LMFDB - Embedded newform 400.6.c.n.49.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,6,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, 6, names="a")

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

chi := DirichletCharacter("400.49");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
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:	$64.1535279252$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{241})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 121x^{2} + 3600 $ x^4 + 121*x^2 + 3600
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 25)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			$-8.26209i$ of defining polynomial
Character	$\chi$	$=$	400.49
Dual form			400.6.c.n.49.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.52417i q^{3} -68.9517i q^{7} +212.483 q^{9} +O(q^{10})$	$q-5.52417i q^{3} -68.9517i q^{7} +212.483 q^{9} +486.104 q^{11} +428.387i q^{13} +1800.64i q^{17} -1046.65 q^{19} -380.901 q^{21} +686.855i q^{23} -2516.17i q^{27} +1339.03 q^{29} -7990.30 q^{31} -2685.33i q^{33} -1970.64i q^{37} +2366.48 q^{39} +10772.2 q^{41} +15017.7i q^{43} -895.337i q^{47} +12052.7 q^{49} +9947.07 q^{51} +19327.1i q^{53} +5781.90i q^{57} +21193.7 q^{59} -27722.2 q^{61} -14651.1i q^{63} +7719.33i q^{67} +3794.31 q^{69} +51410.1 q^{71} +43776.4i q^{73} -33517.7i q^{77} -6225.68 q^{79} +37733.7 q^{81} +52949.9i q^{83} -7397.05i q^{87} -44631.2 q^{89} +29538.0 q^{91} +44139.8i q^{93} -148018. i q^{97} +103289. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 392 q^{9}+O(q^{10}) $ 4 * q - 392 * q^9	$ 4 q - 392 q^{9} + 392 q^{11} - 6360 q^{19} + 5928 q^{21} + 7840 q^{29} + 2192 q^{31} + 8224 q^{39} + 55508 q^{41} + 23372 q^{49} + 35752 q^{51} + 23920 q^{59} - 48792 q^{61} - 36984 q^{69} + 174592 q^{71} + 130960 q^{79} + 92564 q^{81} + 145620 q^{89} + 41152 q^{91} + 443584 q^{99}+O(q^{100}) $ 4 * q - 392 * q^9 + 392 * q^11 - 6360 * q^19 + 5928 * q^21 + 7840 * q^29 + 2192 * q^31 + 8224 * q^39 + 55508 * q^41 + 23372 * q^49 + 35752 * q^51 + 23920 * q^59 - 48792 * q^61 - 36984 * q^69 + 174592 * q^71 + 130960 * q^79 + 92564 * q^81 + 145620 * q^89 + 41152 * q^91 + 443584 * 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
$3$	− 5.52417i	− 0.354376i	−0.984177	−	0.177188i	$-0.943300\pi$
$3$	− 5.52417i	− 0.354376i	0.984177	−	0.177188i	$-0.0567001\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 68.9517i	− 0.531863i	−0.963992	−	0.265931i	$-0.914321\pi$
$7$	− 68.9517i	− 0.531863i	0.963992	−	0.265931i	$-0.0856795\pi$
$8$	0	0
$9$	212.483	0.874418
$10$	0	0
$11$	486.104	1.21129	0.605645	−	0.795735i	$-0.292915\pi$
$11$	486.104	1.21129	0.605645	+	0.795735i	$0.292915\pi$
$12$	0	0
$13$	428.387i	0.703036i	0.936181	+	0.351518i	$0.114334\pi$
$13$	428.387i	0.703036i	−0.936181	+	0.351518i	$0.885666\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1800.64i	1.51114i	0.655066	+	0.755571i	$0.272641\pi$
$17$	1800.64i	1.51114i	−0.655066	+	0.755571i	$0.727359\pi$
$18$	0	0
$19$	−1046.65	−0.665149	−0.332575	−	0.943077i	$-0.607917\pi$
$19$	−1046.65	−0.665149	−0.332575	+	0.943077i	$0.607917\pi$
$20$	0	0
$21$	−380.901	−0.188479
$22$	0	0
$23$	686.855i	0.270736i	0.990795	+	0.135368i	$0.0432216\pi$
$23$	686.855i	0.270736i	−0.990795	+	0.135368i	$0.956778\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 2516.17i	− 0.664249i
$28$	0	0
$29$	1339.03	0.295663	0.147831	−	0.989013i	$-0.452771\pi$
$29$	1339.03	0.295663	0.147831	+	0.989013i	$0.452771\pi$
$30$	0	0
$31$	−7990.30	−1.49334	−0.746670	−	0.665195i	$-0.768349\pi$
$31$	−7990.30	−1.49334	−0.746670	+	0.665195i	$0.768349\pi$
$32$	0	0
$33$	− 2685.33i	− 0.429252i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 1970.64i	− 0.236648i	−0.992975	−	0.118324i	$-0.962248\pi$
$37$	− 1970.64i	− 0.236648i	0.992975	−	0.118324i	$-0.0377522\pi$
$38$	0	0
$39$	2366.48	0.249139
$40$	0	0
$41$	10772.2	1.00079	0.500395	−	0.865797i	$-0.333188\pi$
$41$	10772.2	1.00079	0.500395	+	0.865797i	$0.333188\pi$
$42$	0	0
$43$	15017.7i	1.23861i	0.785152	+	0.619303i	$0.212585\pi$
$43$	15017.7i	1.23861i	−0.785152	+	0.619303i	$0.787415\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 895.337i	− 0.0591210i	−0.999563	−	0.0295605i	$-0.990589\pi$
$47$	− 895.337i	− 0.0591210i	0.999563	−	0.0295605i	$-0.00941077\pi$
$48$	0	0
$49$	12052.7	0.717122
$50$	0	0
$51$	9947.07	0.535513
$52$	0	0
$53$	19327.1i	0.945098i	0.881304	+	0.472549i	$0.156666\pi$
$53$	19327.1i	0.945098i	−0.881304	+	0.472549i	$0.843334\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	5781.90i	0.235713i
$58$	0	0
$59$	21193.7	0.792641	0.396321	−	0.918112i	$-0.370287\pi$
$59$	21193.7	0.792641	0.396321	+	0.918112i	$0.370287\pi$
$60$	0	0
$61$	−27722.2	−0.953900	−0.476950	−	0.878931i	$-0.658258\pi$
$61$	−27722.2	−0.953900	−0.476950	+	0.878931i	$0.658258\pi$
$62$	0	0
$63$	− 14651.1i	− 0.465070i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	7719.33i	0.210084i	0.994468	+	0.105042i	$0.0334977\pi$
$67$	7719.33i	0.210084i	−0.994468	+	0.105042i	$0.966502\pi$
$68$	0	0
$69$	3794.31	0.0959422
$70$	0	0
$71$	51410.1	1.21033	0.605163	−	0.796101i	$-0.293108\pi$
$71$	51410.1	1.21033	0.605163	+	0.796101i	$0.293108\pi$
$72$	0	0
$73$	43776.4i	0.961465i	0.876867	+	0.480732i	$0.159629\pi$
$73$	43776.4i	0.961465i	−0.876867	+	0.480732i	$0.840371\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 33517.7i	− 0.644240i
$78$	0	0
$79$	−6225.68	−0.112233	−0.0561163	−	0.998424i	$-0.517872\pi$
$79$	−6225.68	−0.112233	−0.0561163	+	0.998424i	$0.517872\pi$
$80$	0	0
$81$	37733.7	0.639024
$82$	0	0
$83$	52949.9i	0.843664i	0.906674	+	0.421832i	$0.138613\pi$
$83$	52949.9i	0.843664i	−0.906674	+	0.421832i	$0.861387\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 7397.05i	− 0.104776i
$88$	0	0
$89$	−44631.2	−0.597260	−0.298630	−	0.954369i	$-0.596530\pi$
$89$	−44631.2	−0.597260	−0.298630	+	0.954369i	$0.596530\pi$
$90$	0	0
$91$	29538.0	0.373919
$92$	0	0
$93$	44139.8i	0.529204i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 148018.i	− 1.59730i	−0.601797	−	0.798649i	$-0.705548\pi$
$97$	− 148018.i	− 1.59730i	0.601797	−	0.798649i	$-0.294452\pi$
$98$	0	0
$99$	103289.	1.05917
$100$	0	0
$101$	148476.	1.44828	0.724141	−	0.689652i	$-0.242237\pi$
$101$	148476.	1.44828	0.724141	+	0.689652i	$0.242237\pi$
$102$	0	0
$103$	188391.i	1.74972i	0.484378	+	0.874859i	$0.339046\pi$
$103$	188391.i	1.74972i	−0.484378	+	0.874859i	$0.660954\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 67887.7i	− 0.573234i	−0.958045	−	0.286617i	$-0.907469\pi$
$107$	− 67887.7i	− 0.573234i	0.958045	−	0.286617i	$-0.0925307\pi$
$108$	0	0
$109$	219292.	1.76790	0.883949	−	0.467582i	$-0.154875\pi$
$109$	219292.	1.76790	0.883949	+	0.467582i	$0.154875\pi$
$110$	0	0
$111$	−10886.2	−0.0838625
$112$	0	0
$113$	80783.9i	0.595153i	0.954698	+	0.297577i	$0.0961784\pi$
$113$	80783.9i	0.595153i	−0.954698	+	0.297577i	$0.903822\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	91025.1i	0.614747i
$118$	0	0
$119$	124157.	0.803721
$120$	0	0
$121$	75246.5	0.467221
$122$	0	0
$123$	− 59507.3i	− 0.354656i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 161301.i	− 0.887417i	−0.896171	−	0.443708i	$-0.853663\pi$
$127$	− 161301.i	− 0.887417i	0.896171	−	0.443708i	$-0.146337\pi$
$128$	0	0
$129$	82960.5	0.438932
$130$	0	0
$131$	193006.	0.982636	0.491318	−	0.870980i	$-0.336515\pi$
$131$	193006.	0.982636	0.491318	+	0.870980i	$0.336515\pi$
$132$	0	0
$133$	72168.5i	0.353768i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	250340.i	1.13954i	0.821806	+	0.569768i	$0.192967\pi$
$137$	250340.i	1.13954i	−0.821806	+	0.569768i	$0.807033\pi$
$138$	0	0
$139$	−218650.	−0.959871	−0.479935	−	0.877304i	$-0.659340\pi$
$139$	−218650.	−0.959871	−0.479935	+	0.877304i	$0.659340\pi$
$140$	0	0
$141$	−4946.00	−0.0209511
$142$	0	0
$143$	208241.i	0.851580i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 66581.1i	− 0.254131i
$148$	0	0
$149$	38740.0	0.142953	0.0714766	−	0.997442i	$-0.477229\pi$
$149$	38740.0	0.142953	0.0714766	+	0.997442i	$0.477229\pi$
$150$	0	0
$151$	154945.	0.553013	0.276507	−	0.961012i	$-0.410823\pi$
$151$	154945.	0.553013	0.276507	+	0.961012i	$0.410823\pi$
$152$	0	0
$153$	382607.i	1.32137i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 344442.i	− 1.11523i	−0.830098	−	0.557617i	$-0.811716\pi$
$157$	− 344442.i	− 1.11523i	0.830098	−	0.557617i	$-0.188284\pi$
$158$	0	0
$159$	106766.	0.334920
$160$	0	0
$161$	47359.8	0.143994
$162$	0	0
$163$	366203.i	1.07957i	0.841801	+	0.539787i	$0.181495\pi$
$163$	366203.i	1.07957i	−0.841801	+	0.539787i	$0.818505\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 249272.i	− 0.691644i	−0.938300	−	0.345822i	$-0.887600\pi$
$167$	− 249272.i	− 0.691644i	0.938300	−	0.345822i	$-0.112400\pi$
$168$	0	0
$169$	187778.	0.505740
$170$	0	0
$171$	−222397.	−0.581618
$172$	0	0
$173$	61460.1i	0.156127i	0.996948	+	0.0780635i	$0.0248737\pi$
$173$	61460.1i	0.156127i	−0.996948	+	0.0780635i	$0.975126\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 117078.i	− 0.280893i
$178$	0	0
$179$	606803.	1.41552	0.707759	−	0.706454i	$-0.249706\pi$
$179$	606803.	1.41552	0.707759	+	0.706454i	$0.249706\pi$
$180$	0	0
$181$	153684.	0.348685	0.174343	−	0.984685i	$-0.444220\pi$
$181$	153684.	0.348685	0.174343	+	0.984685i	$0.444220\pi$
$182$	0	0
$183$	153142.i	0.338039i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	875301.i	1.83043i
$188$	0	0
$189$	−173494.	−0.353289
$190$	0	0
$191$	−182315.	−0.361608	−0.180804	−	0.983519i	$-0.557870\pi$
$191$	−182315.	−0.361608	−0.180804	+	0.983519i	$0.557870\pi$
$192$	0	0
$193$	102080.i	0.197265i	0.995124	+	0.0986323i	$0.0314468\pi$
$193$	102080.i	0.197265i	−0.995124	+	0.0986323i	$0.968553\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 404656.i	− 0.742882i	−0.928456	−	0.371441i	$-0.878864\pi$
$197$	− 404656.i	− 0.742882i	0.928456	−	0.371441i	$-0.121136\pi$
$198$	0	0
$199$	−167297.	−0.299472	−0.149736	−	0.988726i	$-0.547842\pi$
$199$	−167297.	−0.299472	−0.149736	+	0.988726i	$0.547842\pi$
$200$	0	0
$201$	42642.9	0.0744487
$202$	0	0
$203$	− 92328.5i	− 0.157252i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	145945.i	0.236736i
$208$	0	0
$209$	−508783.	−0.805688
$210$	0	0
$211$	460778.	0.712502	0.356251	−	0.934390i	$-0.384055\pi$
$211$	460778.	0.712502	0.356251	+	0.934390i	$0.384055\pi$
$212$	0	0
$213$	− 283998.i	− 0.428911i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	550944.i	0.794252i
$218$	0	0
$219$	241829.	0.340720
$220$	0	0
$221$	−771372.	−1.06239
$222$	0	0
$223$	− 1.08298e6i	− 1.45834i	−0.684330	−	0.729172i	$-0.739905\pi$
$223$	− 1.08298e6i	− 1.45834i	0.684330	−	0.729172i	$-0.260095\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 412201.i	− 0.530938i	−0.964119	−	0.265469i	$-0.914473\pi$
$227$	− 412201.i	− 0.530938i	0.964119	−	0.265469i	$-0.0855269\pi$
$228$	0	0
$229$	433163.	0.545836	0.272918	−	0.962037i	$-0.412011\pi$
$229$	433163.	0.545836	0.272918	+	0.962037i	$0.412011\pi$
$230$	0	0
$231$	−185158.	−0.228303
$232$	0	0
$233$	− 760097.i	− 0.917232i	−0.888635	−	0.458616i	$-0.848345\pi$
$233$	− 760097.i	− 0.917232i	0.888635	−	0.458616i	$-0.151655\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	34391.7i	0.0397725i
$238$	0	0
$239$	−988624.	−1.11953	−0.559766	−	0.828651i	$-0.689109\pi$
$239$	−988624.	−1.11953	−0.559766	+	0.828651i	$0.689109\pi$
$240$	0	0
$241$	−358878.	−0.398020	−0.199010	−	0.979997i	$-0.563773\pi$
$241$	−358878.	−0.398020	−0.199010	+	0.979997i	$0.563773\pi$
$242$	0	0
$243$	− 819877.i	− 0.890703i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 448373.i	− 0.467624i
$248$	0	0
$249$	292504.	0.298974
$250$	0	0
$251$	851049.	0.852649	0.426324	−	0.904570i	$-0.359808\pi$
$251$	851049.	0.852649	0.426324	+	0.904570i	$0.359808\pi$
$252$	0	0
$253$	333883.i	0.327939i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 76358.4i	− 0.0721147i	−0.999350	−	0.0360574i	$-0.988520\pi$
$257$	− 76358.4i	− 0.0721147i	0.999350	−	0.0360574i	$-0.0114799\pi$
$258$	0	0
$259$	−135879.	−0.125864
$260$	0	0
$261$	284522.	0.258533
$262$	0	0
$263$	1.19420e6i	1.06460i	0.846555	+	0.532301i	$0.178673\pi$
$263$	1.19420e6i	1.06460i	−0.846555	+	0.532301i	$0.821327\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	246551.i	0.211655i
$268$	0	0
$269$	−1.02930e6	−0.867286	−0.433643	−	0.901085i	$-0.642772\pi$
$269$	−1.02930e6	−0.867286	−0.433643	+	0.901085i	$0.642772\pi$
$270$	0	0
$271$	−2.12144e6	−1.75472	−0.877359	−	0.479834i	$-0.840697\pi$
$271$	−2.12144e6	−1.75472	−0.877359	+	0.479834i	$0.840697\pi$
$272$	0	0
$273$	− 163173.i	− 0.132508i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.85145e6i	1.44982i	0.688845	+	0.724908i	$0.258118\pi$
$277$	1.85145e6i	1.44982i	−0.688845	+	0.724908i	$0.741882\pi$
$278$	0	0
$279$	−1.69781e6	−1.30580
$280$	0	0
$281$	90653.2	0.0684884	0.0342442	−	0.999413i	$-0.489098\pi$
$281$	90653.2	0.0684884	0.0342442	+	0.999413i	$0.489098\pi$
$282$	0	0
$283$	− 929308.i	− 0.689753i	−0.938648	−	0.344877i	$-0.887921\pi$
$283$	− 929308.i	− 0.689753i	0.938648	−	0.344877i	$-0.112079\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 742759.i	− 0.532283i
$288$	0	0
$289$	−1.82246e6	−1.28355
$290$	0	0
$291$	−817679.	−0.566044
$292$	0	0
$293$	− 2.72733e6i	− 1.85596i	−0.372632	−	0.927979i	$-0.621545\pi$
$293$	− 2.72733e6i	− 1.85596i	0.372632	−	0.927979i	$-0.378455\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 1.22312e6i	− 0.804597i
$298$	0	0
$299$	−294240.	−0.190337
$300$	0	0
$301$	1.03550e6	0.658768
$302$	0	0
$303$	− 820208.i	− 0.513236i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	2.29648e6i	1.39064i	0.718698	+	0.695322i	$0.244738\pi$
$307$	2.29648e6i	1.39064i	−0.718698	+	0.695322i	$0.755262\pi$
$308$	0	0
$309$	1.04071e6	0.620058
$310$	0	0
$311$	−984847.	−0.577388	−0.288694	−	0.957421i	$-0.593221\pi$
$311$	−984847.	−0.577388	−0.288694	+	0.957421i	$0.593221\pi$
$312$	0	0
$313$	− 2.06650e6i	− 1.19227i	−0.802884	−	0.596135i	$-0.796702\pi$
$313$	− 2.06650e6i	− 1.19227i	0.802884	−	0.596135i	$-0.203298\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.14349e6i	0.639125i	0.947565	+	0.319563i	$0.103536\pi$
$317$	1.14349e6i	0.639125i	−0.947565	+	0.319563i	$0.896464\pi$
$318$	0	0
$319$	650910.	0.358133
$320$	0	0
$321$	−375024.	−0.203140
$322$	0	0
$323$	− 1.88465e6i	− 1.00514i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 1.21141e6i	− 0.626501i
$328$	0	0
$329$	−61735.0	−0.0314443
$330$	0	0
$331$	−205230.	−0.102961	−0.0514804	−	0.998674i	$-0.516394\pi$
$331$	−205230.	−0.102961	−0.0514804	+	0.998674i	$0.516394\pi$
$332$	0	0
$333$	− 418729.i	− 0.206929i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	488213.i	0.234172i	0.993122	+	0.117086i	$0.0373553\pi$
$337$	488213.i	0.234172i	−0.993122	+	0.117086i	$0.962645\pi$
$338$	0	0
$339$	446265.	0.210908
$340$	0	0
$341$	−3.88412e6	−1.80887
$342$	0	0
$343$	− 1.98992e6i	− 0.913273i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	3.82809e6i	1.70670i	0.521336	+	0.853351i	$0.325434\pi$
$347$	3.82809e6i	1.70670i	−0.521336	+	0.853351i	$0.674566\pi$
$348$	0	0
$349$	−1.45476e6	−0.639333	−0.319667	−	0.947530i	$-0.603571\pi$
$349$	−1.45476e6	−0.639333	−0.319667	+	0.947530i	$0.603571\pi$
$350$	0	0
$351$	1.07789e6	0.466991
$352$	0	0
$353$	778492.i	0.332520i	0.986082	+	0.166260i	$0.0531690\pi$
$353$	778492.i	0.332520i	−0.986082	+	0.166260i	$0.946831\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 685867.i	− 0.284819i
$358$	0	0
$359$	−2.12510e6	−0.870247	−0.435124	−	0.900371i	$-0.643295\pi$
$359$	−2.12510e6	−0.870247	−0.435124	+	0.900371i	$0.643295\pi$
$360$	0	0
$361$	−1.38061e6	−0.557577
$362$	0	0
$363$	− 415675.i	− 0.165572i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 4.10801e6i	− 1.59208i	−0.605242	−	0.796042i	$-0.706923\pi$
$367$	− 4.10801e6i	− 1.59208i	0.605242	−	0.796042i	$-0.293077\pi$
$368$	0	0
$369$	2.28891e6	0.875109
$370$	0	0
$371$	1.33263e6	0.502662
$372$	0	0
$373$	4.54570e6i	1.69172i	0.533405	+	0.845860i	$0.320912\pi$
$373$	4.54570e6i	1.69172i	−0.533405	+	0.845860i	$0.679088\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	573624.i	0.207861i
$378$	0	0
$379$	1.40554e6	0.502626	0.251313	−	0.967906i	$-0.419138\pi$
$379$	1.40554e6	0.502626	0.251313	+	0.967906i	$0.419138\pi$
$380$	0	0
$381$	−891054.	−0.314479
$382$	0	0
$383$	4.64417e6i	1.61775i	0.587982	+	0.808874i	$0.299922\pi$
$383$	4.64417e6i	1.61775i	−0.587982	+	0.808874i	$0.700078\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.19102e6i	1.08306i
$388$	0	0
$389$	−3.53606e6	−1.18480	−0.592400	−	0.805644i	$-0.701819\pi$
$389$	−3.53606e6	−1.18480	−0.592400	+	0.805644i	$0.701819\pi$
$390$	0	0
$391$	−1.23678e6	−0.409120
$392$	0	0
$393$	− 1.06620e6i	− 0.348223i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 2.95611e6i	− 0.941336i	−0.882310	−	0.470668i	$-0.844013\pi$
$397$	− 2.95611e6i	− 0.941336i	0.882310	−	0.470668i	$-0.155987\pi$
$398$	0	0
$399$	398671.	0.125367
$400$	0	0
$401$	−799254.	−0.248213	−0.124106	−	0.992269i	$-0.539606\pi$
$401$	−799254.	−0.248213	−0.124106	+	0.992269i	$0.539606\pi$
$402$	0	0
$403$	− 3.42294e6i	− 1.04987i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 957937.i	− 0.286649i
$408$	0	0
$409$	−898422.	−0.265566	−0.132783	−	0.991145i	$-0.542391\pi$
$409$	−898422.	−0.265566	−0.132783	+	0.991145i	$0.542391\pi$
$410$	0	0
$411$	1.38292e6	0.403824
$412$	0	0
$413$	− 1.46134e6i	− 0.421576i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	1.20786e6i	0.340155i
$418$	0	0
$419$	2.31259e6	0.643523	0.321761	−	0.946821i	$-0.395725\pi$
$419$	2.31259e6	0.643523	0.321761	+	0.946821i	$0.395725\pi$
$420$	0	0
$421$	4.43296e6	1.21896	0.609478	−	0.792803i	$-0.291379\pi$
$421$	4.43296e6	1.21896	0.609478	+	0.792803i	$0.291379\pi$
$422$	0	0
$423$	− 190244.i	− 0.0516965i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1.91149e6i	0.507344i
$428$	0	0
$429$	1.15036e6	0.301780
$430$	0	0
$431$	−5.97999e6	−1.55063	−0.775314	−	0.631576i	$-0.782408\pi$
$431$	−5.97999e6	−1.55063	−0.775314	+	0.631576i	$0.782408\pi$
$432$	0	0
$433$	− 2.06419e6i	− 0.529089i	−0.964373	−	0.264545i	$-0.914778\pi$
$433$	− 2.06419e6i	− 0.529089i	0.964373	−	0.264545i	$-0.0852217\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 718899.i	− 0.180080i
$438$	0	0
$439$	−4.09148e6	−1.01326	−0.506628	−	0.862165i	$-0.669108\pi$
$439$	−4.09148e6	−1.01326	−0.506628	+	0.862165i	$0.669108\pi$
$440$	0	0
$441$	2.56099e6	0.627064
$442$	0	0
$443$	2.75822e6i	0.667759i	0.942616	+	0.333879i	$0.108358\pi$
$443$	2.75822e6i	0.667759i	−0.942616	+	0.333879i	$0.891642\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 214006.i	− 0.0506592i
$448$	0	0
$449$	3.76648e6	0.881698	0.440849	−	0.897581i	$-0.354677\pi$
$449$	3.76648e6	0.881698	0.440849	+	0.897581i	$0.354677\pi$
$450$	0	0
$451$	5.23640e6	1.21225
$452$	0	0
$453$	− 855944.i	− 0.195975i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	480604.i	0.107646i	0.998550	+	0.0538229i	$0.0171407\pi$
$457$	480604.i	0.107646i	−0.998550	+	0.0538229i	$0.982859\pi$
$458$	0	0
$459$	4.53073e6	1.00377
$460$	0	0
$461$	4.52514e6	0.991699	0.495849	−	0.868409i	$-0.334857\pi$
$461$	4.52514e6	0.991699	0.495849	+	0.868409i	$0.334857\pi$
$462$	0	0
$463$	7.39975e6i	1.60422i	0.597175	+	0.802111i	$0.296290\pi$
$463$	7.39975e6i	1.60422i	−0.597175	+	0.802111i	$0.703710\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 1.84711e6i	− 0.391923i	−0.980612	−	0.195962i	$-0.937217\pi$
$467$	− 1.84711e6i	− 0.391923i	0.980612	−	0.195962i	$-0.0627828\pi$
$468$	0	0
$469$	532261.	0.111736
$470$	0	0
$471$	−1.90276e6	−0.395212
$472$	0	0
$473$	7.30018e6i	1.50031i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	4.10669e6i	0.826410i
$478$	0	0
$479$	−3.05088e6	−0.607555	−0.303778	−	0.952743i	$-0.598248\pi$
$479$	−3.05088e6	−0.607555	−0.303778	+	0.952743i	$0.598248\pi$
$480$	0	0
$481$	844197.	0.166372
$482$	0	0
$483$	− 261624.i	− 0.0510281i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 7.28136e6i	− 1.39120i	−0.718429	−	0.695601i	$-0.755138\pi$
$487$	− 7.28136e6i	− 1.39120i	0.718429	−	0.695601i	$-0.244862\pi$
$488$	0	0
$489$	2.02297e6	0.382575
$490$	0	0
$491$	6.60475e6	1.23638	0.618191	−	0.786028i	$-0.287866\pi$
$491$	6.60475e6	1.23638	0.618191	+	0.786028i	$0.287866\pi$
$492$	0	0
$493$	2.41112e6i	0.446788i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 3.54481e6i	− 0.643727i
$498$	0	0
$499$	4.87006e6	0.875555	0.437777	−	0.899083i	$-0.355766\pi$
$499$	4.87006e6	0.875555	0.437777	+	0.899083i	$0.355766\pi$
$500$	0	0
$501$	−1.37702e6	−0.245102
$502$	0	0
$503$	− 1.16752e6i	− 0.205753i	−0.994694	−	0.102876i	$-0.967195\pi$
$503$	− 1.16752e6i	− 0.205753i	0.994694	−	0.102876i	$-0.0328046\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 1.03732e6i	− 0.179222i
$508$	0	0
$509$	7.41468e6	1.26852	0.634261	−	0.773119i	$-0.281305\pi$
$509$	7.41468e6	1.26852	0.634261	+	0.773119i	$0.281305\pi$
$510$	0	0
$511$	3.01846e6	0.511367
$512$	0	0
$513$	2.63356e6i	0.441824i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 435227.i	− 0.0716127i
$518$	0	0
$519$	339516.	0.0553277
$520$	0	0
$521$	−811897.	−0.131041	−0.0655204	−	0.997851i	$-0.520871\pi$
$521$	−811897.	−0.131041	−0.0655204	+	0.997851i	$0.520871\pi$
$522$	0	0
$523$	5.06828e6i	0.810226i	0.914267	+	0.405113i	$0.132768\pi$
$523$	5.06828e6i	0.810226i	−0.914267	+	0.405113i	$0.867232\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 1.43877e7i	− 2.25665i
$528$	0	0
$529$	5.96457e6	0.926702
$530$	0	0
$531$	4.50331e6	0.693099
$532$	0	0
$533$	4.61465e6i	0.703592i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 3.35209e6i	− 0.501626i
$538$	0	0
$539$	5.85886e6	0.868642
$540$	0	0
$541$	1.52830e6	0.224499	0.112250	−	0.993680i	$-0.464194\pi$
$541$	1.52830e6	0.224499	0.112250	+	0.993680i	$0.464194\pi$
$542$	0	0
$543$	− 848980.i	− 0.123566i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	1.23234e7i	1.76101i	0.474036	+	0.880506i	$0.342797\pi$
$547$	1.23234e7i	1.76101i	−0.474036	+	0.880506i	$0.657203\pi$
$548$	0	0
$549$	−5.89050e6	−0.834107
$550$	0	0
$551$	−1.40150e6	−0.196660
$552$	0	0
$553$	429271.i	0.0596923i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 4.08606e6i	− 0.558042i	−0.960285	−	0.279021i	$-0.909990\pi$
$557$	− 4.08606e6i	− 0.558042i	0.960285	−	0.279021i	$-0.0900099\pi$
$558$	0	0
$559$	−6.43339e6	−0.870784
$560$	0	0
$561$	4.83531e6	0.648661
$562$	0	0
$563$	24160.3i	0.00321241i	0.999999	+	0.00160621i	$0.000511272\pi$
$563$	24160.3i	0.00321241i	−0.999999	+	0.00160621i	$0.999489\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 2.60180e6i	− 0.339873i
$568$	0	0
$569$	−1.42000e7	−1.83869	−0.919344	−	0.393454i	$-0.871280\pi$
$569$	−1.42000e7	−1.83869	−0.919344	+	0.393454i	$0.871280\pi$
$570$	0	0
$571$	767642.	0.0985300	0.0492650	−	0.998786i	$-0.484312\pi$
$571$	767642.	0.0985300	0.0492650	+	0.998786i	$0.484312\pi$
$572$	0	0
$573$	1.00714e6i	0.128145i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 1.51488e6i	− 0.189426i	−0.995505	−	0.0947129i	$-0.969807\pi$
$577$	− 1.51488e6i	− 0.189426i	0.995505	−	0.0947129i	$-0.0301933\pi$
$578$	0	0
$579$	563910.	0.0699059
$580$	0	0
$581$	3.65098e6	0.448714
$582$	0	0
$583$	9.39498e6i	1.14479i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.28973e7i	1.54491i	0.635070	+	0.772455i	$0.280971\pi$
$587$	1.28973e7i	1.54491i	−0.635070	+	0.772455i	$0.719029\pi$
$588$	0	0
$589$	8.36307e6	0.993294
$590$	0	0
$591$	−2.23539e6	−0.263260
$592$	0	0
$593$	5.43125e6i	0.634254i	0.948383	+	0.317127i	$0.102718\pi$
$593$	5.43125e6i	0.634254i	−0.948383	+	0.317127i	$0.897282\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	924180.i	0.106126i
$598$	0	0
$599$	3.92217e6	0.446642	0.223321	−	0.974745i	$-0.428310\pi$
$599$	3.92217e6	0.446642	0.223321	+	0.974745i	$0.428310\pi$
$600$	0	0
$601$	−5.64824e6	−0.637863	−0.318931	−	0.947778i	$-0.603324\pi$
$601$	−5.64824e6	−0.637863	−0.318931	+	0.947778i	$0.603324\pi$
$602$	0	0
$603$	1.64023e6i	0.183701i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	1.07148e7i	1.18035i	0.807274	+	0.590177i	$0.200942\pi$
$607$	1.07148e7i	1.18035i	−0.807274	+	0.590177i	$0.799058\pi$
$608$	0	0
$609$	−510039.	−0.0557263
$610$	0	0
$611$	383551.	0.0415642
$612$	0	0
$613$	− 4.08748e6i	− 0.439344i	−0.975574	−	0.219672i	$-0.929501\pi$
$613$	− 4.08748e6i	− 0.439344i	0.975574	−	0.219672i	$-0.0704987\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 7.83395e6i	− 0.828453i	−0.910174	−	0.414227i	$-0.864052\pi$
$617$	− 7.83395e6i	− 0.828453i	0.910174	−	0.414227i	$-0.135948\pi$
$618$	0	0
$619$	1.23423e7	1.29470	0.647352	−	0.762191i	$-0.275876\pi$
$619$	1.23423e7	1.29470	0.647352	+	0.762191i	$0.275876\pi$
$620$	0	0
$621$	1.72824e6	0.179836
$622$	0	0
$623$	3.07739e6i	0.317660i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	2.81061e6i	0.285516i
$628$	0	0
$629$	3.54842e6	0.357609
$630$	0	0
$631$	1.31578e6	0.131556	0.0657780	−	0.997834i	$-0.479047\pi$
$631$	1.31578e6	0.131556	0.0657780	+	0.997834i	$0.479047\pi$
$632$	0	0
$633$	− 2.54542e6i	− 0.252494i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	5.16320e6i	0.504163i
$638$	0	0
$639$	1.09238e7	1.05833
$640$	0	0
$641$	6.55744e6	0.630360	0.315180	−	0.949032i	$-0.397935\pi$
$641$	6.55744e6	0.630360	0.315180	+	0.949032i	$0.397935\pi$
$642$	0	0
$643$	− 4.69954e6i	− 0.448258i	−0.974559	−	0.224129i	$-0.928046\pi$
$643$	− 4.69954e6i	− 0.448258i	0.974559	−	0.224129i	$-0.0719537\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 2.05827e7i	− 1.93305i	−0.256580	−	0.966523i	$-0.582596\pi$
$647$	− 2.05827e7i	− 1.93305i	0.256580	−	0.966523i	$-0.417404\pi$
$648$	0	0
$649$	1.03023e7	0.960117
$650$	0	0
$651$	3.04351e6	0.281464
$652$	0	0
$653$	1.42466e7i	1.30746i	0.756727	+	0.653731i	$0.226797\pi$
$653$	1.42466e7i	1.30746i	−0.756727	+	0.653731i	$0.773203\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	9.30177e6i	0.840722i
$658$	0	0
$659$	−1.35369e7	−1.21425	−0.607123	−	0.794608i	$-0.707676\pi$
$659$	−1.35369e7	−1.21425	−0.607123	+	0.794608i	$0.707676\pi$
$660$	0	0
$661$	−1.30443e7	−1.16122	−0.580612	−	0.814180i	$-0.697187\pi$
$661$	−1.30443e7	−1.16122	−0.580612	+	0.814180i	$0.697187\pi$
$662$	0	0
$663$	4.26119e6i	0.376485i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	919721.i	0.0800464i
$668$	0	0
$669$	−5.98260e6	−0.516802
$670$	0	0
$671$	−1.34759e7	−1.15545
$672$	0	0
$673$	− 4.75951e6i	− 0.405065i	−0.979276	−	0.202532i	$-0.935083\pi$
$673$	− 4.75951e6i	− 0.405065i	0.979276	−	0.202532i	$-0.0649171\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 1.51397e7i	− 1.26954i	−0.772701	−	0.634770i	$-0.781095\pi$
$677$	− 1.51397e7i	− 1.26954i	0.772701	−	0.634770i	$-0.218905\pi$
$678$	0	0
$679$	−1.02061e7	−0.849543
$680$	0	0
$681$	−2.27707e6	−0.188152
$682$	0	0
$683$	− 2.34145e7i	− 1.92058i	−0.278998	−	0.960292i	$-0.590002\pi$
$683$	− 2.34145e7i	− 1.92058i	0.278998	−	0.960292i	$-0.409998\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 2.39287e6i	− 0.193431i
$688$	0	0
$689$	−8.27947e6	−0.664438
$690$	0	0
$691$	1.62194e7	1.29223	0.646113	−	0.763242i	$-0.276394\pi$
$691$	1.62194e7	1.29223	0.646113	+	0.763242i	$0.276394\pi$
$692$	0	0
$693$	− 7.12196e6i	− 0.563334i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1.93968e7i	1.51234i
$698$	0	0
$699$	−4.19891e6	−0.325045
$700$	0	0
$701$	−1.89605e7	−1.45732	−0.728659	−	0.684876i	$-0.759856\pi$
$701$	−1.89605e7	−1.45732	−0.728659	+	0.684876i	$0.759856\pi$
$702$	0	0
$703$	2.06258e6i	0.157406i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 1.02377e7i	− 0.770287i
$708$	0	0
$709$	−128325.	−0.00958732	−0.00479366	−	0.999989i	$-0.501526\pi$
$709$	−128325.	−0.00958732	−0.00479366	+	0.999989i	$0.501526\pi$
$710$	0	0
$711$	−1.32285e6	−0.0981382
$712$	0	0
$713$	− 5.48817e6i	− 0.404300i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	5.46133e6i	0.396735i
$718$	0	0
$719$	−2.41874e7	−1.74489	−0.872444	−	0.488714i	$-0.837466\pi$
$719$	−2.41874e7	−1.74489	−0.872444	+	0.488714i	$0.837466\pi$
$720$	0	0
$721$	1.29899e7	0.930610
$722$	0	0
$723$	1.98251e6i	0.141049i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	513307.i	0.0360198i	0.999838	+	0.0180099i	$0.00573303\pi$
$727$	513307.i	0.0360198i	−0.999838	+	0.0180099i	$0.994267\pi$
$728$	0	0
$729$	4.64015e6	0.323380
$730$	0	0
$731$	−2.70416e7	−1.87171
$732$	0	0
$733$	− 1.64153e7i	− 1.12847i	−0.825615	−	0.564234i	$-0.809172\pi$
$733$	− 1.64153e7i	− 1.12847i	0.825615	−	0.564234i	$-0.190828\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.75240e6i	0.254472i
$738$	0	0
$739$	1.16112e7	0.782109	0.391054	−	0.920368i	$-0.372110\pi$
$739$	1.16112e7	0.782109	0.391054	+	0.920368i	$0.372110\pi$
$740$	0	0
$741$	−2.47689e6	−0.165715
$742$	0	0
$743$	− 5.72590e6i	− 0.380515i	−0.981734	−	0.190257i	$-0.939068\pi$
$743$	− 5.72590e6i	− 0.380515i	0.981734	−	0.190257i	$-0.0609323\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	1.12510e7i	0.737715i
$748$	0	0
$749$	−4.68097e6	−0.304882
$750$	0	0
$751$	−1.15324e7	−0.746137	−0.373069	−	0.927804i	$-0.621694\pi$
$751$	−1.15324e7	−0.746137	−0.373069	+	0.927804i	$0.621694\pi$
$752$	0	0
$753$	− 4.70134e6i	− 0.302158i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	8.63293e6i	0.547544i	0.961795	+	0.273772i	$0.0882713\pi$
$757$	8.63293e6i	0.547544i	−0.961795	+	0.273772i	$0.911729\pi$
$758$	0	0
$759$	1.84443e6	0.116214
$760$	0	0
$761$	−3.52622e6	−0.220723	−0.110361	−	0.993892i	$-0.535201\pi$
$761$	−3.52622e6	−0.220723	−0.110361	+	0.993892i	$0.535201\pi$
$762$	0	0
$763$	− 1.51206e7i	− 0.940280i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	9.07910e6i	0.557255i
$768$	0	0
$769$	−1.40471e7	−0.856585	−0.428293	−	0.903640i	$-0.640885\pi$
$769$	−1.40471e7	−0.856585	−0.428293	+	0.903640i	$0.640885\pi$
$770$	0	0
$771$	−421817.	−0.0255557
$772$	0	0
$773$	− 2.44760e7i	− 1.47330i	−0.676274	−	0.736651i	$-0.736406\pi$
$773$	− 2.44760e7i	− 1.47330i	0.676274	−	0.736651i	$-0.263594\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	750619.i	0.0446033i
$778$	0	0
$779$	−1.12747e7	−0.665675
$780$	0	0
$781$	2.49907e7	1.46606
$782$	0	0
$783$	− 3.36924e6i	− 0.196393i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	4.35977e6i	0.250915i	0.992099	+	0.125458i	$0.0400399\pi$
$787$	4.35977e6i	0.250915i	−0.992099	+	0.125458i	$0.959960\pi$
$788$	0	0
$789$	6.59697e6	0.377270
$790$	0	0
$791$	5.57019e6	0.316540
$792$	0	0
$793$	− 1.18758e7i	− 0.670626i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 1.06887e7i	− 0.596044i	−0.954559	−	0.298022i	$-0.903673\pi$
$797$	− 1.06887e7i	− 0.596044i	0.954559	−	0.298022i	$-0.0963269\pi$
$798$	0	0
$799$	1.61218e6	0.0893403
$800$	0	0
$801$	−9.48339e6	−0.522255
$802$	0	0
$803$	2.12799e7i	1.16461i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	5.68605e6i	0.307345i
$808$	0	0
$809$	−9.12014e6	−0.489926	−0.244963	−	0.969532i	$-0.578776\pi$
$809$	−9.12014e6	−0.489926	−0.244963	+	0.969532i	$0.578776\pi$
$810$	0	0
$811$	−5.22575e6	−0.278995	−0.139497	−	0.990222i	$-0.544549\pi$
$811$	−5.22575e6	−0.278995	−0.139497	+	0.990222i	$0.544549\pi$
$812$	0	0
$813$	1.17192e7i	0.621830i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 1.57184e7i	− 0.823857i
$818$	0	0
$819$	6.27633e6	0.326961
$820$	0	0
$821$	9.00437e6	0.466225	0.233112	−	0.972450i	$-0.425109\pi$
$821$	9.00437e6	0.466225	0.233112	+	0.972450i	$0.425109\pi$
$822$	0	0
$823$	− 2.78867e7i	− 1.43515i	−0.696482	−	0.717574i	$-0.745252\pi$
$823$	− 2.78867e7i	− 1.43515i	0.696482	−	0.717574i	$-0.254748\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	6.64309e6i	0.337758i	0.985637	+	0.168879i	$0.0540148\pi$
$827$	6.64309e6i	0.337758i	−0.985637	+	0.168879i	$0.945985\pi$
$828$	0	0
$829$	−2.17030e7	−1.09682	−0.548408	−	0.836211i	$-0.684766\pi$
$829$	−2.17030e7	−1.09682	−0.548408	+	0.836211i	$0.684766\pi$
$830$	0	0
$831$	1.02277e7	0.513780
$832$	0	0
$833$	2.17026e7i	1.08367i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	2.01049e7i	0.991949i
$838$	0	0
$839$	−1.01238e7	−0.496520	−0.248260	−	0.968693i	$-0.579859\pi$
$839$	−1.01238e7	−0.496520	−0.248260	+	0.968693i	$0.579859\pi$
$840$	0	0
$841$	−1.87181e7	−0.912584
$842$	0	0
$843$	− 500784.i	− 0.0242707i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 5.18837e6i	− 0.248498i
$848$	0	0
$849$	−5.13366e6	−0.244432
$850$	0	0
$851$	1.35354e6	0.0640691
$852$	0	0
$853$	− 1.46326e7i	− 0.688573i	−0.938865	−	0.344286i	$-0.888121\pi$
$853$	− 1.46326e7i	− 0.688573i	0.938865	−	0.344286i	$-0.111879\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 5.52218e6i	− 0.256838i	−0.991720	−	0.128419i	$-0.959010\pi$
$857$	− 5.52218e6i	− 0.256838i	0.991720	−	0.128419i	$-0.0409902\pi$
$858$	0	0
$859$	−3.02260e6	−0.139765	−0.0698824	−	0.997555i	$-0.522262\pi$
$859$	−3.02260e6	−0.139765	−0.0698824	+	0.997555i	$0.522262\pi$
$860$	0	0
$861$	−4.10313e6	−0.188628
$862$	0	0
$863$	− 3.06818e7i	− 1.40234i	−0.712992	−	0.701172i	$-0.752661\pi$
$863$	− 3.06818e7i	− 1.40234i	0.712992	−	0.701172i	$-0.247339\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	1.00676e7i	0.454860i
$868$	0	0
$869$	−3.02633e6	−0.135946
$870$	0	0
$871$	−3.30686e6	−0.147697
$872$	0	0
$873$	− 3.14514e7i	− 1.39671i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 5.17607e6i	− 0.227249i	−0.993524	−	0.113624i	$-0.963754\pi$
$877$	− 5.17607e6i	− 0.227249i	0.993524	−	0.113624i	$-0.0362460\pi$
$878$	0	0
$879$	−1.50662e7	−0.657707
$880$	0	0
$881$	−4.25937e7	−1.84887	−0.924433	−	0.381345i	$-0.875461\pi$
$881$	−4.25937e7	−1.84887	−0.924433	+	0.381345i	$0.875461\pi$
$882$	0	0
$883$	− 1.72076e7i	− 0.742709i	−0.928491	−	0.371354i	$-0.878893\pi$
$883$	− 1.72076e7i	− 0.742709i	0.928491	−	0.371354i	$-0.121107\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 2.53773e6i	− 0.108302i	−0.998533	−	0.0541510i	$-0.982755\pi$
$887$	− 2.53773e6i	− 0.108302i	0.998533	−	0.0541510i	$-0.0172452\pi$
$888$	0	0
$889$	−1.11220e7	−0.471984
$890$	0	0
$891$	1.83425e7	0.774043
$892$	0	0
$893$	937108.i	0.0393243i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	1.62543e6i	0.0674508i
$898$	0	0
$899$	−1.06993e7	−0.441525
$900$	0	0
$901$	−3.48012e7	−1.42818
$902$	0	0
$903$	− 5.72026e6i	− 0.233452i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 2.60899e7i	− 1.05306i	−0.850156	−	0.526531i	$-0.823492\pi$
$907$	− 2.60899e7i	− 1.05306i	0.850156	−	0.526531i	$-0.176508\pi$
$908$	0	0
$909$	3.15487e7	1.26640
$910$	0	0
$911$	−1.44818e7	−0.578130	−0.289065	−	0.957309i	$-0.593344\pi$
$911$	−1.44818e7	−0.578130	−0.289065	+	0.957309i	$0.593344\pi$
$912$	0	0
$913$	2.57392e7i	1.02192i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 1.33081e7i	− 0.522627i
$918$	0	0
$919$	4.61041e7	1.80074	0.900369	−	0.435127i	$-0.143296\pi$
$919$	4.61041e7	1.80074	0.900369	+	0.435127i	$0.143296\pi$
$920$	0	0
$921$	1.26861e7	0.492811
$922$	0	0
$923$	2.20234e7i	0.850903i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	4.00301e7i	1.52998i
$928$	0	0
$929$	1.81557e7	0.690197	0.345098	−	0.938567i	$-0.387846\pi$
$929$	1.81557e7	0.690197	0.345098	+	0.938567i	$0.387846\pi$
$930$	0	0
$931$	−1.26150e7	−0.476993
$932$	0	0
$933$	5.44047e6i	0.204612i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1.78946e7i	0.665844i	0.942954	+	0.332922i	$0.108035\pi$
$937$	1.78946e7i	0.665844i	−0.942954	+	0.332922i	$0.891965\pi$
$938$	0	0
$939$	−1.14157e7	−0.422512
$940$	0	0
$941$	−3.04463e6	−0.112088	−0.0560441	−	0.998428i	$-0.517849\pi$
$941$	−3.04463e6	−0.112088	−0.0560441	+	0.998428i	$0.517849\pi$
$942$	0	0
$943$	7.39891e6i	0.270950i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 3.17110e7i	− 1.14904i	−0.818491	−	0.574519i	$-0.805189\pi$
$947$	− 3.17110e7i	− 1.14904i	0.818491	−	0.574519i	$-0.194811\pi$
$948$	0	0
$949$	−1.87532e7	−0.675944
$950$	0	0
$951$	6.31686e6	0.226491
$952$	0	0
$953$	− 1.01913e7i	− 0.363494i	−0.983345	−	0.181747i	$-0.941825\pi$
$953$	− 1.01913e7i	− 0.363494i	0.983345	−	0.181747i	$-0.0581752\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 3.59574e6i	− 0.126914i
$958$	0	0
$959$	1.72613e7	0.606077
$960$	0	0
$961$	3.52157e7	1.23006
$962$	0	0
$963$	− 1.44250e7i	− 0.501246i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	3.21125e7i	1.10435i	0.833727	+	0.552177i	$0.186203\pi$
$967$	3.21125e7i	1.10435i	−0.833727	+	0.552177i	$0.813797\pi$
$968$	0	0
$969$	−1.04111e7	−0.356196
$970$	0	0
$971$	−2.29867e7	−0.782399	−0.391200	−	0.920306i	$-0.627940\pi$
$971$	−2.29867e7	−0.782399	−0.391200	+	0.920306i	$0.627940\pi$
$972$	0	0
$973$	1.50763e7i	0.510519i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 2.47331e7i	− 0.828978i	−0.910054	−	0.414489i	$-0.863960\pi$
$977$	− 2.47331e7i	− 0.828978i	0.910054	−	0.414489i	$-0.136040\pi$
$978$	0	0
$979$	−2.16954e7	−0.723455
$980$	0	0
$981$	4.65960e7	1.54588
$982$	0	0
$983$	5.57031e7i	1.83863i	0.393518	+	0.919317i	$0.371258\pi$
$983$	5.57031e7i	1.83863i	−0.393518	+	0.919317i	$0.628742\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	341035.i	0.0111431i
$988$	0	0
$989$	−1.03150e7	−0.335335
$990$	0	0
$991$	6.86029e6	0.221901	0.110950	−	0.993826i	$-0.464611\pi$
$991$	6.86029e6	0.221901	0.110950	+	0.993826i	$0.464611\pi$
$992$	0	0
$993$	1.13373e6i	0.0364868i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	6.03725e7i	1.92354i	0.273856	+	0.961771i	$0.411701\pi$
$997$	6.03725e7i	1.92354i	−0.273856	+	0.961771i	$0.588299\pi$
$998$	0	0
$999$	−4.95847e6	−0.157193

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.6.c.n.49.2		4
4.3	odd	2		25.6.b.b.24.4		4
5.2	odd	4		400.6.a.w.1.1		2
5.3	odd	4		400.6.a.o.1.2		2
5.4	even	2	inner	400.6.c.n.49.3		4
12.11	even	2		225.6.b.i.199.1		4
20.3	even	4		25.6.a.d.1.2	yes	2
20.7	even	4		25.6.a.b.1.1	✓	2
20.19	odd	2		25.6.b.b.24.1		4
60.23	odd	4		225.6.a.l.1.1		2
60.47	odd	4		225.6.a.s.1.2		2
60.59	even	2		225.6.b.i.199.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.6.a.b.1.1	✓	2	20.7	even	4
25.6.a.d.1.2	yes	2	20.3	even	4
25.6.b.b.24.1		4	20.19	odd	2
25.6.b.b.24.4		4	4.3	odd	2
225.6.a.l.1.1		2	60.23	odd	4
225.6.a.s.1.2		2	60.47	odd	4
225.6.b.i.199.1		4	12.11	even	2
225.6.b.i.199.4		4	60.59	even	2
400.6.a.o.1.2		2	5.3	odd	4
400.6.a.w.1.1		2	5.2	odd	4
400.6.c.n.49.2		4	1.1	even	1	trivial
400.6.c.n.49.3		4	5.4	even	2	inner

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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	400.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-5.52417i q^{3} -68.9517i q^{7} +212.483 q^{9} +O(q^{10})\)	\(q-5.52417i q^{3} -68.9517i q^{7} +212.483 q^{9} +486.104 q^{11} +428.387i q^{13} +1800.64i q^{17} -1046.65 q^{19} -380.901 q^{21} +686.855i q^{23} -2516.17i q^{27} +1339.03 q^{29} -7990.30 q^{31} -2685.33i q^{33} -1970.64i q^{37} +2366.48 q^{39} +10772.2 q^{41} +15017.7i q^{43} -895.337i q^{47} +12052.7 q^{49} +9947.07 q^{51} +19327.1i q^{53} +5781.90i q^{57} +21193.7 q^{59} -27722.2 q^{61} -14651.1i q^{63} +7719.33i q^{67} +3794.31 q^{69} +51410.1 q^{71} +43776.4i q^{73} -33517.7i q^{77} -6225.68 q^{79} +37733.7 q^{81} +52949.9i q^{83} -7397.05i q^{87} -44631.2 q^{89} +29538.0 q^{91} +44139.8i q^{93} -148018. i q^{97} +103289. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 392 q^{9}+O(q^{10}) \) 4 * q - 392 * q^9	\( 4 q - 392 q^{9} + 392 q^{11} - 6360 q^{19} + 5928 q^{21} + 7840 q^{29} + 2192 q^{31} + 8224 q^{39} + 55508 q^{41} + 23372 q^{49} + 35752 q^{51} + 23920 q^{59} - 48792 q^{61} - 36984 q^{69} + 174592 q^{71} + 130960 q^{79} + 92564 q^{81} + 145620 q^{89} + 41152 q^{91} + 443584 q^{99}+O(q^{100}) \) 4 * q - 392 * q^9 + 392 * q^11 - 6360 * q^19 + 5928 * q^21 + 7840 * q^29 + 2192 * q^31 + 8224 * q^39 + 55508 * q^41 + 23372 * q^49 + 35752 * q^51 + 23920 * q^59 - 48792 * q^61 - 36984 * q^69 + 174592 * q^71 + 130960 * q^79 + 92564 * q^81 + 145620 * q^89 + 41152 * q^91 + 443584 * q^99

Introduction

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