LMFDB - Embedded newform 400.10.c.l.49.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,10,Mod(49,400)]

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, 1]))

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

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

chi := DirichletCharacter("400.49");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	400.c (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$206.014334466$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{79})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 39x^{2} + 400 $ x^4 - 39*x^2 + 400
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{12}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 20)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.4
Root			$-4.44410 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	400.49
Dual form			400.10.c.l.49.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+272.211i q^{3} -10002.6i q^{7} -54415.9 q^{9} +O(q^{10})$	$q+272.211i q^{3} -10002.6i q^{7} -54415.9 q^{9} -47093.7 q^{11} +9362.93i q^{13} -108521. i q^{17} -665173. q^{19} +2.72281e6 q^{21} -576426. i q^{23} -9.45468e6i q^{27} +2.61067e6 q^{29} -3.87896e6 q^{31} -1.28194e7i q^{33} -1.41599e7i q^{37} -2.54869e6 q^{39} +4.62193e6 q^{41} -8.31227e6i q^{43} +2.51923e7i q^{47} -5.96977e7 q^{49} +2.95405e7 q^{51} +3.49333e7i q^{53} -1.81068e8i q^{57} +6.71868e6 q^{59} -4.75982e6 q^{61} +5.44299e8i q^{63} -1.38772e8i q^{67} +1.56909e8 q^{69} -3.54359e8 q^{71} +2.41596e8i q^{73} +4.71058e8i q^{77} +2.61173e8 q^{79} +1.50260e9 q^{81} +6.55898e8i q^{83} +7.10654e8i q^{87} +1.00221e9 q^{89} +9.36534e7 q^{91} -1.05590e9i q^{93} -1.24869e9i q^{97} +2.56264e9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 69764 q^{9}+O(q^{10}) $ 4 * q - 69764 * q^9	$ 4 q - 69764 q^{9} - 205440 q^{11} + 274544 q^{19} + 5680624 q^{21} + 13787496 q^{29} - 583664 q^{31} - 951056 q^{39} + 59546904 q^{41} - 223875828 q^{49} + 54013392 q^{51} - 185861712 q^{59} + 391347848 q^{61} + 344148528 q^{69} - 622414032 q^{71} + 1084523552 q^{79} + 3762476756 q^{81} + 924583704 q^{89} - 3080075536 q^{91} + 2952090240 q^{99}+O(q^{100}) $ 4 * q - 69764 * q^9 - 205440 * q^11 + 274544 * q^19 + 5680624 * q^21 + 13787496 * q^29 - 583664 * q^31 - 951056 * q^39 + 59546904 * q^41 - 223875828 * q^49 + 54013392 * q^51 - 185861712 * q^59 + 391347848 * q^61 + 344148528 * q^69 - 622414032 * q^71 + 1084523552 * q^79 + 3762476756 * q^81 + 924583704 * q^89 - 3080075536 * q^91 + 2952090240 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$351$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	272.211i	1.94026i	0.242583	+	0.970131i	$0.422005\pi$
$3$	272.211i	1.94026i	−0.242583	+	0.970131i	$0.577995\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 10002.6i	− 1.57460i	−0.616570	−	0.787300i	$-0.711478\pi$
$7$	− 10002.6i	− 1.57460i	0.616570	−	0.787300i	$-0.288522\pi$
$8$	0	0
$9$	−54415.9	−2.76461
$10$	0	0
$11$	−47093.7	−0.969830	−0.484915	−	0.874561i	$-0.661149\pi$
$11$	−47093.7	−0.969830	−0.484915	+	0.874561i	$0.661149\pi$
$12$	0	0
$13$	9362.93i	0.0909216i	0.998966	+	0.0454608i	$0.0144756\pi$
$13$	9362.93i	0.0909216i	−0.998966	+	0.0454608i	$0.985524\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 108521.i	− 0.315131i	−0.987509	−	0.157566i	$-0.949635\pi$
$17$	− 108521.i	− 0.315131i	0.987509	−	0.157566i	$-0.0503646\pi$
$18$	0	0
$19$	−665173.	−1.17096	−0.585482	−	0.810685i	$-0.699095\pi$
$19$	−665173.	−1.17096	−0.585482	+	0.810685i	$0.699095\pi$
$20$	0	0
$21$	2.72281e6	3.05514
$22$	0	0
$23$	− 576426.i	− 0.429505i	−0.976669	−	0.214752i	$-0.931106\pi$
$23$	− 576426.i	− 0.429505i	0.976669	−	0.214752i	$-0.0688945\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 9.45468e6i	− 3.42381i
$28$	0	0
$29$	2.61067e6	0.685427	0.342714	−	0.939440i	$-0.388654\pi$
$29$	2.61067e6	0.685427	0.342714	+	0.939440i	$0.388654\pi$
$30$	0	0
$31$	−3.87896e6	−0.754375	−0.377188	−	0.926137i	$-0.623109\pi$
$31$	−3.87896e6	−0.754375	−0.377188	+	0.926137i	$0.623109\pi$
$32$	0	0
$33$	− 1.28194e7i	− 1.88172i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 1.41599e7i	− 1.24209i	−0.783774	−	0.621046i	$-0.786708\pi$
$37$	− 1.41599e7i	− 1.24209i	0.783774	−	0.621046i	$-0.213292\pi$
$38$	0	0
$39$	−2.54869e6	−0.176412
$40$	0	0
$41$	4.62193e6	0.255444	0.127722	−	0.991810i	$-0.459233\pi$
$41$	4.62193e6	0.255444	0.127722	+	0.991810i	$0.459233\pi$
$42$	0	0
$43$	− 8.31227e6i	− 0.370776i	−0.982665	−	0.185388i	$-0.940646\pi$
$43$	− 8.31227e6i	− 0.370776i	0.982665	−	0.185388i	$-0.0593542\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.51923e7i	0.753056i	0.926405	+	0.376528i	$0.122882\pi$
$47$	2.51923e7i	0.753056i	−0.926405	+	0.376528i	$0.877118\pi$
$48$	0	0
$49$	−5.96977e7	−1.47937
$50$	0	0
$51$	2.95405e7	0.611437
$52$	0	0
$53$	3.49333e7i	0.608133i	0.952651	+	0.304066i	$0.0983444\pi$
$53$	3.49333e7i	0.608133i	−0.952651	+	0.304066i	$0.901656\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 1.81068e8i	− 2.27198i
$58$	0	0
$59$	6.71868e6	0.0721855	0.0360928	−	0.999348i	$-0.488509\pi$
$59$	6.71868e6	0.0721855	0.0360928	+	0.999348i	$0.488509\pi$
$60$	0	0
$61$	−4.75982e6	−0.0440156	−0.0220078	−	0.999758i	$-0.507006\pi$
$61$	−4.75982e6	−0.0440156	−0.0220078	+	0.999758i	$0.507006\pi$
$62$	0	0
$63$	5.44299e8i	4.35316i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 1.38772e8i	− 0.841330i	−0.907216	−	0.420665i	$-0.861797\pi$
$67$	− 1.38772e8i	− 0.841330i	0.907216	−	0.420665i	$-0.138203\pi$
$68$	0	0
$69$	1.56909e8	0.833352
$70$	0	0
$71$	−3.54359e8	−1.65494	−0.827468	−	0.561513i	$-0.810219\pi$
$71$	−3.54359e8	−1.65494	−0.827468	+	0.561513i	$0.810219\pi$
$72$	0	0
$73$	2.41596e8i	0.995719i	0.867258	+	0.497859i	$0.165880\pi$
$73$	2.41596e8i	0.995719i	−0.867258	+	0.497859i	$0.834120\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.71058e8i	1.52709i
$78$	0	0
$79$	2.61173e8	0.754409	0.377204	−	0.926130i	$-0.376885\pi$
$79$	2.61173e8	0.754409	0.377204	+	0.926130i	$0.376885\pi$
$80$	0	0
$81$	1.50260e9	3.87847
$82$	0	0
$83$	6.55898e8i	1.51700i	0.651674	+	0.758499i	$0.274067\pi$
$83$	6.55898e8i	1.51700i	−0.651674	+	0.758499i	$0.725933\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	7.10654e8i	1.32991i
$88$	0	0
$89$	1.00221e9	1.69318	0.846590	−	0.532245i	$-0.178652\pi$
$89$	1.00221e9	1.69318	0.846590	+	0.532245i	$0.178652\pi$
$90$	0	0
$91$	9.36534e7	0.143165
$92$	0	0
$93$	− 1.05590e9i	− 1.46368i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 1.24869e9i	− 1.43213i	−0.698034	−	0.716065i	$-0.745942\pi$
$97$	− 1.24869e9i	− 1.43213i	0.698034	−	0.716065i	$-0.254058\pi$
$98$	0	0
$99$	2.56264e9	2.68120
$100$	0	0
$101$	−9.39590e7	−0.0898446	−0.0449223	−	0.998990i	$-0.514304\pi$
$101$	−9.39590e7	−0.0898446	−0.0449223	+	0.998990i	$0.514304\pi$
$102$	0	0
$103$	3.58300e8i	0.313675i	0.987624	+	0.156837i	$0.0501299\pi$
$103$	3.58300e8i	0.313675i	−0.987624	+	0.156837i	$0.949870\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1.02197e9i	− 0.753719i	−0.926270	−	0.376860i	$-0.877004\pi$
$107$	− 1.02197e9i	− 0.753719i	0.926270	−	0.376860i	$-0.122996\pi$
$108$	0	0
$109$	8.94205e8	0.606761	0.303380	−	0.952869i	$-0.401885\pi$
$109$	8.94205e8	0.606761	0.303380	+	0.952869i	$0.401885\pi$
$110$	0	0
$111$	3.85449e9	2.40998
$112$	0	0
$113$	1.11113e9i	0.641077i	0.947235	+	0.320539i	$0.103864\pi$
$113$	1.11113e9i	0.641077i	−0.947235	+	0.320539i	$0.896136\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 5.09492e8i	− 0.251363i
$118$	0	0
$119$	−1.08548e9	−0.496206
$120$	0	0
$121$	−1.40134e8	−0.0594306
$122$	0	0
$123$	1.25814e9i	0.495628i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	5.51824e9i	1.88228i	0.338020	+	0.941139i	$0.390243\pi$
$127$	5.51824e9i	1.88228i	−0.338020	+	0.941139i	$0.609757\pi$
$128$	0	0
$129$	2.26269e9	0.719402
$130$	0	0
$131$	−4.66527e9	−1.38406	−0.692031	−	0.721867i	$-0.743284\pi$
$131$	−4.66527e9	−1.38406	−0.692031	+	0.721867i	$0.743284\pi$
$132$	0	0
$133$	6.65344e9i	1.84380i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	6.97325e9i	1.69119i	0.533825	+	0.845595i	$0.320754\pi$
$137$	6.97325e9i	1.69119i	−0.533825	+	0.845595i	$0.679246\pi$
$138$	0	0
$139$	6.27736e9	1.42630	0.713149	−	0.701013i	$-0.247268\pi$
$139$	6.27736e9	1.42630	0.713149	+	0.701013i	$0.247268\pi$
$140$	0	0
$141$	−6.85762e9	−1.46113
$142$	0	0
$143$	− 4.40935e8i	− 0.0881784i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 1.62504e10i	− 2.87036i
$148$	0	0
$149$	1.62156e9	0.269523	0.134761	−	0.990878i	$-0.456973\pi$
$149$	1.62156e9	0.269523	0.134761	+	0.990878i	$0.456973\pi$
$150$	0	0
$151$	−1.17677e10	−1.84202	−0.921011	−	0.389537i	$-0.872635\pi$
$151$	−1.17677e10	−1.84202	−0.921011	+	0.389537i	$0.872635\pi$
$152$	0	0
$153$	5.90524e9i	0.871217i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	1.71288e9i	0.224998i	0.993652	+	0.112499i	$0.0358855\pi$
$157$	1.71288e9i	0.224998i	−0.993652	+	0.112499i	$0.964114\pi$
$158$	0	0
$159$	−9.50923e9	−1.17994
$160$	0	0
$161$	−5.76574e9	−0.676298
$162$	0	0
$163$	− 9.98636e9i	− 1.10806i	−0.832497	−	0.554030i	$-0.813089\pi$
$163$	− 9.98636e9i	− 1.10806i	0.832497	−	0.554030i	$-0.186911\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.56153e10i	1.55355i	0.629777	+	0.776776i	$0.283146\pi$
$167$	1.56153e10i	1.55355i	−0.629777	+	0.776776i	$0.716854\pi$
$168$	0	0
$169$	1.05168e10	0.991733
$170$	0	0
$171$	3.61960e10	3.23726
$172$	0	0
$173$	4.61727e8i	0.0391902i	0.999808	+	0.0195951i	$0.00623772\pi$
$173$	4.61727e8i	0.0391902i	−0.999808	+	0.0195951i	$0.993762\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1.82890e9i	0.140059i
$178$	0	0
$179$	−1.56860e10	−1.14202	−0.571011	−	0.820943i	$-0.693449\pi$
$179$	−1.56860e10	−1.14202	−0.571011	+	0.820943i	$0.693449\pi$
$180$	0	0
$181$	5.26075e8	0.0364329	0.0182165	−	0.999834i	$-0.494201\pi$
$181$	5.26075e8	0.0364329	0.0182165	+	0.999834i	$0.494201\pi$
$182$	0	0
$183$	− 1.29568e9i	− 0.0854017i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	5.11063e9i	0.305624i
$188$	0	0
$189$	−9.45710e10	−5.39113
$190$	0	0
$191$	2.60641e10	1.41708	0.708538	−	0.705673i	$-0.249355\pi$
$191$	2.60641e10	1.41708	0.708538	+	0.705673i	$0.249355\pi$
$192$	0	0
$193$	4.83514e9i	0.250843i	0.992104	+	0.125421i	$0.0400283\pi$
$193$	4.83514e9i	0.250843i	−0.992104	+	0.125421i	$0.959972\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 1.09275e10i	− 0.516919i	−0.966022	−	0.258460i	$-0.916785\pi$
$197$	− 1.09275e10i	− 0.516919i	0.966022	−	0.258460i	$-0.0832149\pi$
$198$	0	0
$199$	−2.24042e10	−1.01272	−0.506361	−	0.862322i	$-0.669010\pi$
$199$	−2.24042e10	−1.01272	−0.506361	+	0.862322i	$0.669010\pi$
$200$	0	0
$201$	3.77754e10	1.63240
$202$	0	0
$203$	− 2.61134e10i	− 1.07927i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	3.13667e10i	1.18741i
$208$	0	0
$209$	3.13255e10	1.13564
$210$	0	0
$211$	−3.17129e10	−1.10145	−0.550725	−	0.834687i	$-0.685649\pi$
$211$	−3.17129e10	−1.10145	−0.550725	+	0.834687i	$0.685649\pi$
$212$	0	0
$213$	− 9.64605e10i	− 3.21101i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	3.87995e10i	1.18784i
$218$	0	0
$219$	−6.57651e10	−1.93195
$220$	0	0
$221$	1.01607e9	0.0286522
$222$	0	0
$223$	− 1.52662e10i	− 0.413388i	−0.978406	−	0.206694i	$-0.933730\pi$
$223$	− 1.52662e10i	− 0.413388i	0.978406	−	0.206694i	$-0.0662705\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 7.53481e9i	− 0.188346i	−0.995556	−	0.0941729i	$-0.969979\pi$
$227$	− 7.53481e9i	− 0.188346i	0.995556	−	0.0941729i	$-0.0300206\pi$
$228$	0	0
$229$	1.28244e10	0.308160	0.154080	−	0.988058i	$-0.450759\pi$
$229$	1.28244e10	0.308160	0.154080	+	0.988058i	$0.450759\pi$
$230$	0	0
$231$	−1.28227e11	−2.96296
$232$	0	0
$233$	2.06272e10i	0.458500i	0.973368	+	0.229250i	$0.0736274\pi$
$233$	2.06272e10i	0.458500i	−0.973368	+	0.229250i	$0.926373\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	7.10943e10i	1.46375i
$238$	0	0
$239$	3.96664e10	0.786380	0.393190	−	0.919457i	$-0.371372\pi$
$239$	3.96664e10	0.786380	0.393190	+	0.919457i	$0.371372\pi$
$240$	0	0
$241$	−3.75642e9	−0.0717294	−0.0358647	−	0.999357i	$-0.511419\pi$
$241$	−3.75642e9	−0.0717294	−0.0358647	+	0.999357i	$0.511419\pi$
$242$	0	0
$243$	2.22928e11i	4.10144i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 6.22797e9i	− 0.106466i
$248$	0	0
$249$	−1.78543e11	−2.94337
$250$	0	0
$251$	−7.74656e10	−1.23191	−0.615953	−	0.787783i	$-0.711229\pi$
$251$	−7.74656e10	−1.23191	−0.615953	+	0.787783i	$0.711229\pi$
$252$	0	0
$253$	2.71460e10i	0.416546i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 3.31990e10i	− 0.474707i	−0.971423	−	0.237354i	$-0.923720\pi$
$257$	− 3.31990e10i	− 0.474707i	0.971423	−	0.237354i	$-0.0762800\pi$
$258$	0	0
$259$	−1.41636e11	−1.95580
$260$	0	0
$261$	−1.42062e11	−1.89494
$262$	0	0
$263$	− 1.00754e11i	− 1.29856i	−0.760549	−	0.649281i	$-0.775070\pi$
$263$	− 1.00754e11i	− 1.29856i	0.760549	−	0.649281i	$-0.224930\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2.72812e11i	3.28521i
$268$	0	0
$269$	7.49946e10	0.873262	0.436631	−	0.899641i	$-0.356172\pi$
$269$	7.49946e10	0.873262	0.436631	+	0.899641i	$0.356172\pi$
$270$	0	0
$271$	5.80768e10	0.654095	0.327048	−	0.945008i	$-0.393946\pi$
$271$	5.80768e10	0.654095	0.327048	+	0.945008i	$0.393946\pi$
$272$	0	0
$273$	2.54935e10i	0.277778i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 1.44538e11i	− 1.47511i	−0.675289	−	0.737554i	$-0.735981\pi$
$277$	− 1.44538e11i	− 1.47511i	0.675289	−	0.737554i	$-0.264019\pi$
$278$	0	0
$279$	2.11077e11	2.08556
$280$	0	0
$281$	1.73877e11	1.66366	0.831828	−	0.555034i	$-0.187295\pi$
$281$	1.73877e11	1.66366	0.831828	+	0.555034i	$0.187295\pi$
$282$	0	0
$283$	3.86637e8i	0.00358315i	0.999998	+	0.00179157i	$0.000570276\pi$
$283$	3.86637e8i	0.00358315i	−0.999998	+	0.00179157i	$0.999430\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 4.62311e10i	− 0.402222i
$288$	0	0
$289$	1.06811e11	0.900692
$290$	0	0
$291$	3.39908e11	2.77870
$292$	0	0
$293$	6.25206e10i	0.495586i	0.968813	+	0.247793i	$0.0797053\pi$
$293$	6.25206e10i	0.495586i	−0.968813	+	0.247793i	$0.920295\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	4.45255e11i	3.32051i
$298$	0	0
$299$	5.39703e9	0.0390512
$300$	0	0
$301$	−8.31440e10	−0.583824
$302$	0	0
$303$	− 2.55767e10i	− 0.174322i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 1.57636e10i	− 0.101282i	−0.998717	−	0.0506410i	$-0.983874\pi$
$307$	− 1.57636e10i	− 0.101282i	0.998717	−	0.0506410i	$-0.0161264\pi$
$308$	0	0
$309$	−9.75334e10	−0.608611
$310$	0	0
$311$	2.01301e10	0.122018	0.0610091	−	0.998137i	$-0.480568\pi$
$311$	2.01301e10	0.122018	0.0610091	+	0.998137i	$0.480568\pi$
$312$	0	0
$313$	− 6.03790e10i	− 0.355579i	−0.984069	−	0.177790i	$-0.943105\pi$
$313$	− 6.03790e10i	− 0.355579i	0.984069	−	0.177790i	$-0.0568946\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.24292e11i	0.691313i	0.938361	+	0.345657i	$0.112344\pi$
$317$	1.24292e11i	0.691313i	−0.938361	+	0.345657i	$0.887656\pi$
$318$	0	0
$319$	−1.22946e11	−0.664748
$320$	0	0
$321$	2.78191e11	1.46241
$322$	0	0
$323$	7.21850e10i	0.369008i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	2.43412e11i	1.17727i
$328$	0	0
$329$	2.51988e11	1.18576
$330$	0	0
$331$	1.14638e11	0.524930	0.262465	−	0.964941i	$-0.415465\pi$
$331$	1.14638e11	0.524930	0.262465	+	0.964941i	$0.415465\pi$
$332$	0	0
$333$	7.70526e11i	3.43390i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	3.40785e11i	1.43928i	0.694345	+	0.719642i	$0.255694\pi$
$337$	3.40785e11i	1.43928i	−0.694345	+	0.719642i	$0.744306\pi$
$338$	0	0
$339$	−3.02461e11	−1.24386
$340$	0	0
$341$	1.82674e11	0.731615
$342$	0	0
$343$	1.93491e11i	0.754809i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 3.38240e11i	− 1.25240i	−0.779664	−	0.626198i	$-0.784610\pi$
$347$	− 3.38240e11i	− 1.25240i	0.779664	−	0.626198i	$-0.215390\pi$
$348$	0	0
$349$	−2.34359e11	−0.845603	−0.422802	−	0.906222i	$-0.638953\pi$
$349$	−2.34359e11	−0.845603	−0.422802	+	0.906222i	$0.638953\pi$
$350$	0	0
$351$	8.85235e10	0.311298
$352$	0	0
$353$	3.56202e11i	1.22098i	0.792022	+	0.610492i	$0.209028\pi$
$353$	3.56202e11i	1.22098i	−0.792022	+	0.610492i	$0.790972\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 2.95481e11i	− 0.962769i
$358$	0	0
$359$	3.37815e11	1.07338	0.536691	−	0.843779i	$-0.319674\pi$
$359$	3.37815e11	1.07338	0.536691	+	0.843779i	$0.319674\pi$
$360$	0	0
$361$	1.19768e11	0.371157
$362$	0	0
$363$	− 3.81461e10i	− 0.115311i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	1.36383e11i	0.392431i	0.980561	+	0.196215i	$0.0628652\pi$
$367$	1.36383e11i	0.392431i	−0.980561	+	0.196215i	$0.937135\pi$
$368$	0	0
$369$	−2.51506e11	−0.706204
$370$	0	0
$371$	3.49423e11	0.957566
$372$	0	0
$373$	5.08929e11i	1.36134i	0.732589	+	0.680672i	$0.238312\pi$
$373$	5.08929e11i	1.36134i	−0.732589	+	0.680672i	$0.761688\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.44436e10i	0.0623201i
$378$	0	0
$379$	9.70726e10	0.241669	0.120834	−	0.992673i	$-0.461443\pi$
$379$	9.70726e10	0.241669	0.120834	+	0.992673i	$0.461443\pi$
$380$	0	0
$381$	−1.50213e12	−3.65211
$382$	0	0
$383$	7.13792e11i	1.69503i	0.530772	+	0.847514i	$0.321902\pi$
$383$	7.13792e11i	1.69503i	−0.530772	+	0.847514i	$0.678098\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4.52320e11i	1.02505i
$388$	0	0
$389$	−2.48149e11	−0.549465	−0.274732	−	0.961521i	$-0.588589\pi$
$389$	−2.48149e11	−0.549465	−0.274732	+	0.961521i	$0.588589\pi$
$390$	0	0
$391$	−6.25540e10	−0.135350
$392$	0	0
$393$	− 1.26994e12i	− 2.68544i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 7.04445e11i	− 1.42328i	−0.702545	−	0.711640i	$-0.747953\pi$
$397$	− 7.04445e11i	− 1.42328i	0.702545	−	0.711640i	$-0.252047\pi$
$398$	0	0
$399$	−1.81114e12	−3.57745
$400$	0	0
$401$	−2.82127e11	−0.544873	−0.272437	−	0.962174i	$-0.587830\pi$
$401$	−2.82127e11	−0.544873	−0.272437	+	0.962174i	$0.587830\pi$
$402$	0	0
$403$	− 3.63184e10i	− 0.0685890i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6.66844e11i	1.20462i
$408$	0	0
$409$	−9.15011e11	−1.61686	−0.808428	−	0.588595i	$-0.799681\pi$
$409$	−9.15011e11	−1.61686	−0.808428	+	0.588595i	$0.799681\pi$
$410$	0	0
$411$	−1.89820e12	−3.28135
$412$	0	0
$413$	− 6.72041e10i	− 0.113663i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	1.70877e12i	2.76739i
$418$	0	0
$419$	1.19010e12	1.88635	0.943174	−	0.332299i	$-0.107824\pi$
$419$	1.19010e12	1.88635	0.943174	+	0.332299i	$0.107824\pi$
$420$	0	0
$421$	7.44534e11	1.15509	0.577544	−	0.816360i	$-0.304011\pi$
$421$	7.44534e11	1.15509	0.577544	+	0.816360i	$0.304011\pi$
$422$	0	0
$423$	− 1.37086e12i	− 2.08191i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	4.76104e10i	0.0693069i
$428$	0	0
$429$	1.20027e11	0.171089
$430$	0	0
$431$	3.47758e11	0.485433	0.242716	−	0.970097i	$-0.421962\pi$
$431$	3.47758e11	0.485433	0.242716	+	0.970097i	$0.421962\pi$
$432$	0	0
$433$	− 8.13744e11i	− 1.11248i	−0.831022	−	0.556240i	$-0.812244\pi$
$433$	− 8.13744e11i	− 1.11248i	0.831022	−	0.556240i	$-0.187756\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.83423e11i	0.502935i
$438$	0	0
$439$	−1.09730e12	−1.41005	−0.705023	−	0.709184i	$-0.749063\pi$
$439$	−1.09730e12	−1.41005	−0.705023	+	0.709184i	$0.749063\pi$
$440$	0	0
$441$	3.24851e12	4.08987
$442$	0	0
$443$	− 6.59236e10i	− 0.0813251i	−0.999173	−	0.0406625i	$-0.987053\pi$
$443$	− 6.59236e10i	− 0.0813251i	0.999173	−	0.0406625i	$-0.0129469\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	4.41407e11i	0.522945i
$448$	0	0
$449$	−2.15220e11	−0.249905	−0.124952	−	0.992163i	$-0.539878\pi$
$449$	−2.15220e11	−0.249905	−0.124952	+	0.992163i	$0.539878\pi$
$450$	0	0
$451$	−2.17664e11	−0.247737
$452$	0	0
$453$	− 3.20329e12i	− 3.57400i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 7.36259e11i	− 0.789601i	−0.918767	−	0.394801i	$-0.870814\pi$
$457$	− 7.36259e11i	− 0.789601i	0.918767	−	0.394801i	$-0.129186\pi$
$458$	0	0
$459$	−1.02603e12	−1.07895
$460$	0	0
$461$	−1.23730e12	−1.27591	−0.637957	−	0.770072i	$-0.720220\pi$
$461$	−1.23730e12	−1.27591	−0.637957	+	0.770072i	$0.720220\pi$
$462$	0	0
$463$	7.36969e11i	0.745306i	0.927971	+	0.372653i	$0.121552\pi$
$463$	7.36969e11i	0.745306i	−0.927971	+	0.372653i	$0.878448\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 1.22439e12i	− 1.19123i	−0.803271	−	0.595614i	$-0.796909\pi$
$467$	− 1.22439e12i	− 1.19123i	0.803271	−	0.595614i	$-0.203091\pi$
$468$	0	0
$469$	−1.38808e12	−1.32476
$470$	0	0
$471$	−4.66265e11	−0.436555
$472$	0	0
$473$	3.91455e11i	0.359590i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 1.90093e12i	− 1.68125i
$478$	0	0
$479$	1.11970e12	0.971833	0.485916	−	0.874005i	$-0.338486\pi$
$479$	1.11970e12	0.971833	0.485916	+	0.874005i	$0.338486\pi$
$480$	0	0
$481$	1.32579e11	0.112933
$482$	0	0
$483$	− 1.56950e12i	− 1.31220i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 2.25337e11i	− 0.181532i	−0.995872	−	0.0907659i	$-0.971068\pi$
$487$	− 2.25337e11i	− 0.181532i	0.995872	−	0.0907659i	$-0.0289315\pi$
$488$	0	0
$489$	2.71840e12	2.14992
$490$	0	0
$491$	1.60716e11	0.124794	0.0623970	−	0.998051i	$-0.480126\pi$
$491$	1.60716e11	0.124794	0.0623970	+	0.998051i	$0.480126\pi$
$492$	0	0
$493$	− 2.83312e11i	− 0.216000i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3.54450e12i	2.60586i
$498$	0	0
$499$	8.05187e11	0.581359	0.290680	−	0.956820i	$-0.406119\pi$
$499$	8.05187e11	0.581359	0.290680	+	0.956820i	$0.406119\pi$
$500$	0	0
$501$	−4.25065e12	−3.01430
$502$	0	0
$503$	1.68493e12i	1.17361i	0.809727	+	0.586807i	$0.199615\pi$
$503$	1.68493e12i	1.17361i	−0.809727	+	0.586807i	$0.800385\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	2.86280e12i	1.92422i
$508$	0	0
$509$	−1.50208e12	−0.991890	−0.495945	−	0.868354i	$-0.665178\pi$
$509$	−1.50208e12	−0.991890	−0.495945	+	0.868354i	$0.665178\pi$
$510$	0	0
$511$	2.41658e12	1.56786
$512$	0	0
$513$	6.28900e12i	4.00916i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 1.18640e12i	− 0.730336i
$518$	0	0
$519$	−1.25687e11	−0.0760393
$520$	0	0
$521$	9.24724e11	0.549848	0.274924	−	0.961466i	$-0.411347\pi$
$521$	9.24724e11	0.549848	0.274924	+	0.961466i	$0.411347\pi$
$522$	0	0
$523$	3.81782e10i	0.0223130i	0.999938	+	0.0111565i	$0.00355130\pi$
$523$	3.81782e10i	0.0223130i	−0.999938	+	0.0111565i	$0.996449\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.20947e11i	0.237727i
$528$	0	0
$529$	1.46889e12	0.815526
$530$	0	0
$531$	−3.65603e11	−0.199565
$532$	0	0
$533$	4.32748e10i	0.0232254i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 4.26991e12i	− 2.21582i
$538$	0	0
$539$	2.81139e12	1.43473
$540$	0	0
$541$	−1.05812e12	−0.531064	−0.265532	−	0.964102i	$-0.585548\pi$
$541$	−1.05812e12	−0.531064	−0.265532	+	0.964102i	$0.585548\pi$
$542$	0	0
$543$	1.43203e11i	0.0706894i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 1.05156e12i	− 0.502215i	−0.967959	−	0.251108i	$-0.919205\pi$
$547$	− 1.05156e12i	− 0.502215i	0.967959	−	0.251108i	$-0.0807948\pi$
$548$	0	0
$549$	2.59010e11	0.121686
$550$	0	0
$551$	−1.73655e12	−0.802611
$552$	0	0
$553$	− 2.61240e12i	− 1.18789i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	3.92684e11i	0.172860i	0.996258	+	0.0864301i	$0.0275459\pi$
$557$	3.92684e11i	0.172860i	−0.996258	+	0.0864301i	$0.972454\pi$
$558$	0	0
$559$	7.78272e10	0.0337115
$560$	0	0
$561$	−1.39117e12	−0.592990
$562$	0	0
$563$	− 2.36287e12i	− 0.991180i	−0.868557	−	0.495590i	$-0.834952\pi$
$563$	− 2.36287e12i	− 0.991180i	0.868557	−	0.495590i	$-0.165048\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 1.50299e13i	− 6.10705i
$568$	0	0
$569$	2.50603e12	1.00226	0.501130	−	0.865372i	$-0.332918\pi$
$569$	2.50603e12	1.00226	0.501130	+	0.865372i	$0.332918\pi$
$570$	0	0
$571$	2.99195e12	1.17785	0.588927	−	0.808187i	$-0.299551\pi$
$571$	2.99195e12	1.17785	0.588927	+	0.808187i	$0.299551\pi$
$572$	0	0
$573$	7.09494e12i	2.74950i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	3.06622e11i	0.115163i	0.998341	+	0.0575815i	$0.0183389\pi$
$577$	3.06622e11i	0.115163i	−0.998341	+	0.0575815i	$0.981661\pi$
$578$	0	0
$579$	−1.31618e12	−0.486700
$580$	0	0
$581$	6.56067e12	2.38867
$582$	0	0
$583$	− 1.64514e12i	− 0.589785i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 2.80823e12i	− 0.976249i	−0.872774	−	0.488125i	$-0.837681\pi$
$587$	− 2.80823e12i	− 0.976249i	0.872774	−	0.488125i	$-0.162319\pi$
$588$	0	0
$589$	2.58018e12	0.883346
$590$	0	0
$591$	2.97459e12	1.00296
$592$	0	0
$593$	− 1.34954e12i	− 0.448166i	−0.974570	−	0.224083i	$-0.928061\pi$
$593$	− 1.34954e12i	− 0.448166i	0.974570	−	0.224083i	$-0.0719387\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 6.09867e12i	− 1.96494i
$598$	0	0
$599$	3.38598e12	1.07464	0.537321	−	0.843378i	$-0.319436\pi$
$599$	3.38598e12	1.07464	0.537321	+	0.843378i	$0.319436\pi$
$600$	0	0
$601$	4.62837e12	1.44708	0.723540	−	0.690282i	$-0.242514\pi$
$601$	4.62837e12	1.44708	0.723540	+	0.690282i	$0.242514\pi$
$602$	0	0
$603$	7.55142e12i	2.32595i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	4.78828e12i	1.43163i	0.698291	+	0.715814i	$0.253944\pi$
$607$	4.78828e12i	1.43163i	−0.698291	+	0.715814i	$0.746056\pi$
$608$	0	0
$609$	7.10837e12	2.09407
$610$	0	0
$611$	−2.35874e11	−0.0684690
$612$	0	0
$613$	− 1.02749e12i	− 0.293905i	−0.989144	−	0.146953i	$-0.953054\pi$
$613$	− 1.02749e12i	− 0.293905i	0.989144	−	0.146953i	$-0.0469465\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	4.53079e12i	1.25861i	0.777159	+	0.629304i	$0.216660\pi$
$617$	4.53079e12i	1.25861i	−0.777159	+	0.629304i	$0.783340\pi$
$618$	0	0
$619$	−3.42902e12	−0.938777	−0.469388	−	0.882992i	$-0.655526\pi$
$619$	−3.42902e12	−0.938777	−0.469388	+	0.882992i	$0.655526\pi$
$620$	0	0
$621$	−5.44992e12	−1.47054
$622$	0	0
$623$	− 1.00247e13i	− 2.66608i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	8.52714e12i	2.20343i
$628$	0	0
$629$	−1.53664e12	−0.391422
$630$	0	0
$631$	3.52587e12	0.885389	0.442695	−	0.896672i	$-0.354023\pi$
$631$	3.52587e12	0.885389	0.442695	+	0.896672i	$0.354023\pi$
$632$	0	0
$633$	− 8.63260e12i	− 2.13710i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 5.58946e11i	− 0.134506i
$638$	0	0
$639$	1.92828e13	4.57526
$640$	0	0
$641$	−7.00458e12	−1.63878	−0.819390	−	0.573236i	$-0.805688\pi$
$641$	−7.00458e12	−1.63878	−0.819390	+	0.573236i	$0.805688\pi$
$642$	0	0
$643$	4.31047e12i	0.994432i	0.867627	+	0.497216i	$0.165644\pi$
$643$	4.31047e12i	0.994432i	−0.867627	+	0.497216i	$0.834356\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 2.84949e12i	− 0.639290i	−0.947537	−	0.319645i	$-0.896436\pi$
$647$	− 2.84949e12i	− 0.639290i	0.947537	−	0.319645i	$-0.103564\pi$
$648$	0	0
$649$	−3.16407e11	−0.0700077
$650$	0	0
$651$	−1.05617e13	−2.30472
$652$	0	0
$653$	1.93213e12i	0.415841i	0.978146	+	0.207920i	$0.0666695\pi$
$653$	1.93213e12i	0.415841i	−0.978146	+	0.207920i	$0.933331\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 1.31467e13i	− 2.75278i
$658$	0	0
$659$	8.58675e12	1.77355	0.886777	−	0.462198i	$-0.152939\pi$
$659$	8.58675e12	1.77355	0.886777	+	0.462198i	$0.152939\pi$
$660$	0	0
$661$	1.11478e12	0.227133	0.113567	−	0.993530i	$-0.463772\pi$
$661$	1.11478e12	0.227133	0.113567	+	0.993530i	$0.463772\pi$
$662$	0	0
$663$	2.76586e11i	0.0555928i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 1.50486e12i	− 0.294394i
$668$	0	0
$669$	4.15562e12	0.802081
$670$	0	0
$671$	2.24157e11	0.0426876
$672$	0	0
$673$	7.57716e12i	1.42377i	0.702298	+	0.711883i	$0.252157\pi$
$673$	7.57716e12i	1.42377i	−0.702298	+	0.711883i	$0.747843\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	5.08558e12i	0.930447i	0.885193	+	0.465223i	$0.154026\pi$
$677$	5.08558e12i	0.930447i	−0.885193	+	0.465223i	$0.845974\pi$
$678$	0	0
$679$	−1.24901e13	−2.25503
$680$	0	0
$681$	2.05106e12	0.365440
$682$	0	0
$683$	1.78278e11i	0.0313476i	0.999877	+	0.0156738i	$0.00498933\pi$
$683$	1.78278e11i	0.0313476i	−0.999877	+	0.0156738i	$0.995011\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	3.49093e12i	0.597911i
$688$	0	0
$689$	−3.27078e11	−0.0552924
$690$	0	0
$691$	8.66612e11	0.144602	0.0723008	−	0.997383i	$-0.476966\pi$
$691$	8.66612e11	0.144602	0.0723008	+	0.997383i	$0.476966\pi$
$692$	0	0
$693$	− 2.56330e13i	− 4.22182i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 5.01574e11i	− 0.0804985i
$698$	0	0
$699$	−5.61497e12	−0.889610
$700$	0	0
$701$	5.18099e12	0.810367	0.405183	−	0.914235i	$-0.367208\pi$
$701$	5.18099e12	0.810367	0.405183	+	0.914235i	$0.367208\pi$
$702$	0	0
$703$	9.41882e12i	1.45445i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	9.39831e11i	0.141469i
$708$	0	0
$709$	1.22351e13	1.81844	0.909218	−	0.416320i	$-0.136681\pi$
$709$	1.22351e13	1.81844	0.909218	+	0.416320i	$0.136681\pi$
$710$	0	0
$711$	−1.42120e13	−2.08565
$712$	0	0
$713$	2.23593e12i	0.324008i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	1.07976e13i	1.52578i
$718$	0	0
$719$	−1.11881e13	−1.56127	−0.780634	−	0.624989i	$-0.785104\pi$
$719$	−1.11881e13	−1.56127	−0.780634	+	0.624989i	$0.785104\pi$
$720$	0	0
$721$	3.58392e12	0.493913
$722$	0	0
$723$	− 1.02254e12i	− 0.139174i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	7.94265e12i	1.05453i	0.849699	+	0.527267i	$0.176783\pi$
$727$	7.94265e12i	1.05453i	−0.849699	+	0.527267i	$0.823217\pi$
$728$	0	0
$729$	−3.11078e13	−4.07940
$730$	0	0
$731$	−9.02052e11	−0.116843
$732$	0	0
$733$	1.09012e13i	1.39478i	0.716690	+	0.697392i	$0.245656\pi$
$733$	1.09012e13i	1.39478i	−0.716690	+	0.697392i	$0.754344\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6.53530e12i	0.815947i
$738$	0	0
$739$	1.54854e12	0.190996	0.0954978	−	0.995430i	$-0.469556\pi$
$739$	1.54854e12	0.190996	0.0954978	+	0.995430i	$0.469556\pi$
$740$	0	0
$741$	1.69532e12	0.206572
$742$	0	0
$743$	− 5.96254e12i	− 0.717764i	−0.933383	−	0.358882i	$-0.883158\pi$
$743$	− 5.96254e12i	− 0.717764i	0.933383	−	0.358882i	$-0.116842\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 3.56913e13i	− 4.19392i
$748$	0	0
$749$	−1.02223e13	−1.18681
$750$	0	0
$751$	−9.92051e12	−1.13803	−0.569016	−	0.822326i	$-0.692676\pi$
$751$	−9.92051e12	−1.13803	−0.569016	+	0.822326i	$0.692676\pi$
$752$	0	0
$753$	− 2.10870e13i	− 2.39022i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 1.23733e13i	− 1.36947i	−0.728791	−	0.684737i	$-0.759917\pi$
$757$	− 1.23733e13i	− 1.36947i	0.728791	−	0.684737i	$-0.240083\pi$
$758$	0	0
$759$	−7.38944e12	−0.808209
$760$	0	0
$761$	−8.52217e12	−0.921127	−0.460563	−	0.887627i	$-0.652353\pi$
$761$	−8.52217e12	−0.921127	−0.460563	+	0.887627i	$0.652353\pi$
$762$	0	0
$763$	− 8.94434e12i	− 0.955406i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	6.29066e10i	0.00656322i
$768$	0	0
$769$	−2.99595e12	−0.308934	−0.154467	−	0.987998i	$-0.549366\pi$
$769$	−2.99595e12	−0.308934	−0.154467	+	0.987998i	$0.549366\pi$
$770$	0	0
$771$	9.03713e12	0.921056
$772$	0	0
$773$	6.44491e11i	0.0649245i	0.999473	+	0.0324623i	$0.0103349\pi$
$773$	6.44491e11i	0.0649245i	−0.999473	+	0.0324623i	$0.989665\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 3.85548e13i	− 3.79476i
$778$	0	0
$779$	−3.07438e12	−0.299116
$780$	0	0
$781$	1.66881e13	1.60501
$782$	0	0
$783$	− 2.46831e13i	− 2.34677i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	8.05096e12i	0.748103i	0.927408	+	0.374052i	$0.122032\pi$
$787$	8.05096e12i	0.748103i	−0.927408	+	0.374052i	$0.877968\pi$
$788$	0	0
$789$	2.74264e13	2.51955
$790$	0	0
$791$	1.11141e13	1.00944
$792$	0	0
$793$	− 4.45659e10i	− 0.00400196i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.11381e13i	0.977799i	0.872340	+	0.488899i	$0.162601\pi$
$797$	1.11381e13i	0.977799i	−0.872340	+	0.488899i	$0.837399\pi$
$798$	0	0
$799$	2.73388e12	0.237312
$800$	0	0
$801$	−5.45361e13	−4.68099
$802$	0	0
$803$	− 1.13776e13i	− 0.965678i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	2.04144e13i	1.69436i
$808$	0	0
$809$	−1.69716e13	−1.39301	−0.696506	−	0.717550i	$-0.745263\pi$
$809$	−1.69716e13	−1.39301	−0.696506	+	0.717550i	$0.745263\pi$
$810$	0	0
$811$	1.17583e13	0.954444	0.477222	−	0.878783i	$-0.341644\pi$
$811$	1.17583e13	0.954444	0.477222	+	0.878783i	$0.341644\pi$
$812$	0	0
$813$	1.58092e13i	1.26912i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	5.52910e12i	0.434165i
$818$	0	0
$819$	−5.09623e12	−0.395796
$820$	0	0
$821$	1.99657e13	1.53370	0.766851	−	0.641825i	$-0.221823\pi$
$821$	1.99657e13	1.53370	0.766851	+	0.641825i	$0.221823\pi$
$822$	0	0
$823$	− 5.52813e11i	− 0.0420029i	−0.999779	−	0.0210014i	$-0.993315\pi$
$823$	− 5.52813e11i	− 0.0420029i	0.999779	−	0.0210014i	$-0.00668546\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 1.57036e13i	− 1.16741i	−0.811965	−	0.583707i	$-0.801602\pi$
$827$	− 1.57036e13i	− 1.16741i	0.811965	−	0.583707i	$-0.198398\pi$
$828$	0	0
$829$	−1.05928e13	−0.778958	−0.389479	−	0.921035i	$-0.627345\pi$
$829$	−1.05928e13	−0.778958	−0.389479	+	0.921035i	$0.627345\pi$
$830$	0	0
$831$	3.93449e13	2.86209
$832$	0	0
$833$	6.47843e12i	0.466195i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	3.66743e13i	2.58284i
$838$	0	0
$839$	−6.55506e11	−0.0456718	−0.0228359	−	0.999739i	$-0.507270\pi$
$839$	−6.55506e11	−0.0456718	−0.0228359	+	0.999739i	$0.507270\pi$
$840$	0	0
$841$	−7.69153e12	−0.530189
$842$	0	0
$843$	4.73312e13i	3.22793i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1.40170e12i	0.0935794i
$848$	0	0
$849$	−1.05247e11	−0.00695224
$850$	0	0
$851$	−8.16215e12	−0.533484
$852$	0	0
$853$	2.08805e13i	1.35043i	0.737622	+	0.675213i	$0.235949\pi$
$853$	2.08805e13i	1.35043i	−0.737622	+	0.675213i	$0.764051\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2.19074e13i	1.38732i	0.720303	+	0.693660i	$0.244003\pi$
$857$	2.19074e13i	1.38732i	−0.720303	+	0.693660i	$0.755997\pi$
$858$	0	0
$859$	1.47238e13	0.922680	0.461340	−	0.887223i	$-0.347369\pi$
$859$	1.47238e13	0.922680	0.461340	+	0.887223i	$0.347369\pi$
$860$	0	0
$861$	1.25846e13	0.780416
$862$	0	0
$863$	2.74805e12i	0.168646i	0.996438	+	0.0843230i	$0.0268727\pi$
$863$	2.74805e12i	0.168646i	−0.996438	+	0.0843230i	$0.973127\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	2.90752e13i	1.74758i
$868$	0	0
$869$	−1.22996e13	−0.731648
$870$	0	0
$871$	1.29932e12	0.0764950
$872$	0	0
$873$	6.79486e13i	3.95928i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 6.68274e12i	− 0.381466i	−0.981642	−	0.190733i	$-0.938913\pi$
$877$	− 6.68274e12i	− 0.381466i	0.981642	−	0.190733i	$-0.0610865\pi$
$878$	0	0
$879$	−1.70188e13	−0.961567
$880$	0	0
$881$	1.42384e13	0.796289	0.398144	−	0.917323i	$-0.369654\pi$
$881$	1.42384e13	0.796289	0.398144	+	0.917323i	$0.369654\pi$
$882$	0	0
$883$	2.46098e13i	1.36234i	0.732125	+	0.681170i	$0.238529\pi$
$883$	2.46098e13i	1.36234i	−0.732125	+	0.681170i	$0.761471\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 9.12625e12i	− 0.495035i	−0.968883	−	0.247518i	$-0.920385\pi$
$887$	− 9.12625e12i	− 0.495035i	0.968883	−	0.247518i	$-0.0796148\pi$
$888$	0	0
$889$	5.51966e13	2.96384
$890$	0	0
$891$	−7.07630e13	−3.76146
$892$	0	0
$893$	− 1.67572e13i	− 0.881802i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	1.46913e12i	0.0757696i
$898$	0	0
$899$	−1.01267e13	−0.517069
$900$	0	0
$901$	3.79098e12	0.191642
$902$	0	0
$903$	− 2.26327e13i	− 1.13277i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	3.32446e13i	1.63113i	0.578668	+	0.815564i	$0.303573\pi$
$907$	3.32446e13i	1.63113i	−0.578668	+	0.815564i	$0.696427\pi$
$908$	0	0
$909$	5.11286e12	0.248386
$910$	0	0
$911$	1.42660e13	0.686232	0.343116	−	0.939293i	$-0.388518\pi$
$911$	1.42660e13	0.686232	0.343116	+	0.939293i	$0.388518\pi$
$912$	0	0
$913$	− 3.08887e13i	− 1.47123i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.66647e13i	2.17935i
$918$	0	0
$919$	3.23171e13	1.49456	0.747278	−	0.664512i	$-0.231360\pi$
$919$	3.23171e13	1.49456	0.747278	+	0.664512i	$0.231360\pi$
$920$	0	0
$921$	4.29102e12	0.196514
$922$	0	0
$923$	− 3.31784e12i	− 0.150469i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 1.94972e13i	− 0.867190i
$928$	0	0
$929$	3.49040e12	0.153746	0.0768732	−	0.997041i	$-0.475506\pi$
$929$	3.49040e12	0.153746	0.0768732	+	0.997041i	$0.475506\pi$
$930$	0	0
$931$	3.97093e13	1.73228
$932$	0	0
$933$	5.47964e12i	0.236747i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1.48063e13i	0.627507i	0.949504	+	0.313754i	$0.101587\pi$
$937$	1.48063e13i	0.627507i	−0.949504	+	0.313754i	$0.898413\pi$
$938$	0	0
$939$	1.64358e13	0.689916
$940$	0	0
$941$	3.90935e13	1.62537	0.812684	−	0.582705i	$-0.198006\pi$
$941$	3.90935e13	1.62537	0.812684	+	0.582705i	$0.198006\pi$
$942$	0	0
$943$	− 2.66420e12i	− 0.109714i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2.30013e13i	0.929346i	0.885482	+	0.464673i	$0.153828\pi$
$947$	2.30013e13i	0.929346i	−0.885482	+	0.464673i	$0.846172\pi$
$948$	0	0
$949$	−2.26205e12	−0.0905323
$950$	0	0
$951$	−3.38335e13	−1.34133
$952$	0	0
$953$	− 3.32845e13i	− 1.30714i	−0.756864	−	0.653572i	$-0.773270\pi$
$953$	− 3.32845e13i	− 1.30714i	0.756864	−	0.653572i	$-0.226730\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 3.34673e13i	− 1.28978i
$958$	0	0
$959$	6.97504e13	2.66295
$960$	0	0
$961$	−1.13933e13	−0.430918
$962$	0	0
$963$	5.56112e13i	2.08374i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 5.77848e12i	− 0.212518i	−0.994339	−	0.106259i	$-0.966113\pi$
$967$	− 5.77848e12i	− 0.212518i	0.994339	−	0.106259i	$-0.0338872\pi$
$968$	0	0
$969$	−1.96495e13	−0.715971
$970$	0	0
$971$	4.46200e13	1.61080	0.805402	−	0.592728i	$-0.201949\pi$
$971$	4.46200e13	1.61080	0.805402	+	0.592728i	$0.201949\pi$
$972$	0	0
$973$	− 6.27897e13i	− 2.24585i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 3.90697e13i	− 1.37188i	−0.727660	−	0.685938i	$-0.759392\pi$
$977$	− 3.90697e13i	− 1.37188i	0.727660	−	0.685938i	$-0.240608\pi$
$978$	0	0
$979$	−4.71977e13	−1.64210
$980$	0	0
$981$	−4.86589e13	−1.67746
$982$	0	0
$983$	− 1.96137e12i	− 0.0669990i	−0.999439	−	0.0334995i	$-0.989335\pi$
$983$	− 1.96137e12i	− 0.0669990i	0.999439	−	0.0334995i	$-0.0106652\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	6.85938e13i	2.30069i
$988$	0	0
$989$	−4.79141e12	−0.159250
$990$	0	0
$991$	4.33785e13	1.42871	0.714354	−	0.699785i	$-0.246721\pi$
$991$	4.33785e13	1.42871	0.714354	+	0.699785i	$0.246721\pi$
$992$	0	0
$993$	3.12057e13i	1.01850i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	2.79360e12i	0.0895440i	0.998997	+	0.0447720i	$0.0142561\pi$
$997$	2.79360e12i	0.0895440i	−0.998997	+	0.0447720i	$0.985744\pi$
$998$	0	0
$999$	−1.33878e14	−4.25269

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.10.c.l.49.4		4
4.3	odd	2		100.10.c.c.49.1		4
5.2	odd	4		80.10.a.j.1.2		2
5.3	odd	4		400.10.a.l.1.1		2
5.4	even	2	inner	400.10.c.l.49.1		4
20.3	even	4		100.10.a.c.1.2		2
20.7	even	4		20.10.a.b.1.1	✓	2
20.19	odd	2		100.10.c.c.49.4		4
40.27	even	4		320.10.a.t.1.2		2
40.37	odd	4		320.10.a.l.1.1		2
60.47	odd	4		180.10.a.e.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.10.a.b.1.1	✓	2	20.7	even	4
80.10.a.j.1.2		2	5.2	odd	4
100.10.a.c.1.2		2	20.3	even	4
100.10.c.c.49.1		4	4.3	odd	2
100.10.c.c.49.4		4	20.19	odd	2
180.10.a.e.1.1		2	60.47	odd	4
320.10.a.l.1.1		2	40.37	odd	4
320.10.a.t.1.2		2	40.27	even	4
400.10.a.l.1.1		2	5.3	odd	4
400.10.c.l.49.1		4	5.4	even	2	inner
400.10.c.l.49.4		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	400.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+272.211i q^{3} -10002.6i q^{7} -54415.9 q^{9} +O(q^{10})\)	\(q+272.211i q^{3} -10002.6i q^{7} -54415.9 q^{9} -47093.7 q^{11} +9362.93i q^{13} -108521. i q^{17} -665173. q^{19} +2.72281e6 q^{21} -576426. i q^{23} -9.45468e6i q^{27} +2.61067e6 q^{29} -3.87896e6 q^{31} -1.28194e7i q^{33} -1.41599e7i q^{37} -2.54869e6 q^{39} +4.62193e6 q^{41} -8.31227e6i q^{43} +2.51923e7i q^{47} -5.96977e7 q^{49} +2.95405e7 q^{51} +3.49333e7i q^{53} -1.81068e8i q^{57} +6.71868e6 q^{59} -4.75982e6 q^{61} +5.44299e8i q^{63} -1.38772e8i q^{67} +1.56909e8 q^{69} -3.54359e8 q^{71} +2.41596e8i q^{73} +4.71058e8i q^{77} +2.61173e8 q^{79} +1.50260e9 q^{81} +6.55898e8i q^{83} +7.10654e8i q^{87} +1.00221e9 q^{89} +9.36534e7 q^{91} -1.05590e9i q^{93} -1.24869e9i q^{97} +2.56264e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 69764 q^{9}+O(q^{10}) \) 4 * q - 69764 * q^9	\( 4 q - 69764 q^{9} - 205440 q^{11} + 274544 q^{19} + 5680624 q^{21} + 13787496 q^{29} - 583664 q^{31} - 951056 q^{39} + 59546904 q^{41} - 223875828 q^{49} + 54013392 q^{51} - 185861712 q^{59} + 391347848 q^{61} + 344148528 q^{69} - 622414032 q^{71} + 1084523552 q^{79} + 3762476756 q^{81} + 924583704 q^{89} - 3080075536 q^{91} + 2952090240 q^{99}+O(q^{100}) \) 4 * q - 69764 * q^9 - 205440 * q^11 + 274544 * q^19 + 5680624 * q^21 + 13787496 * q^29 - 583664 * q^31 - 951056 * q^39 + 59546904 * q^41 - 223875828 * q^49 + 54013392 * q^51 - 185861712 * q^59 + 391347848 * q^61 + 344148528 * q^69 - 622414032 * q^71 + 1084523552 * q^79 + 3762476756 * q^81 + 924583704 * q^89 - 3080075536 * q^91 + 2952090240 * q^99

Introduction

L-functions

Modular forms

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Database

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