LMFDB - Embedded newform 2790.2.a.bj.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	$-0.374401\pi$
$11$	2.54997	0.768845	0.384422	+	0.923157i	$0.374401\pi$
$12$	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	$-0.321121\pi$
$17$	4.39400	1.06570	0.532851	+	0.846209i	$0.321121\pi$
$18$	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	$-0.459795\pi$
$23$	1.20828	0.251945	0.125972	+	0.992034i	$0.459795\pi$
$24$	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	$-0.691819\pi$
$29$	−6.10463	−1.13360	−0.566801	+	0.823855i	$0.691819\pi$
$30$	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	$-0.658164\pi$
$41$	−6.10463	−0.953383	−0.476692	+	0.879071i	$0.658164\pi$
$42$	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	$-0.562700\pi$
$47$	−2.68337	−0.391410	−0.195705	+	0.980663i	$0.562700\pi$
$48$	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	$-0.489016\pi$
$53$	0.502345	0.0690025	0.0345012	+	0.999405i	$0.489016\pi$
$54$	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	$-0.250795\pi$
$59$	10.8356	1.41068	0.705339	+	0.708870i	$0.250795\pi$
$60$	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	$-0.328328\pi$
$71$	8.65460	1.02711	0.513556	+	0.858056i	$0.328328\pi$
$72$	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	$-0.542341\pi$
$83$	−2.41657	−0.265253	−0.132626	+	0.991166i	$0.542341\pi$
$84$	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	$-0.836663\pi$
$89$	−16.4379	−1.74242	−0.871208	+	0.490915i	$0.836663\pi$
$90$	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	$-0.550664\pi$
$101$	−3.18572	−0.316991	−0.158495	+	0.987360i	$0.550664\pi$
$102$	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	$-0.257380\pi$
$107$	14.2857	1.38105	0.690523	+	0.723310i	$0.257380\pi$
$108$	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	$-0.623105\pi$
$113$	−8.01885	−0.754350	−0.377175	+	0.926142i	$0.623105\pi$
$114$	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	$-0.437523\pi$
$131$	4.46419	0.390038	0.195019	+	0.980799i	$0.437523\pi$
$132$	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	$-0.758924\pi$
$137$	−17.0104	−1.45330	−0.726650	+	0.687008i	$0.758924\pi$
$138$	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	$-0.528475\pi$
$149$	−2.18103	−0.178677	−0.0893383	+	0.996001i	$0.528475\pi$
$150$	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	$-0.218596\pi$
$167$	19.9869	1.54663	0.773317	+	0.634020i	$0.218596\pi$
$168$	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	$-0.313860\pi$
$173$	14.5212	1.10403	0.552013	+	0.833835i	$0.313860\pi$
$174$	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	$-0.287455\pi$
$179$	16.5688	1.23841	0.619206	+	0.785229i	$0.287455\pi$
$180$	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	$-0.623588\pi$
$191$	−10.4642	−0.757162	−0.378581	+	0.925568i	$0.623588\pi$
$192$	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	$-0.360712\pi$
$197$	11.8954	0.847510	0.423755	+	0.905777i	$0.360712\pi$
$198$	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	$-0.647753\pi$
$227$	−13.4902	−0.895378	−0.447689	+	0.894189i	$0.647753\pi$
$228$	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	$-0.244387\pi$
$233$	21.9643	1.43893	0.719466	+	0.694528i	$0.244387\pi$
$234$	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	$-0.633789\pi$
$239$	−12.6164	−0.816090	−0.408045	+	0.912962i	$0.633789\pi$
$240$	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	$-0.675153\pi$
$251$	−16.5688	−1.04581	−0.522907	+	0.852390i	$0.675153\pi$
$252$	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	$-0.119983\pi$
$257$	29.8115	1.85959	0.929797	+	0.368074i	$0.119983\pi$
$258$	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	$-0.620794\pi$
$263$	−12.0151	−0.740885	−0.370443	+	0.928855i	$0.620794\pi$
$264$	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	$-0.249866\pi$
$269$	23.2046	1.41481	0.707404	+	0.706810i	$0.249866\pi$
$270$	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	$-0.381352\pi$
$281$	12.2093	0.728343	0.364172	+	0.931332i	$0.381352\pi$
$282$	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	$-0.408673\pi$
$293$	9.68806	0.565983	0.282991	+	0.959122i	$0.408673\pi$
$294$	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	$-0.569505\pi$
$311$	−7.64044	−0.433250	−0.216625	+	0.976255i	$0.569505\pi$
$312$	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	$-0.537630\pi$
$317$	−4.19988	−0.235889	−0.117944	+	0.993020i	$0.537630\pi$
$318$	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	$-0.187335\pi$
$347$	30.9879	1.66352	0.831758	+	0.555138i	$0.187335\pi$
$348$	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	$-0.264623\pi$
$353$	25.3224	1.34777	0.673887	+	0.738834i	$0.264623\pi$
$354$	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	$-0.656120\pi$
$359$	−17.8498	−0.942076	−0.471038	+	0.882113i	$0.656120\pi$
$360$	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	$-0.183678\pi$
$383$	32.8031	1.67616	0.838081	+	0.545546i	$0.183678\pi$
$384$	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	$-0.401607\pi$
$389$	12.0000	0.608424	0.304212	+	0.952604i	$0.401607\pi$
$390$	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	$-0.743110\pi$
$401$	−27.7000	−1.38327	−0.691637	+	0.722246i	$0.743110\pi$
$402$	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	$-0.690318\pi$
$419$	−23.0449	−1.12582	−0.562908	+	0.826519i	$0.690318\pi$
$420$	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	$-0.595445\pi$
$431$	−12.2643	−0.590751	−0.295376	+	0.955381i	$0.595445\pi$
$432$	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	$-0.180863\pi$
$443$	35.4808	1.68575	0.842873	+	0.538113i	$0.180863\pi$
$444$	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	$-0.227049\pi$
$449$	32.0476	1.51242	0.756210	+	0.654328i	$0.227049\pi$
$450$	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	$-0.535902\pi$
$461$	−4.83314	−0.225102	−0.112551	+	0.993646i	$0.535902\pi$
$462$	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	$-0.334592\pi$
$467$	21.4620	0.993143	0.496571	+	0.867996i	$0.334592\pi$
$468$	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	$-0.601746\pi$
$479$	−13.7545	−0.628461	−0.314230	+	0.949347i	$0.601746\pi$
$480$	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	$-0.824561\pi$
$491$	−37.7545	−1.70384	−0.851919	+	0.523673i	$0.824561\pi$
$492$	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	$-0.455045\pi$
$503$	6.31389	0.281522	0.140761	+	0.990044i	$0.455045\pi$
$504$	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	$-0.657323\pi$
$509$	−21.4044	−0.948736	−0.474368	+	0.880327i	$0.657323\pi$
$510$	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	$-0.614693\pi$
$521$	−16.0952	−0.705146	−0.352573	+	0.935784i	$0.614693\pi$
$522$	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	$-0.474926\pi$
$557$	3.71434	0.157382	0.0786909	+	0.996899i	$0.474926\pi$
$558$	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	$-0.739213\pi$
$563$	−32.3998	−1.36549	−0.682743	+	0.730658i	$0.739213\pi$
$564$	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	$-0.622066\pi$
$569$	−17.8498	−0.748302	−0.374151	+	0.927368i	$0.622066\pi$
$570$	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	$-0.458406\pi$
$587$	6.31389	0.260602	0.130301	+	0.991474i	$0.458406\pi$
$588$	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	$-0.731362\pi$
$593$	−32.3640	−1.32903	−0.664515	+	0.747275i	$0.731362\pi$
$594$	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	$-0.407916\pi$
$599$	13.9638	0.570545	0.285273	+	0.958446i	$0.407916\pi$
$600$	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	$-0.498491\pi$
$617$	0.235541	0.00948253	0.00474126	+	0.999989i	$0.498491\pi$
$618$	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	$-0.702682\pi$
$641$	−30.1071	−1.18916	−0.594580	+	0.804037i	$0.702682\pi$
$642$	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	$-0.813598\pi$
$647$	−42.3960	−1.66676	−0.833380	+	0.552700i	$0.813598\pi$
$648$	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	$-0.554346\pi$
$653$	−8.68337	−0.339807	−0.169903	+	0.985461i	$0.554346\pi$
$654$	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	$-0.265315\pi$
$659$	34.5163	1.34456	0.672281	+	0.740296i	$0.265315\pi$
$660$	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	$-0.744245\pi$
$677$	−36.1254	−1.38841	−0.694207	+	0.719776i	$0.744245\pi$
$678$	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	$-0.780386\pi$
$683$	−40.3140	−1.54257	−0.771286	+	0.636489i	$0.780386\pi$
$684$	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	$-0.273490\pi$
$701$	34.5808	1.30610	0.653049	+	0.757316i	$0.273490\pi$
$702$	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	$-0.521565\pi$
$719$	−3.63052	−0.135396	−0.0676978	+	0.997706i	$0.521565\pi$
$720$	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	$-0.639917\pi$
$743$	−23.1989	−0.851085	−0.425543	+	0.904938i	$0.639917\pi$
$744$	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	$-0.508077\pi$
$761$	−1.39991	−0.0507469	−0.0253734	+	0.999678i	$0.508077\pi$
$762$	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	$-0.232432\pi$
$773$	41.4283	1.49007	0.745036	+	0.667024i	$0.232432\pi$
$774$	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	$-0.256216\pi$
$797$	39.1376	1.38633	0.693163	+	0.720781i	$0.256216\pi$
$798$	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	$-0.396453\pi$
$809$	18.1805	0.639192	0.319596	+	0.947554i	$0.396453\pi$
$810$	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	$-0.438545\pi$
$821$	10.9953	0.383739	0.191869	+	0.981420i	$0.438545\pi$
$822$	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	$-0.402696\pi$
$827$	17.3092	0.601900	0.300950	+	0.953640i	$0.402696\pi$
$828$	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	$-0.602787\pi$
$839$	−18.3834	−0.634665	−0.317333	+	0.948314i	$0.602787\pi$
$840$	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	$-0.269281\pi$
$857$	38.8183	1.32601	0.663004	+	0.748616i	$0.269281\pi$
$858$	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	$-0.582222\pi$
$863$	−15.0084	−0.510892	−0.255446	+	0.966823i	$0.582222\pi$
$864$	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	$-0.882136\pi$
$881$	−55.3400	−1.86445	−0.932226	+	0.361876i	$0.882136\pi$
$882$	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	$-0.405335\pi$
$887$	17.4546	0.586067	0.293033	+	0.956102i	$0.405335\pi$
$888$	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	$-0.377450\pi$
$911$	22.6710	0.751122	0.375561	+	0.926798i	$0.377450\pi$
$912$	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	$-0.741955\pi$
$929$	−42.0015	−1.37802	−0.689012	+	0.724750i	$0.741955\pi$
$930$	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	$-0.613439\pi$
$941$	−21.4044	−0.697765	−0.348883	+	0.937166i	$0.613439\pi$
$942$	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	$-0.597246\pi$
$947$	−18.5118	−0.601553	−0.300777	+	0.953695i	$0.597246\pi$
$948$	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	$-0.753332\pi$
$953$	−44.1123	−1.42894	−0.714469	+	0.699667i	$0.753332\pi$
$954$	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	$-0.423294\pi$
$971$	14.8733	0.477308	0.238654	+	0.971105i	$0.423294\pi$
$972$	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	$-0.476617\pi$
$977$	4.58812	0.146787	0.0733935	+	0.997303i	$0.476617\pi$
$978$	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	$-0.567604\pi$
$983$	−13.2178	−0.421581	−0.210790	+	0.977531i	$0.567604\pi$
$984$	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.bj.1.4	✓	4
3.2	odd	2		2790.2.a.bk.1.4	yes	4

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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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