LMFDB - Embedded newform 2790.2.a.bk.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2790,2,Mod(1,2790)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2790, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2790.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2790 = 2 \cdot 3^{2} \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2790.a (trivial)

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:	yes
Analytic conductor:	$22.2782621639$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.17428.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + 4x + 6 $ x^4 - x^3 - 6x^2 + 4x + 6
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.52616$ of defining polynomial
Character	$\chi$	$=$	2790.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +4.54997 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +4.54997 q^{7} +1.00000 q^{8} +1.00000 q^{10} -2.54997 q^{11} -1.05232 q^{13} +4.54997 q^{14} +1.00000 q^{16} -4.39400 q^{17} +7.60228 q^{19} +1.00000 q^{20} -2.54997 q^{22} -1.20828 q^{23} +1.00000 q^{25} -1.05232 q^{26} +4.54997 q^{28} +6.10463 q^{29} +1.00000 q^{31} +1.00000 q^{32} -4.39400 q^{34} +4.54997 q^{35} +7.73569 q^{37} +7.60228 q^{38} +1.00000 q^{40} +6.10463 q^{41} -6.28566 q^{43} -2.54997 q^{44} -1.20828 q^{46} +2.68337 q^{47} +13.7022 q^{49} +1.00000 q^{50} -1.05232 q^{52} -0.502345 q^{53} -2.54997 q^{55} +4.54997 q^{56} +6.10463 q^{58} -10.8356 q^{59} -9.80588 q^{61} +1.00000 q^{62} +1.00000 q^{64} -1.05232 q^{65} +7.46888 q^{67} -4.39400 q^{68} +4.54997 q^{70} -8.65460 q^{71} -8.70222 q^{73} +7.73569 q^{74} +7.60228 q^{76} -11.6023 q^{77} -0.918913 q^{79} +1.00000 q^{80} +6.10463 q^{82} +2.41657 q^{83} -4.39400 q^{85} -6.28566 q^{86} -2.54997 q^{88} +16.4379 q^{89} -4.78800 q^{91} -1.20828 q^{92} +2.68337 q^{94} +7.60228 q^{95} -9.20457 q^{97} +13.7022 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 4 q^{4} + 4 q^{5} + 5 q^{7} + 4 q^{8}+O(q^{10}) $ 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^5 + 5 * q^7 + 4 * q^8	$ 4 q + 4 q^{2} + 4 q^{4} + 4 q^{5} + 5 q^{7} + 4 q^{8} + 4 q^{10} + 3 q^{11} + 6 q^{13} + 5 q^{14} + 4 q^{16} + 7 q^{19} + 4 q^{20} + 3 q^{22} + q^{23} + 4 q^{25} + 6 q^{26} + 5 q^{28} + 4 q^{29} + 4 q^{31} + 4 q^{32} + 5 q^{35} + 6 q^{37} + 7 q^{38} + 4 q^{40} + 4 q^{41} + 13 q^{43} + 3 q^{44} + q^{46} - 4 q^{47} + 5 q^{49} + 4 q^{50} + 6 q^{52} - 5 q^{53} + 3 q^{55} + 5 q^{56} + 4 q^{58} + 8 q^{59} - 4 q^{61} + 4 q^{62} + 4 q^{64} + 6 q^{65} + 8 q^{67} + 5 q^{70} - q^{71} + 15 q^{73} + 6 q^{74} + 7 q^{76} - 23 q^{77} + 5 q^{79} + 4 q^{80} + 4 q^{82} - 2 q^{83} + 13 q^{86} + 3 q^{88} - 9 q^{89} + 16 q^{91} + q^{92} - 4 q^{94} + 7 q^{95} + 10 q^{97} + 5 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^5 + 5 * q^7 + 4 * q^8 + 4 * q^10 + 3 * q^11 + 6 * q^13 + 5 * q^14 + 4 * q^16 + 7 * q^19 + 4 * q^20 + 3 * q^22 + q^23 + 4 * q^25 + 6 * q^26 + 5 * q^28 + 4 * q^29 + 4 * q^31 + 4 * q^32 + 5 * q^35 + 6 * q^37 + 7 * q^38 + 4 * q^40 + 4 * q^41 + 13 * q^43 + 3 * q^44 + q^46 - 4 * q^47 + 5 * q^49 + 4 * q^50 + 6 * q^52 - 5 * q^53 + 3 * q^55 + 5 * q^56 + 4 * q^58 + 8 * q^59 - 4 * q^61 + 4 * q^62 + 4 * q^64 + 6 * q^65 + 8 * q^67 + 5 * q^70 - q^71 + 15 * q^73 + 6 * q^74 + 7 * q^76 - 23 * q^77 + 5 * q^79 + 4 * q^80 + 4 * q^82 - 2 * q^83 + 13 * q^86 + 3 * q^88 - 9 * q^89 + 16 * q^91 + q^92 - 4 * q^94 + 7 * q^95 + 10 * q^97 + 5 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.54997	1.71973	0.859863	−	0.510524i	$-0.170549\pi$
$7$	4.54997	1.71973	0.859863	+	0.510524i	$0.170549\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	−2.54997	−0.768845	−0.384422	−	0.923157i	$-0.625599\pi$
$11$	−2.54997	−0.768845	−0.384422	+	0.923157i	$0.625599\pi$
$12$	0	0
$13$	−1.05232	−0.291860	−0.145930	−	0.989295i	$-0.546617\pi$
$13$	−1.05232	−0.291860	−0.145930	+	0.989295i	$0.546617\pi$
$14$	4.54997	1.21603
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.39400	−1.06570	−0.532851	−	0.846209i	$-0.678879\pi$
$17$	−4.39400	−1.06570	−0.532851	+	0.846209i	$0.678879\pi$
$18$	0	0
$19$	7.60228	1.74408	0.872042	−	0.489431i	$-0.162796\pi$
$19$	7.60228	1.74408	0.872042	+	0.489431i	$0.162796\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−2.54997	−0.543655
$23$	−1.20828	−0.251945	−0.125972	−	0.992034i	$-0.540205\pi$
$23$	−1.20828	−0.251945	−0.125972	+	0.992034i	$0.540205\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−1.05232	−0.206376
$27$	0	0
$28$	4.54997	0.859863
$29$	6.10463	1.13360	0.566801	−	0.823855i	$-0.308181\pi$
$29$	6.10463	1.13360	0.566801	+	0.823855i	$0.308181\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	1.00000	0.176777
$33$	0	0
$34$	−4.39400	−0.753565
$35$	4.54997	0.769085
$36$	0	0
$37$	7.73569	1.27174	0.635870	−	0.771797i	$-0.280642\pi$
$37$	7.73569	1.27174	0.635870	+	0.771797i	$0.280642\pi$
$38$	7.60228	1.23325
$39$	0	0
$40$	1.00000	0.158114
$41$	6.10463	0.953383	0.476692	−	0.879071i	$-0.341836\pi$
$41$	6.10463	0.953383	0.476692	+	0.879071i	$0.341836\pi$
$42$	0	0
$43$	−6.28566	−0.958554	−0.479277	−	0.877664i	$-0.659101\pi$
$43$	−6.28566	−0.958554	−0.479277	+	0.877664i	$0.659101\pi$
$44$	−2.54997	−0.384422
$45$	0	0
$46$	−1.20828	−0.178152
$47$	2.68337	0.391410	0.195705	−	0.980663i	$-0.437300\pi$
$47$	2.68337	0.391410	0.195705	+	0.980663i	$0.437300\pi$
$48$	0	0
$49$	13.7022	1.95746
$50$	1.00000	0.141421
$51$	0	0
$52$	−1.05232	−0.145930
$53$	−0.502345	−0.0690025	−0.0345012	−	0.999405i	$-0.510984\pi$
$53$	−0.502345	−0.0690025	−0.0345012	+	0.999405i	$0.510984\pi$
$54$	0	0
$55$	−2.54997	−0.343838
$56$	4.54997	0.608015
$57$	0	0
$58$	6.10463	0.801577
$59$	−10.8356	−1.41068	−0.705339	−	0.708870i	$-0.749205\pi$
$59$	−10.8356	−1.41068	−0.705339	+	0.708870i	$0.749205\pi$
$60$	0	0
$61$	−9.80588	−1.25551	−0.627757	−	0.778409i	$-0.716027\pi$
$61$	−9.80588	−1.25551	−0.627757	+	0.778409i	$0.716027\pi$
$62$	1.00000	0.127000
$63$	0	0
$64$	1.00000	0.125000
$65$	−1.05232	−0.130524
$66$	0	0
$67$	7.46888	0.912469	0.456235	−	0.889860i	$-0.349198\pi$
$67$	7.46888	0.912469	0.456235	+	0.889860i	$0.349198\pi$
$68$	−4.39400	−0.532851
$69$	0	0
$70$	4.54997	0.543825
$71$	−8.65460	−1.02711	−0.513556	−	0.858056i	$-0.671672\pi$
$71$	−8.65460	−1.02711	−0.513556	+	0.858056i	$0.671672\pi$
$72$	0	0
$73$	−8.70222	−1.01852	−0.509259	−	0.860613i	$-0.670081\pi$
$73$	−8.70222	−1.01852	−0.509259	+	0.860613i	$0.670081\pi$
$74$	7.73569	0.899255
$75$	0	0
$76$	7.60228	0.872042
$77$	−11.6023	−1.32220
$78$	0	0
$79$	−0.918913	−0.103386	−0.0516929	−	0.998663i	$-0.516462\pi$
$79$	−0.918913	−0.103386	−0.0516929	+	0.998663i	$0.516462\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	6.10463	0.674144
$83$	2.41657	0.265253	0.132626	−	0.991166i	$-0.457659\pi$
$83$	2.41657	0.265253	0.132626	+	0.991166i	$0.457659\pi$
$84$	0	0
$85$	−4.39400	−0.476596
$86$	−6.28566	−0.677800
$87$	0	0
$88$	−2.54997	−0.271828
$89$	16.4379	1.74242	0.871208	−	0.490915i	$-0.163337\pi$
$89$	16.4379	1.74242	0.871208	+	0.490915i	$0.163337\pi$
$90$	0	0
$91$	−4.78800	−0.501919
$92$	−1.20828	−0.125972
$93$	0	0
$94$	2.68337	0.276769
$95$	7.60228	0.779978
$96$	0	0
$97$	−9.20457	−0.934582	−0.467291	−	0.884103i	$-0.654770\pi$
$97$	−9.20457	−0.934582	−0.467291	+	0.884103i	$0.654770\pi$
$98$	13.7022	1.38413
$99$	0	0
$100$	1.00000	0.100000
$101$	3.18572	0.316991	0.158495	−	0.987360i	$-0.449336\pi$
$101$	3.18572	0.316991	0.158495	+	0.987360i	$0.449336\pi$
$102$	0	0
$103$	18.8356	1.85593	0.927965	−	0.372668i	$-0.121557\pi$
$103$	18.8356	1.85593	0.927965	+	0.372668i	$0.121557\pi$
$104$	−1.05232	−0.103188
$105$	0	0
$106$	−0.502345	−0.0487921
$107$	−14.2857	−1.38105	−0.690523	−	0.723310i	$-0.742620\pi$
$107$	−14.2857	−1.38105	−0.690523	+	0.723310i	$0.742620\pi$
$108$	0	0
$109$	−5.78331	−0.553941	−0.276970	−	0.960878i	$-0.589330\pi$
$109$	−5.78331	−0.553941	−0.276970	+	0.960878i	$0.589330\pi$
$110$	−2.54997	−0.243130
$111$	0	0
$112$	4.54997	0.429932
$113$	8.01885	0.754350	0.377175	−	0.926142i	$-0.376895\pi$
$113$	8.01885	0.754350	0.377175	+	0.926142i	$0.376895\pi$
$114$	0	0
$115$	−1.20828	−0.112673
$116$	6.10463	0.566801
$117$	0	0
$118$	−10.8356	−0.997500
$119$	−19.9926	−1.83272
$120$	0	0
$121$	−4.49765	−0.408878
$122$	−9.80588	−0.887782
$123$	0	0
$124$	1.00000	0.0898027
$125$	1.00000	0.0894427
$126$	0	0
$127$	−7.15695	−0.635076	−0.317538	−	0.948246i	$-0.602856\pi$
$127$	−7.15695	−0.635076	−0.317538	+	0.948246i	$0.602856\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−1.05232	−0.0922941
$131$	−4.46419	−0.390038	−0.195019	−	0.980799i	$-0.562477\pi$
$131$	−4.46419	−0.390038	−0.195019	+	0.980799i	$0.562477\pi$
$132$	0	0
$133$	34.5902	2.99935
$134$	7.46888	0.645213
$135$	0	0
$136$	−4.39400	−0.376782
$137$	17.0104	1.45330	0.726650	−	0.687008i	$-0.241076\pi$
$137$	17.0104	1.45330	0.726650	+	0.687008i	$0.241076\pi$
$138$	0	0
$139$	−7.49394	−0.635628	−0.317814	−	0.948153i	$-0.602949\pi$
$139$	−7.49394	−0.635628	−0.317814	+	0.948153i	$0.602949\pi$
$140$	4.54997	0.384543
$141$	0	0
$142$	−8.65460	−0.726278
$143$	2.68337	0.224395
$144$	0	0
$145$	6.10463	0.506962
$146$	−8.70222	−0.720201
$147$	0	0
$148$	7.73569	0.635870
$149$	2.18103	0.178677	0.0893383	−	0.996001i	$-0.471525\pi$
$149$	2.18103	0.178677	0.0893383	+	0.996001i	$0.471525\pi$
$150$	0	0
$151$	4.78800	0.389642	0.194821	−	0.980839i	$-0.437587\pi$
$151$	4.78800	0.389642	0.194821	+	0.980839i	$0.437587\pi$
$152$	7.60228	0.616627
$153$	0	0
$154$	−11.6023	−0.934939
$155$	1.00000	0.0803219
$156$	0	0
$157$	10.0189	0.799591	0.399796	−	0.916604i	$-0.369081\pi$
$157$	10.0189	0.799591	0.399796	+	0.916604i	$0.369081\pi$
$158$	−0.918913	−0.0731048
$159$	0	0
$160$	1.00000	0.0790569
$161$	−5.49765	−0.433276
$162$	0	0
$163$	13.6311	1.06767	0.533833	−	0.845590i	$-0.320751\pi$
$163$	13.6311	1.06767	0.533833	+	0.845590i	$0.320751\pi$
$164$	6.10463	0.476692
$165$	0	0
$166$	2.41657	0.187562
$167$	−19.9869	−1.54663	−0.773317	−	0.634020i	$-0.781404\pi$
$167$	−19.9869	−1.54663	−0.773317	+	0.634020i	$0.781404\pi$
$168$	0	0
$169$	−11.8926	−0.914818
$170$	−4.39400	−0.337005
$171$	0	0
$172$	−6.28566	−0.479277
$173$	−14.5212	−1.10403	−0.552013	−	0.833835i	$-0.686140\pi$
$173$	−14.5212	−1.10403	−0.552013	+	0.833835i	$0.686140\pi$
$174$	0	0
$175$	4.54997	0.343945
$176$	−2.54997	−0.192211
$177$	0	0
$178$	16.4379	1.23207
$179$	−16.5688	−1.23841	−0.619206	−	0.785229i	$-0.712545\pi$
$179$	−16.5688	−1.23841	−0.619206	+	0.785229i	$0.712545\pi$
$180$	0	0
$181$	9.31291	0.692223	0.346112	−	0.938193i	$-0.387502\pi$
$181$	9.31291	0.692223	0.346112	+	0.938193i	$0.387502\pi$
$182$	−4.78800	−0.354910
$183$	0	0
$184$	−1.20828	−0.0890759
$185$	7.73569	0.568739
$186$	0	0
$187$	11.2046	0.819359
$188$	2.68337	0.195705
$189$	0	0
$190$	7.60228	0.551528
$191$	10.4642	0.757162	0.378581	−	0.925568i	$-0.376412\pi$
$191$	10.4642	0.757162	0.378581	+	0.925568i	$0.376412\pi$
$192$	0	0
$193$	−8.19988	−0.590240	−0.295120	−	0.955460i	$-0.595360\pi$
$193$	−8.19988	−0.590240	−0.295120	+	0.955460i	$0.595360\pi$
$194$	−9.20457	−0.660850
$195$	0	0
$196$	13.7022	0.978730
$197$	−11.8954	−0.847510	−0.423755	−	0.905777i	$-0.639288\pi$
$197$	−11.8954	−0.847510	−0.423755	+	0.905777i	$0.639288\pi$
$198$	0	0
$199$	8.60698	0.610132	0.305066	−	0.952331i	$-0.401321\pi$
$199$	8.60698	0.610132	0.305066	+	0.952331i	$0.401321\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	3.18572	0.224146
$203$	27.7759	1.94948
$204$	0	0
$205$	6.10463	0.426366
$206$	18.8356	1.31234
$207$	0	0
$208$	−1.05232	−0.0729649
$209$	−19.3856	−1.34093
$210$	0	0
$211$	−1.92360	−0.132426	−0.0662132	−	0.997805i	$-0.521092\pi$
$211$	−1.92360	−0.132426	−0.0662132	+	0.997805i	$0.521092\pi$
$212$	−0.502345	−0.0345012
$213$	0	0
$214$	−14.2857	−0.976547
$215$	−6.28566	−0.428678
$216$	0	0
$217$	4.54997	0.308872
$218$	−5.78331	−0.391695
$219$	0	0
$220$	−2.54997	−0.171919
$221$	4.62387	0.311035
$222$	0	0
$223$	21.8855	1.46556	0.732779	−	0.680467i	$-0.238223\pi$
$223$	21.8855	1.46556	0.732779	+	0.680467i	$0.238223\pi$
$224$	4.54997	0.304008
$225$	0	0
$226$	8.01885	0.533406
$227$	13.4902	0.895378	0.447689	−	0.894189i	$-0.352247\pi$
$227$	13.4902	0.895378	0.447689	+	0.894189i	$0.352247\pi$
$228$	0	0
$229$	23.4677	1.55079	0.775393	−	0.631479i	$-0.217552\pi$
$229$	23.4677	1.55079	0.775393	+	0.631479i	$0.217552\pi$
$230$	−1.20828	−0.0796719
$231$	0	0
$232$	6.10463	0.400789
$233$	−21.9643	−1.43893	−0.719466	−	0.694528i	$-0.755613\pi$
$233$	−21.9643	−1.43893	−0.719466	+	0.694528i	$0.755613\pi$
$234$	0	0
$235$	2.68337	0.175044
$236$	−10.8356	−0.705339
$237$	0	0
$238$	−19.9926	−1.29593
$239$	12.6164	0.816090	0.408045	−	0.912962i	$-0.366211\pi$
$239$	12.6164	0.816090	0.408045	+	0.912962i	$0.366211\pi$
$240$	0	0
$241$	−18.7880	−1.21024	−0.605121	−	0.796134i	$-0.706875\pi$
$241$	−18.7880	−1.21024	−0.605121	+	0.796134i	$0.706875\pi$
$242$	−4.49765	−0.289120
$243$	0	0
$244$	−9.80588	−0.627757
$245$	13.7022	0.875403
$246$	0	0
$247$	−8.00000	−0.509028
$248$	1.00000	0.0635001
$249$	0	0
$250$	1.00000	0.0632456
$251$	16.5688	1.04581	0.522907	−	0.852390i	$-0.324847\pi$
$251$	16.5688	1.04581	0.522907	+	0.852390i	$0.324847\pi$
$252$	0	0
$253$	3.08109	0.193706
$254$	−7.15695	−0.449067
$255$	0	0
$256$	1.00000	0.0625000
$257$	−29.8115	−1.85959	−0.929797	−	0.368074i	$-0.880017\pi$
$257$	−29.8115	−1.85959	−0.929797	+	0.368074i	$0.880017\pi$
$258$	0	0
$259$	35.1971	2.18704
$260$	−1.05232	−0.0652618
$261$	0	0
$262$	−4.46419	−0.275799
$263$	12.0151	0.740885	0.370443	−	0.928855i	$-0.379206\pi$
$263$	12.0151	0.740885	0.370443	+	0.928855i	$0.379206\pi$
$264$	0	0
$265$	−0.502345	−0.0308588
$266$	34.5902	2.12086
$267$	0	0
$268$	7.46888	0.456235
$269$	−23.2046	−1.41481	−0.707404	−	0.706810i	$-0.750134\pi$
$269$	−23.2046	−1.41481	−0.707404	+	0.706810i	$0.750134\pi$
$270$	0	0
$271$	−10.7116	−0.650684	−0.325342	−	0.945596i	$-0.605479\pi$
$271$	−10.7116	−0.650684	−0.325342	+	0.945596i	$0.605479\pi$
$272$	−4.39400	−0.266425
$273$	0	0
$274$	17.0104	1.02764
$275$	−2.54997	−0.153769
$276$	0	0
$277$	9.14756	0.549624	0.274812	−	0.961498i	$-0.411384\pi$
$277$	9.14756	0.549624	0.274812	+	0.961498i	$0.411384\pi$
$278$	−7.49394	−0.449457
$279$	0	0
$280$	4.54997	0.271913
$281$	−12.2093	−0.728343	−0.364172	−	0.931332i	$-0.618648\pi$
$281$	−12.2093	−0.728343	−0.364172	+	0.931332i	$0.618648\pi$
$282$	0	0
$283$	8.26431	0.491262	0.245631	−	0.969363i	$-0.421005\pi$
$283$	8.26431	0.491262	0.245631	+	0.969363i	$0.421005\pi$
$284$	−8.65460	−0.513556
$285$	0	0
$286$	2.68337	0.158671
$287$	27.7759	1.63956
$288$	0	0
$289$	2.30725	0.135720
$290$	6.10463	0.358476
$291$	0	0
$292$	−8.70222	−0.509259
$293$	−9.68806	−0.565983	−0.282991	−	0.959122i	$-0.591327\pi$
$293$	−9.68806	−0.565983	−0.282991	+	0.959122i	$0.591327\pi$
$294$	0	0
$295$	−10.8356	−0.630875
$296$	7.73569	0.449628
$297$	0	0
$298$	2.18103	0.126343
$299$	1.27150	0.0735325
$300$	0	0
$301$	−28.5995	−1.64845
$302$	4.78800	0.275519
$303$	0	0
$304$	7.60228	0.436021
$305$	−9.80588	−0.561483
$306$	0	0
$307$	−24.6641	−1.40765	−0.703826	−	0.710372i	$-0.748527\pi$
$307$	−24.6641	−1.40765	−0.703826	+	0.710372i	$0.748527\pi$
$308$	−11.6023	−0.661102
$309$	0	0
$310$	1.00000	0.0567962
$311$	7.64044	0.433250	0.216625	−	0.976255i	$-0.430495\pi$
$311$	7.64044	0.433250	0.216625	+	0.976255i	$0.430495\pi$
$312$	0	0
$313$	−13.7927	−0.779609	−0.389805	−	0.920898i	$-0.627457\pi$
$313$	−13.7927	−0.779609	−0.389805	+	0.920898i	$0.627457\pi$
$314$	10.0189	0.565397
$315$	0	0
$316$	−0.918913	−0.0516929
$317$	4.19988	0.235889	0.117944	−	0.993020i	$-0.462370\pi$
$317$	4.19988	0.235889	0.117944	+	0.993020i	$0.462370\pi$
$318$	0	0
$319$	−15.5666	−0.871564
$320$	1.00000	0.0559017
$321$	0	0
$322$	−5.49765	−0.306372
$323$	−33.4044	−1.85867
$324$	0	0
$325$	−1.05232	−0.0583719
$326$	13.6311	0.754954
$327$	0	0
$328$	6.10463	0.337072
$329$	12.2093	0.673118
$330$	0	0
$331$	1.50332	0.0826301	0.0413150	−	0.999146i	$-0.486845\pi$
$331$	1.50332	0.0826301	0.0413150	+	0.999146i	$0.486845\pi$
$332$	2.41657	0.132626
$333$	0	0
$334$	−19.9869	−1.09363
$335$	7.46888	0.408069
$336$	0	0
$337$	−23.7833	−1.29556	−0.647780	−	0.761828i	$-0.724302\pi$
$337$	−23.7833	−1.29556	−0.647780	+	0.761828i	$0.724302\pi$
$338$	−11.8926	−0.646874
$339$	0	0
$340$	−4.39400	−0.238298
$341$	−2.54997	−0.138089
$342$	0	0
$343$	30.4949	1.64657
$344$	−6.28566	−0.338900
$345$	0	0
$346$	−14.5212	−0.780664
$347$	−30.9879	−1.66352	−0.831758	−	0.555138i	$-0.812665\pi$
$347$	−30.9879	−1.66352	−0.831758	+	0.555138i	$0.812665\pi$
$348$	0	0
$349$	−30.6090	−1.63846	−0.819232	−	0.573463i	$-0.805600\pi$
$349$	−30.6090	−1.63846	−0.819232	+	0.573463i	$0.805600\pi$
$350$	4.54997	0.243206
$351$	0	0
$352$	−2.54997	−0.135914
$353$	−25.3224	−1.34777	−0.673887	−	0.738834i	$-0.735377\pi$
$353$	−25.3224	−1.34777	−0.673887	+	0.738834i	$0.735377\pi$
$354$	0	0
$355$	−8.65460	−0.459338
$356$	16.4379	0.871208
$357$	0	0
$358$	−16.5688	−0.875689
$359$	17.8498	0.942076	0.471038	−	0.882113i	$-0.343880\pi$
$359$	17.8498	0.942076	0.471038	+	0.882113i	$0.343880\pi$
$360$	0	0
$361$	38.7947	2.04183
$362$	9.31291	0.489476
$363$	0	0
$364$	−4.78800	−0.250960
$365$	−8.70222	−0.455495
$366$	0	0
$367$	−23.9901	−1.25227	−0.626136	−	0.779714i	$-0.715365\pi$
$367$	−23.9901	−1.25227	−0.626136	+	0.779714i	$0.715365\pi$
$368$	−1.20828	−0.0629861
$369$	0	0
$370$	7.73569	0.402159
$371$	−2.28566	−0.118665
$372$	0	0
$373$	5.92360	0.306713	0.153356	−	0.988171i	$-0.450992\pi$
$373$	5.92360	0.306713	0.153356	+	0.988171i	$0.450992\pi$
$374$	11.2046	0.579375
$375$	0	0
$376$	2.68337	0.138384
$377$	−6.42400	−0.330853
$378$	0	0
$379$	−19.9068	−1.02254	−0.511272	−	0.859419i	$-0.670825\pi$
$379$	−19.9068	−1.02254	−0.511272	+	0.859419i	$0.670825\pi$
$380$	7.60228	0.389989
$381$	0	0
$382$	10.4642	0.535395
$383$	−32.8031	−1.67616	−0.838081	−	0.545546i	$-0.816322\pi$
$383$	−32.8031	−1.67616	−0.838081	+	0.545546i	$0.816322\pi$
$384$	0	0
$385$	−11.6023	−0.591307
$386$	−8.19988	−0.417363
$387$	0	0
$388$	−9.20457	−0.467291
$389$	−12.0000	−0.608424	−0.304212	−	0.952604i	$-0.598393\pi$
$389$	−12.0000	−0.608424	−0.304212	+	0.952604i	$0.598393\pi$
$390$	0	0
$391$	5.30920	0.268498
$392$	13.7022	0.692067
$393$	0	0
$394$	−11.8954	−0.599280
$395$	−0.918913	−0.0462355
$396$	0	0
$397$	11.2903	0.566646	0.283323	−	0.959024i	$-0.408563\pi$
$397$	11.2903	0.566646	0.283323	+	0.959024i	$0.408563\pi$
$398$	8.60698	0.431429
$399$	0	0
$400$	1.00000	0.0500000
$401$	27.7000	1.38327	0.691637	−	0.722246i	$-0.256890\pi$
$401$	27.7000	1.38327	0.691637	+	0.722246i	$0.256890\pi$
$402$	0	0
$403$	−1.05232	−0.0524196
$404$	3.18572	0.158495
$405$	0	0
$406$	27.7759	1.37849
$407$	−19.7258	−0.977770
$408$	0	0
$409$	22.1905	1.09725	0.548625	−	0.836069i	$-0.315152\pi$
$409$	22.1905	1.09725	0.548625	+	0.836069i	$0.315152\pi$
$410$	6.10463	0.301486
$411$	0	0
$412$	18.8356	0.927965
$413$	−49.3018	−2.42598
$414$	0	0
$415$	2.41657	0.118625
$416$	−1.05232	−0.0515940
$417$	0	0
$418$	−19.3856	−0.948181
$419$	23.0449	1.12582	0.562908	−	0.826519i	$-0.309682\pi$
$419$	23.0449	1.12582	0.562908	+	0.826519i	$0.309682\pi$
$420$	0	0
$421$	−26.3621	−1.28481	−0.642404	−	0.766366i	$-0.722063\pi$
$421$	−26.3621	−1.28481	−0.642404	+	0.766366i	$0.722063\pi$
$422$	−1.92360	−0.0936396
$423$	0	0
$424$	−0.502345	−0.0243961
$425$	−4.39400	−0.213140
$426$	0	0
$427$	−44.6164	−2.15914
$428$	−14.2857	−0.690523
$429$	0	0
$430$	−6.28566	−0.303121
$431$	12.2643	0.590751	0.295376	−	0.955381i	$-0.404555\pi$
$431$	12.2643	0.590751	0.295376	+	0.955381i	$0.404555\pi$
$432$	0	0
$433$	−25.0643	−1.20451	−0.602256	−	0.798303i	$-0.705731\pi$
$433$	−25.0643	−1.20451	−0.602256	+	0.798303i	$0.705731\pi$
$434$	4.54997	0.218406
$435$	0	0
$436$	−5.78331	−0.276970
$437$	−9.18572	−0.439412
$438$	0	0
$439$	13.3092	0.635213	0.317607	−	0.948223i	$-0.397121\pi$
$439$	13.3092	0.635213	0.317607	+	0.948223i	$0.397121\pi$
$440$	−2.54997	−0.121565
$441$	0	0
$442$	4.62387	0.219935
$443$	−35.4808	−1.68575	−0.842873	−	0.538113i	$-0.819137\pi$
$443$	−35.4808	−1.68575	−0.842873	+	0.538113i	$0.819137\pi$
$444$	0	0
$445$	16.4379	0.779232
$446$	21.8855	1.03631
$447$	0	0
$448$	4.54997	0.214966
$449$	−32.0476	−1.51242	−0.756210	−	0.654328i	$-0.772951\pi$
$449$	−32.0476	−1.51242	−0.756210	+	0.654328i	$0.772951\pi$
$450$	0	0
$451$	−15.5666	−0.733004
$452$	8.01885	0.377175
$453$	0	0
$454$	13.4902	0.633128
$455$	−4.78800	−0.224465
$456$	0	0
$457$	−3.38355	−0.158276	−0.0791380	−	0.996864i	$-0.525217\pi$
$457$	−3.38355	−0.158276	−0.0791380	+	0.996864i	$0.525217\pi$
$458$	23.4677	1.09657
$459$	0	0
$460$	−1.20828	−0.0563365
$461$	4.83314	0.225102	0.112551	−	0.993646i	$-0.464098\pi$
$461$	4.83314	0.225102	0.112551	+	0.993646i	$0.464098\pi$
$462$	0	0
$463$	26.5143	1.23222	0.616112	−	0.787658i	$-0.288707\pi$
$463$	26.5143	1.23222	0.616112	+	0.787658i	$0.288707\pi$
$464$	6.10463	0.283400
$465$	0	0
$466$	−21.9643	−1.01748
$467$	−21.4620	−0.993143	−0.496571	−	0.867996i	$-0.665408\pi$
$467$	−21.4620	−0.993143	−0.496571	+	0.867996i	$0.665408\pi$
$468$	0	0
$469$	33.9832	1.56920
$470$	2.68337	0.123775
$471$	0	0
$472$	−10.8356	−0.498750
$473$	16.0282	0.736979
$474$	0	0
$475$	7.60228	0.348817
$476$	−19.9926	−0.916358
$477$	0	0
$478$	12.6164	0.577063
$479$	13.7545	0.628461	0.314230	−	0.949347i	$-0.398254\pi$
$479$	13.7545	0.628461	0.314230	+	0.949347i	$0.398254\pi$
$480$	0	0
$481$	−8.14038	−0.371169
$482$	−18.7880	−0.855770
$483$	0	0
$484$	−4.49765	−0.204439
$485$	−9.20457	−0.417958
$486$	0	0
$487$	11.3662	0.515052	0.257526	−	0.966271i	$-0.417093\pi$
$487$	11.3662	0.515052	0.257526	+	0.966271i	$0.417093\pi$
$488$	−9.80588	−0.443891
$489$	0	0
$490$	13.7022	0.619003
$491$	37.7545	1.70384	0.851919	−	0.523673i	$-0.175439\pi$
$491$	37.7545	1.70384	0.851919	+	0.523673i	$0.175439\pi$
$492$	0	0
$493$	−26.8238	−1.20808
$494$	−8.00000	−0.359937
$495$	0	0
$496$	1.00000	0.0449013
$497$	−39.3782	−1.76635
$498$	0	0
$499$	−2.92263	−0.130835	−0.0654174	−	0.997858i	$-0.520838\pi$
$499$	−2.92263	−0.130835	−0.0654174	+	0.997858i	$0.520838\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	16.5688	0.739503
$503$	−6.31389	−0.281522	−0.140761	−	0.990044i	$-0.544955\pi$
$503$	−6.31389	−0.281522	−0.140761	+	0.990044i	$0.544955\pi$
$504$	0	0
$505$	3.18572	0.141763
$506$	3.08109	0.136971
$507$	0	0
$508$	−7.15695	−0.317538
$509$	21.4044	0.948736	0.474368	−	0.880327i	$-0.342677\pi$
$509$	21.4044	0.948736	0.474368	+	0.880327i	$0.342677\pi$
$510$	0	0
$511$	−39.5949	−1.75157
$512$	1.00000	0.0441942
$513$	0	0
$514$	−29.8115	−1.31493
$515$	18.8356	0.829997
$516$	0	0
$517$	−6.84252	−0.300934
$518$	35.1971	1.54647
$519$	0	0
$520$	−1.05232	−0.0461471
$521$	16.0952	0.705146	0.352573	−	0.935784i	$-0.385307\pi$
$521$	16.0952	0.705146	0.352573	+	0.935784i	$0.385307\pi$
$522$	0	0
$523$	1.23085	0.0538213	0.0269107	−	0.999638i	$-0.491433\pi$
$523$	1.23085	0.0538213	0.0269107	+	0.999638i	$0.491433\pi$
$524$	−4.46419	−0.195019
$525$	0	0
$526$	12.0151	0.523885
$527$	−4.39400	−0.191406
$528$	0	0
$529$	−21.5401	−0.936524
$530$	−0.502345	−0.0218205
$531$	0	0
$532$	34.5902	1.49967
$533$	−6.42400	−0.278254
$534$	0	0
$535$	−14.2857	−0.617623
$536$	7.46888	0.322607
$537$	0	0
$538$	−23.2046	−1.00042
$539$	−34.9403	−1.50498
$540$	0	0
$541$	11.7927	0.507007	0.253504	−	0.967334i	$-0.418417\pi$
$541$	11.7927	0.507007	0.253504	+	0.967334i	$0.418417\pi$
$542$	−10.7116	−0.460103
$543$	0	0
$544$	−4.39400	−0.188391
$545$	−5.78331	−0.247730
$546$	0	0
$547$	−4.53112	−0.193737	−0.0968683	−	0.995297i	$-0.530883\pi$
$547$	−4.53112	−0.193737	−0.0968683	+	0.995297i	$0.530883\pi$
$548$	17.0104	0.726650
$549$	0	0
$550$	−2.54997	−0.108731
$551$	46.4091	1.97710
$552$	0	0
$553$	−4.18103	−0.177795
$554$	9.14756	0.388643
$555$	0	0
$556$	−7.49394	−0.317814
$557$	−3.71434	−0.157382	−0.0786909	−	0.996899i	$-0.525074\pi$
$557$	−3.71434	−0.157382	−0.0786909	+	0.996899i	$0.525074\pi$
$558$	0	0
$559$	6.61449	0.279763
$560$	4.54997	0.192271
$561$	0	0
$562$	−12.2093	−0.515017
$563$	32.3998	1.36549	0.682743	−	0.730658i	$-0.260787\pi$
$563$	32.3998	1.36549	0.682743	+	0.730658i	$0.260787\pi$
$564$	0	0
$565$	8.01885	0.337356
$566$	8.26431	0.347375
$567$	0	0
$568$	−8.65460	−0.363139
$569$	17.8498	0.748302	0.374151	−	0.927368i	$-0.377934\pi$
$569$	17.8498	0.748302	0.374151	+	0.927368i	$0.377934\pi$
$570$	0	0
$571$	40.6817	1.70248	0.851238	−	0.524780i	$-0.175852\pi$
$571$	40.6817	1.70248	0.851238	+	0.524780i	$0.175852\pi$
$572$	2.68337	0.112197
$573$	0	0
$574$	27.7759	1.15934
$575$	−1.20828	−0.0503889
$576$	0	0
$577$	−6.17156	−0.256925	−0.128463	−	0.991714i	$-0.541004\pi$
$577$	−6.17156	−0.256925	−0.128463	+	0.991714i	$0.541004\pi$
$578$	2.30725	0.0959688
$579$	0	0
$580$	6.10463	0.253481
$581$	10.9953	0.456162
$582$	0	0
$583$	1.28097	0.0530522
$584$	−8.70222	−0.360101
$585$	0	0
$586$	−9.68806	−0.400210
$587$	−6.31389	−0.260602	−0.130301	−	0.991474i	$-0.541594\pi$
$587$	−6.31389	−0.260602	−0.130301	+	0.991474i	$0.541594\pi$
$588$	0	0
$589$	7.60228	0.313247
$590$	−10.8356	−0.446096
$591$	0	0
$592$	7.73569	0.317935
$593$	32.3640	1.32903	0.664515	−	0.747275i	$-0.268638\pi$
$593$	32.3640	1.32903	0.664515	+	0.747275i	$0.268638\pi$
$594$	0	0
$595$	−19.9926	−0.819616
$596$	2.18103	0.0893383
$597$	0	0
$598$	1.27150	0.0519953
$599$	−13.9638	−0.570545	−0.285273	−	0.958446i	$-0.592084\pi$
$599$	−13.9638	−0.570545	−0.285273	+	0.958446i	$0.592084\pi$
$600$	0	0
$601$	−7.84525	−0.320015	−0.160007	−	0.987116i	$-0.551152\pi$
$601$	−7.84525	−0.320015	−0.160007	+	0.987116i	$0.551152\pi$
$602$	−28.5995	−1.16563
$603$	0	0
$604$	4.78800	0.194821
$605$	−4.49765	−0.182856
$606$	0	0
$607$	−26.8262	−1.08884	−0.544420	−	0.838813i	$-0.683250\pi$
$607$	−26.8262	−1.08884	−0.544420	+	0.838813i	$0.683250\pi$
$608$	7.60228	0.308313
$609$	0	0
$610$	−9.80588	−0.397028
$611$	−2.82375	−0.114237
$612$	0	0
$613$	−31.6330	−1.27765	−0.638823	−	0.769354i	$-0.720578\pi$
$613$	−31.6330	−1.27765	−0.638823	+	0.769354i	$0.720578\pi$
$614$	−24.6641	−0.995361
$615$	0	0
$616$	−11.6023	−0.467469
$617$	−0.235541	−0.00948253	−0.00474126	−	0.999989i	$-0.501509\pi$
$617$	−0.235541	−0.00948253	−0.00474126	+	0.999989i	$0.501509\pi$
$618$	0	0
$619$	−36.3271	−1.46011	−0.730054	−	0.683389i	$-0.760505\pi$
$619$	−36.3271	−1.46011	−0.730054	+	0.683389i	$0.760505\pi$
$620$	1.00000	0.0401610
$621$	0	0
$622$	7.64044	0.306354
$623$	74.7920	2.99648
$624$	0	0
$625$	1.00000	0.0400000
$626$	−13.7927	−0.551267
$627$	0	0
$628$	10.0189	0.399796
$629$	−33.9906	−1.35529
$630$	0	0
$631$	−28.4280	−1.13170	−0.565850	−	0.824508i	$-0.691452\pi$
$631$	−28.4280	−1.13170	−0.565850	+	0.824508i	$0.691452\pi$
$632$	−0.918913	−0.0365524
$633$	0	0
$634$	4.19988	0.166798
$635$	−7.15695	−0.284015
$636$	0	0
$637$	−14.4191	−0.571304
$638$	−15.5666	−0.616288
$639$	0	0
$640$	1.00000	0.0395285
$641$	30.1071	1.18916	0.594580	−	0.804037i	$-0.297318\pi$
$641$	30.1071	1.18916	0.594580	+	0.804037i	$0.297318\pi$
$642$	0	0
$643$	−7.22147	−0.284787	−0.142393	−	0.989810i	$-0.545480\pi$
$643$	−7.22147	−0.284787	−0.142393	+	0.989810i	$0.545480\pi$
$644$	−5.49765	−0.216638
$645$	0	0
$646$	−33.4044	−1.31428
$647$	42.3960	1.66676	0.833380	−	0.552700i	$-0.186402\pi$
$647$	42.3960	1.66676	0.833380	+	0.552700i	$0.186402\pi$
$648$	0	0
$649$	27.6305	1.08459
$650$	−1.05232	−0.0412752
$651$	0	0
$652$	13.6311	0.533833
$653$	8.68337	0.339807	0.169903	−	0.985461i	$-0.445654\pi$
$653$	8.68337	0.339807	0.169903	+	0.985461i	$0.445654\pi$
$654$	0	0
$655$	−4.46419	−0.174430
$656$	6.10463	0.238346
$657$	0	0
$658$	12.2093	0.475967
$659$	−34.5163	−1.34456	−0.672281	−	0.740296i	$-0.734685\pi$
$659$	−34.5163	−1.34456	−0.672281	+	0.740296i	$0.734685\pi$
$660$	0	0
$661$	−38.7806	−1.50839	−0.754195	−	0.656651i	$-0.771973\pi$
$661$	−38.7806	−1.50839	−0.754195	+	0.656651i	$0.771973\pi$
$662$	1.50332	0.0584283
$663$	0	0
$664$	2.41657	0.0937810
$665$	34.5902	1.34135
$666$	0	0
$667$	−7.37613	−0.285605
$668$	−19.9869	−0.773317
$669$	0	0
$670$	7.46888	0.288548
$671$	25.0047	0.965295
$672$	0	0
$673$	−24.5212	−0.945223	−0.472611	−	0.881271i	$-0.656689\pi$
$673$	−24.5212	−0.945223	−0.472611	+	0.881271i	$0.656689\pi$
$674$	−23.7833	−0.916099
$675$	0	0
$676$	−11.8926	−0.457409
$677$	36.1254	1.38841	0.694207	−	0.719776i	$-0.255755\pi$
$677$	36.1254	1.38841	0.694207	+	0.719776i	$0.255755\pi$
$678$	0	0
$679$	−41.8805	−1.60723
$680$	−4.39400	−0.168502
$681$	0	0
$682$	−2.54997	−0.0976434
$683$	40.3140	1.54257	0.771286	−	0.636489i	$-0.219614\pi$
$683$	40.3140	1.54257	0.771286	+	0.636489i	$0.219614\pi$
$684$	0	0
$685$	17.0104	0.649936
$686$	30.4949	1.16430
$687$	0	0
$688$	−6.28566	−0.239638
$689$	0.528626	0.0201390
$690$	0	0
$691$	−9.29973	−0.353778	−0.176889	−	0.984231i	$-0.556603\pi$
$691$	−9.29973	−0.353778	−0.176889	+	0.984231i	$0.556603\pi$
$692$	−14.5212	−0.552013
$693$	0	0
$694$	−30.9879	−1.17628
$695$	−7.49394	−0.284261
$696$	0	0
$697$	−26.8238	−1.01602
$698$	−30.6090	−1.15857
$699$	0	0
$700$	4.54997	0.171973
$701$	−34.5808	−1.30610	−0.653049	−	0.757316i	$-0.726510\pi$
$701$	−34.5808	−1.30610	−0.653049	+	0.757316i	$0.726510\pi$
$702$	0	0
$703$	58.8089	2.21802
$704$	−2.54997	−0.0961056
$705$	0	0
$706$	−25.3224	−0.953021
$707$	14.4949	0.545137
$708$	0	0
$709$	−6.16589	−0.231565	−0.115782	−	0.993275i	$-0.536938\pi$
$709$	−6.16589	−0.231565	−0.115782	+	0.993275i	$0.536938\pi$
$710$	−8.65460	−0.324801
$711$	0	0
$712$	16.4379	0.616037
$713$	−1.20828	−0.0452506
$714$	0	0
$715$	2.68337	0.100352
$716$	−16.5688	−0.619206
$717$	0	0
$718$	17.8498	0.666148
$719$	3.63052	0.135396	0.0676978	−	0.997706i	$-0.478435\pi$
$719$	3.63052	0.135396	0.0676978	+	0.997706i	$0.478435\pi$
$720$	0	0
$721$	85.7015	3.19169
$722$	38.7947	1.44379
$723$	0	0
$724$	9.31291	0.346112
$725$	6.10463	0.226720
$726$	0	0
$727$	31.2636	1.15950	0.579752	−	0.814793i	$-0.303150\pi$
$727$	31.2636	1.15950	0.579752	+	0.814793i	$0.303150\pi$
$728$	−4.78800	−0.177455
$729$	0	0
$730$	−8.70222	−0.322084
$731$	27.6192	1.02153
$732$	0	0
$733$	−9.30920	−0.343843	−0.171922	−	0.985111i	$-0.554998\pi$
$733$	−9.30920	−0.343843	−0.171922	+	0.985111i	$0.554998\pi$
$734$	−23.9901	−0.885490
$735$	0	0
$736$	−1.20828	−0.0445379
$737$	−19.0454	−0.701547
$738$	0	0
$739$	7.13189	0.262351	0.131175	−	0.991359i	$-0.458125\pi$
$739$	7.13189	0.262351	0.131175	+	0.991359i	$0.458125\pi$
$740$	7.73569	0.284370
$741$	0	0
$742$	−2.28566	−0.0839091
$743$	23.1989	0.851085	0.425543	−	0.904938i	$-0.360083\pi$
$743$	23.1989	0.851085	0.425543	+	0.904938i	$0.360083\pi$
$744$	0	0
$745$	2.18103	0.0799066
$746$	5.92360	0.216879
$747$	0	0
$748$	11.2046	0.409680
$749$	−64.9993	−2.37502
$750$	0	0
$751$	−37.6713	−1.37464	−0.687322	−	0.726353i	$-0.741214\pi$
$751$	−37.6713	−1.37464	−0.687322	+	0.726353i	$0.741214\pi$
$752$	2.68337	0.0978525
$753$	0	0
$754$	−6.42400	−0.233948
$755$	4.78800	0.174253
$756$	0	0
$757$	9.41437	0.342171	0.171086	−	0.985256i	$-0.445273\pi$
$757$	9.41437	0.342171	0.171086	+	0.985256i	$0.445273\pi$
$758$	−19.9068	−0.723047
$759$	0	0
$760$	7.60228	0.275764
$761$	1.39991	0.0507469	0.0253734	−	0.999678i	$-0.491923\pi$
$761$	1.39991	0.0507469	0.0253734	+	0.999678i	$0.491923\pi$
$762$	0	0
$763$	−26.3139	−0.952627
$764$	10.4642	0.378581
$765$	0	0
$766$	−32.8031	−1.18523
$767$	11.4025	0.411720
$768$	0	0
$769$	51.4327	1.85471	0.927355	−	0.374183i	$-0.122077\pi$
$769$	51.4327	1.85471	0.927355	+	0.374183i	$0.122077\pi$
$770$	−11.6023	−0.418117
$771$	0	0
$772$	−8.19988	−0.295120
$773$	−41.4283	−1.49007	−0.745036	−	0.667024i	$-0.767568\pi$
$773$	−41.4283	−1.49007	−0.745036	+	0.667024i	$0.767568\pi$
$774$	0	0
$775$	1.00000	0.0359211
$776$	−9.20457	−0.330425
$777$	0	0
$778$	−12.0000	−0.430221
$779$	46.4091	1.66278
$780$	0	0
$781$	22.0690	0.789690
$782$	5.30920	0.189857
$783$	0	0
$784$	13.7022	0.489365
$785$	10.0189	0.357588
$786$	0	0
$787$	−20.0858	−0.715981	−0.357990	−	0.933725i	$-0.616538\pi$
$787$	−20.0858	−0.715981	−0.357990	+	0.933725i	$0.616538\pi$
$788$	−11.8954	−0.423755
$789$	0	0
$790$	−0.918913	−0.0326935
$791$	36.4855	1.29728
$792$	0	0
$793$	10.3189	0.366434
$794$	11.2903	0.400679
$795$	0	0
$796$	8.60698	0.305066
$797$	−39.1376	−1.38633	−0.693163	−	0.720781i	$-0.743784\pi$
$797$	−39.1376	−1.38633	−0.693163	+	0.720781i	$0.743784\pi$
$798$	0	0
$799$	−11.7907	−0.417126
$800$	1.00000	0.0353553
$801$	0	0
$802$	27.7000	0.978122
$803$	22.1904	0.783083
$804$	0	0
$805$	−5.49765	−0.193767
$806$	−1.05232	−0.0370662
$807$	0	0
$808$	3.18572	0.112073
$809$	−18.1805	−0.639192	−0.319596	−	0.947554i	$-0.603547\pi$
$809$	−18.1805	−0.639192	−0.319596	+	0.947554i	$0.603547\pi$
$810$	0	0
$811$	−2.06897	−0.0726513	−0.0363256	−	0.999340i	$-0.511565\pi$
$811$	−2.06897	−0.0726513	−0.0363256	+	0.999340i	$0.511565\pi$
$812$	27.7759	0.974742
$813$	0	0
$814$	−19.7258	−0.691388
$815$	13.6311	0.477475
$816$	0	0
$817$	−47.7854	−1.67180
$818$	22.1905	0.775873
$819$	0	0
$820$	6.10463	0.213183
$821$	−10.9953	−0.383739	−0.191869	−	0.981420i	$-0.561455\pi$
$821$	−10.9953	−0.383739	−0.191869	+	0.981420i	$0.561455\pi$
$822$	0	0
$823$	29.7376	1.03659	0.518295	−	0.855202i	$-0.326567\pi$
$823$	29.7376	1.03659	0.518295	+	0.855202i	$0.326567\pi$
$824$	18.8356	0.656170
$825$	0	0
$826$	−49.3018	−1.71543
$827$	−17.3092	−0.601900	−0.300950	−	0.953640i	$-0.597304\pi$
$827$	−17.3092	−0.601900	−0.300950	+	0.953640i	$0.597304\pi$
$828$	0	0
$829$	39.7722	1.38134	0.690672	−	0.723168i	$-0.257315\pi$
$829$	39.7722	1.38134	0.690672	+	0.723168i	$0.257315\pi$
$830$	2.41657	0.0838803
$831$	0	0
$832$	−1.05232	−0.0364825
$833$	−60.2076	−2.08607
$834$	0	0
$835$	−19.9869	−0.691675
$836$	−19.3856	−0.670465
$837$	0	0
$838$	23.0449	0.796072
$839$	18.3834	0.634665	0.317333	−	0.948314i	$-0.397213\pi$
$839$	18.3834	0.634665	0.317333	+	0.948314i	$0.397213\pi$
$840$	0	0
$841$	8.26651	0.285052
$842$	−26.3621	−0.908496
$843$	0	0
$844$	−1.92360	−0.0662132
$845$	−11.8926	−0.409119
$846$	0	0
$847$	−20.4642	−0.703158
$848$	−0.502345	−0.0172506
$849$	0	0
$850$	−4.39400	−0.150713
$851$	−9.34691	−0.320408
$852$	0	0
$853$	−29.9663	−1.02603	−0.513013	−	0.858381i	$-0.671471\pi$
$853$	−29.9663	−1.02603	−0.513013	+	0.858381i	$0.671471\pi$
$854$	−44.6164	−1.52674
$855$	0	0
$856$	−14.2857	−0.488274
$857$	−38.8183	−1.32601	−0.663004	−	0.748616i	$-0.730719\pi$
$857$	−38.8183	−1.32601	−0.663004	+	0.748616i	$0.730719\pi$
$858$	0	0
$859$	−52.3009	−1.78448	−0.892242	−	0.451558i	$-0.850868\pi$
$859$	−52.3009	−1.78448	−0.892242	+	0.451558i	$0.850868\pi$
$860$	−6.28566	−0.214339
$861$	0	0
$862$	12.2643	0.417724
$863$	15.0084	0.510892	0.255446	−	0.966823i	$-0.417778\pi$
$863$	15.0084	0.510892	0.255446	+	0.966823i	$0.417778\pi$
$864$	0	0
$865$	−14.5212	−0.493736
$866$	−25.0643	−0.851719
$867$	0	0
$868$	4.54997	0.154436
$869$	2.34320	0.0794876
$870$	0	0
$871$	−7.85962	−0.266313
$872$	−5.78331	−0.195848
$873$	0	0
$874$	−9.18572	−0.310712
$875$	4.54997	0.153817
$876$	0	0
$877$	−47.0757	−1.58963	−0.794817	−	0.606850i	$-0.792433\pi$
$877$	−47.0757	−1.58963	−0.794817	+	0.606850i	$0.792433\pi$
$878$	13.3092	0.449164
$879$	0	0
$880$	−2.54997	−0.0859595
$881$	55.3400	1.86445	0.932226	−	0.361876i	$-0.117864\pi$
$881$	55.3400	1.86445	0.932226	+	0.361876i	$0.117864\pi$
$882$	0	0
$883$	−41.3112	−1.39023	−0.695117	−	0.718897i	$-0.744647\pi$
$883$	−41.3112	−1.39023	−0.695117	+	0.718897i	$0.744647\pi$
$884$	4.62387	0.155518
$885$	0	0
$886$	−35.4808	−1.19200
$887$	−17.4546	−0.586067	−0.293033	−	0.956102i	$-0.594665\pi$
$887$	−17.4546	−0.586067	−0.293033	+	0.956102i	$0.594665\pi$
$888$	0	0
$889$	−32.5639	−1.09216
$890$	16.4379	0.551000
$891$	0	0
$892$	21.8855	0.732779
$893$	20.3998	0.682652
$894$	0	0
$895$	−16.5688	−0.553835
$896$	4.54997	0.152004
$897$	0	0
$898$	−32.0476	−1.06944
$899$	6.10463	0.203601
$900$	0	0
$901$	2.20731	0.0735360
$902$	−15.5666	−0.518312
$903$	0	0
$904$	8.01885	0.266703
$905$	9.31291	0.309572
$906$	0	0
$907$	−1.45950	−0.0484619	−0.0242310	−	0.999706i	$-0.507714\pi$
$907$	−1.45950	−0.0484619	−0.0242310	+	0.999706i	$0.507714\pi$
$908$	13.4902	0.447689
$909$	0	0
$910$	−4.78800	−0.158721
$911$	−22.6710	−0.751122	−0.375561	−	0.926798i	$-0.622550\pi$
$911$	−22.6710	−0.751122	−0.375561	+	0.926798i	$0.622550\pi$
$912$	0	0
$913$	−6.16217	−0.203938
$914$	−3.38355	−0.111918
$915$	0	0
$916$	23.4677	0.775393
$917$	−20.3119	−0.670759
$918$	0	0
$919$	−19.2423	−0.634744	−0.317372	−	0.948301i	$-0.602800\pi$
$919$	−19.2423	−0.634744	−0.317372	+	0.948301i	$0.602800\pi$
$920$	−1.20828	−0.0398359
$921$	0	0
$922$	4.83314	0.159171
$923$	9.10737	0.299773
$924$	0	0
$925$	7.73569	0.254348
$926$	26.5143	0.871314
$927$	0	0
$928$	6.10463	0.200394
$929$	42.0015	1.37802	0.689012	−	0.724750i	$-0.258045\pi$
$929$	42.0015	1.37802	0.689012	+	0.724750i	$0.258045\pi$
$930$	0	0
$931$	104.168	3.41398
$932$	−21.9643	−0.719466
$933$	0	0
$934$	−21.4620	−0.702258
$935$	11.2046	0.366429
$936$	0	0
$937$	−8.40914	−0.274715	−0.137357	−	0.990522i	$-0.543861\pi$
$937$	−8.40914	−0.274715	−0.137357	+	0.990522i	$0.543861\pi$
$938$	33.9832	1.10959
$939$	0	0
$940$	2.68337	0.0875219
$941$	21.4044	0.697765	0.348883	−	0.937166i	$-0.386561\pi$
$941$	21.4044	0.697765	0.348883	+	0.937166i	$0.386561\pi$
$942$	0	0
$943$	−7.37613	−0.240200
$944$	−10.8356	−0.352670
$945$	0	0
$946$	16.0282	0.521123
$947$	18.5118	0.601553	0.300777	−	0.953695i	$-0.402754\pi$
$947$	18.5118	0.601553	0.300777	+	0.953695i	$0.402754\pi$
$948$	0	0
$949$	9.15748	0.297264
$950$	7.60228	0.246651
$951$	0	0
$952$	−19.9926	−0.647963
$953$	44.1123	1.42894	0.714469	−	0.699667i	$-0.246668\pi$
$953$	44.1123	1.42894	0.714469	+	0.699667i	$0.246668\pi$
$954$	0	0
$955$	10.4642	0.338613
$956$	12.6164	0.408045
$957$	0	0
$958$	13.7545	0.444389
$959$	77.3970	2.49928
$960$	0	0
$961$	1.00000	0.0322581
$962$	−8.14038	−0.262456
$963$	0	0
$964$	−18.7880	−0.605121
$965$	−8.19988	−0.263963
$966$	0	0
$967$	49.3995	1.58858	0.794291	−	0.607538i	$-0.207843\pi$
$967$	49.3995	1.58858	0.794291	+	0.607538i	$0.207843\pi$
$968$	−4.49765	−0.144560
$969$	0	0
$970$	−9.20457	−0.295541
$971$	−14.8733	−0.477308	−0.238654	−	0.971105i	$-0.576706\pi$
$971$	−14.8733	−0.477308	−0.238654	+	0.971105i	$0.576706\pi$
$972$	0	0
$973$	−34.0972	−1.09311
$974$	11.3662	0.364197
$975$	0	0
$976$	−9.80588	−0.313878
$977$	−4.58812	−0.146787	−0.0733935	−	0.997303i	$-0.523383\pi$
$977$	−4.58812	−0.146787	−0.0733935	+	0.997303i	$0.523383\pi$
$978$	0	0
$979$	−41.9162	−1.33965
$980$	13.7022	0.437702
$981$	0	0
$982$	37.7545	1.20480
$983$	13.2178	0.421581	0.210790	−	0.977531i	$-0.432396\pi$
$983$	13.2178	0.421581	0.210790	+	0.977531i	$0.432396\pi$
$984$	0	0
$985$	−11.8954	−0.379018
$986$	−26.8238	−0.854242
$987$	0	0
$988$	−8.00000	−0.254514
$989$	7.59486	0.241502
$990$	0	0
$991$	6.45721	0.205120	0.102560	−	0.994727i	$-0.467297\pi$
$991$	6.45721	0.205120	0.102560	+	0.994727i	$0.467297\pi$
$992$	1.00000	0.0317500
$993$	0	0
$994$	−39.3782	−1.24900
$995$	8.60698	0.272859
$996$	0	0
$997$	8.26680	0.261812	0.130906	−	0.991395i	$-0.458211\pi$
$997$	8.26680	0.261812	0.130906	+	0.991395i	$0.458211\pi$
$998$	−2.92263	−0.0925141
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2790.2.a.bk.1.4	yes	4
3.2	odd	2		2790.2.a.bj.1.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2790.2.a.bj.1.4	✓	4	3.2	odd	2
2790.2.a.bk.1.4	yes	4	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:	\( N \)	\(=\)	\( 2790 = 2 \cdot 3^{2} \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2790.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +4.54997 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +4.54997 q^{7} +1.00000 q^{8} +1.00000 q^{10} -2.54997 q^{11} -1.05232 q^{13} +4.54997 q^{14} +1.00000 q^{16} -4.39400 q^{17} +7.60228 q^{19} +1.00000 q^{20} -2.54997 q^{22} -1.20828 q^{23} +1.00000 q^{25} -1.05232 q^{26} +4.54997 q^{28} +6.10463 q^{29} +1.00000 q^{31} +1.00000 q^{32} -4.39400 q^{34} +4.54997 q^{35} +7.73569 q^{37} +7.60228 q^{38} +1.00000 q^{40} +6.10463 q^{41} -6.28566 q^{43} -2.54997 q^{44} -1.20828 q^{46} +2.68337 q^{47} +13.7022 q^{49} +1.00000 q^{50} -1.05232 q^{52} -0.502345 q^{53} -2.54997 q^{55} +4.54997 q^{56} +6.10463 q^{58} -10.8356 q^{59} -9.80588 q^{61} +1.00000 q^{62} +1.00000 q^{64} -1.05232 q^{65} +7.46888 q^{67} -4.39400 q^{68} +4.54997 q^{70} -8.65460 q^{71} -8.70222 q^{73} +7.73569 q^{74} +7.60228 q^{76} -11.6023 q^{77} -0.918913 q^{79} +1.00000 q^{80} +6.10463 q^{82} +2.41657 q^{83} -4.39400 q^{85} -6.28566 q^{86} -2.54997 q^{88} +16.4379 q^{89} -4.78800 q^{91} -1.20828 q^{92} +2.68337 q^{94} +7.60228 q^{95} -9.20457 q^{97} +13.7022 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{5} + 5 q^{7} + 4 q^{8}+O(q^{10}) \) 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^5 + 5 * q^7 + 4 * q^8	\( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{5} + 5 q^{7} + 4 q^{8} + 4 q^{10} + 3 q^{11} + 6 q^{13} + 5 q^{14} + 4 q^{16} + 7 q^{19} + 4 q^{20} + 3 q^{22} + q^{23} + 4 q^{25} + 6 q^{26} + 5 q^{28} + 4 q^{29} + 4 q^{31} + 4 q^{32} + 5 q^{35} + 6 q^{37} + 7 q^{38} + 4 q^{40} + 4 q^{41} + 13 q^{43} + 3 q^{44} + q^{46} - 4 q^{47} + 5 q^{49} + 4 q^{50} + 6 q^{52} - 5 q^{53} + 3 q^{55} + 5 q^{56} + 4 q^{58} + 8 q^{59} - 4 q^{61} + 4 q^{62} + 4 q^{64} + 6 q^{65} + 8 q^{67} + 5 q^{70} - q^{71} + 15 q^{73} + 6 q^{74} + 7 q^{76} - 23 q^{77} + 5 q^{79} + 4 q^{80} + 4 q^{82} - 2 q^{83} + 13 q^{86} + 3 q^{88} - 9 q^{89} + 16 q^{91} + q^{92} - 4 q^{94} + 7 q^{95} + 10 q^{97} + 5 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^5 + 5 * q^7 + 4 * q^8 + 4 * q^10 + 3 * q^11 + 6 * q^13 + 5 * q^14 + 4 * q^16 + 7 * q^19 + 4 * q^20 + 3 * q^22 + q^23 + 4 * q^25 + 6 * q^26 + 5 * q^28 + 4 * q^29 + 4 * q^31 + 4 * q^32 + 5 * q^35 + 6 * q^37 + 7 * q^38 + 4 * q^40 + 4 * q^41 + 13 * q^43 + 3 * q^44 + q^46 - 4 * q^47 + 5 * q^49 + 4 * q^50 + 6 * q^52 - 5 * q^53 + 3 * q^55 + 5 * q^56 + 4 * q^58 + 8 * q^59 - 4 * q^61 + 4 * q^62 + 4 * q^64 + 6 * q^65 + 8 * q^67 + 5 * q^70 - q^71 + 15 * q^73 + 6 * q^74 + 7 * q^76 - 23 * q^77 + 5 * q^79 + 4 * q^80 + 4 * q^82 - 2 * q^83 + 13 * q^86 + 3 * q^88 - 9 * q^89 + 16 * q^91 + q^92 - 4 * q^94 + 7 * q^95 + 10 * q^97 + 5 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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