LMFDB - Embedded newform 225.5.c.d.26.3

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:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 38x^{4} - 12x^{3} + 496x^{2} + 336x + 1188 $ x^6 - 38x^4 - 12x^3 + 496x^2 + 336x + 1188
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			26.3
Root			$-0.273659 - 1.41421i$ of defining polynomial
Character	$\chi$	$=$	225.26
Dual form			225.5.c.d.26.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.80123i q^{2} +12.7556 q^{4} -12.1136 q^{7} -51.7953i q^{8} +O(q^{10})$	$q-1.80123i q^{2} +12.7556 q^{4} -12.1136 q^{7} -51.7953i q^{8} -60.4434i q^{11} -168.896 q^{13} +21.8194i q^{14} +110.794 q^{16} -377.366i q^{17} -45.1263 q^{19} -108.872 q^{22} +1.37206i q^{23} +304.221i q^{26} -154.516 q^{28} -1401.24i q^{29} +1605.80 q^{31} -1028.29i q^{32} -679.721 q^{34} -1324.72 q^{37} +81.2826i q^{38} +2354.07i q^{41} -468.707 q^{43} -770.991i q^{44} +2.47139 q^{46} -3910.98i q^{47} -2254.26 q^{49} -2154.37 q^{52} -3228.98i q^{53} +627.429i q^{56} -2523.94 q^{58} -2543.15i q^{59} -254.070 q^{61} -2892.40i q^{62} -79.4731 q^{64} -7493.35 q^{67} -4813.52i q^{68} +4143.85i q^{71} +3711.28 q^{73} +2386.11i q^{74} -575.612 q^{76} +732.189i q^{77} +80.2118 q^{79} +4240.21 q^{82} +5159.51i q^{83} +844.248i q^{86} -3130.68 q^{88} -5870.07i q^{89} +2045.95 q^{91} +17.5015i q^{92} -7044.55 q^{94} +8921.02 q^{97} +4060.43i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 92 q^{4} + 86 q^{7}+O(q^{10}) $ 6 * q - 92 * q^4 + 86 * q^7	$ 6 q - 92 q^{4} + 86 q^{7} + 394 q^{13} + 1816 q^{16} - 250 q^{19} - 1276 q^{22} - 6412 q^{28} + 1846 q^{31} - 3692 q^{34} - 7968 q^{37} - 1382 q^{43} + 9756 q^{46} - 3232 q^{49} - 23268 q^{52} + 2932 q^{58} + 10838 q^{61} - 36144 q^{64} + 6278 q^{67} + 13024 q^{73} - 24780 q^{76} + 3904 q^{79} - 29792 q^{82} - 42744 q^{88} + 17306 q^{91} + 66364 q^{94} - 8262 q^{97}+O(q^{100}) $ 6 * q - 92 * q^4 + 86 * q^7 + 394 * q^13 + 1816 * q^16 - 250 * q^19 - 1276 * q^22 - 6412 * q^28 + 1846 * q^31 - 3692 * q^34 - 7968 * q^37 - 1382 * q^43 + 9756 * q^46 - 3232 * q^49 - 23268 * q^52 + 2932 * q^58 + 10838 * q^61 - 36144 * q^64 + 6278 * q^67 + 13024 * q^73 - 24780 * q^76 + 3904 * q^79 - 29792 * q^82 - 42744 * q^88 + 17306 * q^91 + 66364 * q^94 - 8262 * q^97

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 $

$2$	− 1.80123i	− 0.450306i	−0.974323	−	0.225153i	$-0.927712\pi$
$2$	− 1.80123i	− 0.450306i	0.974323	−	0.225153i	$-0.0722883\pi$
$3$	0	0
$3$	0	0
$4$	12.7556	0.797224
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−12.1136	−0.247217	−0.123609	−	0.992331i	$-0.539447\pi$
$7$	−12.1136	−0.247217	−0.123609	+	0.992331i	$0.539447\pi$
$8$	− 51.7953i	− 0.809302i
$9$	0	0
$10$	0	0
$11$	− 60.4434i	− 0.499532i	−0.968306	−	0.249766i	$-0.919646\pi$
$11$	− 60.4434i	− 0.499532i	0.968306	−	0.249766i	$-0.0803537\pi$
$12$	0	0
$13$	−168.896	−0.999387	−0.499694	−	0.866202i	$-0.666554\pi$
$13$	−168.896	−0.999387	−0.499694	+	0.866202i	$0.666554\pi$
$14$	21.8194i	0.111323i
$15$	0	0
$16$	110.794	0.432790
$17$	− 377.366i	− 1.30576i	−0.757460	−	0.652882i	$-0.773560\pi$
$17$	− 377.366i	− 1.30576i	0.757460	−	0.652882i	$-0.226440\pi$
$18$	0	0
$19$	−45.1263	−0.125004	−0.0625018	−	0.998045i	$-0.519908\pi$
$19$	−45.1263	−0.125004	−0.0625018	+	0.998045i	$0.519908\pi$
$20$	0	0
$21$	0	0
$22$	−108.872	−0.224943
$23$	1.37206i	0.00259369i	0.999999	+	0.00129685i	$0.000412799\pi$
$23$	1.37206i	0.00259369i	−0.999999	+	0.00129685i	$0.999587\pi$
$24$	0	0
$25$	0	0
$26$	304.221i	0.450030i
$27$	0	0
$28$	−154.516	−0.197087
$29$	− 1401.24i	− 1.66615i	−0.553156	−	0.833077i	$-0.686577\pi$
$29$	− 1401.24i	− 1.66615i	0.553156	−	0.833077i	$-0.313423\pi$
$30$	0	0
$31$	1605.80	1.67097	0.835483	−	0.549517i	$-0.185188\pi$
$31$	1605.80	1.67097	0.835483	+	0.549517i	$0.185188\pi$
$32$	− 1028.29i	− 1.00419i
$33$	0	0
$34$	−679.721	−0.587994
$35$	0	0
$36$	0	0
$37$	−1324.72	−0.967652	−0.483826	−	0.875164i	$-0.660753\pi$
$37$	−1324.72	−0.967652	−0.483826	+	0.875164i	$0.660753\pi$
$38$	81.2826i	0.0562899i
$39$	0	0
$40$	0	0
$41$	2354.07i	1.40040i	0.713948	+	0.700199i	$0.246905\pi$
$41$	2354.07i	1.40040i	−0.713948	+	0.700199i	$0.753095\pi$
$42$	0	0
$43$	−468.707	−0.253492	−0.126746	−	0.991935i	$-0.540453\pi$
$43$	−468.707	−0.253492	−0.126746	+	0.991935i	$0.540453\pi$
$44$	− 770.991i	− 0.398239i
$45$	0	0
$46$	2.47139	0.00116796
$47$	− 3910.98i	− 1.77047i	−0.465141	−	0.885237i	$-0.653996\pi$
$47$	− 3910.98i	− 1.77047i	0.465141	−	0.885237i	$-0.346004\pi$
$48$	0	0
$49$	−2254.26	−0.938884
$50$	0	0
$51$	0	0
$52$	−2154.37	−0.796735
$53$	− 3228.98i	− 1.14951i	−0.818325	−	0.574756i	$-0.805097\pi$
$53$	− 3228.98i	− 1.14951i	0.818325	−	0.574756i	$-0.194903\pi$
$54$	0	0
$55$	0	0
$56$	627.429i	0.200073i
$57$	0	0
$58$	−2523.94	−0.750280
$59$	− 2543.15i	− 0.730582i	−0.930893	−	0.365291i	$-0.880970\pi$
$59$	− 2543.15i	− 0.730582i	0.930893	−	0.365291i	$-0.119030\pi$
$60$	0	0
$61$	−254.070	−0.0682800	−0.0341400	−	0.999417i	$-0.510869\pi$
$61$	−254.070	−0.0682800	−0.0341400	+	0.999417i	$0.510869\pi$
$62$	− 2892.40i	− 0.752446i
$63$	0	0
$64$	−79.4731	−0.0194026
$65$	0	0
$66$	0	0
$67$	−7493.35	−1.66927	−0.834634	−	0.550805i	$-0.814321\pi$
$67$	−7493.35	−1.66927	−0.834634	+	0.550805i	$0.814321\pi$
$68$	− 4813.52i	− 1.04099i
$69$	0	0
$70$	0	0
$71$	4143.85i	0.822029i	0.911629	+	0.411014i	$0.134825\pi$
$71$	4143.85i	0.822029i	−0.911629	+	0.411014i	$0.865175\pi$
$72$	0	0
$73$	3711.28	0.696431	0.348215	−	0.937415i	$-0.386788\pi$
$73$	3711.28	0.696431	0.348215	+	0.937415i	$0.386788\pi$
$74$	2386.11i	0.435740i
$75$	0	0
$76$	−575.612	−0.0996559
$77$	732.189i	0.123493i
$78$	0	0
$79$	80.2118	0.0128524	0.00642620	−	0.999979i	$-0.497954\pi$
$79$	80.2118	0.0128524	0.00642620	+	0.999979i	$0.497954\pi$
$80$	0	0
$81$	0	0
$82$	4240.21	0.630608
$83$	5159.51i	0.748949i	0.927237	+	0.374474i	$0.122177\pi$
$83$	5159.51i	0.748949i	−0.927237	+	0.374474i	$0.877823\pi$
$84$	0	0
$85$	0	0
$86$	844.248i	0.114149i
$87$	0	0
$88$	−3130.68	−0.404272
$89$	− 5870.07i	− 0.741076i	−0.928817	−	0.370538i	$-0.879173\pi$
$89$	− 5870.07i	− 0.741076i	0.928817	−	0.370538i	$-0.120827\pi$
$90$	0	0
$91$	2045.95	0.247065
$92$	17.5015i	0.00206775i
$93$	0	0
$94$	−7044.55	−0.797256
$95$	0	0
$96$	0	0
$97$	8921.02	0.948137	0.474068	−	0.880488i	$-0.342785\pi$
$97$	8921.02	0.948137	0.474068	+	0.880488i	$0.342785\pi$
$98$	4060.43i	0.422785i
$99$	0	0
$100$	0	0
$101$	− 2618.57i	− 0.256697i	−0.991729	−	0.128349i	$-0.959032\pi$
$101$	− 2618.57i	− 0.256697i	0.991729	−	0.128349i	$-0.0409676\pi$
$102$	0	0
$103$	9021.12	0.850327	0.425164	−	0.905116i	$-0.360217\pi$
$103$	9021.12	0.850327	0.425164	+	0.905116i	$0.360217\pi$
$104$	8748.04i	0.808805i
$105$	0	0
$106$	−5816.12	−0.517632
$107$	17075.5i	1.49144i	0.666257	+	0.745722i	$0.267895\pi$
$107$	17075.5i	1.49144i	−0.666257	+	0.745722i	$0.732105\pi$
$108$	0	0
$109$	11156.2	0.938997	0.469498	−	0.882933i	$-0.344435\pi$
$109$	11156.2	0.938997	0.469498	+	0.882933i	$0.344435\pi$
$110$	0	0
$111$	0	0
$112$	−1342.12	−0.106993
$113$	15403.5i	1.20632i	0.797621	+	0.603160i	$0.206092\pi$
$113$	15403.5i	1.20632i	−0.797621	+	0.603160i	$0.793908\pi$
$114$	0	0
$115$	0	0
$116$	− 17873.6i	− 1.32830i
$117$	0	0
$118$	−4580.80	−0.328986
$119$	4571.27i	0.322807i
$120$	0	0
$121$	10987.6	0.750468
$122$	457.637i	0.0307469i
$123$	0	0
$124$	20482.9	1.33213
$125$	0	0
$126$	0	0
$127$	24976.5	1.54855	0.774273	−	0.632852i	$-0.218116\pi$
$127$	24976.5	1.54855	0.774273	+	0.632852i	$0.218116\pi$
$128$	− 16309.5i	− 0.995453i
$129$	0	0
$130$	0	0
$131$	− 17627.3i	− 1.02717i	−0.858039	−	0.513585i	$-0.828317\pi$
$131$	− 17627.3i	− 1.02717i	0.858039	−	0.513585i	$-0.171683\pi$
$132$	0	0
$133$	546.643	0.0309030
$134$	13497.2i	0.751682i
$135$	0	0
$136$	−19545.8	−1.05676
$137$	24784.3i	1.32049i	0.751049	+	0.660247i	$0.229548\pi$
$137$	24784.3i	1.32049i	−0.751049	+	0.660247i	$0.770452\pi$
$138$	0	0
$139$	−241.690	−0.0125092	−0.00625460	−	0.999980i	$-0.501991\pi$
$139$	−241.690	−0.0125092	−0.00625460	+	0.999980i	$0.501991\pi$
$140$	0	0
$141$	0	0
$142$	7464.00	0.370165
$143$	10208.7i	0.499226i
$144$	0	0
$145$	0	0
$146$	− 6684.85i	− 0.313607i
$147$	0	0
$148$	−16897.5	−0.771436
$149$	12802.4i	0.576657i	0.957532	+	0.288328i	$0.0930995\pi$
$149$	12802.4i	0.576657i	−0.957532	+	0.288328i	$0.906901\pi$
$150$	0	0
$151$	−14780.0	−0.648218	−0.324109	−	0.946020i	$-0.605065\pi$
$151$	−14780.0	−0.648218	−0.324109	+	0.946020i	$0.605065\pi$
$152$	2337.33i	0.101166i
$153$	0	0
$154$	1318.84	0.0556096
$155$	0	0
$156$	0	0
$157$	9720.94	0.394375	0.197187	−	0.980366i	$-0.436819\pi$
$157$	9720.94	0.394375	0.197187	+	0.980366i	$0.436819\pi$
$158$	− 144.480i	− 0.00578752i
$159$	0	0
$160$	0	0
$161$	− 16.6207i	0 0.000641204i
$162$	0	0
$163$	34436.1	1.29610	0.648051	−	0.761597i	$-0.275584\pi$
$163$	34436.1	1.29610	0.648051	+	0.761597i	$0.275584\pi$
$164$	30027.5i	1.11643i
$165$	0	0
$166$	9293.44	0.337256
$167$	16921.4i	0.606740i	0.952873	+	0.303370i	$0.0981118\pi$
$167$	16921.4i	0.606740i	−0.952873	+	0.303370i	$0.901888\pi$
$168$	0	0
$169$	−35.0031	−0.00122555
$170$	0	0
$171$	0	0
$172$	−5978.64	−0.202090
$173$	39172.9i	1.30886i	0.756122	+	0.654431i	$0.227092\pi$
$173$	39172.9i	1.30886i	−0.756122	+	0.654431i	$0.772908\pi$
$174$	0	0
$175$	0	0
$176$	− 6696.79i	− 0.216193i
$177$	0	0
$178$	−10573.3	−0.333711
$179$	− 51682.6i	− 1.61301i	−0.591224	−	0.806507i	$-0.701355\pi$
$179$	− 51682.6i	− 1.61301i	0.591224	−	0.806507i	$-0.298645\pi$
$180$	0	0
$181$	662.351	0.0202177	0.0101088	−	0.999949i	$-0.496782\pi$
$181$	662.351	0.0202177	0.0101088	+	0.999949i	$0.496782\pi$
$182$	− 3685.22i	− 0.111255i
$183$	0	0
$184$	71.0664	0.00209908
$185$	0	0
$186$	0	0
$187$	−22809.3	−0.652271
$188$	− 49886.8i	− 1.41146i
$189$	0	0
$190$	0	0
$191$	63272.8i	1.73440i	0.497956	+	0.867202i	$0.334084\pi$
$191$	63272.8i	1.73440i	−0.497956	+	0.867202i	$0.665916\pi$
$192$	0	0
$193$	51567.4	1.38440	0.692198	−	0.721707i	$-0.256642\pi$
$193$	51567.4	1.38440	0.692198	+	0.721707i	$0.256642\pi$
$194$	− 16068.8i	− 0.426952i
$195$	0	0
$196$	−28754.4	−0.748501
$197$	11639.8i	0.299926i	0.988692	+	0.149963i	$0.0479154\pi$
$197$	11639.8i	0.299926i	−0.988692	+	0.149963i	$0.952085\pi$
$198$	0	0
$199$	−44340.2	−1.11967	−0.559837	−	0.828603i	$-0.689136\pi$
$199$	−44340.2	−1.11967	−0.559837	+	0.828603i	$0.689136\pi$
$200$	0	0
$201$	0	0
$202$	−4716.63	−0.115592
$203$	16974.1i	0.411902i
$204$	0	0
$205$	0	0
$206$	− 16249.1i	− 0.382908i
$207$	0	0
$208$	−18712.8	−0.432525
$209$	2727.59i	0.0624433i
$210$	0	0
$211$	34425.8	0.773249	0.386624	−	0.922237i	$-0.373641\pi$
$211$	34425.8	0.773249	0.386624	+	0.922237i	$0.373641\pi$
$212$	− 41187.5i	− 0.916418i
$213$	0	0
$214$	30756.9	0.671607
$215$	0	0
$216$	0	0
$217$	−19452.0	−0.413091
$218$	− 20094.9i	− 0.422836i
$219$	0	0
$220$	0	0
$221$	63735.7i	1.30496i
$222$	0	0
$223$	−80710.8	−1.62301	−0.811506	−	0.584344i	$-0.801352\pi$
$223$	−80710.8	−1.62301	−0.811506	+	0.584344i	$0.801352\pi$
$224$	12456.3i	0.248253i
$225$	0	0
$226$	27745.2	0.543213
$227$	− 96827.1i	− 1.87908i	−0.342441	−	0.939539i	$-0.611254\pi$
$227$	− 96827.1i	− 1.87908i	0.342441	−	0.939539i	$-0.388746\pi$
$228$	0	0
$229$	−63732.2	−1.21531	−0.607656	−	0.794200i	$-0.707890\pi$
$229$	−63732.2	−1.21531	−0.607656	+	0.794200i	$0.707890\pi$
$230$	0	0
$231$	0	0
$232$	−72577.5	−1.34842
$233$	− 22886.7i	− 0.421571i	−0.977532	−	0.210786i	$-0.932398\pi$
$233$	− 22886.7i	− 0.421571i	0.977532	−	0.210786i	$-0.0676022\pi$
$234$	0	0
$235$	0	0
$236$	− 32439.4i	− 0.582437i
$237$	0	0
$238$	8233.89	0.145362
$239$	− 26181.9i	− 0.458358i	−0.973384	−	0.229179i	$-0.926396\pi$
$239$	− 26181.9i	− 0.458358i	0.973384	−	0.229179i	$-0.0736041\pi$
$240$	0	0
$241$	5080.09	0.0874657	0.0437328	−	0.999043i	$-0.486075\pi$
$241$	5080.09	0.0874657	0.0437328	+	0.999043i	$0.486075\pi$
$242$	− 19791.1i	− 0.337940i
$243$	0	0
$244$	−3240.81	−0.0544345
$245$	0	0
$246$	0	0
$247$	7621.67	0.124927
$248$	− 83172.8i	− 1.35231i
$249$	0	0
$250$	0	0
$251$	23074.8i	0.366260i	0.983089	+	0.183130i	$0.0586230\pi$
$251$	23074.8i	0.366260i	−0.983089	+	0.183130i	$0.941377\pi$
$252$	0	0
$253$	82.9321	0.00129563
$254$	− 44988.3i	− 0.697320i
$255$	0	0
$256$	−30648.7	−0.467661
$257$	− 51974.3i	− 0.786905i	−0.919345	−	0.393452i	$-0.871281\pi$
$257$	− 51974.3i	− 0.786905i	0.919345	−	0.393452i	$-0.128719\pi$
$258$	0	0
$259$	16047.1	0.239220
$260$	0	0
$261$	0	0
$262$	−31750.7	−0.462541
$263$	24353.6i	0.352089i	0.984382	+	0.176044i	$0.0563302\pi$
$263$	24353.6i	0.352089i	−0.984382	+	0.176044i	$0.943670\pi$
$264$	0	0
$265$	0	0
$266$	− 984.628i	− 0.0139158i
$267$	0	0
$268$	−95582.0	−1.33078
$269$	− 98409.2i	− 1.35998i	−0.733223	−	0.679988i	$-0.761985\pi$
$269$	− 98409.2i	− 1.35998i	0.733223	−	0.679988i	$-0.238015\pi$
$270$	0	0
$271$	135833.	1.84956	0.924778	−	0.380508i	$-0.124251\pi$
$271$	135833.	1.84956	0.924778	+	0.380508i	$0.124251\pi$
$272$	− 41810.0i	− 0.565122i
$273$	0	0
$274$	44642.2	0.594627
$275$	0	0
$276$	0	0
$277$	−19597.6	−0.255413	−0.127706	−	0.991812i	$-0.540762\pi$
$277$	−19597.6	−0.255413	−0.127706	+	0.991812i	$0.540762\pi$
$278$	435.339i	0.00563297i
$279$	0	0
$280$	0	0
$281$	− 25457.8i	− 0.322409i	−0.986921	−	0.161205i	$-0.948462\pi$
$281$	− 25457.8i	− 0.322409i	0.986921	−	0.161205i	$-0.0515379\pi$
$282$	0	0
$283$	88022.8	1.09906	0.549531	−	0.835473i	$-0.314806\pi$
$283$	88022.8	1.09906	0.549531	+	0.835473i	$0.314806\pi$
$284$	52857.2i	0.655341i
$285$	0	0
$286$	18388.1	0.224805
$287$	− 28516.3i	− 0.346202i
$288$	0	0
$289$	−58883.9	−0.705020
$290$	0	0
$291$	0	0
$292$	47339.5	0.555211
$293$	91729.0i	1.06849i	0.845329	+	0.534246i	$0.179404\pi$
$293$	91729.0i	1.06849i	−0.845329	+	0.534246i	$0.820596\pi$
$294$	0	0
$295$	0	0
$296$	68614.1i	0.783123i
$297$	0	0
$298$	23059.9	0.259672
$299$	− 231.736i	− 0.00259210i
$300$	0	0
$301$	5677.75	0.0626676
$302$	26622.2i	0.291897i
$303$	0	0
$304$	−4999.74	−0.0541004
$305$	0	0
$306$	0	0
$307$	−1346.81	−0.0142899	−0.00714494	−	0.999974i	$-0.502274\pi$
$307$	−1346.81	−0.0142899	−0.00714494	+	0.999974i	$0.502274\pi$
$308$	9339.50i	0.0984515i
$309$	0	0
$310$	0	0
$311$	48663.9i	0.503137i	0.967839	+	0.251568i	$0.0809464\pi$
$311$	48663.9i	0.503137i	−0.967839	+	0.251568i	$0.919054\pi$
$312$	0	0
$313$	−106387.	−1.08593	−0.542963	−	0.839757i	$-0.682698\pi$
$313$	−106387.	−1.08593	−0.542963	+	0.839757i	$0.682698\pi$
$314$	− 17509.6i	− 0.177589i
$315$	0	0
$316$	1023.15	0.0102462
$317$	92241.5i	0.917926i	0.888455	+	0.458963i	$0.151779\pi$
$317$	92241.5i	0.917926i	−0.888455	+	0.458963i	$0.848221\pi$
$318$	0	0
$319$	−84695.5	−0.832298
$320$	0	0
$321$	0	0
$322$	−29.9376	−0.000288738 0
$323$	17029.1i	0.163225i
$324$	0	0
$325$	0	0
$326$	− 62027.2i	− 0.583643i
$327$	0	0
$328$	121930.	1.13334
$329$	47376.1i	0.437691i
$330$	0	0
$331$	−160354.	−1.46361	−0.731803	−	0.681516i	$-0.761321\pi$
$331$	−160354.	−1.46361	−0.731803	+	0.681516i	$0.761321\pi$
$332$	65812.5i	0.597080i
$333$	0	0
$334$	30479.2	0.273219
$335$	0	0
$336$	0	0
$337$	213689.	1.88158	0.940788	−	0.338996i	$-0.110087\pi$
$337$	213689.	1.88158	0.940788	+	0.338996i	$0.110087\pi$
$338$	63.0484i	0 0.000551875i
$339$	0	0
$340$	0	0
$341$	− 97059.9i	− 0.834701i
$342$	0	0
$343$	56392.1	0.479325
$344$	24276.8i	0.205152i
$345$	0	0
$346$	70559.3	0.589389
$347$	61402.4i	0.509948i	0.966948	+	0.254974i	$0.0820670\pi$
$347$	61402.4i	0.509948i	−0.966948	+	0.254974i	$0.917933\pi$
$348$	0	0
$349$	215730.	1.77117	0.885585	−	0.464477i	$-0.153758\pi$
$349$	215730.	1.77117	0.885585	+	0.464477i	$0.153758\pi$
$350$	0	0
$351$	0	0
$352$	−62153.4	−0.501625
$353$	− 13643.8i	− 0.109493i	−0.998500	−	0.0547465i	$-0.982565\pi$
$353$	− 13643.8i	− 0.109493i	0.998500	−	0.0547465i	$-0.0174351\pi$
$354$	0	0
$355$	0	0
$356$	− 74876.1i	− 0.590804i
$357$	0	0
$358$	−93092.0	−0.726351
$359$	130543.i	1.01290i	0.862270	+	0.506449i	$0.169042\pi$
$359$	130543.i	1.01290i	−0.862270	+	0.506449i	$0.830958\pi$
$360$	0	0
$361$	−128285.	−0.984374
$362$	− 1193.04i	− 0.00910414i
$363$	0	0
$364$	26097.3	0.196967
$365$	0	0
$366$	0	0
$367$	−168011.	−1.24740	−0.623701	−	0.781663i	$-0.714372\pi$
$367$	−168011.	−1.24740	−0.623701	+	0.781663i	$0.714372\pi$
$368$	152.017i	0.00112252i
$369$	0	0
$370$	0	0
$371$	39114.6i	0.284179i
$372$	0	0
$373$	158259.	1.13750	0.568750	−	0.822510i	$-0.307427\pi$
$373$	158259.	1.13750	0.568750	+	0.822510i	$0.307427\pi$
$374$	41084.6i	0.293722i
$375$	0	0
$376$	−202570.	−1.43285
$377$	236664.i	1.66513i
$378$	0	0
$379$	−18061.8	−0.125743	−0.0628715	−	0.998022i	$-0.520026\pi$
$379$	−18061.8	−0.125743	−0.0628715	+	0.998022i	$0.520026\pi$
$380$	0	0
$381$	0	0
$382$	113969.	0.781013
$383$	− 110035.i	− 0.750126i	−0.926999	−	0.375063i	$-0.877621\pi$
$383$	− 110035.i	− 0.750126i	0.926999	−	0.375063i	$-0.122379\pi$
$384$	0	0
$385$	0	0
$386$	− 92884.5i	− 0.623403i
$387$	0	0
$388$	113793.	0.755877
$389$	247517.i	1.63571i	0.575425	+	0.817855i	$0.304837\pi$
$389$	247517.i	1.63571i	−0.575425	+	0.817855i	$0.695163\pi$
$390$	0	0
$391$	517.769	0.00338675
$392$	116760.i	0.759840i
$393$	0	0
$394$	20966.0	0.135059
$395$	0	0
$396$	0	0
$397$	170927.	1.08450	0.542249	−	0.840218i	$-0.317573\pi$
$397$	170927.	1.08450	0.542249	+	0.840218i	$0.317573\pi$
$398$	79866.7i	0.504196i
$399$	0	0
$400$	0	0
$401$	− 209413.i	− 1.30231i	−0.758944	−	0.651156i	$-0.774284\pi$
$401$	− 209413.i	− 1.30231i	0.758944	−	0.651156i	$-0.225716\pi$
$402$	0	0
$403$	−271213.	−1.66994
$404$	− 33401.4i	− 0.204645i
$405$	0	0
$406$	30574.1	0.185482
$407$	80070.3i	0.483374i
$408$	0	0
$409$	47045.0	0.281233	0.140617	−	0.990064i	$-0.455092\pi$
$409$	47045.0	0.281233	0.140617	+	0.990064i	$0.455092\pi$
$410$	0	0
$411$	0	0
$412$	115070.	0.677902
$413$	30806.8i	0.180612i
$414$	0	0
$415$	0	0
$416$	173675.i	1.00357i
$417$	0	0
$418$	4913.00	0.0281186
$419$	− 88341.4i	− 0.503195i	−0.967832	−	0.251597i	$-0.919044\pi$
$419$	− 88341.4i	− 0.503195i	0.967832	−	0.251597i	$-0.0809559\pi$
$420$	0	0
$421$	−146004.	−0.823759	−0.411879	−	0.911238i	$-0.635127\pi$
$421$	−146004.	−0.823759	−0.411879	+	0.911238i	$0.635127\pi$
$422$	− 62008.6i	− 0.348199i
$423$	0	0
$424$	−167246.	−0.930301
$425$	0	0
$426$	0	0
$427$	3077.71	0.0168800
$428$	217809.i	1.18901i
$429$	0	0
$430$	0	0
$431$	− 38335.8i	− 0.206372i	−0.994662	−	0.103186i	$-0.967096\pi$
$431$	− 38335.8i	− 0.206372i	0.994662	−	0.103186i	$-0.0329036\pi$
$432$	0	0
$433$	236134.	1.25946	0.629728	−	0.776815i	$-0.283166\pi$
$433$	236134.	1.25946	0.629728	+	0.776815i	$0.283166\pi$
$434$	35037.5i	0.186018i
$435$	0	0
$436$	142304.	0.748591
$437$	− 61.9161i	0 0.000324221i
$438$	0	0
$439$	240933.	1.25017	0.625083	−	0.780558i	$-0.285065\pi$
$439$	240933.	1.25017	0.625083	+	0.780558i	$0.285065\pi$
$440$	0	0
$441$	0	0
$442$	114802.	0.587633
$443$	− 70261.5i	− 0.358022i	−0.983847	−	0.179011i	$-0.942710\pi$
$443$	− 70261.5i	− 0.358022i	0.983847	−	0.179011i	$-0.0572898\pi$
$444$	0	0
$445$	0	0
$446$	145378.i	0.730853i
$447$	0	0
$448$	962.708	0.00479666
$449$	93152.3i	0.462062i	0.972946	+	0.231031i	$0.0742100\pi$
$449$	93152.3i	0.462062i	−0.972946	+	0.231031i	$0.925790\pi$
$450$	0	0
$451$	142288.	0.699544
$452$	196481.i	0.961707i
$453$	0	0
$454$	−174407.	−0.846161
$455$	0	0
$456$	0	0
$457$	−145652.	−0.697406	−0.348703	−	0.937233i	$-0.613378\pi$
$457$	−145652.	−0.697406	−0.348703	+	0.937233i	$0.613378\pi$
$458$	114796.i	0.547263i
$459$	0	0
$460$	0	0
$461$	− 128000.i	− 0.602294i	−0.953578	−	0.301147i	$-0.902630\pi$
$461$	− 128000.i	− 0.602294i	0.953578	−	0.301147i	$-0.0973695\pi$
$462$	0	0
$463$	112886.	0.526597	0.263298	−	0.964714i	$-0.415190\pi$
$463$	112886.	0.526597	0.263298	+	0.964714i	$0.415190\pi$
$464$	− 155249.i	− 0.721096i
$465$	0	0
$466$	−41224.1	−0.189836
$467$	− 317214.i	− 1.45452i	−0.686364	−	0.727258i	$-0.740794\pi$
$467$	− 317214.i	− 1.45452i	0.686364	−	0.727258i	$-0.259206\pi$
$468$	0	0
$469$	90771.7	0.412672
$470$	0	0
$471$	0	0
$472$	−131723.	−0.591261
$473$	28330.3i	0.126628i
$474$	0	0
$475$	0	0
$476$	58309.2i	0.257350i
$477$	0	0
$478$	−47159.4	−0.206402
$479$	− 374765.i	− 1.63338i	−0.577073	−	0.816692i	$-0.695805\pi$
$479$	− 374765.i	− 1.63338i	0.577073	−	0.816692i	$-0.304195\pi$
$480$	0	0
$481$	223740.	0.967059
$482$	− 9150.40i	− 0.0393864i
$483$	0	0
$484$	140153.	0.598291
$485$	0	0
$486$	0	0
$487$	47680.4	0.201040	0.100520	−	0.994935i	$-0.467949\pi$
$487$	47680.4	0.201040	0.100520	+	0.994935i	$0.467949\pi$
$488$	13159.6i	0.0552591i
$489$	0	0
$490$	0	0
$491$	− 153766.i	− 0.637817i	−0.947786	−	0.318908i	$-0.896684\pi$
$491$	− 153766.i	− 0.637817i	0.947786	−	0.318908i	$-0.103316\pi$
$492$	0	0
$493$	−528779.	−2.17561
$494$	− 13728.3i	− 0.0562554i
$495$	0	0
$496$	177913.	0.723178
$497$	− 50197.0i	− 0.203220i
$498$	0	0
$499$	−348851.	−1.40100	−0.700501	−	0.713651i	$-0.747040\pi$
$499$	−348851.	−1.40100	−0.700501	+	0.713651i	$0.747040\pi$
$500$	0	0
$501$	0	0
$502$	41562.9	0.164929
$503$	− 269552.i	− 1.06539i	−0.846308	−	0.532693i	$-0.821180\pi$
$503$	− 269552.i	− 1.06539i	0.846308	−	0.532693i	$-0.178820\pi$
$504$	0	0
$505$	0	0
$506$	− 149.379i	0 0.000583431i
$507$	0	0
$508$	318590.	1.23454
$509$	177376.i	0.684636i	0.939584	+	0.342318i	$0.111212\pi$
$509$	177376.i	0.684636i	−0.939584	+	0.342318i	$0.888788\pi$
$510$	0	0
$511$	−44957.1	−0.172170
$512$	− 205747.i	− 0.784862i
$513$	0	0
$514$	−93617.4	−0.354348
$515$	0	0
$516$	0	0
$517$	−236393.	−0.884409
$518$	− 28904.5i	− 0.107722i
$519$	0	0
$520$	0	0
$521$	233062.i	0.858609i	0.903160	+	0.429305i	$0.141241\pi$
$521$	233062.i	0.858609i	−0.903160	+	0.429305i	$0.858759\pi$
$522$	0	0
$523$	−315841.	−1.15469	−0.577345	−	0.816501i	$-0.695911\pi$
$523$	−315841.	−1.15469	−0.577345	+	0.816501i	$0.695911\pi$
$524$	− 224846.i	− 0.818885i
$525$	0	0
$526$	43866.4	0.158548
$527$	− 605973.i	− 2.18189i
$528$	0	0
$529$	279839.	0.999993
$530$	0	0
$531$	0	0
$532$	6972.76	0.0246366
$533$	− 397594.i	− 1.39954i
$534$	0	0
$535$	0	0
$536$	388120.i	1.35094i
$537$	0	0
$538$	−177257.	−0.612406
$539$	136255.i	0.469003i
$540$	0	0
$541$	−186404.	−0.636886	−0.318443	−	0.947942i	$-0.603160\pi$
$541$	−186404.	−0.636886	−0.318443	+	0.947942i	$0.603160\pi$
$542$	− 244666.i	− 0.832867i
$543$	0	0
$544$	−388042.	−1.31123
$545$	0	0
$546$	0	0
$547$	−443355.	−1.48176	−0.740879	−	0.671639i	$-0.765591\pi$
$547$	−443355.	−1.48176	−0.740879	+	0.671639i	$0.765591\pi$
$548$	316139.i	1.05273i
$549$	0	0
$550$	0	0
$551$	63232.6i	0.208275i
$552$	0	0
$553$	−971.657	−0.00317733
$554$	35299.7i	0.115014i
$555$	0	0
$556$	−3082.90	−0.00997263
$557$	82562.1i	0.266116i	0.991108	+	0.133058i	$0.0424796\pi$
$557$	82562.1i	0.266116i	−0.991108	+	0.133058i	$0.957520\pi$
$558$	0	0
$559$	79163.0	0.253337
$560$	0	0
$561$	0	0
$562$	−45855.2	−0.145183
$563$	− 518910.i	− 1.63710i	−0.574435	−	0.818550i	$-0.694778\pi$
$563$	− 518910.i	− 1.63710i	0.574435	−	0.818550i	$-0.305222\pi$
$564$	0	0
$565$	0	0
$566$	− 158549.i	− 0.494915i
$567$	0	0
$568$	214632.	0.665269
$569$	− 152195.i	− 0.470085i	−0.971985	−	0.235042i	$-0.924477\pi$
$569$	− 152195.i	− 0.470085i	0.971985	−	0.235042i	$-0.0755229\pi$
$570$	0	0
$571$	414409.	1.27103	0.635516	−	0.772088i	$-0.280787\pi$
$571$	414409.	1.27103	0.635516	+	0.772088i	$0.280787\pi$
$572$	130218.i	0.397995i
$573$	0	0
$574$	−51364.4	−0.155897
$575$	0	0
$576$	0	0
$577$	31681.0	0.0951585	0.0475793	−	0.998867i	$-0.484849\pi$
$577$	31681.0	0.0951585	0.0475793	+	0.998867i	$0.484849\pi$
$578$	106063.i	0.317475i
$579$	0	0
$580$	0	0
$581$	− 62500.4i	− 0.185153i
$582$	0	0
$583$	−195170.	−0.574218
$584$	− 192227.i	− 0.563622i
$585$	0	0
$586$	165225.	0.481149
$587$	− 347152.i	− 1.00750i	−0.863850	−	0.503748i	$-0.831954\pi$
$587$	− 347152.i	− 1.00750i	0.863850	−	0.503748i	$-0.168046\pi$
$588$	0	0
$589$	−72463.7	−0.208877
$590$	0	0
$591$	0	0
$592$	−146771.	−0.418791
$593$	− 27468.2i	− 0.0781125i	−0.999237	−	0.0390563i	$-0.987565\pi$
$593$	− 27468.2i	− 0.0781125i	0.999237	−	0.0390563i	$-0.0124352\pi$
$594$	0	0
$595$	0	0
$596$	163302.i	0.459725i
$597$	0	0
$598$	−417.409	−0.00116724
$599$	− 600764.i	− 1.67436i	−0.546924	−	0.837182i	$-0.684201\pi$
$599$	− 600764.i	− 1.67436i	0.546924	−	0.837182i	$-0.315799\pi$
$600$	0	0
$601$	−494521.	−1.36910	−0.684551	−	0.728965i	$-0.740002\pi$
$601$	−494521.	−1.36910	−0.684551	+	0.728965i	$0.740002\pi$
$602$	− 10226.9i	− 0.0282196i
$603$	0	0
$604$	−188528.	−0.516775
$605$	0	0
$606$	0	0
$607$	−578761.	−1.57080	−0.785401	−	0.618987i	$-0.787543\pi$
$607$	−578761.	−1.57080	−0.785401	+	0.618987i	$0.787543\pi$
$608$	46402.9i	0.125527i
$609$	0	0
$610$	0	0
$611$	660550.i	1.76939i
$612$	0	0
$613$	538455.	1.43294	0.716471	−	0.697616i	$-0.245756\pi$
$613$	538455.	1.43294	0.716471	+	0.697616i	$0.245756\pi$
$614$	2425.90i	0.00643483i
$615$	0	0
$616$	37924.0	0.0999430
$617$	486339.i	1.27752i	0.769404	+	0.638762i	$0.220553\pi$
$617$	486339.i	1.27752i	−0.769404	+	0.638762i	$0.779447\pi$
$618$	0	0
$619$	−163391.	−0.426430	−0.213215	−	0.977005i	$-0.568394\pi$
$619$	−163391.	−0.426430	−0.213215	+	0.977005i	$0.568394\pi$
$620$	0	0
$621$	0	0
$622$	87654.7	0.226566
$623$	71107.8i	0.183207i
$624$	0	0
$625$	0	0
$626$	191627.i	0.488999i
$627$	0	0
$628$	123996.	0.314405
$629$	499903.i	1.26353i
$630$	0	0
$631$	−132675.	−0.333220	−0.166610	−	0.986023i	$-0.553282\pi$
$631$	−132675.	−0.333220	−0.166610	+	0.986023i	$0.553282\pi$
$632$	− 4154.60i	− 0.0104015i
$633$	0	0
$634$	166148.	0.413348
$635$	0	0
$636$	0	0
$637$	380736.	0.938308
$638$	152556.i	0.374789i
$639$	0	0
$640$	0	0
$641$	− 648011.i	− 1.57713i	−0.614954	−	0.788563i	$-0.710826\pi$
$641$	− 648011.i	− 1.57713i	0.614954	−	0.788563i	$-0.289174\pi$
$642$	0	0
$643$	62712.3	0.151681	0.0758405	−	0.997120i	$-0.475836\pi$
$643$	62712.3	0.151681	0.0758405	+	0.997120i	$0.475836\pi$
$644$	− 212.006i	0 0.000511184i
$645$	0	0
$646$	30673.3	0.0735013
$647$	− 472097.i	− 1.12778i	−0.825852	−	0.563888i	$-0.809305\pi$
$647$	− 472097.i	− 1.12778i	0.825852	−	0.563888i	$-0.190695\pi$
$648$	0	0
$649$	−153717.	−0.364949
$650$	0	0
$651$	0	0
$652$	439253.	1.03328
$653$	233253.i	0.547017i	0.961870	+	0.273509i	$0.0881842\pi$
$653$	233253.i	0.547017i	−0.961870	+	0.273509i	$0.911816\pi$
$654$	0	0
$655$	0	0
$656$	260818.i	0.606079i
$657$	0	0
$658$	85335.1	0.197095
$659$	399872.i	0.920768i	0.887720	+	0.460384i	$0.152288\pi$
$659$	399872.i	0.920768i	−0.887720	+	0.460384i	$0.847712\pi$
$660$	0	0
$661$	112993.	0.258613	0.129306	−	0.991605i	$-0.458725\pi$
$661$	112993.	0.258613	0.129306	+	0.991605i	$0.458725\pi$
$662$	288834.i	0.659071i
$663$	0	0
$664$	267238.	0.606125
$665$	0	0
$666$	0	0
$667$	1922.58	0.00432149
$668$	215842.i	0.483708i
$669$	0	0
$670$	0	0
$671$	15356.9i	0.0341081i
$672$	0	0
$673$	−544760.	−1.20275	−0.601374	−	0.798967i	$-0.705380\pi$
$673$	−544760.	−1.20275	−0.601374	+	0.798967i	$0.705380\pi$
$674$	− 384901.i	− 0.847285i
$675$	0	0
$676$	−446.485	−0.000977042 0
$677$	475103.i	1.03660i	0.855199	+	0.518299i	$0.173435\pi$
$677$	475103.i	1.03660i	−0.855199	+	0.518299i	$0.826565\pi$
$678$	0	0
$679$	−108066.	−0.234396
$680$	0	0
$681$	0	0
$682$	−174827.	−0.375871
$683$	− 53819.1i	− 0.115371i	−0.998335	−	0.0576853i	$-0.981628\pi$
$683$	− 53819.1i	− 0.115371i	0.998335	−	0.0576853i	$-0.0183720\pi$
$684$	0	0
$685$	0	0
$686$	− 101575.i	− 0.215843i
$687$	0	0
$688$	−51930.1	−0.109709
$689$	545363.i	1.14881i
$690$	0	0
$691$	−400653.	−0.839097	−0.419548	−	0.907733i	$-0.637812\pi$
$691$	−400653.	−0.839097	−0.419548	+	0.907733i	$0.637812\pi$
$692$	499673.i	1.04346i
$693$	0	0
$694$	110600.	0.229633
$695$	0	0
$696$	0	0
$697$	888345.	1.82859
$698$	− 388579.i	− 0.797569i
$699$	0	0
$700$	0	0
$701$	− 860604.i	− 1.75133i	−0.482921	−	0.875664i	$-0.660424\pi$
$701$	− 860604.i	− 1.75133i	0.482921	−	0.875664i	$-0.339576\pi$
$702$	0	0
$703$	59779.5	0.120960
$704$	4803.63i	0.00969223i
$705$	0	0
$706$	−24575.6	−0.0493054
$707$	31720.4i	0.0634599i
$708$	0	0
$709$	489913.	0.974601	0.487300	−	0.873234i	$-0.337982\pi$
$709$	489913.	0.974601	0.487300	+	0.873234i	$0.337982\pi$
$710$	0	0
$711$	0	0
$712$	−304042.	−0.599754
$713$	2203.25i	0.00433397i
$714$	0	0
$715$	0	0
$716$	− 659242.i	− 1.28593i
$717$	0	0
$718$	235138.	0.456114
$719$	288471.i	0.558014i	0.960289	+	0.279007i	$0.0900053\pi$
$719$	288471.i	0.558014i	−0.960289	+	0.279007i	$0.909995\pi$
$720$	0	0
$721$	−109279.	−0.210215
$722$	231070.i	0.443270i
$723$	0	0
$724$	8448.67	0.0161180
$725$	0	0
$726$	0	0
$727$	−75173.7	−0.142232	−0.0711160	−	0.997468i	$-0.522656\pi$
$727$	−75173.7	−0.142232	−0.0711160	+	0.997468i	$0.522656\pi$
$728$	− 105971.i	− 0.199950i
$729$	0	0
$730$	0	0
$731$	176874.i	0.331001i
$732$	0	0
$733$	108535.	0.202005	0.101003	−	0.994886i	$-0.467795\pi$
$733$	108535.	0.202005	0.101003	+	0.994886i	$0.467795\pi$
$734$	302627.i	0.561714i
$735$	0	0
$736$	1410.88	0.00260456
$737$	452923.i	0.833853i
$738$	0	0
$739$	654768.	1.19894	0.599472	−	0.800396i	$-0.295377\pi$
$739$	654768.	1.19894	0.599472	+	0.800396i	$0.295377\pi$
$740$	0	0
$741$	0	0
$742$	70454.3	0.127967
$743$	423118.i	0.766450i	0.923655	+	0.383225i	$0.125187\pi$
$743$	423118.i	0.766450i	−0.923655	+	0.383225i	$0.874813\pi$
$744$	0	0
$745$	0	0
$746$	− 285061.i	− 0.512224i
$747$	0	0
$748$	−290946.	−0.520006
$749$	− 206847.i	− 0.368710i
$750$	0	0
$751$	93003.6	0.164900	0.0824498	−	0.996595i	$-0.473726\pi$
$751$	93003.6	0.164900	0.0824498	+	0.996595i	$0.473726\pi$
$752$	− 433314.i	− 0.766244i
$753$	0	0
$754$	426285.	0.749820
$755$	0	0
$756$	0	0
$757$	849613.	1.48262	0.741309	−	0.671164i	$-0.234205\pi$
$757$	849613.	1.48262	0.741309	+	0.671164i	$0.234205\pi$
$758$	32533.4i	0.0566228i
$759$	0	0
$760$	0	0
$761$	− 407917.i	− 0.704373i	−0.935930	−	0.352186i	$-0.885438\pi$
$761$	− 407917.i	− 0.704373i	0.935930	−	0.352186i	$-0.114562\pi$
$762$	0	0
$763$	−135142.	−0.232136
$764$	807082.i	1.38271i
$765$	0	0
$766$	−198198.	−0.337787
$767$	429530.i	0.730134i
$768$	0	0
$769$	569500.	0.963033	0.481517	−	0.876437i	$-0.340086\pi$
$769$	569500.	0.963033	0.481517	+	0.876437i	$0.340086\pi$
$770$	0	0
$771$	0	0
$772$	657772.	1.10367
$773$	711311.i	1.19042i	0.803570	+	0.595211i	$0.202931\pi$
$773$	711311.i	1.19042i	−0.803570	+	0.595211i	$0.797069\pi$
$774$	0	0
$775$	0	0
$776$	− 462067.i	− 0.767329i
$777$	0	0
$778$	445834.	0.736571
$779$	− 106230.i	− 0.175055i
$780$	0	0
$781$	250468.	0.410630
$782$	− 932.619i	− 0.00152507i
$783$	0	0
$784$	−249759.	−0.406340
$785$	0	0
$786$	0	0
$787$	215275.	0.347572	0.173786	−	0.984783i	$-0.444400\pi$
$787$	215275.	0.347572	0.173786	+	0.984783i	$0.444400\pi$
$788$	148473.i	0.239108i
$789$	0	0
$790$	0	0
$791$	− 186592.i	− 0.298223i
$792$	0	0
$793$	42911.5	0.0682382
$794$	− 307878.i	− 0.488357i
$795$	0	0
$796$	−565585.	−0.892630
$797$	623058.i	0.980871i	0.871478	+	0.490435i	$0.163162\pi$
$797$	623058.i	0.980871i	−0.871478	+	0.490435i	$0.836838\pi$
$798$	0	0
$799$	−1.47587e6	−2.31182
$800$	0	0
$801$	0	0
$802$	−377200.	−0.586440
$803$	− 224322.i	− 0.347890i
$804$	0	0
$805$	0	0
$806$	488517.i	0.751985i
$807$	0	0
$808$	−135630.	−0.207745
$809$	385095.i	0.588398i	0.955744	+	0.294199i	$0.0950528\pi$
$809$	385095.i	0.588398i	−0.955744	+	0.294199i	$0.904947\pi$
$810$	0	0
$811$	586828.	0.892215	0.446107	−	0.894979i	$-0.352810\pi$
$811$	586828.	0.892215	0.446107	+	0.894979i	$0.352810\pi$
$812$	216514.i	0.328378i
$813$	0	0
$814$	144225.	0.217666
$815$	0	0
$816$	0	0
$817$	21151.0	0.0316875
$818$	− 84738.6i	− 0.126641i
$819$	0	0
$820$	0	0
$821$	879206.i	1.30438i	0.758055	+	0.652191i	$0.226150\pi$
$821$	879206.i	1.30438i	−0.758055	+	0.652191i	$0.773850\pi$
$822$	0	0
$823$	−56403.1	−0.0832728	−0.0416364	−	0.999133i	$-0.513257\pi$
$823$	−56403.1	−0.0832728	−0.0416364	+	0.999133i	$0.513257\pi$
$824$	− 467252.i	− 0.688171i
$825$	0	0
$826$	55490.1	0.0813308
$827$	− 201627.i	− 0.294807i	−0.989076	−	0.147403i	$-0.952908\pi$
$827$	− 201627.i	− 0.294807i	0.989076	−	0.147403i	$-0.0470916\pi$
$828$	0	0
$829$	−85071.2	−0.123787	−0.0618933	−	0.998083i	$-0.519714\pi$
$829$	−85071.2	−0.123787	−0.0618933	+	0.998083i	$0.519714\pi$
$830$	0	0
$831$	0	0
$832$	13422.7	0.0193907
$833$	850681.i	1.22596i
$834$	0	0
$835$	0	0
$836$	34792.0i	0.0497813i
$837$	0	0
$838$	−159123.	−0.226592
$839$	− 150877.i	− 0.214338i	−0.994241	−	0.107169i	$-0.965821\pi$
$839$	− 150877.i	− 0.214338i	0.994241	−	0.107169i	$-0.0341785\pi$
$840$	0	0
$841$	−1.25618e6	−1.77607
$842$	262986.i	0.370944i
$843$	0	0
$844$	439121.	0.616452
$845$	0	0
$846$	0	0
$847$	−133100.	−0.185528
$848$	− 357752.i	− 0.497497i
$849$	0	0
$850$	0	0
$851$	− 1817.59i	− 0.00250979i
$852$	0	0
$853$	1.17719e6	1.61788	0.808941	−	0.587890i	$-0.200041\pi$
$853$	1.17719e6	1.61788	0.808941	+	0.587890i	$0.200041\pi$
$854$	− 5543.65i	− 0.00760117i
$855$	0	0
$856$	884433.	1.20703
$857$	341140.i	0.464485i	0.972658	+	0.232242i	$0.0746062\pi$
$857$	341140.i	0.464485i	−0.972658	+	0.232242i	$0.925394\pi$
$858$	0	0
$859$	44556.7	0.0603846	0.0301923	−	0.999544i	$-0.490388\pi$
$859$	44556.7	0.0603846	0.0301923	+	0.999544i	$0.490388\pi$
$860$	0	0
$861$	0	0
$862$	−69051.4	−0.0929304
$863$	− 5280.21i	− 0.00708973i	−0.999994	−	0.00354486i	$-0.998872\pi$
$863$	− 5280.21i	− 0.00708973i	0.999994	−	0.00354486i	$-0.00112837\pi$
$864$	0	0
$865$	0	0
$866$	− 425331.i	− 0.567141i
$867$	0	0
$868$	−248122.	−0.329326
$869$	− 4848.27i	− 0.00642019i
$870$	0	0
$871$	1.26560e6	1.66825
$872$	− 577840.i	− 0.759932i
$873$	0	0
$874$	−111.525	−0.000145999 0
$875$	0	0
$876$	0	0
$877$	−1228.71	−0.00159754	−0.000798768	−	1.00000i	$-0.500254\pi$
$877$	−1228.71	−0.00159754	−0.000798768		1.00000i	$0.500254\pi$
$878$	− 433975.i	− 0.562958i
$879$	0	0
$880$	0	0
$881$	513912.i	0.662120i	0.943610	+	0.331060i	$0.107406\pi$
$881$	513912.i	0.662120i	−0.943610	+	0.331060i	$0.892594\pi$
$882$	0	0
$883$	−116500.	−0.149419	−0.0747093	−	0.997205i	$-0.523803\pi$
$883$	−116500.	−0.149419	−0.0747093	+	0.997205i	$0.523803\pi$
$884$	812987.i	1.04035i
$885$	0	0
$886$	−126557.	−0.161220
$887$	− 1.23199e6i	− 1.56588i	−0.622096	−	0.782941i	$-0.713719\pi$
$887$	− 1.23199e6i	− 1.56588i	0.622096	−	0.782941i	$-0.286281\pi$
$888$	0	0
$889$	−302556.	−0.382827
$890$	0	0
$891$	0	0
$892$	−1.02951e6	−1.29390
$893$	176488.i	0.221316i
$894$	0	0
$895$	0	0
$896$	197567.i	0.246093i
$897$	0	0
$898$	167788.	0.208070
$899$	− 2.25010e6i	− 2.78409i
$900$	0	0
$901$	−1.21851e6	−1.50099
$902$	− 256293.i	− 0.315009i
$903$	0	0
$904$	797828.	0.976276
$905$	0	0
$906$	0	0
$907$	−1.02123e6	−1.24139	−0.620694	−	0.784053i	$-0.713149\pi$
$907$	−1.02123e6	−1.24139	−0.620694	+	0.784053i	$0.713149\pi$
$908$	− 1.23509e6i	− 1.49805i
$909$	0	0
$910$	0	0
$911$	1.49023e6i	1.79562i	0.440378	+	0.897812i	$0.354844\pi$
$911$	1.49023e6i	1.79562i	−0.440378	+	0.897812i	$0.645156\pi$
$912$	0	0
$913$	311858.	0.374124
$914$	262353.i	0.314046i
$915$	0	0
$916$	−812942.	−0.968877
$917$	213530.i	0.253934i
$918$	0	0
$919$	−1.16759e6	−1.38248	−0.691240	−	0.722625i	$-0.742935\pi$
$919$	−1.16759e6	−1.38248	−0.691240	+	0.722625i	$0.742935\pi$
$920$	0	0
$921$	0	0
$922$	−230557.	−0.271217
$923$	− 699881.i	− 0.821525i
$924$	0	0
$925$	0	0
$926$	− 203333.i	− 0.237130i
$927$	0	0
$928$	−1.44088e6	−1.67314
$929$	709717.i	0.822345i	0.911558	+	0.411172i	$0.134881\pi$
$929$	709717.i	0.822345i	−0.911558	+	0.411172i	$0.865119\pi$
$930$	0	0
$931$	101726.	0.117364
$932$	− 291933.i	− 0.336087i
$933$	0	0
$934$	−571374.	−0.654978
$935$	0	0
$936$	0	0
$937$	−294273.	−0.335175	−0.167587	−	0.985857i	$-0.553598\pi$
$937$	−294273.	−0.335175	−0.167587	+	0.985857i	$0.553598\pi$
$938$	− 163500.i	− 0.185829i
$939$	0	0
$940$	0	0
$941$	− 783701.i	− 0.885057i	−0.896755	−	0.442528i	$-0.854082\pi$
$941$	− 783701.i	− 0.885057i	0.896755	−	0.442528i	$-0.145918\pi$
$942$	0	0
$943$	−3229.93	−0.00363220
$944$	− 281767.i	− 0.316189i
$945$	0	0
$946$	51029.2	0.0570212
$947$	− 1.35618e6i	− 1.51222i	−0.654442	−	0.756112i	$-0.727096\pi$
$947$	− 1.35618e6i	− 1.51222i	0.654442	−	0.756112i	$-0.272904\pi$
$948$	0	0
$949$	−626822.	−0.696004
$950$	0	0
$951$	0	0
$952$	236770.	0.261248
$953$	− 226626.i	− 0.249531i	−0.992186	−	0.124766i	$-0.960182\pi$
$953$	− 226626.i	− 0.249531i	0.992186	−	0.124766i	$-0.0398179\pi$
$954$	0	0
$955$	0	0
$956$	− 333965.i	− 0.365414i
$957$	0	0
$958$	−675037.	−0.735524
$959$	− 300228.i	− 0.326448i
$960$	0	0
$961$	1.65507e6	1.79212
$962$	− 403006.i	− 0.435473i
$963$	0	0
$964$	64799.6	0.0697298
$965$	0	0
$966$	0	0
$967$	−688811.	−0.736626	−0.368313	−	0.929702i	$-0.620065\pi$
$967$	−688811.	−0.736626	−0.368313	+	0.929702i	$0.620065\pi$
$968$	− 569106.i	− 0.607355i
$969$	0	0
$970$	0	0
$971$	1.30899e6i	1.38835i	0.719807	+	0.694174i	$0.244230\pi$
$971$	1.30899e6i	1.38835i	−0.719807	+	0.694174i	$0.755770\pi$
$972$	0	0
$973$	2927.75	0.00309249
$974$	− 85883.2i	− 0.0905295i
$975$	0	0
$976$	−28149.5	−0.0295509
$977$	− 29454.1i	− 0.0308572i	−0.999881	−	0.0154286i	$-0.995089\pi$
$977$	− 29454.1i	− 0.0308572i	0.999881	−	0.0154286i	$-0.00491127\pi$
$978$	0	0
$979$	−354807.	−0.370192
$980$	0	0
$981$	0	0
$982$	−276966.	−0.287213
$983$	369377.i	0.382264i	0.981564	+	0.191132i	$0.0612158\pi$
$983$	369377.i	0.382264i	−0.981564	+	0.191132i	$0.938784\pi$
$984$	0	0
$985$	0	0
$986$	952450.i	0.979689i
$987$	0	0
$988$	97218.9	0.0995948
$989$	− 643.096i	0 0.000657481i
$990$	0	0
$991$	121176.	0.123387	0.0616933	−	0.998095i	$-0.480350\pi$
$991$	121176.	0.123387	0.0616933	+	0.998095i	$0.480350\pi$
$992$	− 1.65123e6i	− 1.67797i
$993$	0	0
$994$	−90416.2	−0.0915110
$995$	0	0
$996$	0	0
$997$	−277965.	−0.279640	−0.139820	−	0.990177i	$-0.544652\pi$
$997$	−277965.	−0.279640	−0.139820	+	0.990177i	$0.544652\pi$
$998$	628359.i	0.630880i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.5.c.d.26.3	yes	6
3.2	odd	2	inner	225.5.c.d.26.4	yes	6
5.2	odd	4		225.5.d.c.224.7		12
5.3	odd	4		225.5.d.c.224.6		12
5.4	even	2		225.5.c.c.26.4	yes	6
15.2	even	4		225.5.d.c.224.5		12
15.8	even	4		225.5.d.c.224.8		12
15.14	odd	2		225.5.c.c.26.3	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.5.c.c.26.3	✓	6	15.14	odd	2
225.5.c.c.26.4	yes	6	5.4	even	2
225.5.c.d.26.3	yes	6	1.1	even	1	trivial
225.5.c.d.26.4	yes	6	3.2	odd	2	inner
225.5.d.c.224.5		12	15.2	even	4
225.5.d.c.224.6		12	5.3	odd	4
225.5.d.c.224.7		12	5.2	odd	4
225.5.d.c.224.8		12	15.8	even	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)

\(f(q)\)	\(=\)	\(q-1.80123i q^{2} +12.7556 q^{4} -12.1136 q^{7} -51.7953i q^{8} +O(q^{10})\)	\(q-1.80123i q^{2} +12.7556 q^{4} -12.1136 q^{7} -51.7953i q^{8} -60.4434i q^{11} -168.896 q^{13} +21.8194i q^{14} +110.794 q^{16} -377.366i q^{17} -45.1263 q^{19} -108.872 q^{22} +1.37206i q^{23} +304.221i q^{26} -154.516 q^{28} -1401.24i q^{29} +1605.80 q^{31} -1028.29i q^{32} -679.721 q^{34} -1324.72 q^{37} +81.2826i q^{38} +2354.07i q^{41} -468.707 q^{43} -770.991i q^{44} +2.47139 q^{46} -3910.98i q^{47} -2254.26 q^{49} -2154.37 q^{52} -3228.98i q^{53} +627.429i q^{56} -2523.94 q^{58} -2543.15i q^{59} -254.070 q^{61} -2892.40i q^{62} -79.4731 q^{64} -7493.35 q^{67} -4813.52i q^{68} +4143.85i q^{71} +3711.28 q^{73} +2386.11i q^{74} -575.612 q^{76} +732.189i q^{77} +80.2118 q^{79} +4240.21 q^{82} +5159.51i q^{83} +844.248i q^{86} -3130.68 q^{88} -5870.07i q^{89} +2045.95 q^{91} +17.5015i q^{92} -7044.55 q^{94} +8921.02 q^{97} +4060.43i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 92 q^{4} + 86 q^{7}+O(q^{10}) \) 6 * q - 92 * q^4 + 86 * q^7	\( 6 q - 92 q^{4} + 86 q^{7} + 394 q^{13} + 1816 q^{16} - 250 q^{19} - 1276 q^{22} - 6412 q^{28} + 1846 q^{31} - 3692 q^{34} - 7968 q^{37} - 1382 q^{43} + 9756 q^{46} - 3232 q^{49} - 23268 q^{52} + 2932 q^{58} + 10838 q^{61} - 36144 q^{64} + 6278 q^{67} + 13024 q^{73} - 24780 q^{76} + 3904 q^{79} - 29792 q^{82} - 42744 q^{88} + 17306 q^{91} + 66364 q^{94} - 8262 q^{97}+O(q^{100}) \) 6 * q - 92 * q^4 + 86 * q^7 + 394 * q^13 + 1816 * q^16 - 250 * q^19 - 1276 * q^22 - 6412 * q^28 + 1846 * q^31 - 3692 * q^34 - 7968 * q^37 - 1382 * q^43 + 9756 * q^46 - 3232 * q^49 - 23268 * q^52 + 2932 * q^58 + 10838 * q^61 - 36144 * q^64 + 6278 * q^67 + 13024 * q^73 - 24780 * q^76 + 3904 * q^79 - 29792 * q^82 - 42744 * q^88 + 17306 * q^91 + 66364 * q^94 - 8262 * q^97

Introduction

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