LMFDB - Embedded newform 9075.2.a.dt.1.4

Error: no document with id 234107538 found in table mf_hecke_traces.

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9075,2,Mod(1,9075)] 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("9075.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9075, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9075 = 3 \cdot 5^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9075.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,-1,-8,7,0,1,-8,-3,8,0,0,-7,-7,-8,0,-7,1,-1,6,0,8,0,-7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)

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

Self dual:	yes
Analytic conductor:	$72.4642398343$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 11x^{6} + 9x^{5} + 38x^{4} - 25x^{3} - 41x^{2} + 20x + 5 $ x^8 - x^7 - 11x^6 + 9x^5 + 38x^4 - 25x^3 - 41x^2 + 20x + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 825)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.684625$ of defining polynomial
Character	$\chi$	$=$	9075.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.684625 q^{2} -1.00000 q^{3} -1.53129 q^{4} +0.684625 q^{6} -4.22715 q^{7} +2.41761 q^{8} +1.00000 q^{9} +1.53129 q^{12} -6.27371 q^{13} +2.89401 q^{14} +1.40742 q^{16} +6.19454 q^{17} -0.684625 q^{18} +2.81055 q^{19} +4.22715 q^{21} +0.456579 q^{23} -2.41761 q^{24} +4.29514 q^{26} -1.00000 q^{27} +6.47298 q^{28} +7.07254 q^{29} +3.54905 q^{31} -5.79877 q^{32} -4.24094 q^{34} -1.53129 q^{36} -6.32477 q^{37} -1.92418 q^{38} +6.27371 q^{39} -8.77435 q^{41} -2.89401 q^{42} +9.83046 q^{43} -0.312586 q^{46} -11.1325 q^{47} -1.40742 q^{48} +10.8688 q^{49} -6.19454 q^{51} +9.60686 q^{52} +1.89080 q^{53} +0.684625 q^{54} -10.2196 q^{56} -2.81055 q^{57} -4.84204 q^{58} -8.57869 q^{59} +4.19759 q^{61} -2.42977 q^{62} -4.22715 q^{63} +1.15515 q^{64} -10.0915 q^{67} -9.48562 q^{68} -0.456579 q^{69} -1.44029 q^{71} +2.41761 q^{72} -1.40901 q^{73} +4.33010 q^{74} -4.30377 q^{76} -4.29514 q^{78} -7.65396 q^{79} +1.00000 q^{81} +6.00714 q^{82} -1.72142 q^{83} -6.47298 q^{84} -6.73018 q^{86} -7.07254 q^{87} -12.6663 q^{89} +26.5199 q^{91} -0.699154 q^{92} -3.54905 q^{93} +7.62162 q^{94} +5.79877 q^{96} -3.64121 q^{97} -7.44105 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{2} - 8 q^{3} + 7 q^{4} + q^{6} - 8 q^{7} - 3 q^{8} + 8 q^{9} - 7 q^{12} - 7 q^{13} - 8 q^{14} - 7 q^{16} + q^{17} - q^{18} + 6 q^{19} + 8 q^{21} - 7 q^{23} + 3 q^{24} - q^{26} - 8 q^{27} - 11 q^{28}+ \cdots + 27 q^{98}+O(q^{100}) $ 8 * q - q^2 - 8 * q^3 + 7 * q^4 + q^6 - 8 * q^7 - 3 * q^8 + 8 * q^9 - 7 * q^12 - 7 * q^13 - 8 * q^14 - 7 * q^16 + q^17 - q^18 + 6 * q^19 + 8 * q^21 - 7 * q^23 + 3 * q^24 - q^26 - 8 * q^27 - 11 * q^28 + 28 * q^29 + 10 * q^31 - 12 * q^32 - 8 * q^34 + 7 * q^36 - 14 * q^37 - 19 * q^38 + 7 * q^39 + 14 * q^41 + 8 * q^42 - 11 * q^43 - 5 * q^47 + 7 * q^48 + 2 * q^49 - q^51 - q^52 + 5 * q^53 + q^54 - 4 * q^56 - 6 * q^57 + 3 * q^58 - 2 * q^59 + 36 * q^61 + 17 * q^62 - 8 * q^63 - 23 * q^64 - 6 * q^67 + 3 * q^68 + 7 * q^69 - 17 * q^71 - 3 * q^72 - 32 * q^73 + 47 * q^74 + 5 * q^76 + q^78 + 41 * q^79 + 8 * q^81 + 6 * q^83 + 11 * q^84 - 9 * q^86 - 28 * q^87 + 21 * q^89 - 9 * q^91 - 69 * q^92 - 10 * q^93 + 51 * q^94 + 12 * q^96 + 4 * q^97 + 27 * q^98

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 $

$2$	−0.684625	−0.484103	−0.242052	−	0.970263i	$-0.577820\pi$
$2$	−0.684625	−0.484103	−0.242052	+	0.970263i	$0.577820\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.53129	−0.765644
$5$	0	0
$5$	0	0
$6$	0.684625	0.279497
$7$	−4.22715	−1.59771	−0.798856	−	0.601522i	$-0.794561\pi$
$7$	−4.22715	−1.59771	−0.798856	+	0.601522i	$0.794561\pi$
$8$	2.41761	0.854754
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	1.53129	0.442045
$13$	−6.27371	−1.74002	−0.870008	−	0.493038i	$-0.835886\pi$
$13$	−6.27371	−1.74002	−0.870008	+	0.493038i	$0.835886\pi$
$14$	2.89401	0.773458
$15$	0	0
$16$	1.40742	0.351855
$17$	6.19454	1.50240	0.751198	−	0.660077i	$-0.229476\pi$
$17$	6.19454	1.50240	0.751198	+	0.660077i	$0.229476\pi$
$18$	−0.684625	−0.161368
$19$	2.81055	0.644785	0.322392	−	0.946606i	$-0.395513\pi$
$19$	2.81055	0.644785	0.322392	+	0.946606i	$0.395513\pi$
$20$	0	0
$21$	4.22715	0.922440
$22$	0	0
$23$	0.456579	0.0952034	0.0476017	−	0.998866i	$-0.484842\pi$
$23$	0.456579	0.0952034	0.0476017	+	0.998866i	$0.484842\pi$
$24$	−2.41761	−0.493492
$25$	0	0
$26$	4.29514	0.842347
$27$	−1.00000	−0.192450
$28$	6.47298	1.22328
$29$	7.07254	1.31334	0.656669	−	0.754179i	$-0.271965\pi$
$29$	7.07254	1.31334	0.656669	+	0.754179i	$0.271965\pi$
$30$	0	0
$31$	3.54905	0.637428	0.318714	−	0.947851i	$-0.396749\pi$
$31$	3.54905	0.637428	0.318714	+	0.947851i	$0.396749\pi$
$32$	−5.79877	−1.02509
$33$	0	0
$34$	−4.24094	−0.727315
$35$	0	0
$36$	−1.53129	−0.255215
$37$	−6.32477	−1.03979	−0.519893	−	0.854232i	$-0.674028\pi$
$37$	−6.32477	−1.03979	−0.519893	+	0.854232i	$0.674028\pi$
$38$	−1.92418	−0.312143
$39$	6.27371	1.00460
$40$	0	0
$41$	−8.77435	−1.37032	−0.685161	−	0.728391i	$-0.740268\pi$
$41$	−8.77435	−1.37032	−0.685161	+	0.728391i	$0.740268\pi$
$42$	−2.89401	−0.446556
$43$	9.83046	1.49913	0.749565	−	0.661931i	$-0.230263\pi$
$43$	9.83046	1.49913	0.749565	+	0.661931i	$0.230263\pi$
$44$	0	0
$45$	0	0
$46$	−0.312586	−0.0460883
$47$	−11.1325	−1.62385	−0.811924	−	0.583763i	$-0.801580\pi$
$47$	−11.1325	−1.62385	−0.811924	+	0.583763i	$0.801580\pi$
$48$	−1.40742	−0.203143
$49$	10.8688	1.55268
$50$	0	0
$51$	−6.19454	−0.867409
$52$	9.60686	1.33223
$53$	1.89080	0.259722	0.129861	−	0.991532i	$-0.458547\pi$
$53$	1.89080	0.259722	0.129861	+	0.991532i	$0.458547\pi$
$54$	0.684625	0.0931657
$55$	0	0
$56$	−10.2196	−1.36565
$57$	−2.81055	−0.372267
$58$	−4.84204	−0.635791
$59$	−8.57869	−1.11685	−0.558425	−	0.829555i	$-0.688594\pi$
$59$	−8.57869	−1.11685	−0.558425	+	0.829555i	$0.688594\pi$
$60$	0	0
$61$	4.19759	0.537447	0.268723	−	0.963217i	$-0.413398\pi$
$61$	4.19759	0.537447	0.268723	+	0.963217i	$0.413398\pi$
$62$	−2.42977	−0.308581
$63$	−4.22715	−0.532571
$64$	1.15515	0.144394
$65$	0	0
$66$	0	0
$67$	−10.0915	−1.23287	−0.616434	−	0.787406i	$-0.711423\pi$
$67$	−10.0915	−1.23287	−0.616434	+	0.787406i	$0.711423\pi$
$68$	−9.48562	−1.15030
$69$	−0.456579	−0.0549657
$70$	0	0
$71$	−1.44029	−0.170931	−0.0854654	−	0.996341i	$-0.527238\pi$
$71$	−1.44029	−0.170931	−0.0854654	+	0.996341i	$0.527238\pi$
$72$	2.41761	0.284918
$73$	−1.40901	−0.164912	−0.0824562	−	0.996595i	$-0.526276\pi$
$73$	−1.40901	−0.164912	−0.0824562	+	0.996595i	$0.526276\pi$
$74$	4.33010	0.503364
$75$	0	0
$76$	−4.30377	−0.493676
$77$	0	0
$78$	−4.29514	−0.486329
$79$	−7.65396	−0.861138	−0.430569	−	0.902558i	$-0.641687\pi$
$79$	−7.65396	−0.861138	−0.430569	+	0.902558i	$0.641687\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	6.00714	0.663378
$83$	−1.72142	−0.188950	−0.0944751	−	0.995527i	$-0.530117\pi$
$83$	−1.72142	−0.188950	−0.0944751	+	0.995527i	$0.530117\pi$
$84$	−6.47298	−0.706260
$85$	0	0
$86$	−6.73018	−0.725734
$87$	−7.07254	−0.758256
$88$	0	0
$89$	−12.6663	−1.34262	−0.671310	−	0.741176i	$-0.734268\pi$
$89$	−12.6663	−1.34262	−0.671310	+	0.741176i	$0.734268\pi$
$90$	0	0
$91$	26.5199	2.78004
$92$	−0.699154	−0.0728919
$93$	−3.54905	−0.368019
$94$	7.62162	0.786110
$95$	0	0
$96$	5.79877	0.591835
$97$	−3.64121	−0.369709	−0.184855	−	0.982766i	$-0.559181\pi$
$97$	−3.64121	−0.369709	−0.184855	+	0.982766i	$0.559181\pi$
$98$	−7.44105	−0.751660
$99$	0	0
$100$	0	0
$101$	1.16775	0.116196	0.0580980	−	0.998311i	$-0.481496\pi$
$101$	1.16775	0.116196	0.0580980	+	0.998311i	$0.481496\pi$
$102$	4.24094	0.419915
$103$	13.7630	1.35611	0.678055	−	0.735011i	$-0.262823\pi$
$103$	13.7630	1.35611	0.678055	+	0.735011i	$0.262823\pi$
$104$	−15.1674	−1.48729
$105$	0	0
$106$	−1.29449	−0.125732
$107$	−12.8497	−1.24223	−0.621114	−	0.783720i	$-0.713320\pi$
$107$	−12.8497	−1.24223	−0.621114	+	0.783720i	$0.713320\pi$
$108$	1.53129	0.147348
$109$	−1.58733	−0.152039	−0.0760196	−	0.997106i	$-0.524221\pi$
$109$	−1.58733	−0.152039	−0.0760196	+	0.997106i	$0.524221\pi$
$110$	0	0
$111$	6.32477	0.600321
$112$	−5.94937	−0.562163
$113$	2.90973	0.273724	0.136862	−	0.990590i	$-0.456298\pi$
$113$	2.90973	0.273724	0.136862	+	0.990590i	$0.456298\pi$
$114$	1.92418	0.180216
$115$	0	0
$116$	−10.8301	−1.00555
$117$	−6.27371	−0.580005
$118$	5.87319	0.540671
$119$	−26.1852	−2.40040
$120$	0	0
$121$	0	0
$122$	−2.87378	−0.260180
$123$	8.77435	0.791156
$124$	−5.43462	−0.488043
$125$	0	0
$126$	2.89401	0.257819
$127$	−17.1199	−1.51915	−0.759573	−	0.650423i	$-0.774592\pi$
$127$	−17.1199	−1.51915	−0.759573	+	0.650423i	$0.774592\pi$
$128$	10.8067	0.955187
$129$	−9.83046	−0.865523
$130$	0	0
$131$	−8.11280	−0.708818	−0.354409	−	0.935090i	$-0.615318\pi$
$131$	−8.11280	−0.708818	−0.354409	+	0.935090i	$0.615318\pi$
$132$	0	0
$133$	−11.8806	−1.03018
$134$	6.90887	0.596836
$135$	0	0
$136$	14.9760	1.28418
$137$	−9.05297	−0.773448	−0.386724	−	0.922196i	$-0.626393\pi$
$137$	−9.05297	−0.773448	−0.386724	+	0.922196i	$0.626393\pi$
$138$	0.312586	0.0266091
$139$	4.95081	0.419923	0.209961	−	0.977710i	$-0.432666\pi$
$139$	4.95081	0.419923	0.209961	+	0.977710i	$0.432666\pi$
$140$	0	0
$141$	11.1325	0.937529
$142$	0.986058	0.0827482
$143$	0	0
$144$	1.40742	0.117285
$145$	0	0
$146$	0.964646	0.0798347
$147$	−10.8688	−0.896443
$148$	9.68504	0.796106
$149$	19.8361	1.62503	0.812516	−	0.582938i	$-0.198097\pi$
$149$	19.8361	1.62503	0.812516	+	0.582938i	$0.198097\pi$
$150$	0	0
$151$	7.00404	0.569981	0.284990	−	0.958530i	$-0.408010\pi$
$151$	7.00404	0.569981	0.284990	+	0.958530i	$0.408010\pi$
$152$	6.79482	0.551133
$153$	6.19454	0.500799
$154$	0	0
$155$	0	0
$156$	−9.60686	−0.769165
$157$	−3.19676	−0.255129	−0.127564	−	0.991830i	$-0.540716\pi$
$157$	−3.19676	−0.255129	−0.127564	+	0.991830i	$0.540716\pi$
$158$	5.24010	0.416880
$159$	−1.89080	−0.149950
$160$	0	0
$161$	−1.93003	−0.152108
$162$	−0.684625	−0.0537893
$163$	−9.58733	−0.750938	−0.375469	−	0.926835i	$-0.622518\pi$
$163$	−9.58733	−0.750938	−0.375469	+	0.926835i	$0.622518\pi$
$164$	13.4360	1.04918
$165$	0	0
$166$	1.17853	0.0914715
$167$	3.31591	0.256593	0.128296	−	0.991736i	$-0.459049\pi$
$167$	3.31591	0.256593	0.128296	+	0.991736i	$0.459049\pi$
$168$	10.2196	0.788459
$169$	26.3595	2.02765
$170$	0	0
$171$	2.81055	0.214928
$172$	−15.0533	−1.14780
$173$	12.6756	0.963708	0.481854	−	0.876252i	$-0.339964\pi$
$173$	12.6756	0.963708	0.481854	+	0.876252i	$0.339964\pi$
$174$	4.84204	0.367074
$175$	0	0
$176$	0	0
$177$	8.57869	0.644814
$178$	8.67164	0.649967
$179$	−13.3806	−1.00012	−0.500058	−	0.865992i	$-0.666688\pi$
$179$	−13.3806	−1.00012	−0.500058	+	0.865992i	$0.666688\pi$
$180$	0	0
$181$	−3.48177	−0.258798	−0.129399	−	0.991593i	$-0.541305\pi$
$181$	−3.48177	−0.258798	−0.129399	+	0.991593i	$0.541305\pi$
$182$	−18.1562	−1.34583
$183$	−4.19759	−0.310295
$184$	1.10383	0.0813755
$185$	0	0
$186$	2.42977	0.178159
$187$	0	0
$188$	17.0471	1.24329
$189$	4.22715	0.307480
$190$	0	0
$191$	5.81744	0.420935	0.210467	−	0.977601i	$-0.432501\pi$
$191$	5.81744	0.420935	0.210467	+	0.977601i	$0.432501\pi$
$192$	−1.15515	−0.0833658
$193$	−0.0960587	−0.00691445	−0.00345723	−	0.999994i	$-0.501100\pi$
$193$	−0.0960587	−0.00691445	−0.00345723	+	0.999994i	$0.501100\pi$
$194$	2.49287	0.178977
$195$	0	0
$196$	−16.6432	−1.18880
$197$	7.33166	0.522359	0.261179	−	0.965290i	$-0.415889\pi$
$197$	7.33166	0.522359	0.261179	+	0.965290i	$0.415889\pi$
$198$	0	0
$199$	10.5380	0.747020	0.373510	−	0.927626i	$-0.378154\pi$
$199$	10.5380	0.747020	0.373510	+	0.927626i	$0.378154\pi$
$200$	0	0
$201$	10.0915	0.711797
$202$	−0.799475	−0.0562508
$203$	−29.8967	−2.09834
$204$	9.48562	0.664126
$205$	0	0
$206$	−9.42252	−0.656498
$207$	0.456579	0.0317345
$208$	−8.82974	−0.612233
$209$	0	0
$210$	0	0
$211$	−2.53914	−0.174802	−0.0874008	−	0.996173i	$-0.527856\pi$
$211$	−2.53914	−0.174802	−0.0874008	+	0.996173i	$0.527856\pi$
$212$	−2.89536	−0.198854
$213$	1.44029	0.0986869
$214$	8.79724	0.601367
$215$	0	0
$216$	−2.41761	−0.164497
$217$	−15.0024	−1.01843
$218$	1.08673	0.0736026
$219$	1.40901	0.0952123
$220$	0	0
$221$	−38.8628	−2.61419
$222$	−4.33010	−0.290617
$223$	11.9815	0.802338	0.401169	−	0.916004i	$-0.368604\pi$
$223$	11.9815	0.802338	0.401169	+	0.916004i	$0.368604\pi$
$224$	24.5123	1.63780
$225$	0	0
$226$	−1.99207	−0.132511
$227$	8.35198	0.554340	0.277170	−	0.960821i	$-0.410603\pi$
$227$	8.35198	0.554340	0.277170	+	0.960821i	$0.410603\pi$
$228$	4.30377	0.285024
$229$	−15.1046	−0.998141	−0.499071	−	0.866561i	$-0.666325\pi$
$229$	−15.1046	−0.998141	−0.499071	+	0.866561i	$0.666325\pi$
$230$	0	0
$231$	0	0
$232$	17.0986	1.12258
$233$	−15.9543	−1.04520	−0.522602	−	0.852577i	$-0.675038\pi$
$233$	−15.9543	−1.04520	−0.522602	+	0.852577i	$0.675038\pi$
$234$	4.29514	0.280782
$235$	0	0
$236$	13.1364	0.855110
$237$	7.65396	0.497178
$238$	17.9271	1.16204
$239$	−7.19454	−0.465376	−0.232688	−	0.972551i	$-0.574752\pi$
$239$	−7.19454	−0.465376	−0.232688	+	0.972551i	$0.574752\pi$
$240$	0	0
$241$	0.529998	0.0341402	0.0170701	−	0.999854i	$-0.494566\pi$
$241$	0.529998	0.0341402	0.0170701	+	0.999854i	$0.494566\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	−6.42772	−0.411493
$245$	0	0
$246$	−6.00714	−0.383001
$247$	−17.6326	−1.12194
$248$	8.58021	0.544844
$249$	1.72142	0.109090
$250$	0	0
$251$	−9.67975	−0.610981	−0.305490	−	0.952195i	$-0.598820\pi$
$251$	−9.67975	−0.610981	−0.305490	+	0.952195i	$0.598820\pi$
$252$	6.47298	0.407760
$253$	0	0
$254$	11.7207	0.735423
$255$	0	0
$256$	−9.70884	−0.606803
$257$	8.09076	0.504688	0.252344	−	0.967638i	$-0.418799\pi$
$257$	8.09076	0.504688	0.252344	+	0.967638i	$0.418799\pi$
$258$	6.73018	0.419003
$259$	26.7357	1.66128
$260$	0	0
$261$	7.07254	0.437779
$262$	5.55423	0.343141
$263$	12.7851	0.788360	0.394180	−	0.919033i	$-0.371029\pi$
$263$	12.7851	0.788360	0.394180	+	0.919033i	$0.371029\pi$
$264$	0	0
$265$	0	0
$266$	8.13378	0.498714
$267$	12.6663	0.775163
$268$	15.4529	0.943938
$269$	16.9311	1.03231	0.516154	−	0.856496i	$-0.327363\pi$
$269$	16.9311	1.03231	0.516154	+	0.856496i	$0.327363\pi$
$270$	0	0
$271$	23.1087	1.40375	0.701877	−	0.712298i	$-0.252346\pi$
$271$	23.1087	1.40375	0.701877	+	0.712298i	$0.252346\pi$
$272$	8.71831	0.528625
$273$	−26.5199	−1.60506
$274$	6.19790	0.374429
$275$	0	0
$276$	0.699154	0.0420841
$277$	−9.45197	−0.567914	−0.283957	−	0.958837i	$-0.591647\pi$
$277$	−9.45197	−0.567914	−0.283957	+	0.958837i	$0.591647\pi$
$278$	−3.38945	−0.203286
$279$	3.54905	0.212476
$280$	0	0
$281$	22.4288	1.33799	0.668994	−	0.743268i	$-0.266725\pi$
$281$	22.4288	1.33799	0.668994	+	0.743268i	$0.266725\pi$
$282$	−7.62162	−0.453861
$283$	−8.97630	−0.533586	−0.266793	−	0.963754i	$-0.585964\pi$
$283$	−8.97630	−0.533586	−0.266793	+	0.963754i	$0.585964\pi$
$284$	2.20550	0.130872
$285$	0	0
$286$	0	0
$287$	37.0905	2.18938
$288$	−5.79877	−0.341696
$289$	21.3723	1.25719
$290$	0	0
$291$	3.64121	0.213452
$292$	2.15760	0.126264
$293$	−2.69997	−0.157734	−0.0788670	−	0.996885i	$-0.525130\pi$
$293$	−2.69997	−0.157734	−0.0788670	+	0.996885i	$0.525130\pi$
$294$	7.44105	0.433971
$295$	0	0
$296$	−15.2908	−0.888761
$297$	0	0
$298$	−13.5803	−0.786684
$299$	−2.86445	−0.165655
$300$	0	0
$301$	−41.5548	−2.39518
$302$	−4.79515	−0.275930
$303$	−1.16775	−0.0670858
$304$	3.95562	0.226871
$305$	0	0
$306$	−4.24094	−0.242438
$307$	6.23363	0.355772	0.177886	−	0.984051i	$-0.443074\pi$
$307$	6.23363	0.355772	0.177886	+	0.984051i	$0.443074\pi$
$308$	0	0
$309$	−13.7630	−0.782951
$310$	0	0
$311$	−22.5636	−1.27946	−0.639732	−	0.768598i	$-0.720955\pi$
$311$	−22.5636	−1.27946	−0.639732	+	0.768598i	$0.720955\pi$
$312$	15.1674	0.858684
$313$	−1.64746	−0.0931200	−0.0465600	−	0.998915i	$-0.514826\pi$
$313$	−1.64746	−0.0931200	−0.0465600	+	0.998915i	$0.514826\pi$
$314$	2.18858	0.123509
$315$	0	0
$316$	11.7204	0.659325
$317$	3.89355	0.218684	0.109342	−	0.994004i	$-0.465126\pi$
$317$	3.89355	0.218684	0.109342	+	0.994004i	$0.465126\pi$
$318$	1.29449	0.0725914
$319$	0	0
$320$	0	0
$321$	12.8497	0.717201
$322$	1.32135	0.0736358
$323$	17.4101	0.968722
$324$	−1.53129	−0.0850716
$325$	0	0
$326$	6.56373	0.363531
$327$	1.58733	0.0877798
$328$	−21.2129	−1.17129
$329$	47.0589	2.59444
$330$	0	0
$331$	7.67417	0.421811	0.210905	−	0.977506i	$-0.432359\pi$
$331$	7.67417	0.421811	0.210905	+	0.977506i	$0.432359\pi$
$332$	2.63599	0.144669
$333$	−6.32477	−0.346595
$334$	−2.27015	−0.124217
$335$	0	0
$336$	5.94937	0.324565
$337$	−18.4714	−1.00620	−0.503100	−	0.864228i	$-0.667807\pi$
$337$	−18.4714	−1.00620	−0.503100	+	0.864228i	$0.667807\pi$
$338$	−18.0464	−0.981594
$339$	−2.90973	−0.158035
$340$	0	0
$341$	0	0
$342$	−1.92418	−0.104048
$343$	−16.3539	−0.883030
$344$	23.7662	1.28139
$345$	0	0
$346$	−8.67804	−0.466534
$347$	−20.9153	−1.12279	−0.561395	−	0.827548i	$-0.689735\pi$
$347$	−20.9153	−1.12279	−0.561395	+	0.827548i	$0.689735\pi$
$348$	10.8301	0.580554
$349$	−8.34373	−0.446630	−0.223315	−	0.974746i	$-0.571688\pi$
$349$	−8.34373	−0.446630	−0.223315	+	0.974746i	$0.571688\pi$
$350$	0	0
$351$	6.27371	0.334866
$352$	0	0
$353$	−22.8476	−1.21606	−0.608028	−	0.793915i	$-0.708039\pi$
$353$	−22.8476	−1.21606	−0.608028	+	0.793915i	$0.708039\pi$
$354$	−5.87319	−0.312157
$355$	0	0
$356$	19.3957	1.02797
$357$	26.1852	1.38587
$358$	9.16073	0.484160
$359$	31.9092	1.68410	0.842052	−	0.539396i	$-0.181347\pi$
$359$	31.9092	1.68410	0.842052	+	0.539396i	$0.181347\pi$
$360$	0	0
$361$	−11.1008	−0.584252
$362$	2.38371	0.125285
$363$	0	0
$364$	−40.6096	−2.12852
$365$	0	0
$366$	2.87378	0.150215
$367$	−26.1946	−1.36735	−0.683674	−	0.729788i	$-0.739619\pi$
$367$	−26.1946	−1.36735	−0.683674	+	0.729788i	$0.739619\pi$
$368$	0.642598	0.0334977
$369$	−8.77435	−0.456774
$370$	0	0
$371$	−7.99270	−0.414960
$372$	5.43462	0.281772
$373$	27.5995	1.42905	0.714523	−	0.699612i	$-0.246644\pi$
$373$	27.5995	1.42905	0.714523	+	0.699612i	$0.246644\pi$
$374$	0	0
$375$	0	0
$376$	−26.9141	−1.38799
$377$	−44.3711	−2.28523
$378$	−2.89401	−0.148852
$379$	−22.8685	−1.17468	−0.587338	−	0.809342i	$-0.699824\pi$
$379$	−22.8685	−1.17468	−0.587338	+	0.809342i	$0.699824\pi$
$380$	0	0
$381$	17.1199	0.877079
$382$	−3.98276	−0.203776
$383$	−2.08508	−0.106543	−0.0532714	−	0.998580i	$-0.516965\pi$
$383$	−2.08508	−0.106543	−0.0532714	+	0.998580i	$0.516965\pi$
$384$	−10.8067	−0.551477
$385$	0	0
$386$	0.0657642	0.00334731
$387$	9.83046	0.499710
$388$	5.57575	0.283066
$389$	23.3511	1.18395	0.591974	−	0.805957i	$-0.298349\pi$
$389$	23.3511	1.18395	0.591974	+	0.805957i	$0.298349\pi$
$390$	0	0
$391$	2.82830	0.143033
$392$	26.2765	1.32716
$393$	8.11280	0.409236
$394$	−5.01944	−0.252876
$395$	0	0
$396$	0	0
$397$	27.5109	1.38073	0.690365	−	0.723461i	$-0.257450\pi$
$397$	27.5109	1.38073	0.690365	+	0.723461i	$0.257450\pi$
$398$	−7.21459	−0.361635
$399$	11.8806	0.594775
$400$	0	0
$401$	−8.67427	−0.433173	−0.216586	−	0.976263i	$-0.569492\pi$
$401$	−8.67427	−0.433173	−0.216586	+	0.976263i	$0.569492\pi$
$402$	−6.90887	−0.344583
$403$	−22.2657	−1.10913
$404$	−1.78817	−0.0889647
$405$	0	0
$406$	20.4680	1.01581
$407$	0	0
$408$	−14.9760	−0.741421
$409$	−5.62220	−0.278000	−0.139000	−	0.990292i	$-0.544389\pi$
$409$	−5.62220	−0.278000	−0.139000	+	0.990292i	$0.544389\pi$
$410$	0	0
$411$	9.05297	0.446550
$412$	−21.0752	−1.03830
$413$	36.2634	1.78441
$414$	−0.312586	−0.0153628
$415$	0	0
$416$	36.3799	1.78367
$417$	−4.95081	−0.242442
$418$	0	0
$419$	−19.3628	−0.945937	−0.472968	−	0.881079i	$-0.656817\pi$
$419$	−19.3628	−0.945937	−0.472968	+	0.881079i	$0.656817\pi$
$420$	0	0
$421$	−28.5563	−1.39175	−0.695874	−	0.718164i	$-0.744983\pi$
$421$	−28.5563	−1.39175	−0.695874	+	0.718164i	$0.744983\pi$
$422$	1.73836	0.0846220
$423$	−11.1325	−0.541283
$424$	4.57122	0.221998
$425$	0	0
$426$	−0.986058	−0.0477747
$427$	−17.7439	−0.858685
$428$	19.6766	0.951105
$429$	0	0
$430$	0	0
$431$	5.24275	0.252534	0.126267	−	0.991996i	$-0.459700\pi$
$431$	5.24275	0.252534	0.126267	+	0.991996i	$0.459700\pi$
$432$	−1.40742	−0.0677145
$433$	−24.2829	−1.16696	−0.583480	−	0.812128i	$-0.698309\pi$
$433$	−24.2829	−1.16696	−0.583480	+	0.812128i	$0.698309\pi$
$434$	10.2710	0.493024
$435$	0	0
$436$	2.43067	0.116408
$437$	1.28324	0.0613857
$438$	−0.964646	−0.0460926
$439$	12.2291	0.583661	0.291831	−	0.956470i	$-0.405736\pi$
$439$	12.2291	0.583661	0.291831	+	0.956470i	$0.405736\pi$
$440$	0	0
$441$	10.8688	0.517561
$442$	26.6064	1.26554
$443$	−18.1312	−0.861439	−0.430720	−	0.902486i	$-0.641740\pi$
$443$	−18.1312	−0.861439	−0.430720	+	0.902486i	$0.641740\pi$
$444$	−9.68504	−0.459632
$445$	0	0
$446$	−8.20281	−0.388414
$447$	−19.8361	−0.938213
$448$	−4.88299	−0.230700
$449$	40.3757	1.90545	0.952723	−	0.303839i	$-0.0982685\pi$
$449$	40.3757	1.90545	0.952723	+	0.303839i	$0.0982685\pi$
$450$	0	0
$451$	0	0
$452$	−4.45563	−0.209575
$453$	−7.00404	−0.329079
$454$	−5.71798	−0.268358
$455$	0	0
$456$	−6.79482	−0.318197
$457$	5.85095	0.273696	0.136848	−	0.990592i	$-0.456303\pi$
$457$	5.85095	0.273696	0.136848	+	0.990592i	$0.456303\pi$
$458$	10.3410	0.483204
$459$	−6.19454	−0.289136
$460$	0	0
$461$	11.3973	0.530824	0.265412	−	0.964135i	$-0.414492\pi$
$461$	11.3973	0.530824	0.265412	+	0.964135i	$0.414492\pi$
$462$	0	0
$463$	13.3271	0.619363	0.309681	−	0.950840i	$-0.399778\pi$
$463$	13.3271	0.619363	0.309681	+	0.950840i	$0.399778\pi$
$464$	9.95403	0.462104
$465$	0	0
$466$	10.9227	0.505986
$467$	−16.8436	−0.779430	−0.389715	−	0.920936i	$-0.627426\pi$
$467$	−16.8436	−0.779430	−0.389715	+	0.920936i	$0.627426\pi$
$468$	9.60686	0.444077
$469$	42.6581	1.96977
$470$	0	0
$471$	3.19676	0.147299
$472$	−20.7399	−0.954632
$473$	0	0
$474$	−5.24010	−0.240686
$475$	0	0
$476$	40.0971	1.83785
$477$	1.89080	0.0865738
$478$	4.92556	0.225290
$479$	25.8720	1.18212	0.591062	−	0.806626i	$-0.298709\pi$
$479$	25.8720	1.18212	0.591062	+	0.806626i	$0.298709\pi$
$480$	0	0
$481$	39.6798	1.80924
$482$	−0.362850	−0.0165274
$483$	1.93003	0.0878193
$484$	0	0
$485$	0	0
$486$	0.684625	0.0310552
$487$	5.45864	0.247354	0.123677	−	0.992323i	$-0.460531\pi$
$487$	5.45864	0.247354	0.123677	+	0.992323i	$0.460531\pi$
$488$	10.1481	0.459385
$489$	9.58733	0.433554
$490$	0	0
$491$	−13.0003	−0.586697	−0.293348	−	0.956006i	$-0.594770\pi$
$491$	−13.0003	−0.586697	−0.293348	+	0.956006i	$0.594770\pi$
$492$	−13.4360	−0.605744
$493$	43.8111	1.97315
$494$	12.0717	0.543133
$495$	0	0
$496$	4.99500	0.224282
$497$	6.08831	0.273098
$498$	−1.17853	−0.0528111
$499$	5.09826	0.228230	0.114115	−	0.993468i	$-0.463597\pi$
$499$	5.09826	0.228230	0.114115	+	0.993468i	$0.463597\pi$
$500$	0	0
$501$	−3.31591	−0.148144
$502$	6.62700	0.295778
$503$	−14.4915	−0.646146	−0.323073	−	0.946374i	$-0.604716\pi$
$503$	−14.4915	−0.646146	−0.323073	+	0.946374i	$0.604716\pi$
$504$	−10.2196	−0.455217
$505$	0	0
$506$	0	0
$507$	−26.3595	−1.17067
$508$	26.2155	1.16312
$509$	−31.2521	−1.38522	−0.692612	−	0.721311i	$-0.743540\pi$
$509$	−31.2521	−1.38522	−0.692612	+	0.721311i	$0.743540\pi$
$510$	0	0
$511$	5.95611	0.263483
$512$	−14.9665	−0.661431
$513$	−2.81055	−0.124089
$514$	−5.53914	−0.244321
$515$	0	0
$516$	15.0533	0.662683
$517$	0	0
$518$	−18.3040	−0.804230
$519$	−12.6756	−0.556397
$520$	0	0
$521$	−5.85045	−0.256313	−0.128156	−	0.991754i	$-0.540906\pi$
$521$	−5.85045	−0.256313	−0.128156	+	0.991754i	$0.540906\pi$
$522$	−4.84204	−0.211930
$523$	29.0728	1.27126	0.635631	−	0.771993i	$-0.280740\pi$
$523$	29.0728	1.27126	0.635631	+	0.771993i	$0.280740\pi$
$524$	12.4230	0.542703
$525$	0	0
$526$	−8.75298	−0.381648
$527$	21.9847	0.957669
$528$	0	0
$529$	−22.7915	−0.990936
$530$	0	0
$531$	−8.57869	−0.372283
$532$	18.1927	0.788752
$533$	55.0477	2.38438
$534$	−8.67164	−0.375259
$535$	0	0
$536$	−24.3972	−1.05380
$537$	13.3806	0.577418
$538$	−11.5915	−0.499744
$539$	0	0
$540$	0	0
$541$	25.6908	1.10453	0.552266	−	0.833668i	$-0.313763\pi$
$541$	25.6908	1.10453	0.552266	+	0.833668i	$0.313763\pi$
$542$	−15.8208	−0.679562
$543$	3.48177	0.149417
$544$	−35.9207	−1.54009
$545$	0	0
$546$	18.1562	0.777014
$547$	19.5385	0.835404	0.417702	−	0.908584i	$-0.362836\pi$
$547$	19.5385	0.835404	0.417702	+	0.908584i	$0.362836\pi$
$548$	13.8627	0.592186
$549$	4.19759	0.179149
$550$	0	0
$551$	19.8777	0.846820
$552$	−1.10383	−0.0469821
$553$	32.3544	1.37585
$554$	6.47106	0.274929
$555$	0	0
$556$	−7.58112	−0.321511
$557$	7.08277	0.300106	0.150053	−	0.988678i	$-0.452055\pi$
$557$	7.08277	0.300106	0.150053	+	0.988678i	$0.452055\pi$
$558$	−2.42977	−0.102860
$559$	−61.6735	−2.60851
$560$	0	0
$561$	0	0
$562$	−15.3553	−0.647724
$563$	3.78253	0.159415	0.0797074	−	0.996818i	$-0.474601\pi$
$563$	3.78253	0.159415	0.0797074	+	0.996818i	$0.474601\pi$
$564$	−17.0471	−0.717813
$565$	0	0
$566$	6.14540	0.258311
$567$	−4.22715	−0.177524
$568$	−3.48206	−0.146104
$569$	43.0892	1.80639	0.903196	−	0.429229i	$-0.141215\pi$
$569$	43.0892	1.80639	0.903196	+	0.429229i	$0.141215\pi$
$570$	0	0
$571$	23.2838	0.974398	0.487199	−	0.873291i	$-0.338019\pi$
$571$	23.2838	0.974398	0.487199	+	0.873291i	$0.338019\pi$
$572$	0	0
$573$	−5.81744	−0.243027
$574$	−25.3931	−1.05989
$575$	0	0
$576$	1.15515	0.0481313
$577$	40.5589	1.68849	0.844245	−	0.535957i	$-0.180049\pi$
$577$	40.5589	1.68849	0.844245	+	0.535957i	$0.180049\pi$
$578$	−14.6320	−0.608612
$579$	0.0960587	0.00399206
$580$	0	0
$581$	7.27669	0.301888
$582$	−2.49287	−0.103333
$583$	0	0
$584$	−3.40644	−0.140960
$585$	0	0
$586$	1.84847	0.0763595
$587$	−41.6256	−1.71807	−0.859037	−	0.511913i	$-0.828937\pi$
$587$	−41.6256	−1.71807	−0.859037	+	0.511913i	$0.828937\pi$
$588$	16.6432	0.686356
$589$	9.97479	0.411004
$590$	0	0
$591$	−7.33166	−0.301584
$592$	−8.90160	−0.365853
$593$	−38.6530	−1.58729	−0.793644	−	0.608383i	$-0.791818\pi$
$593$	−38.6530	−1.58729	−0.793644	+	0.608383i	$0.791818\pi$
$594$	0	0
$595$	0	0
$596$	−30.3747	−1.24420
$597$	−10.5380	−0.431292
$598$	1.96107	0.0801943
$599$	−2.17895	−0.0890293	−0.0445146	−	0.999009i	$-0.514174\pi$
$599$	−2.17895	−0.0890293	−0.0445146	+	0.999009i	$0.514174\pi$
$600$	0	0
$601$	39.1748	1.59797	0.798987	−	0.601348i	$-0.205369\pi$
$601$	39.1748	1.59797	0.798987	+	0.601348i	$0.205369\pi$
$602$	28.4495	1.15951
$603$	−10.0915	−0.410956
$604$	−10.7252	−0.436402
$605$	0	0
$606$	0.799475	0.0324764
$607$	0.253714	0.0102979	0.00514896	−	0.999987i	$-0.498361\pi$
$607$	0.253714	0.0102979	0.00514896	+	0.999987i	$0.498361\pi$
$608$	−16.2978	−0.660961
$609$	29.8967	1.21147
$610$	0	0
$611$	69.8424	2.82552
$612$	−9.48562	−0.383434
$613$	2.26495	0.0914805	0.0457402	−	0.998953i	$-0.485435\pi$
$613$	2.26495	0.0914805	0.0457402	+	0.998953i	$0.485435\pi$
$614$	−4.26770	−0.172231
$615$	0	0
$616$	0	0
$617$	46.7095	1.88045	0.940226	−	0.340550i	$-0.110613\pi$
$617$	46.7095	1.88045	0.940226	+	0.340550i	$0.110613\pi$
$618$	9.42252	0.379029
$619$	26.2499	1.05507	0.527537	−	0.849532i	$-0.323116\pi$
$619$	26.2499	1.05507	0.527537	+	0.849532i	$0.323116\pi$
$620$	0	0
$621$	−0.456579	−0.0183219
$622$	15.4476	0.619393
$623$	53.5422	2.14512
$624$	8.82974	0.353473
$625$	0	0
$626$	1.12789	0.0450797
$627$	0	0
$628$	4.89515	0.195338
$629$	−39.1790	−1.56217
$630$	0	0
$631$	43.0754	1.71481	0.857403	−	0.514646i	$-0.172077\pi$
$631$	43.0754	1.71481	0.857403	+	0.514646i	$0.172077\pi$
$632$	−18.5043	−0.736061
$633$	2.53914	0.100922
$634$	−2.66563	−0.105866
$635$	0	0
$636$	2.89536	0.114809
$637$	−68.1877	−2.70169
$638$	0	0
$639$	−1.44029	−0.0569769
$640$	0	0
$641$	18.1698	0.717664	0.358832	−	0.933402i	$-0.383175\pi$
$641$	18.1698	0.717664	0.358832	+	0.933402i	$0.383175\pi$
$642$	−8.79724	−0.347199
$643$	18.7674	0.740115	0.370057	−	0.929009i	$-0.379338\pi$
$643$	18.7674	0.740115	0.370057	+	0.929009i	$0.379338\pi$
$644$	2.95543	0.116460
$645$	0	0
$646$	−11.9194	−0.468962
$647$	48.6561	1.91287	0.956435	−	0.291947i	$-0.0943031\pi$
$647$	48.6561	1.91287	0.956435	+	0.291947i	$0.0943031\pi$
$648$	2.41761	0.0949727
$649$	0	0
$650$	0	0
$651$	15.0024	0.587989
$652$	14.6810	0.574951
$653$	2.83694	0.111018	0.0555090	−	0.998458i	$-0.482322\pi$
$653$	2.83694	0.111018	0.0555090	+	0.998458i	$0.482322\pi$
$654$	−1.08673	−0.0424945
$655$	0	0
$656$	−12.3492	−0.482154
$657$	−1.40901	−0.0549708
$658$	−32.2177	−1.25598
$659$	−26.9059	−1.04811	−0.524053	−	0.851686i	$-0.675580\pi$
$659$	−26.9059	−1.04811	−0.524053	+	0.851686i	$0.675580\pi$
$660$	0	0
$661$	−21.5973	−0.840037	−0.420018	−	0.907516i	$-0.637976\pi$
$661$	−21.5973	−0.840037	−0.420018	+	0.907516i	$0.637976\pi$
$662$	−5.25393	−0.204200
$663$	38.8628	1.50930
$664$	−4.16172	−0.161506
$665$	0	0
$666$	4.33010	0.167788
$667$	3.22917	0.125034
$668$	−5.07761	−0.196459
$669$	−11.9815	−0.463230
$670$	0	0
$671$	0	0
$672$	−24.5123	−0.945582
$673$	1.86330	0.0718248	0.0359124	−	0.999355i	$-0.488566\pi$
$673$	1.86330	0.0718248	0.0359124	+	0.999355i	$0.488566\pi$
$674$	12.6460	0.487105
$675$	0	0
$676$	−40.3640	−1.55246
$677$	−11.0904	−0.426241	−0.213120	−	0.977026i	$-0.568363\pi$
$677$	−11.0904	−0.426241	−0.213120	+	0.977026i	$0.568363\pi$
$678$	1.99207	0.0765051
$679$	15.3920	0.590689
$680$	0	0
$681$	−8.35198	−0.320049
$682$	0	0
$683$	4.37404	0.167368	0.0836840	−	0.996492i	$-0.473331\pi$
$683$	4.37404	0.167368	0.0836840	+	0.996492i	$0.473331\pi$
$684$	−4.30377	−0.164559
$685$	0	0
$686$	11.1963	0.427478
$687$	15.1046	0.576277
$688$	13.8356	0.527476
$689$	−11.8623	−0.451919
$690$	0	0
$691$	33.6219	1.27904	0.639518	−	0.768776i	$-0.279134\pi$
$691$	33.6219	1.27904	0.639518	+	0.768776i	$0.279134\pi$
$692$	−19.4100	−0.737857
$693$	0	0
$694$	14.3191	0.543546
$695$	0	0
$696$	−17.0986	−0.648122
$697$	−54.3530	−2.05877
$698$	5.71233	0.216215
$699$	15.9543	0.603448
$700$	0	0
$701$	22.1962	0.838339	0.419170	−	0.907908i	$-0.362321\pi$
$701$	22.1962	0.838339	0.419170	+	0.907908i	$0.362321\pi$
$702$	−4.29514	−0.162110
$703$	−17.7761	−0.670438
$704$	0	0
$705$	0	0
$706$	15.6421	0.588697
$707$	−4.93627	−0.185648
$708$	−13.1364	−0.493698
$709$	17.7098	0.665104	0.332552	−	0.943085i	$-0.392090\pi$
$709$	17.7098	0.665104	0.332552	+	0.943085i	$0.392090\pi$
$710$	0	0
$711$	−7.65396	−0.287046
$712$	−30.6221	−1.14761
$713$	1.62042	0.0606853
$714$	−17.9271	−0.670904
$715$	0	0
$716$	20.4896	0.765733
$717$	7.19454	0.268685
$718$	−21.8459	−0.815280
$719$	−28.8459	−1.07577	−0.537886	−	0.843018i	$-0.680777\pi$
$719$	−28.8459	−1.07577	−0.537886	+	0.843018i	$0.680777\pi$
$720$	0	0
$721$	−58.1783	−2.16667
$722$	7.59989	0.282839
$723$	−0.529998	−0.0197108
$724$	5.33159	0.198147
$725$	0	0
$726$	0	0
$727$	−21.7927	−0.808245	−0.404122	−	0.914705i	$-0.632423\pi$
$727$	−21.7927	−0.808245	−0.404122	+	0.914705i	$0.632423\pi$
$728$	64.1148	2.37625
$729$	1.00000	0.0370370
$730$	0	0
$731$	60.8951	2.25229
$732$	6.42772	0.237575
$733$	−16.0166	−0.591586	−0.295793	−	0.955252i	$-0.595584\pi$
$733$	−16.0166	−0.591586	−0.295793	+	0.955252i	$0.595584\pi$
$734$	17.9335	0.661937
$735$	0	0
$736$	−2.64760	−0.0975918
$737$	0	0
$738$	6.00714	0.221126
$739$	38.4904	1.41589	0.707946	−	0.706267i	$-0.249622\pi$
$739$	38.4904	1.41589	0.707946	+	0.706267i	$0.249622\pi$
$740$	0	0
$741$	17.6326	0.647750
$742$	5.47200	0.200884
$743$	−3.71014	−0.136112	−0.0680559	−	0.997682i	$-0.521680\pi$
$743$	−3.71014	−0.136112	−0.0680559	+	0.997682i	$0.521680\pi$
$744$	−8.58021	−0.314566
$745$	0	0
$746$	−18.8953	−0.691806
$747$	−1.72142	−0.0629834
$748$	0	0
$749$	54.3177	1.98472
$750$	0	0
$751$	−11.6536	−0.425245	−0.212623	−	0.977134i	$-0.568200\pi$
$751$	−11.6536	−0.425245	−0.212623	+	0.977134i	$0.568200\pi$
$752$	−15.6681	−0.571359
$753$	9.67975	0.352750
$754$	30.3776	1.10629
$755$	0	0
$756$	−6.47298	−0.235420
$757$	20.6237	0.749580	0.374790	−	0.927110i	$-0.377715\pi$
$757$	20.6237	0.749580	0.374790	+	0.927110i	$0.377715\pi$
$758$	15.6563	0.568664
$759$	0	0
$760$	0	0
$761$	−2.32909	−0.0844295	−0.0422148	−	0.999109i	$-0.513441\pi$
$761$	−2.32909	−0.0844295	−0.0422148	+	0.999109i	$0.513441\pi$
$762$	−11.7207	−0.424597
$763$	6.70990	0.242915
$764$	−8.90817	−0.322286
$765$	0	0
$766$	1.42750	0.0515777
$767$	53.8203	1.94334
$768$	9.70884	0.350338
$769$	−29.3154	−1.05714	−0.528571	−	0.848889i	$-0.677272\pi$
$769$	−29.3154	−1.05714	−0.528571	+	0.848889i	$0.677272\pi$
$770$	0	0
$771$	−8.09076	−0.291382
$772$	0.147093	0.00529401
$773$	11.2143	0.403352	0.201676	−	0.979452i	$-0.435361\pi$
$773$	11.2143	0.403352	0.201676	+	0.979452i	$0.435361\pi$
$774$	−6.73018	−0.241911
$775$	0	0
$776$	−8.80303	−0.316010
$777$	−26.7357	−0.959139
$778$	−15.9868	−0.573153
$779$	−24.6608	−0.883563
$780$	0	0
$781$	0	0
$782$	−1.93632	−0.0692428
$783$	−7.07254	−0.252752
$784$	15.2969	0.546319
$785$	0	0
$786$	−5.55423	−0.198113
$787$	−30.1281	−1.07395	−0.536975	−	0.843598i	$-0.680433\pi$
$787$	−30.1281	−1.07395	−0.536975	+	0.843598i	$0.680433\pi$
$788$	−11.2269	−0.399941
$789$	−12.7851	−0.455160
$790$	0	0
$791$	−12.2998	−0.437332
$792$	0	0
$793$	−26.3345	−0.935165
$794$	−18.8346	−0.668416
$795$	0	0
$796$	−16.1367	−0.571952
$797$	35.4999	1.25747	0.628736	−	0.777619i	$-0.283573\pi$
$797$	35.4999	1.25747	0.628736	+	0.777619i	$0.283573\pi$
$798$	−8.13378	−0.287933
$799$	−68.9609	−2.43966
$800$	0	0
$801$	−12.6663	−0.447540
$802$	5.93863	0.209700
$803$	0	0
$804$	−15.4529	−0.544983
$805$	0	0
$806$	15.2437	0.536936
$807$	−16.9311	−0.596003
$808$	2.82318	0.0993190
$809$	3.72091	0.130820	0.0654101	−	0.997858i	$-0.479164\pi$
$809$	3.72091	0.130820	0.0654101	+	0.997858i	$0.479164\pi$
$810$	0	0
$811$	−51.2319	−1.79900	−0.899498	−	0.436925i	$-0.856067\pi$
$811$	−51.2319	−1.79900	−0.899498	+	0.436925i	$0.856067\pi$
$812$	45.7804	1.60658
$813$	−23.1087	−0.810457
$814$	0	0
$815$	0	0
$816$	−8.71831	−0.305202
$817$	27.6290	0.966617
$818$	3.84910	0.134581
$819$	26.5199	0.926681
$820$	0	0
$821$	−31.8527	−1.11167	−0.555833	−	0.831294i	$-0.687601\pi$
$821$	−31.8527	−1.11167	−0.555833	+	0.831294i	$0.687601\pi$
$822$	−6.19790	−0.216176
$823$	5.67092	0.197676	0.0988380	−	0.995104i	$-0.468487\pi$
$823$	5.67092	0.197676	0.0988380	+	0.995104i	$0.468487\pi$
$824$	33.2736	1.15914
$825$	0	0
$826$	−24.8268	−0.863837
$827$	−8.73903	−0.303886	−0.151943	−	0.988389i	$-0.548553\pi$
$827$	−8.73903	−0.303886	−0.151943	+	0.988389i	$0.548553\pi$
$828$	−0.699154	−0.0242973
$829$	−7.17389	−0.249159	−0.124580	−	0.992210i	$-0.539758\pi$
$829$	−7.17389	−0.249159	−0.124580	+	0.992210i	$0.539758\pi$
$830$	0	0
$831$	9.45197	0.327885
$832$	−7.24708	−0.251247
$833$	67.3271	2.33275
$834$	3.38945	0.117367
$835$	0	0
$836$	0	0
$837$	−3.54905	−0.122673
$838$	13.2563	0.457931
$839$	−10.6586	−0.367975	−0.183987	−	0.982929i	$-0.558901\pi$
$839$	−10.6586	−0.367975	−0.183987	+	0.982929i	$0.558901\pi$
$840$	0	0
$841$	21.0208	0.724856
$842$	19.5504	0.673750
$843$	−22.4288	−0.772487
$844$	3.88816	0.133836
$845$	0	0
$846$	7.62162	0.262037
$847$	0	0
$848$	2.66115	0.0913842
$849$	8.97630	0.308066
$850$	0	0
$851$	−2.88776	−0.0989911
$852$	−2.20550	−0.0755591
$853$	40.6266	1.39103	0.695514	−	0.718513i	$-0.255177\pi$
$853$	40.6266	1.39103	0.695514	+	0.718513i	$0.255177\pi$
$854$	12.1479	0.415692
$855$	0	0
$856$	−31.0656	−1.06180
$857$	56.5172	1.93059	0.965295	−	0.261162i	$-0.0841056\pi$
$857$	56.5172	1.93059	0.965295	+	0.261162i	$0.0841056\pi$
$858$	0	0
$859$	−34.5324	−1.17823	−0.589116	−	0.808048i	$-0.700524\pi$
$859$	−34.5324	−1.17823	−0.589116	+	0.808048i	$0.700524\pi$
$860$	0	0
$861$	−37.0905	−1.26404
$862$	−3.58932	−0.122253
$863$	−51.7681	−1.76221	−0.881103	−	0.472925i	$-0.843198\pi$
$863$	−51.7681	−1.76221	−0.881103	+	0.472925i	$0.843198\pi$
$864$	5.79877	0.197278
$865$	0	0
$866$	16.6247	0.564929
$867$	−21.3723	−0.725842
$868$	22.9729	0.779752
$869$	0	0
$870$	0	0
$871$	63.3110	2.14521
$872$	−3.83756	−0.129956
$873$	−3.64121	−0.123236
$874$	−0.878539	−0.0297170
$875$	0	0
$876$	−2.15760	−0.0728987
$877$	−21.0449	−0.710636	−0.355318	−	0.934746i	$-0.615627\pi$
$877$	−21.0449	−0.710636	−0.355318	+	0.934746i	$0.615627\pi$
$878$	−8.37233	−0.282552
$879$	2.69997	0.0910677
$880$	0	0
$881$	38.9312	1.31162	0.655812	−	0.754924i	$-0.272326\pi$
$881$	38.9312	1.31162	0.655812	+	0.754924i	$0.272326\pi$
$882$	−7.44105	−0.250553
$883$	−0.382192	−0.0128618	−0.00643089	−	0.999979i	$-0.502047\pi$
$883$	−0.382192	−0.0128618	−0.00643089	+	0.999979i	$0.502047\pi$
$884$	59.5101	2.00154
$885$	0	0
$886$	12.4131	0.417025
$887$	41.7936	1.40329	0.701646	−	0.712526i	$-0.252449\pi$
$887$	41.7936	1.40329	0.701646	+	0.712526i	$0.252449\pi$
$888$	15.2908	0.513126
$889$	72.3684	2.42716
$890$	0	0
$891$	0	0
$892$	−18.3471	−0.614305
$893$	−31.2886	−1.04703
$894$	13.5803	0.454192
$895$	0	0
$896$	−45.6815	−1.52611
$897$	2.86445	0.0956411
$898$	−27.6422	−0.922433
$899$	25.1008	0.837158
$900$	0	0
$901$	11.7126	0.390205
$902$	0	0
$903$	41.5548	1.38286
$904$	7.03458	0.233967
$905$	0	0
$906$	4.79515	0.159308
$907$	21.9074	0.727422	0.363711	−	0.931512i	$-0.381510\pi$
$907$	21.9074	0.727422	0.363711	+	0.931512i	$0.381510\pi$
$908$	−12.7893	−0.424427
$909$	1.16775	0.0387320
$910$	0	0
$911$	−12.7852	−0.423593	−0.211796	−	0.977314i	$-0.567931\pi$
$911$	−12.7852	−0.423593	−0.211796	+	0.977314i	$0.567931\pi$
$912$	−3.95562	−0.130984
$913$	0	0
$914$	−4.00571	−0.132497
$915$	0	0
$916$	23.1295	0.764221
$917$	34.2940	1.13249
$918$	4.24094	0.139972
$919$	−52.6601	−1.73710	−0.868549	−	0.495604i	$-0.834947\pi$
$919$	−52.6601	−1.73710	−0.868549	+	0.495604i	$0.834947\pi$
$920$	0	0
$921$	−6.23363	−0.205405
$922$	−7.80286	−0.256974
$923$	9.03596	0.297422
$924$	0	0
$925$	0	0
$926$	−9.12407	−0.299836
$927$	13.7630	0.452037
$928$	−41.0121	−1.34629
$929$	−34.8932	−1.14481	−0.572404	−	0.819972i	$-0.693989\pi$
$929$	−34.8932	−1.14481	−0.572404	+	0.819972i	$0.693989\pi$
$930$	0	0
$931$	30.5473	1.00115
$932$	24.4307	0.800254
$933$	22.5636	0.738699
$934$	11.5316	0.377325
$935$	0	0
$936$	−15.1674	−0.495762
$937$	21.7147	0.709387	0.354694	−	0.934983i	$-0.384585\pi$
$937$	21.7147	0.709387	0.354694	+	0.934983i	$0.384585\pi$
$938$	−29.2048	−0.953572
$939$	1.64746	0.0537629
$940$	0	0
$941$	40.0882	1.30684	0.653419	−	0.756997i	$-0.273334\pi$
$941$	40.0882	1.30684	0.653419	+	0.756997i	$0.273334\pi$
$942$	−2.18858	−0.0713078
$943$	−4.00618	−0.130459
$944$	−12.0738	−0.392969
$945$	0	0
$946$	0	0
$947$	52.3489	1.70111	0.850555	−	0.525887i	$-0.176266\pi$
$947$	52.3489	1.70111	0.850555	+	0.525887i	$0.176266\pi$
$948$	−11.7204	−0.380661
$949$	8.83974	0.286950
$950$	0	0
$951$	−3.89355	−0.126257
$952$	−63.3057	−2.05175
$953$	32.4575	1.05140	0.525701	−	0.850669i	$-0.323803\pi$
$953$	32.4575	1.05140	0.525701	+	0.850669i	$0.323803\pi$
$954$	−1.29449	−0.0419107
$955$	0	0
$956$	11.0169	0.356312
$957$	0	0
$958$	−17.7126	−0.572270
$959$	38.2683	1.23575
$960$	0	0
$961$	−18.4043	−0.593686
$962$	−27.1658	−0.875860
$963$	−12.8497	−0.414076
$964$	−0.811579	−0.0261392
$965$	0	0
$966$	−1.32135	−0.0425136
$967$	−18.8006	−0.604587	−0.302294	−	0.953215i	$-0.597752\pi$
$967$	−18.8006	−0.604587	−0.302294	+	0.953215i	$0.597752\pi$
$968$	0	0
$969$	−17.4101	−0.559292
$970$	0	0
$971$	−7.03293	−0.225698	−0.112849	−	0.993612i	$-0.535998\pi$
$971$	−7.03293	−0.225698	−0.112849	+	0.993612i	$0.535998\pi$
$972$	1.53129	0.0491161
$973$	−20.9278	−0.670915
$974$	−3.73712	−0.119745
$975$	0	0
$976$	5.90777	0.189103
$977$	4.63681	0.148345	0.0741724	−	0.997245i	$-0.476368\pi$
$977$	4.63681	0.148345	0.0741724	+	0.997245i	$0.476368\pi$
$978$	−6.56373	−0.209885
$979$	0	0
$980$	0	0
$981$	−1.58733	−0.0506797
$982$	8.90036	0.284022
$983$	3.07853	0.0981899	0.0490950	−	0.998794i	$-0.484366\pi$
$983$	3.07853	0.0981899	0.0490950	+	0.998794i	$0.484366\pi$
$984$	21.2129	0.676244
$985$	0	0
$986$	−29.9942	−0.955210
$987$	−47.0589	−1.49790
$988$	27.0006	0.859003
$989$	4.48838	0.142722
$990$	0	0
$991$	35.0952	1.11484	0.557418	−	0.830232i	$-0.311792\pi$
$991$	35.0952	1.11484	0.557418	+	0.830232i	$0.311792\pi$
$992$	−20.5801	−0.653420
$993$	−7.67417	−0.243533
$994$	−4.16822	−0.132208
$995$	0	0
$996$	−2.63599	−0.0835245
$997$	24.2164	0.766942	0.383471	−	0.923553i	$-0.374729\pi$
$997$	24.2164	0.766942	0.383471	+	0.923553i	$0.374729\pi$
$998$	−3.49040	−0.110487
$999$	6.32477	0.200107

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9075.2.a.dt.1.4		8
5.4	even	2		9075.2.a.dw.1.5		8
11.2	odd	10		825.2.n.n.301.2	yes	16
11.6	odd	10		825.2.n.n.751.2	yes	16
11.10	odd	2		9075.2.a.dv.1.5		8
55.2	even	20		825.2.bx.j.499.5		32
55.13	even	20		825.2.bx.j.499.4		32
55.17	even	20		825.2.bx.j.124.4		32
55.24	odd	10		825.2.n.m.301.3	✓	16
55.28	even	20		825.2.bx.j.124.5		32
55.39	odd	10		825.2.n.m.751.3	yes	16
55.54	odd	2		9075.2.a.du.1.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.2.n.m.301.3	✓	16	55.24	odd	10
825.2.n.m.751.3	yes	16	55.39	odd	10
825.2.n.n.301.2	yes	16	11.2	odd	10
825.2.n.n.751.2	yes	16	11.6	odd	10
825.2.bx.j.124.4		32	55.17	even	20
825.2.bx.j.124.5		32	55.28	even	20
825.2.bx.j.499.4		32	55.13	even	20
825.2.bx.j.499.5		32	55.2	even	20
9075.2.a.dt.1.4		8	1.1	even	1	trivial
9075.2.a.du.1.4		8	55.54	odd	2
9075.2.a.dv.1.5		8	11.10	odd	2
9075.2.a.dw.1.5		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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9075.a (trivial)

\(f(q)\)	\(=\)	\(q-0.684625 q^{2} -1.00000 q^{3} -1.53129 q^{4} +0.684625 q^{6} -4.22715 q^{7} +2.41761 q^{8} +1.00000 q^{9} +1.53129 q^{12} -6.27371 q^{13} +2.89401 q^{14} +1.40742 q^{16} +6.19454 q^{17} -0.684625 q^{18} +2.81055 q^{19} +4.22715 q^{21} +0.456579 q^{23} -2.41761 q^{24} +4.29514 q^{26} -1.00000 q^{27} +6.47298 q^{28} +7.07254 q^{29} +3.54905 q^{31} -5.79877 q^{32} -4.24094 q^{34} -1.53129 q^{36} -6.32477 q^{37} -1.92418 q^{38} +6.27371 q^{39} -8.77435 q^{41} -2.89401 q^{42} +9.83046 q^{43} -0.312586 q^{46} -11.1325 q^{47} -1.40742 q^{48} +10.8688 q^{49} -6.19454 q^{51} +9.60686 q^{52} +1.89080 q^{53} +0.684625 q^{54} -10.2196 q^{56} -2.81055 q^{57} -4.84204 q^{58} -8.57869 q^{59} +4.19759 q^{61} -2.42977 q^{62} -4.22715 q^{63} +1.15515 q^{64} -10.0915 q^{67} -9.48562 q^{68} -0.456579 q^{69} -1.44029 q^{71} +2.41761 q^{72} -1.40901 q^{73} +4.33010 q^{74} -4.30377 q^{76} -4.29514 q^{78} -7.65396 q^{79} +1.00000 q^{81} +6.00714 q^{82} -1.72142 q^{83} -6.47298 q^{84} -6.73018 q^{86} -7.07254 q^{87} -12.6663 q^{89} +26.5199 q^{91} -0.699154 q^{92} -3.54905 q^{93} +7.62162 q^{94} +5.79877 q^{96} -3.64121 q^{97} -7.44105 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{2} - 8 q^{3} + 7 q^{4} + q^{6} - 8 q^{7} - 3 q^{8} + 8 q^{9} - 7 q^{12} - 7 q^{13} - 8 q^{14} - 7 q^{16} + q^{17} - q^{18} + 6 q^{19} + 8 q^{21} - 7 q^{23} + 3 q^{24} - q^{26} - 8 q^{27} - 11 q^{28}+ \cdots + 27 q^{98}+O(q^{100}) \) 8 * q - q^2 - 8 * q^3 + 7 * q^4 + q^6 - 8 * q^7 - 3 * q^8 + 8 * q^9 - 7 * q^12 - 7 * q^13 - 8 * q^14 - 7 * q^16 + q^17 - q^18 + 6 * q^19 + 8 * q^21 - 7 * q^23 + 3 * q^24 - q^26 - 8 * q^27 - 11 * q^28 + 28 * q^29 + 10 * q^31 - 12 * q^32 - 8 * q^34 + 7 * q^36 - 14 * q^37 - 19 * q^38 + 7 * q^39 + 14 * q^41 + 8 * q^42 - 11 * q^43 - 5 * q^47 + 7 * q^48 + 2 * q^49 - q^51 - q^52 + 5 * q^53 + q^54 - 4 * q^56 - 6 * q^57 + 3 * q^58 - 2 * q^59 + 36 * q^61 + 17 * q^62 - 8 * q^63 - 23 * q^64 - 6 * q^67 + 3 * q^68 + 7 * q^69 - 17 * q^71 - 3 * q^72 - 32 * q^73 + 47 * q^74 + 5 * q^76 + q^78 + 41 * q^79 + 8 * q^81 + 6 * q^83 + 11 * q^84 - 9 * q^86 - 28 * q^87 + 21 * q^89 - 9 * q^91 - 69 * q^92 - 10 * q^93 + 51 * q^94 + 12 * q^96 + 4 * q^97 + 27 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more