0	0
$721$	−5.97329e9	−0.593526
$722$	4.54888e9	0.449805
$723$	0	0
$724$	1.01116e10	0.990227
$725$	0	0
$726$	0	0
$727$	−1.61701e10	−1.56078	−0.780392	−	0.625291i	$-0.784980\pi$
$727$	−1.61701e10	−1.56078	−0.780392	+	0.625291i	$0.784980\pi$
$728$	1.29391e10	1.24292
$729$	0	0
$730$	0	0
$731$	−9.16629e9	−0.867926
$732$	0	0
$733$	−6.59871e9	−0.618864	−0.309432	−	0.950922i	$-0.600139\pi$
$733$	−6.59871e9	−0.618864	−0.309432	+	0.950922i	$0.600139\pi$
$734$	1.18331e9	0.110449
$735$	0	0
$736$	1.41221e10	1.30565
$737$	−4.32626e9	−0.398086
$738$	0	0
$739$	−4.52765e8	−0.0412683	−0.0206342	−	0.999787i	$-0.506569\pi$
$739$	−4.52765e8	−0.0412683	−0.0206342	+	0.999787i	$0.506569\pi$
$740$	0	0
$741$	0	0
$742$	2.52638e9	0.227031
$743$	2.03152e10	1.81702	0.908509	−	0.417864i	$-0.137221\pi$
$743$	2.03152e10	1.81702	0.908509	+	0.417864i	$0.137221\pi$
$744$	0	0
$745$	0	0
$746$	4.29038e9	0.378364
$747$	0	0
$748$	6.68085e9	0.583682
$749$	−6.82654e9	−0.593628
$750$	0	0
$751$	1.40167e10	1.20755	0.603776	−	0.797154i	$-0.293662\pi$
$751$	1.40167e10	1.20755	0.603776	+	0.797154i	$0.293662\pi$
$752$	7.05086e9	0.604616
$753$	0	0
$754$	−1.43757e10	−1.22132
$755$	0	0
$756$	0	0
$757$	1.74979e10	1.46605	0.733026	−	0.680200i	$-0.238107\pi$
$757$	1.74979e10	1.46605	0.733026	+	0.680200i	$0.238107\pi$
$758$	9.76086e8	0.0814040
$759$	0	0
$760$	0	0
$761$	−7.69298e9	−0.632774	−0.316387	−	0.948630i	$-0.602470\pi$
$761$	−7.69298e9	−0.632774	−0.316387	+	0.948630i	$0.602470\pi$
$762$	0	0
$763$	−1.19442e9	−0.0973466
$764$	2.29397e9	0.186106
$765$	0	0
$766$	−7.53390e9	−0.605646
$767$	3.01600e10	2.41350
$768$	0	0
$769$	−6.85942e9	−0.543933	−0.271967	−	0.962307i	$-0.587674\pi$
$769$	−6.85942e9	−0.543933	−0.271967	+	0.962307i	$0.587674\pi$
$770$	0	0
$771$	0	0
$772$	−1.93338e8	−0.0151236
$773$	−8.93170e9	−0.695514	−0.347757	−	0.937585i	$-0.613057\pi$
$773$	−8.93170e9	−0.695514	−0.347757	+	0.937585i	$0.613057\pi$
$774$	0	0
$775$	0	0
$776$	1.21194e10	0.931034
$777$	0	0
$778$	−5.25728e9	−0.400251
$779$	−1.44956e8	−0.0109864
$780$	0	0
$781$	−9.25175e9	−0.694937
$782$	4.88440e9	0.365248
$783$	0	0
$784$	−1.52301e9	−0.112875
$785$	0	0
$786$	0	0
$787$	5.20674e9	0.380762	0.190381	−	0.981710i	$-0.439028\pi$
$787$	5.20674e9	0.380762	0.190381	+	0.981710i	$0.439028\pi$
$788$	1.99240e9	0.145055
$789$	0	0
$790$	0	0
$791$	1.38834e10	0.997419
$792$	0	0
$793$	3.61448e10	2.57389
$794$	−4.74113e9	−0.336132
$795$	0	0
$796$	1.14716e10	0.806176
$797$	1.96111e10	1.37214	0.686068	−	0.727538i	$-0.259335\pi$
$797$	1.96111e10	1.37214	0.686068	+	0.727538i	$0.259335\pi$
$798$	0	0
$799$	1.32456e10	0.918666
$800$	0	0
$801$	0	0
$802$	1.07854e10	0.738287
$803$	−9.34798e9	−0.637109
$804$	0	0
$805$	0	0
$806$	−6.87933e9	−0.462779
$807$	0	0
$808$	9.97477e9	0.665217
$809$	−1.99512e10	−1.32479	−0.662397	−	0.749153i	$-0.730461\pi$
$809$	−1.99512e10	−1.32479	−0.662397	+	0.749153i	$0.730461\pi$
$810$	0	0
$811$	2.18039e9	0.143536	0.0717679	−	0.997421i	$-0.477136\pi$
$811$	2.18039e9	0.143536	0.0717679	+	0.997421i	$0.477136\pi$
$812$	−1.51403e10	−0.992407
$813$	0	0
$814$	−4.53250e9	−0.294545
$815$	0	0
$816$	0	0
$817$	3.72412e9	0.238917
$818$	3.19425e9	0.204048
$819$	0	0
$820$	0	0
$821$	2.52847e10	1.59462	0.797310	−	0.603570i	$-0.206256\pi$
$821$	2.52847e10	1.59462	0.797310	+	0.603570i	$0.206256\pi$
$822$	0	0
$823$	−1.86851e10	−1.16841	−0.584205	−	0.811606i	$-0.698594\pi$
$823$	−1.86851e10	−1.16841	−0.584205	+	0.811606i	$0.698594\pi$
$824$	9.28742e9	0.578296
$825$	0	0
$826$	−8.66512e9	−0.534989
$827$	−7.19901e9	−0.442592	−0.221296	−	0.975207i	$-0.571029\pi$
$827$	−7.19901e9	−0.442592	−0.221296	+	0.975207i	$0.571029\pi$
$828$	0	0
$829$	1.83330e10	1.11762	0.558808	−	0.829297i	$-0.311259\pi$
$829$	1.83330e10	1.11762	0.558808	+	0.829297i	$0.311259\pi$
$830$	0	0
$831$	0	0
$832$	−1.94668e9	−0.117183
$833$	−2.86109e9	−0.171504
$834$	0	0
$835$	0	0
$836$	−2.71433e9	−0.160672
$837$	0	0
$838$	1.44890e10	0.850517
$839$	−2.42360e10	−1.41675	−0.708376	−	0.705835i	$-0.750572\pi$
$839$	−2.42360e10	−1.41675	−0.708376	+	0.705835i	$0.750572\pi$
$840$	0	0
$841$	2.09816e10	1.21633
$842$	−5.81568e9	−0.335744
$843$	0	0
$844$	−9.55664e8	−0.0547150
$845$	0	0
$846$	0	0
$847$	−7.08698e9	−0.400746
$848$	4.13577e9	0.232901
$849$	0	0
$850$	0	0
$851$	1.21473e10	0.675658
$852$	0	0
$853$	−3.21411e10	−1.77312	−0.886561	−	0.462612i	$-0.846912\pi$
$853$	−3.21411e10	−1.77312	−0.886561	+	0.462612i	$0.846912\pi$
$854$	−1.03846e10	−0.570541
$855$	0	0
$856$	1.06141e10	0.578395
$857$	9.48584e9	0.514805	0.257403	−	0.966304i	$-0.417133\pi$
$857$	9.48584e9	0.514805	0.257403	+	0.966304i	$0.417133\pi$
$858$	0	0
$859$	−1.33721e10	−0.719819	−0.359910	−	0.932987i	$-0.617192\pi$
$859$	−1.33721e10	−0.719819	−0.359910	+	0.932987i	$0.617192\pi$
$860$	0	0
$861$	0	0
$862$	−8.12550e9	−0.432091
$863$	1.13928e10	0.603381	0.301690	−	0.953406i	$-0.402449\pi$
$863$	1.13928e10	0.603381	0.301690	+	0.953406i	$0.402449\pi$
$864$	0	0
$865$	0	0
$866$	−6.26088e9	−0.327584
$867$	0	0
$868$	−7.24525e9	−0.376040
$869$	1.32243e10	0.683604
$870$	0	0
$871$	−1.13373e10	−0.581364
$872$	1.85711e9	0.0948486
$873$	0	0
$874$	−1.98446e9	−0.100543
$875$	0	0
$876$	0	0
$877$	−1.45509e10	−0.728435	−0.364218	−	0.931314i	$-0.618664\pi$
$877$	−1.45509e10	−0.728435	−0.364218	+	0.931314i	$0.618664\pi$
$878$	1.47297e9	0.0734452
$879$	0	0
$880$	0	0
$881$	5.46095e9	0.269062	0.134531	−	0.990909i	$-0.457047\pi$
$881$	5.46095e9	0.269062	0.134531	+	0.990909i	$0.457047\pi$
$882$	0	0
$883$	1.95772e10	0.956948	0.478474	−	0.878102i	$-0.341190\pi$
$883$	1.95772e10	0.956948	0.478474	+	0.878102i	$0.341190\pi$
$884$	1.75078e10	0.852408
$885$	0	0
$886$	1.08830e10	0.525691
$887$	1.35654e10	0.652678	0.326339	−	0.945253i	$-0.394185\pi$
$887$	1.35654e10	0.652678	0.326339	+	0.945253i	$0.394185\pi$
$888$	0	0
$889$	−1.53081e10	−0.730742
$890$	0	0
$891$	0	0
$892$	1.93387e10	0.912328
$893$	−5.38148e9	−0.252884
$894$	0	0
$895$	0	0
$896$	−1.79513e10	−0.833714
$897$	0	0
$898$	−9.07579e9	−0.418232
$899$	1.82953e10	0.839807
$900$	0	0
$901$	7.76936e9	0.353874
$902$	8.07090e8	0.0366184
$903$	0	0
$904$	−2.15862e10	−0.971824
$905$	0	0
$906$	0	0
$907$	7.63541e9	0.339787	0.169894	−	0.985462i	$-0.445658\pi$
$907$	7.63541e9	0.339787	0.169894	+	0.985462i	$0.445658\pi$
$908$	1.61096e10	0.714141
$909$	0	0
$910$	0	0
$911$	2.03069e10	0.889875	0.444937	−	0.895562i	$-0.353226\pi$
$911$	2.03069e10	0.889875	0.444937	+	0.895562i	$0.353226\pi$
$912$	0	0
$913$	−3.70049e10	−1.60921
$914$	−9.36748e9	−0.405799
$915$	0	0
$916$	−1.98779e10	−0.854547
$917$	−4.48115e9	−0.191910
$918$	0	0
$919$	1.93832e10	0.823797	0.411899	−	0.911230i	$-0.364866\pi$
$919$	1.93832e10	0.823797	0.411899	+	0.911230i	$0.364866\pi$
$920$	0	0
$921$	0	0
$922$	−2.05145e9	−0.0861990
$923$	−2.42450e10	−1.01488
$924$	0	0
$925$	0	0
$926$	−5.52629e9	−0.228715
$927$	0	0
$928$	3.67237e10	1.50844
$929$	−1.17298e10	−0.479993	−0.239997	−	0.970774i	$-0.577146\pi$
$929$	−1.17298e10	−0.479993	−0.239997	+	0.970774i	$0.577146\pi$
$930$	0	0
$931$	1.16242e9	0.0472104
$932$	2.00307e10	0.810478
$933$	0	0
$934$	9.08964e9	0.365033
$935$	0	0
$936$	0	0
$937$	1.41104e10	0.560340	0.280170	−	0.959950i	$-0.409609\pi$
$937$	1.41104e10	0.560340	0.280170	+	0.959950i	$0.409609\pi$
$938$	3.25728e9	0.128868
$939$	0	0
$940$	0	0
$941$	−1.98332e10	−0.775942	−0.387971	−	0.921672i	$-0.626824\pi$
$941$	−1.98332e10	−0.775942	−0.387971	+	0.921672i	$0.626824\pi$
$942$	0	0
$943$	−2.16304e9	−0.0839990
$944$	−1.41851e10	−0.548821
$945$	0	0
$946$	−2.07352e10	−0.796324
$947$	−4.96249e10	−1.89878	−0.949390	−	0.314098i	$-0.898298\pi$
$947$	−4.96249e10	−1.89878	−0.949390	+	0.314098i	$0.898298\pi$
$948$	0	0
$949$	−2.44972e10	−0.930432
$950$	0	0
$951$	0	0
$952$	−1.14323e10	−0.429442
$953$	−3.24475e10	−1.21438	−0.607192	−	0.794555i	$-0.707704\pi$
$953$	−3.24475e10	−1.21438	−0.607192	+	0.794555i	$0.707704\pi$
$954$	0	0
$955$	0	0
$956$	−3.32985e10	−1.23260
$957$	0	0
$958$	1.28895e9	0.0473648
$959$	−2.95151e10	−1.08064
$960$	0	0
$961$	−1.87576e10	−0.681782
$962$	−1.18778e10	−0.430153
$963$	0	0
$964$	3.70355e10	1.33152
$965$	0	0
$966$	0	0
$967$	−6.43301e9	−0.228782	−0.114391	−	0.993436i	$-0.536492\pi$
$967$	−6.43301e9	−0.228782	−0.114391	+	0.993436i	$0.536492\pi$
$968$	1.10190e10	0.390462
$969$	0	0
$970$	0	0
$971$	−7.99885e9	−0.280389	−0.140194	−	0.990124i	$-0.544773\pi$
$971$	−7.99885e9	−0.280389	−0.140194	+	0.990124i	$0.544773\pi$
$972$	0	0
$973$	2.77825e10	0.966889
$974$	1.36228e10	0.472400
$975$	0	0
$976$	−1.69999e10	−0.585293
$977$	−2.43029e10	−0.833733	−0.416866	−	0.908968i	$-0.636872\pi$
$977$	−2.43029e10	−0.833733	−0.416866	+	0.908968i	$0.636872\pi$
$978$	0	0
$979$	1.41178e10	0.480871
$980$	0	0
$981$	0	0
$982$	1.66942e10	0.562568
$983$	−2.05054e10	−0.688544	−0.344272	−	0.938870i	$-0.611874\pi$
$983$	−2.05054e10	−0.688544	−0.344272	+	0.938870i	$0.611874\pi$
$984$	0	0
$985$	0	0
$986$	1.27016e10	0.421978
$987$	0	0
$988$	−7.11313e9	−0.234645
$989$	5.55714e10	1.82669
$990$	0	0
$991$	−3.93587e10	−1.28464	−0.642322	−	0.766435i	$-0.722029\pi$
$991$	−3.93587e10	−1.28464	−0.642322	+	0.766435i	$0.722029\pi$
$992$	1.75737e10	0.571574
$993$	0	0
$994$	6.96571e9	0.224964
$995$	0	0
$996$	0	0
$997$	−2.18052e10	−0.696831	−0.348416	−	0.937340i	$-0.613280\pi$
$997$	−2.18052e10	−0.696831	−0.348416	+	0.937340i	$0.613280\pi$
$998$	1.61374e10	0.513897
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.8.a.v.1.1		2
3.2	odd	2		25.8.a.c.1.2	✓	2
5.2	odd	4		225.8.b.l.199.2		4
5.3	odd	4		225.8.b.l.199.3		4
5.4	even	2		225.8.a.k.1.2		2
12.11	even	2		400.8.a.bd.1.1		2
15.2	even	4		25.8.b.b.24.3		4
15.8	even	4		25.8.b.b.24.2		4
15.14	odd	2		25.8.a.e.1.1	yes	2
60.23	odd	4		400.8.c.s.49.2		4
60.47	odd	4		400.8.c.s.49.3		4
60.59	even	2		400.8.a.v.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.8.a.c.1.2	✓	2	3.2	odd	2
25.8.a.e.1.1	yes	2	15.14	odd	2
25.8.b.b.24.2		4	15.8	even	4
25.8.b.b.24.3		4	15.2	even	4
225.8.a.k.1.2		2	5.4	even	2
225.8.a.v.1.1		2	1.1	even	1	trivial
225.8.b.l.199.2		4	5.2	odd	4
225.8.b.l.199.3		4	5.3	odd	4
400.8.a.v.1.2		2	60.59	even	2
400.8.a.bd.1.1		2	12.11	even	2
400.8.c.s.49.2		4	60.23	odd	4
400.8.c.s.49.3		4	60.47	odd	4

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:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	225.a (trivial)

\(f(q)\)	\(=\)	\(q-5.23774 q^{2} -100.566 q^{4} -769.970 q^{7} +1197.17 q^{8} +O(q^{10})\)	\(q-5.23774 q^{2} -100.566 q^{4} -769.970 q^{7} +1197.17 q^{8} -5356.43 q^{11} -14037.0 q^{13} +4032.90 q^{14} +6602.00 q^{16} +12402.4 q^{17} -5038.89 q^{19} +28055.6 q^{22} -75190.5 q^{23} +73522.1 q^{26} +77432.9 q^{28} -195529. q^{29} -93568.2 q^{31} -187817. q^{32} -64960.4 q^{34} -161554. q^{37} +26392.4 q^{38} +28767.5 q^{41} -739076. q^{43} +538676. q^{44} +393828. q^{46} +1.06799e6 q^{47} -230689. q^{49} +1.41165e6 q^{52} +626442. q^{53} -921785. q^{56} +1.02413e6 q^{58} -2.14861e6 q^{59} -2.57497e6 q^{61} +490086. q^{62} +138682. q^{64} +807676. q^{67} -1.24726e6 q^{68} +1.72722e6 q^{71} +1.74519e6 q^{73} +846178. q^{74} +506742. q^{76} +4.12429e6 q^{77} -2.46887e6 q^{79} -150677. q^{82} +6.90850e6 q^{83} +3.87109e6 q^{86} -6.41256e6 q^{88} -2.63567e6 q^{89} +1.08081e7 q^{91} +7.56161e6 q^{92} -5.59385e6 q^{94} +1.01234e7 q^{97} +1.20829e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 15 q^{2} + 181 q^{4} + 600 q^{7} + 4305 q^{8}+O(q^{10}) \) 2 * q + 15 * q^2 + 181 * q^4 + 600 * q^7 + 4305 * q^8	\( 2 q + 15 q^{2} + 181 q^{4} + 600 q^{7} + 4305 q^{8} - 4344 q^{11} - 17680 q^{13} + 31758 q^{14} + 33457 q^{16} + 6870 q^{17} + 18200 q^{19} + 48545 q^{22} + 21120 q^{23} - 204 q^{26} + 463170 q^{28} - 55800 q^{29} - 301776 q^{31} - 42135 q^{32} - 176923 q^{34} - 609860 q^{37} + 496695 q^{38} + 108486 q^{41} - 966400 q^{43} + 823743 q^{44} + 2342934 q^{46} + 1787880 q^{47} + 822586 q^{49} + 385900 q^{52} + 130740 q^{53} + 3335850 q^{56} + 3851920 q^{58} - 2067600 q^{59} + 582044 q^{61} - 3723570 q^{62} - 350479 q^{64} - 255720 q^{67} - 2804985 q^{68} + 4728216 q^{71} + 1339430 q^{73} - 8226522 q^{74} + 7050025 q^{76} + 5511300 q^{77} - 7186200 q^{79} + 1462645 q^{82} + 12049560 q^{83} - 729444 q^{86} - 3266085 q^{88} + 5990850 q^{89} + 5817264 q^{91} + 34679370 q^{92} + 8975132 q^{94} + 17120020 q^{97} + 22524195 q^{98}+O(q^{100}) \) 2 * q + 15 * q^2 + 181 * q^4 + 600 * q^7 + 4305 * q^8 - 4344 * q^11 - 17680 * q^13 + 31758 * q^14 + 33457 * q^16 + 6870 * q^17 + 18200 * q^19 + 48545 * q^22 + 21120 * q^23 - 204 * q^26 + 463170 * q^28 - 55800 * q^29 - 301776 * q^31 - 42135 * q^32 - 176923 * q^34 - 609860 * q^37 + 496695 * q^38 + 108486 * q^41 - 966400 * q^43 + 823743 * q^44 + 2342934 * q^46 + 1787880 * q^47 + 822586 * q^49 + 385900 * q^52 + 130740 * q^53 + 3335850 * q^56 + 3851920 * q^58 - 2067600 * q^59 + 582044 * q^61 - 3723570 * q^62 - 350479 * q^64 - 255720 * q^67 - 2804985 * q^68 + 4728216 * q^71 + 1339430 * q^73 - 8226522 * q^74 + 7050025 * q^76 + 5511300 * q^77 - 7186200 * q^79 + 1462645 * q^82 + 12049560 * q^83 - 729444 * q^86 - 3266085 * q^88 + 5990850 * q^89 + 5817264 * q^91 + 34679370 * q^92 + 8975132 * q^94 + 17120020 * q^97 + 22524195 * q^98

Introduction

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