LMFDB - Embedded newform 400.10.a.y.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,10,Mod(1,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("400.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	400.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:	$206.014334466$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 652x + 4000 $ x^3 - x^2 - 652*x + 4000
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 5 $
Twist minimal:	no (minimal twist has level 25)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$6.48955$ of defining polynomial
Character	$\chi$	$=$	400.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+30.5073 q^{3} -4010.25 q^{7} -18752.3 q^{9} +O(q^{10})$	$q+30.5073 q^{3} -4010.25 q^{7} -18752.3 q^{9} +42110.0 q^{11} -123743. q^{13} -319945. q^{17} -1.08733e6 q^{19} -122342. q^{21} +1.50672e6 q^{23} -1.17256e6 q^{27} -2.62160e6 q^{29} -3.27023e6 q^{31} +1.28466e6 q^{33} +2.51034e6 q^{37} -3.77508e6 q^{39} +2.95349e7 q^{41} -1.42413e7 q^{43} +1.35318e6 q^{47} -2.42715e7 q^{49} -9.76067e6 q^{51} -9.73342e7 q^{53} -3.31714e7 q^{57} +7.48924e6 q^{59} -9.11752e7 q^{61} +7.52015e7 q^{63} +2.94376e8 q^{67} +4.59659e7 q^{69} -1.56193e8 q^{71} +2.82539e8 q^{73} -1.68872e8 q^{77} +5.55294e8 q^{79} +3.33330e8 q^{81} +6.48378e6 q^{83} -7.99778e7 q^{87} -5.99001e8 q^{89} +4.96242e8 q^{91} -9.97660e7 q^{93} -9.25317e8 q^{97} -7.89660e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 89 q^{3} + 5258 q^{7} + 58234 q^{9}+O(q^{10}) $ 3 * q + 89 * q^3 + 5258 * q^7 + 58234 * q^9	$ 3 q + 89 q^{3} + 5258 q^{7} + 58234 q^{9} + 54699 q^{11} - 215884 q^{13} - 334983 q^{17} - 818845 q^{19} - 2375394 q^{21} + 3526854 q^{23} + 6633395 q^{27} + 2175480 q^{29} - 4274066 q^{31} + 22122137 q^{33} + 10305042 q^{37} + 50414092 q^{39} + 5926311 q^{41} - 24429956 q^{43} + 66858708 q^{47} - 6453929 q^{49} + 25634699 q^{51} - 132620514 q^{53} - 252946415 q^{57} - 5670960 q^{59} + 125306926 q^{61} + 284323404 q^{63} + 88829483 q^{67} - 314274942 q^{69} - 297550596 q^{71} - 181321729 q^{73} - 561214086 q^{77} + 310025170 q^{79} + 1398847363 q^{81} - 731088801 q^{83} + 1046385560 q^{87} - 1103860035 q^{89} - 1183187656 q^{91} - 107386758 q^{93} + 332236842 q^{97} + 892234522 q^{99}+O(q^{100}) $ 3 * q + 89 * q^3 + 5258 * q^7 + 58234 * q^9 + 54699 * q^11 - 215884 * q^13 - 334983 * q^17 - 818845 * q^19 - 2375394 * q^21 + 3526854 * q^23 + 6633395 * q^27 + 2175480 * q^29 - 4274066 * q^31 + 22122137 * q^33 + 10305042 * q^37 + 50414092 * q^39 + 5926311 * q^41 - 24429956 * q^43 + 66858708 * q^47 - 6453929 * q^49 + 25634699 * q^51 - 132620514 * q^53 - 252946415 * q^57 - 5670960 * q^59 + 125306926 * q^61 + 284323404 * q^63 + 88829483 * q^67 - 314274942 * q^69 - 297550596 * q^71 - 181321729 * q^73 - 561214086 * q^77 + 310025170 * q^79 + 1398847363 * q^81 - 731088801 * q^83 + 1046385560 * q^87 - 1103860035 * q^89 - 1183187656 * q^91 - 107386758 * q^93 + 332236842 * q^97 + 892234522 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	30.5073	0.217449	0.108725	−	0.994072i	$-0.465323\pi$
$3$	30.5073	0.217449	0.108725	+	0.994072i	$0.465323\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4010.25	−0.631292	−0.315646	−	0.948877i	$-0.602221\pi$
$7$	−4010.25	−0.631292	−0.315646	+	0.948877i	$0.602221\pi$
$8$	0	0
$9$	−18752.3	−0.952716
$10$	0	0
$11$	42110.0	0.867198	0.433599	−	0.901106i	$-0.357243\pi$
$11$	42110.0	0.867198	0.433599	+	0.901106i	$0.357243\pi$
$12$	0	0
$13$	−123743.	−1.20165	−0.600824	−	0.799382i	$-0.705161\pi$
$13$	−123743.	−1.20165	−0.600824	+	0.799382i	$0.705161\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−319945.	−0.929085	−0.464543	−	0.885551i	$-0.653781\pi$
$17$	−319945.	−0.929085	−0.464543	+	0.885551i	$0.653781\pi$
$18$	0	0
$19$	−1.08733e6	−1.91412	−0.957059	−	0.289893i	$-0.906380\pi$
$19$	−1.08733e6	−1.91412	−0.957059	+	0.289893i	$0.906380\pi$
$20$	0	0
$21$	−122342.	−0.137274
$22$	0	0
$23$	1.50672e6	1.12268	0.561341	−	0.827585i	$-0.310286\pi$
$23$	1.50672e6	1.12268	0.561341	+	0.827585i	$0.310286\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.17256e6	−0.424617
$28$	0	0
$29$	−2.62160e6	−0.688295	−0.344148	−	0.938916i	$-0.611832\pi$
$29$	−2.62160e6	−0.688295	−0.344148	+	0.938916i	$0.611832\pi$
$30$	0	0
$31$	−3.27023e6	−0.635991	−0.317996	−	0.948092i	$-0.603010\pi$
$31$	−3.27023e6	−0.635991	−0.317996	+	0.948092i	$0.603010\pi$
$32$	0	0
$33$	1.28466e6	0.188572
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.51034e6	0.220204	0.110102	−	0.993920i	$-0.464882\pi$
$37$	2.51034e6	0.220204	0.110102	+	0.993920i	$0.464882\pi$
$38$	0	0
$39$	−3.77508e6	−0.261297
$40$	0	0
$41$	2.95349e7	1.63233	0.816165	−	0.577819i	$-0.196096\pi$
$41$	2.95349e7	1.63233	0.816165	+	0.577819i	$0.196096\pi$
$42$	0	0
$43$	−1.42413e7	−0.635244	−0.317622	−	0.948217i	$-0.602884\pi$
$43$	−1.42413e7	−0.635244	−0.317622	+	0.948217i	$0.602884\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.35318e6	0.0404496	0.0202248	−	0.999795i	$-0.493562\pi$
$47$	1.35318e6	0.0404496	0.0202248	+	0.999795i	$0.493562\pi$
$48$	0	0
$49$	−2.42715e7	−0.601470
$50$	0	0
$51$	−9.76067e6	−0.202029
$52$	0	0
$53$	−9.73342e7	−1.69443	−0.847216	−	0.531249i	$-0.821723\pi$
$53$	−9.73342e7	−1.69443	−0.847216	+	0.531249i	$0.821723\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−3.31714e7	−0.416224
$58$	0	0
$59$	7.48924e6	0.0804644	0.0402322	−	0.999190i	$-0.487190\pi$
$59$	7.48924e6	0.0804644	0.0402322	+	0.999190i	$0.487190\pi$
$60$	0	0
$61$	−9.11752e7	−0.843126	−0.421563	−	0.906799i	$-0.638518\pi$
$61$	−9.11752e7	−0.843126	−0.421563	+	0.906799i	$0.638518\pi$
$62$	0	0
$63$	7.52015e7	0.601442
$64$	0	0
$65$	0	0
$66$	0	0
$67$	2.94376e8	1.78470	0.892350	−	0.451344i	$-0.149055\pi$
$67$	2.94376e8	1.78470	0.892350	+	0.451344i	$0.149055\pi$
$68$	0	0
$69$	4.59659e7	0.244127
$70$	0	0
$71$	−1.56193e8	−0.729455	−0.364728	−	0.931114i	$-0.618838\pi$
$71$	−1.56193e8	−0.729455	−0.364728	+	0.931114i	$0.618838\pi$
$72$	0	0
$73$	2.82539e8	1.16446	0.582232	−	0.813023i	$-0.302180\pi$
$73$	2.82539e8	1.16446	0.582232	+	0.813023i	$0.302180\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.68872e8	−0.547455
$78$	0	0
$79$	5.55294e8	1.60399	0.801994	−	0.597332i	$-0.203772\pi$
$79$	5.55294e8	1.60399	0.801994	+	0.597332i	$0.203772\pi$
$80$	0	0
$81$	3.33330e8	0.860383
$82$	0	0
$83$	6.48378e6	0.0149960	0.00749802	−	0.999972i	$-0.497613\pi$
$83$	6.48378e6	0.0149960	0.00749802	+	0.999972i	$0.497613\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−7.99778e7	−0.149669
$88$	0	0
$89$	−5.99001e8	−1.01198	−0.505990	−	0.862539i	$-0.668873\pi$
$89$	−5.99001e8	−1.01198	−0.505990	+	0.862539i	$0.668873\pi$
$90$	0	0
$91$	4.96242e8	0.758591
$92$	0	0
$93$	−9.97660e7	−0.138296
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−9.25317e8	−1.06125	−0.530625	−	0.847606i	$-0.678043\pi$
$97$	−9.25317e8	−1.06125	−0.530625	+	0.847606i	$0.678043\pi$
$98$	0	0
$99$	−7.89660e8	−0.826193
$100$	0	0
$101$	9.58959e8	0.916967	0.458483	−	0.888703i	$-0.348393\pi$
$101$	9.58959e8	0.916967	0.458483	+	0.888703i	$0.348393\pi$
$102$	0	0
$103$	1.60441e8	0.140458	0.0702292	−	0.997531i	$-0.477627\pi$
$103$	1.60441e8	0.140458	0.0702292	+	0.997531i	$0.477627\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	9.60457e8	0.708355	0.354178	−	0.935178i	$-0.384761\pi$
$107$	9.60457e8	0.708355	0.354178	+	0.935178i	$0.384761\pi$
$108$	0	0
$109$	9.98912e8	0.677810	0.338905	−	0.940821i	$-0.389943\pi$
$109$	9.98912e8	0.677810	0.338905	+	0.940821i	$0.389943\pi$
$110$	0	0
$111$	7.65836e7	0.0478831
$112$	0	0
$113$	−2.50705e9	−1.44647	−0.723236	−	0.690601i	$-0.757346\pi$
$113$	−2.50705e9	−1.44647	−0.723236	+	0.690601i	$0.757346\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.32047e9	1.14483
$118$	0	0
$119$	1.28306e9	0.586524
$120$	0	0
$121$	−5.84695e8	−0.247968
$122$	0	0
$123$	9.01030e8	0.354949
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−2.47541e9	−0.844364	−0.422182	−	0.906511i	$-0.638736\pi$
$127$	−2.47541e9	−0.844364	−0.422182	+	0.906511i	$0.638736\pi$
$128$	0	0
$129$	−4.34463e8	−0.138134
$130$	0	0
$131$	−1.92402e9	−0.570808	−0.285404	−	0.958407i	$-0.592128\pi$
$131$	−1.92402e9	−0.570808	−0.285404	+	0.958407i	$0.592128\pi$
$132$	0	0
$133$	4.36045e9	1.20837
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−4.48594e8	−0.108796	−0.0543978	−	0.998519i	$-0.517324\pi$
$137$	−4.48594e8	−0.108796	−0.0543978	+	0.998519i	$0.517324\pi$
$138$	0	0
$139$	−4.48415e9	−1.01886	−0.509429	−	0.860513i	$-0.670143\pi$
$139$	−4.48415e9	−1.01886	−0.509429	+	0.860513i	$0.670143\pi$
$140$	0	0
$141$	4.12818e7	0.00879575
$142$	0	0
$143$	−5.21084e9	−1.04207
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−7.40458e8	−0.130789
$148$	0	0
$149$	−2.20480e9	−0.366463	−0.183232	−	0.983070i	$-0.558656\pi$
$149$	−2.20480e9	−0.366463	−0.183232	+	0.983070i	$0.558656\pi$
$150$	0	0
$151$	3.21248e9	0.502857	0.251428	−	0.967876i	$-0.419100\pi$
$151$	3.21248e9	0.502857	0.251428	+	0.967876i	$0.419100\pi$
$152$	0	0
$153$	5.99971e9	0.885154
$154$	0	0
$155$	0	0
$156$	0	0
$157$	1.08870e10	1.43007	0.715036	−	0.699088i	$-0.246410\pi$
$157$	1.08870e10	1.43007	0.715036	+	0.699088i	$0.246410\pi$
$158$	0	0
$159$	−2.96940e9	−0.368453
$160$	0	0
$161$	−6.04232e9	−0.708740
$162$	0	0
$163$	1.19994e10	1.33142	0.665708	−	0.746212i	$-0.268130\pi$
$163$	1.19994e10	1.33142	0.665708	+	0.746212i	$0.268130\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	9.68608e9	0.963660	0.481830	−	0.876265i	$-0.339972\pi$
$167$	9.68608e9	0.963660	0.481830	+	0.876265i	$0.339972\pi$
$168$	0	0
$169$	4.70793e9	0.443956
$170$	0	0
$171$	2.03899e10	1.82361
$172$	0	0
$173$	−7.35665e9	−0.624414	−0.312207	−	0.950014i	$-0.601068\pi$
$173$	−7.35665e9	−0.624414	−0.312207	+	0.950014i	$0.601068\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2.28477e8	0.0174969
$178$	0	0
$179$	2.00351e9	0.145866	0.0729329	−	0.997337i	$-0.476764\pi$
$179$	2.00351e9	0.145866	0.0729329	+	0.997337i	$0.476764\pi$
$180$	0	0
$181$	5.63414e9	0.390188	0.195094	−	0.980785i	$-0.437499\pi$
$181$	5.63414e9	0.390188	0.195094	+	0.980785i	$0.437499\pi$
$182$	0	0
$183$	−2.78151e9	−0.183337
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−1.34729e10	−0.805701
$188$	0	0
$189$	4.70225e9	0.268057
$190$	0	0
$191$	−9.16925e9	−0.498521	−0.249261	−	0.968436i	$-0.580188\pi$
$191$	−9.16925e9	−0.498521	−0.249261	+	0.968436i	$0.580188\pi$
$192$	0	0
$193$	3.16327e10	1.64107	0.820536	−	0.571594i	$-0.193675\pi$
$193$	3.16327e10	1.64107	0.820536	+	0.571594i	$0.193675\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2.59858e10	−1.22924	−0.614621	−	0.788822i	$-0.710691\pi$
$197$	−2.59858e10	−1.22924	−0.614621	+	0.788822i	$0.710691\pi$
$198$	0	0
$199$	−1.05766e10	−0.478088	−0.239044	−	0.971009i	$-0.576834\pi$
$199$	−1.05766e10	−0.478088	−0.239044	+	0.971009i	$0.576834\pi$
$200$	0	0
$201$	8.98061e9	0.388082
$202$	0	0
$203$	1.05133e10	0.434515
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−2.82544e10	−1.06960
$208$	0	0
$209$	−4.57873e10	−1.65992
$210$	0	0
$211$	−1.80228e10	−0.625965	−0.312983	−	0.949759i	$-0.601328\pi$
$211$	−1.80228e10	−0.625965	−0.312983	+	0.949759i	$0.601328\pi$
$212$	0	0
$213$	−4.76502e9	−0.158620
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1.31145e10	0.401496
$218$	0	0
$219$	8.61951e9	0.253212
$220$	0	0
$221$	3.95911e10	1.11643
$222$	0	0
$223$	4.44522e10	1.20371	0.601855	−	0.798606i	$-0.294429\pi$
$223$	4.44522e10	1.20371	0.601855	+	0.798606i	$0.294429\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5.59677e10	1.39901	0.699505	−	0.714627i	$-0.253404\pi$
$227$	5.59677e10	1.39901	0.699505	+	0.714627i	$0.253404\pi$
$228$	0	0
$229$	−1.47705e9	−0.0354923	−0.0177462	−	0.999843i	$-0.505649\pi$
$229$	−1.47705e9	−0.0354923	−0.0177462	+	0.999843i	$0.505649\pi$
$230$	0	0
$231$	−5.15182e9	−0.119044
$232$	0	0
$233$	7.83279e9	0.174106	0.0870532	−	0.996204i	$-0.472255\pi$
$233$	7.83279e9	0.174106	0.0870532	+	0.996204i	$0.472255\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.69405e10	0.348786
$238$	0	0
$239$	−5.56371e10	−1.10300	−0.551498	−	0.834176i	$-0.685943\pi$
$239$	−5.56371e10	−1.10300	−0.551498	+	0.834176i	$0.685943\pi$
$240$	0	0
$241$	−1.16053e10	−0.221606	−0.110803	−	0.993842i	$-0.535342\pi$
$241$	−1.16053e10	−0.221606	−0.110803	+	0.993842i	$0.535342\pi$
$242$	0	0
$243$	3.32485e10	0.611707
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.34549e11	2.30009
$248$	0	0
$249$	1.97803e8	0.00326088
$250$	0	0
$251$	3.45974e10	0.550189	0.275094	−	0.961417i	$-0.411291\pi$
$251$	3.45974e10	0.550189	0.275094	+	0.961417i	$0.411291\pi$
$252$	0	0
$253$	6.34479e10	0.973587
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3.67735e10	0.525818	0.262909	−	0.964821i	$-0.415318\pi$
$257$	3.67735e10	0.525818	0.262909	+	0.964821i	$0.415318\pi$
$258$	0	0
$259$	−1.00671e10	−0.139013
$260$	0	0
$261$	4.91609e10	0.655750
$262$	0	0
$263$	1.33758e11	1.72392	0.861962	−	0.506974i	$-0.169236\pi$
$263$	1.33758e11	1.72392	0.861962	+	0.506974i	$0.169236\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−1.82739e10	−0.220055
$268$	0	0
$269$	−1.42461e11	−1.65886	−0.829429	−	0.558612i	$-0.811334\pi$
$269$	−1.42461e11	−1.65886	−0.829429	+	0.558612i	$0.811334\pi$
$270$	0	0
$271$	1.11046e11	1.25067	0.625333	−	0.780358i	$-0.284963\pi$
$271$	1.11046e11	1.25067	0.625333	+	0.780358i	$0.284963\pi$
$272$	0	0
$273$	1.51390e10	0.164955
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−2.80726e10	−0.286500	−0.143250	−	0.989687i	$-0.545755\pi$
$277$	−2.80726e10	−0.286500	−0.143250	+	0.989687i	$0.545755\pi$
$278$	0	0
$279$	6.13244e10	0.605919
$280$	0	0
$281$	5.47143e10	0.523507	0.261753	−	0.965135i	$-0.415699\pi$
$281$	5.47143e10	0.523507	0.261753	+	0.965135i	$0.415699\pi$
$282$	0	0
$283$	−1.09950e11	−1.01895	−0.509477	−	0.860484i	$-0.670161\pi$
$283$	−1.09950e11	−1.01895	−0.509477	+	0.860484i	$0.670161\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.18442e11	−1.03048
$288$	0	0
$289$	−1.62229e10	−0.136801
$290$	0	0
$291$	−2.82289e10	−0.230768
$292$	0	0
$293$	2.30453e11	1.82675	0.913373	−	0.407124i	$-0.133468\pi$
$293$	2.30453e11	1.82675	0.913373	+	0.407124i	$0.133468\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−4.93764e10	−0.368227
$298$	0	0
$299$	−1.86446e11	−1.34907
$300$	0	0
$301$	5.71111e10	0.401025
$302$	0	0
$303$	2.92552e10	0.199394
$304$	0	0
$305$	0	0
$306$	0	0
$307$	6.85036e10	0.440140	0.220070	−	0.975484i	$-0.429371\pi$
$307$	6.85036e10	0.440140	0.220070	+	0.975484i	$0.429371\pi$
$308$	0	0
$309$	4.89462e9	0.0305426
$310$	0	0
$311$	1.98797e11	1.20500	0.602502	−	0.798118i	$-0.294171\pi$
$311$	1.98797e11	1.20500	0.602502	+	0.798118i	$0.294171\pi$
$312$	0	0
$313$	6.15202e10	0.362300	0.181150	−	0.983455i	$-0.442018\pi$
$313$	6.15202e10	0.362300	0.181150	+	0.983455i	$0.442018\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.25932e11	1.25664	0.628320	−	0.777955i	$-0.283743\pi$
$317$	2.25932e11	1.25664	0.628320	+	0.777955i	$0.283743\pi$
$318$	0	0
$319$	−1.10395e11	−0.596888
$320$	0	0
$321$	2.93010e10	0.154031
$322$	0	0
$323$	3.47885e11	1.77838
$324$	0	0
$325$	0	0
$326$	0	0
$327$	3.04741e10	0.147389
$328$	0	0
$329$	−5.42658e9	−0.0255355
$330$	0	0
$331$	8.38825e10	0.384101	0.192050	−	0.981385i	$-0.438486\pi$
$331$	8.38825e10	0.384101	0.192050	+	0.981385i	$0.438486\pi$
$332$	0	0
$333$	−4.70746e10	−0.209791
$334$	0	0
$335$	0	0
$336$	0	0
$337$	2.19457e11	0.926862	0.463431	−	0.886133i	$-0.346618\pi$
$337$	2.19457e11	0.926862	0.463431	+	0.886133i	$0.346618\pi$
$338$	0	0
$339$	−7.64834e10	−0.314535
$340$	0	0
$341$	−1.37710e11	−0.551530
$342$	0	0
$343$	2.59163e11	1.01100
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.85960e11	−0.688551	−0.344276	−	0.938869i	$-0.611875\pi$
$347$	−1.85960e11	−0.688551	−0.344276	+	0.938869i	$0.611875\pi$
$348$	0	0
$349$	2.73237e11	0.985881	0.492940	−	0.870063i	$-0.335922\pi$
$349$	2.73237e11	0.985881	0.492940	+	0.870063i	$0.335922\pi$
$350$	0	0
$351$	1.45096e11	0.510240
$352$	0	0
$353$	3.02861e11	1.03814	0.519072	−	0.854731i	$-0.326278\pi$
$353$	3.02861e11	1.03814	0.519072	+	0.854731i	$0.326278\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	3.91427e10	0.127539
$358$	0	0
$359$	−2.47660e11	−0.786919	−0.393460	−	0.919342i	$-0.628722\pi$
$359$	−2.47660e11	−0.786919	−0.393460	+	0.919342i	$0.628722\pi$
$360$	0	0
$361$	8.59591e11	2.66385
$362$	0	0
$363$	−1.78375e10	−0.0539205
$364$	0	0
$365$	0	0
$366$	0	0
$367$	2.66354e11	0.766412	0.383206	−	0.923663i	$-0.374820\pi$
$367$	2.66354e11	0.766412	0.383206	+	0.923663i	$0.374820\pi$
$368$	0	0
$369$	−5.53847e11	−1.55515
$370$	0	0
$371$	3.90335e11	1.06968
$372$	0	0
$373$	5.20850e11	1.39323	0.696616	−	0.717445i	$-0.254688\pi$
$373$	5.20850e11	1.39323	0.696616	+	0.717445i	$0.254688\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.24405e11	0.827088
$378$	0	0
$379$	−3.28308e10	−0.0817344	−0.0408672	−	0.999165i	$-0.513012\pi$
$379$	−3.28308e10	−0.0817344	−0.0408672	+	0.999165i	$0.513012\pi$
$380$	0	0
$381$	−7.55180e10	−0.183607
$382$	0	0
$383$	−7.15293e10	−0.169859	−0.0849297	−	0.996387i	$-0.527067\pi$
$383$	−7.15293e10	−0.169859	−0.0849297	+	0.996387i	$0.527067\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2.67057e11	0.605207
$388$	0	0
$389$	−2.02681e11	−0.448787	−0.224394	−	0.974499i	$-0.572040\pi$
$389$	−2.02681e11	−0.448787	−0.224394	+	0.974499i	$0.572040\pi$
$390$	0	0
$391$	−4.82067e11	−1.04307
$392$	0	0
$393$	−5.86968e10	−0.124122
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−1.63266e10	−0.0329867	−0.0164934	−	0.999864i	$-0.505250\pi$
$397$	−1.63266e10	−0.0329867	−0.0164934	+	0.999864i	$0.505250\pi$
$398$	0	0
$399$	1.33026e11	0.262759
$400$	0	0
$401$	−8.20766e11	−1.58515	−0.792574	−	0.609776i	$-0.791259\pi$
$401$	−8.20766e11	−1.58515	−0.792574	+	0.609776i	$0.791259\pi$
$402$	0	0
$403$	4.04670e11	0.764237
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.05710e11	0.190960
$408$	0	0
$409$	−4.24628e11	−0.750332	−0.375166	−	0.926958i	$-0.622414\pi$
$409$	−4.24628e11	−0.750332	−0.375166	+	0.926958i	$0.622414\pi$
$410$	0	0
$411$	−1.36854e10	−0.0236575
$412$	0	0
$413$	−3.00337e10	−0.0507966
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−1.36799e11	−0.221550
$418$	0	0
$419$	1.26375e10	0.0200308	0.0100154	−	0.999950i	$-0.496812\pi$
$419$	1.26375e10	0.0200308	0.0100154	+	0.999950i	$0.496812\pi$
$420$	0	0
$421$	−3.04545e10	−0.0472479	−0.0236240	−	0.999721i	$-0.507520\pi$
$421$	−3.04545e10	−0.0472479	−0.0236240	+	0.999721i	$0.507520\pi$
$422$	0	0
$423$	−2.53752e10	−0.0385370
$424$	0	0
$425$	0	0
$426$	0	0
$427$	3.65635e11	0.532259
$428$	0	0
$429$	−1.58969e11	−0.226597
$430$	0	0
$431$	−4.05830e11	−0.566495	−0.283248	−	0.959047i	$-0.591412\pi$
$431$	−4.05830e11	−0.566495	−0.283248	+	0.959047i	$0.591412\pi$
$432$	0	0
$433$	−1.36978e11	−0.187265	−0.0936324	−	0.995607i	$-0.529848\pi$
$433$	−1.36978e11	−0.187265	−0.0936324	+	0.995607i	$0.529848\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.63829e12	−2.14895
$438$	0	0
$439$	−7.84981e11	−1.00872	−0.504358	−	0.863495i	$-0.668271\pi$
$439$	−7.84981e11	−1.00872	−0.504358	+	0.863495i	$0.668271\pi$
$440$	0	0
$441$	4.55146e11	0.573030
$442$	0	0
$443$	8.87799e11	1.09521	0.547605	−	0.836737i	$-0.315540\pi$
$443$	8.87799e11	1.09521	0.547605	+	0.836737i	$0.315540\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−6.72624e10	−0.0796872
$448$	0	0
$449$	8.35477e11	0.970122	0.485061	−	0.874480i	$-0.338797\pi$
$449$	8.35477e11	0.970122	0.485061	+	0.874480i	$0.338797\pi$
$450$	0	0
$451$	1.24371e12	1.41555
$452$	0	0
$453$	9.80041e10	0.109346
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−6.01172e11	−0.644727	−0.322364	−	0.946616i	$-0.604477\pi$
$457$	−6.01172e11	−0.644727	−0.322364	+	0.946616i	$0.604477\pi$
$458$	0	0
$459$	3.75154e11	0.394505
$460$	0	0
$461$	1.52807e12	1.57576	0.787879	−	0.615829i	$-0.211179\pi$
$461$	1.52807e12	1.57576	0.787879	+	0.615829i	$0.211179\pi$
$462$	0	0
$463$	−7.80402e11	−0.789231	−0.394615	−	0.918846i	$-0.629122\pi$
$463$	−7.80402e11	−0.789231	−0.394615	+	0.918846i	$0.629122\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−4.04751e11	−0.393788	−0.196894	−	0.980425i	$-0.563085\pi$
$467$	−4.04751e11	−0.393788	−0.196894	+	0.980425i	$0.563085\pi$
$468$	0	0
$469$	−1.18052e12	−1.12667
$470$	0	0
$471$	3.32132e11	0.310968
$472$	0	0
$473$	−5.99700e11	−0.550883
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1.82524e12	1.61431
$478$	0	0
$479$	2.06972e12	1.79640	0.898199	−	0.439588i	$-0.144876\pi$
$479$	2.06972e12	1.79640	0.898199	+	0.439588i	$0.144876\pi$
$480$	0	0
$481$	−3.10638e11	−0.264607
$482$	0	0
$483$	−1.84335e11	−0.154115
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−2.41184e11	−0.194298	−0.0971490	−	0.995270i	$-0.530972\pi$
$487$	−2.41184e11	−0.194298	−0.0971490	+	0.995270i	$0.530972\pi$
$488$	0	0
$489$	3.66068e11	0.289516
$490$	0	0
$491$	−2.27883e12	−1.76948	−0.884739	−	0.466088i	$-0.845663\pi$
$491$	−2.27883e12	−1.76948	−0.884739	+	0.466088i	$0.845663\pi$
$492$	0	0
$493$	8.38767e11	0.639485
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6.26373e11	0.460499
$498$	0	0
$499$	−9.88752e11	−0.713896	−0.356948	−	0.934124i	$-0.616183\pi$
$499$	−9.88752e11	−0.713896	−0.356948	+	0.934124i	$0.616183\pi$
$500$	0	0
$501$	2.95496e11	0.209547
$502$	0	0
$503$	1.22385e12	0.852455	0.426228	−	0.904616i	$-0.359842\pi$
$503$	1.22385e12	0.852455	0.426228	+	0.904616i	$0.359842\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.43626e11	0.0965380
$508$	0	0
$509$	1.58447e12	1.04629	0.523146	−	0.852243i	$-0.324758\pi$
$509$	1.58447e12	1.04629	0.523146	+	0.852243i	$0.324758\pi$
$510$	0	0
$511$	−1.13305e12	−0.735117
$512$	0	0
$513$	1.27495e12	0.812767
$514$	0	0
$515$	0	0
$516$	0	0
$517$	5.69823e10	0.0350778
$518$	0	0
$519$	−2.24432e11	−0.135778
$520$	0	0
$521$	−2.71561e12	−1.61472	−0.807362	−	0.590057i	$-0.799105\pi$
$521$	−2.71561e12	−1.61472	−0.807362	+	0.590057i	$0.799105\pi$
$522$	0	0
$523$	−2.16171e12	−1.26340	−0.631698	−	0.775214i	$-0.717642\pi$
$523$	−2.16171e12	−1.26340	−0.631698	+	0.775214i	$0.717642\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.04630e12	0.590890
$528$	0	0
$529$	4.69046e11	0.260415
$530$	0	0
$531$	−1.40441e11	−0.0766597
$532$	0	0
$533$	−3.65475e12	−1.96149
$534$	0	0
$535$	0	0
$536$	0	0
$537$	6.11217e10	0.0317184
$538$	0	0
$539$	−1.02207e12	−0.521594
$540$	0	0
$541$	−2.29090e12	−1.14979	−0.574895	−	0.818227i	$-0.694957\pi$
$541$	−2.29090e12	−1.14979	−0.574895	+	0.818227i	$0.694957\pi$
$542$	0	0
$543$	1.71882e11	0.0848462
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−3.39447e12	−1.62117	−0.810586	−	0.585620i	$-0.800851\pi$
$547$	−3.39447e12	−1.62117	−0.810586	+	0.585620i	$0.800851\pi$
$548$	0	0
$549$	1.70974e12	0.803259
$550$	0	0
$551$	2.85053e12	1.31748
$552$	0	0
$553$	−2.22687e12	−1.01259
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1.74706e12	−0.769060	−0.384530	−	0.923113i	$-0.625636\pi$
$557$	−1.74706e12	−0.769060	−0.384530	+	0.923113i	$0.625636\pi$
$558$	0	0
$559$	1.76226e12	0.763340
$560$	0	0
$561$	−4.11022e11	−0.175199
$562$	0	0
$563$	−2.58864e12	−1.08588	−0.542942	−	0.839770i	$-0.682690\pi$
$563$	−2.58864e12	−1.08588	−0.542942	+	0.839770i	$0.682690\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−1.33674e12	−0.543153
$568$	0	0
$569$	−1.99294e12	−0.797055	−0.398527	−	0.917156i	$-0.630479\pi$
$569$	−1.99294e12	−0.797055	−0.398527	+	0.917156i	$0.630479\pi$
$570$	0	0
$571$	−3.50761e10	−0.0138086	−0.00690428	−	0.999976i	$-0.502198\pi$
$571$	−3.50761e10	−0.0138086	−0.00690428	+	0.999976i	$0.502198\pi$
$572$	0	0
$573$	−2.79729e11	−0.108403
$574$	0	0
$575$	0	0
$576$	0	0
$577$	7.75900e11	0.291417	0.145708	−	0.989328i	$-0.453454\pi$
$577$	7.75900e11	0.291417	0.145708	+	0.989328i	$0.453454\pi$
$578$	0	0
$579$	9.65027e11	0.356850
$580$	0	0
$581$	−2.60016e10	−0.00946689
$582$	0	0
$583$	−4.09874e12	−1.46941
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.24477e12	0.432730	0.216365	−	0.976313i	$-0.430580\pi$
$587$	1.24477e12	0.432730	0.216365	+	0.976313i	$0.430580\pi$
$588$	0	0
$589$	3.55581e12	1.21736
$590$	0	0
$591$	−7.92756e11	−0.267298
$592$	0	0
$593$	2.57794e11	0.0856103	0.0428052	−	0.999083i	$-0.486371\pi$
$593$	2.57794e11	0.0856103	0.0428052	+	0.999083i	$0.486371\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−3.22664e11	−0.103960
$598$	0	0
$599$	−2.32089e12	−0.736604	−0.368302	−	0.929706i	$-0.620061\pi$
$599$	−2.32089e12	−0.736604	−0.368302	+	0.929706i	$0.620061\pi$
$600$	0	0
$601$	1.78665e12	0.558605	0.279302	−	0.960203i	$-0.409897\pi$
$601$	1.78665e12	0.558605	0.279302	+	0.960203i	$0.409897\pi$
$602$	0	0
$603$	−5.52022e12	−1.70031
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−1.45787e12	−0.435883	−0.217941	−	0.975962i	$-0.569934\pi$
$607$	−1.45787e12	−0.435883	−0.217941	+	0.975962i	$0.569934\pi$
$608$	0	0
$609$	3.20731e11	0.0944851
$610$	0	0
$611$	−1.67447e11	−0.0486062
$612$	0	0
$613$	−1.42075e12	−0.406394	−0.203197	−	0.979138i	$-0.565133\pi$
$613$	−1.42075e12	−0.406394	−0.203197	+	0.979138i	$0.565133\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1.20441e12	−0.334573	−0.167286	−	0.985908i	$-0.553500\pi$
$617$	−1.20441e12	−0.334573	−0.167286	+	0.985908i	$0.553500\pi$
$618$	0	0
$619$	4.91349e12	1.34519	0.672593	−	0.740013i	$-0.265181\pi$
$619$	4.91349e12	1.34519	0.672593	+	0.740013i	$0.265181\pi$
$620$	0	0
$621$	−1.76671e12	−0.476710
$622$	0	0
$623$	2.40214e12	0.638856
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−1.39685e12	−0.360948
$628$	0	0
$629$	−8.03171e11	−0.204588
$630$	0	0
$631$	−4.58663e12	−1.15176	−0.575879	−	0.817535i	$-0.695340\pi$
$631$	−4.58663e12	−1.15176	−0.575879	+	0.817535i	$0.695340\pi$
$632$	0	0
$633$	−5.49826e11	−0.136116
$634$	0	0
$635$	0	0
$636$	0	0
$637$	3.00344e12	0.722755
$638$	0	0
$639$	2.92898e12	0.694963
$640$	0	0
$641$	−2.30636e12	−0.539593	−0.269796	−	0.962917i	$-0.586956\pi$
$641$	−2.30636e12	−0.539593	−0.269796	+	0.962917i	$0.586956\pi$
$642$	0	0
$643$	2.59493e12	0.598654	0.299327	−	0.954151i	$-0.403238\pi$
$643$	2.59493e12	0.598654	0.299327	+	0.954151i	$0.403238\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	5.14811e12	1.15499	0.577495	−	0.816394i	$-0.304030\pi$
$647$	5.14811e12	1.15499	0.577495	+	0.816394i	$0.304030\pi$
$648$	0	0
$649$	3.15372e11	0.0697786
$650$	0	0
$651$	4.00087e11	0.0873051
$652$	0	0
$653$	−4.42559e12	−0.952492	−0.476246	−	0.879312i	$-0.658003\pi$
$653$	−4.42559e12	−0.952492	−0.476246	+	0.879312i	$0.658003\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−5.29826e12	−1.10940
$658$	0	0
$659$	−1.09827e12	−0.226842	−0.113421	−	0.993547i	$-0.536181\pi$
$659$	−1.09827e12	−0.226842	−0.113421	+	0.993547i	$0.536181\pi$
$660$	0	0
$661$	7.99232e11	0.162842	0.0814209	−	0.996680i	$-0.474054\pi$
$661$	7.99232e11	0.162842	0.0814209	+	0.996680i	$0.474054\pi$
$662$	0	0
$663$	1.20782e12	0.242768
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−3.95000e12	−0.772736
$668$	0	0
$669$	1.35612e12	0.261746
$670$	0	0
$671$	−3.83939e12	−0.731157
$672$	0	0
$673$	−7.68445e12	−1.44393	−0.721963	−	0.691931i	$-0.756760\pi$
$673$	−7.68445e12	−1.44393	−0.721963	+	0.691931i	$0.756760\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−1.28854e12	−0.235749	−0.117874	−	0.993029i	$-0.537608\pi$
$677$	−1.28854e12	−0.235749	−0.117874	+	0.993029i	$0.537608\pi$
$678$	0	0
$679$	3.71076e12	0.669959
$680$	0	0
$681$	1.70742e12	0.304214
$682$	0	0
$683$	−8.11444e11	−0.142681	−0.0713404	−	0.997452i	$-0.522728\pi$
$683$	−8.11444e11	−0.142681	−0.0713404	+	0.997452i	$0.522728\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−4.50607e10	−0.00771778
$688$	0	0
$689$	1.20445e13	2.03611
$690$	0	0
$691$	−2.26741e12	−0.378338	−0.189169	−	0.981945i	$-0.560579\pi$
$691$	−2.26741e12	−0.378338	−0.189169	+	0.981945i	$0.560579\pi$
$692$	0	0
$693$	3.16673e12	0.521569
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−9.44955e12	−1.51657
$698$	0	0
$699$	2.38957e11	0.0378594
$700$	0	0
$701$	−4.92113e12	−0.769721	−0.384861	−	0.922975i	$-0.625751\pi$
$701$	−4.92113e12	−0.769721	−0.384861	+	0.922975i	$0.625751\pi$
$702$	0	0
$703$	−2.72956e12	−0.421496
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−3.84566e12	−0.578874
$708$	0	0
$709$	−6.12354e12	−0.910112	−0.455056	−	0.890463i	$-0.650381\pi$
$709$	−6.12354e12	−0.910112	−0.455056	+	0.890463i	$0.650381\pi$
$710$	0	0
$711$	−1.04130e13	−1.52815
$712$	0	0
$713$	−4.92732e12	−0.714016
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−1.69734e12	−0.239846
$718$	0	0
$719$	2.30376e11	0.0321483	0.0160741	−	0.999871i	$-0.494883\pi$
$719$	2.30376e11	0.0321483	0.0160741	+	0.999871i	$0.494883\pi$
$720$	0	0
$721$	−6.43408e11	−0.0886702
$722$	0	0
$723$	−3.54048e11	−0.0481880
$724$	0	0
$725$	0	0
$726$	0	0
$727$	9.25894e12	1.22930	0.614648	−	0.788801i	$-0.289298\pi$
$727$	9.25894e12	1.22930	0.614648	+	0.788801i	$0.289298\pi$
$728$	0	0
$729$	−5.54661e12	−0.727368
$730$	0	0
$731$	4.55643e12	0.590196
$732$	0	0
$733$	7.22531e12	0.924461	0.462231	−	0.886760i	$-0.347049\pi$
$733$	7.22531e12	0.924461	0.462231	+	0.886760i	$0.347049\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.23962e13	1.54769
$738$	0	0
$739$	1.24725e13	1.53834	0.769170	−	0.639044i	$-0.220670\pi$
$739$	1.24725e13	1.53834	0.769170	+	0.639044i	$0.220670\pi$
$740$	0	0
$741$	4.10474e12	0.500154
$742$	0	0
$743$	−1.15645e13	−1.39212	−0.696058	−	0.717986i	$-0.745064\pi$
$743$	−1.15645e13	−1.39212	−0.696058	+	0.717986i	$0.745064\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−1.21586e11	−0.0142870
$748$	0	0
$749$	−3.85167e12	−0.447179
$750$	0	0
$751$	3.70732e12	0.425285	0.212642	−	0.977130i	$-0.431793\pi$
$751$	3.70732e12	0.425285	0.212642	+	0.977130i	$0.431793\pi$
$752$	0	0
$753$	1.05547e12	0.119638
$754$	0	0
$755$	0	0
$756$	0	0
$757$	4.86635e11	0.0538607	0.0269303	−	0.999637i	$-0.491427\pi$
$757$	4.86635e11	0.0538607	0.0269303	+	0.999637i	$0.491427\pi$
$758$	0	0
$759$	1.93562e12	0.211706
$760$	0	0
$761$	−4.28395e12	−0.463035	−0.231518	−	0.972831i	$-0.574369\pi$
$761$	−4.28395e12	−0.463035	−0.231518	+	0.972831i	$0.574369\pi$
$762$	0	0
$763$	−4.00589e12	−0.427896
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−9.26745e11	−0.0966899
$768$	0	0
$769$	−4.03625e12	−0.416207	−0.208103	−	0.978107i	$-0.566729\pi$
$769$	−4.03625e12	−0.416207	−0.208103	+	0.978107i	$0.566729\pi$
$770$	0	0
$771$	1.12186e12	0.114339
$772$	0	0
$773$	−1.21916e13	−1.22815	−0.614076	−	0.789247i	$-0.710471\pi$
$773$	−1.21916e13	−1.22815	−0.614076	+	0.789247i	$0.710471\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−3.07120e11	−0.0302282
$778$	0	0
$779$	−3.21141e13	−3.12447
$780$	0	0
$781$	−6.57728e12	−0.632582
$782$	0	0
$783$	3.07397e12	0.292262
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−1.20299e12	−0.111783	−0.0558913	−	0.998437i	$-0.517800\pi$
$787$	−1.20299e12	−0.111783	−0.0558913	+	0.998437i	$0.517800\pi$
$788$	0	0
$789$	4.08059e12	0.374866
$790$	0	0
$791$	1.00539e13	0.913147
$792$	0	0
$793$	1.12823e13	1.01314
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−1.72611e13	−1.51533	−0.757663	−	0.652646i	$-0.773659\pi$
$797$	−1.72611e13	−1.51533	−0.757663	+	0.652646i	$0.773659\pi$
$798$	0	0
$799$	−4.32943e11	−0.0375811
$800$	0	0
$801$	1.12326e13	0.964130
$802$	0	0
$803$	1.18977e13	1.00982
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−4.34609e12	−0.360718
$808$	0	0
$809$	7.44408e12	0.611002	0.305501	−	0.952192i	$-0.401176\pi$
$809$	7.44408e12	0.611002	0.305501	+	0.952192i	$0.401176\pi$
$810$	0	0
$811$	9.59955e12	0.779214	0.389607	−	0.920981i	$-0.372611\pi$
$811$	9.59955e12	0.779214	0.389607	+	0.920981i	$0.372611\pi$
$812$	0	0
$813$	3.38772e12	0.271957
$814$	0	0
$815$	0	0
$816$	0	0
$817$	1.54849e13	1.21593
$818$	0	0
$819$	−9.30568e12	−0.722721
$820$	0	0
$821$	−5.04043e12	−0.387189	−0.193595	−	0.981082i	$-0.562015\pi$
$821$	−5.04043e12	−0.387189	−0.193595	+	0.981082i	$0.562015\pi$
$822$	0	0
$823$	1.62323e13	1.23333	0.616666	−	0.787225i	$-0.288483\pi$
$823$	1.62323e13	1.23333	0.616666	+	0.787225i	$0.288483\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−1.74898e13	−1.30020	−0.650099	−	0.759850i	$-0.725273\pi$
$827$	−1.74898e13	−1.30020	−0.650099	+	0.759850i	$0.725273\pi$
$828$	0	0
$829$	−1.04102e13	−0.765535	−0.382768	−	0.923845i	$-0.625029\pi$
$829$	−1.04102e13	−0.765535	−0.382768	+	0.923845i	$0.625029\pi$
$830$	0	0
$831$	−8.56420e11	−0.0622992
$832$	0	0
$833$	7.76555e12	0.558817
$834$	0	0
$835$	0	0
$836$	0	0
$837$	3.83454e12	0.270053
$838$	0	0
$839$	2.66219e13	1.85485	0.927426	−	0.374006i	$-0.122016\pi$
$839$	2.66219e13	1.85485	0.927426	+	0.374006i	$0.122016\pi$
$840$	0	0
$841$	−7.63439e12	−0.526250
$842$	0	0
$843$	1.66919e12	0.113836
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2.34477e12	0.156540
$848$	0	0
$849$	−3.35427e12	−0.221571
$850$	0	0
$851$	3.78237e12	0.247219
$852$	0	0
$853$	7.30064e12	0.472161	0.236080	−	0.971734i	$-0.424137\pi$
$853$	7.30064e12	0.472161	0.236080	+	0.971734i	$0.424137\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.18598e13	0.751041	0.375520	−	0.926814i	$-0.377464\pi$
$857$	1.18598e13	0.751041	0.375520	+	0.926814i	$0.377464\pi$
$858$	0	0
$859$	−1.67692e13	−1.05085	−0.525427	−	0.850839i	$-0.676094\pi$
$859$	−1.67692e13	−1.05085	−0.525427	+	0.850839i	$0.676094\pi$
$860$	0	0
$861$	−3.61336e12	−0.224077
$862$	0	0
$863$	6.71596e12	0.412154	0.206077	−	0.978536i	$-0.433930\pi$
$863$	6.71596e12	0.412154	0.206077	+	0.978536i	$0.433930\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−4.94917e11	−0.0297472
$868$	0	0
$869$	2.33834e13	1.39098
$870$	0	0
$871$	−3.64270e13	−2.14458
$872$	0	0
$873$	1.73518e13	1.01107
$874$	0	0
$875$	0	0
$876$	0	0
$877$	2.79757e13	1.59692	0.798458	−	0.602051i	$-0.205649\pi$
$877$	2.79757e13	1.59692	0.798458	+	0.602051i	$0.205649\pi$
$878$	0	0
$879$	7.03050e12	0.397225
$880$	0	0
$881$	2.06225e13	1.15332	0.576660	−	0.816984i	$-0.304356\pi$
$881$	2.06225e13	1.15332	0.576660	+	0.816984i	$0.304356\pi$
$882$	0	0
$883$	−2.02048e13	−1.11849	−0.559244	−	0.829003i	$-0.688909\pi$
$883$	−2.02048e13	−1.11849	−0.559244	+	0.829003i	$0.688909\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	3.19954e13	1.73553	0.867763	−	0.496978i	$-0.165557\pi$
$887$	3.19954e13	1.73553	0.867763	+	0.496978i	$0.165557\pi$
$888$	0	0
$889$	9.92701e12	0.533041
$890$	0	0
$891$	1.40365e13	0.746122
$892$	0	0
$893$	−1.47135e12	−0.0774254
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−5.68798e12	−0.293354
$898$	0	0
$899$	8.57323e12	0.437750
$900$	0	0
$901$	3.11416e13	1.57427
$902$	0	0
$903$	1.74231e12	0.0872026
$904$	0	0
$905$	0	0
$906$	0	0
$907$	2.40537e12	0.118018	0.0590091	−	0.998257i	$-0.481206\pi$
$907$	2.40537e12	0.118018	0.0590091	+	0.998257i	$0.481206\pi$
$908$	0	0
$909$	−1.79827e13	−0.873609
$910$	0	0
$911$	3.11773e13	1.49970	0.749852	−	0.661606i	$-0.230125\pi$
$911$	3.11773e13	1.49970	0.749852	+	0.661606i	$0.230125\pi$
$912$	0	0
$913$	2.73032e11	0.0130045
$914$	0	0
$915$	0	0
$916$	0	0
$917$	7.71582e12	0.360346
$918$	0	0
$919$	−3.42677e13	−1.58477	−0.792383	−	0.610024i	$-0.791160\pi$
$919$	−3.42677e13	−1.58477	−0.792383	+	0.610024i	$0.791160\pi$
$920$	0	0
$921$	2.08986e12	0.0957082
$922$	0	0
$923$	1.93278e13	0.876548
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−3.00864e12	−0.133817
$928$	0	0
$929$	−2.52782e13	−1.11346	−0.556730	−	0.830693i	$-0.687944\pi$
$929$	−2.52782e13	−1.11346	−0.556730	+	0.830693i	$0.687944\pi$
$930$	0	0
$931$	2.63910e13	1.15129
$932$	0	0
$933$	6.06476e12	0.262027
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−2.72636e13	−1.15546	−0.577731	−	0.816227i	$-0.696062\pi$
$937$	−2.72636e13	−1.15546	−0.577731	+	0.816227i	$0.696062\pi$
$938$	0	0
$939$	1.87682e12	0.0787819
$940$	0	0
$941$	1.57383e13	0.654340	0.327170	−	0.944966i	$-0.393905\pi$
$941$	1.57383e13	0.654340	0.327170	+	0.944966i	$0.393905\pi$
$942$	0	0
$943$	4.45008e13	1.83259
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−9.71128e12	−0.392375	−0.196188	−	0.980566i	$-0.562856\pi$
$947$	−9.71128e12	−0.392375	−0.196188	+	0.980566i	$0.562856\pi$
$948$	0	0
$949$	−3.49624e13	−1.39927
$950$	0	0
$951$	6.89257e12	0.273256
$952$	0	0
$953$	−4.15385e12	−0.163130	−0.0815648	−	0.996668i	$-0.525992\pi$
$953$	−4.15385e12	−0.163130	−0.0815648	+	0.996668i	$0.525992\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−3.36787e12	−0.129793
$958$	0	0
$959$	1.79897e12	0.0686818
$960$	0	0
$961$	−1.57452e13	−0.595515
$962$	0	0
$963$	−1.80108e13	−0.674861
$964$	0	0
$965$	0	0
$966$	0	0
$967$	1.84544e13	0.678706	0.339353	−	0.940659i	$-0.389792\pi$
$967$	1.84544e13	0.678706	0.339353	+	0.940659i	$0.389792\pi$
$968$	0	0
$969$	1.06130e13	0.386707
$970$	0	0
$971$	2.95973e12	0.106848	0.0534238	−	0.998572i	$-0.482987\pi$
$971$	2.95973e12	0.106848	0.0534238	+	0.998572i	$0.482987\pi$
$972$	0	0
$973$	1.79826e13	0.643197
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−4.44178e13	−1.55967	−0.779833	−	0.625988i	$-0.784696\pi$
$977$	−4.44178e13	−1.55967	−0.779833	+	0.625988i	$0.784696\pi$
$978$	0	0
$979$	−2.52239e13	−0.877588
$980$	0	0
$981$	−1.87319e13	−0.645761
$982$	0	0
$983$	4.57558e13	1.56299	0.781493	−	0.623914i	$-0.214459\pi$
$983$	4.57558e13	1.56299	0.781493	+	0.623914i	$0.214459\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−1.65550e11	−0.00555269
$988$	0	0
$989$	−2.14576e13	−0.713177
$990$	0	0
$991$	−1.30469e13	−0.429710	−0.214855	−	0.976646i	$-0.568928\pi$
$991$	−1.30469e13	−0.429710	−0.214855	+	0.976646i	$0.568928\pi$
$992$	0	0
$993$	2.55903e12	0.0835225
$994$	0	0
$995$	0	0
$996$	0	0
$997$	5.07578e12	0.162695	0.0813476	−	0.996686i	$-0.474078\pi$
$997$	5.07578e12	0.162695	0.0813476	+	0.996686i	$0.474078\pi$
$998$	0	0
$999$	−2.94352e12	−0.0935022

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.10.a.y.1.2		3
4.3	odd	2		25.10.a.c.1.3	✓	3
5.2	odd	4		400.10.c.q.49.3		6
5.3	odd	4		400.10.c.q.49.4		6
5.4	even	2		400.10.a.u.1.2		3
12.11	even	2		225.10.a.p.1.1		3
20.3	even	4		25.10.b.c.24.3		6
20.7	even	4		25.10.b.c.24.4		6
20.19	odd	2		25.10.a.d.1.1	yes	3
60.23	odd	4		225.10.b.m.199.4		6
60.47	odd	4		225.10.b.m.199.3		6
60.59	even	2		225.10.a.m.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.10.a.c.1.3	✓	3	4.3	odd	2
25.10.a.d.1.1	yes	3	20.19	odd	2
25.10.b.c.24.3		6	20.3	even	4
25.10.b.c.24.4		6	20.7	even	4
225.10.a.m.1.3		3	60.59	even	2
225.10.a.p.1.1		3	12.11	even	2
225.10.b.m.199.3		6	60.47	odd	4
225.10.b.m.199.4		6	60.23	odd	4
400.10.a.u.1.2		3	5.4	even	2
400.10.a.y.1.2		3	1.1	even	1	trivial
400.10.c.q.49.3		6	5.2	odd	4
400.10.c.q.49.4		6	5.3	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:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	400.a (trivial)

\(f(q)\)	\(=\)	\(q+30.5073 q^{3} -4010.25 q^{7} -18752.3 q^{9} +O(q^{10})\)	\(q+30.5073 q^{3} -4010.25 q^{7} -18752.3 q^{9} +42110.0 q^{11} -123743. q^{13} -319945. q^{17} -1.08733e6 q^{19} -122342. q^{21} +1.50672e6 q^{23} -1.17256e6 q^{27} -2.62160e6 q^{29} -3.27023e6 q^{31} +1.28466e6 q^{33} +2.51034e6 q^{37} -3.77508e6 q^{39} +2.95349e7 q^{41} -1.42413e7 q^{43} +1.35318e6 q^{47} -2.42715e7 q^{49} -9.76067e6 q^{51} -9.73342e7 q^{53} -3.31714e7 q^{57} +7.48924e6 q^{59} -9.11752e7 q^{61} +7.52015e7 q^{63} +2.94376e8 q^{67} +4.59659e7 q^{69} -1.56193e8 q^{71} +2.82539e8 q^{73} -1.68872e8 q^{77} +5.55294e8 q^{79} +3.33330e8 q^{81} +6.48378e6 q^{83} -7.99778e7 q^{87} -5.99001e8 q^{89} +4.96242e8 q^{91} -9.97660e7 q^{93} -9.25317e8 q^{97} -7.89660e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 89 q^{3} + 5258 q^{7} + 58234 q^{9}+O(q^{10}) \) 3 * q + 89 * q^3 + 5258 * q^7 + 58234 * q^9	\( 3 q + 89 q^{3} + 5258 q^{7} + 58234 q^{9} + 54699 q^{11} - 215884 q^{13} - 334983 q^{17} - 818845 q^{19} - 2375394 q^{21} + 3526854 q^{23} + 6633395 q^{27} + 2175480 q^{29} - 4274066 q^{31} + 22122137 q^{33} + 10305042 q^{37} + 50414092 q^{39} + 5926311 q^{41} - 24429956 q^{43} + 66858708 q^{47} - 6453929 q^{49} + 25634699 q^{51} - 132620514 q^{53} - 252946415 q^{57} - 5670960 q^{59} + 125306926 q^{61} + 284323404 q^{63} + 88829483 q^{67} - 314274942 q^{69} - 297550596 q^{71} - 181321729 q^{73} - 561214086 q^{77} + 310025170 q^{79} + 1398847363 q^{81} - 731088801 q^{83} + 1046385560 q^{87} - 1103860035 q^{89} - 1183187656 q^{91} - 107386758 q^{93} + 332236842 q^{97} + 892234522 q^{99}+O(q^{100}) \) 3 * q + 89 * q^3 + 5258 * q^7 + 58234 * q^9 + 54699 * q^11 - 215884 * q^13 - 334983 * q^17 - 818845 * q^19 - 2375394 * q^21 + 3526854 * q^23 + 6633395 * q^27 + 2175480 * q^29 - 4274066 * q^31 + 22122137 * q^33 + 10305042 * q^37 + 50414092 * q^39 + 5926311 * q^41 - 24429956 * q^43 + 66858708 * q^47 - 6453929 * q^49 + 25634699 * q^51 - 132620514 * q^53 - 252946415 * q^57 - 5670960 * q^59 + 125306926 * q^61 + 284323404 * q^63 + 88829483 * q^67 - 314274942 * q^69 - 297550596 * q^71 - 181321729 * q^73 - 561214086 * q^77 + 310025170 * q^79 + 1398847363 * q^81 - 731088801 * q^83 + 1046385560 * q^87 - 1103860035 * q^89 - 1183187656 * q^91 - 107386758 * q^93 + 332236842 * q^97 + 892234522 * q^99

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