LMFDB - Embedded newform 850.2.c.i.749.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [850,2,Mod(749,850)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(850, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("850.749"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 850 = 2 \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	850.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,-4,0,-2,0,0,-6,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$6.78728417181$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 170)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			749.4
Root			$2.56155i$ of defining polynomial
Character	$\chi$	$=$	850.749
Dual form			850.2.c.i.749.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} +2.56155i q^{3} -1.00000 q^{4} -2.56155 q^{6} +5.12311i q^{7} -1.00000i q^{8} -3.56155 q^{9} -4.00000 q^{11} -2.56155i q^{12} -4.56155i q^{13} -5.12311 q^{14} +1.00000 q^{16} +1.00000i q^{17} -3.56155i q^{18} +2.56155 q^{19} -13.1231 q^{21} -4.00000i q^{22} +5.12311i q^{23} +2.56155 q^{24} +4.56155 q^{26} -1.43845i q^{27} -5.12311i q^{28} +5.68466 q^{29} -6.56155 q^{31} +1.00000i q^{32} -10.2462i q^{33} -1.00000 q^{34} +3.56155 q^{36} -7.12311i q^{37} +2.56155i q^{38} +11.6847 q^{39} +4.24621 q^{41} -13.1231i q^{42} +1.12311i q^{43} +4.00000 q^{44} -5.12311 q^{46} +6.56155i q^{47} +2.56155i q^{48} -19.2462 q^{49} -2.56155 q^{51} +4.56155i q^{52} +0.561553i q^{53} +1.43845 q^{54} +5.12311 q^{56} +6.56155i q^{57} +5.68466i q^{58} +0.315342 q^{59} +7.43845 q^{61} -6.56155i q^{62} -18.2462i q^{63} -1.00000 q^{64} +10.2462 q^{66} -9.12311i q^{67} -1.00000i q^{68} -13.1231 q^{69} -4.31534 q^{71} +3.56155i q^{72} +6.80776i q^{73} +7.12311 q^{74} -2.56155 q^{76} -20.4924i q^{77} +11.6847i q^{78} -7.00000 q^{81} +4.24621i q^{82} +6.24621i q^{83} +13.1231 q^{84} -1.12311 q^{86} +14.5616i q^{87} +4.00000i q^{88} +9.68466 q^{89} +23.3693 q^{91} -5.12311i q^{92} -16.8078i q^{93} -6.56155 q^{94} -2.56155 q^{96} -1.68466i q^{97} -19.2462i q^{98} +14.2462 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} - 2 q^{6} - 6 q^{9} - 16 q^{11} - 4 q^{14} + 4 q^{16} + 2 q^{19} - 36 q^{21} + 2 q^{24} + 10 q^{26} - 2 q^{29} - 18 q^{31} - 4 q^{34} + 6 q^{36} + 22 q^{39} - 16 q^{41} + 16 q^{44} - 4 q^{46}+ \cdots + 24 q^{99}+O(q^{100}) $ 4 * q - 4 * q^4 - 2 * q^6 - 6 * q^9 - 16 * q^11 - 4 * q^14 + 4 * q^16 + 2 * q^19 - 36 * q^21 + 2 * q^24 + 10 * q^26 - 2 * q^29 - 18 * q^31 - 4 * q^34 + 6 * q^36 + 22 * q^39 - 16 * q^41 + 16 * q^44 - 4 * q^46 - 44 * q^49 - 2 * q^51 + 14 * q^54 + 4 * q^56 + 26 * q^59 + 38 * q^61 - 4 * q^64 + 8 * q^66 - 36 * q^69 - 42 * q^71 + 12 * q^74 - 2 * q^76 - 28 * q^81 + 36 * q^84 + 12 * q^86 + 14 * q^89 + 44 * q^91 - 18 * q^94 - 2 * q^96 + 24 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/850\mathbb{Z}\right)^\times$.

$n$	$477$	$751$
$\chi(n)$	$-1$	$1$

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 $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	2.56155i	1.47891i	0.673204	+	0.739457i	$0.264917\pi$
$3$	2.56155i	1.47891i	−0.673204	+	0.739457i	$0.735083\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−2.56155	−1.04575
$7$	5.12311i	1.93635i	0.250270	+	0.968176i	$0.419480\pi$
$7$	5.12311i	1.93635i	−0.250270	+	0.968176i	$0.580520\pi$
$8$	− 1.00000i	− 0.353553i
$9$	−3.56155	−1.18718
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	− 2.56155i	− 0.739457i
$13$	− 4.56155i	− 1.26515i	−0.774500	−	0.632574i	$-0.781999\pi$
$13$	− 4.56155i	− 1.26515i	0.774500	−	0.632574i	$-0.218001\pi$
$14$	−5.12311	−1.36921
$15$	0	0
$16$	1.00000	0.250000
$17$	1.00000i	0.242536i
$17$	1.00000i	0.242536i
$18$	− 3.56155i	− 0.839466i
$19$	2.56155	0.587661	0.293830	−	0.955858i	$-0.405070\pi$
$19$	2.56155	0.587661	0.293830	+	0.955858i	$0.405070\pi$
$20$	0	0
$21$	−13.1231	−2.86370
$22$	− 4.00000i	− 0.852803i
$23$	5.12311i	1.06824i	0.845408	+	0.534121i	$0.179357\pi$
$23$	5.12311i	1.06824i	−0.845408	+	0.534121i	$0.820643\pi$
$24$	2.56155	0.522875
$25$	0	0
$26$	4.56155	0.894594
$27$	− 1.43845i	− 0.276829i
$28$	− 5.12311i	− 0.968176i
$29$	5.68466	1.05561	0.527807	−	0.849364i	$-0.323014\pi$
$29$	5.68466	1.05561	0.527807	+	0.849364i	$0.323014\pi$
$30$	0	0
$31$	−6.56155	−1.17849	−0.589245	−	0.807955i	$-0.700575\pi$
$31$	−6.56155	−1.17849	−0.589245	+	0.807955i	$0.700575\pi$
$32$	1.00000i	0.176777i
$33$	− 10.2462i	− 1.78364i
$34$	−1.00000	−0.171499
$35$	0	0
$36$	3.56155	0.593592
$37$	− 7.12311i	− 1.17103i	−0.810661	−	0.585516i	$-0.800892\pi$
$37$	− 7.12311i	− 1.17103i	0.810661	−	0.585516i	$-0.199108\pi$
$38$	2.56155i	0.415539i
$39$	11.6847	1.87104
$40$	0	0
$41$	4.24621	0.663147	0.331573	−	0.943429i	$-0.392421\pi$
$41$	4.24621	0.663147	0.331573	+	0.943429i	$0.392421\pi$
$42$	− 13.1231i	− 2.02494i
$43$	1.12311i	0.171272i	0.996326	+	0.0856360i	$0.0272922\pi$
$43$	1.12311i	0.171272i	−0.996326	+	0.0856360i	$0.972708\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	−5.12311	−0.755361
$47$	6.56155i	0.957101i	0.878060	+	0.478550i	$0.158838\pi$
$47$	6.56155i	0.957101i	−0.878060	+	0.478550i	$0.841162\pi$
$48$	2.56155i	0.369728i
$49$	−19.2462	−2.74946
$50$	0	0
$51$	−2.56155	−0.358689
$52$	4.56155i	0.632574i
$53$	0.561553i	0.0771352i	0.999256	+	0.0385676i	$0.0122795\pi$
$53$	0.561553i	0.0771352i	−0.999256	+	0.0385676i	$0.987721\pi$
$54$	1.43845	0.195748
$55$	0	0
$56$	5.12311	0.684604
$57$	6.56155i	0.869099i
$58$	5.68466i	0.746432i
$59$	0.315342	0.0410540	0.0205270	−	0.999789i	$-0.493466\pi$
$59$	0.315342	0.0410540	0.0205270	+	0.999789i	$0.493466\pi$
$60$	0	0
$61$	7.43845	0.952396	0.476198	−	0.879338i	$-0.342015\pi$
$61$	7.43845	0.952396	0.476198	+	0.879338i	$0.342015\pi$
$62$	− 6.56155i	− 0.833318i
$63$	− 18.2462i	− 2.29881i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	10.2462	1.26122
$67$	− 9.12311i	− 1.11456i	−0.830323	−	0.557282i	$-0.811844\pi$
$67$	− 9.12311i	− 1.11456i	0.830323	−	0.557282i	$-0.188156\pi$
$68$	− 1.00000i	− 0.121268i
$69$	−13.1231	−1.57984
$70$	0	0
$71$	−4.31534	−0.512137	−0.256068	−	0.966659i	$-0.582427\pi$
$71$	−4.31534	−0.512137	−0.256068	+	0.966659i	$0.582427\pi$
$72$	3.56155i	0.419733i
$73$	6.80776i	0.796789i	0.917214	+	0.398394i	$0.130432\pi$
$73$	6.80776i	0.796789i	−0.917214	+	0.398394i	$0.869568\pi$
$74$	7.12311	0.828044
$75$	0	0
$76$	−2.56155	−0.293830
$77$	− 20.4924i	− 2.33533i
$78$	11.6847i	1.32303i
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	4.24621i	0.468916i
$83$	6.24621i	0.685611i	0.939406	+	0.342805i	$0.111377\pi$
$83$	6.24621i	0.685611i	−0.939406	+	0.342805i	$0.888623\pi$
$84$	13.1231	1.43185
$85$	0	0
$86$	−1.12311	−0.121108
$87$	14.5616i	1.56116i
$88$	4.00000i	0.426401i
$89$	9.68466	1.02657	0.513286	−	0.858218i	$-0.328428\pi$
$89$	9.68466	1.02657	0.513286	+	0.858218i	$0.328428\pi$
$90$	0	0
$91$	23.3693	2.44977
$92$	− 5.12311i	− 0.534121i
$93$	− 16.8078i	− 1.74288i
$94$	−6.56155	−0.676772
$95$	0	0
$96$	−2.56155	−0.261437
$97$	− 1.68466i	− 0.171051i	−0.996336	−	0.0855256i	$-0.972743\pi$
$97$	− 1.68466i	− 0.171051i	0.996336	−	0.0855256i	$-0.0272569\pi$
$98$	− 19.2462i	− 1.94416i
$99$	14.2462	1.43180
$100$	0	0
$101$	−14.4924	−1.44205	−0.721025	−	0.692909i	$-0.756329\pi$
$101$	−14.4924	−1.44205	−0.721025	+	0.692909i	$0.756329\pi$
$102$	− 2.56155i	− 0.253632i
$103$	− 8.00000i	− 0.788263i	−0.919054	−	0.394132i	$-0.871045\pi$
$103$	− 8.00000i	− 0.788263i	0.919054	−	0.394132i	$-0.128955\pi$
$104$	−4.56155	−0.447297
$105$	0	0
$106$	−0.561553	−0.0545428
$107$	16.4924i	1.59438i	0.603727	+	0.797191i	$0.293682\pi$
$107$	16.4924i	1.59438i	−0.603727	+	0.797191i	$0.706318\pi$
$108$	1.43845i	0.138415i
$109$	8.56155	0.820048	0.410024	−	0.912075i	$-0.365520\pi$
$109$	8.56155	0.820048	0.410024	+	0.912075i	$0.365520\pi$
$110$	0	0
$111$	18.2462	1.73185
$112$	5.12311i	0.484088i
$113$	− 2.80776i	− 0.264132i	−0.991241	−	0.132066i	$-0.957839\pi$
$113$	− 2.80776i	− 0.264132i	0.991241	−	0.132066i	$-0.0421611\pi$
$114$	−6.56155	−0.614546
$115$	0	0
$116$	−5.68466	−0.527807
$117$	16.2462i	1.50196i
$118$	0.315342i	0.0290295i
$119$	−5.12311	−0.469634
$120$	0	0
$121$	5.00000	0.454545
$122$	7.43845i	0.673445i
$123$	10.8769i	0.980737i
$124$	6.56155	0.589245
$125$	0	0
$126$	18.2462	1.62550
$127$	9.43845i	0.837527i	0.908095	+	0.418763i	$0.137536\pi$
$127$	9.43845i	0.837527i	−0.908095	+	0.418763i	$0.862464\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	−2.87689	−0.253296
$130$	0	0
$131$	1.12311	0.0981262	0.0490631	−	0.998796i	$-0.484376\pi$
$131$	1.12311	0.0981262	0.0490631	+	0.998796i	$0.484376\pi$
$132$	10.2462i	0.891818i
$133$	13.1231i	1.13792i
$134$	9.12311	0.788116
$135$	0	0
$136$	1.00000	0.0857493
$137$	10.0000i	0.854358i	0.904167	+	0.427179i	$0.140493\pi$
$137$	10.0000i	0.854358i	−0.904167	+	0.427179i	$0.859507\pi$
$138$	− 13.1231i	− 1.11711i
$139$	−16.4924	−1.39887	−0.699435	−	0.714697i	$-0.746565\pi$
$139$	−16.4924	−1.39887	−0.699435	+	0.714697i	$0.746565\pi$
$140$	0	0
$141$	−16.8078	−1.41547
$142$	− 4.31534i	− 0.362135i
$143$	18.2462i	1.52582i
$144$	−3.56155	−0.296796
$145$	0	0
$146$	−6.80776	−0.563415
$147$	− 49.3002i	− 4.06621i
$148$	7.12311i	0.585516i
$149$	−3.12311	−0.255855	−0.127927	−	0.991784i	$-0.540832\pi$
$149$	−3.12311	−0.255855	−0.127927	+	0.991784i	$0.540832\pi$
$150$	0	0
$151$	−15.3693	−1.25074	−0.625369	−	0.780329i	$-0.715051\pi$
$151$	−15.3693	−1.25074	−0.625369	+	0.780329i	$0.715051\pi$
$152$	− 2.56155i	− 0.207769i
$153$	− 3.56155i	− 0.287934i
$154$	20.4924	1.65133
$155$	0	0
$156$	−11.6847	−0.935521
$157$	18.4924i	1.47586i	0.674879	+	0.737928i	$0.264196\pi$
$157$	18.4924i	1.47586i	−0.674879	+	0.737928i	$0.735804\pi$
$158$	0	0
$159$	−1.43845	−0.114076
$160$	0	0
$161$	−26.2462	−2.06849
$162$	− 7.00000i	− 0.549972i
$163$	− 14.2462i	− 1.11585i	−0.829892	−	0.557925i	$-0.811598\pi$
$163$	− 14.2462i	− 1.11585i	0.829892	−	0.557925i	$-0.188402\pi$
$164$	−4.24621	−0.331573
$165$	0	0
$166$	−6.24621	−0.484800
$167$	5.12311i	0.396438i	0.980158	+	0.198219i	$0.0635157\pi$
$167$	5.12311i	0.396438i	−0.980158	+	0.198219i	$0.936484\pi$
$168$	13.1231i	1.01247i
$169$	−7.80776	−0.600597
$170$	0	0
$171$	−9.12311	−0.697661
$172$	− 1.12311i	− 0.0856360i
$173$	18.0000i	1.36851i	0.729241	+	0.684257i	$0.239873\pi$
$173$	18.0000i	1.36851i	−0.729241	+	0.684257i	$0.760127\pi$
$174$	−14.5616	−1.10391
$175$	0	0
$176$	−4.00000	−0.301511
$177$	0.807764i	0.0607153i
$178$	9.68466i	0.725896i
$179$	−8.49242	−0.634753	−0.317377	−	0.948300i	$-0.602802\pi$
$179$	−8.49242	−0.634753	−0.317377	+	0.948300i	$0.602802\pi$
$180$	0	0
$181$	16.2462	1.20757	0.603786	−	0.797147i	$-0.293658\pi$
$181$	16.2462	1.20757	0.603786	+	0.797147i	$0.293658\pi$
$182$	23.3693i	1.73225i
$183$	19.0540i	1.40851i
$184$	5.12311	0.377680
$185$	0	0
$186$	16.8078	1.23241
$187$	− 4.00000i	− 0.292509i
$188$	− 6.56155i	− 0.478550i
$189$	7.36932	0.536039
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	− 2.56155i	− 0.184864i
$193$	24.2462i	1.74528i	0.488364	+	0.872640i	$0.337594\pi$
$193$	24.2462i	1.74528i	−0.488364	+	0.872640i	$0.662406\pi$
$194$	1.68466	0.120951
$195$	0	0
$196$	19.2462	1.37473
$197$	13.3693i	0.952524i	0.879303	+	0.476262i	$0.158009\pi$
$197$	13.3693i	0.952524i	−0.879303	+	0.476262i	$0.841991\pi$
$198$	14.2462i	1.01243i
$199$	−14.5616	−1.03224	−0.516121	−	0.856516i	$-0.672624\pi$
$199$	−14.5616	−1.03224	−0.516121	+	0.856516i	$0.672624\pi$
$200$	0	0
$201$	23.3693	1.64834
$202$	− 14.4924i	− 1.01968i
$203$	29.1231i	2.04404i
$204$	2.56155	0.179345
$205$	0	0
$206$	8.00000	0.557386
$207$	− 18.2462i	− 1.26820i
$208$	− 4.56155i	− 0.316287i
$209$	−10.2462	−0.708745
$210$	0	0
$211$	−3.36932	−0.231953	−0.115977	−	0.993252i	$-0.537000\pi$
$211$	−3.36932	−0.231953	−0.115977	+	0.993252i	$0.537000\pi$
$212$	− 0.561553i	− 0.0385676i
$213$	− 11.0540i	− 0.757406i
$214$	−16.4924	−1.12740
$215$	0	0
$216$	−1.43845	−0.0978739
$217$	− 33.6155i	− 2.28197i
$218$	8.56155i	0.579862i
$219$	−17.4384	−1.17838
$220$	0	0
$221$	4.56155	0.306843
$222$	18.2462i	1.22461i
$223$	− 3.68466i	− 0.246743i	−0.992361	−	0.123371i	$-0.960629\pi$
$223$	− 3.68466i	− 0.246743i	0.992361	−	0.123371i	$-0.0393707\pi$
$224$	−5.12311	−0.342302
$225$	0	0
$226$	2.80776	0.186770
$227$	16.3153i	1.08289i	0.840737	+	0.541444i	$0.182122\pi$
$227$	16.3153i	1.08289i	−0.840737	+	0.541444i	$0.817878\pi$
$228$	− 6.56155i	− 0.434549i
$229$	14.4924	0.957686	0.478843	−	0.877900i	$-0.341056\pi$
$229$	14.4924	0.957686	0.478843	+	0.877900i	$0.341056\pi$
$230$	0	0
$231$	52.4924	3.45375
$232$	− 5.68466i	− 0.373216i
$233$	− 6.31534i	− 0.413732i	−0.978369	−	0.206866i	$-0.933674\pi$
$233$	− 6.31534i	− 0.413732i	0.978369	−	0.206866i	$-0.0663264\pi$
$234$	−16.2462	−1.06205
$235$	0	0
$236$	−0.315342	−0.0205270
$237$	0	0
$238$	− 5.12311i	− 0.332082i
$239$	23.3693	1.51164	0.755818	−	0.654782i	$-0.227240\pi$
$239$	23.3693	1.51164	0.755818	+	0.654782i	$0.227240\pi$
$240$	0	0
$241$	12.2462	0.788848	0.394424	−	0.918929i	$-0.370944\pi$
$241$	12.2462	0.788848	0.394424	+	0.918929i	$0.370944\pi$
$242$	5.00000i	0.321412i
$243$	− 22.2462i	− 1.42710i
$244$	−7.43845	−0.476198
$245$	0	0
$246$	−10.8769	−0.693485
$247$	− 11.6847i	− 0.743477i
$248$	6.56155i	0.416659i
$249$	−16.0000	−1.01396
$250$	0	0
$251$	−4.00000	−0.252478	−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000	−0.252478	−0.126239	+	0.992000i	$0.540291\pi$
$252$	18.2462i	1.14940i
$253$	− 20.4924i	− 1.28835i
$254$	−9.43845	−0.592221
$255$	0	0
$256$	1.00000	0.0625000
$257$	2.00000i	0.124757i	0.998053	+	0.0623783i	$0.0198685\pi$
$257$	2.00000i	0.124757i	−0.998053	+	0.0623783i	$0.980131\pi$
$258$	− 2.87689i	− 0.179108i
$259$	36.4924	2.26753
$260$	0	0
$261$	−20.2462	−1.25321
$262$	1.12311i	0.0693857i
$263$	− 24.8078i	− 1.52971i	−0.644201	−	0.764856i	$-0.722810\pi$
$263$	− 24.8078i	− 1.52971i	0.644201	−	0.764856i	$-0.277190\pi$
$264$	−10.2462	−0.630611
$265$	0	0
$266$	−13.1231	−0.804629
$267$	24.8078i	1.51821i
$268$	9.12311i	0.557282i
$269$	2.80776	0.171192	0.0855962	−	0.996330i	$-0.472721\pi$
$269$	2.80776	0.171192	0.0855962	+	0.996330i	$0.472721\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	1.00000i	0.0606339i
$273$	59.8617i	3.62300i
$274$	−10.0000	−0.604122
$275$	0	0
$276$	13.1231	0.789918
$277$	− 15.7538i	− 0.946553i	−0.880914	−	0.473277i	$-0.843071\pi$
$277$	− 15.7538i	− 0.946553i	0.880914	−	0.473277i	$-0.156929\pi$
$278$	− 16.4924i	− 0.989150i
$279$	23.3693	1.39908
$280$	0	0
$281$	16.5616	0.987979	0.493990	−	0.869468i	$-0.335538\pi$
$281$	16.5616	0.987979	0.493990	+	0.869468i	$0.335538\pi$
$282$	− 16.8078i	− 1.00089i
$283$	− 8.31534i	− 0.494296i	−0.968978	−	0.247148i	$-0.920507\pi$
$283$	− 8.31534i	− 0.494296i	0.968978	−	0.247148i	$-0.0794933\pi$
$284$	4.31534	0.256068
$285$	0	0
$286$	−18.2462	−1.07892
$287$	21.7538i	1.28409i
$288$	− 3.56155i	− 0.209867i
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	4.31534	0.252970
$292$	− 6.80776i	− 0.398394i
$293$	16.5616i	0.967536i	0.875196	+	0.483768i	$0.160732\pi$
$293$	16.5616i	0.967536i	−0.875196	+	0.483768i	$0.839268\pi$
$294$	49.3002	2.87525
$295$	0	0
$296$	−7.12311	−0.414022
$297$	5.75379i	0.333869i
$298$	− 3.12311i	− 0.180917i
$299$	23.3693	1.35148
$300$	0	0
$301$	−5.75379	−0.331643
$302$	− 15.3693i	− 0.884405i
$303$	− 37.1231i	− 2.13267i
$304$	2.56155	0.146915
$305$	0	0
$306$	3.56155	0.203600
$307$	− 17.7538i	− 1.01326i	−0.862163	−	0.506631i	$-0.830891\pi$
$307$	− 17.7538i	− 1.01326i	0.862163	−	0.506631i	$-0.169109\pi$
$308$	20.4924i	1.16766i
$309$	20.4924	1.16577
$310$	0	0
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	− 11.6847i	− 0.661514i
$313$	6.00000i	0.339140i	0.985518	+	0.169570i	$0.0542379\pi$
$313$	6.00000i	0.339140i	−0.985518	+	0.169570i	$0.945762\pi$
$314$	−18.4924	−1.04359
$315$	0	0
$316$	0	0
$317$	31.6155i	1.77570i	0.460128	+	0.887852i	$0.347803\pi$
$317$	31.6155i	1.77570i	−0.460128	+	0.887852i	$0.652197\pi$
$318$	− 1.43845i	− 0.0806641i
$319$	−22.7386	−1.27312
$320$	0	0
$321$	−42.2462	−2.35795
$322$	− 26.2462i	− 1.46264i
$323$	2.56155i	0.142529i
$324$	7.00000	0.388889
$325$	0	0
$326$	14.2462	0.789025
$327$	21.9309i	1.21278i
$328$	− 4.24621i	− 0.234458i
$329$	−33.6155	−1.85328
$330$	0	0
$331$	23.0540	1.26716	0.633581	−	0.773677i	$-0.281585\pi$
$331$	23.0540	1.26716	0.633581	+	0.773677i	$0.281585\pi$
$332$	− 6.24621i	− 0.342805i
$333$	25.3693i	1.39023i
$334$	−5.12311	−0.280324
$335$	0	0
$336$	−13.1231	−0.715924
$337$	− 10.3153i	− 0.561912i	−0.959721	−	0.280956i	$-0.909348\pi$
$337$	− 10.3153i	− 0.561912i	0.959721	−	0.280956i	$-0.0906516\pi$
$338$	− 7.80776i	− 0.424686i
$339$	7.19224	0.390629
$340$	0	0
$341$	26.2462	1.42131
$342$	− 9.12311i	− 0.493321i
$343$	− 62.7386i	− 3.38757i
$344$	1.12311	0.0605538
$345$	0	0
$346$	−18.0000	−0.967686
$347$	− 10.5616i	− 0.566974i	−0.958976	−	0.283487i	$-0.908509\pi$
$347$	− 10.5616i	− 0.566974i	0.958976	−	0.283487i	$-0.0914913\pi$
$348$	− 14.5616i	− 0.780581i
$349$	15.1231	0.809521	0.404761	−	0.914423i	$-0.367355\pi$
$349$	15.1231	0.809521	0.404761	+	0.914423i	$0.367355\pi$
$350$	0	0
$351$	−6.56155	−0.350230
$352$	− 4.00000i	− 0.213201i
$353$	− 22.4924i	− 1.19715i	−0.801066	−	0.598575i	$-0.795734\pi$
$353$	− 22.4924i	− 1.19715i	0.801066	−	0.598575i	$-0.204266\pi$
$354$	−0.807764	−0.0429322
$355$	0	0
$356$	−9.68466	−0.513286
$357$	− 13.1231i	− 0.694548i
$358$	− 8.49242i	− 0.448838i
$359$	10.8769	0.574061	0.287030	−	0.957922i	$-0.407332\pi$
$359$	10.8769	0.574061	0.287030	+	0.957922i	$0.407332\pi$
$360$	0	0
$361$	−12.4384	−0.654655
$362$	16.2462i	0.853882i
$363$	12.8078i	0.672233i
$364$	−23.3693	−1.22489
$365$	0	0
$366$	−19.0540	−0.995967
$367$	− 33.6155i	− 1.75472i	−0.479836	−	0.877358i	$-0.659304\pi$
$367$	− 33.6155i	− 1.75472i	0.479836	−	0.877358i	$-0.340696\pi$
$368$	5.12311i	0.267060i
$369$	−15.1231	−0.787277
$370$	0	0
$371$	−2.87689	−0.149361
$372$	16.8078i	0.871442i
$373$	24.7386i	1.28092i	0.767992	+	0.640459i	$0.221256\pi$
$373$	24.7386i	1.28092i	−0.767992	+	0.640459i	$0.778744\pi$
$374$	4.00000	0.206835
$375$	0	0
$376$	6.56155	0.338386
$377$	− 25.9309i	− 1.33551i
$378$	7.36932i	0.379037i
$379$	14.2462	0.731779	0.365889	−	0.930658i	$-0.380765\pi$
$379$	14.2462	0.731779	0.365889	+	0.930658i	$0.380765\pi$
$380$	0	0
$381$	−24.1771	−1.23863
$382$	0	0
$383$	16.8078i	0.858837i	0.903106	+	0.429418i	$0.141281\pi$
$383$	16.8078i	0.858837i	−0.903106	+	0.429418i	$0.858719\pi$
$384$	2.56155	0.130719
$385$	0	0
$386$	−24.2462	−1.23410
$387$	− 4.00000i	− 0.203331i
$388$	1.68466i	0.0855256i
$389$	−13.3693	−0.677851	−0.338926	−	0.940813i	$-0.610064\pi$
$389$	−13.3693	−0.677851	−0.338926	+	0.940813i	$0.610064\pi$
$390$	0	0
$391$	−5.12311	−0.259087
$392$	19.2462i	0.972080i
$393$	2.87689i	0.145120i
$394$	−13.3693	−0.673536
$395$	0	0
$396$	−14.2462	−0.715899
$397$	21.3693i	1.07250i	0.844061	+	0.536248i	$0.180159\pi$
$397$	21.3693i	1.07250i	−0.844061	+	0.536248i	$0.819841\pi$
$398$	− 14.5616i	− 0.729905i
$399$	−33.6155	−1.68288
$400$	0	0
$401$	10.6307	0.530871	0.265435	−	0.964129i	$-0.414484\pi$
$401$	10.6307	0.530871	0.265435	+	0.964129i	$0.414484\pi$
$402$	23.3693i	1.16556i
$403$	29.9309i	1.49096i
$404$	14.4924	0.721025
$405$	0	0
$406$	−29.1231	−1.44536
$407$	28.4924i	1.41232i
$408$	2.56155i	0.126816i
$409$	2.31534	0.114486	0.0572431	−	0.998360i	$-0.481769\pi$
$409$	2.31534	0.114486	0.0572431	+	0.998360i	$0.481769\pi$
$410$	0	0
$411$	−25.6155	−1.26352
$412$	8.00000i	0.394132i
$413$	1.61553i	0.0794949i
$414$	18.2462	0.896752
$415$	0	0
$416$	4.56155	0.223649
$417$	− 42.2462i	− 2.06881i
$418$	− 10.2462i	− 0.501159i
$419$	25.1231	1.22734	0.613672	−	0.789561i	$-0.289692\pi$
$419$	25.1231	1.22734	0.613672	+	0.789561i	$0.289692\pi$
$420$	0	0
$421$	7.61553	0.371158	0.185579	−	0.982629i	$-0.440584\pi$
$421$	7.61553	0.371158	0.185579	+	0.982629i	$0.440584\pi$
$422$	− 3.36932i	− 0.164016i
$423$	− 23.3693i	− 1.13626i
$424$	0.561553	0.0272714
$425$	0	0
$426$	11.0540	0.535567
$427$	38.1080i	1.84417i
$428$	− 16.4924i	− 0.797191i
$429$	−46.7386	−2.25656
$430$	0	0
$431$	−20.4924	−0.987085	−0.493543	−	0.869722i	$-0.664298\pi$
$431$	−20.4924	−0.987085	−0.493543	+	0.869722i	$0.664298\pi$
$432$	− 1.43845i	− 0.0692073i
$433$	2.49242i	0.119778i	0.998205	+	0.0598891i	$0.0190747\pi$
$433$	2.49242i	0.119778i	−0.998205	+	0.0598891i	$0.980925\pi$
$434$	33.6155	1.61360
$435$	0	0
$436$	−8.56155	−0.410024
$437$	13.1231i	0.627763i
$438$	− 17.4384i	− 0.833241i
$439$	32.9848	1.57428	0.787140	−	0.616774i	$-0.211561\pi$
$439$	32.9848	1.57428	0.787140	+	0.616774i	$0.211561\pi$
$440$	0	0
$441$	68.5464	3.26411
$442$	4.56155i	0.216971i
$443$	− 28.0000i	− 1.33032i	−0.746701	−	0.665160i	$-0.768363\pi$
$443$	− 28.0000i	− 1.33032i	0.746701	−	0.665160i	$-0.231637\pi$
$444$	−18.2462	−0.865927
$445$	0	0
$446$	3.68466	0.174474
$447$	− 8.00000i	− 0.378387i
$448$	− 5.12311i	− 0.242044i
$449$	16.8769	0.796470	0.398235	−	0.917283i	$-0.369623\pi$
$449$	16.8769	0.796470	0.398235	+	0.917283i	$0.369623\pi$
$450$	0	0
$451$	−16.9848	−0.799785
$452$	2.80776i	0.132066i
$453$	− 39.3693i	− 1.84973i
$454$	−16.3153	−0.765717
$455$	0	0
$456$	6.56155	0.307273
$457$	− 16.2462i	− 0.759966i	−0.924994	−	0.379983i	$-0.875930\pi$
$457$	− 16.2462i	− 0.759966i	0.924994	−	0.379983i	$-0.124070\pi$
$458$	14.4924i	0.677186i
$459$	1.43845	0.0671410
$460$	0	0
$461$	2.49242	0.116084	0.0580418	−	0.998314i	$-0.481514\pi$
$461$	2.49242	0.116084	0.0580418	+	0.998314i	$0.481514\pi$
$462$	52.4924i	2.44217i
$463$	13.9309i	0.647422i	0.946156	+	0.323711i	$0.104931\pi$
$463$	13.9309i	0.647422i	−0.946156	+	0.323711i	$0.895069\pi$
$464$	5.68466	0.263904
$465$	0	0
$466$	6.31534	0.292553
$467$	− 32.4924i	− 1.50357i	−0.659408	−	0.751785i	$-0.729193\pi$
$467$	− 32.4924i	− 1.50357i	0.659408	−	0.751785i	$-0.270807\pi$
$468$	− 16.2462i	− 0.750981i
$469$	46.7386	2.15819
$470$	0	0
$471$	−47.3693	−2.18266
$472$	− 0.315342i	− 0.0145148i
$473$	− 4.49242i	− 0.206562i
$474$	0	0
$475$	0	0
$476$	5.12311	0.234817
$477$	− 2.00000i	− 0.0915737i
$478$	23.3693i	1.06889i
$479$	−38.5616	−1.76192	−0.880961	−	0.473189i	$-0.843103\pi$
$479$	−38.5616	−1.76192	−0.880961	+	0.473189i	$0.843103\pi$
$480$	0	0
$481$	−32.4924	−1.48153
$482$	12.2462i	0.557800i
$483$	− 67.2311i	− 3.05912i
$484$	−5.00000	−0.227273
$485$	0	0
$486$	22.2462	1.00911
$487$	25.6155i	1.16075i	0.814349	+	0.580375i	$0.197094\pi$
$487$	25.6155i	1.16075i	−0.814349	+	0.580375i	$0.802906\pi$
$488$	− 7.43845i	− 0.336723i
$489$	36.4924	1.65024
$490$	0	0
$491$	2.56155	0.115601	0.0578006	−	0.998328i	$-0.481591\pi$
$491$	2.56155	0.115601	0.0578006	+	0.998328i	$0.481591\pi$
$492$	− 10.8769i	− 0.490368i
$493$	5.68466i	0.256024i
$494$	11.6847	0.525718
$495$	0	0
$496$	−6.56155	−0.294622
$497$	− 22.1080i	− 0.991677i
$498$	− 16.0000i	− 0.716977i
$499$	9.12311	0.408406	0.204203	−	0.978929i	$-0.434540\pi$
$499$	9.12311	0.408406	0.204203	+	0.978929i	$0.434540\pi$
$500$	0	0
$501$	−13.1231	−0.586297
$502$	− 4.00000i	− 0.178529i
$503$	− 22.7386i	− 1.01387i	−0.861986	−	0.506933i	$-0.830779\pi$
$503$	− 22.7386i	− 1.01387i	0.861986	−	0.506933i	$-0.169221\pi$
$504$	−18.2462	−0.812751
$505$	0	0
$506$	20.4924	0.910999
$507$	− 20.0000i	− 0.888231i
$508$	− 9.43845i	− 0.418763i
$509$	−16.8769	−0.748055	−0.374028	−	0.927418i	$-0.622023\pi$
$509$	−16.8769	−0.748055	−0.374028	+	0.927418i	$0.622023\pi$
$510$	0	0
$511$	−34.8769	−1.54286
$512$	1.00000i	0.0441942i
$513$	− 3.68466i	− 0.162682i
$514$	−2.00000	−0.0882162
$515$	0	0
$516$	2.87689	0.126648
$517$	− 26.2462i	− 1.15431i
$518$	36.4924i	1.60338i
$519$	−46.1080	−2.02391
$520$	0	0
$521$	43.6155	1.91083	0.955415	−	0.295265i	$-0.0954078\pi$
$521$	43.6155	1.91083	0.955415	+	0.295265i	$0.0954078\pi$
$522$	− 20.2462i	− 0.886153i
$523$	− 39.8617i	− 1.74303i	−0.490367	−	0.871516i	$-0.663137\pi$
$523$	− 39.8617i	− 1.74303i	0.490367	−	0.871516i	$-0.336863\pi$
$524$	−1.12311	−0.0490631
$525$	0	0
$526$	24.8078	1.08167
$527$	− 6.56155i	− 0.285826i
$528$	− 10.2462i	− 0.445909i
$529$	−3.24621	−0.141140
$530$	0	0
$531$	−1.12311	−0.0487386
$532$	− 13.1231i	− 0.568959i
$533$	− 19.3693i	− 0.838978i
$534$	−24.8078	−1.07354
$535$	0	0
$536$	−9.12311	−0.394058
$537$	− 21.7538i	− 0.938745i
$538$	2.80776i	0.121051i
$539$	76.9848	3.31597
$540$	0	0
$541$	30.0000	1.28980	0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000	1.28980	0.644900	+	0.764267i	$0.276899\pi$
$542$	16.0000i	0.687259i
$543$	41.6155i	1.78589i
$544$	−1.00000	−0.0428746
$545$	0	0
$546$	−59.8617	−2.56185
$547$	29.4384i	1.25870i	0.777123	+	0.629349i	$0.216678\pi$
$547$	29.4384i	1.25870i	−0.777123	+	0.629349i	$0.783322\pi$
$548$	− 10.0000i	− 0.427179i
$549$	−26.4924	−1.13067
$550$	0	0
$551$	14.5616	0.620343
$552$	13.1231i	0.558556i
$553$	0	0
$554$	15.7538	0.669314
$555$	0	0
$556$	16.4924	0.699435
$557$	36.5616i	1.54916i	0.632474	+	0.774581i	$0.282039\pi$
$557$	36.5616i	1.54916i	−0.632474	+	0.774581i	$0.717961\pi$
$558$	23.3693i	0.989302i
$559$	5.12311	0.216684
$560$	0	0
$561$	10.2462	0.432595
$562$	16.5616i	0.698607i
$563$	4.63068i	0.195160i	0.995228	+	0.0975800i	$0.0311102\pi$
$563$	4.63068i	0.195160i	−0.995228	+	0.0975800i	$0.968890\pi$
$564$	16.8078	0.707735
$565$	0	0
$566$	8.31534	0.349520
$567$	− 35.8617i	− 1.50605i
$568$	4.31534i	0.181068i
$569$	3.93087	0.164791	0.0823953	−	0.996600i	$-0.473743\pi$
$569$	3.93087	0.164791	0.0823953	+	0.996600i	$0.473743\pi$
$570$	0	0
$571$	25.1231	1.05137	0.525685	−	0.850680i	$-0.323809\pi$
$571$	25.1231	1.05137	0.525685	+	0.850680i	$0.323809\pi$
$572$	− 18.2462i	− 0.762912i
$573$	0	0
$574$	−21.7538	−0.907986
$575$	0	0
$576$	3.56155	0.148398
$577$	− 5.36932i	− 0.223528i	−0.993735	−	0.111764i	$-0.964350\pi$
$577$	− 5.36932i	− 0.223528i	0.993735	−	0.111764i	$-0.0356500\pi$
$578$	− 1.00000i	− 0.0415945i
$579$	−62.1080	−2.58112
$580$	0	0
$581$	−32.0000	−1.32758
$582$	4.31534i	0.178877i
$583$	− 2.24621i	− 0.0930286i
$584$	6.80776	0.281707
$585$	0	0
$586$	−16.5616	−0.684151
$587$	− 6.87689i	− 0.283840i	−0.989878	−	0.141920i	$-0.954672\pi$
$587$	− 6.87689i	− 0.283840i	0.989878	−	0.141920i	$-0.0453275\pi$
$588$	49.3002i	2.03311i
$589$	−16.8078	−0.692552
$590$	0	0
$591$	−34.2462	−1.40870
$592$	− 7.12311i	− 0.292758i
$593$	8.24621i	0.338631i	0.985562	+	0.169316i	$0.0541557\pi$
$593$	8.24621i	0.338631i	−0.985562	+	0.169316i	$0.945844\pi$
$594$	−5.75379	−0.236081
$595$	0	0
$596$	3.12311	0.127927
$597$	− 37.3002i	− 1.52660i
$598$	23.3693i	0.955642i
$599$	9.61553	0.392880	0.196440	−	0.980516i	$-0.437062\pi$
$599$	9.61553	0.392880	0.196440	+	0.980516i	$0.437062\pi$
$600$	0	0
$601$	−7.61553	−0.310644	−0.155322	−	0.987864i	$-0.549641\pi$
$601$	−7.61553	−0.310644	−0.155322	+	0.987864i	$0.549641\pi$
$602$	− 5.75379i	− 0.234507i
$603$	32.4924i	1.32319i
$604$	15.3693	0.625369
$605$	0	0
$606$	37.1231	1.50802
$607$	− 43.8617i	− 1.78029i	−0.455674	−	0.890147i	$-0.650602\pi$
$607$	− 43.8617i	− 1.78029i	0.455674	−	0.890147i	$-0.349398\pi$
$608$	2.56155i	0.103885i
$609$	−74.6004	−3.02296
$610$	0	0
$611$	29.9309	1.21087
$612$	3.56155i	0.143967i
$613$	23.9309i	0.966559i	0.875466	+	0.483279i	$0.160554\pi$
$613$	23.9309i	0.966559i	−0.875466	+	0.483279i	$0.839446\pi$
$614$	17.7538	0.716485
$615$	0	0
$616$	−20.4924	−0.825663
$617$	6.31534i	0.254246i	0.991887	+	0.127123i	$0.0405743\pi$
$617$	6.31534i	0.254246i	−0.991887	+	0.127123i	$0.959426\pi$
$618$	20.4924i	0.824326i
$619$	−9.12311	−0.366689	−0.183344	−	0.983049i	$-0.558692\pi$
$619$	−9.12311	−0.366689	−0.183344	+	0.983049i	$0.558692\pi$
$620$	0	0
$621$	7.36932	0.295720
$622$	8.00000i	0.320771i
$623$	49.6155i	1.98780i
$624$	11.6847	0.467761
$625$	0	0
$626$	−6.00000	−0.239808
$627$	− 26.2462i	− 1.04817i
$628$	− 18.4924i	− 0.737928i
$629$	7.12311	0.284017
$630$	0	0
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	0	0
$633$	− 8.63068i	− 0.343039i
$634$	−31.6155	−1.25561
$635$	0	0
$636$	1.43845	0.0570381
$637$	87.7926i	3.47847i
$638$	− 22.7386i	− 0.900231i
$639$	15.3693	0.608001
$640$	0	0
$641$	−24.2462	−0.957668	−0.478834	−	0.877906i	$-0.658940\pi$
$641$	−24.2462	−0.957668	−0.478834	+	0.877906i	$0.658940\pi$
$642$	− 42.2462i	− 1.66732i
$643$	− 40.4924i	− 1.59687i	−0.602084	−	0.798433i	$-0.705663\pi$
$643$	− 40.4924i	− 1.59687i	0.602084	−	0.798433i	$-0.294337\pi$
$644$	26.2462	1.03425
$645$	0	0
$646$	−2.56155	−0.100783
$647$	11.6847i	0.459371i	0.973265	+	0.229686i	$0.0737698\pi$
$647$	11.6847i	0.459371i	−0.973265	+	0.229686i	$0.926230\pi$
$648$	7.00000i	0.274986i
$649$	−1.26137	−0.0495130
$650$	0	0
$651$	86.1080	3.37484
$652$	14.2462i	0.557925i
$653$	38.4924i	1.50632i	0.657835	+	0.753162i	$0.271473\pi$
$653$	38.4924i	1.50632i	−0.657835	+	0.753162i	$0.728527\pi$
$654$	−21.9309	−0.857565
$655$	0	0
$656$	4.24621	0.165787
$657$	− 24.2462i	− 0.945935i
$658$	− 33.6155i	− 1.31047i
$659$	41.9309	1.63339	0.816697	−	0.577066i	$-0.195803\pi$
$659$	41.9309	1.63339	0.816697	+	0.577066i	$0.195803\pi$
$660$	0	0
$661$	17.8617	0.694741	0.347371	−	0.937728i	$-0.387075\pi$
$661$	17.8617	0.694741	0.347371	+	0.937728i	$0.387075\pi$
$662$	23.0540i	0.896018i
$663$	11.6847i	0.453795i
$664$	6.24621	0.242400
$665$	0	0
$666$	−25.3693	−0.983041
$667$	29.1231i	1.12765i
$668$	− 5.12311i	− 0.198219i
$669$	9.43845	0.364911
$670$	0	0
$671$	−29.7538	−1.14863
$672$	− 13.1231i	− 0.506235i
$673$	− 20.4233i	− 0.787260i	−0.919269	−	0.393630i	$-0.871219\pi$
$673$	− 20.4233i	− 0.787260i	0.919269	−	0.393630i	$-0.128781\pi$
$674$	10.3153	0.397332
$675$	0	0
$676$	7.80776	0.300299
$677$	− 40.7386i	− 1.56571i	−0.622202	−	0.782856i	$-0.713762\pi$
$677$	− 40.7386i	− 1.56571i	0.622202	−	0.782856i	$-0.286238\pi$
$678$	7.19224i	0.276216i
$679$	8.63068	0.331215
$680$	0	0
$681$	−41.7926	−1.60150
$682$	26.2462i	1.00502i
$683$	41.3002i	1.58031i	0.612909	+	0.790154i	$0.289999\pi$
$683$	41.3002i	1.58031i	−0.612909	+	0.790154i	$0.710001\pi$
$684$	9.12311	0.348831
$685$	0	0
$686$	62.7386	2.39537
$687$	37.1231i	1.41633i
$688$	1.12311i	0.0428180i
$689$	2.56155	0.0975874
$690$	0	0
$691$	9.75379	0.371052	0.185526	−	0.982639i	$-0.440601\pi$
$691$	9.75379	0.371052	0.185526	+	0.982639i	$0.440601\pi$
$692$	− 18.0000i	− 0.684257i
$693$	72.9848i	2.77247i
$694$	10.5616	0.400911
$695$	0	0
$696$	14.5616	0.551954
$697$	4.24621i	0.160837i
$698$	15.1231i	0.572418i
$699$	16.1771	0.611873
$700$	0	0
$701$	46.3542	1.75077	0.875386	−	0.483424i	$-0.160607\pi$
$701$	46.3542	1.75077	0.875386	+	0.483424i	$0.160607\pi$
$702$	− 6.56155i	− 0.247650i
$703$	− 18.2462i	− 0.688169i
$704$	4.00000	0.150756
$705$	0	0
$706$	22.4924	0.846513
$707$	− 74.2462i	− 2.79232i
$708$	− 0.807764i	− 0.0303576i
$709$	−37.5464	−1.41008	−0.705042	−	0.709165i	$-0.749072\pi$
$709$	−37.5464	−1.41008	−0.705042	+	0.709165i	$0.749072\pi$
$710$	0	0
$711$	0	0
$712$	− 9.68466i	− 0.362948i
$713$	− 33.6155i	− 1.25891i
$714$	13.1231	0.491120
$715$	0	0
$716$	8.49242	0.317377
$717$	59.8617i	2.23558i
$718$	10.8769i	0.405922i
$719$	−15.1922	−0.566575	−0.283287	−	0.959035i	$-0.591425\pi$
$719$	−15.1922	−0.566575	−0.283287	+	0.959035i	$0.591425\pi$
$720$	0	0
$721$	40.9848	1.52636
$722$	− 12.4384i	− 0.462911i
$723$	31.3693i	1.16664i
$724$	−16.2462	−0.603786
$725$	0	0
$726$	−12.8078	−0.475341
$727$	46.5616i	1.72687i	0.504458	+	0.863436i	$0.331692\pi$
$727$	46.5616i	1.72687i	−0.504458	+	0.863436i	$0.668308\pi$
$728$	− 23.3693i	− 0.866125i
$729$	35.9848	1.33277
$730$	0	0
$731$	−1.12311	−0.0415396
$732$	− 19.0540i	− 0.704255i
$733$	22.4924i	0.830777i	0.909644	+	0.415388i	$0.136354\pi$
$733$	22.4924i	0.830777i	−0.909644	+	0.415388i	$0.863646\pi$
$734$	33.6155	1.24077
$735$	0	0
$736$	−5.12311	−0.188840
$737$	36.4924i	1.34422i
$738$	− 15.1231i	− 0.556689i
$739$	20.1771	0.742226	0.371113	−	0.928588i	$-0.378976\pi$
$739$	20.1771	0.742226	0.371113	+	0.928588i	$0.378976\pi$
$740$	0	0
$741$	29.9309	1.09954
$742$	− 2.87689i	− 0.105614i
$743$	21.1231i	0.774932i	0.921884	+	0.387466i	$0.126649\pi$
$743$	21.1231i	0.774932i	−0.921884	+	0.387466i	$0.873351\pi$
$744$	−16.8078	−0.616203
$745$	0	0
$746$	−24.7386	−0.905746
$747$	− 22.2462i	− 0.813946i
$748$	4.00000i	0.146254i
$749$	−84.4924	−3.08729
$750$	0	0
$751$	−24.1771	−0.882234	−0.441117	−	0.897450i	$-0.645418\pi$
$751$	−24.1771	−0.882234	−0.441117	+	0.897450i	$0.645418\pi$
$752$	6.56155i	0.239275i
$753$	− 10.2462i	− 0.373393i
$754$	25.9309	0.944347
$755$	0	0
$756$	−7.36932	−0.268019
$757$	47.7926i	1.73705i	0.495644	+	0.868526i	$0.334932\pi$
$757$	47.7926i	1.73705i	−0.495644	+	0.868526i	$0.665068\pi$
$758$	14.2462i	0.517446i
$759$	52.4924	1.90535
$760$	0	0
$761$	−20.7386	−0.751775	−0.375887	−	0.926665i	$-0.622662\pi$
$761$	−20.7386	−0.751775	−0.375887	+	0.926665i	$0.622662\pi$
$762$	− 24.1771i	− 0.875843i
$763$	43.8617i	1.58790i
$764$	0	0
$765$	0	0
$766$	−16.8078	−0.607289
$767$	− 1.43845i	− 0.0519393i
$768$	2.56155i	0.0924321i
$769$	−11.4384	−0.412481	−0.206240	−	0.978501i	$-0.566123\pi$
$769$	−11.4384	−0.412481	−0.206240	+	0.978501i	$0.566123\pi$
$770$	0	0
$771$	−5.12311	−0.184504
$772$	− 24.2462i	− 0.872640i
$773$	− 20.7386i	− 0.745917i	−0.927848	−	0.372958i	$-0.878343\pi$
$773$	− 20.7386i	− 0.745917i	0.927848	−	0.372958i	$-0.121657\pi$
$774$	4.00000	0.143777
$775$	0	0
$776$	−1.68466	−0.0604757
$777$	93.4773i	3.35348i
$778$	− 13.3693i	− 0.479313i
$779$	10.8769	0.389705
$780$	0	0
$781$	17.2614	0.617660
$782$	− 5.12311i	− 0.183202i
$783$	− 8.17708i	− 0.292225i
$784$	−19.2462	−0.687365
$785$	0	0
$786$	−2.87689	−0.102615
$787$	− 27.5464i	− 0.981923i	−0.871181	−	0.490962i	$-0.836646\pi$
$787$	− 27.5464i	− 0.981923i	0.871181	−	0.490962i	$-0.163354\pi$
$788$	− 13.3693i	− 0.476262i
$789$	63.5464	2.26231
$790$	0	0
$791$	14.3845	0.511453
$792$	− 14.2462i	− 0.506217i
$793$	− 33.9309i	− 1.20492i
$794$	−21.3693	−0.758369
$795$	0	0
$796$	14.5616	0.516121
$797$	18.4924i	0.655035i	0.944845	+	0.327518i	$0.106212\pi$
$797$	18.4924i	0.655035i	−0.944845	+	0.327518i	$0.893788\pi$
$798$	− 33.6155i	− 1.18998i
$799$	−6.56155	−0.232131
$800$	0	0
$801$	−34.4924	−1.21873
$802$	10.6307i	0.375382i
$803$	− 27.2311i	− 0.960963i
$804$	−23.3693	−0.824172
$805$	0	0
$806$	−29.9309	−1.05427
$807$	7.19224i	0.253179i
$808$	14.4924i	0.509842i
$809$	−18.9848	−0.667472	−0.333736	−	0.942667i	$-0.608309\pi$
$809$	−18.9848	−0.667472	−0.333736	+	0.942667i	$0.608309\pi$
$810$	0	0
$811$	3.36932	0.118313	0.0591564	−	0.998249i	$-0.481159\pi$
$811$	3.36932	0.118313	0.0591564	+	0.998249i	$0.481159\pi$
$812$	− 29.1231i	− 1.02202i
$813$	40.9848i	1.43740i
$814$	−28.4924	−0.998659
$815$	0	0
$816$	−2.56155	−0.0896723
$817$	2.87689i	0.100650i
$818$	2.31534i	0.0809540i
$819$	−83.2311	−2.90833
$820$	0	0
$821$	−15.3002	−0.533980	−0.266990	−	0.963699i	$-0.586029\pi$
$821$	−15.3002	−0.533980	−0.266990	+	0.963699i	$0.586029\pi$
$822$	− 25.6155i	− 0.893444i
$823$	− 0.630683i	− 0.0219842i	−0.999940	−	0.0109921i	$-0.996501\pi$
$823$	− 0.630683i	− 0.0219842i	0.999940	−	0.0109921i	$-0.00349897\pi$
$824$	−8.00000	−0.278693
$825$	0	0
$826$	−1.61553	−0.0562114
$827$	0.492423i	0.0171232i	0.999963	+	0.00856160i	$0.00272528\pi$
$827$	0.492423i	0.0171232i	−0.999963	+	0.00856160i	$0.997275\pi$
$828$	18.2462i	0.634100i
$829$	9.36932	0.325410	0.162705	−	0.986675i	$-0.447978\pi$
$829$	9.36932	0.325410	0.162705	+	0.986675i	$0.447978\pi$
$830$	0	0
$831$	40.3542	1.39987
$832$	4.56155i	0.158143i
$833$	− 19.2462i	− 0.666842i
$834$	42.2462	1.46287
$835$	0	0
$836$	10.2462	0.354373
$837$	9.43845i	0.326240i
$838$	25.1231i	0.867863i
$839$	−35.0540	−1.21020	−0.605099	−	0.796150i	$-0.706866\pi$
$839$	−35.0540	−1.21020	−0.605099	+	0.796150i	$0.706866\pi$
$840$	0	0
$841$	3.31534	0.114322
$842$	7.61553i	0.262448i
$843$	42.4233i	1.46114i
$844$	3.36932	0.115977
$845$	0	0
$846$	23.3693	0.803454
$847$	25.6155i	0.880160i
$848$	0.561553i	0.0192838i
$849$	21.3002	0.731021
$850$	0	0
$851$	36.4924	1.25094
$852$	11.0540i	0.378703i
$853$	− 48.2462i	− 1.65192i	−0.563730	−	0.825959i	$-0.690634\pi$
$853$	− 48.2462i	− 1.65192i	0.563730	−	0.825959i	$-0.309366\pi$
$854$	−38.1080	−1.30403
$855$	0	0
$856$	16.4924	0.563699
$857$	1.82292i	0.0622697i	0.999515	+	0.0311349i	$0.00991214\pi$
$857$	1.82292i	0.0622697i	−0.999515	+	0.0311349i	$0.990088\pi$
$858$	− 46.7386i	− 1.59563i
$859$	35.5464	1.21283	0.606414	−	0.795149i	$-0.292608\pi$
$859$	35.5464	1.21283	0.606414	+	0.795149i	$0.292608\pi$
$860$	0	0
$861$	−55.7235	−1.89905
$862$	− 20.4924i	− 0.697975i
$863$	− 11.5076i	− 0.391722i	−0.980632	−	0.195861i	$-0.937250\pi$
$863$	− 11.5076i	− 0.391722i	0.980632	−	0.195861i	$-0.0627502\pi$
$864$	1.43845	0.0489370
$865$	0	0
$866$	−2.49242	−0.0846960
$867$	− 2.56155i	− 0.0869949i
$868$	33.6155i	1.14099i
$869$	0	0
$870$	0	0
$871$	−41.6155	−1.41009
$872$	− 8.56155i	− 0.289931i
$873$	6.00000i	0.203069i
$874$	−13.1231	−0.443896
$875$	0	0
$876$	17.4384	0.589191
$877$	− 51.6155i	− 1.74293i	−0.490455	−	0.871466i	$-0.663170\pi$
$877$	− 51.6155i	− 1.74293i	0.490455	−	0.871466i	$-0.336830\pi$
$878$	32.9848i	1.11318i
$879$	−42.4233	−1.43090
$880$	0	0
$881$	−28.7386	−0.968229	−0.484115	−	0.875005i	$-0.660858\pi$
$881$	−28.7386	−0.968229	−0.484115	+	0.875005i	$0.660858\pi$
$882$	68.5464i	2.30808i
$883$	− 27.3693i	− 0.921051i	−0.887647	−	0.460525i	$-0.847661\pi$
$883$	− 27.3693i	− 0.921051i	0.887647	−	0.460525i	$-0.152339\pi$
$884$	−4.56155	−0.153422
$885$	0	0
$886$	28.0000	0.940678
$887$	50.6004i	1.69899i	0.527593	+	0.849497i	$0.323095\pi$
$887$	50.6004i	1.69899i	−0.527593	+	0.849497i	$0.676905\pi$
$888$	− 18.2462i	− 0.612303i
$889$	−48.3542	−1.62175
$890$	0	0
$891$	28.0000	0.938035
$892$	3.68466i	0.123371i
$893$	16.8078i	0.562450i
$894$	8.00000	0.267560
$895$	0	0
$896$	5.12311	0.171151
$897$	59.8617i	1.99873i
$898$	16.8769i	0.563189i
$899$	−37.3002	−1.24403
$900$	0	0
$901$	−0.561553	−0.0187080
$902$	− 16.9848i	− 0.565533i
$903$	− 14.7386i	− 0.490471i
$904$	−2.80776	−0.0933848
$905$	0	0
$906$	39.3693	1.30796
$907$	18.5616i	0.616326i	0.951334	+	0.308163i	$0.0997143\pi$
$907$	18.5616i	0.616326i	−0.951334	+	0.308163i	$0.900286\pi$
$908$	− 16.3153i	− 0.541444i
$909$	51.6155	1.71198
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	6.56155i	0.217275i
$913$	− 24.9848i	− 0.826878i
$914$	16.2462	0.537377
$915$	0	0
$916$	−14.4924	−0.478843
$917$	5.75379i	0.190007i
$918$	1.43845i	0.0474758i
$919$	21.1231	0.696787	0.348393	−	0.937348i	$-0.386727\pi$
$919$	21.1231	0.696787	0.348393	+	0.937348i	$0.386727\pi$
$920$	0	0
$921$	45.4773	1.49853
$922$	2.49242i	0.0820836i
$923$	19.6847i	0.647928i
$924$	−52.4924	−1.72687
$925$	0	0
$926$	−13.9309	−0.457797
$927$	28.4924i	0.935814i
$928$	5.68466i	0.186608i
$929$	59.4773	1.95139	0.975693	−	0.219142i	$-0.0703259\pi$
$929$	59.4773	1.95139	0.975693	+	0.219142i	$0.0703259\pi$
$930$	0	0
$931$	−49.3002	−1.61575
$932$	6.31534i	0.206866i
$933$	20.4924i	0.670892i
$934$	32.4924	1.06318
$935$	0	0
$936$	16.2462	0.531024
$937$	− 44.1080i	− 1.44094i	−0.693484	−	0.720472i	$-0.743925\pi$
$937$	− 44.1080i	− 1.44094i	0.693484	−	0.720472i	$-0.256075\pi$
$938$	46.7386i	1.52607i
$939$	−15.3693	−0.501559
$940$	0	0
$941$	−27.4384	−0.894468	−0.447234	−	0.894417i	$-0.647591\pi$
$941$	−27.4384	−0.894468	−0.447234	+	0.894417i	$0.647591\pi$
$942$	− 47.3693i	− 1.54338i
$943$	21.7538i	0.708401i
$944$	0.315342	0.0102635
$945$	0	0
$946$	4.49242	0.146061
$947$	52.8078i	1.71602i	0.513632	+	0.858011i	$0.328300\pi$
$947$	52.8078i	1.71602i	−0.513632	+	0.858011i	$0.671700\pi$
$948$	0	0
$949$	31.0540	1.00805
$950$	0	0
$951$	−80.9848	−2.62611
$952$	5.12311i	0.166041i
$953$	− 39.1231i	− 1.26732i	−0.773611	−	0.633661i	$-0.781551\pi$
$953$	− 39.1231i	− 1.26732i	0.773611	−	0.633661i	$-0.218449\pi$
$954$	2.00000	0.0647524
$955$	0	0
$956$	−23.3693	−0.755818
$957$	− 58.2462i	− 1.88283i
$958$	− 38.5616i	− 1.24587i
$959$	−51.2311	−1.65434
$960$	0	0
$961$	12.0540	0.388838
$962$	− 32.4924i	− 1.04760i
$963$	− 58.7386i	− 1.89283i
$964$	−12.2462	−0.394424
$965$	0	0
$966$	67.2311	2.16312
$967$	19.5076i	0.627321i	0.949535	+	0.313661i	$0.101555\pi$
$967$	19.5076i	0.627321i	−0.949535	+	0.313661i	$0.898445\pi$
$968$	− 5.00000i	− 0.160706i
$969$	−6.56155	−0.210787
$970$	0	0
$971$	15.6847	0.503345	0.251672	−	0.967813i	$-0.419019\pi$
$971$	15.6847	0.503345	0.251672	+	0.967813i	$0.419019\pi$
$972$	22.2462i	0.713548i
$973$	− 84.4924i	− 2.70870i
$974$	−25.6155	−0.820774
$975$	0	0
$976$	7.43845	0.238099
$977$	− 27.1231i	− 0.867745i	−0.900974	−	0.433873i	$-0.857147\pi$
$977$	− 27.1231i	− 0.867745i	0.900974	−	0.433873i	$-0.142853\pi$
$978$	36.4924i	1.16690i
$979$	−38.7386	−1.23809
$980$	0	0
$981$	−30.4924	−0.973548
$982$	2.56155i	0.0817424i
$983$	37.1231i	1.18404i	0.805922	+	0.592022i	$0.201670\pi$
$983$	37.1231i	1.18404i	−0.805922	+	0.592022i	$0.798330\pi$
$984$	10.8769	0.346743
$985$	0	0
$986$	−5.68466	−0.181036
$987$	− 86.1080i	− 2.74085i
$988$	11.6847i	0.371739i
$989$	−5.75379	−0.182960
$990$	0	0
$991$	34.4233	1.09349	0.546746	−	0.837299i	$-0.315866\pi$
$991$	34.4233	1.09349	0.546746	+	0.837299i	$0.315866\pi$
$992$	− 6.56155i	− 0.208330i
$993$	59.0540i	1.87402i
$994$	22.1080	0.701222
$995$	0	0
$996$	16.0000	0.506979
$997$	11.7538i	0.372246i	0.982526	+	0.186123i	$0.0595923\pi$
$997$	11.7538i	0.372246i	−0.982526	+	0.186123i	$0.940408\pi$
$998$	9.12311i	0.288787i
$999$	−10.2462	−0.324176

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	850.2.c.i.749.4		4
5.2	odd	4		850.2.a.n.1.2		2
5.3	odd	4		170.2.a.f.1.1	✓	2
5.4	even	2	inner	850.2.c.i.749.1		4
15.2	even	4		7650.2.a.de.1.1		2
15.8	even	4		1530.2.a.r.1.2		2
20.3	even	4		1360.2.a.m.1.2		2
20.7	even	4		6800.2.a.be.1.1		2
35.13	even	4		8330.2.a.bq.1.2		2
40.3	even	4		5440.2.a.bd.1.1		2
40.13	odd	4		5440.2.a.bj.1.2		2
85.13	odd	4		2890.2.b.i.2311.1		4
85.33	odd	4		2890.2.a.u.1.2		2
85.38	odd	4		2890.2.b.i.2311.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
170.2.a.f.1.1	✓	2	5.3	odd	4
850.2.a.n.1.2		2	5.2	odd	4
850.2.c.i.749.1		4	5.4	even	2	inner
850.2.c.i.749.4		4	1.1	even	1	trivial
1360.2.a.m.1.2		2	20.3	even	4
1530.2.a.r.1.2		2	15.8	even	4
2890.2.a.u.1.2		2	85.33	odd	4
2890.2.b.i.2311.1		4	85.13	odd	4
2890.2.b.i.2311.4		4	85.38	odd	4
5440.2.a.bd.1.1		2	40.3	even	4
5440.2.a.bj.1.2		2	40.13	odd	4
6800.2.a.be.1.1		2	20.7	even	4
7650.2.a.de.1.1		2	15.2	even	4
8330.2.a.bq.1.2		2	35.13	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 850 = 2 \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	850.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +2.56155i q^{3} -1.00000 q^{4} -2.56155 q^{6} +5.12311i q^{7} -1.00000i q^{8} -3.56155 q^{9} -4.00000 q^{11} -2.56155i q^{12} -4.56155i q^{13} -5.12311 q^{14} +1.00000 q^{16} +1.00000i q^{17} -3.56155i q^{18} +2.56155 q^{19} -13.1231 q^{21} -4.00000i q^{22} +5.12311i q^{23} +2.56155 q^{24} +4.56155 q^{26} -1.43845i q^{27} -5.12311i q^{28} +5.68466 q^{29} -6.56155 q^{31} +1.00000i q^{32} -10.2462i q^{33} -1.00000 q^{34} +3.56155 q^{36} -7.12311i q^{37} +2.56155i q^{38} +11.6847 q^{39} +4.24621 q^{41} -13.1231i q^{42} +1.12311i q^{43} +4.00000 q^{44} -5.12311 q^{46} +6.56155i q^{47} +2.56155i q^{48} -19.2462 q^{49} -2.56155 q^{51} +4.56155i q^{52} +0.561553i q^{53} +1.43845 q^{54} +5.12311 q^{56} +6.56155i q^{57} +5.68466i q^{58} +0.315342 q^{59} +7.43845 q^{61} -6.56155i q^{62} -18.2462i q^{63} -1.00000 q^{64} +10.2462 q^{66} -9.12311i q^{67} -1.00000i q^{68} -13.1231 q^{69} -4.31534 q^{71} +3.56155i q^{72} +6.80776i q^{73} +7.12311 q^{74} -2.56155 q^{76} -20.4924i q^{77} +11.6847i q^{78} -7.00000 q^{81} +4.24621i q^{82} +6.24621i q^{83} +13.1231 q^{84} -1.12311 q^{86} +14.5616i q^{87} +4.00000i q^{88} +9.68466 q^{89} +23.3693 q^{91} -5.12311i q^{92} -16.8078i q^{93} -6.56155 q^{94} -2.56155 q^{96} -1.68466i q^{97} -19.2462i q^{98} +14.2462 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} - 2 q^{6} - 6 q^{9} - 16 q^{11} - 4 q^{14} + 4 q^{16} + 2 q^{19} - 36 q^{21} + 2 q^{24} + 10 q^{26} - 2 q^{29} - 18 q^{31} - 4 q^{34} + 6 q^{36} + 22 q^{39} - 16 q^{41} + 16 q^{44} - 4 q^{46}+ \cdots + 24 q^{99}+O(q^{100}) \) 4 * q - 4 * q^4 - 2 * q^6 - 6 * q^9 - 16 * q^11 - 4 * q^14 + 4 * q^16 + 2 * q^19 - 36 * q^21 + 2 * q^24 + 10 * q^26 - 2 * q^29 - 18 * q^31 - 4 * q^34 + 6 * q^36 + 22 * q^39 - 16 * q^41 + 16 * q^44 - 4 * q^46 - 44 * q^49 - 2 * q^51 + 14 * q^54 + 4 * q^56 + 26 * q^59 + 38 * q^61 - 4 * q^64 + 8 * q^66 - 36 * q^69 - 42 * q^71 + 12 * q^74 - 2 * q^76 - 28 * q^81 + 36 * q^84 + 12 * q^86 + 14 * q^89 + 44 * q^91 - 18 * q^94 - 2 * q^96 + 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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