LMFDB - Embedded newform 1805.2.b.h.1084.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1805,2,Mod(1084,1805)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1805, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1805.1084");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1805 = 5 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1805.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:	$14.4129975648$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.280944640000.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 16x^{6} + 80x^{4} + 128x^{2} + 19 $ x^8 + 16x^6 + 80x^4 + 128*x^2 + 19
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			1084.6
Root			$1.69217i$ of defining polynomial
Character	$\chi$	$=$	1805.1084
Dual form			1805.2.b.h.1084.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.69217i q^{2} +1.52380i q^{3} -0.863428 q^{4} -2.23607 q^{5} -2.57853 q^{6} +1.92327i q^{8} +0.678024 q^{9} +O(q^{10})$	$q+1.69217i q^{2} +1.52380i q^{3} -0.863428 q^{4} -2.23607 q^{5} -2.57853 q^{6} +1.92327i q^{8} +0.678024 q^{9} -3.78380i q^{10} -5.95117 q^{11} -1.31569i q^{12} -6.79166i q^{13} -3.40733i q^{15} -4.98135 q^{16} +1.14733i q^{18} +1.93068 q^{20} -10.0704i q^{22} -2.93068 q^{24} +5.00000 q^{25} +11.4926 q^{26} +5.60458i q^{27} +5.76577 q^{30} -4.58273i q^{32} -9.06841i q^{33} -0.585425 q^{36} -12.1390i q^{37} +10.3492 q^{39} -4.30056i q^{40} +5.13841 q^{44} -1.51611 q^{45} -7.59059i q^{48} +7.00000 q^{49} +8.46083i q^{50} +5.86411i q^{52} -6.04380i q^{53} -9.48389 q^{54} +13.3072 q^{55} +2.94198i q^{60} -15.5804 q^{61} -2.20795 q^{64} +15.1866i q^{65} +15.3453 q^{66} -3.36134i q^{67} +1.30402i q^{72} +20.5412 q^{74} +7.61901i q^{75} +17.5125i q^{78} +11.1386 q^{80} -6.50621 q^{81} -11.4457i q^{88} -2.56551i q^{90} +6.98318 q^{96} -2.99619i q^{97} +11.8452i q^{98} -4.03504 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{4} - 24 q^{9}+O(q^{10}) $ 8 * q - 16 * q^4 - 24 * q^9	$ 8 q - 16 q^{4} - 24 q^{9} + 32 q^{16} - 8 q^{24} + 40 q^{25} + 24 q^{26} - 40 q^{30} - 8 q^{36} - 72 q^{44} + 56 q^{49} - 88 q^{54} - 64 q^{64} + 104 q^{66} + 120 q^{80} + 72 q^{81} - 120 q^{96} - 32 q^{99}+O(q^{100}) $ 8 * q - 16 * q^4 - 24 * q^9 + 32 * q^16 - 8 * q^24 + 40 * q^25 + 24 * q^26 - 40 * q^30 - 8 * q^36 - 72 * q^44 + 56 * q^49 - 88 * q^54 - 64 * q^64 + 104 * q^66 + 120 * q^80 + 72 * q^81 - 120 * q^96 - 32 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1805\mathbb{Z}\right)^\times$.

$n$	$362$	$1446$
$\chi(n)$	$-1$	$1$

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 $

$2$	1.69217i	1.19654i	0.801294	+	0.598271i	$0.204145\pi$
$2$	1.69217i	1.19654i	−0.801294	+	0.598271i	$0.795855\pi$
$3$	1.52380i	0.879768i	0.898055	+	0.439884i	$0.144980\pi$
$3$	1.52380i	0.879768i	−0.898055	+	0.439884i	$0.855020\pi$
$4$	−0.863428	−0.431714
$5$	−2.23607	−1.00000
$5$	−2.23607	−1.00000
$6$	−2.57853	−1.05268
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	1.92327i	0.679978i
$9$	0.678024	0.226008
$10$	− 3.78380i	− 1.19654i
$11$	−5.95117	−1.79434	−0.897172	−	0.441680i	$-0.854382\pi$
$11$	−5.95117	−1.79434	−0.897172	+	0.441680i	$0.854382\pi$
$12$	− 1.31569i	− 0.379808i
$13$	− 6.79166i	− 1.88367i	−0.336079	−	0.941834i	$-0.609101\pi$
$13$	− 6.79166i	− 1.88367i	0.336079	−	0.941834i	$-0.390899\pi$
$14$	0	0
$15$	− 3.40733i	− 0.879768i
$16$	−4.98135	−1.24534
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	1.14733i	0.270428i
$19$	0	0
$19$	0	0
$20$	1.93068	0.431714
$21$	0	0
$22$	− 10.0704i	− 2.14701i
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	−2.93068	−0.598223
$25$	5.00000	1.00000
$26$	11.4926	2.25389
$27$	5.60458i	1.07860i
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	5.76577	1.05268
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	− 4.58273i	− 0.810120i
$33$	− 9.06841i	− 1.57861i
$34$	0	0
$35$	0	0
$36$	−0.585425	−0.0975709
$37$	− 12.1390i	− 1.99564i	−0.0659893	−	0.997820i	$-0.521020\pi$
$37$	− 12.1390i	− 1.99564i	0.0659893	−	0.997820i	$-0.478980\pi$
$38$	0	0
$39$	10.3492	1.65719
$40$	− 4.30056i	− 0.679978i
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	5.13841	0.774644
$45$	−1.51611	−0.226008
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	− 7.59059i	− 1.09561i
$49$	7.00000	1.00000
$50$	8.46083i	1.19654i
$51$	0	0
$52$	5.86411i	0.813206i
$53$	− 6.04380i	− 0.830179i	−0.909781	−	0.415090i	$-0.863750\pi$
$53$	− 6.04380i	− 0.830179i	0.909781	−	0.415090i	$-0.136250\pi$
$54$	−9.48389	−1.29059
$55$	13.3072	1.79434
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	2.94198i	0.379808i
$61$	−15.5804	−1.99486	−0.997430	−	0.0716414i	$-0.977176\pi$
$61$	−15.5804	−1.99486	−0.997430	+	0.0716414i	$0.977176\pi$
$62$	0	0
$63$	0	0
$64$	−2.20795	−0.275994
$65$	15.1866i	1.88367i
$66$	15.3453	1.88887
$67$	− 3.36134i	− 0.410653i	−0.978694	−	0.205326i	$-0.934174\pi$
$67$	− 3.36134i	− 0.410653i	0.978694	−	0.205326i	$-0.0658256\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	1.30402i	0.153681i
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	20.5412	2.38787
$75$	7.61901i	0.879768i
$76$	0	0
$77$	0	0
$78$	17.5125i	1.98290i
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	11.1386	1.24534
$81$	−6.50621	−0.722912
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	− 11.4457i	− 1.22012i
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	− 2.56551i	− 0.270428i
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	6.98318	0.712718
$97$	− 2.99619i	− 0.304217i	−0.988364	−	0.152109i	$-0.951394\pi$
$97$	− 2.99619i	− 0.304217i	0.988364	−	0.152109i	$-0.0486063\pi$
$98$	11.8452i	1.19654i
$99$	−4.03504	−0.405537
$100$	−4.31714	−0.431714
$101$	−0.868264	−0.0863955	−0.0431977	−	0.999067i	$-0.513755\pi$
$101$	−0.868264	−0.0863955	−0.0431977	+	0.999067i	$0.513755\pi$
$102$	0	0
$103$	− 16.9447i	− 1.66961i	−0.550548	−	0.834803i	$-0.685581\pi$
$103$	− 16.9447i	− 1.66961i	0.550548	−	0.834803i	$-0.314419\pi$
$104$	13.0622	1.28085
$105$	0	0
$106$	10.2271	0.993345
$107$	16.9906i	1.64255i	0.570534	+	0.821274i	$0.306736\pi$
$107$	16.9906i	1.64255i	−0.570534	+	0.821274i	$0.693264\pi$
$108$	− 4.83915i	− 0.465648i
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	22.5180i	2.14701i
$111$	18.4975	1.75570
$112$	0	0
$113$	− 13.6063i	− 1.27997i	−0.768386	−	0.639987i	$-0.778940\pi$
$113$	− 13.6063i	− 1.27997i	0.768386	−	0.639987i	$-0.221060\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 4.60491i	− 0.425724i
$118$	0	0
$119$	0	0
$120$	6.55321	0.598223
$121$	24.4164	2.21967
$122$	− 26.3646i	− 2.38694i
$123$	0	0
$124$	0	0
$125$	−11.1803	−1.00000
$126$	0	0
$127$	19.7066i	1.74868i	0.485315	+	0.874339i	$0.338705\pi$
$127$	19.7066i	1.74868i	−0.485315	+	0.874339i	$0.661295\pi$
$128$	− 12.9017i	− 1.14036i
$129$	0	0
$130$	−25.6983	−2.25389
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	7.82992i	0.681507i
$133$	0	0
$134$	5.68795	0.491364
$135$	− 12.5322i	− 1.07860i
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	−8.76093	−0.743092	−0.371546	−	0.928414i	$-0.621172\pi$
$139$	−8.76093	−0.743092	−0.371546	+	0.928414i	$0.621172\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	40.4183i	3.37995i
$144$	−3.37748	−0.281456
$145$	0	0
$146$	0	0
$147$	10.6666i	0.879768i
$148$	10.4812i	0.861546i
$149$	−22.9364	−1.87902	−0.939512	−	0.342516i	$-0.888721\pi$
$149$	−22.9364	−1.87902	−0.939512	+	0.342516i	$0.888721\pi$
$150$	−12.8926	−1.05268
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	−8.93575	−0.715432
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	9.20955	0.730365
$160$	10.2473i	0.810120i
$161$	0	0
$162$	− 11.0096i	− 0.864995i
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	20.2776i	1.57861i
$166$	0	0
$167$	− 25.8018i	− 1.99660i	−0.0582442	−	0.998302i	$-0.518550\pi$
$167$	− 25.8018i	− 1.99660i	0.0582442	−	0.998302i	$-0.481450\pi$
$168$	0	0
$169$	−33.1266	−2.54820
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 6.83765i	− 0.519857i	−0.965628	−	0.259928i	$-0.916301\pi$
$173$	− 6.83765i	− 0.519857i	0.965628	−	0.259928i	$-0.0836990\pi$
$174$	0	0
$175$	0	0
$176$	29.6448	2.23456
$177$	0	0
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	1.30905	0.0975709
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	− 23.7414i	− 1.75501i
$184$	0	0
$185$	27.1436i	1.99564i
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	8.94427	0.647185	0.323592	−	0.946197i	$-0.395109\pi$
$191$	8.94427	0.647185	0.323592	+	0.946197i	$0.395109\pi$
$192$	− 3.36448i	− 0.242810i
$193$	21.2818i	1.53190i	0.642901	+	0.765950i	$0.277731\pi$
$193$	21.2818i	1.53190i	−0.642901	+	0.765950i	$0.722269\pi$
$194$	5.07005	0.364009
$195$	−23.1414	−1.65719
$196$	−6.04400	−0.431714
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	− 6.82796i	− 0.485242i
$199$	−26.8328	−1.90213	−0.951064	−	0.308994i	$-0.900008\pi$
$199$	−26.8328	−1.90213	−0.951064	+	0.308994i	$0.900008\pi$
$200$	9.61635i	0.679978i
$201$	5.12202	0.361279
$202$	− 1.46925i	− 0.103376i
$203$	0	0
$204$	0	0
$205$	0	0
$206$	28.6732	1.99776
$207$	0	0
$208$	33.8316i	2.34580i
$209$	0	0
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	5.21838i	0.358400i
$213$	0	0
$214$	−28.7510	−1.96538
$215$	0	0
$216$	−10.7791	−0.733427
$217$	0	0
$218$	0	0
$219$	0	0
$220$	−11.4898	−0.774644
$221$	0	0
$222$	31.3008i	2.10077i
$223$	− 28.8494i	− 1.93190i	−0.258728	−	0.965950i	$-0.583303\pi$
$223$	− 28.8494i	− 1.93190i	0.258728	−	0.965950i	$-0.416697\pi$
$224$	0	0
$225$	3.39012	0.226008
$226$	23.0242	1.53154
$227$	3.45331i	0.229205i	0.993411	+	0.114602i	$0.0365594\pi$
$227$	3.45331i	0.229205i	−0.993411	+	0.114602i	$0.963441\pi$
$228$	0	0
$229$	−6.48779	−0.428725	−0.214362	−	0.976754i	$-0.568767\pi$
$229$	−6.48779	−0.428725	−0.214362	+	0.976754i	$0.568767\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	7.79228	0.509397
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	16.9731i	1.09561i
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	41.3166i	2.65593i
$243$	6.89957i	0.442608i
$244$	13.4525	0.861209
$245$	−15.6525	−1.00000
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	− 18.9190i	− 1.19654i
$251$	−17.8885	−1.12911	−0.564557	−	0.825394i	$-0.690953\pi$
$251$	−17.8885	−1.12911	−0.564557	+	0.825394i	$0.690953\pi$
$252$	0	0
$253$	0	0
$254$	−33.3469	−2.09237
$255$	0	0
$256$	17.4159	1.08849
$257$	− 3.09902i	− 0.193312i	−0.995318	−	0.0966558i	$-0.969185\pi$
$257$	− 3.09902i	− 0.193312i	0.995318	−	0.0966558i	$-0.0308146\pi$
$258$	0	0
$259$	0	0
$260$	− 13.1125i	− 0.813206i
$261$	0	0
$262$	20.3060i	1.25451i
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	17.4410	1.07342
$265$	13.5143i	0.830179i
$266$	0	0
$267$	0	0
$268$	2.90227i	0.177285i
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	21.2066	1.29059
$271$	22.3998	1.36069	0.680345	−	0.732892i	$-0.261830\pi$
$271$	22.3998	1.36069	0.680345	+	0.732892i	$0.261830\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−29.7558	−1.79434
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	− 14.8250i	− 0.889142i
$279$	0	0
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	0	0
$286$	−68.3945	−4.04425
$287$	0	0
$288$	− 3.10720i	− 0.183094i
$289$	17.0000	1.00000
$290$	0	0
$291$	4.56560	0.267640
$292$	0	0
$293$	− 24.3294i	− 1.42134i	−0.703525	−	0.710670i	$-0.748392\pi$
$293$	− 24.3294i	− 1.42134i	0.703525	−	0.710670i	$-0.251608\pi$
$294$	−18.0497	−1.05268
$295$	0	0
$296$	23.3466	1.35699
$297$	− 33.3538i	− 1.93539i
$298$	− 38.8122i	− 2.24833i
$299$	0	0
$300$	− 6.57847i	− 0.379808i
$301$	0	0
$302$	0	0
$303$	− 1.32306i	− 0.0760080i
$304$	0	0
$305$	34.8387	1.99486
$306$	0	0
$307$	− 23.8053i	− 1.35864i	−0.733842	−	0.679320i	$-0.762275\pi$
$307$	− 23.8053i	− 1.35864i	0.733842	−	0.679320i	$-0.237725\pi$
$308$	0	0
$309$	25.8203	1.46887
$310$	0	0
$311$	−16.1170	−0.913910	−0.456955	−	0.889490i	$-0.651060\pi$
$311$	−16.1170	−0.913910	−0.456955	+	0.889490i	$0.651060\pi$
$312$	19.9042i	1.12685i
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 33.9123i	− 1.90471i	−0.304999	−	0.952353i	$-0.598656\pi$
$317$	− 33.9123i	− 1.90471i	0.304999	−	0.952353i	$-0.401344\pi$
$318$	15.5841i	0.873913i
$319$	0	0
$320$	4.93712	0.275994
$321$	−25.8904	−1.44506
$322$	0	0
$323$	0	0
$324$	5.61764	0.312091
$325$	− 33.9583i	− 1.88367i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	−34.3130	−1.88887
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	0	0
$333$	− 8.23054i	− 0.451031i
$334$	43.6610	2.38902
$335$	7.51618i	0.410653i
$336$	0	0
$337$	− 27.0977i	− 1.47610i	−0.674744	−	0.738052i	$-0.735746\pi$
$337$	− 27.0977i	− 1.47610i	0.674744	−	0.738052i	$-0.264254\pi$
$338$	− 56.0558i	− 3.04903i
$339$	20.7333	1.12608
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	11.5704	0.622031
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	13.4164	0.718164	0.359082	−	0.933306i	$-0.383090\pi$
$349$	13.4164	0.718164	0.359082	+	0.933306i	$0.383090\pi$
$350$	0	0
$351$	38.0644	2.03173
$352$	27.2726i	1.45364i
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−28.0193	−1.47880	−0.739401	−	0.673265i	$-0.764891\pi$
$359$	−28.0193	−1.47880	−0.739401	+	0.673265i	$0.764891\pi$
$360$	− 2.91589i	− 0.153681i
$361$	0	0
$362$	0	0
$363$	37.2058i	1.95280i
$364$	0	0
$365$	0	0
$366$	40.1744	2.09995
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	0	0
$370$	−45.9316	−2.38787
$371$	0	0
$372$	0	0
$373$	6.14663i	0.318260i	0.987258	+	0.159130i	$0.0508689\pi$
$373$	6.14663i	0.318260i	−0.987258	+	0.159130i	$0.949131\pi$
$374$	0	0
$375$	− 17.0366i	− 0.879768i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	0	0
$381$	−30.0290	−1.53843
$382$	15.1352i	0.774384i
$383$	3.31535i	0.169407i	0.996406	+	0.0847033i	$0.0269942\pi$
$383$	3.31535i	0.169407i	−0.996406	+	0.0847033i	$0.973006\pi$
$384$	19.6596	1.00325
$385$	0	0
$386$	−36.0124	−1.83298
$387$	0	0
$388$	2.58699i	0.131335i
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	− 39.1591i	− 1.98290i
$391$	0	0
$392$	13.4629i	0.679978i
$393$	18.2856i	0.922388i
$394$	0	0
$395$	0	0
$396$	3.48396	0.175076
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	− 45.4056i	− 2.27598i
$399$	0	0
$400$	−24.9067	−1.24534
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	8.66731i	0.432286i
$403$	0	0
$404$	0.749683	0.0372981
$405$	14.5483	0.722912
$406$	0	0
$407$	72.2413i	3.58087i
$408$	0	0
$409$	0	0		−	1.00000i	$-0.5\pi$
$409$	0	0			1.00000i	$0.5\pi$
$410$	0	0
$411$	0	0
$412$	14.6305i	0.720793i
$413$	0	0
$414$	0	0
$415$	0	0
$416$	−31.1244	−1.52600
$417$	− 13.3499i	− 0.653749i
$418$	0	0
$419$	−36.0000	−1.75872	−0.879358	−	0.476162i	$-0.842028\pi$
$419$	−36.0000	−1.75872	−0.879358	+	0.476162i	$0.842028\pi$
$420$	0	0
$421$	0	0		−	1.00000i	$-0.5\pi$
$421$	0	0			1.00000i	$0.5\pi$
$422$	0	0
$423$	0	0
$424$	11.6238	0.564504
$425$	0	0
$426$	0	0
$427$	0	0
$428$	− 14.6702i	− 0.709111i
$429$	−61.5896	−2.97357
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	− 27.9184i	− 1.34322i
$433$	13.6523i	0.656088i	0.944662	+	0.328044i	$0.106389\pi$
$433$	13.6523i	0.656088i	−0.944662	+	0.328044i	$0.893611\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	25.5934i	1.22012i
$441$	4.74617	0.226008
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	−15.9712	−0.757961
$445$	0	0
$446$	48.8180	2.31160
$447$	− 34.9506i	− 1.65311i
$448$	0	0
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	5.73665i	0.270428i
$451$	0	0
$452$	11.7481i	0.552583i
$453$	0	0
$454$	−5.84358	−0.274253
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	− 10.9784i	− 0.512988i
$459$	0	0
$460$	0	0
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	3.97601i	0.183791i
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 4.09784i	− 0.187627i
$478$	− 40.6120i	− 1.85755i
$479$	−7.68770	−0.351260	−0.175630	−	0.984456i	$-0.556196\pi$
$479$	−7.68770	−0.351260	−0.175630	+	0.984456i	$0.556196\pi$
$480$	−15.6149	−0.712718
$481$	−82.4440	−3.75912
$482$	0	0
$483$	0	0
$484$	−21.0818	−0.958264
$485$	6.69969i	0.304217i
$486$	−11.6752	−0.529599
$487$	44.1113i	1.99887i	0.0335531	+	0.999437i	$0.489318\pi$
$487$	44.1113i	1.99887i	−0.0335531	+	0.999437i	$0.510682\pi$
$488$	− 29.9652i	− 1.35646i
$489$	0	0
$490$	− 26.4866i	− 1.19654i
$491$	35.7771	1.61460	0.807299	−	0.590143i	$-0.200929\pi$
$491$	35.7771	1.61460	0.807299	+	0.590143i	$0.200929\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	9.02262	0.405537
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−44.4679	−1.99066	−0.995329	−	0.0965389i	$-0.969223\pi$
$499$	−44.4679	−1.99066	−0.995329	+	0.0965389i	$0.969223\pi$
$500$	9.65342	0.431714
$501$	39.3169	1.75655
$502$	− 30.2704i	− 1.35103i
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	1.94150	0.0863955
$506$	0	0
$507$	− 50.4785i	− 2.24183i
$508$	− 17.0152i	− 0.754929i
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	3.66724i	0.162071i
$513$	0	0
$514$	5.24406	0.231306
$515$	37.8894i	1.66961i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	10.4192	0.457353
$520$	−29.2079	−1.28085
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	− 43.9846i	− 1.92331i	−0.274256	−	0.961657i	$-0.588432\pi$
$523$	− 43.9846i	− 1.92331i	0.274256	−	0.961657i	$-0.411568\pi$
$524$	−10.3611	−0.452628
$525$	0	0
$526$	0	0
$527$	0	0
$528$	45.1729i	1.96590i
$529$	23.0000	1.00000
$530$	−22.8685	−0.993345
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	− 37.9922i	− 1.64255i
$536$	6.46476	0.279235
$537$	0	0
$538$	0	0
$539$	−41.6582	−1.79434
$540$	10.8207i	0.465648i
$541$	−37.6485	−1.61864	−0.809318	−	0.587371i	$-0.800163\pi$
$541$	−37.6485	−1.61864	−0.809318	+	0.587371i	$0.800163\pi$
$542$	37.9042i	1.62812i
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	34.8418i	1.48973i	0.667216	+	0.744864i	$0.267486\pi$
$547$	34.8418i	1.48973i	−0.667216	+	0.744864i	$0.732514\pi$
$548$	0	0
$549$	−10.5639	−0.450855
$550$	− 50.3518i	− 2.14701i
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−41.3616	−1.75570
$556$	7.56443	0.320803
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	47.0322i	1.98217i	0.133223	+	0.991086i	$0.457467\pi$
$563$	47.0322i	1.98217i	−0.133223	+	0.991086i	$0.542533\pi$
$564$	0	0
$565$	30.4246i	1.27997i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	26.9461	1.12766	0.563829	−	0.825891i	$-0.309328\pi$
$571$	26.9461	1.12766	0.563829	+	0.825891i	$0.309328\pi$
$572$	− 34.8983i	− 1.45917i
$573$	13.6293i	0.569373i
$574$	0	0
$575$	0	0
$576$	−1.49704	−0.0623768
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	28.7668i	1.19654i
$579$	−32.4293	−1.34772
$580$	0	0
$581$	0	0
$582$	7.72576i	0.320243i
$583$	35.9677i	1.48963i
$584$	0	0
$585$	10.2969i	0.425724i
$586$	41.1695	1.70069
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	− 9.20986i	− 0.379808i
$589$	0	0
$590$	0	0
$591$	0	0
$592$	60.4686i	2.48525i
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	56.4402	2.31577
$595$	0	0
$596$	19.8039	0.811201
$597$	− 40.8879i	− 1.67343i
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	−14.6534	−0.598223
$601$	0	0		−	1.00000i	$-0.5\pi$
$601$	0	0			1.00000i	$0.5\pi$
$602$	0	0
$603$	− 2.27907i	− 0.0928109i
$604$	0	0
$605$	−54.5967	−2.21967
$606$	2.23884	0.0909468
$607$	− 23.6673i	− 0.960628i	−0.877097	−	0.480314i	$-0.840523\pi$
$607$	− 23.6673i	− 0.960628i	0.877097	−	0.480314i	$-0.159477\pi$
$608$	0	0
$609$	0	0
$610$	58.9530i	2.38694i
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	40.2825	1.62567
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	43.6923i	1.75756i
$619$	35.3754	1.42186	0.710928	−	0.703265i	$-0.248275\pi$
$619$	35.3754	1.42186	0.710928	+	0.703265i	$0.248275\pi$
$620$	0	0
$621$	0	0
$622$	− 27.2726i	− 1.09353i
$623$	0	0
$624$	−51.5527	−2.06376
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−49.0142	−1.95123	−0.975613	−	0.219499i	$-0.929558\pi$
$631$	−49.0142	−1.95123	−0.975613	+	0.219499i	$0.929558\pi$
$632$	0	0
$633$	0	0
$634$	57.3853	2.27906
$635$	− 44.0653i	− 1.74868i
$636$	−7.95179	−0.315309
$637$	− 47.5416i	− 1.88367i
$638$	0	0
$639$	0	0
$640$	28.8490i	1.14036i
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	− 43.8109i	− 1.72908i
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	− 12.5132i	− 0.491565i
$649$	0	0
$650$	57.4631	2.25389
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	−26.8328	−1.04844
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	− 17.5082i	− 0.681507i
$661$	0	0		−	1.00000i	$-0.5\pi$
$661$	0	0			1.00000i	$0.5\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	13.9275	0.539678
$667$	0	0
$668$	22.2780i	0.861962i
$669$	43.9608	1.69962
$670$	−12.7186	−0.491364
$671$	92.7214	3.57947
$672$	0	0
$673$	27.2356i	1.04986i	0.851147	+	0.524928i	$0.175908\pi$
$673$	27.2356i	1.04986i	−0.851147	+	0.524928i	$0.824092\pi$
$674$	45.8538	1.76622
$675$	28.0229i	1.07860i
$676$	28.6025	1.10010
$677$	− 12.2418i	− 0.470492i	−0.971936	−	0.235246i	$-0.924410\pi$
$677$	− 12.2418i	− 0.470492i	0.971936	−	0.235246i	$-0.0755896\pi$
$678$	35.0843i	1.34740i
$679$	0	0
$680$	0	0
$681$	−5.26217	−0.201647
$682$	0	0
$683$	28.7466i	1.09996i	0.835179	+	0.549979i	$0.185364\pi$
$683$	28.7466i	1.09996i	−0.835179	+	0.549979i	$0.814636\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 9.88611i	− 0.377178i
$688$	0	0
$689$	−41.0474	−1.56378
$690$	0	0
$691$	−0.331647	−0.0126165	−0.00630823	−	0.999980i	$-0.502008\pi$
$691$	−0.331647	−0.0126165	−0.00630823	+	0.999980i	$0.502008\pi$
$692$	5.90382i	0.224429i
$693$	0	0
$694$	0	0
$695$	19.5900	0.743092
$696$	0	0
$697$	0	0
$698$	22.7028i	0.859314i
$699$	0	0
$700$	0	0
$701$	21.1999	0.800709	0.400354	−	0.916360i	$-0.368887\pi$
$701$	21.1999	0.800709	0.400354	+	0.916360i	$0.368887\pi$
$702$	64.4114i	2.43105i
$703$	0	0
$704$	13.1399	0.495228
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−40.2492	−1.51159	−0.755796	−	0.654808i	$-0.772750\pi$
$709$	−40.2492	−1.51159	−0.755796	+	0.654808i	$0.772750\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	− 90.3781i	− 3.37995i
$716$	0	0
$717$	− 36.5713i	− 1.36578i
$718$	− 47.4134i	− 1.76945i
$719$	51.8240	1.93271	0.966354	−	0.257214i	$-0.0828047\pi$
$719$	51.8240	1.93271	0.966354	+	0.257214i	$0.0828047\pi$
$720$	7.55227	0.281456
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	−62.9584	−2.33661
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	−30.0322	−1.11230
$730$	0	0
$731$	0	0
$732$	20.4990i	0.757664i
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	− 23.8513i	− 0.879768i
$736$	0	0
$737$	20.0039i	0.736853i
$738$	0	0
$739$	53.6656	1.97412	0.987061	−	0.160345i	$-0.0512606\pi$
$739$	53.6656	1.97412	0.987061	+	0.160345i	$0.0512606\pi$
$740$	− 23.4366i	− 0.861546i
$741$	0	0
$742$	0	0
$743$	− 1.62663i	− 0.0596754i	−0.999555	−	0.0298377i	$-0.990501\pi$
$743$	− 1.62663i	− 0.0596754i	0.999555	−	0.0298377i	$-0.00949905\pi$
$744$	0	0
$745$	51.2874	1.87902
$746$	−10.4011	−0.380812
$747$	0	0
$748$	0	0
$749$	0	0
$750$	28.8288	1.05268
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	0	0
$753$	− 27.2586i	− 0.993359i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	9.29755	0.337036	0.168518	−	0.985699i	$-0.446102\pi$
$761$	9.29755	0.337036	0.168518	+	0.985699i	$0.446102\pi$
$762$	− 50.8141i	− 1.84080i
$763$	0	0
$764$	−7.72273	−0.279399
$765$	0	0
$766$	−5.61013	−0.202702
$767$	0	0
$768$	26.5384i	0.957622i
$769$	29.2192	1.05367	0.526836	−	0.849967i	$-0.323378\pi$
$769$	29.2192	1.05367	0.526836	+	0.849967i	$0.323378\pi$
$770$	0	0
$771$	4.72230	0.170069
$772$	− 18.3753i	− 0.661342i
$773$	47.4496i	1.70665i	0.521383	+	0.853323i	$0.325416\pi$
$773$	47.4496i	1.70665i	−0.521383	+	0.853323i	$0.674584\pi$
$774$	0	0
$775$	0	0
$776$	5.76248	0.206861
$777$	0	0
$778$	− 10.1530i	− 0.364003i
$779$	0	0
$780$	19.9809	0.715432
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−34.8694	−1.24534
$785$	0	0
$786$	−30.9423	−1.10368
$787$	− 25.6990i	− 0.916070i	−0.888934	−	0.458035i	$-0.848553\pi$
$787$	− 25.6990i	− 0.916070i	0.888934	−	0.458035i	$-0.151447\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	− 7.76046i	− 0.275756i
$793$	105.817i	3.75765i
$794$	0	0
$795$	−20.5932	−0.730365
$796$	23.1682	0.821175
$797$	48.5046i	1.71812i	0.511873	+	0.859061i	$0.328952\pi$
$797$	48.5046i	1.71812i	−0.511873	+	0.859061i	$0.671048\pi$
$798$	0	0
$799$	0	0
$800$	− 22.9137i	− 0.810120i
$801$	0	0
$802$	0	0
$803$	0	0
$804$	−4.42249	−0.155969
$805$	0	0
$806$	0	0
$807$	0	0
$808$	− 1.66991i	− 0.0587471i
$809$	−22.3607	−0.786160	−0.393080	−	0.919504i	$-0.628590\pi$
$809$	−22.3607	−0.786160	−0.393080	+	0.919504i	$0.628590\pi$
$810$	24.6182i	0.864995i
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	34.1329i	1.19709i
$814$	−122.244	−4.28466
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−42.0000	−1.46581	−0.732905	−	0.680331i	$-0.761836\pi$
$821$	−42.0000	−1.46581	−0.732905	+	0.680331i	$0.761836\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	32.5891	1.13530
$825$	− 45.3420i	− 1.57861i
$826$	0	0
$827$	56.1750i	1.95340i	0.214614	+	0.976699i	$0.431151\pi$
$827$	56.1750i	1.95340i	−0.214614	+	0.976699i	$0.568849\pi$
$828$	0	0
$829$	0	0		−	1.00000i	$-0.5\pi$
$829$	0	0			1.00000i	$0.5\pi$
$830$	0	0
$831$	0	0
$832$	14.9956i	0.519880i
$833$	0	0
$834$	22.5903	0.782238
$835$	57.6946i	1.99660i
$836$	0	0
$837$	0	0
$838$	− 60.9180i	− 2.10438i
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	74.0734	2.54820
$846$	0	0
$847$	0	0
$848$	30.1063i	1.03385i
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	−32.6776	−1.11690
$857$	57.6474i	1.96920i	0.174825	+	0.984599i	$0.444064\pi$
$857$	57.6474i	1.96920i	−0.174825	+	0.984599i	$0.555936\pi$
$858$	− 104.220i	− 3.55800i
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 44.0875i	− 1.50075i	−0.661010	−	0.750377i	$-0.729872\pi$
$863$	− 44.0875i	− 1.50075i	0.661010	−	0.750377i	$-0.270128\pi$
$864$	25.6843	0.873798
$865$	15.2894i	0.519857i
$866$	−23.1020	−0.785037
$867$	25.9047i	0.879768i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−22.8291	−0.773534
$872$	0	0
$873$	− 2.03149i	− 0.0687555i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	42.4094i	1.43206i	0.698068	+	0.716032i	$0.254044\pi$
$877$	42.4094i	1.43206i	−0.698068	+	0.716032i	$0.745956\pi$
$878$	0	0
$879$	37.0733	1.25045
$880$	−66.2879	−2.23456
$881$	58.6434	1.97575	0.987874	−	0.155261i	$-0.0496217\pi$
$881$	58.6434	1.97575	0.987874	+	0.155261i	$0.0496217\pi$
$882$	8.03131i	0.270428i
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 16.8527i	− 0.565858i	−0.959141	−	0.282929i	$-0.908694\pi$
$887$	− 16.8527i	− 0.565858i	0.959141	−	0.282929i	$-0.0913060\pi$
$888$	35.5756i	1.19384i
$889$	0	0
$890$	0	0
$891$	38.7195	1.29715
$892$	24.9094i	0.834028i
$893$	0	0
$894$	59.1422	1.97801
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	−2.92713	−0.0975709
$901$	0	0
$902$	0	0
$903$	0	0
$904$	26.1686	0.870355
$905$	0	0
$906$	0	0
$907$	17.0826i	0.567219i	0.958940	+	0.283610i	$0.0915320\pi$
$907$	17.0826i	0.567219i	−0.958940	+	0.283610i	$0.908468\pi$
$908$	− 2.98169i	− 0.0989508i
$909$	−0.588704	−0.0195261
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	53.0874i	1.75501i
$916$	5.60173	0.185087
$917$	0	0
$918$	0	0
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	36.2746	1.19529
$922$	30.4590i	1.00311i
$923$	0	0
$924$	0	0
$925$	− 60.6950i	− 1.99564i
$926$	0	0
$927$	− 11.4889i	− 0.377345i
$928$	0	0
$929$	−31.3050	−1.02708	−0.513541	−	0.858065i	$-0.671667\pi$
$929$	−31.3050	−1.02708	−0.513541	+	0.858065i	$0.671667\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	− 24.5591i	− 0.804029i
$934$	0	0
$935$	0	0
$936$	8.85649	0.289483
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	51.6757	1.67570
$952$	0	0
$953$	− 39.3618i	− 1.27505i	−0.770428	−	0.637527i	$-0.779957\pi$
$953$	− 39.3618i	− 1.27505i	0.770428	−	0.637527i	$-0.220043\pi$
$954$	6.93423	0.224504
$955$	−20.0000	−0.647185
$956$	20.7223	0.670206
$957$	0	0
$958$	− 13.0089i	− 0.420297i
$959$	0	0
$960$	7.52321i	0.242810i
$961$	−31.0000	−1.00000
$962$	− 139.509i	− 4.49795i
$963$	11.5201i	0.371229i
$964$	0	0
$965$	− 47.5876i	− 1.53190i
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	46.9593i	1.50933i
$969$	0	0
$970$	−11.3370	−0.364009
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	− 5.95728i	− 0.191080i
$973$	0	0
$974$	−74.6437	−2.39174
$975$	51.7458	1.65719
$976$	77.6112	2.48427
$977$	54.3563i	1.73901i	0.493923	+	0.869506i	$0.335562\pi$
$977$	54.3563i	1.73901i	−0.493923	+	0.869506i	$0.664438\pi$
$978$	0	0
$979$	0	0
$980$	13.5148	0.431714
$981$	0	0
$982$	60.5408i	1.93193i
$983$	44.2033i	1.40987i	0.709274	+	0.704933i	$0.249023\pi$
$983$	44.2033i	1.40987i	−0.709274	+	0.704933i	$0.750977\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	15.2678i	0.485242i
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	60.0000	1.90213
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	− 75.2472i	− 2.38191i
$999$	68.0341	2.15250

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1805.2.b.h.1084.6	yes	8
5.2	odd	4		9025.2.a.cb.1.3		8
5.3	odd	4		9025.2.a.cb.1.6		8
5.4	even	2	inner	1805.2.b.h.1084.3	✓	8
19.18	odd	2	inner	1805.2.b.h.1084.3	✓	8
95.18	even	4		9025.2.a.cb.1.3		8
95.37	even	4		9025.2.a.cb.1.6		8
95.94	odd	2	CM	1805.2.b.h.1084.6	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1805.2.b.h.1084.3	✓	8	5.4	even	2	inner
1805.2.b.h.1084.3	✓	8	19.18	odd	2	inner
1805.2.b.h.1084.6	yes	8	1.1	even	1	trivial
1805.2.b.h.1084.6	yes	8	95.94	odd	2	CM
9025.2.a.cb.1.3		8	5.2	odd	4
9025.2.a.cb.1.3		8	95.18	even	4
9025.2.a.cb.1.6		8	5.3	odd	4
9025.2.a.cb.1.6		8	95.37	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1805 = 5 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1805.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.69217i q^{2} +1.52380i q^{3} -0.863428 q^{4} -2.23607 q^{5} -2.57853 q^{6} +1.92327i q^{8} +0.678024 q^{9} +O(q^{10})\)	\(q+1.69217i q^{2} +1.52380i q^{3} -0.863428 q^{4} -2.23607 q^{5} -2.57853 q^{6} +1.92327i q^{8} +0.678024 q^{9} -3.78380i q^{10} -5.95117 q^{11} -1.31569i q^{12} -6.79166i q^{13} -3.40733i q^{15} -4.98135 q^{16} +1.14733i q^{18} +1.93068 q^{20} -10.0704i q^{22} -2.93068 q^{24} +5.00000 q^{25} +11.4926 q^{26} +5.60458i q^{27} +5.76577 q^{30} -4.58273i q^{32} -9.06841i q^{33} -0.585425 q^{36} -12.1390i q^{37} +10.3492 q^{39} -4.30056i q^{40} +5.13841 q^{44} -1.51611 q^{45} -7.59059i q^{48} +7.00000 q^{49} +8.46083i q^{50} +5.86411i q^{52} -6.04380i q^{53} -9.48389 q^{54} +13.3072 q^{55} +2.94198i q^{60} -15.5804 q^{61} -2.20795 q^{64} +15.1866i q^{65} +15.3453 q^{66} -3.36134i q^{67} +1.30402i q^{72} +20.5412 q^{74} +7.61901i q^{75} +17.5125i q^{78} +11.1386 q^{80} -6.50621 q^{81} -11.4457i q^{88} -2.56551i q^{90} +6.98318 q^{96} -2.99619i q^{97} +11.8452i q^{98} -4.03504 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{4} - 24 q^{9}+O(q^{10}) \) 8 * q - 16 * q^4 - 24 * q^9	\( 8 q - 16 q^{4} - 24 q^{9} + 32 q^{16} - 8 q^{24} + 40 q^{25} + 24 q^{26} - 40 q^{30} - 8 q^{36} - 72 q^{44} + 56 q^{49} - 88 q^{54} - 64 q^{64} + 104 q^{66} + 120 q^{80} + 72 q^{81} - 120 q^{96} - 32 q^{99}+O(q^{100}) \) 8 * q - 16 * q^4 - 24 * q^9 + 32 * q^16 - 8 * q^24 + 40 * q^25 + 24 * q^26 - 40 * q^30 - 8 * q^36 - 72 * q^44 + 56 * q^49 - 88 * q^54 - 64 * q^64 + 104 * q^66 + 120 * q^80 + 72 * q^81 - 120 * q^96 - 32 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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