LMFDB - Embedded newform 5625.2.a.t.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5625,2,Mod(1,5625)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5625, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("5625.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5625 = 3^{2} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5625.a (trivial)

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:	yes
Analytic conductor:	$44.9158511370$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.5444000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 2x^{6} + 20x^{5} - 4x^{4} - 30x^{3} + 7x^{2} + 12x + 1 $ x^8 - 4x^7 - 2x^6 + 20x^5 - 4x^4 - 30x^3 + 7x^2 + 12*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 75)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.35083$ of defining polynomial
Character	$\chi$	$=$	5625.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.35083 q^{2} +3.52640 q^{4} +3.48189 q^{7} -3.58831 q^{8} +O(q^{10})$	$q-2.35083 q^{2} +3.52640 q^{4} +3.48189 q^{7} -3.58831 q^{8} -2.93111 q^{11} -1.87575 q^{13} -8.18532 q^{14} +1.38270 q^{16} -6.78566 q^{17} -2.94950 q^{19} +6.89053 q^{22} +5.49019 q^{23} +4.40956 q^{26} +12.2785 q^{28} -2.55593 q^{29} +0.418084 q^{31} +3.92613 q^{32} +15.9519 q^{34} +5.23959 q^{37} +6.93377 q^{38} -1.67869 q^{41} +10.9233 q^{43} -10.3363 q^{44} -12.9065 q^{46} +7.49178 q^{47} +5.12353 q^{49} -6.61463 q^{52} -3.70953 q^{53} -12.4941 q^{56} +6.00857 q^{58} +7.10854 q^{59} -6.43710 q^{61} -0.982844 q^{62} -11.9951 q^{64} +10.0415 q^{67} -23.9290 q^{68} -0.728602 q^{71} -3.59269 q^{73} -12.3174 q^{74} -10.4011 q^{76} -10.2058 q^{77} +3.07265 q^{79} +3.94632 q^{82} -10.1152 q^{83} -25.6788 q^{86} +10.5177 q^{88} -0.287512 q^{89} -6.53114 q^{91} +19.3606 q^{92} -17.6119 q^{94} -10.4090 q^{97} -12.0446 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{2} + 4 q^{4} + 8 q^{7} - 12 q^{8}+O(q^{10}) $ 8 * q - 4 * q^2 + 4 * q^4 + 8 * q^7 - 12 * q^8	$ 8 q - 4 q^{2} + 4 q^{4} + 8 q^{7} - 12 q^{8} - 2 q^{11} + 16 q^{13} - 6 q^{14} - 16 q^{17} - 14 q^{19} + 12 q^{22} - 14 q^{23} - 6 q^{26} + 16 q^{28} - 2 q^{29} - 22 q^{31} + 2 q^{32} - 12 q^{34} + 28 q^{37} + 16 q^{38} - 8 q^{41} + 20 q^{43} - 22 q^{44} - 2 q^{46} - 10 q^{47} + 16 q^{52} - 44 q^{53} - 30 q^{56} + 8 q^{58} - 14 q^{59} - 20 q^{61} - 16 q^{62} + 6 q^{64} + 16 q^{67} + 2 q^{68} - 16 q^{71} + 24 q^{73} - 26 q^{74} - 16 q^{76} - 46 q^{77} - 30 q^{79} + 16 q^{82} - 12 q^{83} - 32 q^{86} + 32 q^{88} - 16 q^{89} - 12 q^{91} + 2 q^{92} + 14 q^{94} + 16 q^{97} - 4 q^{98}+O(q^{100}) $ 8 * q - 4 * q^2 + 4 * q^4 + 8 * q^7 - 12 * q^8 - 2 * q^11 + 16 * q^13 - 6 * q^14 - 16 * q^17 - 14 * q^19 + 12 * q^22 - 14 * q^23 - 6 * q^26 + 16 * q^28 - 2 * q^29 - 22 * q^31 + 2 * q^32 - 12 * q^34 + 28 * q^37 + 16 * q^38 - 8 * q^41 + 20 * q^43 - 22 * q^44 - 2 * q^46 - 10 * q^47 + 16 * q^52 - 44 * q^53 - 30 * q^56 + 8 * q^58 - 14 * q^59 - 20 * q^61 - 16 * q^62 + 6 * q^64 + 16 * q^67 + 2 * q^68 - 16 * q^71 + 24 * q^73 - 26 * q^74 - 16 * q^76 - 46 * q^77 - 30 * q^79 + 16 * q^82 - 12 * q^83 - 32 * q^86 + 32 * q^88 - 16 * q^89 - 12 * q^91 + 2 * q^92 + 14 * q^94 + 16 * q^97 - 4 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	−2.35083	−1.66229	−0.831144	−	0.556058i	$-0.812313\pi$
$2$	−2.35083	−1.66229	−0.831144	+	0.556058i	$0.812313\pi$
$3$	0	0
$3$	0	0
$4$	3.52640	1.76320
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.48189	1.31603	0.658015	−	0.753005i	$-0.271396\pi$
$7$	3.48189	1.31603	0.658015	+	0.753005i	$0.271396\pi$
$8$	−3.58831	−1.26866
$9$	0	0
$10$	0	0
$11$	−2.93111	−0.883762	−0.441881	−	0.897074i	$-0.645689\pi$
$11$	−2.93111	−0.883762	−0.441881	+	0.897074i	$0.645689\pi$
$12$	0	0
$13$	−1.87575	−0.520239	−0.260119	−	0.965576i	$-0.583762\pi$
$13$	−1.87575	−0.520239	−0.260119	+	0.965576i	$0.583762\pi$
$14$	−8.18532	−2.18762
$15$	0	0
$16$	1.38270	0.345674
$17$	−6.78566	−1.64576	−0.822882	−	0.568212i	$-0.807635\pi$
$17$	−6.78566	−1.64576	−0.822882	+	0.568212i	$0.807635\pi$
$18$	0	0
$19$	−2.94950	−0.676662	−0.338331	−	0.941027i	$-0.609862\pi$
$19$	−2.94950	−0.676662	−0.338331	+	0.941027i	$0.609862\pi$
$20$	0	0
$21$	0	0
$22$	6.89053	1.46907
$23$	5.49019	1.14478	0.572392	−	0.819980i	$-0.306015\pi$
$23$	5.49019	1.14478	0.572392	+	0.819980i	$0.306015\pi$
$24$	0	0
$25$	0	0
$26$	4.40956	0.864786
$27$	0	0
$28$	12.2785	2.32042
$29$	−2.55593	−0.474625	−0.237313	−	0.971433i	$-0.576267\pi$
$29$	−2.55593	−0.474625	−0.237313	+	0.971433i	$0.576267\pi$
$30$	0	0
$31$	0.418084	0.0750901	0.0375451	−	0.999295i	$-0.488046\pi$
$31$	0.418084	0.0750901	0.0375451	+	0.999295i	$0.488046\pi$
$32$	3.92613	0.694048
$33$	0	0
$34$	15.9519	2.73573
$35$	0	0
$36$	0	0
$37$	5.23959	0.861383	0.430691	−	0.902499i	$-0.358270\pi$
$37$	5.23959	0.861383	0.430691	+	0.902499i	$0.358270\pi$
$38$	6.93377	1.12481
$39$	0	0
$40$	0	0
$41$	−1.67869	−0.262167	−0.131084	−	0.991371i	$-0.541846\pi$
$41$	−1.67869	−0.262167	−0.131084	+	0.991371i	$0.541846\pi$
$42$	0	0
$43$	10.9233	1.66578	0.832892	−	0.553436i	$-0.186684\pi$
$43$	10.9233	1.66578	0.832892	+	0.553436i	$0.186684\pi$
$44$	−10.3363	−1.55825
$45$	0	0
$46$	−12.9065	−1.90296
$47$	7.49178	1.09279	0.546394	−	0.837528i	$-0.316000\pi$
$47$	7.49178	1.09279	0.546394	+	0.837528i	$0.316000\pi$
$48$	0	0
$49$	5.12353	0.731933
$50$	0	0
$51$	0	0
$52$	−6.61463	−0.917285
$53$	−3.70953	−0.509543	−0.254771	−	0.967001i	$-0.582000\pi$
$53$	−3.70953	−0.509543	−0.254771	+	0.967001i	$0.582000\pi$
$54$	0	0
$55$	0	0
$56$	−12.4941	−1.66959
$57$	0	0
$58$	6.00857	0.788964
$59$	7.10854	0.925453	0.462727	−	0.886501i	$-0.346871\pi$
$59$	7.10854	0.925453	0.462727	+	0.886501i	$0.346871\pi$
$60$	0	0
$61$	−6.43710	−0.824186	−0.412093	−	0.911142i	$-0.635202\pi$
$61$	−6.43710	−0.824186	−0.412093	+	0.911142i	$0.635202\pi$
$62$	−0.982844	−0.124821
$63$	0	0
$64$	−11.9951	−1.49938
$65$	0	0
$66$	0	0
$67$	10.0415	1.22677	0.613383	−	0.789786i	$-0.289808\pi$
$67$	10.0415	1.22677	0.613383	+	0.789786i	$0.289808\pi$
$68$	−23.9290	−2.90181
$69$	0	0
$70$	0	0
$71$	−0.728602	−0.0864691	−0.0432346	−	0.999065i	$-0.513766\pi$
$71$	−0.728602	−0.0864691	−0.0432346	+	0.999065i	$0.513766\pi$
$72$	0	0
$73$	−3.59269	−0.420492	−0.210246	−	0.977648i	$-0.567427\pi$
$73$	−3.59269	−0.420492	−0.210246	+	0.977648i	$0.567427\pi$
$74$	−12.3174	−1.43187
$75$	0	0
$76$	−10.4011	−1.19309
$77$	−10.2058	−1.16306
$78$	0	0
$79$	3.07265	0.345700	0.172850	−	0.984948i	$-0.444702\pi$
$79$	3.07265	0.345700	0.172850	+	0.984948i	$0.444702\pi$
$80$	0	0
$81$	0	0
$82$	3.94632	0.435798
$83$	−10.1152	−1.11029	−0.555146	−	0.831753i	$-0.687338\pi$
$83$	−10.1152	−1.11029	−0.555146	+	0.831753i	$0.687338\pi$
$84$	0	0
$85$	0	0
$86$	−25.6788	−2.76901
$87$	0	0
$88$	10.5177	1.12119
$89$	−0.287512	−0.0304762	−0.0152381	−	0.999884i	$-0.504851\pi$
$89$	−0.287512	−0.0304762	−0.0152381	+	0.999884i	$0.504851\pi$
$90$	0	0
$91$	−6.53114	−0.684649
$92$	19.3606	2.01848
$93$	0	0
$94$	−17.6119	−1.81653
$95$	0	0
$96$	0	0
$97$	−10.4090	−1.05688	−0.528439	−	0.848971i	$-0.677222\pi$
$97$	−10.4090	−1.05688	−0.528439	+	0.848971i	$0.677222\pi$
$98$	−12.0446	−1.21668
$99$	0	0
$100$	0	0
$101$	−7.65744	−0.761943	−0.380972	−	0.924587i	$-0.624410\pi$
$101$	−7.65744	−0.761943	−0.380972	+	0.924587i	$0.624410\pi$
$102$	0	0
$103$	−2.98602	−0.294221	−0.147111	−	0.989120i	$-0.546997\pi$
$103$	−2.98602	−0.294221	−0.147111	+	0.989120i	$0.546997\pi$
$104$	6.73076	0.660005
$105$	0	0
$106$	8.72047	0.847007
$107$	7.07213	0.683689	0.341844	−	0.939757i	$-0.388948\pi$
$107$	7.07213	0.683689	0.341844	+	0.939757i	$0.388948\pi$
$108$	0	0
$109$	−13.3022	−1.27412	−0.637058	−	0.770816i	$-0.719849\pi$
$109$	−13.3022	−1.27412	−0.637058	+	0.770816i	$0.719849\pi$
$110$	0	0
$111$	0	0
$112$	4.81440	0.454918
$113$	−10.0233	−0.942911	−0.471456	−	0.881890i	$-0.656271\pi$
$113$	−10.0233	−0.942911	−0.471456	+	0.881890i	$0.656271\pi$
$114$	0	0
$115$	0	0
$116$	−9.01325	−0.836859
$117$	0	0
$118$	−16.7110	−1.53837
$119$	−23.6269	−2.16587
$120$	0	0
$121$	−2.40861	−0.218965
$122$	15.1325	1.37003
$123$	0	0
$124$	1.47433	0.132399
$125$	0	0
$126$	0	0
$127$	10.5730	0.938205	0.469103	−	0.883144i	$-0.344577\pi$
$127$	10.5730	0.938205	0.469103	+	0.883144i	$0.344577\pi$
$128$	20.3461	1.79836
$129$	0	0
$130$	0	0
$131$	9.02608	0.788612	0.394306	−	0.918979i	$-0.370985\pi$
$131$	9.02608	0.788612	0.394306	+	0.918979i	$0.370985\pi$
$132$	0	0
$133$	−10.2698	−0.890507
$134$	−23.6059	−2.03924
$135$	0	0
$136$	24.3490	2.08791
$137$	19.6646	1.68006	0.840029	−	0.542541i	$-0.182538\pi$
$137$	19.6646	1.68006	0.840029	+	0.542541i	$0.182538\pi$
$138$	0	0
$139$	−11.6520	−0.988310	−0.494155	−	0.869374i	$-0.664522\pi$
$139$	−11.6520	−0.988310	−0.494155	+	0.869374i	$0.664522\pi$
$140$	0	0
$141$	0	0
$142$	1.71282	0.143737
$143$	5.49801	0.459767
$144$	0	0
$145$	0	0
$146$	8.44580	0.698979
$147$	0	0
$148$	18.4769	1.51879
$149$	7.33020	0.600513	0.300257	−	0.953858i	$-0.402928\pi$
$149$	7.33020	0.600513	0.300257	+	0.953858i	$0.402928\pi$
$150$	0	0
$151$	−16.7358	−1.36194	−0.680968	−	0.732313i	$-0.738441\pi$
$151$	−16.7358	−1.36194	−0.680968	+	0.732313i	$0.738441\pi$
$152$	10.5837	0.858453
$153$	0	0
$154$	23.9921	1.93333
$155$	0	0
$156$	0	0
$157$	7.88635	0.629399	0.314700	−	0.949191i	$-0.398096\pi$
$157$	7.88635	0.629399	0.314700	+	0.949191i	$0.398096\pi$
$158$	−7.22328	−0.574653
$159$	0	0
$160$	0	0
$161$	19.1162	1.50657
$162$	0	0
$163$	−9.93992	−0.778555	−0.389277	−	0.921121i	$-0.627275\pi$
$163$	−9.93992	−0.778555	−0.389277	+	0.921121i	$0.627275\pi$
$164$	−5.91973	−0.462254
$165$	0	0
$166$	23.7792	1.84563
$167$	−5.57767	−0.431613	−0.215806	−	0.976436i	$-0.569238\pi$
$167$	−5.57767	−0.431613	−0.215806	+	0.976436i	$0.569238\pi$
$168$	0	0
$169$	−9.48157	−0.729352
$170$	0	0
$171$	0	0
$172$	38.5198	2.93711
$173$	−16.5682	−1.25966	−0.629828	−	0.776734i	$-0.716875\pi$
$173$	−16.5682	−1.25966	−0.629828	+	0.776734i	$0.716875\pi$
$174$	0	0
$175$	0	0
$176$	−4.05284	−0.305494
$177$	0	0
$178$	0.675892	0.0506602
$179$	5.36021	0.400641	0.200321	−	0.979730i	$-0.435802\pi$
$179$	5.36021	0.400641	0.200321	+	0.979730i	$0.435802\pi$
$180$	0	0
$181$	−0.327745	−0.0243611	−0.0121805	−	0.999926i	$-0.503877\pi$
$181$	−0.327745	−0.0243611	−0.0121805	+	0.999926i	$0.503877\pi$
$182$	15.3536	1.13808
$183$	0	0
$184$	−19.7005	−1.45234
$185$	0	0
$186$	0	0
$187$	19.8895	1.45446
$188$	26.4190	1.92680
$189$	0	0
$190$	0	0
$191$	3.46992	0.251074	0.125537	−	0.992089i	$-0.459935\pi$
$191$	3.46992	0.251074	0.125537	+	0.992089i	$0.459935\pi$
$192$	0	0
$193$	−24.3134	−1.75012	−0.875058	−	0.484018i	$-0.839177\pi$
$193$	−24.3134	−1.75012	−0.875058	+	0.484018i	$0.839177\pi$
$194$	24.4699	1.75684
$195$	0	0
$196$	18.0676	1.29055
$197$	−14.9561	−1.06558	−0.532789	−	0.846248i	$-0.678856\pi$
$197$	−14.9561	−1.06558	−0.532789	+	0.846248i	$0.678856\pi$
$198$	0	0
$199$	−11.3251	−0.802817	−0.401408	−	0.915899i	$-0.631479\pi$
$199$	−11.3251	−0.802817	−0.401408	+	0.915899i	$0.631479\pi$
$200$	0	0
$201$	0	0
$202$	18.0013	1.26657
$203$	−8.89948	−0.624621
$204$	0	0
$205$	0	0
$206$	7.01963	0.489081
$207$	0	0
$208$	−2.59359	−0.179833
$209$	8.64530	0.598008
$210$	0	0
$211$	7.54283	0.519270	0.259635	−	0.965707i	$-0.416398\pi$
$211$	7.54283	0.519270	0.259635	+	0.965707i	$0.416398\pi$
$212$	−13.0813	−0.898426
$213$	0	0
$214$	−16.6254	−1.13649
$215$	0	0
$216$	0	0
$217$	1.45572	0.0988208
$218$	31.2711	2.11795
$219$	0	0
$220$	0	0
$221$	12.7282	0.856190
$222$	0	0
$223$	−7.79539	−0.522018	−0.261009	−	0.965336i	$-0.584055\pi$
$223$	−7.79539	−0.522018	−0.261009	+	0.965336i	$0.584055\pi$
$224$	13.6703	0.913387
$225$	0	0
$226$	23.5630	1.56739
$227$	12.8319	0.851686	0.425843	−	0.904797i	$-0.359978\pi$
$227$	12.8319	0.851686	0.425843	+	0.904797i	$0.359978\pi$
$228$	0	0
$229$	−14.5033	−0.958405	−0.479203	−	0.877704i	$-0.659074\pi$
$229$	−14.5033	−0.958405	−0.479203	+	0.877704i	$0.659074\pi$
$230$	0	0
$231$	0	0
$232$	9.17148	0.602137
$233$	−11.3970	−0.746646	−0.373323	−	0.927702i	$-0.621782\pi$
$233$	−11.3970	−0.746646	−0.373323	+	0.927702i	$0.621782\pi$
$234$	0	0
$235$	0	0
$236$	25.0676	1.63176
$237$	0	0
$238$	55.5428	3.60031
$239$	−6.90072	−0.446371	−0.223185	−	0.974776i	$-0.571646\pi$
$239$	−6.90072	−0.446371	−0.223185	+	0.974776i	$0.571646\pi$
$240$	0	0
$241$	−28.4467	−1.83242	−0.916208	−	0.400704i	$-0.868766\pi$
$241$	−28.4467	−1.83242	−0.916208	+	0.400704i	$0.868766\pi$
$242$	5.66224	0.363982
$243$	0	0
$244$	−22.6998	−1.45320
$245$	0	0
$246$	0	0
$247$	5.53252	0.352026
$248$	−1.50021	−0.0952637
$249$	0	0
$250$	0	0
$251$	12.3258	0.777999	0.389000	−	0.921238i	$-0.372821\pi$
$251$	12.3258	0.777999	0.389000	+	0.921238i	$0.372821\pi$
$252$	0	0
$253$	−16.0923	−1.01172
$254$	−24.8554	−1.55957
$255$	0	0
$256$	−23.8400	−1.49000
$257$	−20.8274	−1.29918	−0.649590	−	0.760285i	$-0.725059\pi$
$257$	−20.8274	−1.29918	−0.649590	+	0.760285i	$0.725059\pi$
$258$	0	0
$259$	18.2437	1.13361
$260$	0	0
$261$	0	0
$262$	−21.2188	−1.31090
$263$	−2.23517	−0.137826	−0.0689131	−	0.997623i	$-0.521953\pi$
$263$	−2.23517	−0.137826	−0.0689131	+	0.997623i	$0.521953\pi$
$264$	0	0
$265$	0	0
$266$	24.1426	1.48028
$267$	0	0
$268$	35.4104	2.16303
$269$	19.3036	1.17696	0.588482	−	0.808511i	$-0.299726\pi$
$269$	19.3036	1.17696	0.588482	+	0.808511i	$0.299726\pi$
$270$	0	0
$271$	−11.9411	−0.725372	−0.362686	−	0.931911i	$-0.618140\pi$
$271$	−11.9411	−0.725372	−0.362686	+	0.931911i	$0.618140\pi$
$272$	−9.38252	−0.568899
$273$	0	0
$274$	−46.2281	−2.79274
$275$	0	0
$276$	0	0
$277$	26.1466	1.57100	0.785499	−	0.618863i	$-0.212406\pi$
$277$	26.1466	1.57100	0.785499	+	0.618863i	$0.212406\pi$
$278$	27.3919	1.64285
$279$	0	0
$280$	0	0
$281$	−10.1219	−0.603820	−0.301910	−	0.953336i	$-0.597624\pi$
$281$	−10.1219	−0.603820	−0.301910	+	0.953336i	$0.597624\pi$
$282$	0	0
$283$	−31.8638	−1.89410	−0.947052	−	0.321081i	$-0.895954\pi$
$283$	−31.8638	−1.89410	−0.947052	+	0.321081i	$0.895954\pi$
$284$	−2.56934	−0.152462
$285$	0	0
$286$	−12.9249	−0.764265
$287$	−5.84501	−0.345020
$288$	0	0
$289$	29.0452	1.70854
$290$	0	0
$291$	0	0
$292$	−12.6693	−0.741412
$293$	−16.6235	−0.971153	−0.485576	−	0.874194i	$-0.661390\pi$
$293$	−16.6235	−0.971153	−0.485576	+	0.874194i	$0.661390\pi$
$294$	0	0
$295$	0	0
$296$	−18.8012	−1.09280
$297$	0	0
$298$	−17.2320	−0.998225
$299$	−10.2982	−0.595561
$300$	0	0
$301$	38.0336	2.19222
$302$	39.3429	2.26393
$303$	0	0
$304$	−4.07827	−0.233905
$305$	0	0
$306$	0	0
$307$	−14.1643	−0.808402	−0.404201	−	0.914670i	$-0.632450\pi$
$307$	−14.1643	−0.808402	−0.404201	+	0.914670i	$0.632450\pi$
$308$	−35.9897	−2.05070
$309$	0	0
$310$	0	0
$311$	−15.0500	−0.853406	−0.426703	−	0.904392i	$-0.640325\pi$
$311$	−15.0500	−0.853406	−0.426703	+	0.904392i	$0.640325\pi$
$312$	0	0
$313$	−3.57476	−0.202057	−0.101029	−	0.994884i	$-0.532213\pi$
$313$	−3.57476	−0.202057	−0.101029	+	0.994884i	$0.532213\pi$
$314$	−18.5395	−1.04624
$315$	0	0
$316$	10.8354	0.609539
$317$	−12.7820	−0.717907	−0.358954	−	0.933355i	$-0.616866\pi$
$317$	−12.7820	−0.717907	−0.358954	+	0.933355i	$0.616866\pi$
$318$	0	0
$319$	7.49172	0.419456
$320$	0	0
$321$	0	0
$322$	−44.9390	−2.50435
$323$	20.0143	1.11363
$324$	0	0
$325$	0	0
$326$	23.3671	1.29418
$327$	0	0
$328$	6.02366	0.332601
$329$	26.0855	1.43814
$330$	0	0
$331$	4.70504	0.258612	0.129306	−	0.991605i	$-0.458725\pi$
$331$	4.70504	0.258612	0.129306	+	0.991605i	$0.458725\pi$
$332$	−35.6704	−1.95767
$333$	0	0
$334$	13.1121	0.717465
$335$	0	0
$336$	0	0
$337$	−17.4048	−0.948098	−0.474049	−	0.880499i	$-0.657208\pi$
$337$	−17.4048	−0.948098	−0.474049	+	0.880499i	$0.657208\pi$
$338$	22.2896	1.21239
$339$	0	0
$340$	0	0
$341$	−1.22545	−0.0663618
$342$	0	0
$343$	−6.53364	−0.352783
$344$	−39.1961	−2.11331
$345$	0	0
$346$	38.9490	2.09391
$347$	−14.0106	−0.752130	−0.376065	−	0.926593i	$-0.622723\pi$
$347$	−14.0106	−0.752130	−0.376065	+	0.926593i	$0.622723\pi$
$348$	0	0
$349$	35.9459	1.92414	0.962069	−	0.272806i	$-0.0879518\pi$
$349$	35.9459	1.92414	0.962069	+	0.272806i	$0.0879518\pi$
$350$	0	0
$351$	0	0
$352$	−11.5079	−0.613373
$353$	−8.80198	−0.468482	−0.234241	−	0.972179i	$-0.575260\pi$
$353$	−8.80198	−0.468482	−0.234241	+	0.972179i	$0.575260\pi$
$354$	0	0
$355$	0	0
$356$	−1.01388	−0.0537357
$357$	0	0
$358$	−12.6009	−0.665981
$359$	−6.02962	−0.318231	−0.159116	−	0.987260i	$-0.550864\pi$
$359$	−6.02962	−0.318231	−0.159116	+	0.987260i	$0.550864\pi$
$360$	0	0
$361$	−10.3004	−0.542129
$362$	0.770472	0.0404951
$363$	0	0
$364$	−23.0314	−1.20717
$365$	0	0
$366$	0	0
$367$	28.7662	1.50158	0.750792	−	0.660539i	$-0.229672\pi$
$367$	28.7662	1.50158	0.750792	+	0.660539i	$0.229672\pi$
$368$	7.59128	0.395723
$369$	0	0
$370$	0	0
$371$	−12.9162	−0.670574
$372$	0	0
$373$	33.6000	1.73974	0.869872	−	0.493278i	$-0.164201\pi$
$373$	33.6000	1.73974	0.869872	+	0.493278i	$0.164201\pi$
$374$	−46.7568	−2.41774
$375$	0	0
$376$	−26.8828	−1.38637
$377$	4.79429	0.246918
$378$	0	0
$379$	4.63403	0.238034	0.119017	−	0.992892i	$-0.462026\pi$
$379$	4.63403	0.238034	0.119017	+	0.992892i	$0.462026\pi$
$380$	0	0
$381$	0	0
$382$	−8.15718	−0.417358
$383$	12.9058	0.659453	0.329727	−	0.944076i	$-0.393043\pi$
$383$	12.9058	0.659453	0.329727	+	0.944076i	$0.393043\pi$
$384$	0	0
$385$	0	0
$386$	57.1567	2.90920
$387$	0	0
$388$	−36.7064	−1.86349
$389$	17.5246	0.888532	0.444266	−	0.895895i	$-0.353465\pi$
$389$	17.5246	0.888532	0.444266	+	0.895895i	$0.353465\pi$
$390$	0	0
$391$	−37.2546	−1.88405
$392$	−18.3848	−0.928573
$393$	0	0
$394$	35.1593	1.77130
$395$	0	0
$396$	0	0
$397$	−27.0176	−1.35597	−0.677987	−	0.735074i	$-0.737147\pi$
$397$	−27.0176	−1.35597	−0.677987	+	0.735074i	$0.737147\pi$
$398$	26.6234	1.33451
$399$	0	0
$400$	0	0
$401$	15.9792	0.797965	0.398983	−	0.916958i	$-0.369363\pi$
$401$	15.9792	0.797965	0.398983	+	0.916958i	$0.369363\pi$
$402$	0	0
$403$	−0.784220	−0.0390648
$404$	−27.0032	−1.34346
$405$	0	0
$406$	20.9212	1.03830
$407$	−15.3578	−0.761258
$408$	0	0
$409$	−30.9962	−1.53266	−0.766331	−	0.642446i	$-0.777920\pi$
$409$	−30.9962	−1.53266	−0.766331	+	0.642446i	$0.777920\pi$
$410$	0	0
$411$	0	0
$412$	−10.5299	−0.518771
$413$	24.7511	1.21792
$414$	0	0
$415$	0	0
$416$	−7.36442	−0.361070
$417$	0	0
$418$	−20.3236	−0.994061
$419$	−25.9153	−1.26605	−0.633024	−	0.774132i	$-0.718186\pi$
$419$	−25.9153	−1.26605	−0.633024	+	0.774132i	$0.718186\pi$
$420$	0	0
$421$	3.92646	0.191364	0.0956821	−	0.995412i	$-0.469497\pi$
$421$	3.92646	0.191364	0.0956821	+	0.995412i	$0.469497\pi$
$422$	−17.7319	−0.863176
$423$	0	0
$424$	13.3109	0.646436
$425$	0	0
$426$	0	0
$427$	−22.4132	−1.08465
$428$	24.9392	1.20548
$429$	0	0
$430$	0	0
$431$	35.4632	1.70820	0.854101	−	0.520108i	$-0.174108\pi$
$431$	35.4632	1.70820	0.854101	+	0.520108i	$0.174108\pi$
$432$	0	0
$433$	10.0959	0.485178	0.242589	−	0.970129i	$-0.422003\pi$
$433$	10.0959	0.485178	0.242589	+	0.970129i	$0.422003\pi$
$434$	−3.42215	−0.164269
$435$	0	0
$436$	−46.9088	−2.24652
$437$	−16.1933	−0.774632
$438$	0	0
$439$	7.10560	0.339132	0.169566	−	0.985519i	$-0.445763\pi$
$439$	7.10560	0.339132	0.169566	+	0.985519i	$0.445763\pi$
$440$	0	0
$441$	0	0
$442$	−29.9218	−1.42323
$443$	−20.6841	−0.982733	−0.491366	−	0.870953i	$-0.663502\pi$
$443$	−20.6841	−0.982733	−0.491366	+	0.870953i	$0.663502\pi$
$444$	0	0
$445$	0	0
$446$	18.3256	0.867744
$447$	0	0
$448$	−41.7654	−1.97323
$449$	−19.4940	−0.919980	−0.459990	−	0.887924i	$-0.652147\pi$
$449$	−19.4940	−0.919980	−0.459990	+	0.887924i	$0.652147\pi$
$450$	0	0
$451$	4.92042	0.231694
$452$	−35.3461	−1.66254
$453$	0	0
$454$	−30.1657	−1.41575
$455$	0	0
$456$	0	0
$457$	−4.34194	−0.203107	−0.101554	−	0.994830i	$-0.532381\pi$
$457$	−4.34194	−0.203107	−0.101554	+	0.994830i	$0.532381\pi$
$458$	34.0948	1.59315
$459$	0	0
$460$	0	0
$461$	−11.7897	−0.549102	−0.274551	−	0.961573i	$-0.588529\pi$
$461$	−11.7897	−0.549102	−0.274551	+	0.961573i	$0.588529\pi$
$462$	0	0
$463$	−9.07870	−0.421923	−0.210962	−	0.977494i	$-0.567659\pi$
$463$	−9.07870	−0.421923	−0.210962	+	0.977494i	$0.567659\pi$
$464$	−3.53409	−0.164066
$465$	0	0
$466$	26.7925	1.24114
$467$	34.9181	1.61582	0.807910	−	0.589307i	$-0.200599\pi$
$467$	34.9181	1.61582	0.807910	+	0.589307i	$0.200599\pi$
$468$	0	0
$469$	34.9634	1.61446
$470$	0	0
$471$	0	0
$472$	−25.5076	−1.17408
$473$	−32.0173	−1.47216
$474$	0	0
$475$	0	0
$476$	−83.3179	−3.81887
$477$	0	0
$478$	16.2224	0.741996
$479$	8.64649	0.395068	0.197534	−	0.980296i	$-0.436707\pi$
$479$	8.64649	0.395068	0.197534	+	0.980296i	$0.436707\pi$
$480$	0	0
$481$	−9.82814	−0.448125
$482$	66.8734	3.04600
$483$	0	0
$484$	−8.49373	−0.386079
$485$	0	0
$486$	0	0
$487$	2.84462	0.128902	0.0644510	−	0.997921i	$-0.479470\pi$
$487$	2.84462	0.128902	0.0644510	+	0.997921i	$0.479470\pi$
$488$	23.0983	1.04561
$489$	0	0
$490$	0	0
$491$	−36.8041	−1.66095	−0.830473	−	0.557059i	$-0.811930\pi$
$491$	−36.8041	−1.66095	−0.830473	+	0.557059i	$0.811930\pi$
$492$	0	0
$493$	17.3437	0.781121
$494$	−13.0060	−0.585168
$495$	0	0
$496$	0.578084	0.0259567
$497$	−2.53691	−0.113796
$498$	0	0
$499$	5.85775	0.262229	0.131114	−	0.991367i	$-0.458144\pi$
$499$	5.85775	0.262229	0.131114	+	0.991367i	$0.458144\pi$
$500$	0	0
$501$	0	0
$502$	−28.9759	−1.29326
$503$	30.2874	1.35045	0.675223	−	0.737613i	$-0.264047\pi$
$503$	30.2874	1.35045	0.675223	+	0.737613i	$0.264047\pi$
$504$	0	0
$505$	0	0
$506$	37.8304	1.68177
$507$	0	0
$508$	37.2848	1.65424
$509$	−15.1430	−0.671201	−0.335601	−	0.942004i	$-0.608939\pi$
$509$	−15.1430	−0.671201	−0.335601	+	0.942004i	$0.608939\pi$
$510$	0	0
$511$	−12.5093	−0.553380
$512$	15.3517	0.678457
$513$	0	0
$514$	48.9618	2.15961
$515$	0	0
$516$	0	0
$517$	−21.9592	−0.965765
$518$	−42.8877	−1.88438
$519$	0	0
$520$	0	0
$521$	−39.0832	−1.71227	−0.856134	−	0.516754i	$-0.827140\pi$
$521$	−39.0832	−1.71227	−0.856134	+	0.516754i	$0.827140\pi$
$522$	0	0
$523$	−38.7976	−1.69650	−0.848251	−	0.529594i	$-0.822344\pi$
$523$	−38.7976	−1.69650	−0.848251	+	0.529594i	$0.822344\pi$
$524$	31.8296	1.39048
$525$	0	0
$526$	5.25449	0.229107
$527$	−2.83698	−0.123581
$528$	0	0
$529$	7.14224	0.310532
$530$	0	0
$531$	0	0
$532$	−36.2155	−1.57014
$533$	3.14880	0.136390
$534$	0	0
$535$	0	0
$536$	−36.0320	−1.55635
$537$	0	0
$538$	−45.3796	−1.95645
$539$	−15.0176	−0.646855
$540$	0	0
$541$	14.2280	0.611710	0.305855	−	0.952078i	$-0.401058\pi$
$541$	14.2280	0.611710	0.305855	+	0.952078i	$0.401058\pi$
$542$	28.0716	1.20578
$543$	0	0
$544$	−26.6414	−1.14224
$545$	0	0
$546$	0	0
$547$	39.3229	1.68133	0.840663	−	0.541559i	$-0.182166\pi$
$547$	39.3229	1.68133	0.840663	+	0.541559i	$0.182166\pi$
$548$	69.3452	2.96228
$549$	0	0
$550$	0	0
$551$	7.53873	0.321161
$552$	0	0
$553$	10.6986	0.454952
$554$	−61.4662	−2.61145
$555$	0	0
$556$	−41.0896	−1.74259
$557$	−35.3849	−1.49931	−0.749654	−	0.661830i	$-0.769780\pi$
$557$	−35.3849	−1.49931	−0.749654	+	0.661830i	$0.769780\pi$
$558$	0	0
$559$	−20.4893	−0.866605
$560$	0	0
$561$	0	0
$562$	23.7948	1.00372
$563$	−15.0386	−0.633800	−0.316900	−	0.948459i	$-0.602642\pi$
$563$	−15.0386	−0.633800	−0.316900	+	0.948459i	$0.602642\pi$
$564$	0	0
$565$	0	0
$566$	74.9063	3.14855
$567$	0	0
$568$	2.61445	0.109700
$569$	2.72409	0.114200	0.0570998	−	0.998368i	$-0.481815\pi$
$569$	2.72409	0.114200	0.0570998	+	0.998368i	$0.481815\pi$
$570$	0	0
$571$	12.6236	0.528282	0.264141	−	0.964484i	$-0.414912\pi$
$571$	12.6236	0.528282	0.264141	+	0.964484i	$0.414912\pi$
$572$	19.3882	0.810661
$573$	0	0
$574$	13.7406	0.573522
$575$	0	0
$576$	0	0
$577$	−23.5844	−0.981831	−0.490915	−	0.871207i	$-0.663338\pi$
$577$	−23.5844	−0.981831	−0.490915	+	0.871207i	$0.663338\pi$
$578$	−68.2803	−2.84009
$579$	0	0
$580$	0	0
$581$	−35.2201	−1.46118
$582$	0	0
$583$	10.8730	0.450315
$584$	12.8917	0.533461
$585$	0	0
$586$	39.0789	1.61434
$587$	22.6772	0.935988	0.467994	−	0.883732i	$-0.344977\pi$
$587$	22.6772	0.935988	0.467994	+	0.883732i	$0.344977\pi$
$588$	0	0
$589$	−1.23314	−0.0508106
$590$	0	0
$591$	0	0
$592$	7.24477	0.297758
$593$	8.74287	0.359027	0.179513	−	0.983756i	$-0.442548\pi$
$593$	8.74287	0.359027	0.179513	+	0.983756i	$0.442548\pi$
$594$	0	0
$595$	0	0
$596$	25.8492	1.05882
$597$	0	0
$598$	24.2094	0.989994
$599$	−16.3209	−0.666854	−0.333427	−	0.942776i	$-0.608205\pi$
$599$	−16.3209	−0.666854	−0.333427	+	0.942776i	$0.608205\pi$
$600$	0	0
$601$	36.2713	1.47954	0.739768	−	0.672862i	$-0.234935\pi$
$601$	36.2713	1.47954	0.739768	+	0.672862i	$0.234935\pi$
$602$	−89.4105	−3.64410
$603$	0	0
$604$	−59.0170	−2.40137
$605$	0	0
$606$	0	0
$607$	−16.6820	−0.677102	−0.338551	−	0.940948i	$-0.609937\pi$
$607$	−16.6820	−0.677102	−0.338551	+	0.940948i	$0.609937\pi$
$608$	−11.5801	−0.469636
$609$	0	0
$610$	0	0
$611$	−14.0527	−0.568511
$612$	0	0
$613$	24.7967	1.00153	0.500765	−	0.865583i	$-0.333052\pi$
$613$	24.7967	1.00153	0.500765	+	0.865583i	$0.333052\pi$
$614$	33.2980	1.34380
$615$	0	0
$616$	36.6215	1.47552
$617$	−27.9669	−1.12591	−0.562953	−	0.826489i	$-0.690335\pi$
$617$	−27.9669	−1.12591	−0.562953	+	0.826489i	$0.690335\pi$
$618$	0	0
$619$	27.2686	1.09602	0.548009	−	0.836473i	$-0.315386\pi$
$619$	27.2686	1.09602	0.548009	+	0.836473i	$0.315386\pi$
$620$	0	0
$621$	0	0
$622$	35.3799	1.41861
$623$	−1.00108	−0.0401076
$624$	0	0
$625$	0	0
$626$	8.40365	0.335878
$627$	0	0
$628$	27.8104	1.10976
$629$	−35.5541	−1.41763
$630$	0	0
$631$	42.2603	1.68235	0.841177	−	0.540759i	$-0.181863\pi$
$631$	42.2603	1.68235	0.841177	+	0.540759i	$0.181863\pi$
$632$	−11.0256	−0.438575
$633$	0	0
$634$	30.0482	1.19337
$635$	0	0
$636$	0	0
$637$	−9.61045	−0.380780
$638$	−17.6118	−0.697256
$639$	0	0
$640$	0	0
$641$	−45.8456	−1.81079	−0.905396	−	0.424569i	$-0.860426\pi$
$641$	−45.8456	−1.81079	−0.905396	+	0.424569i	$0.860426\pi$
$642$	0	0
$643$	−46.6710	−1.84052	−0.920261	−	0.391304i	$-0.872024\pi$
$643$	−46.6710	−1.84052	−0.920261	+	0.391304i	$0.872024\pi$
$644$	67.4115	2.65639
$645$	0	0
$646$	−47.0502	−1.85117
$647$	12.1264	0.476740	0.238370	−	0.971174i	$-0.423387\pi$
$647$	12.1264	0.476740	0.238370	+	0.971174i	$0.423387\pi$
$648$	0	0
$649$	−20.8359	−0.817880
$650$	0	0
$651$	0	0
$652$	−35.0521	−1.37275
$653$	1.72017	0.0673154	0.0336577	−	0.999433i	$-0.489284\pi$
$653$	1.72017	0.0673154	0.0336577	+	0.999433i	$0.489284\pi$
$654$	0	0
$655$	0	0
$656$	−2.32112	−0.0906246
$657$	0	0
$658$	−61.3226	−2.39060
$659$	−5.47947	−0.213450	−0.106725	−	0.994289i	$-0.534036\pi$
$659$	−5.47947	−0.213450	−0.106725	+	0.994289i	$0.534036\pi$
$660$	0	0
$661$	0.797448	0.0310171	0.0155086	−	0.999880i	$-0.495063\pi$
$661$	0.797448	0.0310171	0.0155086	+	0.999880i	$0.495063\pi$
$662$	−11.0607	−0.429888
$663$	0	0
$664$	36.2966	1.40858
$665$	0	0
$666$	0	0
$667$	−14.0326	−0.543344
$668$	−19.6691	−0.761020
$669$	0	0
$670$	0	0
$671$	18.8678	0.728384
$672$	0	0
$673$	−51.3280	−1.97855	−0.989274	−	0.146074i	$-0.953336\pi$
$673$	−51.3280	−1.97855	−0.989274	+	0.146074i	$0.953336\pi$
$674$	40.9156	1.57601
$675$	0	0
$676$	−33.4358	−1.28599
$677$	−15.0929	−0.580069	−0.290034	−	0.957016i	$-0.593667\pi$
$677$	−15.0929	−0.580069	−0.290034	+	0.957016i	$0.593667\pi$
$678$	0	0
$679$	−36.2431	−1.39088
$680$	0	0
$681$	0	0
$682$	2.88082	0.110312
$683$	10.8602	0.415555	0.207777	−	0.978176i	$-0.433377\pi$
$683$	10.8602	0.415555	0.207777	+	0.978176i	$0.433377\pi$
$684$	0	0
$685$	0	0
$686$	15.3595	0.586428
$687$	0	0
$688$	15.1036	0.575819
$689$	6.95814	0.265084
$690$	0	0
$691$	8.44552	0.321283	0.160641	−	0.987013i	$-0.448644\pi$
$691$	8.44552	0.321283	0.160641	+	0.987013i	$0.448644\pi$
$692$	−58.4261	−2.22103
$693$	0	0
$694$	32.9366	1.25026
$695$	0	0
$696$	0	0
$697$	11.3910	0.431466
$698$	−84.5026	−3.19847
$699$	0	0
$700$	0	0
$701$	−48.9399	−1.84843	−0.924216	−	0.381869i	$-0.875281\pi$
$701$	−48.9399	−1.84843	−0.924216	+	0.381869i	$0.875281\pi$
$702$	0	0
$703$	−15.4542	−0.582865
$704$	35.1588	1.32510
$705$	0	0
$706$	20.6920	0.778752
$707$	−26.6623	−1.00274
$708$	0	0
$709$	24.7834	0.930762	0.465381	−	0.885111i	$-0.345917\pi$
$709$	24.7834	0.930762	0.465381	+	0.885111i	$0.345917\pi$
$710$	0	0
$711$	0	0
$712$	1.03168	0.0386639
$713$	2.29536	0.0859620
$714$	0	0
$715$	0	0
$716$	18.9023	0.706410
$717$	0	0
$718$	14.1746	0.528992
$719$	18.2171	0.679382	0.339691	−	0.940537i	$-0.389678\pi$
$719$	18.2171	0.679382	0.339691	+	0.940537i	$0.389678\pi$
$720$	0	0
$721$	−10.3970	−0.387204
$722$	24.2146	0.901174
$723$	0	0
$724$	−1.15576	−0.0429534
$725$	0	0
$726$	0	0
$727$	16.6699	0.618253	0.309127	−	0.951021i	$-0.399963\pi$
$727$	16.6699	0.618253	0.309127	+	0.951021i	$0.399963\pi$
$728$	23.4357	0.868586
$729$	0	0
$730$	0	0
$731$	−74.1216	−2.74149
$732$	0	0
$733$	20.0644	0.741094	0.370547	−	0.928814i	$-0.379170\pi$
$733$	20.0644	0.741094	0.370547	+	0.928814i	$0.379170\pi$
$734$	−67.6245	−2.49606
$735$	0	0
$736$	21.5552	0.794535
$737$	−29.4328	−1.08417
$738$	0	0
$739$	−34.0571	−1.25281	−0.626406	−	0.779497i	$-0.715475\pi$
$739$	−34.0571	−1.25281	−0.626406	+	0.779497i	$0.715475\pi$
$740$	0	0
$741$	0	0
$742$	30.3637	1.11469
$743$	38.5355	1.41373	0.706865	−	0.707348i	$-0.250109\pi$
$743$	38.5355	1.41373	0.706865	+	0.707348i	$0.250109\pi$
$744$	0	0
$745$	0	0
$746$	−78.9880	−2.89195
$747$	0	0
$748$	70.1383	2.56451
$749$	24.6244	0.899754
$750$	0	0
$751$	−36.0351	−1.31494	−0.657470	−	0.753481i	$-0.728373\pi$
$751$	−36.0351	−1.31494	−0.657470	+	0.753481i	$0.728373\pi$
$752$	10.3589	0.377749
$753$	0	0
$754$	−11.2706	−0.410449
$755$	0	0
$756$	0	0
$757$	35.0131	1.27257	0.636287	−	0.771453i	$-0.280470\pi$
$757$	35.0131	1.27257	0.636287	+	0.771453i	$0.280470\pi$
$758$	−10.8938	−0.395681
$759$	0	0
$760$	0	0
$761$	−31.3577	−1.13672	−0.568358	−	0.822781i	$-0.692421\pi$
$761$	−31.3577	−1.13672	−0.568358	+	0.822781i	$0.692421\pi$
$762$	0	0
$763$	−46.3166	−1.67678
$764$	12.2363	0.442694
$765$	0	0
$766$	−30.3392	−1.09620
$767$	−13.3338	−0.481456
$768$	0	0
$769$	2.39815	0.0864793	0.0432397	−	0.999065i	$-0.486232\pi$
$769$	2.39815	0.0864793	0.0432397	+	0.999065i	$0.486232\pi$
$770$	0	0
$771$	0	0
$772$	−85.7388	−3.08581
$773$	10.9226	0.392858	0.196429	−	0.980518i	$-0.437065\pi$
$773$	10.9226	0.392858	0.196429	+	0.980518i	$0.437065\pi$
$774$	0	0
$775$	0	0
$776$	37.3508	1.34082
$777$	0	0
$778$	−41.1973	−1.47700
$779$	4.95130	0.177399
$780$	0	0
$781$	2.13561	0.0764181
$782$	87.5792	3.13183
$783$	0	0
$784$	7.08430	0.253011
$785$	0	0
$786$	0	0
$787$	11.5724	0.412511	0.206256	−	0.978498i	$-0.433872\pi$
$787$	11.5724	0.412511	0.206256	+	0.978498i	$0.433872\pi$
$788$	−52.7412	−1.87883
$789$	0	0
$790$	0	0
$791$	−34.8999	−1.24090
$792$	0	0
$793$	12.0744	0.428773
$794$	63.5137	2.25402
$795$	0	0
$796$	−39.9369	−1.41553
$797$	−21.8923	−0.775467	−0.387733	−	0.921772i	$-0.626742\pi$
$797$	−21.8923	−0.775467	−0.387733	+	0.921772i	$0.626742\pi$
$798$	0	0
$799$	−50.8367	−1.79847
$800$	0	0
$801$	0	0
$802$	−37.5645	−1.32645
$803$	10.5306	0.371615
$804$	0	0
$805$	0	0
$806$	1.84357	0.0649369
$807$	0	0
$808$	27.4772	0.966646
$809$	21.5498	0.757652	0.378826	−	0.925468i	$-0.376328\pi$
$809$	21.5498	0.757652	0.378826	+	0.925468i	$0.376328\pi$
$810$	0	0
$811$	−43.9615	−1.54370	−0.771848	−	0.635807i	$-0.780667\pi$
$811$	−43.9615	−1.54370	−0.771848	+	0.635807i	$0.780667\pi$
$812$	−31.3831	−1.10133
$813$	0	0
$814$	36.1036	1.26543
$815$	0	0
$816$	0	0
$817$	−32.2182	−1.12717
$818$	72.8667	2.54772
$819$	0	0
$820$	0	0
$821$	−33.5061	−1.16937	−0.584686	−	0.811260i	$-0.698782\pi$
$821$	−33.5061	−1.16937	−0.584686	+	0.811260i	$0.698782\pi$
$822$	0	0
$823$	55.9860	1.95155	0.975774	−	0.218781i	$-0.0702080\pi$
$823$	55.9860	1.95155	0.975774	+	0.218781i	$0.0702080\pi$
$824$	10.7148	0.373266
$825$	0	0
$826$	−58.1857	−2.02454
$827$	40.8100	1.41910	0.709552	−	0.704653i	$-0.248897\pi$
$827$	40.8100	1.41910	0.709552	+	0.704653i	$0.248897\pi$
$828$	0	0
$829$	0.122914	0.00426896	0.00213448	−	0.999998i	$-0.499321\pi$
$829$	0.122914	0.00426896	0.00213448	+	0.999998i	$0.499321\pi$
$830$	0	0
$831$	0	0
$832$	22.4997	0.780036
$833$	−34.7666	−1.20459
$834$	0	0
$835$	0	0
$836$	30.4868	1.05441
$837$	0	0
$838$	60.9226	2.10454
$839$	−14.8083	−0.511241	−0.255620	−	0.966777i	$-0.582280\pi$
$839$	−14.8083	−0.511241	−0.255620	+	0.966777i	$0.582280\pi$
$840$	0	0
$841$	−22.4672	−0.774731
$842$	−9.23045	−0.318102
$843$	0	0
$844$	26.5990	0.915577
$845$	0	0
$846$	0	0
$847$	−8.38651	−0.288164
$848$	−5.12916	−0.176136
$849$	0	0
$850$	0	0
$851$	28.7664	0.986098
$852$	0	0
$853$	7.26474	0.248740	0.124370	−	0.992236i	$-0.460309\pi$
$853$	7.26474	0.248740	0.124370	+	0.992236i	$0.460309\pi$
$854$	52.6897	1.80301
$855$	0	0
$856$	−25.3770	−0.867367
$857$	26.8175	0.916068	0.458034	−	0.888935i	$-0.348554\pi$
$857$	26.8175	0.916068	0.458034	+	0.888935i	$0.348554\pi$
$858$	0	0
$859$	−20.6038	−0.702991	−0.351496	−	0.936190i	$-0.614327\pi$
$859$	−20.6038	−0.702991	−0.351496	+	0.936190i	$0.614327\pi$
$860$	0	0
$861$	0	0
$862$	−83.3679	−2.83952
$863$	−13.7488	−0.468015	−0.234008	−	0.972235i	$-0.575184\pi$
$863$	−13.7488	−0.468015	−0.234008	+	0.972235i	$0.575184\pi$
$864$	0	0
$865$	0	0
$866$	−23.7338	−0.806506
$867$	0	0
$868$	5.13346	0.174241
$869$	−9.00627	−0.305517
$870$	0	0
$871$	−18.8353	−0.638211
$872$	47.7323	1.61642
$873$	0	0
$874$	38.0678	1.28766
$875$	0	0
$876$	0	0
$877$	22.0430	0.744339	0.372170	−	0.928165i	$-0.378614\pi$
$877$	22.0430	0.744339	0.372170	+	0.928165i	$0.378614\pi$
$878$	−16.7041	−0.563735
$879$	0	0
$880$	0	0
$881$	9.03929	0.304541	0.152271	−	0.988339i	$-0.451341\pi$
$881$	9.03929	0.304541	0.152271	+	0.988339i	$0.451341\pi$
$882$	0	0
$883$	−1.60120	−0.0538848	−0.0269424	−	0.999637i	$-0.508577\pi$
$883$	−1.60120	−0.0538848	−0.0269424	+	0.999637i	$0.508577\pi$
$884$	44.8847	1.50963
$885$	0	0
$886$	48.6249	1.63358
$887$	−17.4253	−0.585083	−0.292541	−	0.956253i	$-0.594501\pi$
$887$	−17.4253	−0.585083	−0.292541	+	0.956253i	$0.594501\pi$
$888$	0	0
$889$	36.8141	1.23471
$890$	0	0
$891$	0	0
$892$	−27.4897	−0.920423
$893$	−22.0970	−0.739448
$894$	0	0
$895$	0	0
$896$	70.8427	2.36669
$897$	0	0
$898$	45.8272	1.52927
$899$	−1.06860	−0.0356397
$900$	0	0
$901$	25.1716	0.838588
$902$	−11.5671	−0.385141
$903$	0	0
$904$	35.9666	1.19623
$905$	0	0
$906$	0	0
$907$	−20.8690	−0.692942	−0.346471	−	0.938061i	$-0.612620\pi$
$907$	−20.8690	−0.692942	−0.346471	+	0.938061i	$0.612620\pi$
$908$	45.2506	1.50169
$909$	0	0
$910$	0	0
$911$	−49.1748	−1.62923	−0.814617	−	0.580000i	$-0.803053\pi$
$911$	−49.1748	−1.62923	−0.814617	+	0.580000i	$0.803053\pi$
$912$	0	0
$913$	29.6489	0.981234
$914$	10.2072	0.337623
$915$	0	0
$916$	−51.1445	−1.68986
$917$	31.4278	1.03784
$918$	0	0
$919$	−11.1959	−0.369318	−0.184659	−	0.982803i	$-0.559118\pi$
$919$	−11.1959	−0.369318	−0.184659	+	0.982803i	$0.559118\pi$
$920$	0	0
$921$	0	0
$922$	27.7156	0.912765
$923$	1.36667	0.0449846
$924$	0	0
$925$	0	0
$926$	21.3425	0.701357
$927$	0	0
$928$	−10.0349	−0.329413
$929$	−48.2290	−1.58234	−0.791171	−	0.611596i	$-0.790528\pi$
$929$	−48.2290	−1.58234	−0.791171	+	0.611596i	$0.790528\pi$
$930$	0	0
$931$	−15.1119	−0.495271
$932$	−40.1906	−1.31649
$933$	0	0
$934$	−82.0866	−2.68596
$935$	0	0
$936$	0	0
$937$	50.5397	1.65106	0.825529	−	0.564359i	$-0.190877\pi$
$937$	50.5397	1.65106	0.825529	+	0.564359i	$0.190877\pi$
$938$	−82.1930	−2.68370
$939$	0	0
$940$	0	0
$941$	38.1644	1.24412	0.622061	−	0.782969i	$-0.286295\pi$
$941$	38.1644	1.24412	0.622061	+	0.782969i	$0.286295\pi$
$942$	0	0
$943$	−9.21634	−0.300125
$944$	9.82896	0.319906
$945$	0	0
$946$	75.2672	2.44715
$947$	−24.5190	−0.796759	−0.398380	−	0.917221i	$-0.630427\pi$
$947$	−24.5190	−0.796759	−0.398380	+	0.917221i	$0.630427\pi$
$948$	0	0
$949$	6.73897	0.218756
$950$	0	0
$951$	0	0
$952$	84.7806	2.74775
$953$	−13.0019	−0.421173	−0.210586	−	0.977575i	$-0.567537\pi$
$953$	−13.0019	−0.421173	−0.210586	+	0.977575i	$0.567537\pi$
$954$	0	0
$955$	0	0
$956$	−24.3347	−0.787041
$957$	0	0
$958$	−20.3264	−0.656717
$959$	68.4698	2.21101
$960$	0	0
$961$	−30.8252	−0.994361
$962$	23.1043	0.744912
$963$	0	0
$964$	−100.315	−3.23091
$965$	0	0
$966$	0	0
$967$	−43.3212	−1.39312	−0.696559	−	0.717500i	$-0.745286\pi$
$967$	−43.3212	−1.39312	−0.696559	+	0.717500i	$0.745286\pi$
$968$	8.64284	0.277791
$969$	0	0
$970$	0	0
$971$	29.5324	0.947741	0.473870	−	0.880595i	$-0.342857\pi$
$971$	29.5324	0.947741	0.473870	+	0.880595i	$0.342857\pi$
$972$	0	0
$973$	−40.5709	−1.30064
$974$	−6.68721	−0.214272
$975$	0	0
$976$	−8.90056	−0.284900
$977$	60.6153	1.93926	0.969628	−	0.244583i	$-0.0786512\pi$
$977$	60.6153	1.93926	0.969628	+	0.244583i	$0.0786512\pi$
$978$	0	0
$979$	0.842729	0.0269337
$980$	0	0
$981$	0	0
$982$	86.5202	2.76097
$983$	40.2796	1.28472	0.642359	−	0.766404i	$-0.277956\pi$
$983$	40.2796	1.28472	0.642359	+	0.766404i	$0.277956\pi$
$984$	0	0
$985$	0	0
$986$	−40.7721	−1.29845
$987$	0	0
$988$	19.5099	0.620692
$989$	59.9709	1.90696
$990$	0	0
$991$	38.1495	1.21186	0.605930	−	0.795518i	$-0.292801\pi$
$991$	38.1495	1.21186	0.605930	+	0.795518i	$0.292801\pi$
$992$	1.64145	0.0521161
$993$	0	0
$994$	5.96384	0.189162
$995$	0	0
$996$	0	0
$997$	26.5777	0.841724	0.420862	−	0.907125i	$-0.361728\pi$
$997$	26.5777	0.841724	0.420862	+	0.907125i	$0.361728\pi$
$998$	−13.7706	−0.435900
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.t.1.2		8
3.2	odd	2		1875.2.a.p.1.7		8
5.4	even	2		5625.2.a.bd.1.7		8
15.2	even	4		1875.2.b.h.1249.15		16
15.8	even	4		1875.2.b.h.1249.2		16
15.14	odd	2		1875.2.a.m.1.2		8
25.12	odd	20		225.2.m.b.19.4		16
25.23	odd	20		225.2.m.b.154.4		16
75.2	even	20		375.2.i.c.274.4		16
75.11	odd	10		375.2.g.d.226.4		16
75.14	odd	10		375.2.g.e.226.1		16
75.23	even	20		75.2.i.a.4.1	✓	16
75.38	even	20		375.2.i.c.349.4		16
75.41	odd	10		375.2.g.d.151.4		16
75.59	odd	10		375.2.g.e.151.1		16
75.62	even	20		75.2.i.a.19.1	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.2.i.a.4.1	✓	16	75.23	even	20
75.2.i.a.19.1	yes	16	75.62	even	20
225.2.m.b.19.4		16	25.12	odd	20
225.2.m.b.154.4		16	25.23	odd	20
375.2.g.d.151.4		16	75.41	odd	10
375.2.g.d.226.4		16	75.11	odd	10
375.2.g.e.151.1		16	75.59	odd	10
375.2.g.e.226.1		16	75.14	odd	10
375.2.i.c.274.4		16	75.2	even	20
375.2.i.c.349.4		16	75.38	even	20
1875.2.a.m.1.2		8	15.14	odd	2
1875.2.a.p.1.7		8	3.2	odd	2
1875.2.b.h.1249.2		16	15.8	even	4
1875.2.b.h.1249.15		16	15.2	even	4
5625.2.a.t.1.2		8	1.1	even	1	trivial
5625.2.a.bd.1.7		8	5.4	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:	\( N \)	\(=\)	\( 5625 = 3^{2} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5625.a (trivial)

\(f(q)\)	\(=\)	\(q-2.35083 q^{2} +3.52640 q^{4} +3.48189 q^{7} -3.58831 q^{8} +O(q^{10})\)	\(q-2.35083 q^{2} +3.52640 q^{4} +3.48189 q^{7} -3.58831 q^{8} -2.93111 q^{11} -1.87575 q^{13} -8.18532 q^{14} +1.38270 q^{16} -6.78566 q^{17} -2.94950 q^{19} +6.89053 q^{22} +5.49019 q^{23} +4.40956 q^{26} +12.2785 q^{28} -2.55593 q^{29} +0.418084 q^{31} +3.92613 q^{32} +15.9519 q^{34} +5.23959 q^{37} +6.93377 q^{38} -1.67869 q^{41} +10.9233 q^{43} -10.3363 q^{44} -12.9065 q^{46} +7.49178 q^{47} +5.12353 q^{49} -6.61463 q^{52} -3.70953 q^{53} -12.4941 q^{56} +6.00857 q^{58} +7.10854 q^{59} -6.43710 q^{61} -0.982844 q^{62} -11.9951 q^{64} +10.0415 q^{67} -23.9290 q^{68} -0.728602 q^{71} -3.59269 q^{73} -12.3174 q^{74} -10.4011 q^{76} -10.2058 q^{77} +3.07265 q^{79} +3.94632 q^{82} -10.1152 q^{83} -25.6788 q^{86} +10.5177 q^{88} -0.287512 q^{89} -6.53114 q^{91} +19.3606 q^{92} -17.6119 q^{94} -10.4090 q^{97} -12.0446 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{2} + 4 q^{4} + 8 q^{7} - 12 q^{8}+O(q^{10}) \) 8 * q - 4 * q^2 + 4 * q^4 + 8 * q^7 - 12 * q^8	\( 8 q - 4 q^{2} + 4 q^{4} + 8 q^{7} - 12 q^{8} - 2 q^{11} + 16 q^{13} - 6 q^{14} - 16 q^{17} - 14 q^{19} + 12 q^{22} - 14 q^{23} - 6 q^{26} + 16 q^{28} - 2 q^{29} - 22 q^{31} + 2 q^{32} - 12 q^{34} + 28 q^{37} + 16 q^{38} - 8 q^{41} + 20 q^{43} - 22 q^{44} - 2 q^{46} - 10 q^{47} + 16 q^{52} - 44 q^{53} - 30 q^{56} + 8 q^{58} - 14 q^{59} - 20 q^{61} - 16 q^{62} + 6 q^{64} + 16 q^{67} + 2 q^{68} - 16 q^{71} + 24 q^{73} - 26 q^{74} - 16 q^{76} - 46 q^{77} - 30 q^{79} + 16 q^{82} - 12 q^{83} - 32 q^{86} + 32 q^{88} - 16 q^{89} - 12 q^{91} + 2 q^{92} + 14 q^{94} + 16 q^{97} - 4 q^{98}+O(q^{100}) \) 8 * q - 4 * q^2 + 4 * q^4 + 8 * q^7 - 12 * q^8 - 2 * q^11 + 16 * q^13 - 6 * q^14 - 16 * q^17 - 14 * q^19 + 12 * q^22 - 14 * q^23 - 6 * q^26 + 16 * q^28 - 2 * q^29 - 22 * q^31 + 2 * q^32 - 12 * q^34 + 28 * q^37 + 16 * q^38 - 8 * q^41 + 20 * q^43 - 22 * q^44 - 2 * q^46 - 10 * q^47 + 16 * q^52 - 44 * q^53 - 30 * q^56 + 8 * q^58 - 14 * q^59 - 20 * q^61 - 16 * q^62 + 6 * q^64 + 16 * q^67 + 2 * q^68 - 16 * q^71 + 24 * q^73 - 26 * q^74 - 16 * q^76 - 46 * q^77 - 30 * q^79 + 16 * q^82 - 12 * q^83 - 32 * q^86 + 32 * q^88 - 16 * q^89 - 12 * q^91 + 2 * q^92 + 14 * q^94 + 16 * q^97 - 4 * q^98

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