LMFDB - Embedded newform 425.2.b.f.324.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [425,2,Mod(324,425)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("425.324"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(425, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 425 = 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	425.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [10,0,0,-22,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$3.39364208590$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	10.0.229451239931904.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{7} + 64x^{6} - 30x^{5} + 2x^{4} + 136x^{3} + 324x^{2} + 180x + 50 $ x^10 - 2x^7 + 64x^6 - 30x^5 + 2x^4 + 136x^3 + 324x^2 + 180*x + 50
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			324.3
Root			$-2.08367 - 2.08367i$ of defining polynomial
Character	$\chi$	$=$	425.324
Dual form			425.2.b.f.324.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.19447i q^{2} +2.48887i q^{3} -2.81568 q^{4} +5.46174 q^{6} +3.05725i q^{7} +1.78998i q^{8} -3.19447 q^{9} -5.36180 q^{11} -7.00786i q^{12} +4.59895i q^{13} +6.70903 q^{14} -1.70331 q^{16} +1.00000i q^{17} +7.01015i q^{18} -4.57325 q^{19} -7.60910 q^{21} +11.7663i q^{22} -1.24730i q^{23} -4.45503 q^{24} +10.0922 q^{26} -0.483999i q^{27} -8.60824i q^{28} +5.93018 q^{29} +9.84580 q^{31} +7.31781i q^{32} -13.3448i q^{33} +2.19447 q^{34} +8.99459 q^{36} +4.20461i q^{37} +10.0358i q^{38} -11.4462 q^{39} +0.404485 q^{41} +16.6979i q^{42} +5.76142i q^{43} +15.0971 q^{44} -2.73715 q^{46} -3.35693i q^{47} -4.23931i q^{48} -2.34678 q^{49} -2.48887 q^{51} -12.9492i q^{52} -4.81568i q^{53} -1.06212 q^{54} -5.47242 q^{56} -11.3822i q^{57} -13.0136i q^{58} -12.7392 q^{59} +4.97774 q^{61} -21.6063i q^{62} -9.76628i q^{63} +12.6521 q^{64} -29.2847 q^{66} +6.82926i q^{67} -2.81568i q^{68} +3.10436 q^{69} +11.9408 q^{71} -5.71803i q^{72} -10.8876i q^{73} +9.22687 q^{74} +12.8768 q^{76} -16.3924i q^{77} +25.1183i q^{78} -16.7139 q^{79} -8.37879 q^{81} -0.887628i q^{82} -4.11450i q^{83} +21.4248 q^{84} +12.6432 q^{86} +14.7594i q^{87} -9.59752i q^{88} +10.4142 q^{89} -14.0601 q^{91} +3.51199i q^{92} +24.5049i q^{93} -7.36667 q^{94} -18.2131 q^{96} -2.27443i q^{97} +5.14994i q^{98} +17.1281 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 22 q^{4} + 6 q^{6} - 12 q^{9} + 8 q^{11} + 14 q^{14} + 54 q^{16} - 12 q^{19} - 10 q^{21} + 38 q^{24} - 10 q^{26} - 4 q^{29} + 42 q^{31} + 2 q^{34} + 44 q^{36} - 46 q^{39} - 16 q^{41} + 8 q^{44} - 12 q^{46}+ \cdots + 28 q^{99}+O(q^{100}) $ 10 * q - 22 * q^4 + 6 * q^6 - 12 * q^9 + 8 * q^11 + 14 * q^14 + 54 * q^16 - 12 * q^19 - 10 * q^21 + 38 * q^24 - 10 * q^26 - 4 * q^29 + 42 * q^31 + 2 * q^34 + 44 * q^36 - 46 * q^39 - 16 * q^41 + 8 * q^44 - 12 * q^46 - 20 * q^49 + 2 * q^51 + 50 * q^54 - 58 * q^56 - 24 * q^59 - 4 * q^61 - 86 * q^64 - 56 * q^66 + 42 * q^71 + 100 * q^74 - 28 * q^76 - 82 * q^79 - 70 * q^81 + 142 * q^84 - 72 * q^86 + 20 * q^89 + 26 * q^91 + 20 * q^94 - 170 * q^96 + 28 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/425\mathbb{Z}\right)^\times$.

$n$	$52$	$326$
$\chi(n)$	$-1$	$1$

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 $

$2$	− 2.19447i	− 1.55172i	−0.630904	−	0.775861i	$-0.717316\pi$
$2$	− 2.19447i	− 1.55172i	0.630904	−	0.775861i	$-0.282684\pi$
$3$	2.48887i	1.43695i	0.695553	+	0.718474i	$0.255159\pi$
$3$	2.48887i	1.43695i	−0.695553	+	0.718474i	$0.744841\pi$
$4$	−2.81568	−1.40784
$5$	0	0
$5$	0	0
$6$	5.46174	2.22974
$7$	3.05725i	1.15553i	0.816202	+	0.577766i	$0.196075\pi$
$7$	3.05725i	1.15553i	−0.816202	+	0.577766i	$0.803925\pi$
$8$	1.78998i	0.632854i
$9$	−3.19447	−1.06482
$10$	0	0
$11$	−5.36180	−1.61664	−0.808322	−	0.588741i	$-0.799624\pi$
$11$	−5.36180	−1.61664	−0.808322	+	0.588741i	$0.799624\pi$
$12$	− 7.00786i	− 2.02299i
$13$	4.59895i	1.27552i	0.770235	+	0.637760i	$0.220139\pi$
$13$	4.59895i	1.27552i	−0.770235	+	0.637760i	$0.779861\pi$
$14$	6.70903	1.79306
$15$	0	0
$16$	−1.70331	−0.425827
$17$	1.00000i	0.242536i
$17$	1.00000i	0.242536i
$18$	7.01015i	1.65231i
$19$	−4.57325	−1.04918	−0.524588	−	0.851356i	$-0.675781\pi$
$19$	−4.57325	−1.04918	−0.524588	+	0.851356i	$0.675781\pi$
$20$	0	0
$21$	−7.60910	−1.66044
$22$	11.7663i	2.50858i
$23$	− 1.24730i	− 0.260079i	−0.991509	−	0.130040i	$-0.958490\pi$
$23$	− 1.24730i	− 0.260079i	0.991509	−	0.130040i	$-0.0415105\pi$
$24$	−4.45503	−0.909378
$25$	0	0
$26$	10.0922	1.97925
$27$	− 0.483999i	− 0.0931457i
$28$	− 8.60824i	− 1.62680i
$29$	5.93018	1.10121	0.550604	−	0.834767i	$-0.314398\pi$
$29$	5.93018	1.10121	0.550604	+	0.834767i	$0.314398\pi$
$30$	0	0
$31$	9.84580	1.76836	0.884179	−	0.467149i	$-0.154719\pi$
$31$	9.84580	1.76836	0.884179	+	0.467149i	$0.154719\pi$
$32$	7.31781i	1.29362i
$33$	− 13.3448i	− 2.32303i
$34$	2.19447	0.376348
$35$	0	0
$36$	8.99459	1.49910
$37$	4.20461i	0.691234i	0.938376	+	0.345617i	$0.112330\pi$
$37$	4.20461i	0.691234i	−0.938376	+	0.345617i	$0.887670\pi$
$38$	10.0358i	1.62803i
$39$	−11.4462	−1.83286
$40$	0	0
$41$	0.404485	0.0631699	0.0315850	−	0.999501i	$-0.489945\pi$
$41$	0.404485	0.0631699	0.0315850	+	0.999501i	$0.489945\pi$
$42$	16.6979i	2.57654i
$43$	5.76142i	0.878608i	0.898339	+	0.439304i	$0.144775\pi$
$43$	5.76142i	0.878608i	−0.898339	+	0.439304i	$0.855225\pi$
$44$	15.0971	2.27597
$45$	0	0
$46$	−2.73715	−0.403571
$47$	− 3.35693i	− 0.489659i	−0.969566	−	0.244829i	$-0.921268\pi$
$47$	− 3.35693i	− 0.489659i	0.969566	−	0.244829i	$-0.0787319\pi$
$48$	− 4.23931i	− 0.611892i
$49$	−2.34678	−0.335255
$50$	0	0
$51$	−2.48887	−0.348511
$52$	− 12.9492i	− 1.79573i
$53$	− 4.81568i	− 0.661484i	−0.943721	−	0.330742i	$-0.892701\pi$
$53$	− 4.81568i	− 0.661484i	0.943721	−	0.330742i	$-0.107299\pi$
$54$	−1.06212	−0.144536
$55$	0	0
$56$	−5.47242	−0.731283
$57$	− 11.3822i	− 1.50761i
$58$	− 13.0136i	− 1.70877i
$59$	−12.7392	−1.65850	−0.829248	−	0.558881i	$-0.811231\pi$
$59$	−12.7392	−1.65850	−0.829248	+	0.558881i	$0.811231\pi$
$60$	0	0
$61$	4.97774	0.637334	0.318667	−	0.947867i	$-0.396765\pi$
$61$	4.97774	0.637334	0.318667	+	0.947867i	$0.396765\pi$
$62$	− 21.6063i	− 2.74400i
$63$	− 9.76628i	− 1.23044i
$64$	12.6521	1.58151
$65$	0	0
$66$	−29.2847	−3.60470
$67$	6.82926i	0.834327i	0.908831	+	0.417163i	$0.136976\pi$
$67$	6.82926i	0.834327i	−0.908831	+	0.417163i	$0.863024\pi$
$68$	− 2.81568i	− 0.341451i
$69$	3.10436	0.373721
$70$	0	0
$71$	11.9408	1.41711	0.708555	−	0.705656i	$-0.249348\pi$
$71$	11.9408	1.41711	0.708555	+	0.705656i	$0.249348\pi$
$72$	− 5.71803i	− 0.673877i
$73$	− 10.8876i	− 1.27430i	−0.770740	−	0.637150i	$-0.780113\pi$
$73$	− 10.8876i	− 1.27430i	0.770740	−	0.637150i	$-0.219887\pi$
$74$	9.22687	1.07260
$75$	0	0
$76$	12.8768	1.47707
$77$	− 16.3924i	− 1.86808i
$78$	25.1183i	2.84408i
$79$	−16.7139	−1.88046	−0.940230	−	0.340539i	$-0.889391\pi$
$79$	−16.7139	−1.88046	−0.940230	+	0.340539i	$0.889391\pi$
$80$	0	0
$81$	−8.37879	−0.930976
$82$	− 0.887628i	− 0.0980221i
$83$	− 4.11450i	− 0.451625i	−0.974171	−	0.225813i	$-0.927496\pi$
$83$	− 4.11450i	− 0.451625i	0.974171	−	0.225813i	$-0.0725037\pi$
$84$	21.4248	2.33763
$85$	0	0
$86$	12.6432	1.36335
$87$	14.7594i	1.58238i
$88$	− 9.59752i	− 1.02310i
$89$	10.4142	1.10391	0.551953	−	0.833875i	$-0.313883\pi$
$89$	10.4142	1.10391	0.551953	+	0.833875i	$0.313883\pi$
$90$	0	0
$91$	−14.0601	−1.47390
$92$	3.51199i	0.366150i
$93$	24.5049i	2.54104i
$94$	−7.36667	−0.759814
$95$	0	0
$96$	−18.2131	−1.85886
$97$	− 2.27443i	− 0.230933i	−0.993311	−	0.115467i	$-0.963164\pi$
$97$	− 2.27443i	− 0.230933i	0.993311	−	0.115467i	$-0.0368363\pi$
$98$	5.14994i	0.520222i
$99$	17.1281	1.72144
$100$	0	0
$101$	8.12465	0.808433	0.404216	−	0.914663i	$-0.367544\pi$
$101$	8.12465	0.808433	0.404216	+	0.914663i	$0.367544\pi$
$102$	5.46174i	0.540792i
$103$	5.15706i	0.508140i	0.967186	+	0.254070i	$0.0817693\pi$
$103$	5.15706i	0.508140i	−0.967186	+	0.254070i	$0.918231\pi$
$104$	−8.23203	−0.807217
$105$	0	0
$106$	−10.5678	−1.02644
$107$	5.61021i	0.542360i	0.962529	+	0.271180i	$0.0874138\pi$
$107$	5.61021i	0.542360i	−0.962529	+	0.271180i	$0.912586\pi$
$108$	1.36279i	0.131134i
$109$	3.98158	0.381366	0.190683	−	0.981652i	$-0.438930\pi$
$109$	3.98158	0.381366	0.190683	+	0.981652i	$0.438930\pi$
$110$	0	0
$111$	−10.4647	−0.993268
$112$	− 5.20744i	− 0.492057i
$113$	0.913734i	0.0859569i	0.999076	+	0.0429785i	$0.0136847\pi$
$113$	0.913734i	0.0859569i	−0.999076	+	0.0429785i	$0.986315\pi$
$114$	−24.9779	−2.33939
$115$	0	0
$116$	−16.6975	−1.55032
$117$	− 14.6912i	− 1.35820i
$118$	27.9556i	2.57352i
$119$	−3.05725	−0.280258
$120$	0	0
$121$	17.7489	1.61354
$122$	− 10.9235i	− 0.988965i
$123$	1.00671i	0.0907720i
$124$	−27.7226	−2.48956
$125$	0	0
$126$	−21.4318	−1.90929
$127$	8.41422i	0.746642i	0.927702	+	0.373321i	$0.121781\pi$
$127$	8.41422i	0.746642i	−0.927702	+	0.373321i	$0.878219\pi$
$128$	− 13.1289i	− 1.16044i
$129$	−14.3394	−1.26251
$130$	0	0
$131$	−1.34723	−0.117708	−0.0588542	−	0.998267i	$-0.518745\pi$
$131$	−1.34723	−0.117708	−0.0588542	+	0.998267i	$0.518745\pi$
$132$	37.5747i	3.27046i
$133$	− 13.9816i	− 1.21236i
$134$	14.9866	1.29464
$135$	0	0
$136$	−1.78998	−0.153490
$137$	12.5650i	1.07350i	0.843742	+	0.536749i	$0.180348\pi$
$137$	12.5650i	1.07350i	−0.843742	+	0.536749i	$0.819652\pi$
$138$	− 6.81241i	− 0.579910i
$139$	9.83024	0.833790	0.416895	−	0.908955i	$-0.363118\pi$
$139$	9.83024	0.833790	0.416895	+	0.908955i	$0.363118\pi$
$140$	0	0
$141$	8.35496	0.703614
$142$	− 26.2036i	− 2.19896i
$143$	− 24.6586i	− 2.06206i
$144$	5.44116	0.453430
$145$	0	0
$146$	−23.8925	−1.97736
$147$	− 5.84084i	− 0.481744i
$148$	− 11.8388i	− 0.973146i
$149$	9.03003	0.739769	0.369884	−	0.929078i	$-0.379397\pi$
$149$	9.03003	0.739769	0.369884	+	0.929078i	$0.379397\pi$
$150$	0	0
$151$	−5.97576	−0.486301	−0.243150	−	0.969989i	$-0.578181\pi$
$151$	−5.97576	−0.486301	−0.243150	+	0.969989i	$0.578181\pi$
$152$	− 8.18603i	− 0.663975i
$153$	− 3.19447i	− 0.258257i
$154$	−35.9725	−2.89875
$155$	0	0
$156$	32.2288	2.58037
$157$	− 8.71002i	− 0.695135i	−0.937655	−	0.347568i	$-0.887008\pi$
$157$	− 8.71002i	− 0.695135i	0.937655	−	0.347568i	$-0.112992\pi$
$158$	36.6781i	2.91795i
$159$	11.9856	0.950519
$160$	0	0
$161$	3.81330	0.300530
$162$	18.3870i	1.44462i
$163$	− 0.852508i	− 0.0667736i	−0.999443	−	0.0333868i	$-0.989371\pi$
$163$	− 0.852508i	− 0.0667736i	0.999443	−	0.0333868i	$-0.0106293\pi$
$164$	−1.13890	−0.0889331
$165$	0	0
$166$	−9.02913	−0.700797
$167$	1.32122i	0.102239i	0.998693	+	0.0511195i	$0.0162789\pi$
$167$	1.32122i	0.102239i	−0.998693	+	0.0511195i	$0.983721\pi$
$168$	− 13.6201i	− 1.05082i
$169$	−8.15035	−0.626950
$170$	0	0
$171$	14.6091	1.11719
$172$	− 16.2223i	− 1.23694i
$173$	7.49672i	0.569965i	0.958533	+	0.284983i	$0.0919879\pi$
$173$	7.49672i	0.569965i	−0.958533	+	0.284983i	$0.908012\pi$
$174$	32.3891	2.45541
$175$	0	0
$176$	9.13279	0.688410
$177$	− 31.7061i	− 2.38317i
$178$	− 22.8537i	− 1.71295i
$179$	−15.9913	−1.19525	−0.597624	−	0.801777i	$-0.703888\pi$
$179$	−15.9913	−1.19525	−0.597624	+	0.801777i	$0.703888\pi$
$180$	0	0
$181$	−11.0136	−0.818633	−0.409317	−	0.912392i	$-0.634233\pi$
$181$	−11.0136	−0.818633	−0.409317	+	0.912392i	$0.634233\pi$
$182$	30.8545i	2.28709i
$183$	12.3889i	0.915816i
$184$	2.23264	0.164592
$185$	0	0
$186$	53.7752	3.94299
$187$	− 5.36180i	− 0.392094i
$188$	9.45204i	0.689361i
$189$	1.47971	0.107633
$190$	0	0
$191$	8.49870	0.614944	0.307472	−	0.951557i	$-0.400517\pi$
$191$	8.49870	0.614944	0.307472	+	0.951557i	$0.400517\pi$
$192$	31.4893i	2.27255i
$193$	− 18.8634i	− 1.35782i	−0.734223	−	0.678908i	$-0.762453\pi$
$193$	− 18.8634i	− 1.35782i	0.734223	−	0.678908i	$-0.237547\pi$
$194$	−4.99116	−0.358344
$195$	0	0
$196$	6.60779	0.471985
$197$	10.7750i	0.767687i	0.923398	+	0.383843i	$0.125400\pi$
$197$	10.7750i	0.767687i	−0.923398	+	0.383843i	$0.874600\pi$
$198$	− 37.5870i	− 2.67119i
$199$	12.7488	0.903735	0.451868	−	0.892085i	$-0.350758\pi$
$199$	12.7488	0.903735	0.451868	+	0.892085i	$0.350758\pi$
$200$	0	0
$201$	−16.9971	−1.19889
$202$	− 17.8293i	− 1.25446i
$203$	18.1301i	1.27248i
$204$	7.00786	0.490648
$205$	0	0
$206$	11.3170	0.788492
$207$	3.98445i	0.276938i
$208$	− 7.83343i	− 0.543151i
$209$	24.5209	1.69614
$210$	0	0
$211$	26.3541	1.81429	0.907146	−	0.420817i	$-0.138257\pi$
$211$	26.3541	1.81429	0.907146	+	0.420817i	$0.138257\pi$
$212$	13.5594i	0.931264i
$213$	29.7190i	2.03631i
$214$	12.3114	0.841591
$215$	0	0
$216$	0.866350	0.0589476
$217$	30.1011i	2.04339i
$218$	− 8.73744i	− 0.591774i
$219$	27.0979	1.83110
$220$	0	0
$221$	−4.59895	−0.309359
$222$	22.9645i	1.54127i
$223$	20.0300i	1.34131i	0.741769	+	0.670655i	$0.233987\pi$
$223$	20.0300i	1.34131i	−0.741769	+	0.670655i	$0.766013\pi$
$224$	−22.3724	−1.49482
$225$	0	0
$226$	2.00516	0.133381
$227$	13.2928i	0.882277i	0.897439	+	0.441138i	$0.145425\pi$
$227$	13.2928i	0.882277i	−0.897439	+	0.441138i	$0.854575\pi$
$228$	32.0487i	2.12248i
$229$	−10.0240	−0.662404	−0.331202	−	0.943560i	$-0.607454\pi$
$229$	−10.0240	−0.662404	−0.331202	+	0.943560i	$0.607454\pi$
$230$	0	0
$231$	40.7984	2.68434
$232$	10.6149i	0.696903i
$233$	5.12121i	0.335502i	0.985829	+	0.167751i	$0.0536504\pi$
$233$	5.12121i	0.335502i	−0.985829	+	0.167751i	$0.946350\pi$
$234$	−32.2393	−2.10755
$235$	0	0
$236$	35.8694	2.33490
$237$	− 41.5987i	− 2.70213i
$238$	6.70903i	0.434882i
$239$	13.3667	0.864618	0.432309	−	0.901726i	$-0.357699\pi$
$239$	13.3667	0.864618	0.432309	+	0.901726i	$0.357699\pi$
$240$	0	0
$241$	−4.12595	−0.265776	−0.132888	−	0.991131i	$-0.542425\pi$
$241$	−4.12595	−0.265776	−0.132888	+	0.991131i	$0.542425\pi$
$242$	− 38.9493i	− 2.50376i
$243$	− 22.3057i	− 1.43091i
$244$	−14.0157	−0.897264
$245$	0	0
$246$	2.20919	0.140853
$247$	− 21.0322i	− 1.33824i
$248$	17.6238i	1.11911i
$249$	10.2405	0.648962
$250$	0	0
$251$	−7.71502	−0.486968	−0.243484	−	0.969905i	$-0.578290\pi$
$251$	−7.71502	−0.486968	−0.243484	+	0.969905i	$0.578290\pi$
$252$	27.4987i	1.73226i
$253$	6.68775i	0.420456i
$254$	18.4647	1.15858
$255$	0	0
$256$	−3.50680	−0.219175
$257$	− 18.4651i	− 1.15182i	−0.817512	−	0.575912i	$-0.804647\pi$
$257$	− 18.4651i	− 1.15182i	0.817512	−	0.575912i	$-0.195353\pi$
$258$	31.4673i	1.95907i
$259$	−12.8546	−0.798743
$260$	0	0
$261$	−18.9438	−1.17259
$262$	2.95646i	0.182651i
$263$	18.7644i	1.15707i	0.815659	+	0.578533i	$0.196374\pi$
$263$	18.7644i	1.15707i	−0.815659	+	0.578533i	$0.803626\pi$
$264$	23.8870	1.47014
$265$	0	0
$266$	−30.6821	−1.88124
$267$	25.9196i	1.58626i
$268$	− 19.2290i	− 1.17460i
$269$	−14.7847	−0.901441	−0.450721	−	0.892665i	$-0.648833\pi$
$269$	−14.7847	−0.901441	−0.450721	+	0.892665i	$0.648833\pi$
$270$	0	0
$271$	−24.2920	−1.47563	−0.737817	−	0.675001i	$-0.764143\pi$
$271$	−24.2920	−1.47563	−0.737817	+	0.675001i	$0.764143\pi$
$272$	− 1.70331i	− 0.103278i
$273$	− 34.9939i	− 2.11792i
$274$	27.5734	1.66577
$275$	0	0
$276$	−8.74088	−0.526139
$277$	− 5.84481i	− 0.351181i	−0.984463	−	0.175590i	$-0.943817\pi$
$277$	− 5.84481i	− 0.351181i	0.984463	−	0.175590i	$-0.0561834\pi$
$278$	− 21.5721i	− 1.29381i
$279$	−31.4521	−1.88299
$280$	0	0
$281$	−0.651914	−0.0388899	−0.0194450	−	0.999811i	$-0.506190\pi$
$281$	−0.651914	−0.0388899	−0.0194450	+	0.999811i	$0.506190\pi$
$282$	− 18.3347i	− 1.09181i
$283$	− 4.62452i	− 0.274899i	−0.990509	−	0.137450i	$-0.956110\pi$
$283$	− 4.62452i	− 0.274899i	0.990509	−	0.137450i	$-0.0438905\pi$
$284$	−33.6214	−1.99506
$285$	0	0
$286$	−54.1126	−3.19974
$287$	1.23661i	0.0729949i
$288$	− 23.3765i	− 1.37747i
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	5.66075	0.331839
$292$	30.6561i	1.79401i
$293$	− 3.46169i	− 0.202234i	−0.994875	−	0.101117i	$-0.967758\pi$
$293$	− 3.46169i	− 0.202234i	0.994875	−	0.101117i	$-0.0322417\pi$
$294$	−12.8175	−0.747533
$295$	0	0
$296$	−7.52617	−0.437450
$297$	2.59511i	0.150583i
$298$	− 19.8161i	− 1.14792i
$299$	5.73626	0.331736
$300$	0	0
$301$	−17.6141	−1.01526
$302$	13.1136i	0.754603i
$303$	20.2212i	1.16168i
$304$	7.78966	0.446767
$305$	0	0
$306$	−7.01015	−0.400743
$307$	− 19.7625i	− 1.12790i	−0.825808	−	0.563952i	$-0.809280\pi$
$307$	− 19.7625i	− 1.12790i	0.825808	−	0.563952i	$-0.190720\pi$
$308$	46.1557i	2.62996i
$309$	−12.8352	−0.730171
$310$	0	0
$311$	26.2366	1.48774	0.743871	−	0.668324i	$-0.232988\pi$
$311$	26.2366	1.48774	0.743871	+	0.668324i	$0.232988\pi$
$312$	− 20.4884i	− 1.15993i
$313$	22.9825i	1.29905i	0.760342	+	0.649523i	$0.225031\pi$
$313$	22.9825i	1.29905i	−0.760342	+	0.649523i	$0.774969\pi$
$314$	−19.1138	−1.07866
$315$	0	0
$316$	47.0610	2.64739
$317$	10.9419i	0.614558i	0.951619	+	0.307279i	$0.0994185\pi$
$317$	10.9419i	0.614558i	−0.951619	+	0.307279i	$0.900582\pi$
$318$	− 26.3020i	− 1.47494i
$319$	−31.7964	−1.78026
$320$	0	0
$321$	−13.9631	−0.779343
$322$	− 8.36815i	− 0.466339i
$323$	− 4.57325i	− 0.254463i
$324$	23.5920	1.31067
$325$	0	0
$326$	−1.87080	−0.103614
$327$	9.90963i	0.548004i
$328$	0.724020i	0.0399773i
$329$	10.2630	0.565816
$330$	0	0
$331$	11.1037	0.610314	0.305157	−	0.952302i	$-0.401291\pi$
$331$	11.1037	0.610314	0.305157	+	0.952302i	$0.401291\pi$
$332$	11.5851i	0.635816i
$333$	− 13.4315i	− 0.736041i
$334$	2.89937	0.158646
$335$	0	0
$336$	12.9606	0.707060
$337$	30.0360i	1.63617i	0.575101	+	0.818083i	$0.304963\pi$
$337$	30.0360i	1.63617i	−0.575101	+	0.818083i	$0.695037\pi$
$338$	17.8857i	0.972851i
$339$	−2.27416	−0.123516
$340$	0	0
$341$	−52.7912	−2.85880
$342$	− 32.0592i	− 1.73356i
$343$	14.2260i	0.768134i
$344$	−10.3128	−0.556030
$345$	0	0
$346$	16.4513	0.884428
$347$	26.4956i	1.42236i	0.703011	+	0.711179i	$0.251839\pi$
$347$	26.4956i	1.42236i	−0.703011	+	0.711179i	$0.748161\pi$
$348$	− 41.5579i	− 2.22774i
$349$	−11.8225	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$349$	−11.8225	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$350$	0	0
$351$	2.22589	0.118809
$352$	− 39.2366i	− 2.09132i
$353$	− 11.3987i	− 0.606690i	−0.952881	−	0.303345i	$-0.901897\pi$
$353$	− 11.3987i	− 0.606690i	0.952881	−	0.303345i	$-0.0981035\pi$
$354$	−69.5779	−3.69802
$355$	0	0
$356$	−29.3231	−1.55412
$357$	− 7.60910i	− 0.402716i
$358$	35.0924i	1.85469i
$359$	−12.2135	−0.644601	−0.322301	−	0.946637i	$-0.604456\pi$
$359$	−12.2135	−0.644601	−0.322301	+	0.946637i	$0.604456\pi$
$360$	0	0
$361$	1.91463	0.100770
$362$	24.1689i	1.27029i
$363$	44.1746i	2.31857i
$364$	39.5889	2.07502
$365$	0	0
$366$	27.1871	1.42109
$367$	− 6.91651i	− 0.361039i	−0.983571	−	0.180519i	$-0.942222\pi$
$367$	− 6.91651i	− 0.361039i	0.983571	−	0.180519i	$-0.0577779\pi$
$368$	2.12453i	0.110749i
$369$	−1.29211	−0.0672647
$370$	0	0
$371$	14.7227	0.764367
$372$	− 68.9979i	− 3.57738i
$373$	27.3487i	1.41606i	0.706183	+	0.708030i	$0.250416\pi$
$373$	27.3487i	1.41606i	−0.706183	+	0.708030i	$0.749584\pi$
$374$	−11.7663	−0.608420
$375$	0	0
$376$	6.00884	0.309882
$377$	27.2726i	1.40461i
$378$	− 3.24717i	− 0.167016i
$379$	−14.9563	−0.768255	−0.384127	−	0.923280i	$-0.625498\pi$
$379$	−14.9563	−0.768255	−0.384127	+	0.923280i	$0.625498\pi$
$380$	0	0
$381$	−20.9419	−1.07289
$382$	− 18.6501i	− 0.954222i
$383$	− 28.4094i	− 1.45165i	−0.687879	−	0.725826i	$-0.741458\pi$
$383$	− 28.4094i	− 1.45165i	0.687879	−	0.725826i	$-0.258542\pi$
$384$	32.6761	1.66750
$385$	0	0
$386$	−41.3951	−2.10695
$387$	− 18.4046i	− 0.935561i
$388$	6.40406i	0.325117i
$389$	22.2240	1.12680	0.563401	−	0.826184i	$-0.309493\pi$
$389$	22.2240	1.12680	0.563401	+	0.826184i	$0.309493\pi$
$390$	0	0
$391$	1.24730	0.0630785
$392$	− 4.20070i	− 0.212167i
$393$	− 3.35309i	− 0.169141i
$394$	23.6454	1.19124
$395$	0	0
$396$	−48.2272	−2.42351
$397$	1.76615i	0.0886406i	0.999017	+	0.0443203i	$0.0141122\pi$
$397$	1.76615i	0.0886406i	−0.999017	+	0.0443203i	$0.985888\pi$
$398$	− 27.9767i	− 1.40235i
$399$	34.7983	1.74209
$400$	0	0
$401$	−4.79423	−0.239412	−0.119706	−	0.992809i	$-0.538195\pi$
$401$	−4.79423	−0.239412	−0.119706	+	0.992809i	$0.538195\pi$
$402$	37.2996i	1.86034i
$403$	45.2803i	2.25557i
$404$	−22.8764	−1.13814
$405$	0	0
$406$	39.7858	1.97454
$407$	− 22.5443i	− 1.11748i
$408$	− 4.45503i	− 0.220557i
$409$	36.2834	1.79410	0.897050	−	0.441929i	$-0.145706\pi$
$409$	36.2834	1.79410	0.897050	+	0.441929i	$0.145706\pi$
$410$	0	0
$411$	−31.2726	−1.54256
$412$	− 14.5206i	− 0.715379i
$413$	− 38.9468i	− 1.91645i
$414$	8.74373	0.429731
$415$	0	0
$416$	−33.6543	−1.65004
$417$	24.4662i	1.19811i
$418$	− 53.8102i	− 2.63194i
$419$	24.4452	1.19423	0.597113	−	0.802157i	$-0.296314\pi$
$419$	24.4452	1.19423	0.597113	+	0.802157i	$0.296314\pi$
$420$	0	0
$421$	−14.0909	−0.686750	−0.343375	−	0.939198i	$-0.611570\pi$
$421$	−14.0909	−0.686750	−0.343375	+	0.939198i	$0.611570\pi$
$422$	− 57.8332i	− 2.81527i
$423$	10.7236i	0.521399i
$424$	8.61997	0.418623
$425$	0	0
$426$	65.2174	3.15979
$427$	15.2182i	0.736460i
$428$	− 15.7966i	− 0.763556i
$429$	61.3721	2.96307
$430$	0	0
$431$	−17.5277	−0.844282	−0.422141	−	0.906530i	$-0.638721\pi$
$431$	−17.5277	−0.844282	−0.422141	+	0.906530i	$0.638721\pi$
$432$	0.824400i	0.0396640i
$433$	11.1755i	0.537059i	0.963271	+	0.268530i	$0.0865377\pi$
$433$	11.1755i	0.537059i	−0.963271	+	0.268530i	$0.913462\pi$
$434$	66.0558	3.17078
$435$	0	0
$436$	−11.2109	−0.536902
$437$	5.70420i	0.272869i
$438$	− 59.4654i	− 2.84136i
$439$	12.0419	0.574727	0.287363	−	0.957822i	$-0.407221\pi$
$439$	12.0419	0.574727	0.287363	+	0.957822i	$0.407221\pi$
$440$	0	0
$441$	7.49672	0.356987
$442$	10.0922i	0.480039i
$443$	− 39.5984i	− 1.88138i	−0.339269	−	0.940689i	$-0.610180\pi$
$443$	− 39.5984i	− 1.88138i	0.339269	−	0.940689i	$-0.389820\pi$
$444$	29.4653	1.39836
$445$	0	0
$446$	43.9552	2.08134
$447$	22.4746i	1.06301i
$448$	38.6806i	1.82748i
$449$	−11.6778	−0.551107	−0.275554	−	0.961286i	$-0.588861\pi$
$449$	−11.6778	−0.551107	−0.275554	+	0.961286i	$0.588861\pi$
$450$	0	0
$451$	−2.16877	−0.102123
$452$	− 2.57278i	− 0.121014i
$453$	− 14.8729i	− 0.698789i
$454$	29.1707	1.36905
$455$	0	0
$456$	20.3740	0.954098
$457$	20.2148i	0.945606i	0.881168	+	0.472803i	$0.156758\pi$
$457$	20.2148i	0.945606i	−0.881168	+	0.472803i	$0.843242\pi$
$458$	21.9973i	1.02787i
$459$	0.483999	0.0225912
$460$	0	0
$461$	2.77786	0.129378	0.0646890	−	0.997905i	$-0.479394\pi$
$461$	2.77786	0.129378	0.0646890	+	0.997905i	$0.479394\pi$
$462$	− 89.5308i	− 4.16535i
$463$	− 33.8186i	− 1.57168i	−0.618427	−	0.785842i	$-0.712230\pi$
$463$	− 33.8186i	− 1.57168i	0.618427	−	0.785842i	$-0.287770\pi$
$464$	−10.1009	−0.468924
$465$	0	0
$466$	11.2383	0.520605
$467$	41.1417i	1.90381i	0.306395	+	0.951904i	$0.400877\pi$
$467$	41.1417i	1.90381i	−0.306395	+	0.951904i	$0.599123\pi$
$468$	41.3657i	1.91213i
$469$	−20.8788	−0.964092
$470$	0	0
$471$	21.6781	0.998874
$472$	− 22.8028i	− 1.04959i
$473$	− 30.8916i	− 1.42039i
$474$	−91.2869	−4.19295
$475$	0	0
$476$	8.60824	0.394558
$477$	15.3835i	0.704363i
$478$	− 29.3327i	− 1.34165i
$479$	10.8273	0.494710	0.247355	−	0.968925i	$-0.420439\pi$
$479$	10.8273	0.494710	0.247355	+	0.968925i	$0.420439\pi$
$480$	0	0
$481$	−19.3368	−0.881682
$482$	9.05426i	0.412410i
$483$	9.49080i	0.431846i
$484$	−49.9752	−2.27160
$485$	0	0
$486$	−48.9491	−2.22038
$487$	− 17.2533i	− 0.781820i	−0.920429	−	0.390910i	$-0.872160\pi$
$487$	− 17.2533i	− 0.781820i	0.920429	−	0.390910i	$-0.127840\pi$
$488$	8.91005i	0.403339i
$489$	2.12178	0.0959502
$490$	0	0
$491$	−28.6442	−1.29269	−0.646347	−	0.763043i	$-0.723704\pi$
$491$	−28.6442	−1.29269	−0.646347	+	0.763043i	$0.723704\pi$
$492$	− 2.83457i	− 0.127792i
$493$	5.93018i	0.267082i
$494$	−46.1544	−2.07658
$495$	0	0
$496$	−16.7704	−0.753014
$497$	36.5060i	1.63752i
$498$	− 22.4723i	− 1.00701i
$499$	10.5083	0.470418	0.235209	−	0.971945i	$-0.424423\pi$
$499$	10.5083	0.470418	0.235209	+	0.971945i	$0.424423\pi$
$500$	0	0
$501$	−3.28834	−0.146912
$502$	16.9303i	0.755638i
$503$	− 14.7844i	− 0.659206i	−0.944120	−	0.329603i	$-0.893085\pi$
$503$	− 14.7844i	− 0.659206i	0.944120	−	0.329603i	$-0.106915\pi$
$504$	17.4815	0.778686
$505$	0	0
$506$	14.6760	0.652430
$507$	− 20.2851i	− 0.900895i
$508$	− 23.6918i	− 1.05115i
$509$	−32.5112	−1.44103	−0.720517	−	0.693438i	$-0.756095\pi$
$509$	−32.5112	−1.44103	−0.720517	+	0.693438i	$0.756095\pi$
$510$	0	0
$511$	33.2862	1.47250
$512$	− 18.5623i	− 0.820344i
$513$	2.21345i	0.0977263i
$514$	−40.5211	−1.78731
$515$	0	0
$516$	40.3752	1.77742
$517$	17.9992i	0.791603i
$518$	28.2089i	1.23943i
$519$	−18.6584	−0.819011
$520$	0	0
$521$	40.4949	1.77411	0.887057	−	0.461660i	$-0.152746\pi$
$521$	40.4949	1.77411	0.887057	+	0.461660i	$0.152746\pi$
$522$	41.5714i	1.81953i
$523$	19.7050i	0.861640i	0.902438	+	0.430820i	$0.141776\pi$
$523$	19.7050i	0.861640i	−0.902438	+	0.430820i	$0.858224\pi$
$524$	3.79338	0.165715
$525$	0	0
$526$	41.1779	1.79544
$527$	9.84580i	0.428890i
$528$	22.7303i	0.989210i
$529$	21.4443	0.932359
$530$	0	0
$531$	40.6948	1.76600
$532$	39.3676i	1.70680i
$533$	1.86021i	0.0805745i
$534$	56.8797	2.46143
$535$	0	0
$536$	−12.2242	−0.528007
$537$	− 39.8003i	− 1.71751i
$538$	32.4446i	1.39879i
$539$	12.5830	0.541988
$540$	0	0
$541$	21.0224	0.903824	0.451912	−	0.892062i	$-0.350742\pi$
$541$	21.0224	0.903824	0.451912	+	0.892062i	$0.350742\pi$
$542$	53.3080i	2.28977i
$543$	− 27.4114i	− 1.17633i
$544$	−7.31781	−0.313749
$545$	0	0
$546$	−76.7928	−3.28643
$547$	− 25.1379i	− 1.07482i	−0.843321	−	0.537410i	$-0.819403\pi$
$547$	− 25.1379i	− 1.07482i	0.843321	−	0.537410i	$-0.180597\pi$
$548$	− 35.3789i	− 1.51131i
$549$	−15.9012	−0.678647
$550$	0	0
$551$	−27.1202	−1.15536
$552$	5.55674i	0.236511i
$553$	− 51.0986i	− 2.17293i
$554$	−12.8262	−0.544935
$555$	0	0
$556$	−27.6788	−1.17384
$557$	− 22.9470i	− 0.972297i	−0.873876	−	0.486149i	$-0.838401\pi$
$557$	− 22.9470i	− 0.972297i	0.873876	−	0.486149i	$-0.161599\pi$
$558$	69.0205i	2.92187i
$559$	−26.4965	−1.12068
$560$	0	0
$561$	13.3448	0.563418
$562$	1.43060i	0.0603463i
$563$	34.8974i	1.47075i	0.677661	+	0.735374i	$0.262994\pi$
$563$	34.8974i	1.47075i	−0.677661	+	0.735374i	$0.737006\pi$
$564$	−23.5249	−0.990576
$565$	0	0
$566$	−10.1483	−0.426567
$567$	− 25.6161i	− 1.07577i
$568$	21.3738i	0.896823i
$569$	−6.87405	−0.288175	−0.144088	−	0.989565i	$-0.546025\pi$
$569$	−6.87405	−0.288175	−0.144088	+	0.989565i	$0.546025\pi$
$570$	0	0
$571$	24.7188	1.03445	0.517224	−	0.855850i	$-0.326965\pi$
$571$	24.7188	1.03445	0.517224	+	0.855850i	$0.326965\pi$
$572$	69.4309i	2.90305i
$573$	21.1521i	0.883643i
$574$	2.71370	0.113268
$575$	0	0
$576$	−40.4166	−1.68403
$577$	− 4.89908i	− 0.203951i	−0.994787	−	0.101976i	$-0.967484\pi$
$577$	− 4.89908i	− 0.203951i	0.994787	−	0.101976i	$-0.0325164\pi$
$578$	2.19447i	0.0912777i
$579$	46.9485	1.95111
$580$	0	0
$581$	12.5791	0.521868
$582$	− 12.4223i	− 0.514922i
$583$	25.8207i	1.06938i
$584$	19.4886	0.806446
$585$	0	0
$586$	−7.59657	−0.313811
$587$	− 23.8761i	− 0.985471i	−0.870179	−	0.492736i	$-0.835997\pi$
$587$	− 23.8761i	− 0.985471i	0.870179	−	0.492736i	$-0.164003\pi$
$588$	16.4459i	0.678219i
$589$	−45.0273	−1.85532
$590$	0	0
$591$	−26.8175	−1.10313
$592$	− 7.16175i	− 0.294346i
$593$	− 37.8742i	− 1.55531i	−0.628693	−	0.777654i	$-0.716410\pi$
$593$	− 37.8742i	− 1.55531i	0.628693	−	0.777654i	$-0.283590\pi$
$594$	5.69487	0.233664
$595$	0	0
$596$	−25.4257	−1.04148
$597$	31.7300i	1.29862i
$598$	− 12.5880i	− 0.514762i
$599$	−43.9867	−1.79725	−0.898625	−	0.438718i	$-0.855433\pi$
$599$	−43.9867	−1.79725	−0.898625	+	0.438718i	$0.855433\pi$
$600$	0	0
$601$	11.4083	0.465355	0.232678	−	0.972554i	$-0.425251\pi$
$601$	11.4083	0.465355	0.232678	+	0.972554i	$0.425251\pi$
$602$	38.6535i	1.57540i
$603$	− 21.8158i	− 0.888410i
$604$	16.8258	0.684634
$605$	0	0
$606$	44.3747	1.80260
$607$	17.4364i	0.707721i	0.935298	+	0.353861i	$0.115131\pi$
$607$	17.4364i	0.707721i	−0.935298	+	0.353861i	$0.884869\pi$
$608$	− 33.4662i	− 1.35723i
$609$	−45.1233	−1.82849
$610$	0	0
$611$	15.4384	0.624569
$612$	8.99459i	0.363585i
$613$	− 0.620807i	− 0.0250741i	−0.999921	−	0.0125371i	$-0.996009\pi$
$613$	− 0.620807i	− 0.0250741i	0.999921	−	0.0125371i	$-0.00399078\pi$
$614$	−43.3681	−1.75019
$615$	0	0
$616$	29.3420	1.18222
$617$	− 45.0359i	− 1.81308i	−0.422123	−	0.906539i	$-0.638715\pi$
$617$	− 45.0359i	− 1.81308i	0.422123	−	0.906539i	$-0.361285\pi$
$618$	28.1665i	1.13302i
$619$	11.4780	0.461340	0.230670	−	0.973032i	$-0.425908\pi$
$619$	11.4780	0.461340	0.230670	+	0.973032i	$0.425908\pi$
$620$	0	0
$621$	−0.603691	−0.0242253
$622$	− 57.5753i	− 2.30856i
$623$	31.8389i	1.27560i
$624$	19.4964	0.780480
$625$	0	0
$626$	50.4342	2.01576
$627$	61.0292i	2.43727i
$628$	24.5246i	0.978639i
$629$	−4.20461	−0.167649
$630$	0	0
$631$	−10.9485	−0.435853	−0.217926	−	0.975965i	$-0.569929\pi$
$631$	−10.9485	−0.435853	−0.217926	+	0.975965i	$0.569929\pi$
$632$	− 29.9176i	− 1.19006i
$633$	65.5919i	2.60704i
$634$	24.0116	0.953623
$635$	0	0
$636$	−33.7476	−1.33818
$637$	− 10.7927i	− 0.427624i
$638$	69.7762i	2.76247i
$639$	−38.1444	−1.50897
$640$	0	0
$641$	−1.51186	−0.0597147	−0.0298574	−	0.999554i	$-0.509505\pi$
$641$	−1.51186	−0.0597147	−0.0298574	+	0.999554i	$0.509505\pi$
$642$	30.6415i	1.20932i
$643$	17.7547i	0.700179i	0.936716	+	0.350089i	$0.113849\pi$
$643$	17.7547i	0.700179i	−0.936716	+	0.350089i	$0.886151\pi$
$644$	−10.7370	−0.423098
$645$	0	0
$646$	−10.0358	−0.394855
$647$	− 35.1787i	− 1.38302i	−0.722369	−	0.691508i	$-0.756947\pi$
$647$	− 35.1787i	− 1.38302i	0.722369	−	0.691508i	$-0.243053\pi$
$648$	− 14.9979i	− 0.589172i
$649$	68.3048	2.68120
$650$	0	0
$651$	−74.9176	−2.93625
$652$	2.40039i	0.0940065i
$653$	− 9.99410i	− 0.391099i	−0.980694	−	0.195550i	$-0.937351\pi$
$653$	− 9.99410i	− 0.391099i	0.980694	−	0.195550i	$-0.0626491\pi$
$654$	21.7463	0.850349
$655$	0	0
$656$	−0.688962	−0.0268995
$657$	34.7802i	1.35690i
$658$	− 22.5218i	− 0.877989i
$659$	23.9165	0.931655	0.465827	−	0.884876i	$-0.345757\pi$
$659$	23.9165	0.931655	0.465827	+	0.884876i	$0.345757\pi$
$660$	0	0
$661$	28.6063	1.11266	0.556328	−	0.830963i	$-0.312210\pi$
$661$	28.6063	1.11266	0.556328	+	0.830963i	$0.312210\pi$
$662$	− 24.3667i	− 0.947037i
$663$	− 11.4462i	− 0.444533i
$664$	7.36488	0.285813
$665$	0	0
$666$	−29.4749	−1.14213
$667$	− 7.39670i	− 0.286401i
$668$	− 3.72013i	− 0.143936i
$669$	−49.8521	−1.92739
$670$	0	0
$671$	−26.6896	−1.03034
$672$	− 55.6819i	− 2.14798i
$673$	− 12.9564i	− 0.499431i	−0.968319	−	0.249716i	$-0.919663\pi$
$673$	− 12.9564i	− 0.499431i	0.968319	−	0.249716i	$-0.0803371\pi$
$674$	65.9130	2.53887
$675$	0	0
$676$	22.9488	0.882645
$677$	− 23.5326i	− 0.904430i	−0.891909	−	0.452215i	$-0.850634\pi$
$677$	− 23.5326i	− 0.904430i	0.891909	−	0.452215i	$-0.149366\pi$
$678$	4.99058i	0.191662i
$679$	6.95350	0.266851
$680$	0	0
$681$	−33.0841	−1.26779
$682$	115.848i	4.43607i
$683$	− 25.4677i	− 0.974495i	−0.873264	−	0.487247i	$-0.838001\pi$
$683$	− 25.4677i	− 0.974495i	0.873264	−	0.487247i	$-0.161999\pi$
$684$	−41.1345	−1.57282
$685$	0	0
$686$	31.2186	1.19193
$687$	− 24.9484i	− 0.951840i
$688$	− 9.81346i	− 0.374135i
$689$	22.1471	0.843736
$690$	0	0
$691$	47.8923	1.82191	0.910955	−	0.412507i	$-0.135347\pi$
$691$	47.8923	1.82191	0.910955	+	0.412507i	$0.135347\pi$
$692$	− 21.1084i	− 0.802420i
$693$	52.3649i	1.98918i
$694$	58.1437	2.20710
$695$	0	0
$696$	−26.4191	−1.00141
$697$	0.404485i	0.0153210i
$698$	25.9442i	0.982001i
$699$	−12.7460	−0.482099
$700$	0	0
$701$	−5.52783	−0.208783	−0.104392	−	0.994536i	$-0.533290\pi$
$701$	−5.52783	−0.208783	−0.104392	+	0.994536i	$0.533290\pi$
$702$	− 4.88464i	− 0.184359i
$703$	− 19.2287i	− 0.725226i
$704$	−67.8379	−2.55674
$705$	0	0
$706$	−25.0140	−0.941414
$707$	24.8391i	0.934170i
$708$	89.2741i	3.35513i
$709$	−22.8895	−0.859633	−0.429817	−	0.902916i	$-0.641422\pi$
$709$	−22.8895	−0.859633	−0.429817	+	0.902916i	$0.641422\pi$
$710$	0	0
$711$	53.3920	2.00236
$712$	18.6413i	0.698611i
$713$	− 12.2806i	− 0.459913i
$714$	−16.6979	−0.624903
$715$	0	0
$716$	45.0264	1.68272
$717$	33.2679i	1.24241i
$718$	26.8020i	1.00024i
$719$	−0.748823	−0.0279264	−0.0139632	−	0.999903i	$-0.504445\pi$
$719$	−0.748823	−0.0279264	−0.0139632	+	0.999903i	$0.504445\pi$
$720$	0	0
$721$	−15.7664	−0.587172
$722$	− 4.20159i	− 0.156367i
$723$	− 10.2690i	− 0.381906i
$724$	31.0107	1.15250
$725$	0	0
$726$	96.9398	3.59777
$727$	− 51.8312i	− 1.92231i	−0.276004	−	0.961157i	$-0.589010\pi$
$727$	− 51.8312i	− 1.92231i	0.276004	−	0.961157i	$-0.410990\pi$
$728$	− 25.1674i	− 0.932766i
$729$	30.3796	1.12517
$730$	0	0
$731$	−5.76142	−0.213094
$732$	− 34.8833i	− 1.28932i
$733$	0.440590i	0.0162735i	0.999967	+	0.00813677i	$0.00259004\pi$
$733$	0.440590i	0.0162735i	−0.999967	+	0.00813677i	$0.997410\pi$
$734$	−15.1780	−0.560232
$735$	0	0
$736$	9.12748	0.336443
$737$	− 36.6171i	− 1.34881i
$738$	2.83550i	0.104376i
$739$	30.4658	1.12070	0.560351	−	0.828255i	$-0.310666\pi$
$739$	30.4658	1.12070	0.560351	+	0.828255i	$0.310666\pi$
$740$	0	0
$741$	52.3463	1.92299
$742$	− 32.3086i	− 1.18608i
$743$	− 15.1434i	− 0.555557i	−0.960645	−	0.277779i	$-0.910402\pi$
$743$	− 15.1434i	− 0.555557i	0.960645	−	0.277779i	$-0.0895981\pi$
$744$	−43.8633	−1.60811
$745$	0	0
$746$	60.0157	2.19733
$747$	13.1436i	0.480901i
$748$	15.0971i	0.552005i
$749$	−17.1518	−0.626714
$750$	0	0
$751$	34.1610	1.24655	0.623277	−	0.782001i	$-0.285801\pi$
$751$	34.1610	1.24655	0.623277	+	0.782001i	$0.285801\pi$
$752$	5.71789i	0.208510i
$753$	− 19.2017i	− 0.699748i
$754$	59.8488	2.17957
$755$	0	0
$756$	−4.16638	−0.151530
$757$	20.5792i	0.747965i	0.927436	+	0.373982i	$0.122008\pi$
$757$	20.5792i	0.747965i	−0.927436	+	0.373982i	$0.877992\pi$
$758$	32.8211i	1.19212i
$759$	−16.6449	−0.604173
$760$	0	0
$761$	−18.8045	−0.681661	−0.340831	−	0.940125i	$-0.610708\pi$
$761$	−18.8045	−0.681661	−0.340831	+	0.940125i	$0.610708\pi$
$762$	45.9563i	1.66482i
$763$	12.1727i	0.440681i
$764$	−23.9296	−0.865743
$765$	0	0
$766$	−62.3434	−2.25256
$767$	− 58.5867i	− 2.11544i
$768$	− 8.72798i	− 0.314944i
$769$	36.5974	1.31974	0.659868	−	0.751382i	$-0.270612\pi$
$769$	36.5974	1.31974	0.659868	+	0.751382i	$0.270612\pi$
$770$	0	0
$771$	45.9573	1.65511
$772$	53.1133i	1.91159i
$773$	− 19.4040i	− 0.697912i	−0.937139	−	0.348956i	$-0.886536\pi$
$773$	− 19.4040i	− 0.697912i	0.937139	−	0.348956i	$-0.113464\pi$
$774$	−40.3884	−1.45173
$775$	0	0
$776$	4.07118	0.146147
$777$	− 31.9933i	− 1.14775i
$778$	− 48.7698i	− 1.74848i
$779$	−1.84981	−0.0662764
$780$	0	0
$781$	−64.0240	−2.29096
$782$	− 2.73715i	− 0.0978803i
$783$	− 2.87020i	− 0.102573i
$784$	3.99730	0.142761
$785$	0	0
$786$	−7.35823	−0.262460
$787$	28.7760i	1.02575i	0.858462	+	0.512876i	$0.171420\pi$
$787$	28.7760i	1.02575i	−0.858462	+	0.512876i	$0.828580\pi$
$788$	− 30.3389i	− 1.08078i
$789$	−46.7022	−1.66264
$790$	0	0
$791$	−2.79352	−0.0993260
$792$	30.6589i	1.08942i
$793$	22.8924i	0.812932i
$794$	3.87576	0.137546
$795$	0	0
$796$	−35.8964	−1.27231
$797$	4.55967i	0.161512i	0.996734	+	0.0807559i	$0.0257334\pi$
$797$	4.55967i	0.161512i	−0.996734	+	0.0807559i	$0.974267\pi$
$798$	− 76.3637i	− 2.70325i
$799$	3.35693	0.118760
$800$	0	0
$801$	−33.2679	−1.17546
$802$	10.5208i	0.371501i
$803$	58.3773i	2.06009i
$804$	47.8585	1.68784
$805$	0	0
$806$	99.3662	3.50002
$807$	− 36.7973i	− 1.29532i
$808$	14.5430i	0.511620i
$809$	−37.0052	−1.30103	−0.650516	−	0.759493i	$-0.725447\pi$
$809$	−37.0052	−1.30103	−0.650516	+	0.759493i	$0.725447\pi$
$810$	0	0
$811$	−30.4268	−1.06843	−0.534215	−	0.845349i	$-0.679393\pi$
$811$	−30.4268	−1.06843	−0.534215	+	0.845349i	$0.679393\pi$
$812$	− 51.0484i	− 1.79145i
$813$	− 60.4596i	− 2.12041i
$814$	−49.4726	−1.73402
$815$	0	0
$816$	4.23931	0.148405
$817$	− 26.3484i	− 0.921814i
$818$	− 79.6227i	− 2.78394i
$819$	44.9147	1.56945
$820$	0	0
$821$	2.43436	0.0849596	0.0424798	−	0.999097i	$-0.486474\pi$
$821$	2.43436	0.0849596	0.0424798	+	0.999097i	$0.486474\pi$
$822$	68.6266i	2.39363i
$823$	30.1177i	1.04984i	0.851153	+	0.524918i	$0.175904\pi$
$823$	30.1177i	1.04984i	−0.851153	+	0.524918i	$0.824096\pi$
$824$	−9.23103	−0.321578
$825$	0	0
$826$	−85.4674	−2.97379
$827$	− 17.7944i	− 0.618771i	−0.950937	−	0.309385i	$-0.899877\pi$
$827$	− 17.7944i	− 0.618771i	0.950937	−	0.309385i	$-0.100123\pi$
$828$	− 11.2189i	− 0.389885i
$829$	−27.9359	−0.970254	−0.485127	−	0.874444i	$-0.661227\pi$
$829$	−27.9359	−0.970254	−0.485127	+	0.874444i	$0.661227\pi$
$830$	0	0
$831$	14.5470	0.504629
$832$	58.1863i	2.01725i
$833$	− 2.34678i	− 0.0813113i
$834$	53.6902	1.85914
$835$	0	0
$836$	−69.0429	−2.38790
$837$	− 4.76536i	− 0.164715i
$838$	− 53.6441i	− 1.85311i
$839$	40.4717	1.39724	0.698618	−	0.715494i	$-0.253799\pi$
$839$	40.4717	1.39724	0.698618	+	0.715494i	$0.253799\pi$
$840$	0	0
$841$	6.16706	0.212657
$842$	30.9221i	1.06565i
$843$	− 1.62253i	− 0.0558828i
$844$	−74.2047	−2.55423
$845$	0	0
$846$	23.5326	0.809066
$847$	54.2628i	1.86449i
$848$	8.20258i	0.281678i
$849$	11.5098	0.395016
$850$	0	0
$851$	5.24440	0.179776
$852$	− 83.6792i	− 2.86680i
$853$	− 7.32217i	− 0.250706i	−0.992112	−	0.125353i	$-0.959994\pi$
$853$	− 7.32217i	− 0.250706i	0.992112	−	0.125353i	$-0.0400064\pi$
$854$	33.3958	1.14278
$855$	0	0
$856$	−10.0422	−0.343234
$857$	4.74086i	0.161945i	0.996716	+	0.0809724i	$0.0258026\pi$
$857$	4.74086i	0.161945i	−0.996716	+	0.0809724i	$0.974197\pi$
$858$	− 134.679i	− 4.59787i
$859$	−8.14829	−0.278016	−0.139008	−	0.990291i	$-0.544391\pi$
$859$	−8.14829	−0.278016	−0.139008	+	0.990291i	$0.544391\pi$
$860$	0	0
$861$	−3.07776	−0.104890
$862$	38.4640i	1.31009i
$863$	− 15.8044i	− 0.537988i	−0.963142	−	0.268994i	$-0.913309\pi$
$863$	− 15.8044i	− 0.537988i	0.963142	−	0.268994i	$-0.0866911\pi$
$864$	3.54182	0.120495
$865$	0	0
$866$	24.5242	0.833366
$867$	− 2.48887i	− 0.0845264i
$868$	− 84.7550i	− 2.87677i
$869$	89.6166	3.04003
$870$	0	0
$871$	−31.4074	−1.06420
$872$	7.12695i	0.241349i
$873$	7.26559i	0.245903i
$874$	12.5177	0.423417
$875$	0	0
$876$	−76.2989	−2.57790
$877$	− 22.3063i	− 0.753231i	−0.926370	−	0.376616i	$-0.877088\pi$
$877$	− 22.3063i	− 0.753231i	0.926370	−	0.376616i	$-0.122912\pi$
$878$	− 26.4254i	− 0.891816i
$879$	8.61570	0.290600
$880$	0	0
$881$	9.67865	0.326082	0.163041	−	0.986619i	$-0.447870\pi$
$881$	9.67865	0.326082	0.163041	+	0.986619i	$0.447870\pi$
$882$	− 16.4513i	− 0.553944i
$883$	− 21.7255i	− 0.731120i	−0.930788	−	0.365560i	$-0.880877\pi$
$883$	− 21.7255i	− 0.731120i	0.930788	−	0.365560i	$-0.119123\pi$
$884$	12.9492	0.435528
$885$	0	0
$886$	−86.8974	−2.91938
$887$	2.17244i	0.0729434i	0.999335	+	0.0364717i	$0.0116119\pi$
$887$	2.17244i	0.0729434i	−0.999335	+	0.0364717i	$0.988388\pi$
$888$	− 18.7317i	− 0.628593i
$889$	−25.7244	−0.862768
$890$	0	0
$891$	44.9254	1.50506
$892$	− 56.3981i	− 1.88835i
$893$	15.3521i	0.513738i
$894$	49.3196	1.64950
$895$	0	0
$896$	40.1384	1.34093
$897$	14.2768i	0.476688i
$898$	25.6264i	0.855165i
$899$	58.3874	1.94733
$900$	0	0
$901$	4.81568	0.160434
$902$	4.75928i	0.158467i
$903$	− 43.8392i	− 1.45888i
$904$	−1.63557	−0.0543982
$905$	0	0
$906$	−32.6380	−1.08433
$907$	25.8704i	0.859012i	0.903064	+	0.429506i	$0.141312\pi$
$907$	25.8704i	0.859012i	−0.903064	+	0.429506i	$0.858688\pi$
$908$	− 37.4284i	− 1.24210i
$909$	−25.9539	−0.860837
$910$	0	0
$911$	−36.1769	−1.19859	−0.599297	−	0.800527i	$-0.704553\pi$
$911$	−36.1769	−1.19859	−0.599297	+	0.800527i	$0.704553\pi$
$912$	19.3874i	0.641982i
$913$	22.0611i	0.730117i
$914$	44.3606	1.46732
$915$	0	0
$916$	28.2243	0.932558
$917$	− 4.11883i	− 0.136016i
$918$	− 1.06212i	− 0.0350552i
$919$	−52.2475	−1.72349	−0.861743	−	0.507346i	$-0.830627\pi$
$919$	−52.2475	−1.72349	−0.861743	+	0.507346i	$0.830627\pi$
$920$	0	0
$921$	49.1862	1.62074
$922$	− 6.09592i	− 0.200759i
$923$	54.9150i	1.80755i
$924$	−114.875	−3.77912
$925$	0	0
$926$	−74.2138	−2.43882
$927$	− 16.4740i	− 0.541078i
$928$	43.3960i	1.42454i
$929$	−19.6884	−0.645955	−0.322978	−	0.946407i	$-0.604684\pi$
$929$	−19.6884	−0.645955	−0.322978	+	0.946407i	$0.604684\pi$
$930$	0	0
$931$	10.7324	0.351741
$932$	− 14.4197i	− 0.472333i
$933$	65.2994i	2.13781i
$934$	90.2840	2.95418
$935$	0	0
$936$	26.2969	0.859543
$937$	15.0805i	0.492657i	0.969186	+	0.246328i	$0.0792242\pi$
$937$	15.0805i	0.492657i	−0.969186	+	0.246328i	$0.920776\pi$
$938$	45.8177i	1.49600i
$939$	−57.2004	−1.86666
$940$	0	0
$941$	−12.2054	−0.397886	−0.198943	−	0.980011i	$-0.563751\pi$
$941$	−12.2054	−0.397886	−0.198943	+	0.980011i	$0.563751\pi$
$942$	− 47.5718i	− 1.54997i
$943$	− 0.504513i	− 0.0164292i
$944$	21.6987	0.706232
$945$	0	0
$946$	−67.7904	−2.20406
$947$	− 23.4424i	− 0.761776i	−0.924621	−	0.380888i	$-0.875618\pi$
$947$	− 23.4424i	− 0.761776i	0.924621	−	0.380888i	$-0.124382\pi$
$948$	117.129i	3.80416i
$949$	50.0717	1.62540
$950$	0	0
$951$	−27.2329	−0.883088
$952$	− 5.47242i	− 0.177362i
$953$	50.7165i	1.64287i	0.570302	+	0.821435i	$0.306826\pi$
$953$	50.7165i	1.64287i	−0.570302	+	0.821435i	$0.693174\pi$
$954$	33.7586	1.09298
$955$	0	0
$956$	−37.6363	−1.21724
$957$	− 79.1372i	− 2.55814i
$958$	− 23.7600i	− 0.767652i
$959$	−38.4143	−1.24046
$960$	0	0
$961$	65.9397	2.12709
$962$	42.4339i	1.36813i
$963$	− 17.9216i	− 0.577517i
$964$	11.6174	0.374170
$965$	0	0
$966$	20.8272	0.670105
$967$	41.6170i	1.33831i	0.743121	+	0.669157i	$0.233344\pi$
$967$	41.6170i	1.33831i	−0.743121	+	0.669157i	$0.766656\pi$
$968$	31.7702i	1.02113i
$969$	11.3822	0.365650
$970$	0	0
$971$	19.2045	0.616302	0.308151	−	0.951337i	$-0.400290\pi$
$971$	19.2045	0.616302	0.308151	+	0.951337i	$0.400290\pi$
$972$	62.8057i	2.01449i
$973$	30.0535i	0.963472i
$974$	−37.8617	−1.21317
$975$	0	0
$976$	−8.47862	−0.271394
$977$	− 21.0264i	− 0.672696i	−0.941738	−	0.336348i	$-0.890808\pi$
$977$	− 21.0264i	− 0.672696i	0.941738	−	0.336348i	$-0.109192\pi$
$978$	− 4.65618i	− 0.148888i
$979$	−55.8390	−1.78462
$980$	0	0
$981$	−12.7190	−0.406087
$982$	62.8587i	2.00590i
$983$	− 3.77964i	− 0.120552i	−0.998182	−	0.0602758i	$-0.980802\pi$
$983$	− 3.77964i	− 0.120552i	0.998182	−	0.0602758i	$-0.0191980\pi$
$984$	−1.80199	−0.0574454
$985$	0	0
$986$	13.0136	0.414437
$987$	25.5432i	0.813049i
$988$	59.2198i	1.88403i
$989$	7.18619	0.228508
$990$	0	0
$991$	35.7886	1.13686	0.568431	−	0.822731i	$-0.307551\pi$
$991$	35.7886	1.13686	0.568431	+	0.822731i	$0.307551\pi$
$992$	72.0497i	2.28758i
$993$	27.6356i	0.876990i
$994$	80.1111	2.54097
$995$	0	0
$996$	−28.8338	−0.913635
$997$	4.92887i	0.156099i	0.996949	+	0.0780494i	$0.0248692\pi$
$997$	4.92887i	0.156099i	−0.996949	+	0.0780494i	$0.975131\pi$
$998$	− 23.0602i	− 0.729958i
$999$	2.03503	0.0643855

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	425.2.b.f.324.3		10
5.2	odd	4		425.2.a.j.1.4	yes	5
5.3	odd	4		425.2.a.i.1.2	✓	5
5.4	even	2	inner	425.2.b.f.324.8		10
15.2	even	4		3825.2.a.bl.1.2		5
15.8	even	4		3825.2.a.bq.1.4		5
20.3	even	4		6800.2.a.bz.1.5		5
20.7	even	4		6800.2.a.cd.1.1		5
85.33	odd	4		7225.2.a.x.1.2		5
85.67	odd	4		7225.2.a.y.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
425.2.a.i.1.2	✓	5	5.3	odd	4
425.2.a.j.1.4	yes	5	5.2	odd	4
425.2.b.f.324.3		10	1.1	even	1	trivial
425.2.b.f.324.8		10	5.4	even	2	inner
3825.2.a.bl.1.2		5	15.2	even	4
3825.2.a.bq.1.4		5	15.8	even	4
6800.2.a.bz.1.5		5	20.3	even	4
6800.2.a.cd.1.1		5	20.7	even	4
7225.2.a.x.1.2		5	85.33	odd	4
7225.2.a.y.1.4		5	85.67	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 425 = 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	425.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.19447i q^{2} +2.48887i q^{3} -2.81568 q^{4} +5.46174 q^{6} +3.05725i q^{7} +1.78998i q^{8} -3.19447 q^{9} -5.36180 q^{11} -7.00786i q^{12} +4.59895i q^{13} +6.70903 q^{14} -1.70331 q^{16} +1.00000i q^{17} +7.01015i q^{18} -4.57325 q^{19} -7.60910 q^{21} +11.7663i q^{22} -1.24730i q^{23} -4.45503 q^{24} +10.0922 q^{26} -0.483999i q^{27} -8.60824i q^{28} +5.93018 q^{29} +9.84580 q^{31} +7.31781i q^{32} -13.3448i q^{33} +2.19447 q^{34} +8.99459 q^{36} +4.20461i q^{37} +10.0358i q^{38} -11.4462 q^{39} +0.404485 q^{41} +16.6979i q^{42} +5.76142i q^{43} +15.0971 q^{44} -2.73715 q^{46} -3.35693i q^{47} -4.23931i q^{48} -2.34678 q^{49} -2.48887 q^{51} -12.9492i q^{52} -4.81568i q^{53} -1.06212 q^{54} -5.47242 q^{56} -11.3822i q^{57} -13.0136i q^{58} -12.7392 q^{59} +4.97774 q^{61} -21.6063i q^{62} -9.76628i q^{63} +12.6521 q^{64} -29.2847 q^{66} +6.82926i q^{67} -2.81568i q^{68} +3.10436 q^{69} +11.9408 q^{71} -5.71803i q^{72} -10.8876i q^{73} +9.22687 q^{74} +12.8768 q^{76} -16.3924i q^{77} +25.1183i q^{78} -16.7139 q^{79} -8.37879 q^{81} -0.887628i q^{82} -4.11450i q^{83} +21.4248 q^{84} +12.6432 q^{86} +14.7594i q^{87} -9.59752i q^{88} +10.4142 q^{89} -14.0601 q^{91} +3.51199i q^{92} +24.5049i q^{93} -7.36667 q^{94} -18.2131 q^{96} -2.27443i q^{97} +5.14994i q^{98} +17.1281 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 22 q^{4} + 6 q^{6} - 12 q^{9} + 8 q^{11} + 14 q^{14} + 54 q^{16} - 12 q^{19} - 10 q^{21} + 38 q^{24} - 10 q^{26} - 4 q^{29} + 42 q^{31} + 2 q^{34} + 44 q^{36} - 46 q^{39} - 16 q^{41} + 8 q^{44} - 12 q^{46}+ \cdots + 28 q^{99}+O(q^{100}) \) 10 * q - 22 * q^4 + 6 * q^6 - 12 * q^9 + 8 * q^11 + 14 * q^14 + 54 * q^16 - 12 * q^19 - 10 * q^21 + 38 * q^24 - 10 * q^26 - 4 * q^29 + 42 * q^31 + 2 * q^34 + 44 * q^36 - 46 * q^39 - 16 * q^41 + 8 * q^44 - 12 * q^46 - 20 * q^49 + 2 * q^51 + 50 * q^54 - 58 * q^56 - 24 * q^59 - 4 * q^61 - 86 * q^64 - 56 * q^66 + 42 * q^71 + 100 * q^74 - 28 * q^76 - 82 * q^79 - 70 * q^81 + 142 * q^84 - 72 * q^86 + 20 * q^89 + 26 * q^91 + 20 * q^94 - 170 * q^96 + 28 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

Properties

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