LMFDB - Embedded newform 1089.3.b.f.485.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,3,Mod(485,1089)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1089, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1089.485");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1089 = 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1089.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$29.6731007888$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.1105425137664.10
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 14x^{6} + 71x^{4} - 80x^{2} + 121 $ x^8 - 14x^6 + 71x^4 - 80*x^2 + 121
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			485.6
Root			$-0.979091 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1089.485
Dual form			1089.3.b.f.485.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.38464i q^{2} +2.08276 q^{4} +0.765629i q^{5} -8.89285 q^{7} +8.42246i q^{8} +O(q^{10})$	$q+1.38464i q^{2} +2.08276 q^{4} +0.765629i q^{5} -8.89285 q^{7} +8.42246i q^{8} -1.06012 q^{10} -10.8510 q^{13} -12.3134i q^{14} -3.33105 q^{16} -20.8842i q^{17} -9.79091 q^{19} +1.59462i q^{20} -25.0413i q^{23} +24.4138 q^{25} -15.0248i q^{26} -18.5217 q^{28} -13.8464i q^{29} -18.9172 q^{31} +29.0775i q^{32} +28.9172 q^{34} -6.80862i q^{35} +15.4138 q^{37} -13.5569i q^{38} -6.44848 q^{40} -62.8819i q^{41} +74.8971 q^{43} +34.6733 q^{46} -67.4678i q^{47} +30.0828 q^{49} +33.8044i q^{50} -22.6001 q^{52} +96.3372i q^{53} -74.8996i q^{56} +19.1724 q^{58} -8.48528i q^{59} -93.4188 q^{61} -26.1936i q^{62} -53.5862 q^{64} -8.30786i q^{65} +80.8276 q^{67} -43.4969i q^{68} +9.42751 q^{70} -91.2656i q^{71} -15.9896 q^{73} +21.3426i q^{74} -20.3921 q^{76} -83.6279 q^{79} -2.55035i q^{80} +87.0691 q^{82} -98.8827i q^{83} +15.9896 q^{85} +103.706i q^{86} -113.722i q^{89} +96.4966 q^{91} -52.1552i q^{92} +93.4188 q^{94} -7.49620i q^{95} -146.241 q^{97} +41.6539i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 32 q^{4}+O(q^{10}) $ 8 * q - 32 * q^4	$ 8 q - 32 q^{4} + 168 q^{16} - 48 q^{25} - 200 q^{31} + 280 q^{34} - 120 q^{37} + 192 q^{49} + 640 q^{58} - 672 q^{64} + 160 q^{67} + 1000 q^{70} - 520 q^{82} + 480 q^{91} - 440 q^{97}+O(q^{100}) $ 8 * q - 32 * q^4 + 168 * q^16 - 48 * q^25 - 200 * q^31 + 280 * q^34 - 120 * q^37 + 192 * q^49 + 640 * q^58 - 672 * q^64 + 160 * q^67 + 1000 * q^70 - 520 * q^82 + 480 * q^91 - 440 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times$.

$n$	$244$	$848$
$\chi(n)$	$1$	$-1$

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 $

$2$	1.38464i	0.692322i	0.938175	+	0.346161i	$0.112515\pi$
$2$	1.38464i	0.692322i	−0.938175	+	0.346161i	$0.887485\pi$
$3$	0	0
$3$	0	0
$4$	2.08276	0.520691
$5$	0.765629i	0.153126i	0.997065	+	0.0765629i	$0.0243946\pi$
$5$	0.765629i	0.153126i	−0.997065	+	0.0765629i	$0.975605\pi$
$6$	0	0
$7$	−8.89285	−1.27041	−0.635203	−	0.772345i	$-0.719084\pi$
$7$	−8.89285	−1.27041	−0.635203	+	0.772345i	$0.719084\pi$
$8$	8.42246i	1.05281i
$9$	0	0
$10$	−1.06012	−0.106012
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−10.8510	−0.834695	−0.417347	−	0.908747i	$-0.637040\pi$
$13$	−10.8510	−0.834695	−0.417347	+	0.908747i	$0.637040\pi$
$14$	− 12.3134i	− 0.879530i
$15$	0	0
$16$	−3.33105	−0.208191
$17$	− 20.8842i	− 1.22849i	−0.789117	−	0.614243i	$-0.789462\pi$
$17$	− 20.8842i	− 1.22849i	0.789117	−	0.614243i	$-0.210538\pi$
$18$	0	0
$19$	−9.79091	−0.515311	−0.257655	−	0.966237i	$-0.582950\pi$
$19$	−9.79091	−0.515311	−0.257655	+	0.966237i	$0.582950\pi$
$20$	1.59462i	0.0797311i
$21$	0	0
$22$	0	0
$23$	− 25.0413i	− 1.08875i	−0.838841	−	0.544377i	$-0.816766\pi$
$23$	− 25.0413i	− 1.08875i	0.838841	−	0.544377i	$-0.183234\pi$
$24$	0	0
$25$	24.4138	0.976553
$26$	− 15.0248i	− 0.577877i
$27$	0	0
$28$	−18.5217	−0.661489
$29$	− 13.8464i	− 0.477463i	−0.971086	−	0.238732i	$-0.923268\pi$
$29$	− 13.8464i	− 0.477463i	0.971086	−	0.238732i	$-0.0767316\pi$
$30$	0	0
$31$	−18.9172	−0.610233	−0.305117	−	0.952315i	$-0.598695\pi$
$31$	−18.9172	−0.610233	−0.305117	+	0.952315i	$0.598695\pi$
$32$	29.0775i	0.908672i
$33$	0	0
$34$	28.9172	0.850507
$35$	− 6.80862i	− 0.194532i
$36$	0	0
$37$	15.4138	0.416590	0.208295	−	0.978066i	$-0.433209\pi$
$37$	15.4138	0.416590	0.208295	+	0.978066i	$0.433209\pi$
$38$	− 13.5569i	− 0.356761i
$39$	0	0
$40$	−6.44848	−0.161212
$41$	− 62.8819i	− 1.53371i	−0.641823	−	0.766853i	$-0.721822\pi$
$41$	− 62.8819i	− 1.53371i	0.641823	−	0.766853i	$-0.278178\pi$
$42$	0	0
$43$	74.8971	1.74179	0.870896	−	0.491467i	$-0.163539\pi$
$43$	74.8971	1.74179	0.870896	+	0.491467i	$0.163539\pi$
$44$	0	0
$45$	0	0
$46$	34.6733	0.753768
$47$	− 67.4678i	− 1.43548i	−0.696309	−	0.717742i	$-0.745176\pi$
$47$	− 67.4678i	− 1.43548i	0.696309	−	0.717742i	$-0.254824\pi$
$48$	0	0
$49$	30.0828	0.613934
$50$	33.8044i	0.676089i
$51$	0	0
$52$	−22.6001	−0.434618
$53$	96.3372i	1.81768i	0.417141	+	0.908842i	$0.363032\pi$
$53$	96.3372i	1.81768i	−0.417141	+	0.908842i	$0.636968\pi$
$54$	0	0
$55$	0	0
$56$	− 74.8996i	− 1.33749i
$57$	0	0
$58$	19.1724	0.330558
$59$	− 8.48528i	− 0.143818i	−0.997411	−	0.0719092i	$-0.977091\pi$
$59$	− 8.48528i	− 0.143818i	0.997411	−	0.0719092i	$-0.0229092\pi$
$60$	0	0
$61$	−93.4188	−1.53146	−0.765728	−	0.643165i	$-0.777621\pi$
$61$	−93.4188	−1.53146	−0.765728	+	0.643165i	$0.777621\pi$
$62$	− 26.1936i	− 0.422478i
$63$	0	0
$64$	−53.5862	−0.837284
$65$	− 8.30786i	− 0.127813i
$66$	0	0
$67$	80.8276	1.20638	0.603191	−	0.797597i	$-0.293896\pi$
$67$	80.8276	1.20638	0.603191	+	0.797597i	$0.293896\pi$
$68$	− 43.4969i	− 0.639661i
$69$	0	0
$70$	9.42751	0.134679
$71$	− 91.2656i	− 1.28543i	−0.766105	−	0.642716i	$-0.777808\pi$
$71$	− 91.2656i	− 1.28543i	0.766105	−	0.642716i	$-0.222192\pi$
$72$	0	0
$73$	−15.9896	−0.219035	−0.109518	−	0.993985i	$-0.534931\pi$
$73$	−15.9896	−0.219035	−0.109518	+	0.993985i	$0.534931\pi$
$74$	21.3426i	0.288414i
$75$	0	0
$76$	−20.3921	−0.268318
$77$	0	0
$78$	0	0
$79$	−83.6279	−1.05858	−0.529290	−	0.848441i	$-0.677542\pi$
$79$	−83.6279	−1.05858	−0.529290	+	0.848441i	$0.677542\pi$
$80$	− 2.55035i	− 0.0318793i
$81$	0	0
$82$	87.0691	1.06182
$83$	− 98.8827i	− 1.19136i	−0.803223	−	0.595679i	$-0.796883\pi$
$83$	− 98.8827i	− 1.19136i	0.803223	−	0.595679i	$-0.203117\pi$
$84$	0	0
$85$	15.9896	0.188113
$86$	103.706i	1.20588i
$87$	0	0
$88$	0	0
$89$	− 113.722i	− 1.27778i	−0.769299	−	0.638889i	$-0.779394\pi$
$89$	− 113.722i	− 1.27778i	0.769299	−	0.638889i	$-0.220606\pi$
$90$	0	0
$91$	96.4966	1.06040
$92$	− 52.1552i	− 0.566904i
$93$	0	0
$94$	93.4188	0.993817
$95$	− 7.49620i	− 0.0789074i
$96$	0	0
$97$	−146.241	−1.50764	−0.753822	−	0.657079i	$-0.771792\pi$
$97$	−146.241	−1.50764	−0.753822	+	0.657079i	$0.771792\pi$
$98$	41.6539i	0.425040i
$99$	0	0
$100$	50.8482	0.508482
$101$	62.8819i	0.622593i	0.950313	+	0.311297i	$0.100763\pi$
$101$	62.8819i	0.622593i	−0.950313	+	0.311297i	$0.899237\pi$
$102$	0	0
$103$	−45.4138	−0.440911	−0.220455	−	0.975397i	$-0.570754\pi$
$103$	−45.4138	−0.440911	−0.220455	+	0.975397i	$0.570754\pi$
$104$	− 91.3923i	− 0.878773i
$105$	0	0
$106$	−133.393	−1.25842
$107$	117.456i	1.09772i	0.835914	+	0.548860i	$0.184938\pi$
$107$	117.456i	1.09772i	−0.835914	+	0.548860i	$0.815062\pi$
$108$	0	0
$109$	−113.001	−1.03670	−0.518351	−	0.855168i	$-0.673454\pi$
$109$	−113.001	−1.03670	−0.518351	+	0.855168i	$0.673454\pi$
$110$	0	0
$111$	0	0
$112$	29.6225	0.264487
$113$	− 148.492i	− 1.31409i	−0.753850	−	0.657046i	$-0.771806\pi$
$113$	− 148.492i	− 1.31409i	0.753850	−	0.657046i	$-0.228194\pi$
$114$	0	0
$115$	19.1724	0.166716
$116$	− 28.8388i	− 0.248611i
$117$	0	0
$118$	11.7491	0.0995686
$119$	185.720i	1.56068i
$120$	0	0
$121$	0	0
$122$	− 129.352i	− 1.06026i
$123$	0	0
$124$	−39.4001	−0.317743
$125$	37.8326i	0.302661i
$126$	0	0
$127$	89.9886	0.708572	0.354286	−	0.935137i	$-0.384724\pi$
$127$	89.9886	0.708572	0.354286	+	0.935137i	$0.384724\pi$
$128$	42.1123i	0.329002i
$129$	0	0
$130$	11.5034	0.0884879
$131$	− 62.8819i	− 0.480015i	−0.970771	−	0.240007i	$-0.922850\pi$
$131$	− 62.8819i	− 0.480015i	0.970771	−	0.240007i	$-0.0771499\pi$
$132$	0	0
$133$	87.0691	0.654655
$134$	111.917i	0.835205i
$135$	0	0
$136$	175.897	1.29336
$137$	− 28.8695i	− 0.210726i	−0.994434	−	0.105363i	$-0.966400\pi$
$137$	− 28.8695i	− 0.210726i	0.994434	−	0.105363i	$-0.0336005\pi$
$138$	0	0
$139$	246.393	1.77261	0.886307	−	0.463098i	$-0.153262\pi$
$139$	246.393	1.77261	0.886307	+	0.463098i	$0.153262\pi$
$140$	− 14.1807i	− 0.101291i
$141$	0	0
$142$	126.370	0.889932
$143$	0	0
$144$	0	0
$145$	10.6012	0.0731119
$146$	− 22.1399i	− 0.151643i
$147$	0	0
$148$	32.1033	0.216914
$149$	216.339i	1.45194i	0.687728	+	0.725969i	$0.258608\pi$
$149$	216.339i	1.45194i	−0.687728	+	0.725969i	$0.741392\pi$
$150$	0	0
$151$	−74.6473	−0.494353	−0.247176	−	0.968970i	$-0.579503\pi$
$151$	−74.6473	−0.494353	−0.247176	+	0.968970i	$0.579503\pi$
$152$	− 82.4635i	− 0.542523i
$153$	0	0
$154$	0	0
$155$	− 14.4836i	− 0.0934425i
$156$	0	0
$157$	151.655	0.965957	0.482979	−	0.875632i	$-0.339555\pi$
$157$	151.655	0.965957	0.482979	+	0.875632i	$0.339555\pi$
$158$	− 115.795i	− 0.732878i
$159$	0	0
$160$	−22.2626	−0.139141
$161$	222.689i	1.38316i
$162$	0	0
$163$	−89.1724	−0.547070	−0.273535	−	0.961862i	$-0.588193\pi$
$163$	−89.1724	−0.547070	−0.273535	+	0.961862i	$0.588193\pi$
$164$	− 130.968i	− 0.798586i
$165$	0	0
$166$	136.917	0.824803
$167$	− 249.570i	− 1.49443i	−0.664581	−	0.747216i	$-0.731390\pi$
$167$	− 249.570i	− 1.49443i	0.664581	−	0.747216i	$-0.268610\pi$
$168$	0	0
$169$	−51.2551	−0.303285
$170$	22.1399i	0.130235i
$171$	0	0
$172$	155.993	0.906935
$173$	− 166.845i	− 0.964421i	−0.876055	−	0.482210i	$-0.839834\pi$
$173$	− 166.845i	− 0.964421i	0.876055	−	0.482210i	$-0.160166\pi$
$174$	0	0
$175$	−217.108	−1.24062
$176$	0	0
$177$	0	0
$178$	157.465	0.884634
$179$	− 198.800i	− 1.11061i	−0.831646	−	0.555306i	$-0.812601\pi$
$179$	− 198.800i	− 1.11061i	0.831646	−	0.555306i	$-0.187399\pi$
$180$	0	0
$181$	−47.2620	−0.261116	−0.130558	−	0.991441i	$-0.541677\pi$
$181$	−47.2620	−0.261116	−0.130558	+	0.991441i	$0.541677\pi$
$182$	133.613i	0.734139i
$183$	0	0
$184$	210.910	1.14625
$185$	11.8013i	0.0637906i
$186$	0	0
$187$	0	0
$188$	− 140.519i	− 0.747443i
$189$	0	0
$190$	10.3796	0.0546293
$191$	94.5816i	0.495192i	0.968863	+	0.247596i	$0.0796405\pi$
$191$	94.5816i	0.495192i	−0.968863	+	0.247596i	$0.920360\pi$
$192$	0	0
$193$	62.9859	0.326352	0.163176	−	0.986597i	$-0.447826\pi$
$193$	62.9859	0.326352	0.163176	+	0.986597i	$0.447826\pi$
$194$	− 202.492i	− 1.04377i
$195$	0	0
$196$	62.6553	0.319670
$197$	35.6475i	0.180952i	0.995899	+	0.0904758i	$0.0288388\pi$
$197$	35.6475i	0.180952i	−0.995899	+	0.0904758i	$0.971161\pi$
$198$	0	0
$199$	−284.138	−1.42783	−0.713915	−	0.700232i	$-0.753080\pi$
$199$	−284.138	−1.42783	−0.713915	+	0.700232i	$0.753080\pi$
$200$	205.624i	1.02812i
$201$	0	0
$202$	−87.0691	−0.431035
$203$	123.134i	0.606573i
$204$	0	0
$205$	48.1442	0.234850
$206$	− 62.8819i	− 0.305252i
$207$	0	0
$208$	36.1453	0.173776
$209$	0	0
$210$	0	0
$211$	89.7388	0.425302	0.212651	−	0.977128i	$-0.431790\pi$
$211$	89.7388	0.425302	0.212651	+	0.977128i	$0.431790\pi$
$212$	200.648i	0.946451i
$213$	0	0
$214$	−162.635	−0.759975
$215$	57.3434i	0.266713i
$216$	0	0
$217$	168.228	0.775245
$218$	− 156.466i	− 0.717732i
$219$	0	0
$220$	0	0
$221$	226.616i	1.02541i
$222$	0	0
$223$	17.5171	0.0785521	0.0392761	−	0.999228i	$-0.487495\pi$
$223$	17.5171	0.0785521	0.0392761	+	0.999228i	$0.487495\pi$
$224$	− 258.582i	− 1.15438i
$225$	0	0
$226$	205.609	0.909775
$227$	− 28.3804i	− 0.125024i	−0.998044	−	0.0625120i	$-0.980089\pi$
$227$	− 28.3804i	− 0.125024i	0.998044	−	0.0625120i	$-0.0199112\pi$
$228$	0	0
$229$	190.000	0.829694	0.414847	−	0.909891i	$-0.363835\pi$
$229$	190.000	0.829694	0.414847	+	0.909891i	$0.363835\pi$
$230$	26.5469i	0.115421i
$231$	0	0
$232$	116.621	0.502677
$233$	− 179.192i	− 0.769064i	−0.923112	−	0.384532i	$-0.874363\pi$
$233$	− 179.192i	− 0.769064i	0.923112	−	0.384532i	$-0.125637\pi$
$234$	0	0
$235$	51.6553	0.219810
$236$	− 17.6728i	− 0.0748849i
$237$	0	0
$238$	−257.157	−1.08049
$239$	397.488i	1.66313i	0.555427	+	0.831565i	$0.312555\pi$
$239$	397.488i	1.66313i	−0.555427	+	0.831565i	$0.687445\pi$
$240$	0	0
$241$	−247.204	−1.02574	−0.512871	−	0.858466i	$-0.671418\pi$
$241$	−247.204	−1.02574	−0.512871	+	0.858466i	$0.671418\pi$
$242$	0	0
$243$	0	0
$244$	−194.569	−0.797415
$245$	23.0322i	0.0940091i
$246$	0	0
$247$	106.241	0.430127
$248$	− 159.330i	− 0.642458i
$249$	0	0
$250$	−52.3847	−0.209539
$251$	− 76.3675i	− 0.304253i	−0.988361	−	0.152127i	$-0.951388\pi$
$251$	− 76.3675i	− 0.304253i	0.988361	−	0.152127i	$-0.0486121\pi$
$252$	0	0
$253$	0	0
$254$	124.602i	0.490560i
$255$	0	0
$256$	−272.655	−1.06506
$257$	391.250i	1.52237i	0.648534	+	0.761186i	$0.275383\pi$
$257$	391.250i	1.52237i	−0.648534	+	0.761186i	$0.724617\pi$
$258$	0	0
$259$	−137.073	−0.529238
$260$	− 17.3033i	− 0.0665512i
$261$	0	0
$262$	87.0691	0.332325
$263$	− 271.266i	− 1.03143i	−0.856760	−	0.515715i	$-0.827526\pi$
$263$	− 271.266i	− 1.03143i	0.856760	−	0.515715i	$-0.172474\pi$
$264$	0	0
$265$	−73.7586	−0.278334
$266$	120.560i	0.453232i
$267$	0	0
$268$	168.345	0.628152
$269$	282.184i	1.04901i	0.851407	+	0.524506i	$0.175750\pi$
$269$	282.184i	1.04901i	−0.851407	+	0.524506i	$0.824250\pi$
$270$	0	0
$271$	−74.6473	−0.275451	−0.137726	−	0.990470i	$-0.543979\pi$
$271$	−74.6473	−0.275451	−0.137726	+	0.990470i	$0.543979\pi$
$272$	69.5665i	0.255759i
$273$	0	0
$274$	39.9740	0.145890
$275$	0	0
$276$	0	0
$277$	−265.887	−0.959882	−0.479941	−	0.877301i	$-0.659342\pi$
$277$	−265.887	−0.959882	−0.479941	+	0.877301i	$0.659342\pi$
$278$	341.167i	1.22722i
$279$	0	0
$280$	57.3453	0.204805
$281$	− 368.649i	− 1.31192i	−0.754796	−	0.655960i	$-0.772264\pi$
$281$	− 368.649i	− 1.31192i	0.754796	−	0.655960i	$-0.227736\pi$
$282$	0	0
$283$	−310.527	−1.09727	−0.548635	−	0.836062i	$-0.684852\pi$
$283$	−310.527	−1.09727	−0.548635	+	0.836062i	$0.684852\pi$
$284$	− 190.085i	− 0.669312i
$285$	0	0
$286$	0	0
$287$	559.200i	1.94843i
$288$	0	0
$289$	−147.152	−0.509176
$290$	14.6789i	0.0506170i
$291$	0	0
$292$	−33.3025	−0.114050
$293$	396.677i	1.35385i	0.736054	+	0.676923i	$0.236687\pi$
$293$	396.677i	1.35385i	−0.736054	+	0.676923i	$0.763313\pi$
$294$	0	0
$295$	6.49658	0.0220223
$296$	129.822i	0.438588i
$297$	0	0
$298$	−299.552	−1.00521
$299$	271.724i	0.908777i
$300$	0	0
$301$	−666.049	−2.21279
$302$	− 103.360i	− 0.342251i
$303$	0	0
$304$	32.6140	0.107283
$305$	− 71.5241i	− 0.234505i
$306$	0	0
$307$	370.893	1.20812	0.604061	−	0.796938i	$-0.293548\pi$
$307$	370.893	1.20812	0.604061	+	0.796938i	$0.293548\pi$
$308$	0	0
$309$	0	0
$310$	20.0546	0.0646922
$311$	155.003i	0.498402i	0.968452	+	0.249201i	$0.0801679\pi$
$311$	155.003i	0.498402i	−0.968452	+	0.249201i	$0.919832\pi$
$312$	0	0
$313$	−390.828	−1.24865	−0.624325	−	0.781164i	$-0.714626\pi$
$313$	−390.828	−1.24865	−0.624325	+	0.781164i	$0.714626\pi$
$314$	209.988i	0.668753i
$315$	0	0
$316$	−174.177	−0.551193
$317$	− 184.701i	− 0.582654i	−0.956624	−	0.291327i	$-0.905903\pi$
$317$	− 184.701i	− 0.582654i	0.956624	−	0.291327i	$-0.0940968\pi$
$318$	0	0
$319$	0	0
$320$	− 41.0271i	− 0.128210i
$321$	0	0
$322$	−308.345	−0.957592
$323$	204.476i	0.633052i
$324$	0	0
$325$	−264.915	−0.815123
$326$	− 123.472i	− 0.378748i
$327$	0	0
$328$	529.620	1.61470
$329$	599.981i	1.82365i
$330$	0	0
$331$	−567.642	−1.71493	−0.857465	−	0.514543i	$-0.827962\pi$
$331$	−567.642	−1.71493	−0.857465	+	0.514543i	$0.827962\pi$
$332$	− 205.949i	− 0.620329i
$333$	0	0
$334$	345.566	1.03463
$335$	61.8840i	0.184728i
$336$	0	0
$337$	−82.5678	−0.245008	−0.122504	−	0.992468i	$-0.539092\pi$
$337$	−82.5678	−0.245008	−0.122504	+	0.992468i	$0.539092\pi$
$338$	− 70.9701i	− 0.209971i
$339$	0	0
$340$	33.3025	0.0979485
$341$	0	0
$342$	0	0
$343$	168.228	0.490461
$344$	630.818i	1.83377i
$345$	0	0
$346$	231.021	0.667689
$347$	− 132.572i	− 0.382053i	−0.981585	−	0.191027i	$-0.938818\pi$
$347$	− 132.572i	− 0.382053i	0.981585	−	0.191027i	$-0.0611817\pi$
$348$	0	0
$349$	63.2357	0.181191	0.0905956	−	0.995888i	$-0.471123\pi$
$349$	63.2357	0.181191	0.0905956	+	0.995888i	$0.471123\pi$
$350$	− 300.618i	− 0.858908i
$351$	0	0
$352$	0	0
$353$	487.270i	1.38037i	0.723634	+	0.690184i	$0.242470\pi$
$353$	487.270i	1.38037i	−0.723634	+	0.690184i	$0.757530\pi$
$354$	0	0
$355$	69.8756	0.196833
$356$	− 236.857i	− 0.665327i
$357$	0	0
$358$	275.266	0.768901
$359$	− 626.527i	− 1.74520i	−0.488434	−	0.872601i	$-0.662432\pi$
$359$	− 626.527i	− 1.74520i	0.488434	−	0.872601i	$-0.337568\pi$
$360$	0	0
$361$	−265.138	−0.734455
$362$	− 65.4410i	− 0.180776i
$363$	0	0
$364$	200.979	0.552141
$365$	− 12.2421i	− 0.0335400i
$366$	0	0
$367$	−194.966	−0.531242	−0.265621	−	0.964078i	$-0.585577\pi$
$367$	−194.966	−0.531242	−0.265621	+	0.964078i	$0.585577\pi$
$368$	83.4140i	0.226668i
$369$	0	0
$370$	−16.3405	−0.0441636
$371$	− 856.713i	− 2.30920i
$372$	0	0
$373$	602.195	1.61446	0.807232	−	0.590234i	$-0.200965\pi$
$373$	602.195	1.61446	0.807232	+	0.590234i	$0.200965\pi$
$374$	0	0
$375$	0	0
$376$	568.244	1.51129
$377$	150.248i	0.398536i
$378$	0	0
$379$	−334.138	−0.881631	−0.440815	−	0.897598i	$-0.645311\pi$
$379$	−334.138	−0.881631	−0.440815	+	0.897598i	$0.645311\pi$
$380$	− 15.6128i	− 0.0410863i
$381$	0	0
$382$	−130.962	−0.342832
$383$	− 310.225i	− 0.809987i	−0.914320	−	0.404993i	$-0.867274\pi$
$383$	− 310.225i	− 0.809987i	0.914320	−	0.404993i	$-0.132726\pi$
$384$	0	0
$385$	0	0
$386$	87.2131i	0.225941i
$387$	0	0
$388$	−304.586	−0.785016
$389$	413.609i	1.06326i	0.846976	+	0.531631i	$0.178421\pi$
$389$	413.609i	1.06326i	−0.846976	+	0.531631i	$0.821579\pi$
$390$	0	0
$391$	−522.970	−1.33752
$392$	253.371i	0.646354i
$393$	0	0
$394$	−49.3590	−0.125277
$395$	− 64.0279i	− 0.162096i
$396$	0	0
$397$	−375.414	−0.945627	−0.472813	−	0.881163i	$-0.656762\pi$
$397$	−375.414	−0.945627	−0.472813	+	0.881163i	$0.656762\pi$
$398$	− 393.430i	− 0.988518i
$399$	0	0
$400$	−81.3236	−0.203309
$401$	415.750i	1.03678i	0.855144	+	0.518391i	$0.173469\pi$
$401$	415.750i	1.03678i	−0.855144	+	0.518391i	$0.826531\pi$
$402$	0	0
$403$	205.272	0.509359
$404$	130.968i	0.324179i
$405$	0	0
$406$	−170.497	−0.419943
$407$	0	0
$408$	0	0
$409$	−492.787	−1.20486	−0.602429	−	0.798173i	$-0.705800\pi$
$409$	−492.787	−1.20486	−0.602429	+	0.798173i	$0.705800\pi$
$410$	66.6626i	0.162592i
$411$	0	0
$412$	−94.5862	−0.229578
$413$	75.4583i	0.182708i
$414$	0	0
$415$	75.7074	0.182427
$416$	− 315.521i	− 0.758464i
$417$	0	0
$418$	0	0
$419$	703.323i	1.67857i	0.543688	+	0.839287i	$0.317027\pi$
$419$	703.323i	1.67857i	−0.543688	+	0.839287i	$0.682973\pi$
$420$	0	0
$421$	−220.255	−0.523171	−0.261586	−	0.965180i	$-0.584245\pi$
$421$	−220.255	−0.523171	−0.261586	+	0.965180i	$0.584245\pi$
$422$	124.256i	0.294446i
$423$	0	0
$424$	−811.396	−1.91367
$425$	− 509.864i	− 1.19968i
$426$	0	0
$427$	830.759	1.94557
$428$	244.633i	0.571572i
$429$	0	0
$430$	−79.4001	−0.184651
$431$	− 244.032i	− 0.566198i	−0.959091	−	0.283099i	$-0.908637\pi$
$431$	− 244.032i	− 0.566198i	0.959091	−	0.283099i	$-0.0913626\pi$
$432$	0	0
$433$	226.621	0.523374	0.261687	−	0.965153i	$-0.415721\pi$
$433$	226.621	0.523374	0.261687	+	0.965153i	$0.415721\pi$
$434$	232.936i	0.536719i
$435$	0	0
$436$	−235.353	−0.539801
$437$	245.178i	0.561047i
$438$	0	0
$439$	−192.138	−0.437672	−0.218836	−	0.975762i	$-0.570226\pi$
$439$	−192.138	−0.437672	−0.218836	+	0.975762i	$0.570226\pi$
$440$	0	0
$441$	0	0
$442$	−313.782	−0.709914
$443$	− 535.597i	− 1.20902i	−0.796597	−	0.604511i	$-0.793368\pi$
$443$	− 535.597i	− 1.20902i	0.796597	−	0.604511i	$-0.206632\pi$
$444$	0	0
$445$	87.0691	0.195661
$446$	24.2550i	0.0543833i
$447$	0	0
$448$	476.534	1.06369
$449$	83.0972i	0.185072i	0.995709	+	0.0925358i	$0.0294973\pi$
$449$	83.0972i	0.185072i	−0.995709	+	0.0925358i	$0.970503\pi$
$450$	0	0
$451$	0	0
$452$	− 309.274i	− 0.684236i
$453$	0	0
$454$	39.2968	0.0865568
$455$	73.8806i	0.162375i
$456$	0	0
$457$	430.449	0.941901	0.470951	−	0.882160i	$-0.343911\pi$
$457$	430.449	0.941901	0.470951	+	0.882160i	$0.343911\pi$
$458$	263.082i	0.574415i
$459$	0	0
$460$	39.9315	0.0868076
$461$	− 713.044i	− 1.54673i	−0.633959	−	0.773367i	$-0.718571\pi$
$461$	− 713.044i	− 1.54673i	0.633959	−	0.773367i	$-0.281429\pi$
$462$	0	0
$463$	25.4138	0.0548894	0.0274447	−	0.999623i	$-0.491263\pi$
$463$	25.4138	0.0548894	0.0274447	+	0.999623i	$0.491263\pi$
$464$	46.1232i	0.0994034i
$465$	0	0
$466$	248.117	0.532440
$467$	− 242.757i	− 0.519823i	−0.965632	−	0.259911i	$-0.916307\pi$
$467$	− 242.757i	− 0.519823i	0.965632	−	0.259911i	$-0.0836933\pi$
$468$	0	0
$469$	−718.788	−1.53260
$470$	71.5241i	0.152179i
$471$	0	0
$472$	71.4669	0.151413
$473$	0	0
$474$	0	0
$475$	−239.033	−0.503228
$476$	386.812i	0.812629i
$477$	0	0
$478$	−550.380	−1.15142
$479$	− 160.953i	− 0.336019i	−0.985785	−	0.168009i	$-0.946266\pi$
$479$	− 160.953i	− 0.336019i	0.985785	−	0.168009i	$-0.0537339\pi$
$480$	0	0
$481$	−167.256	−0.347725
$482$	− 342.289i	− 0.710143i
$483$	0	0
$484$	0	0
$485$	− 111.967i	− 0.230859i
$486$	0	0
$487$	244.138	0.501310	0.250655	−	0.968076i	$-0.419354\pi$
$487$	244.138	0.501310	0.250655	+	0.968076i	$0.419354\pi$
$488$	− 786.816i	− 1.61233i
$489$	0	0
$490$	−31.8914	−0.0650845
$491$	207.697i	0.423007i	0.977377	+	0.211504i	$0.0678360\pi$
$491$	207.697i	0.423007i	−0.977377	+	0.211504i	$0.932164\pi$
$492$	0	0
$493$	−289.172	−0.586557
$494$	147.107i	0.297786i
$495$	0	0
$496$	63.0143	0.127045
$497$	811.611i	1.63302i
$498$	0	0
$499$	−548.000	−1.09820	−0.549098	−	0.835758i	$-0.685029\pi$
$499$	−548.000	−1.09820	−0.549098	+	0.835758i	$0.685029\pi$
$500$	78.7964i	0.157593i
$501$	0	0
$502$	105.742	0.210641
$503$	229.498i	0.456258i	0.973631	+	0.228129i	$0.0732607\pi$
$503$	229.498i	0.456258i	−0.973631	+	0.228129i	$0.926739\pi$
$504$	0	0
$505$	−48.1442	−0.0953351
$506$	0	0
$507$	0	0
$508$	187.425	0.368947
$509$	6.51047i	0.0127907i	0.999980	+	0.00639535i	$0.00203572\pi$
$509$	6.51047i	0.0127907i	−0.999980	+	0.00639535i	$0.997964\pi$
$510$	0	0
$511$	142.193	0.278264
$512$	− 209.081i	− 0.408362i
$513$	0	0
$514$	−541.741	−1.05397
$515$	− 34.7701i	− 0.0675148i
$516$	0	0
$517$	0	0
$518$	− 189.797i	− 0.366403i
$519$	0	0
$520$	69.9726	0.134563
$521$	478.716i	0.918841i	0.888219	+	0.459420i	$0.151943\pi$
$521$	478.716i	0.918841i	−0.888219	+	0.459420i	$0.848057\pi$
$522$	0	0
$523$	781.936	1.49510	0.747549	−	0.664207i	$-0.231231\pi$
$523$	781.936	1.49510	0.747549	+	0.664207i	$0.231231\pi$
$524$	− 130.968i	− 0.249939i
$525$	0	0
$526$	375.607	0.714081
$527$	395.072i	0.749663i
$528$	0	0
$529$	−98.0691	−0.185386
$530$	− 102.129i	− 0.192697i
$531$	0	0
$532$	181.344	0.340873
$533$	682.334i	1.28018i
$534$	0	0
$535$	−89.9277	−0.168089
$536$	680.767i	1.27009i
$537$	0	0
$538$	−390.725	−0.726254
$539$	0	0
$540$	0	0
$541$	646.572	1.19514	0.597571	−	0.801816i	$-0.296133\pi$
$541$	646.572	1.19514	0.597571	+	0.801816i	$0.296133\pi$
$542$	− 103.360i	− 0.190701i
$543$	0	0
$544$	607.262	1.11629
$545$	− 86.5165i	− 0.158746i
$546$	0	0
$547$	269.743	0.493131	0.246566	−	0.969126i	$-0.420698\pi$
$547$	269.743	0.493131	0.246566	+	0.969126i	$0.420698\pi$
$548$	− 60.1283i	− 0.109723i
$549$	0	0
$550$	0	0
$551$	135.569i	0.246042i
$552$	0	0
$553$	743.690	1.34483
$554$	− 368.159i	− 0.664547i
$555$	0	0
$556$	513.179	0.922983
$557$	302.521i	0.543125i	0.962421	+	0.271563i	$0.0875404\pi$
$557$	302.521i	0.543125i	−0.962421	+	0.271563i	$0.912460\pi$
$558$	0	0
$559$	−812.711	−1.45387
$560$	22.6799i	0.0404997i
$561$	0	0
$562$	510.448	0.908271
$563$	− 9.45383i	− 0.0167919i	−0.999965	−	0.00839594i	$-0.997327\pi$
$563$	− 9.45383i	− 0.0167919i	0.999965	−	0.00839594i	$-0.00267254\pi$
$564$	0	0
$565$	113.690	0.201221
$566$	− 429.969i	− 0.759663i
$567$	0	0
$568$	768.681	1.35331
$569$	91.7208i	0.161196i	0.996747	+	0.0805982i	$0.0256831\pi$
$569$	91.7208i	0.161196i	−0.996747	+	0.0805982i	$0.974317\pi$
$570$	0	0
$571$	38.3533	0.0671687	0.0335843	−	0.999436i	$-0.489308\pi$
$571$	38.3533	0.0671687	0.0335843	+	0.999436i	$0.489308\pi$
$572$	0	0
$573$	0	0
$574$	−774.292	−1.34894
$575$	− 611.355i	− 1.06323i
$576$	0	0
$577$	−486.241	−0.842706	−0.421353	−	0.906897i	$-0.638445\pi$
$577$	−486.241	−0.842706	−0.421353	+	0.906897i	$0.638445\pi$
$578$	− 203.753i	− 0.352514i
$579$	0	0
$580$	22.0798	0.0380687
$581$	879.349i	1.51351i
$582$	0	0
$583$	0	0
$584$	− 134.672i	− 0.230602i
$585$	0	0
$586$	−549.256	−0.937296
$587$	− 23.6025i	− 0.0402087i	−0.999798	−	0.0201044i	$-0.993600\pi$
$587$	− 23.6025i	− 0.0402087i	0.999798	−	0.0201044i	$-0.00639985\pi$
$588$	0	0
$589$	185.217	0.314460
$590$	8.99544i	0.0152465i
$591$	0	0
$592$	−51.3442	−0.0867300
$593$	58.4893i	0.0986329i	0.998783	+	0.0493165i	$0.0157043\pi$
$593$	58.4893i	0.0986329i	−0.998783	+	0.0493165i	$0.984296\pi$
$594$	0	0
$595$	−142.193	−0.238980
$596$	450.582i	0.756010i
$597$	0	0
$598$	−376.241	−0.629166
$599$	− 570.460i	− 0.952353i	−0.879350	−	0.476177i	$-0.842022\pi$
$599$	− 570.460i	− 0.952353i	0.879350	−	0.476177i	$-0.157978\pi$
$600$	0	0
$601$	1017.38	1.69281	0.846404	−	0.532542i	$-0.178763\pi$
$601$	1017.38	1.69281	0.846404	+	0.532542i	$0.178763\pi$
$602$	− 922.240i	− 1.53196i
$603$	0	0
$604$	−155.473	−0.257405
$605$	0	0
$606$	0	0
$607$	481.125	0.792628	0.396314	−	0.918115i	$-0.370289\pi$
$607$	481.125	0.792628	0.396314	+	0.918115i	$0.370289\pi$
$608$	− 284.695i	− 0.468249i
$609$	0	0
$610$	99.0354	0.162353
$611$	732.095i	1.19819i
$612$	0	0
$613$	−179.727	−0.293193	−0.146597	−	0.989196i	$-0.546832\pi$
$613$	−179.727	−0.293193	−0.146597	+	0.989196i	$0.546832\pi$
$614$	513.555i	0.836409i
$615$	0	0
$616$	0	0
$617$	− 972.784i	− 1.57664i	−0.615268	−	0.788318i	$-0.710952\pi$
$617$	− 972.784i	− 1.57664i	0.615268	−	0.788318i	$-0.289048\pi$
$618$	0	0
$619$	236.497	0.382062	0.191031	−	0.981584i	$-0.438817\pi$
$619$	236.497	0.382062	0.191031	+	0.981584i	$0.438817\pi$
$620$	− 30.1659i	− 0.0486546i
$621$	0	0
$622$	−214.624	−0.345054
$623$	1011.32i	1.62330i
$624$	0	0
$625$	581.380	0.930207
$626$	− 541.157i	− 0.864468i
$627$	0	0
$628$	315.862	0.502965
$629$	− 321.906i	− 0.511774i
$630$	0	0
$631$	612.931	0.971364	0.485682	−	0.874135i	$-0.338571\pi$
$631$	612.931	0.971364	0.485682	+	0.874135i	$0.338571\pi$
$632$	− 704.352i	− 1.11448i
$633$	0	0
$634$	255.746	0.403384
$635$	68.8979i	0.108501i
$636$	0	0
$637$	−326.429	−0.512447
$638$	0	0
$639$	0	0
$640$	−32.2424	−0.0503787
$641$	100.390i	0.156614i	0.996929	+	0.0783072i	$0.0249515\pi$
$641$	100.390i	0.156614i	−0.996929	+	0.0783072i	$0.975049\pi$
$642$	0	0
$643$	1129.10	1.75599	0.877997	−	0.478667i	$-0.158880\pi$
$643$	1129.10	1.75599	0.877997	+	0.478667i	$0.158880\pi$
$644$	463.808i	0.720199i
$645$	0	0
$646$	−283.126	−0.438276
$647$	607.210i	0.938500i	0.883065	+	0.469250i	$0.155476\pi$
$647$	607.210i	0.938500i	−0.883065	+	0.469250i	$0.844524\pi$
$648$	0	0
$649$	0	0
$650$	− 366.813i	− 0.564327i
$651$	0	0
$652$	−185.725	−0.284854
$653$	219.471i	0.336097i	0.985779	+	0.168049i	$0.0537466\pi$
$653$	219.471i	0.336097i	−0.985779	+	0.168049i	$0.946253\pi$
$654$	0	0
$655$	48.1442	0.0735026
$656$	209.463i	0.319303i
$657$	0	0
$658$	−830.759	−1.26255
$659$	− 304.622i	− 0.462248i	−0.972924	−	0.231124i	$-0.925760\pi$
$659$	− 304.622i	− 0.462248i	0.972924	−	0.231124i	$-0.0742403\pi$
$660$	0	0
$661$	1084.14	1.64015	0.820074	−	0.572257i	$-0.193932\pi$
$661$	1084.14	1.64015	0.820074	+	0.572257i	$0.193932\pi$
$662$	− 785.981i	− 1.18728i
$663$	0	0
$664$	832.835	1.25427
$665$	66.6626i	0.100244i
$666$	0	0
$667$	−346.733	−0.519840
$668$	− 519.795i	− 0.778137i
$669$	0	0
$670$	−85.6872	−0.127891
$671$	0	0
$672$	0	0
$673$	1135.22	1.68680	0.843402	−	0.537283i	$-0.180549\pi$
$673$	1135.22	1.68680	0.843402	+	0.537283i	$0.180549\pi$
$674$	− 114.327i	− 0.169624i
$675$	0	0
$676$	−106.752	−0.157918
$677$	837.328i	1.23682i	0.785855	+	0.618410i	$0.212223\pi$
$677$	837.328i	1.23682i	−0.785855	+	0.618410i	$0.787777\pi$
$678$	0	0
$679$	1300.50	1.91532
$680$	134.672i	0.198046i
$681$	0	0
$682$	0	0
$683$	− 50.7163i	− 0.0742553i	−0.999311	−	0.0371276i	$-0.988179\pi$
$683$	− 50.7163i	− 0.0742553i	0.999311	−	0.0371276i	$-0.0118208\pi$
$684$	0	0
$685$	22.1033	0.0322676
$686$	232.936i	0.339557i
$687$	0	0
$688$	−249.486	−0.362625
$689$	− 1045.36i	− 1.51721i
$690$	0	0
$691$	−1090.28	−1.57782	−0.788912	−	0.614506i	$-0.789355\pi$
$691$	−1090.28	−1.57782	−0.788912	+	0.614506i	$0.789355\pi$
$692$	− 347.498i	− 0.502165i
$693$	0	0
$694$	183.566	0.264504
$695$	188.646i	0.271433i
$696$	0	0
$697$	−1313.24	−1.88413
$698$	87.5590i	0.125443i
$699$	0	0
$700$	−452.185	−0.645979
$701$	919.595i	1.31183i	0.754834	+	0.655916i	$0.227718\pi$
$701$	919.595i	1.31183i	−0.754834	+	0.655916i	$0.772282\pi$
$702$	0	0
$703$	−150.915	−0.214673
$704$	0	0
$705$	0	0
$706$	−674.695	−0.955659
$707$	− 559.200i	− 0.790947i
$708$	0	0
$709$	−185.159	−0.261155	−0.130577	−	0.991438i	$-0.541683\pi$
$709$	−185.159	−0.261155	−0.130577	+	0.991438i	$0.541683\pi$
$710$	96.7528i	0.136271i
$711$	0	0
$712$	957.821	1.34525
$713$	473.713i	0.664394i
$714$	0	0
$715$	0	0
$716$	− 414.052i	− 0.578285i
$717$	0	0
$718$	867.517	1.20824
$719$	− 891.760i	− 1.24028i	−0.784492	−	0.620139i	$-0.787076\pi$
$719$	− 891.760i	− 1.24028i	0.784492	−	0.620139i	$-0.212924\pi$
$720$	0	0
$721$	403.858	0.560136
$722$	− 367.122i	− 0.508479i
$723$	0	0
$724$	−98.4355	−0.135961
$725$	− 338.044i	− 0.466268i
$726$	0	0
$727$	−410.000	−0.563961	−0.281981	−	0.959420i	$-0.590991\pi$
$727$	−410.000	−0.563961	−0.281981	+	0.959420i	$0.590991\pi$
$728$	812.738i	1.11640i
$729$	0	0
$730$	16.9509	0.0232204
$731$	− 1564.17i	− 2.13977i
$732$	0	0
$733$	196.716	0.268371	0.134186	−	0.990956i	$-0.457158\pi$
$733$	196.716	0.268371	0.134186	+	0.990956i	$0.457158\pi$
$734$	− 269.958i	− 0.367790i
$735$	0	0
$736$	728.140	0.989321
$737$	0	0
$738$	0	0
$739$	−30.9934	−0.0419396	−0.0209698	−	0.999780i	$-0.506675\pi$
$739$	−30.9934	−0.0419396	−0.0209698	+	0.999780i	$0.506675\pi$
$740$	24.5792i	0.0332152i
$741$	0	0
$742$	1186.24	1.59871
$743$	712.356i	0.958757i	0.877608	+	0.479378i	$0.159138\pi$
$743$	712.356i	0.958757i	−0.877608	+	0.479378i	$0.840862\pi$
$744$	0	0
$745$	−165.635	−0.222329
$746$	833.825i	1.11773i
$747$	0	0
$748$	0	0
$749$	− 1044.52i	− 1.39455i
$750$	0	0
$751$	787.345	1.04840	0.524198	−	0.851596i	$-0.324365\pi$
$751$	787.345	1.04840	0.524198	+	0.851596i	$0.324365\pi$
$752$	224.738i	0.298854i
$753$	0	0
$754$	−208.040	−0.275915
$755$	− 57.1521i	− 0.0756982i
$756$	0	0
$757$	−1232.86	−1.62862	−0.814308	−	0.580433i	$-0.802883\pi$
$757$	−1232.86	−1.62862	−0.814308	+	0.580433i	$0.802883\pi$
$758$	− 462.662i	− 0.610372i
$759$	0	0
$760$	63.1364	0.0830742
$761$	− 763.225i	− 1.00292i	−0.865180	−	0.501462i	$-0.832796\pi$
$761$	− 763.225i	− 1.00292i	0.865180	−	0.501462i	$-0.167204\pi$
$762$	0	0
$763$	1004.90	1.31703
$764$	196.991i	0.257842i
$765$	0	0
$766$	429.551	0.560771
$767$	92.0740i	0.120044i
$768$	0	0
$769$	−1106.31	−1.43863	−0.719314	−	0.694685i	$-0.755544\pi$
$769$	−1106.31	−1.43863	−0.719314	+	0.694685i	$0.755544\pi$
$770$	0	0
$771$	0	0
$772$	131.185	0.169928
$773$	− 908.828i	− 1.17572i	−0.808964	−	0.587858i	$-0.799972\pi$
$773$	− 908.828i	− 1.17572i	0.808964	−	0.587858i	$-0.200028\pi$
$774$	0	0
$775$	−461.842	−0.595925
$776$	− 1231.71i	− 1.58726i
$777$	0	0
$778$	−572.700	−0.736119
$779$	615.671i	0.790335i
$780$	0	0
$781$	0	0
$782$	− 724.127i	− 0.925993i
$783$	0	0
$784$	−100.207	−0.127815
$785$	116.112i	0.147913i
$786$	0	0
$787$	434.230	0.551754	0.275877	−	0.961193i	$-0.411032\pi$
$787$	434.230	0.551754	0.275877	+	0.961193i	$0.411032\pi$
$788$	74.2452i	0.0942198i
$789$	0	0
$790$	88.6558	0.112223
$791$	1320.52i	1.66943i
$792$	0	0
$793$	1013.69	1.27830
$794$	− 519.814i	− 0.654678i
$795$	0	0
$796$	−591.792	−0.743458
$797$	− 423.314i	− 0.531134i	−0.964092	−	0.265567i	$-0.914441\pi$
$797$	− 423.314i	− 0.531134i	0.964092	−	0.265567i	$-0.0855591\pi$
$798$	0	0
$799$	−1409.01	−1.76347
$800$	709.893i	0.887366i
$801$	0	0
$802$	−575.665	−0.717787
$803$	0	0
$804$	0	0
$805$	−170.497	−0.211798
$806$	284.228i	0.352640i
$807$	0	0
$808$	−529.620	−0.655471
$809$	− 1192.46i	− 1.47400i	−0.675893	−	0.736999i	$-0.736242\pi$
$809$	− 1192.46i	− 1.47400i	0.675893	−	0.736999i	$-0.263758\pi$
$810$	0	0
$811$	−244.773	−0.301816	−0.150908	−	0.988548i	$-0.548220\pi$
$811$	−244.773	−0.301816	−0.150908	+	0.988548i	$0.548220\pi$
$812$	256.459i	0.315837i
$813$	0	0
$814$	0	0
$815$	− 68.2729i	− 0.0837705i
$816$	0	0
$817$	−733.311	−0.897565
$818$	− 682.334i	− 0.834149i
$819$	0	0
$820$	100.273	0.122284
$821$	− 229.039i	− 0.278976i	−0.990224	−	0.139488i	$-0.955454\pi$
$821$	− 229.039i	− 0.278976i	0.990224	−	0.139488i	$-0.0445456\pi$
$822$	0	0
$823$	1353.31	1.64436	0.822181	−	0.569226i	$-0.192757\pi$
$823$	1353.31	1.64436	0.822181	+	0.569226i	$0.192757\pi$
$824$	− 382.496i	− 0.464194i
$825$	0	0
$826$	−104.483	−0.126493
$827$	804.077i	0.972282i	0.873880	+	0.486141i	$0.161596\pi$
$827$	804.077i	0.972282i	−0.873880	+	0.486141i	$0.838404\pi$
$828$	0	0
$829$	463.503	0.559111	0.279556	−	0.960129i	$-0.409813\pi$
$829$	463.503	0.559111	0.279556	+	0.960129i	$0.409813\pi$
$830$	104.828i	0.126299i
$831$	0	0
$832$	581.465	0.698877
$833$	− 628.256i	− 0.754209i
$834$	0	0
$835$	191.078	0.228836
$836$	0	0
$837$	0	0
$838$	−973.851	−1.16211
$839$	− 583.412i	− 0.695366i	−0.937612	−	0.347683i	$-0.886969\pi$
$839$	− 583.412i	− 0.695366i	0.937612	−	0.347683i	$-0.113031\pi$
$840$	0	0
$841$	649.276	0.772029
$842$	− 304.975i	− 0.362203i
$843$	0	0
$844$	186.905	0.221451
$845$	− 39.2424i	− 0.0464407i
$846$	0	0
$847$	0	0
$848$	− 320.904i	− 0.378425i
$849$	0	0
$850$	705.980	0.830565
$851$	− 385.983i	− 0.453564i
$852$	0	0
$853$	−989.713	−1.16027	−0.580136	−	0.814519i	$-0.697001\pi$
$853$	−989.713	−1.16027	−0.580136	+	0.814519i	$0.697001\pi$
$854$	1150.31i	1.34696i
$855$	0	0
$856$	−989.268	−1.15569
$857$	615.785i	0.718535i	0.933235	+	0.359268i	$0.116973\pi$
$857$	615.785i	0.718535i	−0.933235	+	0.359268i	$0.883027\pi$
$858$	0	0
$859$	892.221	1.03867	0.519337	−	0.854569i	$-0.326179\pi$
$859$	892.221	1.03867	0.519337	+	0.854569i	$0.326179\pi$
$860$	119.433i	0.138875i
$861$	0	0
$862$	337.897	0.391992
$863$	440.382i	0.510292i	0.966903	+	0.255146i	$0.0821235\pi$
$863$	440.382i	0.510292i	−0.966903	+	0.255146i	$0.917876\pi$
$864$	0	0
$865$	127.741	0.147678
$866$	313.789i	0.362343i
$867$	0	0
$868$	350.379	0.403663
$869$	0	0
$870$	0	0
$871$	−877.063	−1.00696
$872$	− 951.743i	− 1.09145i
$873$	0	0
$874$	−339.483	−0.388425
$875$	− 336.440i	− 0.384503i
$876$	0	0
$877$	−1472.87	−1.67944	−0.839721	−	0.543018i	$-0.817281\pi$
$877$	−1472.87	−1.67944	−0.839721	+	0.543018i	$0.817281\pi$
$878$	− 266.043i	− 0.303010i
$879$	0	0
$880$	0	0
$881$	− 581.242i	− 0.659752i	−0.944024	−	0.329876i	$-0.892993\pi$
$881$	− 581.242i	− 0.659752i	0.944024	−	0.329876i	$-0.107007\pi$
$882$	0	0
$883$	441.724	0.500253	0.250127	−	0.968213i	$-0.419528\pi$
$883$	441.724	0.500253	0.250127	+	0.968213i	$0.419528\pi$
$884$	471.987i	0.533921i
$885$	0	0
$886$	741.611	0.837033
$887$	336.440i	0.379301i	0.981852	+	0.189650i	$0.0607355\pi$
$887$	336.440i	0.379301i	−0.981852	+	0.189650i	$0.939265\pi$
$888$	0	0
$889$	−800.255	−0.900175
$890$	120.560i	0.135460i
$891$	0	0
$892$	36.4840	0.0409014
$893$	660.571i	0.739721i
$894$	0	0
$895$	152.207	0.170063
$896$	− 374.498i	− 0.417967i
$897$	0	0
$898$	−115.060	−0.128129
$899$	261.936i	0.291364i
$900$	0	0
$901$	2011.93	2.23300
$902$	0	0
$903$	0	0
$904$	1250.67	1.38349
$905$	− 36.1851i	− 0.0399836i
$906$	0	0
$907$	1221.59	1.34684	0.673422	−	0.739259i	$-0.264824\pi$
$907$	1221.59	1.34684	0.673422	+	0.739259i	$0.264824\pi$
$908$	− 59.1097i	− 0.0650988i
$909$	0	0
$910$	−102.298	−0.112416
$911$	220.514i	0.242058i	0.992649	+	0.121029i	$0.0386193\pi$
$911$	220.514i	0.242058i	−0.992649	+	0.121029i	$0.961381\pi$
$912$	0	0
$913$	0	0
$914$	596.018i	0.652099i
$915$	0	0
$916$	395.725	0.432014
$917$	559.200i	0.609814i
$918$	0	0
$919$	85.2485	0.0927623	0.0463811	−	0.998924i	$-0.485231\pi$
$919$	85.2485	0.0927623	0.0463811	+	0.998924i	$0.485231\pi$
$920$	161.479i	0.175520i
$921$	0	0
$922$	987.312	1.07084
$923$	990.326i	1.07294i
$924$	0	0
$925$	376.310	0.406822
$926$	35.1891i	0.0380012i
$927$	0	0
$928$	402.620	0.433858
$929$	908.218i	0.977630i	0.872388	+	0.488815i	$0.162571\pi$
$929$	908.218i	0.977630i	−0.872388	+	0.488815i	$0.837429\pi$
$930$	0	0
$931$	−294.538	−0.316367
$932$	− 373.214i	− 0.400445i
$933$	0	0
$934$	336.132	0.359885
$935$	0	0
$936$	0	0
$937$	548.912	0.585819	0.292909	−	0.956140i	$-0.405377\pi$
$937$	548.912	0.585819	0.292909	+	0.956140i	$0.405377\pi$
$938$	− 995.265i	− 1.06105i
$939$	0	0
$940$	107.586	0.114453
$941$	902.836i	0.959443i	0.877421	+	0.479721i	$0.159262\pi$
$941$	902.836i	0.959443i	−0.877421	+	0.479721i	$0.840738\pi$
$942$	0	0
$943$	−1574.65	−1.66983
$944$	28.2649i	0.0299416i
$945$	0	0
$946$	0	0
$947$	315.809i	0.333483i	0.986001	+	0.166742i	$0.0533246\pi$
$947$	315.809i	0.333483i	−0.986001	+	0.166742i	$0.946675\pi$
$948$	0	0
$949$	173.503	0.182828
$950$	− 330.976i	− 0.348396i
$951$	0	0
$952$	−1564.22	−1.64309
$953$	− 1332.76i	− 1.39849i	−0.714881	−	0.699246i	$-0.753519\pi$
$953$	− 1332.76i	− 1.39849i	0.714881	−	0.699246i	$-0.246481\pi$
$954$	0	0
$955$	−72.4144	−0.0758266
$956$	827.874i	0.865977i
$957$	0	0
$958$	222.862	0.232633
$959$	256.732i	0.267708i
$960$	0	0
$961$	−603.138	−0.627615
$962$	− 231.590i	− 0.240738i
$963$	0	0
$964$	−514.867	−0.534094
$965$	48.2238i	0.0499729i
$966$	0	0
$967$	244.523	0.252868	0.126434	−	0.991975i	$-0.459647\pi$
$967$	244.523	0.252868	0.126434	+	0.991975i	$0.459647\pi$
$968$	0	0
$969$	0	0
$970$	155.034	0.159829
$971$	1014.99i	1.04530i	0.852547	+	0.522650i	$0.175057\pi$
$971$	1014.99i	1.04530i	−0.852547	+	0.522650i	$0.824943\pi$
$972$	0	0
$973$	−2191.14	−2.25194
$974$	338.044i	0.347068i
$975$	0	0
$976$	311.183	0.318835
$977$	− 40.6707i	− 0.0416282i	−0.999783	−	0.0208141i	$-0.993374\pi$
$977$	− 40.6707i	− 0.0416282i	0.999783	−	0.0208141i	$-0.00662581\pi$
$978$	0	0
$979$	0	0
$980$	47.9707i	0.0489497i
$981$	0	0
$982$	−287.586	−0.292857
$983$	− 1713.81i	− 1.74345i	−0.489999	−	0.871723i	$-0.663003\pi$
$983$	− 1713.81i	− 1.74345i	0.489999	−	0.871723i	$-0.336997\pi$
$984$	0	0
$985$	−27.2927	−0.0277083
$986$	− 400.401i	− 0.406086i
$987$	0	0
$988$	221.276	0.223963
$989$	− 1875.52i	− 1.89638i
$990$	0	0
$991$	−1465.15	−1.47846	−0.739229	−	0.673454i	$-0.764810\pi$
$991$	−1465.15	−1.47846	−0.739229	+	0.673454i	$0.764810\pi$
$992$	− 550.066i	− 0.554502i
$993$	0	0
$994$	−1123.79	−1.13058
$995$	− 217.544i	− 0.218637i
$996$	0	0
$997$	−639.090	−0.641013	−0.320506	−	0.947246i	$-0.603853\pi$
$997$	−639.090	−0.641013	−0.320506	+	0.947246i	$0.603853\pi$
$998$	− 758.785i	− 0.760305i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.3.b.f.485.6	yes	8
3.2	odd	2	inner	1089.3.b.f.485.3	✓	8
11.10	odd	2	inner	1089.3.b.f.485.4	yes	8
33.32	even	2	inner	1089.3.b.f.485.5	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1089.3.b.f.485.3	✓	8	3.2	odd	2	inner
1089.3.b.f.485.4	yes	8	11.10	odd	2	inner
1089.3.b.f.485.5	yes	8	33.32	even	2	inner
1089.3.b.f.485.6	yes	8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1089 = 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1089.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.38464i q^{2} +2.08276 q^{4} +0.765629i q^{5} -8.89285 q^{7} +8.42246i q^{8} +O(q^{10})\)	\(q+1.38464i q^{2} +2.08276 q^{4} +0.765629i q^{5} -8.89285 q^{7} +8.42246i q^{8} -1.06012 q^{10} -10.8510 q^{13} -12.3134i q^{14} -3.33105 q^{16} -20.8842i q^{17} -9.79091 q^{19} +1.59462i q^{20} -25.0413i q^{23} +24.4138 q^{25} -15.0248i q^{26} -18.5217 q^{28} -13.8464i q^{29} -18.9172 q^{31} +29.0775i q^{32} +28.9172 q^{34} -6.80862i q^{35} +15.4138 q^{37} -13.5569i q^{38} -6.44848 q^{40} -62.8819i q^{41} +74.8971 q^{43} +34.6733 q^{46} -67.4678i q^{47} +30.0828 q^{49} +33.8044i q^{50} -22.6001 q^{52} +96.3372i q^{53} -74.8996i q^{56} +19.1724 q^{58} -8.48528i q^{59} -93.4188 q^{61} -26.1936i q^{62} -53.5862 q^{64} -8.30786i q^{65} +80.8276 q^{67} -43.4969i q^{68} +9.42751 q^{70} -91.2656i q^{71} -15.9896 q^{73} +21.3426i q^{74} -20.3921 q^{76} -83.6279 q^{79} -2.55035i q^{80} +87.0691 q^{82} -98.8827i q^{83} +15.9896 q^{85} +103.706i q^{86} -113.722i q^{89} +96.4966 q^{91} -52.1552i q^{92} +93.4188 q^{94} -7.49620i q^{95} -146.241 q^{97} +41.6539i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 32 q^{4}+O(q^{10}) \) 8 * q - 32 * q^4	\( 8 q - 32 q^{4} + 168 q^{16} - 48 q^{25} - 200 q^{31} + 280 q^{34} - 120 q^{37} + 192 q^{49} + 640 q^{58} - 672 q^{64} + 160 q^{67} + 1000 q^{70} - 520 q^{82} + 480 q^{91} - 440 q^{97}+O(q^{100}) \) 8 * q - 32 * q^4 + 168 * q^16 - 48 * q^25 - 200 * q^31 + 280 * q^34 - 120 * q^37 + 192 * q^49 + 640 * q^58 - 672 * q^64 + 160 * q^67 + 1000 * q^70 - 520 * q^82 + 480 * q^91 - 440 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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