LMFDB - Embedded newform 225.6.b.i.199.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,6,Mod(199,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("225.199");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	225.b (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:	$36.0863594579$
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:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 5^{2} $
Twist minimal:	no (minimal twist has level 25)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.1
Root			$-8.26209i$ of defining polynomial
Character	$\chi$	$=$	225.199
Dual form			225.6.b.i.199.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-10.2621i q^{2} -73.3104 q^{4} +68.9517i q^{7} +423.931i q^{8} +O(q^{10})$	$q-10.2621i q^{2} -73.3104 q^{4} +68.9517i q^{7} +423.931i q^{8} +486.104 q^{11} +428.387i q^{13} +707.588 q^{14} +2004.49 q^{16} -1800.64i q^{17} +1046.65 q^{19} -4988.45i q^{22} +686.855i q^{23} +4396.14 q^{26} -5054.88i q^{28} -1339.03 q^{29} +7990.30 q^{31} -7004.41i q^{32} -18478.4 q^{34} -1970.64i q^{37} -10740.9i q^{38} -10772.2 q^{41} -15017.7i q^{43} -35636.5 q^{44} +7048.57 q^{46} -895.337i q^{47} +12052.7 q^{49} -31405.2i q^{52} -19327.1i q^{53} -29230.8 q^{56} +13741.3i q^{58} +21193.7 q^{59} -27722.2 q^{61} -81997.1i q^{62} -7736.31 q^{64} -7719.33i q^{67} +132006. i q^{68} +51410.1 q^{71} +43776.4i q^{73} -20222.9 q^{74} -76730.7 q^{76} +33517.7i q^{77} +6225.68 q^{79} +110545. i q^{82} +52949.9i q^{83} -154113. q^{86} +206075. i q^{88} +44631.2 q^{89} -29538.0 q^{91} -50353.6i q^{92} -9188.03 q^{94} -148018. i q^{97} -123686. i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 138 q^{4}+O(q^{10}) $ 4 * q - 138 * q^4	$ 4 q - 138 q^{4} + 392 q^{11} + 36 q^{14} + 2274 q^{16} + 6360 q^{19} + 9512 q^{26} - 7840 q^{29} - 2192 q^{31} - 40226 q^{34} - 55508 q^{41} - 73774 q^{44} + 4908 q^{46} + 23372 q^{49} - 108540 q^{56} + 23920 q^{59} - 48792 q^{61} - 87298 q^{64} + 174592 q^{71} - 82444 q^{74} - 135070 q^{76} - 130960 q^{79} - 497848 q^{86} - 145620 q^{89} - 41152 q^{91} + 243304 q^{94}+O(q^{100}) $ 4 * q - 138 * q^4 + 392 * q^11 + 36 * q^14 + 2274 * q^16 + 6360 * q^19 + 9512 * q^26 - 7840 * q^29 - 2192 * q^31 - 40226 * q^34 - 55508 * q^41 - 73774 * q^44 + 4908 * q^46 + 23372 * q^49 - 108540 * q^56 + 23920 * q^59 - 48792 * q^61 - 87298 * q^64 + 174592 * q^71 - 82444 * q^74 - 135070 * q^76 - 130960 * q^79 - 497848 * q^86 - 145620 * q^89 - 41152 * q^91 + 243304 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 10.2621i	− 1.81410i	−0.421025	−	0.907049i	$-0.638330\pi$
$2$	− 10.2621i	− 1.81410i	0.421025	−	0.907049i	$-0.361670\pi$
$3$	0	0
$3$	0	0
$4$	−73.3104	−2.29095
$5$	0	0
$5$	0	0
$6$	0	0
$7$	68.9517i	0.531863i	0.963992	+	0.265931i	$0.0856795\pi$
$7$	68.9517i	0.531863i	−0.963992	+	0.265931i	$0.914321\pi$
$8$	423.931i	2.34191i
$9$	0	0
$10$	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	$0.885666\pi$
$14$	707.588	0.964851
$15$	0	0
$16$	2004.49	1.95751
$17$	− 1800.64i	− 1.51114i	−0.655066	−	0.755571i	$-0.727359\pi$
$17$	− 1800.64i	− 1.51114i	0.655066	−	0.755571i	$-0.272641\pi$
$18$	0	0
$19$	1046.65	0.665149	0.332575	−	0.943077i	$-0.392083\pi$
$19$	1046.65	0.665149	0.332575	+	0.943077i	$0.392083\pi$
$20$	0	0
$21$	0	0
$22$	− 4988.45i	− 2.19740i
$23$	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
$26$	4396.14	1.27538
$27$	0	0
$28$	− 5054.88i	− 1.21847i
$29$	−1339.03	−0.295663	−0.147831	−	0.989013i	$-0.547229\pi$
$29$	−1339.03	−0.295663	−0.147831	+	0.989013i	$0.547229\pi$
$30$	0	0
$31$	7990.30	1.49334	0.746670	−	0.665195i	$-0.231651\pi$
$31$	7990.30	1.49334	0.746670	+	0.665195i	$0.231651\pi$
$32$	− 7004.41i	− 1.20920i
$33$	0	0
$34$	−18478.4	−2.74136
$35$	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	$-0.0377522\pi$
$38$	− 10740.9i	− 1.20665i
$39$	0	0
$40$	0	0
$41$	−10772.2	−1.00079	−0.500395	−	0.865797i	$-0.666812\pi$
$41$	−10772.2	−1.00079	−0.500395	+	0.865797i	$0.666812\pi$
$42$	0	0
$43$	− 15017.7i	− 1.23861i	−0.785152	−	0.619303i	$-0.787415\pi$
$43$	− 15017.7i	− 1.23861i	0.785152	−	0.619303i	$-0.212585\pi$
$44$	−35636.5	−2.77500
$45$	0	0
$46$	7048.57	0.491141
$47$	− 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
$51$	0	0
$52$	− 31405.2i	− 1.61062i
$53$	− 19327.1i	− 0.945098i	−0.881304	−	0.472549i	$-0.843334\pi$
$53$	− 19327.1i	− 0.945098i	0.881304	−	0.472549i	$-0.156666\pi$
$54$	0	0
$55$	0	0
$56$	−29230.8	−1.24558
$57$	0	0
$58$	13741.3i	0.536361i
$59$	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	$0.658258\pi$
$62$	− 81997.1i	− 2.70906i
$63$	0	0
$64$	−7736.31	−0.236093
$65$	0	0
$66$	0	0
$67$	− 7719.33i	− 0.210084i	−0.994468	−	0.105042i	$-0.966502\pi$
$67$	− 7719.33i	− 0.210084i	0.994468	−	0.105042i	$-0.0334977\pi$
$68$	132006.i	3.46195i
$69$	0	0
$70$	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	$0.840371\pi$
$74$	−20222.9	−0.429303
$75$	0	0
$76$	−76730.7	−1.52382
$77$	33517.7i	0.644240i
$78$	0	0
$79$	6225.68	0.112233	0.0561163	−	0.998424i	$-0.482128\pi$
$79$	6225.68	0.112233	0.0561163	+	0.998424i	$0.482128\pi$
$80$	0	0
$81$	0	0
$82$	110545.i	1.81553i
$83$	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
$86$	−154113.	−2.24695
$87$	0	0
$88$	206075.i	2.83673i
$89$	44631.2	0.597260	0.298630	−	0.954369i	$-0.403470\pi$
$89$	44631.2	0.597260	0.298630	+	0.954369i	$0.403470\pi$
$90$	0	0
$91$	−29538.0	−0.373919
$92$	− 50353.6i	− 0.620242i
$93$	0	0
$94$	−9188.03	−0.107251
$95$	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	$-0.294452\pi$
$98$	− 123686.i	− 1.30093i
$99$	0	0
$100$	0	0
$101$	−148476.	−1.44828	−0.724141	−	0.689652i	$-0.757763\pi$
$101$	−148476.	−1.44828	−0.724141	+	0.689652i	$0.757763\pi$
$102$	0	0
$103$	− 188391.i	− 1.74972i	−0.484378	−	0.874859i	$-0.660954\pi$
$103$	− 188391.i	− 1.74972i	0.484378	−	0.874859i	$-0.339046\pi$
$104$	−181607.	−1.64645
$105$	0	0
$106$	−198336.	−1.71450
$107$	− 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
$111$	0	0
$112$	138213.i	1.04112i
$113$	− 80783.9i	− 0.595153i	−0.954698	−	0.297577i	$-0.903822\pi$
$113$	− 80783.9i	− 0.595153i	0.954698	−	0.297577i	$-0.0961784\pi$
$114$	0	0
$115$	0	0
$116$	98165.1	0.677348
$117$	0	0
$118$	− 217492.i	− 1.43793i
$119$	124157.	0.803721
$120$	0	0
$121$	75246.5	0.467221
$122$	284487.i	1.73047i
$123$	0	0
$124$	−585772.	−3.42117
$125$	0	0
$126$	0	0
$127$	161301.i	0.887417i	0.896171	+	0.443708i	$0.146337\pi$
$127$	161301.i	0.887417i	−0.896171	+	0.443708i	$0.853663\pi$
$128$	− 144750.i	− 0.780899i
$129$	0	0
$130$	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
$134$	−79216.4	−0.381113
$135$	0	0
$136$	763349.	3.53896
$137$	− 250340.i	− 1.13954i	−0.821806	−	0.569768i	$-0.807033\pi$
$137$	− 250340.i	− 1.13954i	0.821806	−	0.569768i	$-0.192967\pi$
$138$	0	0
$139$	218650.	0.959871	0.479935	−	0.877304i	$-0.340660\pi$
$139$	218650.	0.959871	0.479935	+	0.877304i	$0.340660\pi$
$140$	0	0
$141$	0	0
$142$	− 527575.i	− 2.19565i
$143$	208241.i	0.851580i
$144$	0	0
$145$	0	0
$146$	449238.	1.74419
$147$	0	0
$148$	144469.i	0.542150i
$149$	−38740.0	−0.142953	−0.0714766	−	0.997442i	$-0.522771\pi$
$149$	−38740.0	−0.142953	−0.0714766	+	0.997442i	$0.522771\pi$
$150$	0	0
$151$	−154945.	−0.553013	−0.276507	−	0.961012i	$-0.589177\pi$
$151$	−154945.	−0.553013	−0.276507	+	0.961012i	$0.589177\pi$
$152$	443709.i	1.55772i
$153$	0	0
$154$	343962.	1.16871
$155$	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	$-0.188284\pi$
$158$	− 63888.5i	− 0.203601i
$159$	0	0
$160$	0	0
$161$	−47359.8	−0.143994
$162$	0	0
$163$	− 366203.i	− 1.07957i	−0.841801	−	0.539787i	$-0.818505\pi$
$163$	− 366203.i	− 1.07957i	0.841801	−	0.539787i	$-0.181495\pi$
$164$	789712.	2.29276
$165$	0	0
$166$	543376.	1.53049
$167$	− 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
$171$	0	0
$172$	1.10096e6i	2.83758i
$173$	− 61460.1i	− 0.156127i	−0.996948	−	0.0780635i	$-0.975126\pi$
$173$	− 61460.1i	− 0.156127i	0.996948	−	0.0780635i	$-0.0248737\pi$
$174$	0	0
$175$	0	0
$176$	974389.	2.37111
$177$	0	0
$178$	− 458009.i	− 1.08349i
$179$	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	$0.444220\pi$
$182$	303121.i	0.678325i
$183$	0	0
$184$	−291179.	−0.634039
$185$	0	0
$186$	0	0
$187$	− 875301.i	− 1.83043i
$188$	65637.6i	0.135443i
$189$	0	0
$190$	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	$0.968553\pi$
$194$	−1.51898e6	−2.89766
$195$	0	0
$196$	−883586.	−1.64289
$197$	404656.i	0.742882i	0.928456	+	0.371441i	$0.121136\pi$
$197$	404656.i	0.742882i	−0.928456	+	0.371441i	$0.878864\pi$
$198$	0	0
$199$	167297.	0.299472	0.149736	−	0.988726i	$-0.452158\pi$
$199$	167297.	0.299472	0.149736	+	0.988726i	$0.452158\pi$
$200$	0	0
$201$	0	0
$202$	1.52367e6i	2.62732i
$203$	− 92328.5i	− 0.157252i
$204$	0	0
$205$	0	0
$206$	−1.93329e6	−3.17416
$207$	0	0
$208$	858695.i	1.37620i
$209$	508783.	0.805688
$210$	0	0
$211$	−460778.	−0.712502	−0.356251	−	0.934390i	$-0.615945\pi$
$211$	−460778.	−0.712502	−0.356251	+	0.934390i	$0.615945\pi$
$212$	1.41688e6i	2.16517i
$213$	0	0
$214$	−696670.	−1.03990
$215$	0	0
$216$	0	0
$217$	550944.i	0.794252i
$218$	− 2.25040e6i	− 3.20714i
$219$	0	0
$220$	0	0
$221$	771372.	1.06239
$222$	0	0
$223$	1.08298e6i	1.45834i	0.684330	+	0.729172i	$0.260095\pi$
$223$	1.08298e6i	1.45834i	−0.684330	+	0.729172i	$0.739905\pi$
$224$	482966.	0.643126
$225$	0	0
$226$	−829012.	−1.07967
$227$	− 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
$231$	0	0
$232$	− 567658.i	− 0.692416i
$233$	760097.i	0.917232i	0.888635	+	0.458616i	$0.151655\pi$
$233$	760097.i	0.917232i	−0.888635	+	0.458616i	$0.848345\pi$
$234$	0	0
$235$	0	0
$236$	−1.55372e6	−1.81590
$237$	0	0
$238$	− 1.27411e6i	− 1.45803i
$239$	−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	$0.563773\pi$
$242$	− 772186.i	− 0.847585i
$243$	0	0
$244$	2.03232e6	2.18534
$245$	0	0
$246$	0	0
$247$	448373.i	0.467624i
$248$	3.38734e6i	3.49727i
$249$	0	0
$250$	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
$254$	1.65528e6	1.60986
$255$	0	0
$256$	−1.73300e6	−1.65272
$257$	76358.4i	0.0721147i	0.999350	+	0.0360574i	$0.0114799\pi$
$257$	76358.4i	0.0721147i	−0.999350	+	0.0360574i	$0.988520\pi$
$258$	0	0
$259$	135879.	0.125864
$260$	0	0
$261$	0	0
$262$	− 1.98064e6i	− 1.78260i
$263$	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
$266$	740600.	0.641770
$267$	0	0
$268$	565907.i	0.481292i
$269$	1.02930e6	0.867286	0.433643	−	0.901085i	$-0.357228\pi$
$269$	1.02930e6	0.867286	0.433643	+	0.901085i	$0.357228\pi$
$270$	0	0
$271$	2.12144e6	1.75472	0.877359	−	0.479834i	$-0.159303\pi$
$271$	2.12144e6	1.75472	0.877359	+	0.479834i	$0.159303\pi$
$272$	− 3.60937e6i	− 2.95807i
$273$	0	0
$274$	−2.56901e6	−2.06723
$275$	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	$0.741882\pi$
$278$	− 2.24381e6i	− 1.74130i
$279$	0	0
$280$	0	0
$281$	−90653.2	−0.0684884	−0.0342442	−	0.999413i	$-0.510902\pi$
$281$	−90653.2	−0.0684884	−0.0342442	+	0.999413i	$0.510902\pi$
$282$	0	0
$283$	929308.i	0.689753i	0.938648	+	0.344877i	$0.112079\pi$
$283$	929308.i	0.689753i	−0.938648	+	0.344877i	$0.887921\pi$
$284$	−3.76890e6	−2.77280
$285$	0	0
$286$	2.13698e6	1.54485
$287$	− 742759.i	− 0.532283i
$288$	0	0
$289$	−1.82246e6	−1.28355
$290$	0	0
$291$	0	0
$292$	− 3.20927e6i	− 2.20267i
$293$	2.72733e6i	1.85596i	0.372632	+	0.927979i	$0.378455\pi$
$293$	2.72733e6i	1.85596i	−0.372632	+	0.927979i	$0.621545\pi$
$294$	0	0
$295$	0	0
$296$	835417.	0.554209
$297$	0	0
$298$	397553.i	0.259331i
$299$	−294240.	−0.190337
$300$	0	0
$301$	1.03550e6	0.658768
$302$	1.59006e6i	1.00322i
$303$	0	0
$304$	2.09800e6	1.30203
$305$	0	0
$306$	0	0
$307$	− 2.29648e6i	− 1.39064i	−0.718698	−	0.695322i	$-0.755262\pi$
$307$	− 2.29648e6i	− 1.39064i	0.718698	−	0.695322i	$-0.244738\pi$
$308$	− 2.45720e6i	− 1.47592i
$309$	0	0
$310$	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	$-0.203298\pi$
$314$	−3.53469e6	−2.02315
$315$	0	0
$316$	−456407.	−0.257119
$317$	− 1.14349e6i	− 0.639125i	−0.947565	−	0.319563i	$-0.896464\pi$
$317$	− 1.14349e6i	− 0.639125i	0.947565	−	0.319563i	$-0.103536\pi$
$318$	0	0
$319$	−650910.	−0.358133
$320$	0	0
$321$	0	0
$322$	486010.i	0.261220i
$323$	− 1.88465e6i	− 1.00514i
$324$	0	0
$325$	0	0
$326$	−3.75801e6	−1.95845
$327$	0	0
$328$	− 4.56666e6i	− 2.34376i
$329$	61735.0	0.0314443
$330$	0	0
$331$	205230.	0.102961	0.0514804	−	0.998674i	$-0.483606\pi$
$331$	205230.	0.102961	0.0514804	+	0.998674i	$0.483606\pi$
$332$	− 3.88178e6i	− 1.93279i
$333$	0	0
$334$	−2.55805e6	−1.25471
$335$	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	$0.962645\pi$
$338$	− 1.92699e6i	− 0.917462i
$339$	0	0
$340$	0	0
$341$	3.88412e6	1.80887
$342$	0	0
$343$	1.98992e6i	0.913273i
$344$	6.36648e6	2.90070
$345$	0	0
$346$	−630709.	−0.283230
$347$	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
$351$	0	0
$352$	− 3.40487e6i	− 1.46469i
$353$	− 778492.i	− 0.332520i	−0.986082	−	0.166260i	$-0.946831\pi$
$353$	− 778492.i	− 0.332520i	0.986082	−	0.166260i	$-0.0531690\pi$
$354$	0	0
$355$	0	0
$356$	−3.27193e6	−1.36829
$357$	0	0
$358$	− 6.22707e6i	− 2.56789i
$359$	−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
$362$	− 1.57712e6i	− 0.632549i
$363$	0	0
$364$	2.16544e6	0.856630
$365$	0	0
$366$	0	0
$367$	4.10801e6i	1.59208i	0.605242	+	0.796042i	$0.293077\pi$
$367$	4.10801e6i	1.59208i	−0.605242	+	0.796042i	$0.706923\pi$
$368$	1.37679e6i	0.529967i
$369$	0	0
$370$	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	$0.679088\pi$
$374$	−8.98241e6	−3.32058
$375$	0	0
$376$	379561.	0.138456
$377$	− 573624.i	− 0.207861i
$378$	0	0
$379$	−1.40554e6	−0.502626	−0.251313	−	0.967906i	$-0.580862\pi$
$379$	−1.40554e6	−0.502626	−0.251313	+	0.967906i	$0.580862\pi$
$380$	0	0
$381$	0	0
$382$	1.87093e6i	0.655993i
$383$	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
$386$	1.04756e6	0.357857
$387$	0	0
$388$	1.08513e7i	3.65933i
$389$	3.53606e6	1.18480	0.592400	−	0.805644i	$-0.298181\pi$
$389$	3.53606e6	1.18480	0.592400	+	0.805644i	$0.298181\pi$
$390$	0	0
$391$	1.23678e6	0.409120
$392$	5.10950e6i	1.67944i
$393$	0	0
$394$	4.15261e6	1.34766
$395$	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	$-0.155987\pi$
$398$	− 1.71682e6i	− 0.543272i
$399$	0	0
$400$	0	0
$401$	799254.	0.248213	0.124106	−	0.992269i	$-0.460394\pi$
$401$	799254.	0.248213	0.124106	+	0.992269i	$0.460394\pi$
$402$	0	0
$403$	3.42294e6i	1.04987i
$404$	1.08848e7	3.31794
$405$	0	0
$406$	−947484.	−0.285270
$407$	− 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
$411$	0	0
$412$	1.38111e7i	4.00852i
$413$	1.46134e6i	0.421576i
$414$	0	0
$415$	0	0
$416$	3.00060e6	0.850108
$417$	0	0
$418$	− 5.22118e6i	− 1.46160i
$419$	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	$0.291379\pi$
$422$	4.72855e6i	1.29255i
$423$	0	0
$424$	8.19336e6	2.21334
$425$	0	0
$426$	0	0
$427$	− 1.91149e6i	− 0.507344i
$428$	4.97688e6i	1.31325i
$429$	0	0
$430$	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	$-0.0852217\pi$
$434$	5.65384e6	1.44085
$435$	0	0
$436$	−1.60764e7	−4.05017
$437$	718899.i	0.180080i
$438$	0	0
$439$	4.09148e6	1.01326	0.506628	−	0.862165i	$-0.330892\pi$
$439$	4.09148e6	1.01326	0.506628	+	0.862165i	$0.330892\pi$
$440$	0	0
$441$	0	0
$442$	− 7.91589e6i	− 1.92728i
$443$	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
$446$	1.11137e7	2.64558
$447$	0	0
$448$	− 533431.i	− 0.125569i
$449$	−3.76648e6	−0.881698	−0.440849	−	0.897581i	$-0.645323\pi$
$449$	−3.76648e6	−0.881698	−0.440849	+	0.897581i	$0.645323\pi$
$450$	0	0
$451$	−5.23640e6	−1.21225
$452$	5.92231e6i	1.36347i
$453$	0	0
$454$	−4.23004e6	−0.963174
$455$	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	$0.982859\pi$
$458$	− 4.44515e6i	− 0.990200i
$459$	0	0
$460$	0	0
$461$	−4.52514e6	−0.991699	−0.495849	−	0.868409i	$-0.665143\pi$
$461$	−4.52514e6	−0.991699	−0.495849	+	0.868409i	$0.665143\pi$
$462$	0	0
$463$	− 7.39975e6i	− 1.60422i	−0.597175	−	0.802111i	$-0.703710\pi$
$463$	− 7.39975e6i	− 1.60422i	0.597175	−	0.802111i	$-0.296290\pi$
$464$	−2.68407e6	−0.578761
$465$	0	0
$466$	7.80018e6	1.66395
$467$	− 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
$471$	0	0
$472$	8.98467e6i	1.85630i
$473$	− 7.30018e6i	− 1.50031i
$474$	0	0
$475$	0	0
$476$	−9.10203e6	−1.84128
$477$	0	0
$478$	1.01453e7i	2.03094i
$479$	−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
$482$	3.68284e6i	0.722047i
$483$	0	0
$484$	−5.51635e6	−1.07038
$485$	0	0
$486$	0	0
$487$	7.28136e6i	1.39120i	0.718429	+	0.695601i	$0.244862\pi$
$487$	7.28136e6i	1.39120i	−0.718429	+	0.695601i	$0.755138\pi$
$488$	− 1.17523e7i	− 2.23395i
$489$	0	0
$490$	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
$494$	4.60124e6	0.848316
$495$	0	0
$496$	1.60164e7	2.92322
$497$	3.54481e6i	0.643727i
$498$	0	0
$499$	−4.87006e6	−0.875555	−0.437777	−	0.899083i	$-0.644234\pi$
$499$	−4.87006e6	−0.875555	−0.437777	+	0.899083i	$0.644234\pi$
$500$	0	0
$501$	0	0
$502$	− 8.73354e6i	− 1.54679i
$503$	− 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
$506$	3.42634e6	0.594914
$507$	0	0
$508$	− 1.18250e7i	− 2.03303i
$509$	−7.41468e6	−1.26852	−0.634261	−	0.773119i	$-0.718695\pi$
$509$	−7.41468e6	−1.26852	−0.634261	+	0.773119i	$0.718695\pi$
$510$	0	0
$511$	−3.01846e6	−0.511367
$512$	1.31522e7i	2.21730i
$513$	0	0
$514$	783596.	0.130823
$515$	0	0
$516$	0	0
$517$	− 435227.i	− 0.0716127i
$518$	− 1.39440e6i	− 0.228330i
$519$	0	0
$520$	0	0
$521$	811897.	0.131041	0.0655204	−	0.997851i	$-0.479129\pi$
$521$	811897.	0.131041	0.0655204	+	0.997851i	$0.479129\pi$
$522$	0	0
$523$	− 5.06828e6i	− 0.810226i	−0.914267	−	0.405113i	$-0.867232\pi$
$523$	− 5.06828e6i	− 0.810226i	0.914267	−	0.405113i	$-0.132768\pi$
$524$	−1.41494e7	−2.25117
$525$	0	0
$526$	1.22550e7	1.93129
$527$	− 1.43877e7i	− 2.25665i
$528$	0	0
$529$	5.96457e6	0.926702
$530$	0	0
$531$	0	0
$532$	− 5.29071e6i	− 0.810465i
$533$	− 4.61465e6i	− 0.703592i
$534$	0	0
$535$	0	0
$536$	3.27247e6	0.491998
$537$	0	0
$538$	− 1.05628e7i	− 1.57334i
$539$	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	$0.464194\pi$
$542$	− 2.17704e7i	− 3.18323i
$543$	0	0
$544$	−1.26124e7	−1.82727
$545$	0	0
$546$	0	0
$547$	− 1.23234e7i	− 1.76101i	−0.474036	−	0.880506i	$-0.657203\pi$
$547$	− 1.23234e7i	− 1.76101i	0.474036	−	0.880506i	$-0.342797\pi$
$548$	1.83525e7i	2.61062i
$549$	0	0
$550$	0	0
$551$	−1.40150e6	−0.196660
$552$	0	0
$553$	429271.i	0.0596923i
$554$	1.89998e7	2.63011
$555$	0	0
$556$	−1.60293e7	−2.19902
$557$	4.08606e6i	0.558042i	0.960285	+	0.279021i	$0.0900099\pi$
$557$	4.08606e6i	0.558042i	−0.960285	+	0.279021i	$0.909990\pi$
$558$	0	0
$559$	6.43339e6	0.870784
$560$	0	0
$561$	0	0
$562$	930291.i	0.124245i
$563$	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
$566$	9.53664e6	1.25128
$567$	0	0
$568$	2.17943e7i	2.83448i
$569$	1.42000e7	1.83869	0.919344	−	0.393454i	$-0.128720\pi$
$569$	1.42000e7	1.83869	0.919344	+	0.393454i	$0.128720\pi$
$570$	0	0
$571$	−767642.	−0.0985300	−0.0492650	−	0.998786i	$-0.515688\pi$
$571$	−767642.	−0.0985300	−0.0492650	+	0.998786i	$0.515688\pi$
$572$	− 1.52662e7i	− 1.95093i
$573$	0	0
$574$	−7.62225e6	−0.965614
$575$	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	$-0.0301933\pi$
$578$	1.87023e7i	2.32849i
$579$	0	0
$580$	0	0
$581$	−3.65098e6	−0.448714
$582$	0	0
$583$	− 9.39498e6i	− 1.14479i
$584$	−1.85582e7	−2.25167
$585$	0	0
$586$	2.79881e7	3.36689
$587$	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
$591$	0	0
$592$	− 3.95012e6i	− 0.463240i
$593$	− 5.43125e6i	− 0.634254i	−0.948383	−	0.317127i	$-0.897282\pi$
$593$	− 5.43125e6i	− 0.634254i	0.948383	−	0.317127i	$-0.102718\pi$
$594$	0	0
$595$	0	0
$596$	2.84004e6	0.327499
$597$	0	0
$598$	3.01951e6i	0.345290i
$599$	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	$0.603324\pi$
$602$	− 1.06264e7i	− 1.19507i
$603$	0	0
$604$	1.13591e7	1.26693
$605$	0	0
$606$	0	0
$607$	− 1.07148e7i	− 1.18035i	−0.807274	−	0.590177i	$-0.799058\pi$
$607$	− 1.07148e7i	− 1.18035i	0.807274	−	0.590177i	$-0.200942\pi$
$608$	− 7.33119e6i	− 0.804296i
$609$	0	0
$610$	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	$-0.0704987\pi$
$614$	−2.35666e7	−2.52276
$615$	0	0
$616$	−1.42092e7	−1.50875
$617$	7.83395e6i	0.828453i	0.910174	+	0.414227i	$0.135948\pi$
$617$	7.83395e6i	0.828453i	−0.910174	+	0.414227i	$0.864052\pi$
$618$	0	0
$619$	−1.23423e7	−1.29470	−0.647352	−	0.762191i	$-0.724124\pi$
$619$	−1.23423e7	−1.29470	−0.647352	+	0.762191i	$0.724124\pi$
$620$	0	0
$621$	0	0
$622$	1.01066e7i	1.04744i
$623$	3.07739e6i	0.317660i
$624$	0	0
$625$	0	0
$626$	−2.12066e7	−2.16290
$627$	0	0
$628$	2.52512e7i	2.55495i
$629$	−3.54842e6	−0.357609
$630$	0	0
$631$	−1.31578e6	−0.131556	−0.0657780	−	0.997834i	$-0.520953\pi$
$631$	−1.31578e6	−0.131556	−0.0657780	+	0.997834i	$0.520953\pi$
$632$	2.63926e6i	0.262839i
$633$	0	0
$634$	−1.17346e7	−1.15944
$635$	0	0
$636$	0	0
$637$	5.16320e6i	0.504163i
$638$	6.67969e6i	0.649688i
$639$	0	0
$640$	0	0
$641$	−6.55744e6	−0.630360	−0.315180	−	0.949032i	$-0.602065\pi$
$641$	−6.55744e6	−0.630360	−0.315180	+	0.949032i	$0.602065\pi$
$642$	0	0
$643$	4.69954e6i	0.448258i	0.974559	+	0.224129i	$0.0719537\pi$
$643$	4.69954e6i	0.448258i	−0.974559	+	0.224129i	$0.928046\pi$
$644$	3.47197e6	0.329884
$645$	0	0
$646$	−1.93405e7	−1.82341
$647$	− 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
$651$	0	0
$652$	2.68465e7i	2.47325i
$653$	− 1.42466e7i	− 1.30746i	−0.756727	−	0.653731i	$-0.773203\pi$
$653$	− 1.42466e7i	− 1.30746i	0.756727	−	0.653731i	$-0.226797\pi$
$654$	0	0
$655$	0	0
$656$	−2.15927e7	−1.95905
$657$	0	0
$658$	− 633530.i	− 0.0570430i
$659$	−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	$0.697187\pi$
$662$	− 2.10609e6i	− 0.186781i
$663$	0	0
$664$	−2.24471e7	−1.97579
$665$	0	0
$666$	0	0
$667$	− 919721.i	− 0.0800464i
$668$	1.82743e7i	1.58452i
$669$	0	0
$670$	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	$-0.0649171\pi$
$674$	5.01008e6	0.424810
$675$	0	0
$676$	−1.37661e7	−1.15863
$677$	1.51397e7i	1.26954i	0.772701	+	0.634770i	$0.218905\pi$
$677$	1.51397e7i	1.26954i	−0.772701	+	0.634770i	$0.781095\pi$
$678$	0	0
$679$	1.02061e7	0.849543
$680$	0	0
$681$	0	0
$682$	− 3.98592e7i	− 3.28146i
$683$	− 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
$686$	2.04208e7	1.65677
$687$	0	0
$688$	− 3.01028e7i	− 2.42458i
$689$	8.27947e6	0.664438
$690$	0	0
$691$	−1.62194e7	−1.29223	−0.646113	−	0.763242i	$-0.723606\pi$
$691$	−1.62194e7	−1.29223	−0.646113	+	0.763242i	$0.723606\pi$
$692$	4.50567e6i	0.357679i
$693$	0	0
$694$	3.92841e7	3.09613
$695$	0	0
$696$	0	0
$697$	1.93968e7i	1.51234i
$698$	1.49289e7i	1.15981i
$699$	0	0
$700$	0	0
$701$	1.89605e7	1.45732	0.728659	−	0.684876i	$-0.240144\pi$
$701$	1.89605e7	1.45732	0.728659	+	0.684876i	$0.240144\pi$
$702$	0	0
$703$	− 2.06258e6i	− 0.157406i
$704$	−3.76065e6	−0.285977
$705$	0	0
$706$	−7.98895e6	−0.603223
$707$	− 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
$711$	0	0
$712$	1.89206e7i	1.39873i
$713$	5.48817e6i	0.404300i
$714$	0	0
$715$	0	0
$716$	−4.44850e7	−3.24288
$717$	0	0
$718$	2.18079e7i	1.57871i
$719$	−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
$722$	1.41680e7i	1.01150i
$723$	0	0
$724$	−1.12667e7	−0.798821
$725$	0	0
$726$	0	0
$727$	− 513307.i	− 0.0360198i	−0.999838	−	0.0180099i	$-0.994267\pi$
$727$	− 513307.i	− 0.0360198i	0.999838	−	0.0180099i	$-0.00573303\pi$
$728$	− 1.25221e7i	− 0.875685i
$729$	0	0
$730$	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	$-0.190828\pi$
$734$	4.21567e7	2.88820
$735$	0	0
$736$	4.81101e6	0.327372
$737$	− 3.75240e6i	− 0.254472i
$738$	0	0
$739$	−1.16112e7	−0.782109	−0.391054	−	0.920368i	$-0.627890\pi$
$739$	−1.16112e7	−0.782109	−0.391054	+	0.920368i	$0.627890\pi$
$740$	0	0
$741$	0	0
$742$	− 1.36756e7i	− 0.911879i
$743$	− 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
$746$	4.66483e7	3.06894
$747$	0	0
$748$	6.41687e7i	4.19343i
$749$	4.68097e6	0.304882
$750$	0	0
$751$	1.15324e7	0.746137	0.373069	−	0.927804i	$-0.378306\pi$
$751$	1.15324e7	0.746137	0.373069	+	0.927804i	$0.378306\pi$
$752$	− 1.79469e6i	− 0.115730i
$753$	0	0
$754$	−5.88658e6	−0.377081
$755$	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	$0.911729\pi$
$758$	1.44238e7i	0.911812i
$759$	0	0
$760$	0	0
$761$	3.52622e6	0.220723	0.110361	−	0.993892i	$-0.464799\pi$
$761$	3.52622e6	0.220723	0.110361	+	0.993892i	$0.464799\pi$
$762$	0	0
$763$	1.51206e7i	0.940280i
$764$	1.33656e7	0.828427
$765$	0	0
$766$	4.76588e7	2.93475
$767$	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
$771$	0	0
$772$	− 7.48356e6i	− 0.451924i
$773$	2.44760e7i	1.47330i	0.676274	+	0.736651i	$0.263594\pi$
$773$	2.44760e7i	1.47330i	−0.676274	+	0.736651i	$0.736406\pi$
$774$	0	0
$775$	0	0
$776$	6.27496e7	3.74073
$777$	0	0
$778$	− 3.62873e7i	− 2.14934i
$779$	−1.12747e7	−0.665675
$780$	0	0
$781$	2.49907e7	1.46606
$782$	− 1.26920e7i	− 0.742184i
$783$	0	0
$784$	2.41594e7	1.40377
$785$	0	0
$786$	0	0
$787$	− 4.35977e6i	− 0.250915i	−0.992099	−	0.125458i	$-0.959960\pi$
$787$	− 4.35977e6i	− 0.250915i	0.992099	−	0.125458i	$-0.0400399\pi$
$788$	− 2.96655e7i	− 1.70191i
$789$	0	0
$790$	0	0
$791$	5.57019e6	0.316540
$792$	0	0
$793$	− 1.18758e7i	− 0.670626i
$794$	−3.03359e7	−1.70768
$795$	0	0
$796$	−1.22647e7	−0.686076
$797$	1.06887e7i	0.596044i	0.954559	+	0.298022i	$0.0963269\pi$
$797$	1.06887e7i	0.596044i	−0.954559	+	0.298022i	$0.903673\pi$
$798$	0	0
$799$	−1.61218e6	−0.0893403
$800$	0	0
$801$	0	0
$802$	− 8.20202e6i	− 0.450282i
$803$	2.12799e7i	1.16461i
$804$	0	0
$805$	0	0
$806$	3.51265e7	1.90457
$807$	0	0
$808$	− 6.29436e7i	− 3.39175i
$809$	9.12014e6	0.489926	0.244963	−	0.969532i	$-0.421224\pi$
$809$	9.12014e6	0.489926	0.244963	+	0.969532i	$0.421224\pi$
$810$	0	0
$811$	5.22575e6	0.278995	0.139497	−	0.990222i	$-0.455451\pi$
$811$	5.22575e6	0.278995	0.139497	+	0.990222i	$0.455451\pi$
$812$	6.76865e6i	0.360256i
$813$	0	0
$814$	−9.83044e6	−0.520010
$815$	0	0
$816$	0	0
$817$	− 1.57184e7i	− 0.823857i
$818$	9.21968e6i	0.481762i
$819$	0	0
$820$	0	0
$821$	−9.00437e6	−0.466225	−0.233112	−	0.972450i	$-0.574891\pi$
$821$	−9.00437e6	−0.466225	−0.233112	+	0.972450i	$0.574891\pi$
$822$	0	0
$823$	2.78867e7i	1.43515i	0.696482	+	0.717574i	$0.254748\pi$
$823$	2.78867e7i	1.43515i	−0.696482	+	0.717574i	$0.745252\pi$
$824$	7.98650e7	4.09768
$825$	0	0
$826$	1.49964e7	0.764781
$827$	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
$831$	0	0
$832$	− 3.31413e6i	− 0.165982i
$833$	− 2.17026e7i	− 1.08367i
$834$	0	0
$835$	0	0
$836$	−3.72991e7	−1.84579
$837$	0	0
$838$	− 2.37320e7i	− 1.16741i
$839$	−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
$842$	− 4.54914e7i	− 2.21131i
$843$	0	0
$844$	3.37799e7	1.63231
$845$	0	0
$846$	0	0
$847$	5.18837e6i	0.248498i
$848$	− 3.87409e7i	− 1.85003i
$849$	0	0
$850$	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	$-0.111879\pi$
$854$	−1.96159e7	−0.920371
$855$	0	0
$856$	2.87797e7	1.34246
$857$	5.52218e6i	0.256838i	0.991720	+	0.128419i	$0.0409902\pi$
$857$	5.52218e6i	0.256838i	−0.991720	+	0.128419i	$0.959010\pi$
$858$	0	0
$859$	3.02260e6	0.139765	0.0698824	−	0.997555i	$-0.477738\pi$
$859$	3.02260e6	0.139765	0.0698824	+	0.997555i	$0.477738\pi$
$860$	0	0
$861$	0	0
$862$	6.13672e7i	2.81299i
$863$	− 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
$866$	−2.11829e7	−0.959820
$867$	0	0
$868$	− 4.03900e7i	− 1.81959i
$869$	3.02633e6	0.135946
$870$	0	0
$871$	3.30686e6	0.147697
$872$	9.29649e7i	4.14026i
$873$	0	0
$874$	7.37741e6	0.326682
$875$	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	$-0.0362460\pi$
$878$	− 4.19871e7i	− 1.83815i
$879$	0	0
$880$	0	0
$881$	4.25937e7	1.84887	0.924433	−	0.381345i	$-0.124539\pi$
$881$	4.25937e7	1.84887	0.924433	+	0.381345i	$0.124539\pi$
$882$	0	0
$883$	1.72076e7i	0.742709i	0.928491	+	0.371354i	$0.121107\pi$
$883$	1.72076e7i	0.742709i	−0.928491	+	0.371354i	$0.878893\pi$
$884$	−5.65496e7	−2.43388
$885$	0	0
$886$	2.83051e7	1.21138
$887$	− 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
$891$	0	0
$892$	− 7.93941e7i	− 3.34100i
$893$	− 937108.i	− 0.0393243i
$894$	0	0
$895$	0	0
$896$	9.98078e6	0.415331
$897$	0	0
$898$	3.86519e7i	1.59949i
$899$	−1.06993e7	−0.441525
$900$	0	0
$901$	−3.48012e7	−1.42818
$902$	5.37364e7i	2.19913i
$903$	0	0
$904$	3.42468e7	1.39380
$905$	0	0
$906$	0	0
$907$	2.60899e7i	1.05306i	0.850156	+	0.526531i	$0.176508\pi$
$907$	2.60899e7i	1.05306i	−0.850156	+	0.526531i	$0.823492\pi$
$908$	3.02186e7i	1.21635i
$909$	0	0
$910$	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
$914$	4.93200e6	0.195280
$915$	0	0
$916$	−3.17553e7	−1.25048
$917$	1.33081e7i	0.522627i
$918$	0	0
$919$	−4.61041e7	−1.80074	−0.900369	−	0.435127i	$-0.856704\pi$
$919$	−4.61041e7	−1.80074	−0.900369	+	0.435127i	$0.856704\pi$
$920$	0	0
$921$	0	0
$922$	4.64374e7i	1.79904i
$923$	2.20234e7i	0.850903i
$924$	0	0
$925$	0	0
$926$	−7.59369e7	−2.91022
$927$	0	0
$928$	9.37914e6i	0.357514i
$929$	−1.81557e7	−0.690197	−0.345098	−	0.938567i	$-0.612154\pi$
$929$	−1.81557e7	−0.690197	−0.345098	+	0.938567i	$0.612154\pi$
$930$	0	0
$931$	1.26150e7	0.476993
$932$	− 5.57230e7i	− 2.10133i
$933$	0	0
$934$	−1.89552e7	−0.710987
$935$	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	$0.891965\pi$
$938$	− 5.46210e6i	− 0.202700i
$939$	0	0
$940$	0	0
$941$	3.04463e6	0.112088	0.0560441	−	0.998428i	$-0.482151\pi$
$941$	3.04463e6	0.112088	0.0560441	+	0.998428i	$0.482151\pi$
$942$	0	0
$943$	− 7.39891e6i	− 0.270950i
$944$	4.24825e7	1.55160
$945$	0	0
$946$	−7.49151e7	−2.72171
$947$	− 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
$951$	0	0
$952$	5.26342e7i	1.88224i
$953$	1.01913e7i	0.363494i	0.983345	+	0.181747i	$0.0581752\pi$
$953$	1.01913e7i	0.363494i	−0.983345	+	0.181747i	$0.941825\pi$
$954$	0	0
$955$	0	0
$956$	7.24764e7	2.56479
$957$	0	0
$958$	3.13084e7i	1.10216i
$959$	1.72613e7	0.606077
$960$	0	0
$961$	3.52157e7	1.23006
$962$	− 8.66322e6i	− 0.301816i
$963$	0	0
$964$	2.63095e7	0.911844
$965$	0	0
$966$	0	0
$967$	− 3.21125e7i	− 1.10435i	−0.833727	−	0.552177i	$-0.813797\pi$
$967$	− 3.21125e7i	− 1.10435i	0.833727	−	0.552177i	$-0.186203\pi$
$968$	3.18993e7i	1.09419i
$969$	0	0
$970$	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
$974$	7.47219e7	2.52378
$975$	0	0
$976$	−5.55687e7	−1.86726
$977$	2.47331e7i	0.828978i	0.910054	+	0.414489i	$0.136040\pi$
$977$	2.47331e7i	0.828978i	−0.910054	+	0.414489i	$0.863960\pi$
$978$	0	0
$979$	2.16954e7	0.723455
$980$	0	0
$981$	0	0
$982$	− 6.77785e7i	− 2.24292i
$983$	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
$986$	2.47431e7	0.810518
$987$	0	0
$988$	− 3.28704e7i	− 1.07130i
$989$	1.03150e7	0.335335
$990$	0	0
$991$	−6.86029e6	−0.221901	−0.110950	−	0.993826i	$-0.535389\pi$
$991$	−6.86029e6	−0.221901	−0.110950	+	0.993826i	$0.535389\pi$
$992$	− 5.59673e7i	− 1.80574i
$993$	0	0
$994$	3.63772e7	1.16778
$995$	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	$0.588299\pi$
$998$	4.99770e7i	1.58834i
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.6.a.b.1.1	✓	2	15.2	even	4
25.6.a.d.1.2	yes	2	15.8	even	4
25.6.b.b.24.1		4	15.14	odd	2
25.6.b.b.24.4		4	3.2	odd	2
225.6.a.l.1.1		2	5.3	odd	4
225.6.a.s.1.2		2	5.2	odd	4
225.6.b.i.199.1		4	1.1	even	1	trivial
225.6.b.i.199.4		4	5.4	even	2	inner
400.6.a.o.1.2		2	60.23	odd	4
400.6.a.w.1.1		2	60.47	odd	4
400.6.c.n.49.2		4	12.11	even	2
400.6.c.n.49.3		4	60.59	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	225.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-10.2621i q^{2} -73.3104 q^{4} +68.9517i q^{7} +423.931i q^{8} +O(q^{10})\)	\(q-10.2621i q^{2} -73.3104 q^{4} +68.9517i q^{7} +423.931i q^{8} +486.104 q^{11} +428.387i q^{13} +707.588 q^{14} +2004.49 q^{16} -1800.64i q^{17} +1046.65 q^{19} -4988.45i q^{22} +686.855i q^{23} +4396.14 q^{26} -5054.88i q^{28} -1339.03 q^{29} +7990.30 q^{31} -7004.41i q^{32} -18478.4 q^{34} -1970.64i q^{37} -10740.9i q^{38} -10772.2 q^{41} -15017.7i q^{43} -35636.5 q^{44} +7048.57 q^{46} -895.337i q^{47} +12052.7 q^{49} -31405.2i q^{52} -19327.1i q^{53} -29230.8 q^{56} +13741.3i q^{58} +21193.7 q^{59} -27722.2 q^{61} -81997.1i q^{62} -7736.31 q^{64} -7719.33i q^{67} +132006. i q^{68} +51410.1 q^{71} +43776.4i q^{73} -20222.9 q^{74} -76730.7 q^{76} +33517.7i q^{77} +6225.68 q^{79} +110545. i q^{82} +52949.9i q^{83} -154113. q^{86} +206075. i q^{88} +44631.2 q^{89} -29538.0 q^{91} -50353.6i q^{92} -9188.03 q^{94} -148018. i q^{97} -123686. i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 138 q^{4}+O(q^{10}) \) 4 * q - 138 * q^4	\( 4 q - 138 q^{4} + 392 q^{11} + 36 q^{14} + 2274 q^{16} + 6360 q^{19} + 9512 q^{26} - 7840 q^{29} - 2192 q^{31} - 40226 q^{34} - 55508 q^{41} - 73774 q^{44} + 4908 q^{46} + 23372 q^{49} - 108540 q^{56} + 23920 q^{59} - 48792 q^{61} - 87298 q^{64} + 174592 q^{71} - 82444 q^{74} - 135070 q^{76} - 130960 q^{79} - 497848 q^{86} - 145620 q^{89} - 41152 q^{91} + 243304 q^{94}+O(q^{100}) \) 4 * q - 138 * q^4 + 392 * q^11 + 36 * q^14 + 2274 * q^16 + 6360 * q^19 + 9512 * q^26 - 7840 * q^29 - 2192 * q^31 - 40226 * q^34 - 55508 * q^41 - 73774 * q^44 + 4908 * q^46 + 23372 * q^49 - 108540 * q^56 + 23920 * q^59 - 48792 * q^61 - 87298 * q^64 + 174592 * q^71 - 82444 * q^74 - 135070 * q^76 - 130960 * q^79 - 497848 * q^86 - 145620 * q^89 - 41152 * q^91 + 243304 * q^94

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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