LMFDB - Embedded newform 225.5.c.e.26.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,5,Mod(26,225)]

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([1, 0]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("225.26");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	225.c (of order $2$, degree $1$, 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:	$23.2582416939$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 90x^{6} + 2057x^{4} + 11880x^{2} + 256 $ x^8 + 90x^6 + 2057x^4 + 11880*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{10} $
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			26.6
Root			$2.97550i$ of defining polynomial
Character	$\chi$	$=$	225.26
Dual form			225.5.c.e.26.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.56128i q^{2} +13.5624 q^{4} +82.9374 q^{7} +46.1553i q^{8} +O(q^{10})$	$q+1.56128i q^{2} +13.5624 q^{4} +82.9374 q^{7} +46.1553i q^{8} +142.746i q^{11} -106.259 q^{13} +129.489i q^{14} +144.937 q^{16} -277.420i q^{17} +262.250 q^{19} -222.867 q^{22} -414.325i q^{23} -165.900i q^{26} +1124.83 q^{28} +1143.83i q^{29} -26.3726 q^{31} +964.772i q^{32} +433.132 q^{34} +1363.29 q^{37} +409.446i q^{38} +678.999i q^{41} -2104.48 q^{43} +1935.98i q^{44} +646.879 q^{46} -2368.85i q^{47} +4477.60 q^{49} -1441.13 q^{52} +1644.51i q^{53} +3828.00i q^{56} -1785.85 q^{58} +5753.62i q^{59} +2163.24 q^{61} -41.1752i q^{62} +812.704 q^{64} -2280.85 q^{67} -3762.48i q^{68} +975.793i q^{71} +8648.81 q^{73} +2128.49i q^{74} +3556.73 q^{76} +11839.0i q^{77} -1708.36 q^{79} -1060.11 q^{82} +1251.37i q^{83} -3285.69i q^{86} -6588.49 q^{88} -6067.49i q^{89} -8812.84 q^{91} -5619.23i q^{92} +3698.45 q^{94} -619.192 q^{97} +6990.81i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 36 q^{4}+O(q^{10}) $ 8 * q - 36 * q^4	$ 8 q - 36 q^{4} + 148 q^{16} + 1520 q^{19} + 2968 q^{31} + 12424 q^{34} + 10088 q^{46} + 8944 q^{49} + 544 q^{61} + 29188 q^{64} + 3600 q^{76} + 12632 q^{79} - 24552 q^{91} - 40928 q^{94}+O(q^{100}) $ 8 * q - 36 * q^4 + 148 * q^16 + 1520 * q^19 + 2968 * q^31 + 12424 * q^34 + 10088 * q^46 + 8944 * q^49 + 544 * q^61 + 29188 * q^64 + 3600 * q^76 + 12632 * q^79 - 24552 * q^91 - 40928 * 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 $

$2$	1.56128i	0.390321i	0.980771	+	0.195161i	$0.0625228\pi$
$2$	1.56128i	0.390321i	−0.980771	+	0.195161i	$0.937477\pi$
$3$	0	0
$3$	0	0
$4$	13.5624	0.847649
$5$	0	0
$5$	0	0
$6$	0	0
$7$	82.9374	1.69260	0.846300	−	0.532707i	$-0.178825\pi$
$7$	82.9374	1.69260	0.846300	+	0.532707i	$0.178825\pi$
$8$	46.1553i	0.721176i
$9$	0	0
$10$	0	0
$11$	142.746i	1.17972i	0.807506	+	0.589860i	$0.200817\pi$
$11$	142.746i	1.17972i	−0.807506	+	0.589860i	$0.799183\pi$
$12$	0	0
$13$	−106.259	−0.628751	−0.314376	−	0.949299i	$-0.601795\pi$
$13$	−106.259	−0.628751	−0.314376	+	0.949299i	$0.601795\pi$
$14$	129.489i	0.660657i
$15$	0	0
$16$	144.937	0.566159
$17$	− 277.420i	− 0.959931i	−0.877287	−	0.479966i	$-0.840649\pi$
$17$	− 277.420i	− 0.959931i	0.877287	−	0.479966i	$-0.159351\pi$
$18$	0	0
$19$	262.250	0.726453	0.363227	−	0.931701i	$-0.381675\pi$
$19$	262.250	0.726453	0.363227	+	0.931701i	$0.381675\pi$
$20$	0	0
$21$	0	0
$22$	−222.867	−0.460469
$23$	− 414.325i	− 0.783223i	−0.920131	−	0.391611i	$-0.871918\pi$
$23$	− 414.325i	− 0.783223i	0.920131	−	0.391611i	$-0.128082\pi$
$24$	0	0
$25$	0	0
$26$	− 165.900i	− 0.245415i
$27$	0	0
$28$	1124.83	1.43473
$29$	1143.83i	1.36009i	0.733172	+	0.680043i	$0.238039\pi$
$29$	1143.83i	1.36009i	−0.733172	+	0.680043i	$0.761961\pi$
$30$	0	0
$31$	−26.3726	−0.0274429	−0.0137214	−	0.999906i	$-0.504368\pi$
$31$	−26.3726	−0.0274429	−0.0137214	+	0.999906i	$0.504368\pi$
$32$	964.772i	0.942160i
$33$	0	0
$34$	433.132	0.374681
$35$	0	0
$36$	0	0
$37$	1363.29	0.995830	0.497915	−	0.867226i	$-0.334099\pi$
$37$	1363.29	0.995830	0.497915	+	0.867226i	$0.334099\pi$
$38$	409.446i	0.283550i
$39$	0	0
$40$	0	0
$41$	678.999i	0.403926i	0.979393	+	0.201963i	$0.0647320\pi$
$41$	678.999i	0.403926i	−0.979393	+	0.201963i	$0.935268\pi$
$42$	0	0
$43$	−2104.48	−1.13817	−0.569086	−	0.822278i	$-0.692703\pi$
$43$	−2104.48	−1.13817	−0.569086	+	0.822278i	$0.692703\pi$
$44$	1935.98i	0.999989i
$45$	0	0
$46$	646.879	0.305708
$47$	− 2368.85i	− 1.07236i	−0.844103	−	0.536181i	$-0.819866\pi$
$47$	− 2368.85i	− 1.07236i	0.844103	−	0.536181i	$-0.180134\pi$
$48$	0	0
$49$	4477.60	1.86489
$50$	0	0
$51$	0	0
$52$	−1441.13	−0.532961
$53$	1644.51i	0.585444i	0.956198	+	0.292722i	$0.0945611\pi$
$53$	1644.51i	0.585444i	−0.956198	+	0.292722i	$0.905439\pi$
$54$	0	0
$55$	0	0
$56$	3828.00i	1.22066i
$57$	0	0
$58$	−1785.85	−0.530870
$59$	5753.62i	1.65286i	0.563036	+	0.826432i	$0.309633\pi$
$59$	5753.62i	1.65286i	−0.563036	+	0.826432i	$0.690367\pi$
$60$	0	0
$61$	2163.24	0.581359	0.290680	−	0.956820i	$-0.406119\pi$
$61$	2163.24	0.581359	0.290680	+	0.956820i	$0.406119\pi$
$62$	− 41.1752i	− 0.0107115i
$63$	0	0
$64$	812.704	0.198414
$65$	0	0
$66$	0	0
$67$	−2280.85	−0.508097	−0.254049	−	0.967191i	$-0.581762\pi$
$67$	−2280.85	−0.508097	−0.254049	+	0.967191i	$0.581762\pi$
$68$	− 3762.48i	− 0.813685i
$69$	0	0
$70$	0	0
$71$	975.793i	0.193571i	0.995305	+	0.0967856i	$0.0308561\pi$
$71$	975.793i	0.193571i	−0.995305	+	0.0967856i	$0.969144\pi$
$72$	0	0
$73$	8648.81	1.62297	0.811485	−	0.584373i	$-0.198660\pi$
$73$	8648.81	1.62297	0.811485	+	0.584373i	$0.198660\pi$
$74$	2128.49i	0.388694i
$75$	0	0
$76$	3556.73	0.615778
$77$	11839.0i	1.99679i
$78$	0	0
$79$	−1708.36	−0.273731	−0.136866	−	0.990590i	$-0.543703\pi$
$79$	−1708.36	−0.273731	−0.136866	+	0.990590i	$0.543703\pi$
$80$	0	0
$81$	0	0
$82$	−1060.11	−0.157661
$83$	1251.37i	0.181647i	0.995867	+	0.0908234i	$0.0289499\pi$
$83$	1251.37i	0.181647i	−0.995867	+	0.0908234i	$0.971050\pi$
$84$	0	0
$85$	0	0
$86$	− 3285.69i	− 0.444253i
$87$	0	0
$88$	−6588.49	−0.850786
$89$	− 6067.49i	− 0.766001i	−0.923748	−	0.383000i	$-0.874891\pi$
$89$	− 6067.49i	− 0.766001i	0.923748	−	0.383000i	$-0.125109\pi$
$90$	0	0
$91$	−8812.84	−1.06422
$92$	− 5619.23i	− 0.663898i
$93$	0	0
$94$	3698.45	0.418566
$95$	0	0
$96$	0	0
$97$	−619.192	−0.0658085	−0.0329042	−	0.999459i	$-0.510476\pi$
$97$	−619.192	−0.0658085	−0.0329042	+	0.999459i	$0.510476\pi$
$98$	6990.81i	0.727906i
$99$	0	0
$100$	0	0
$101$	− 13484.6i	− 1.32189i	−0.750435	−	0.660944i	$-0.770156\pi$
$101$	− 13484.6i	− 1.32189i	0.750435	−	0.660944i	$-0.229844\pi$
$102$	0	0
$103$	−2373.71	−0.223744	−0.111872	−	0.993723i	$-0.535685\pi$
$103$	−2373.71	−0.223744	−0.111872	+	0.993723i	$0.535685\pi$
$104$	− 4904.42i	− 0.453441i
$105$	0	0
$106$	−2567.55	−0.228511
$107$	− 21130.0i	− 1.84557i	−0.385311	−	0.922787i	$-0.625906\pi$
$107$	− 21130.0i	− 1.84557i	0.385311	−	0.922787i	$-0.374094\pi$
$108$	0	0
$109$	−18595.3	−1.56513	−0.782565	−	0.622569i	$-0.786089\pi$
$109$	−18595.3	−1.56513	−0.782565	+	0.622569i	$0.786089\pi$
$110$	0	0
$111$	0	0
$112$	12020.7	0.958280
$113$	− 13972.2i	− 1.09423i	−0.837058	−	0.547113i	$-0.815727\pi$
$113$	− 13972.2i	− 1.09423i	0.837058	−	0.547113i	$-0.184273\pi$
$114$	0	0
$115$	0	0
$116$	15513.1i	1.15288i
$117$	0	0
$118$	−8983.04	−0.645148
$119$	− 23008.5i	− 1.62478i
$120$	0	0
$121$	−5735.45	−0.391739
$122$	3377.43i	0.226917i
$123$	0	0
$124$	−357.676	−0.0232620
$125$	0	0
$126$	0	0
$127$	6749.40	0.418464	0.209232	−	0.977866i	$-0.432904\pi$
$127$	6749.40	0.418464	0.209232	+	0.977866i	$0.432904\pi$
$128$	16705.2i	1.01961i
$129$	0	0
$130$	0	0
$131$	− 16379.3i	− 0.954451i	−0.878781	−	0.477226i	$-0.841642\pi$
$131$	− 16379.3i	− 0.954451i	0.878781	−	0.477226i	$-0.158358\pi$
$132$	0	0
$133$	21750.3	1.22959
$134$	− 3561.05i	− 0.198321i
$135$	0	0
$136$	12804.4	0.692280
$137$	23405.8i	1.24705i	0.781805	+	0.623523i	$0.214299\pi$
$137$	23405.8i	1.24705i	−0.781805	+	0.623523i	$0.785701\pi$
$138$	0	0
$139$	−8086.69	−0.418544	−0.209272	−	0.977857i	$-0.567109\pi$
$139$	−8086.69	−0.418544	−0.209272	+	0.977857i	$0.567109\pi$
$140$	0	0
$141$	0	0
$142$	−1523.49	−0.0755549
$143$	− 15168.1i	− 0.741751i
$144$	0	0
$145$	0	0
$146$	13503.2i	0.633479i
$147$	0	0
$148$	18489.5	0.844115
$149$	− 19374.0i	− 0.872663i	−0.899786	−	0.436331i	$-0.856278\pi$
$149$	− 19374.0i	− 0.872663i	0.899786	−	0.436331i	$-0.143722\pi$
$150$	0	0
$151$	−3225.62	−0.141468	−0.0707342	−	0.997495i	$-0.522534\pi$
$151$	−3225.62	−0.141468	−0.0707342	+	0.997495i	$0.522534\pi$
$152$	12104.2i	0.523901i
$153$	0	0
$154$	−18484.0	−0.779390
$155$	0	0
$156$	0	0
$157$	−21942.0	−0.890177	−0.445088	−	0.895487i	$-0.646828\pi$
$157$	−21942.0	−0.890177	−0.445088	+	0.895487i	$0.646828\pi$
$158$	− 2667.23i	− 0.106843i
$159$	0	0
$160$	0	0
$161$	− 34363.0i	− 1.32568i
$162$	0	0
$163$	−36844.9	−1.38676	−0.693381	−	0.720571i	$-0.743880\pi$
$163$	−36844.9	−1.38676	−0.693381	+	0.720571i	$0.743880\pi$
$164$	9208.85i	0.342387i
$165$	0	0
$166$	−1953.74	−0.0709006
$167$	− 32169.9i	− 1.15350i	−0.816921	−	0.576750i	$-0.804321\pi$
$167$	− 32169.9i	− 1.15350i	0.816921	−	0.576750i	$-0.195679\pi$
$168$	0	0
$169$	−17270.0	−0.604672
$170$	0	0
$171$	0	0
$172$	−28541.8	−0.964772
$173$	9487.64i	0.317005i	0.987359	+	0.158502i	$0.0506666\pi$
$173$	9487.64i	0.317005i	−0.987359	+	0.158502i	$0.949333\pi$
$174$	0	0
$175$	0	0
$176$	20689.2i	0.667909i
$177$	0	0
$178$	9473.08	0.298986
$179$	− 22636.1i	− 0.706474i	−0.935534	−	0.353237i	$-0.885081\pi$
$179$	− 22636.1i	− 0.706474i	0.935534	−	0.353237i	$-0.114919\pi$
$180$	0	0
$181$	−48877.4	−1.49194	−0.745970	−	0.665980i	$-0.768014\pi$
$181$	−48877.4	−1.49194	−0.745970	+	0.665980i	$0.768014\pi$
$182$	− 13759.3i	− 0.415389i
$183$	0	0
$184$	19123.3	0.564842
$185$	0	0
$186$	0	0
$187$	39600.6	1.13245
$188$	− 32127.3i	− 0.908988i
$189$	0	0
$190$	0	0
$191$	− 16675.9i	− 0.457113i	−0.973531	−	0.228556i	$-0.926600\pi$
$191$	− 16675.9i	− 0.457113i	0.973531	−	0.228556i	$-0.0734005\pi$
$192$	0	0
$193$	−16981.6	−0.455894	−0.227947	−	0.973674i	$-0.573201\pi$
$193$	−16981.6	−0.455894	−0.227947	+	0.973674i	$0.573201\pi$
$194$	− 966.734i	− 0.0256864i
$195$	0	0
$196$	60727.0	1.58077
$197$	15439.8i	0.397841i	0.980016	+	0.198920i	$0.0637435\pi$
$197$	15439.8i	0.397841i	−0.980016	+	0.198920i	$0.936257\pi$
$198$	0	0
$199$	33555.4	0.847337	0.423668	−	0.905817i	$-0.360742\pi$
$199$	33555.4	0.847337	0.423668	+	0.905817i	$0.360742\pi$
$200$	0	0
$201$	0	0
$202$	21053.3	0.515961
$203$	94866.5i	2.30208i
$204$	0	0
$205$	0	0
$206$	− 3706.03i	− 0.0873322i
$207$	0	0
$208$	−15400.8	−0.355973
$209$	37435.1i	0.857011i
$210$	0	0
$211$	−45174.7	−1.01468	−0.507342	−	0.861745i	$-0.669372\pi$
$211$	−45174.7	−1.01468	−0.507342	+	0.861745i	$0.669372\pi$
$212$	22303.5i	0.496251i
$213$	0	0
$214$	32989.9	0.720366
$215$	0	0
$216$	0	0
$217$	−2187.28	−0.0464498
$218$	− 29032.6i	− 0.610903i
$219$	0	0
$220$	0	0
$221$	29478.4i	0.603558i
$222$	0	0
$223$	5287.44	0.106325	0.0531625	−	0.998586i	$-0.483070\pi$
$223$	5287.44	0.106325	0.0531625	+	0.998586i	$0.483070\pi$
$224$	80015.6i	1.59470i
$225$	0	0
$226$	21814.5	0.427100
$227$	48260.3i	0.936566i	0.883578	+	0.468283i	$0.155127\pi$
$227$	48260.3i	0.936566i	−0.883578	+	0.468283i	$0.844873\pi$
$228$	0	0
$229$	−20011.2	−0.381594	−0.190797	−	0.981629i	$-0.561107\pi$
$229$	−20011.2	−0.381594	−0.190797	+	0.981629i	$0.561107\pi$
$230$	0	0
$231$	0	0
$232$	−52793.9	−0.980862
$233$	− 49175.9i	− 0.905817i	−0.891557	−	0.452909i	$-0.850386\pi$
$233$	− 49175.9i	− 0.905817i	0.891557	−	0.452909i	$-0.149614\pi$
$234$	0	0
$235$	0	0
$236$	78032.9i	1.40105i
$237$	0	0
$238$	35922.8	0.634185
$239$	73316.8i	1.28353i	0.766899	+	0.641767i	$0.221799\pi$
$239$	73316.8i	1.28353i	−0.766899	+	0.641767i	$0.778201\pi$
$240$	0	0
$241$	103493.	1.78187	0.890936	−	0.454128i	$-0.150049\pi$
$241$	103493.	1.78187	0.890936	+	0.454128i	$0.150049\pi$
$242$	− 8954.67i	− 0.152904i
$243$	0	0
$244$	29338.7	0.492789
$245$	0	0
$246$	0	0
$247$	−27866.4	−0.456758
$248$	− 1217.24i	− 0.0197912i
$249$	0	0
$250$	0	0
$251$	2650.77i	0.0420751i	0.999779	+	0.0210375i	$0.00669695\pi$
$251$	2650.77i	0.0420751i	−0.999779	+	0.0210375i	$0.993303\pi$
$252$	0	0
$253$	59143.2	0.923983
$254$	10537.7i	0.163335i
$255$	0	0
$256$	−13078.3	−0.199559
$257$	− 92657.4i	− 1.40286i	−0.712739	−	0.701429i	$-0.752546\pi$
$257$	− 92657.4i	− 1.40286i	0.712739	−	0.701429i	$-0.247454\pi$
$258$	0	0
$259$	113068.	1.68554
$260$	0	0
$261$	0	0
$262$	25572.8	0.372542
$263$	− 38269.8i	− 0.553279i	−0.960974	−	0.276640i	$-0.910779\pi$
$263$	− 38269.8i	− 0.553279i	0.960974	−	0.276640i	$-0.0892209\pi$
$264$	0	0
$265$	0	0
$266$	33958.4i	0.479936i
$267$	0	0
$268$	−30933.8	−0.430688
$269$	8391.53i	0.115968i	0.998318	+	0.0579838i	$0.0184672\pi$
$269$	8391.53i	0.115968i	−0.998318	+	0.0579838i	$0.981533\pi$
$270$	0	0
$271$	−488.528	−0.00665198	−0.00332599	−	0.999994i	$-0.501059\pi$
$271$	−488.528	−0.00665198	−0.00332599	+	0.999994i	$0.501059\pi$
$272$	− 40208.4i	− 0.543474i
$273$	0	0
$274$	−36543.1	−0.486749
$275$	0	0
$276$	0	0
$277$	−385.258	−0.00502102	−0.00251051	−	0.999997i	$-0.500799\pi$
$277$	−385.258	−0.00502102	−0.00251051	+	0.999997i	$0.500799\pi$
$278$	− 12625.6i	− 0.163367i
$279$	0	0
$280$	0	0
$281$	− 99259.3i	− 1.25707i	−0.777782	−	0.628534i	$-0.783655\pi$
$281$	− 99259.3i	− 1.25707i	0.777782	−	0.628534i	$-0.216345\pi$
$282$	0	0
$283$	−44860.2	−0.560129	−0.280065	−	0.959981i	$-0.590356\pi$
$283$	−44860.2	−0.560129	−0.280065	+	0.959981i	$0.590356\pi$
$284$	13234.1i	0.164081i
$285$	0	0
$286$	23681.7	0.289521
$287$	56314.4i	0.683684i
$288$	0	0
$289$	6559.05	0.0785318
$290$	0	0
$291$	0	0
$292$	117299.	1.37571
$293$	49598.8i	0.577744i	0.957368	+	0.288872i	$0.0932803\pi$
$293$	49598.8i	0.577744i	−0.957368	+	0.288872i	$0.906720\pi$
$294$	0	0
$295$	0	0
$296$	62923.1i	0.718169i
$297$	0	0
$298$	30248.3	0.340619
$299$	44025.7i	0.492452i
$300$	0	0
$301$	−174540.	−1.92647
$302$	− 5036.11i	− 0.0552181i
$303$	0	0
$304$	38009.6	0.411288
$305$	0	0
$306$	0	0
$307$	80916.5	0.858540	0.429270	−	0.903176i	$-0.358771\pi$
$307$	80916.5	0.858540	0.429270	+	0.903176i	$0.358771\pi$
$308$	160565.i	1.69258i
$309$	0	0
$310$	0	0
$311$	− 39337.3i	− 0.406709i	−0.979105	−	0.203354i	$-0.934816\pi$
$311$	− 39337.3i	− 0.406709i	0.979105	−	0.203354i	$-0.0651843\pi$
$312$	0	0
$313$	−106390.	−1.08595	−0.542975	−	0.839749i	$-0.682702\pi$
$313$	−106390.	−1.08595	−0.542975	+	0.839749i	$0.682702\pi$
$314$	− 34257.6i	− 0.347455i
$315$	0	0
$316$	−23169.4	−0.232028
$317$	8729.59i	0.0868711i	0.999056	+	0.0434355i	$0.0138303\pi$
$317$	8729.59i	0.0868711i	−0.999056	+	0.0434355i	$0.986170\pi$
$318$	0	0
$319$	−163278.	−1.60452
$320$	0	0
$321$	0	0
$322$	53650.4	0.517442
$323$	− 72753.3i	− 0.697345i
$324$	0	0
$325$	0	0
$326$	− 57525.3i	− 0.541282i
$327$	0	0
$328$	−31339.4	−0.291302
$329$	− 196466.i	− 1.81508i
$330$	0	0
$331$	144483.	1.31875	0.659374	−	0.751815i	$-0.270821\pi$
$331$	144483.	1.31875	0.659374	+	0.751815i	$0.270821\pi$
$332$	16971.5i	0.153973i
$333$	0	0
$334$	50226.4	0.450235
$335$	0	0
$336$	0	0
$337$	−202953.	−1.78705	−0.893523	−	0.449017i	$-0.851774\pi$
$337$	−202953.	−1.78705	−0.893523	+	0.449017i	$0.851774\pi$
$338$	− 26963.4i	− 0.236016i
$339$	0	0
$340$	0	0
$341$	− 3764.59i	− 0.0323749i
$342$	0	0
$343$	172228.	1.46391
$344$	− 97133.0i	− 0.820823i
$345$	0	0
$346$	−14812.9	−0.123734
$347$	67772.8i	0.562855i	0.959582	+	0.281427i	$0.0908079\pi$
$347$	67772.8i	0.562855i	−0.959582	+	0.281427i	$0.909192\pi$
$348$	0	0
$349$	−11678.9	−0.0958849	−0.0479424	−	0.998850i	$-0.515266\pi$
$349$	−11678.9	−0.0958849	−0.0479424	+	0.998850i	$0.515266\pi$
$350$	0	0
$351$	0	0
$352$	−137717.	−1.11149
$353$	− 129628.i	− 1.04028i	−0.854081	−	0.520140i	$-0.825880\pi$
$353$	− 129628.i	− 1.04028i	0.854081	−	0.520140i	$-0.174120\pi$
$354$	0	0
$355$	0	0
$356$	− 82289.7i	− 0.649300i
$357$	0	0
$358$	35341.5	0.275752
$359$	47869.9i	0.371427i	0.982604	+	0.185714i	$0.0594596\pi$
$359$	47869.9i	0.371427i	−0.982604	+	0.185714i	$0.940540\pi$
$360$	0	0
$361$	−61546.2	−0.472266
$362$	− 76311.6i	− 0.582335i
$363$	0	0
$364$	−119523.	−0.902089
$365$	0	0
$366$	0	0
$367$	−142380.	−1.05710	−0.528550	−	0.848902i	$-0.677264\pi$
$367$	−142380.	−1.05710	−0.528550	+	0.848902i	$0.677264\pi$
$368$	− 60050.9i	− 0.443429i
$369$	0	0
$370$	0	0
$371$	136392.i	0.990922i
$372$	0	0
$373$	−38609.9	−0.277512	−0.138756	−	0.990327i	$-0.544310\pi$
$373$	−38609.9	−0.277512	−0.138756	+	0.990327i	$0.544310\pi$
$374$	61827.9i	0.442019i
$375$	0	0
$376$	109335.	0.773363
$377$	− 121543.i	− 0.855156i
$378$	0	0
$379$	−145465.	−1.01270	−0.506348	−	0.862329i	$-0.669005\pi$
$379$	−145465.	−1.01270	−0.506348	+	0.862329i	$0.669005\pi$
$380$	0	0
$381$	0	0
$382$	26035.9	0.178421
$383$	394.309i	0.00268806i	0.999999	+	0.00134403i	$0.000427819\pi$
$383$	394.309i	0.00268806i	−0.999999	+	0.00134403i	$0.999572\pi$
$384$	0	0
$385$	0	0
$386$	− 26513.1i	− 0.177945i
$387$	0	0
$388$	−8397.72	−0.0557825
$389$	− 189831.i	− 1.25449i	−0.778821	−	0.627247i	$-0.784182\pi$
$389$	− 189831.i	− 1.25449i	0.778821	−	0.627247i	$-0.215818\pi$
$390$	0	0
$391$	−114942.	−0.751840
$392$	206665.i	1.34492i
$393$	0	0
$394$	−24105.9	−0.155286
$395$	0	0
$396$	0	0
$397$	212776.	1.35002	0.675011	−	0.737808i	$-0.264139\pi$
$397$	212776.	1.35002	0.675011	+	0.737808i	$0.264139\pi$
$398$	52389.5i	0.330733i
$399$	0	0
$400$	0	0
$401$	− 46951.1i	− 0.291983i	−0.989286	−	0.145991i	$-0.953363\pi$
$401$	− 46951.1i	− 0.291983i	0.989286	−	0.145991i	$-0.0466372\pi$
$402$	0	0
$403$	2802.33	0.0172548
$404$	− 182883.i	− 1.12050i
$405$	0	0
$406$	−148114.	−0.898551
$407$	194605.i	1.17480i
$408$	0	0
$409$	−33511.3	−0.200330	−0.100165	−	0.994971i	$-0.531937\pi$
$409$	−33511.3	−0.200330	−0.100165	+	0.994971i	$0.531937\pi$
$410$	0	0
$411$	0	0
$412$	−32193.1	−0.189657
$413$	477190.i	2.79764i
$414$	0	0
$415$	0	0
$416$	− 102516.i	− 0.592385i
$417$	0	0
$418$	−58446.8	−0.334509
$419$	− 75612.8i	− 0.430692i	−0.976538	−	0.215346i	$-0.930912\pi$
$419$	− 75612.8i	− 0.430692i	0.976538	−	0.215346i	$-0.0690880\pi$
$420$	0	0
$421$	268477.	1.51476	0.757378	−	0.652977i	$-0.226480\pi$
$421$	268477.	1.51476	0.757378	+	0.652977i	$0.226480\pi$
$422$	− 70530.6i	− 0.396052i
$423$	0	0
$424$	−75903.0	−0.422209
$425$	0	0
$426$	0	0
$427$	179413.	0.984008
$428$	− 286573.i	− 1.56440i
$429$	0	0
$430$	0	0
$431$	− 33271.4i	− 0.179109i	−0.995982	−	0.0895544i	$-0.971456\pi$
$431$	− 33271.4i	− 0.179109i	0.995982	−	0.0895544i	$-0.0285443\pi$
$432$	0	0
$433$	−308075.	−1.64317	−0.821583	−	0.570089i	$-0.806909\pi$
$433$	−308075.	−1.64317	−0.821583	+	0.570089i	$0.806909\pi$
$434$	− 3414.96i	− 0.0181303i
$435$	0	0
$436$	−252197.	−1.32668
$437$	− 108656.i	− 0.568974i
$438$	0	0
$439$	265525.	1.37777	0.688886	−	0.724870i	$-0.258100\pi$
$439$	265525.	1.37777	0.688886	+	0.724870i	$0.258100\pi$
$440$	0	0
$441$	0	0
$442$	−46024.1	−0.235581
$443$	40625.8i	0.207012i	0.994629	+	0.103506i	$0.0330060\pi$
$443$	40625.8i	0.207012i	−0.994629	+	0.103506i	$0.966994\pi$
$444$	0	0
$445$	0	0
$446$	8255.19i	0.0415009i
$447$	0	0
$448$	67403.6	0.335836
$449$	− 215290.i	− 1.06790i	−0.845516	−	0.533950i	$-0.820707\pi$
$449$	− 215290.i	− 1.06790i	0.845516	−	0.533950i	$-0.179293\pi$
$450$	0	0
$451$	−96924.5	−0.476519
$452$	− 189496.i	− 0.927521i
$453$	0	0
$454$	−75348.1	−0.365562
$455$	0	0
$456$	0	0
$457$	324639.	1.55442	0.777210	−	0.629241i	$-0.216634\pi$
$457$	324639.	1.55442	0.777210	+	0.629241i	$0.216634\pi$
$458$	− 31243.1i	− 0.148944i
$459$	0	0
$460$	0	0
$461$	57912.7i	0.272503i	0.990674	+	0.136252i	$0.0435056\pi$
$461$	57912.7i	0.272503i	−0.990674	+	0.136252i	$0.956494\pi$
$462$	0	0
$463$	301265.	1.40536	0.702678	−	0.711508i	$-0.251988\pi$
$463$	301265.	1.40536	0.702678	+	0.711508i	$0.251988\pi$
$464$	165783.i	0.770025i
$465$	0	0
$466$	76777.6	0.353560
$467$	− 294798.i	− 1.35173i	−0.737024	−	0.675866i	$-0.763770\pi$
$467$	− 294798.i	− 1.35173i	0.737024	−	0.675866i	$-0.236230\pi$
$468$	0	0
$469$	−189168.	−0.860005
$470$	0	0
$471$	0	0
$472$	−265560.	−1.19201
$473$	− 300407.i	− 1.34273i
$474$	0	0
$475$	0	0
$476$	− 312050.i	− 1.37724i
$477$	0	0
$478$	−114468.	−0.500991
$479$	428664.i	1.86830i	0.356884	+	0.934149i	$0.383839\pi$
$479$	428664.i	1.86830i	−0.356884	+	0.934149i	$0.616161\pi$
$480$	0	0
$481$	−144862.	−0.626130
$482$	161582.i	0.695502i
$483$	0	0
$484$	−77786.5	−0.332057
$485$	0	0
$486$	0	0
$487$	−457302.	−1.92817	−0.964085	−	0.265596i	$-0.914431\pi$
$487$	−457302.	−1.92817	−0.964085	+	0.265596i	$0.914431\pi$
$488$	99844.9i	0.419263i
$489$	0	0
$490$	0	0
$491$	154537.i	0.641018i	0.947245	+	0.320509i	$0.103854\pi$
$491$	154537.i	0.641018i	−0.947245	+	0.320509i	$0.896146\pi$
$492$	0	0
$493$	317322.	1.30559
$494$	− 43507.3i	− 0.178282i
$495$	0	0
$496$	−3822.36	−0.0155370
$497$	80929.7i	0.327639i
$498$	0	0
$499$	362181.	1.45454	0.727269	−	0.686353i	$-0.240789\pi$
$499$	362181.	1.45454	0.727269	+	0.686353i	$0.240789\pi$
$500$	0	0
$501$	0	0
$502$	−4138.61	−0.0164228
$503$	− 193565.i	− 0.765050i	−0.923945	−	0.382525i	$-0.875055\pi$
$503$	− 193565.i	− 0.765050i	0.923945	−	0.382525i	$-0.124945\pi$
$504$	0	0
$505$	0	0
$506$	92339.4i	0.360650i
$507$	0	0
$508$	91538.1	0.354711
$509$	331435.i	1.27927i	0.768677	+	0.639637i	$0.220915\pi$
$509$	331435.i	1.27927i	−0.768677	+	0.639637i	$0.779085\pi$
$510$	0	0
$511$	717309.	2.74704
$512$	246864.i	0.941713i
$513$	0	0
$514$	144665.	0.547565
$515$	0	0
$516$	0	0
$517$	338144.	1.26509
$518$	176531.i	0.657902i
$519$	0	0
$520$	0	0
$521$	503767.i	1.85590i	0.372707	+	0.927949i	$0.378430\pi$
$521$	503767.i	1.85590i	−0.372707	+	0.927949i	$0.621570\pi$
$522$	0	0
$523$	−171122.	−0.625608	−0.312804	−	0.949818i	$-0.601268\pi$
$523$	−171122.	−0.625608	−0.312804	+	0.949818i	$0.601268\pi$
$524$	− 222143.i	− 0.809040i
$525$	0	0
$526$	59750.0	0.215957
$527$	7316.30i	0.0263433i
$528$	0	0
$529$	108176.	0.386562
$530$	0	0
$531$	0	0
$532$	294986.	1.04226
$533$	− 72149.8i	− 0.253969i
$534$	0	0
$535$	0	0
$536$	− 105273.i	− 0.366428i
$537$	0	0
$538$	−13101.6	−0.0452646
$539$	639161.i	2.20005i
$540$	0	0
$541$	−170873.	−0.583819	−0.291910	−	0.956446i	$-0.594291\pi$
$541$	−170873.	−0.583819	−0.291910	+	0.956446i	$0.594291\pi$
$542$	− 762.732i	− 0.00259641i
$543$	0	0
$544$	267647.	0.904409
$545$	0	0
$546$	0	0
$547$	−187183.	−0.625594	−0.312797	−	0.949820i	$-0.601266\pi$
$547$	−187183.	−0.625594	−0.312797	+	0.949820i	$0.601266\pi$
$548$	317439.i	1.05706i
$549$	0	0
$550$	0	0
$551$	299970.i	0.988039i
$552$	0	0
$553$	−141686.	−0.463317
$554$	− 601.497i	− 0.00195981i
$555$	0	0
$556$	−109675.	−0.354779
$557$	292929.i	0.944175i	0.881552	+	0.472088i	$0.156499\pi$
$557$	292929.i	0.944175i	−0.881552	+	0.472088i	$0.843501\pi$
$558$	0	0
$559$	223620.	0.715628
$560$	0	0
$561$	0	0
$562$	154972.	0.490660
$563$	− 16709.7i	− 0.0527171i	−0.999653	−	0.0263586i	$-0.991609\pi$
$563$	− 16709.7i	− 0.0527171i	0.999653	−	0.0263586i	$-0.00839116\pi$
$564$	0	0
$565$	0	0
$566$	− 70039.5i	− 0.218630i
$567$	0	0
$568$	−45038.0	−0.139599
$569$	47426.8i	0.146487i	0.997314	+	0.0732435i	$0.0233350\pi$
$569$	47426.8i	0.146487i	−0.997314	+	0.0732435i	$0.976665\pi$
$570$	0	0
$571$	124644.	0.382296	0.191148	−	0.981561i	$-0.438779\pi$
$571$	124644.	0.382296	0.191148	+	0.981561i	$0.438779\pi$
$572$	− 205715.i	− 0.628745i
$573$	0	0
$574$	−87922.7	−0.266856
$575$	0	0
$576$	0	0
$577$	−209682.	−0.629810	−0.314905	−	0.949123i	$-0.601973\pi$
$577$	−209682.	−0.629810	−0.314905	+	0.949123i	$0.601973\pi$
$578$	10240.5i	0.0306526i
$579$	0	0
$580$	0	0
$581$	103785.i	0.307455i
$582$	0	0
$583$	−234748.	−0.690660
$584$	399188.i	1.17045i
$585$	0	0
$586$	−77437.8	−0.225506
$587$	− 417823.i	− 1.21260i	−0.795237	−	0.606299i	$-0.792654\pi$
$587$	− 417823.i	− 1.21260i	0.795237	−	0.606299i	$-0.207346\pi$
$588$	0	0
$589$	−6916.21	−0.0199360
$590$	0	0
$591$	0	0
$592$	197591.	0.563798
$593$	25723.9i	0.0731522i	0.999331	+	0.0365761i	$0.0116451\pi$
$593$	25723.9i	0.0731522i	−0.999331	+	0.0365761i	$0.988355\pi$
$594$	0	0
$595$	0	0
$596$	− 262758.i	− 0.739712i
$597$	0	0
$598$	−68736.7	−0.192215
$599$	− 57586.1i	− 0.160496i	−0.996775	−	0.0802479i	$-0.974429\pi$
$599$	− 57586.1i	− 0.160496i	0.996775	−	0.0802479i	$-0.0255712\pi$
$600$	0	0
$601$	174151.	0.482146	0.241073	−	0.970507i	$-0.422501\pi$
$601$	174151.	0.482146	0.241073	+	0.970507i	$0.422501\pi$
$602$	− 272507.i	− 0.751942i
$603$	0	0
$604$	−43747.1	−0.119916
$605$	0	0
$606$	0	0
$607$	−250149.	−0.678923	−0.339462	−	0.940620i	$-0.610245\pi$
$607$	−250149.	−0.678923	−0.339462	+	0.940620i	$0.610245\pi$
$608$	253011.i	0.684435i
$609$	0	0
$610$	0	0
$611$	251712.i	0.674250i
$612$	0	0
$613$	274395.	0.730222	0.365111	−	0.930964i	$-0.381031\pi$
$613$	274395.	0.730222	0.365111	+	0.930964i	$0.381031\pi$
$614$	126334.i	0.335106i
$615$	0	0
$616$	−546432.	−1.44004
$617$	58332.7i	0.153229i	0.997061	+	0.0766147i	$0.0244111\pi$
$617$	58332.7i	0.153229i	−0.997061	+	0.0766147i	$0.975589\pi$
$618$	0	0
$619$	464866.	1.21324	0.606620	−	0.794992i	$-0.292525\pi$
$619$	464866.	1.21324	0.606620	+	0.794992i	$0.292525\pi$
$620$	0	0
$621$	0	0
$622$	61416.6	0.158747
$623$	− 503222.i	− 1.29653i
$624$	0	0
$625$	0	0
$626$	− 166104.i	− 0.423869i
$627$	0	0
$628$	−297585.	−0.754558
$629$	− 378205.i	− 0.955929i
$630$	0	0
$631$	150594.	0.378224	0.189112	−	0.981956i	$-0.439439\pi$
$631$	150594.	0.378224	0.189112	+	0.981956i	$0.439439\pi$
$632$	− 78849.6i	− 0.197408i
$633$	0	0
$634$	−13629.4	−0.0339076
$635$	0	0
$636$	0	0
$637$	−475786.	−1.17255
$638$	− 254923.i	− 0.626278i
$639$	0	0
$640$	0	0
$641$	− 740861.i	− 1.80310i	−0.432672	−	0.901552i	$-0.642429\pi$
$641$	− 740861.i	− 1.80310i	0.432672	−	0.901552i	$-0.357571\pi$
$642$	0	0
$643$	28233.2	0.0682870	0.0341435	−	0.999417i	$-0.489130\pi$
$643$	28233.2	0.0682870	0.0341435	+	0.999417i	$0.489130\pi$
$644$	− 466044.i	− 1.12371i
$645$	0	0
$646$	113589.	0.272188
$647$	759101.i	1.81339i	0.421788	+	0.906694i	$0.361402\pi$
$647$	759101.i	1.81339i	−0.421788	+	0.906694i	$0.638598\pi$
$648$	0	0
$649$	−821307.	−1.94992
$650$	0	0
$651$	0	0
$652$	−499705.	−1.17549
$653$	480373.i	1.12655i	0.826268	+	0.563277i	$0.190459\pi$
$653$	480373.i	1.12655i	−0.826268	+	0.563277i	$0.809541\pi$
$654$	0	0
$655$	0	0
$656$	98411.9i	0.228686i
$657$	0	0
$658$	306739.	0.708464
$659$	257235.i	0.592323i	0.955138	+	0.296162i	$0.0957067\pi$
$659$	257235.i	0.592323i	−0.955138	+	0.296162i	$0.904293\pi$
$660$	0	0
$661$	175393.	0.401430	0.200715	−	0.979650i	$-0.435674\pi$
$661$	175393.	0.401430	0.200715	+	0.979650i	$0.435674\pi$
$662$	225580.i	0.514735i
$663$	0	0
$664$	−57757.1	−0.130999
$665$	0	0
$666$	0	0
$667$	473918.	1.06525
$668$	− 436301.i	− 0.977763i
$669$	0	0
$670$	0	0
$671$	308794.i	0.685841i
$672$	0	0
$673$	462605.	1.02136	0.510681	−	0.859770i	$-0.329393\pi$
$673$	462605.	1.02136	0.510681	+	0.859770i	$0.329393\pi$
$674$	− 316867.i	− 0.697522i
$675$	0	0
$676$	−234223.	−0.512550
$677$	671725.i	1.46560i	0.680446	+	0.732798i	$0.261786\pi$
$677$	671725.i	1.46560i	−0.680446	+	0.732798i	$0.738214\pi$
$678$	0	0
$679$	−51354.1	−0.111387
$680$	0	0
$681$	0	0
$682$	5877.59	0.0126366
$683$	475276.i	1.01884i	0.860519	+	0.509418i	$0.170139\pi$
$683$	475276.i	1.01884i	−0.860519	+	0.509418i	$0.829861\pi$
$684$	0	0
$685$	0	0
$686$	268897.i	0.571397i
$687$	0	0
$688$	−305017.	−0.644387
$689$	− 174744.i	− 0.368099i
$690$	0	0
$691$	−115300.	−0.241475	−0.120737	−	0.992684i	$-0.538526\pi$
$691$	−115300.	−0.241475	−0.120737	+	0.992684i	$0.538526\pi$
$692$	128675.i	0.268709i
$693$	0	0
$694$	−105813.	−0.219694
$695$	0	0
$696$	0	0
$697$	188368.	0.387741
$698$	− 18234.0i	− 0.0374259i
$699$	0	0
$700$	0	0
$701$	− 514039.i	− 1.04607i	−0.852312	−	0.523034i	$-0.824800\pi$
$701$	− 514039.i	− 1.04607i	0.852312	−	0.523034i	$-0.175200\pi$
$702$	0	0
$703$	357523.	0.723424
$704$	116010.i	0.234073i
$705$	0	0
$706$	202387.	0.406044
$707$	− 1.11838e6i	− 2.23743i
$708$	0	0
$709$	−43609.5	−0.0867539	−0.0433769	−	0.999059i	$-0.513812\pi$
$709$	−43609.5	−0.0867539	−0.0433769	+	0.999059i	$0.513812\pi$
$710$	0	0
$711$	0	0
$712$	280047.	0.552422
$713$	10926.8i	0.0214939i
$714$	0	0
$715$	0	0
$716$	− 307000.i	− 0.598843i
$717$	0	0
$718$	−74738.5	−0.144976
$719$	36282.5i	0.0701842i	0.999384	+	0.0350921i	$0.0111725\pi$
$719$	36282.5i	0.0701842i	−0.999384	+	0.0350921i	$0.988828\pi$
$720$	0	0
$721$	−196869.	−0.378710
$722$	− 96091.0i	− 0.184335i
$723$	0	0
$724$	−662895.	−1.26464
$725$	0	0
$726$	0	0
$727$	565757.	1.07044	0.535218	−	0.844714i	$-0.320229\pi$
$727$	565757.	1.07044	0.535218	+	0.844714i	$0.320229\pi$
$728$	− 406759.i	− 0.767493i
$729$	0	0
$730$	0	0
$731$	583826.i	1.09257i
$732$	0	0
$733$	−394264.	−0.733802	−0.366901	−	0.930260i	$-0.619581\pi$
$733$	−394264.	−0.733802	−0.366901	+	0.930260i	$0.619581\pi$
$734$	− 222295.i	− 0.412609i
$735$	0	0
$736$	399729.	0.737921
$737$	− 325582.i	− 0.599413i
$738$	0	0
$739$	252058.	0.461542	0.230771	−	0.973008i	$-0.425875\pi$
$739$	252058.	0.461542	0.230771	+	0.973008i	$0.425875\pi$
$740$	0	0
$741$	0	0
$742$	−212946.	−0.386778
$743$	291552.i	0.528127i	0.964505	+	0.264064i	$0.0850629\pi$
$743$	291552.i	0.528127i	−0.964505	+	0.264064i	$0.914937\pi$
$744$	0	0
$745$	0	0
$746$	− 60281.1i	− 0.108319i
$747$	0	0
$748$	537080.	0.959921
$749$	− 1.75246e6i	− 3.12382i
$750$	0	0
$751$	−694419.	−1.23124	−0.615618	−	0.788044i	$-0.711094\pi$
$751$	−694419.	−1.23124	−0.615618	+	0.788044i	$0.711094\pi$
$752$	− 343333.i	− 0.607128i
$753$	0	0
$754$	189762.	0.333786
$755$	0	0
$756$	0	0
$757$	−254690.	−0.444447	−0.222223	−	0.974996i	$-0.571331\pi$
$757$	−254690.	−0.444447	−0.222223	+	0.974996i	$0.571331\pi$
$758$	− 227111.i	− 0.395276i
$759$	0	0
$760$	0	0
$761$	379005.i	0.654448i	0.944947	+	0.327224i	$0.106113\pi$
$761$	379005.i	0.654448i	−0.944947	+	0.327224i	$0.893887\pi$
$762$	0	0
$763$	−1.54225e6	−2.64914
$764$	− 226165.i	− 0.387471i
$765$	0	0
$766$	−615.629	−0.00104921
$767$	− 611374.i	− 1.03924i
$768$	0	0
$769$	−729722.	−1.23397	−0.616985	−	0.786975i	$-0.711646\pi$
$769$	−729722.	−1.23397	−0.616985	+	0.786975i	$0.711646\pi$
$770$	0	0
$771$	0	0
$772$	−230311.	−0.386438
$773$	− 312829.i	− 0.523537i	−0.965131	−	0.261769i	$-0.915694\pi$
$773$	− 312829.i	− 0.523537i	0.965131	−	0.261769i	$-0.0843058\pi$
$774$	0	0
$775$	0	0
$776$	− 28579.0i	− 0.0474595i
$777$	0	0
$778$	296380.	0.489655
$779$	178067.i	0.293433i
$780$	0	0
$781$	−139291.	−0.228360
$782$	− 179457.i	− 0.293459i
$783$	0	0
$784$	648969.	1.05583
$785$	0	0
$786$	0	0
$787$	1.03100e6	1.66460	0.832302	−	0.554323i	$-0.187023\pi$
$787$	1.03100e6	1.66460	0.832302	+	0.554323i	$0.187023\pi$
$788$	209401.i	0.337229i
$789$	0	0
$790$	0	0
$791$	− 1.15882e6i	− 1.85209i
$792$	0	0
$793$	−229863.	−0.365530
$794$	332203.i	0.526942i
$795$	0	0
$796$	455091.	0.718245
$797$	− 125784.i	− 0.198020i	−0.995086	−	0.0990098i	$-0.968432\pi$
$797$	− 125784.i	− 0.198020i	0.995086	−	0.0990098i	$-0.0315675\pi$
$798$	0	0
$799$	−657167.	−1.02939
$800$	0	0
$801$	0	0
$802$	73304.1	0.113967
$803$	1.23458e6i	1.91465i
$804$	0	0
$805$	0	0
$806$	4375.23i	0.00673490i
$807$	0	0
$808$	622385.	0.953314
$809$	− 567816.i	− 0.867582i	−0.901014	−	0.433791i	$-0.857176\pi$
$809$	− 567816.i	− 0.867582i	0.901014	−	0.433791i	$-0.142824\pi$
$810$	0	0
$811$	−170998.	−0.259985	−0.129993	−	0.991515i	$-0.541495\pi$
$811$	−170998.	−0.259985	−0.129993	+	0.991515i	$0.541495\pi$
$812$	1.28662e6i	1.95136i
$813$	0	0
$814$	−303833.	−0.458550
$815$	0	0
$816$	0	0
$817$	−551899.	−0.826829
$818$	− 52320.7i	− 0.0781929i
$819$	0	0
$820$	0	0
$821$	− 443785.i	− 0.658394i	−0.944261	−	0.329197i	$-0.893222\pi$
$821$	− 443785.i	− 0.658394i	0.944261	−	0.329197i	$-0.106778\pi$
$822$	0	0
$823$	583186.	0.861008	0.430504	−	0.902589i	$-0.358336\pi$
$823$	583186.	0.861008	0.430504	+	0.902589i	$0.358336\pi$
$824$	− 109559.i	− 0.161359i
$825$	0	0
$826$	−745029.	−1.09198
$827$	1.16796e6i	1.70773i	0.520498	+	0.853863i	$0.325746\pi$
$827$	1.16796e6i	1.70773i	−0.520498	+	0.853863i	$0.674254\pi$
$828$	0	0
$829$	−199102.	−0.289713	−0.144856	−	0.989453i	$-0.546272\pi$
$829$	−199102.	−0.289713	−0.144856	+	0.989453i	$0.546272\pi$
$830$	0	0
$831$	0	0
$832$	−86357.2	−0.124753
$833$	− 1.24218e6i	− 1.79017i
$834$	0	0
$835$	0	0
$836$	507710.i	0.726445i
$837$	0	0
$838$	118053.	0.168108
$839$	804194.i	1.14245i	0.820794	+	0.571225i	$0.193532\pi$
$839$	804194.i	1.14245i	−0.820794	+	0.571225i	$0.806468\pi$
$840$	0	0
$841$	−601073.	−0.849836
$842$	419168.i	0.591241i
$843$	0	0
$844$	−612678.	−0.860096
$845$	0	0
$846$	0	0
$847$	−475683.	−0.663057
$848$	238350.i	0.331455i
$849$	0	0
$850$	0	0
$851$	− 564846.i	− 0.779957i
$852$	0	0
$853$	767685.	1.05508	0.527540	−	0.849530i	$-0.323115\pi$
$853$	767685.	1.05508	0.527540	+	0.849530i	$0.323115\pi$
$854$	280115.i	0.384079i
$855$	0	0
$856$	975260.	1.33098
$857$	− 391943.i	− 0.533656i	−0.963744	−	0.266828i	$-0.914024\pi$
$857$	− 391943.i	− 0.533656i	0.963744	−	0.266828i	$-0.0859755\pi$
$858$	0	0
$859$	822992.	1.11534	0.557672	−	0.830061i	$-0.311695\pi$
$859$	822992.	1.11534	0.557672	+	0.830061i	$0.311695\pi$
$860$	0	0
$861$	0	0
$862$	51946.2	0.0699099
$863$	− 486383.i	− 0.653066i	−0.945186	−	0.326533i	$-0.894120\pi$
$863$	− 486383.i	− 0.653066i	0.945186	−	0.326533i	$-0.105880\pi$
$864$	0	0
$865$	0	0
$866$	− 480993.i	− 0.641362i
$867$	0	0
$868$	−29664.7	−0.0393732
$869$	− 243861.i	− 0.322926i
$870$	0	0
$871$	242361.	0.319467
$872$	− 858272.i	− 1.12873i
$873$	0	0
$874$	169644.	0.222083
$875$	0	0
$876$	0	0
$877$	−784033.	−1.01938	−0.509689	−	0.860359i	$-0.670240\pi$
$877$	−784033.	−1.01938	−0.509689	+	0.860359i	$0.670240\pi$
$878$	414561.i	0.537773i
$879$	0	0
$880$	0	0
$881$	95885.0i	0.123537i	0.998090	+	0.0617687i	$0.0196741\pi$
$881$	95885.0i	0.123537i	−0.998090	+	0.0617687i	$0.980326\pi$
$882$	0	0
$883$	−917120.	−1.17626	−0.588132	−	0.808765i	$-0.700137\pi$
$883$	−917120.	−1.17626	−0.588132	+	0.808765i	$0.700137\pi$
$884$	399797.i	0.511606i
$885$	0	0
$886$	−63428.5	−0.0808010
$887$	1.08304e6i	1.37657i	0.725439	+	0.688286i	$0.241637\pi$
$887$	1.08304e6i	1.37657i	−0.725439	+	0.688286i	$0.758363\pi$
$888$	0	0
$889$	559778.	0.708292
$890$	0	0
$891$	0	0
$892$	71710.3	0.0901263
$893$	− 621230.i	− 0.779021i
$894$	0	0
$895$	0	0
$896$	1.38549e6i	1.72578i
$897$	0	0
$898$	336128.	0.416824
$899$	− 30165.9i	− 0.0373247i
$900$	0	0
$901$	456221.	0.561986
$902$	− 151327.i	− 0.185995i
$903$	0	0
$904$	644890.	0.789131
$905$	0	0
$906$	0	0
$907$	114196.	0.138815	0.0694074	−	0.997588i	$-0.477889\pi$
$907$	114196.	0.138815	0.0694074	+	0.997588i	$0.477889\pi$
$908$	654526.i	0.793880i
$909$	0	0
$910$	0	0
$911$	1.03714e6i	1.24968i	0.780751	+	0.624842i	$0.214837\pi$
$911$	1.03714e6i	1.24968i	−0.780751	+	0.624842i	$0.785163\pi$
$912$	0	0
$913$	−178628.	−0.214292
$914$	506854.i	0.606723i
$915$	0	0
$916$	−271400.	−0.323458
$917$	− 1.35846e6i	− 1.61550i
$918$	0	0
$919$	513316.	0.607790	0.303895	−	0.952705i	$-0.401713\pi$
$919$	513316.	0.607790	0.303895	+	0.952705i	$0.401713\pi$
$920$	0	0
$921$	0	0
$922$	−90418.2	−0.106364
$923$	− 103687.i	− 0.121708i
$924$	0	0
$925$	0	0
$926$	470360.i	0.548540i
$927$	0	0
$928$	−1.10354e6	−1.28142
$929$	517344.i	0.599444i	0.954027	+	0.299722i	$0.0968938\pi$
$929$	517344.i	0.599444i	−0.954027	+	0.299722i	$0.903106\pi$
$930$	0	0
$931$	1.17425e6	1.35476
$932$	− 666943.i	− 0.767816i
$933$	0	0
$934$	460264.	0.527610
$935$	0	0
$936$	0	0
$937$	−514142.	−0.585604	−0.292802	−	0.956173i	$-0.594588\pi$
$937$	−514142.	−0.585604	−0.292802	+	0.956173i	$0.594588\pi$
$938$	− 295344.i	− 0.335678i
$939$	0	0
$940$	0	0
$941$	− 1.53216e6i	− 1.73032i	−0.501497	−	0.865160i	$-0.667217\pi$
$941$	− 1.53216e6i	− 1.73032i	0.501497	−	0.865160i	$-0.332783\pi$
$942$	0	0
$943$	281326.	0.316364
$944$	833911.i	0.935784i
$945$	0	0
$946$	469020.	0.524094
$947$	22247.6i	0.0248075i	0.999923	+	0.0124037i	$0.00394833\pi$
$947$	22247.6i	0.0248075i	−0.999923	+	0.0124037i	$0.996052\pi$
$948$	0	0
$949$	−919014.	−1.02044
$950$	0	0
$951$	0	0
$952$	1.06196e6	1.17175
$953$	1.22370e6i	1.34737i	0.739018	+	0.673686i	$0.235290\pi$
$953$	1.22370e6i	1.34737i	−0.739018	+	0.673686i	$0.764710\pi$
$954$	0	0
$955$	0	0
$956$	994351.i	1.08799i
$957$	0	0
$958$	−669266.	−0.729236
$959$	1.94122e6i	2.11075i
$960$	0	0
$961$	−922825.	−0.999247
$962$	− 226171.i	− 0.244392i
$963$	0	0
$964$	1.40361e6	1.51040
$965$	0	0
$966$	0	0
$967$	184358.	0.197156	0.0985780	−	0.995129i	$-0.468571\pi$
$967$	184358.	0.197156	0.0985780	+	0.995129i	$0.468571\pi$
$968$	− 264721.i	− 0.282513i
$969$	0	0
$970$	0	0
$971$	1.64182e6i	1.74135i	0.491858	+	0.870675i	$0.336318\pi$
$971$	1.64182e6i	1.74135i	−0.491858	+	0.870675i	$0.663682\pi$
$972$	0	0
$973$	−670689.	−0.708427
$974$	− 713978.i	− 0.752605i
$975$	0	0
$976$	313533.	0.329142
$977$	1.32358e6i	1.38664i	0.720632	+	0.693318i	$0.243852\pi$
$977$	1.32358e6i	1.38664i	−0.720632	+	0.693318i	$0.756148\pi$
$978$	0	0
$979$	866111.	0.903666
$980$	0	0
$981$	0	0
$982$	−241277.	−0.250203
$983$	1.21359e6i	1.25593i	0.778240	+	0.627967i	$0.216113\pi$
$983$	1.21359e6i	1.25593i	−0.778240	+	0.627967i	$0.783887\pi$
$984$	0	0
$985$	0	0
$986$	495430.i	0.509599i
$987$	0	0
$988$	−377935.	−0.387171
$989$	871939.i	0.891443i
$990$	0	0
$991$	−1.11765e6	−1.13804	−0.569021	−	0.822323i	$-0.692678\pi$
$991$	−1.11765e6	−1.13804	−0.569021	+	0.822323i	$0.692678\pi$
$992$	− 25443.6i	− 0.0258556i
$993$	0	0
$994$	−126354.	−0.127884
$995$	0	0
$996$	0	0
$997$	533846.	0.537063	0.268532	−	0.963271i	$-0.413462\pi$
$997$	533846.	0.537063	0.268532	+	0.963271i	$0.413462\pi$
$998$	565468.i	0.567737i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.5.c.e.26.6		8
3.2	odd	2	inner	225.5.c.e.26.4		8
5.2	odd	4		45.5.d.a.44.3	✓	8
5.3	odd	4		45.5.d.a.44.5	yes	8
5.4	even	2	inner	225.5.c.e.26.3		8
15.2	even	4		45.5.d.a.44.6	yes	8
15.8	even	4		45.5.d.a.44.4	yes	8
15.14	odd	2	inner	225.5.c.e.26.5		8
20.3	even	4		720.5.c.c.449.1		8
20.7	even	4		720.5.c.c.449.7		8
60.23	odd	4		720.5.c.c.449.8		8
60.47	odd	4		720.5.c.c.449.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.5.d.a.44.3	✓	8	5.2	odd	4
45.5.d.a.44.4	yes	8	15.8	even	4
45.5.d.a.44.5	yes	8	5.3	odd	4
45.5.d.a.44.6	yes	8	15.2	even	4
225.5.c.e.26.3		8	5.4	even	2	inner
225.5.c.e.26.4		8	3.2	odd	2	inner
225.5.c.e.26.5		8	15.14	odd	2	inner
225.5.c.e.26.6		8	1.1	even	1	trivial
720.5.c.c.449.1		8	20.3	even	4
720.5.c.c.449.2		8	60.47	odd	4
720.5.c.c.449.7		8	20.7	even	4
720.5.c.c.449.8		8	60.23	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	225.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.56128i q^{2} +13.5624 q^{4} +82.9374 q^{7} +46.1553i q^{8} +O(q^{10})\)	\(q+1.56128i q^{2} +13.5624 q^{4} +82.9374 q^{7} +46.1553i q^{8} +142.746i q^{11} -106.259 q^{13} +129.489i q^{14} +144.937 q^{16} -277.420i q^{17} +262.250 q^{19} -222.867 q^{22} -414.325i q^{23} -165.900i q^{26} +1124.83 q^{28} +1143.83i q^{29} -26.3726 q^{31} +964.772i q^{32} +433.132 q^{34} +1363.29 q^{37} +409.446i q^{38} +678.999i q^{41} -2104.48 q^{43} +1935.98i q^{44} +646.879 q^{46} -2368.85i q^{47} +4477.60 q^{49} -1441.13 q^{52} +1644.51i q^{53} +3828.00i q^{56} -1785.85 q^{58} +5753.62i q^{59} +2163.24 q^{61} -41.1752i q^{62} +812.704 q^{64} -2280.85 q^{67} -3762.48i q^{68} +975.793i q^{71} +8648.81 q^{73} +2128.49i q^{74} +3556.73 q^{76} +11839.0i q^{77} -1708.36 q^{79} -1060.11 q^{82} +1251.37i q^{83} -3285.69i q^{86} -6588.49 q^{88} -6067.49i q^{89} -8812.84 q^{91} -5619.23i q^{92} +3698.45 q^{94} -619.192 q^{97} +6990.81i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 36 q^{4}+O(q^{10}) \) 8 * q - 36 * q^4	\( 8 q - 36 q^{4} + 148 q^{16} + 1520 q^{19} + 2968 q^{31} + 12424 q^{34} + 10088 q^{46} + 8944 q^{49} + 544 q^{61} + 29188 q^{64} + 3600 q^{76} + 12632 q^{79} - 24552 q^{91} - 40928 q^{94}+O(q^{100}) \) 8 * q - 36 * q^4 + 148 * q^16 + 1520 * q^19 + 2968 * q^31 + 12424 * q^34 + 10088 * q^46 + 8944 * q^49 + 544 * q^61 + 29188 * q^64 + 3600 * q^76 + 12632 * q^79 - 24552 * q^91 - 40928 * q^94

Introduction

L-functions

Modular forms

Varieties

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Database

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