LMFDB - Embedded newform 1850.2.b.o.149.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1850,2,Mod(149,1850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1850, 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("1850.149");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1850 = 2 \cdot 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1850.b (of order $2$, degree $1$, not 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:	$14.7723243739$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.3182656.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{3} + 25x^{2} - 10x + 2 $ x^6 - 2x^3 + 25x^2 - 10*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 370)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			149.6
Root			$-1.67298 - 1.67298i$ of defining polynomial
Character	$\chi$	$=$	1850.149
Dual form			1850.2.b.o.149.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} +3.34596i q^{3} -1.00000 q^{4} -3.34596 q^{6} -2.59774i q^{7} -1.00000i q^{8} -8.19547 q^{9} +4.74823 q^{11} -3.34596i q^{12} -6.69193i q^{13} +2.59774 q^{14} +1.00000 q^{16} -0.748228i q^{17} -8.19547i q^{18} +3.34596 q^{19} +8.69193 q^{21} +4.74823i q^{22} -1.49646i q^{23} +3.34596 q^{24} +6.69193 q^{26} -17.3839i q^{27} +2.59774i q^{28} -3.94370 q^{29} +7.79321 q^{31} +1.00000i q^{32} +15.8874i q^{33} +0.748228 q^{34} +8.19547 q^{36} -1.00000i q^{37} +3.34596i q^{38} +22.3909 q^{39} -6.44724 q^{41} +8.69193i q^{42} -1.94370i q^{43} -4.74823 q^{44} +1.49646 q^{46} +1.84951i q^{47} +3.34596i q^{48} +0.251772 q^{49} +2.50354 q^{51} +6.69193i q^{52} -10.4472i q^{53} +17.3839 q^{54} -2.59774 q^{56} +11.1955i q^{57} -3.94370i q^{58} -5.84951 q^{59} +7.94370 q^{61} +7.79321i q^{62} +21.2897i q^{63} -1.00000 q^{64} -15.8874 q^{66} +1.84951i q^{67} +0.748228i q^{68} +5.00709 q^{69} -3.88740 q^{71} +8.19547i q^{72} +7.49646i q^{73} +1.00000 q^{74} -3.34596 q^{76} -12.3346i q^{77} +22.3909i q^{78} +16.5414 q^{79} +33.5793 q^{81} -6.44724i q^{82} -15.2334i q^{83} -8.69193 q^{84} +1.94370 q^{86} -13.1955i q^{87} -4.74823i q^{88} -6.00000 q^{89} -17.3839 q^{91} +1.49646i q^{92} +26.0758i q^{93} -1.84951 q^{94} -3.34596 q^{96} -10.4472i q^{97} +0.251772i q^{98} -38.9140 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{4} - 22 q^{9} + 22 q^{11} + 2 q^{14} + 6 q^{16} + 12 q^{21} + 10 q^{29} + 6 q^{31} - 2 q^{34} + 22 q^{36} + 80 q^{39} - 18 q^{41} - 22 q^{44} - 4 q^{46} + 8 q^{49} + 28 q^{51} + 24 q^{54} - 2 q^{56}+ \cdots - 38 q^{99}+O(q^{100}) $ 6 * q - 6 * q^4 - 22 * q^9 + 22 * q^11 + 2 * q^14 + 6 * q^16 + 12 * q^21 + 10 * q^29 + 6 * q^31 - 2 * q^34 + 22 * q^36 + 80 * q^39 - 18 * q^41 - 22 * q^44 - 4 * q^46 + 8 * q^49 + 28 * q^51 + 24 * q^54 - 2 * q^56 - 28 * q^59 + 14 * q^61 - 6 * q^64 - 28 * q^66 + 56 * q^69 + 44 * q^71 + 6 * q^74 + 52 * q^79 + 94 * q^81 - 12 * q^84 - 22 * q^86 - 36 * q^89 - 24 * q^91 - 4 * q^94 - 38 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1850\mathbb{Z}\right)^\times$.

$n$	$1001$	$1777$
$\chi(n)$	$1$	$-1$

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 $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	3.34596i	1.93179i	0.258929	+	0.965896i	$0.416630\pi$
$3$	3.34596i	1.93179i	−0.258929	+	0.965896i	$0.583370\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−3.34596	−1.36598
$7$	− 2.59774i	− 0.981852i	−0.871201	−	0.490926i	$-0.836659\pi$
$7$	− 2.59774i	− 0.981852i	0.871201	−	0.490926i	$-0.163341\pi$
$8$	− 1.00000i	− 0.353553i
$9$	−8.19547	−2.73182
$10$	0	0
$11$	4.74823	1.43164	0.715822	−	0.698282i	$-0.246052\pi$
$11$	4.74823	1.43164	0.715822	+	0.698282i	$0.246052\pi$
$12$	− 3.34596i	− 0.965896i
$13$	− 6.69193i	− 1.85601i	−0.372572	−	0.928003i	$-0.621524\pi$
$13$	− 6.69193i	− 1.85601i	0.372572	−	0.928003i	$-0.378476\pi$
$14$	2.59774	0.694274
$15$	0	0
$16$	1.00000	0.250000
$17$	− 0.748228i	− 0.181472i	−0.995875	−	0.0907360i	$-0.971078\pi$
$17$	− 0.748228i	− 0.181472i	0.995875	−	0.0907360i	$-0.0289219\pi$
$18$	− 8.19547i	− 1.93169i
$19$	3.34596	0.767617	0.383808	−	0.923413i	$-0.374612\pi$
$19$	3.34596	0.767617	0.383808	+	0.923413i	$0.374612\pi$
$20$	0	0
$21$	8.69193	1.89673
$22$	4.74823i	1.01233i
$23$	− 1.49646i	− 0.312033i	−0.987754	−	0.156016i	$-0.950135\pi$
$23$	− 1.49646i	− 0.312033i	0.987754	−	0.156016i	$-0.0498653\pi$
$24$	3.34596	0.682992
$25$	0	0
$26$	6.69193	1.31239
$27$	− 17.3839i	− 3.34552i
$28$	2.59774i	0.490926i
$29$	−3.94370	−0.732326	−0.366163	−	0.930551i	$-0.619329\pi$
$29$	−3.94370	−0.732326	−0.366163	+	0.930551i	$0.619329\pi$
$30$	0	0
$31$	7.79321	1.39970	0.699851	−	0.714289i	$-0.253250\pi$
$31$	7.79321	1.39970	0.699851	+	0.714289i	$0.253250\pi$
$32$	1.00000i	0.176777i
$33$	15.8874i	2.76564i
$34$	0.748228	0.128320
$35$	0	0
$36$	8.19547	1.36591
$37$	− 1.00000i	− 0.164399i
$37$	− 1.00000i	− 0.164399i
$38$	3.34596i	0.542787i
$39$	22.3909	3.58542
$40$	0	0
$41$	−6.44724	−1.00689	−0.503445	−	0.864027i	$-0.667934\pi$
$41$	−6.44724	−1.00689	−0.503445	+	0.864027i	$0.667934\pi$
$42$	8.69193i	1.34119i
$43$	− 1.94370i	− 0.296411i	−0.988957	−	0.148206i	$-0.952650\pi$
$43$	− 1.94370i	− 0.296411i	0.988957	−	0.148206i	$-0.0473497\pi$
$44$	−4.74823	−0.715822
$45$	0	0
$46$	1.49646	0.220640
$47$	1.84951i	0.269778i	0.990861	+	0.134889i	$0.0430678\pi$
$47$	1.84951i	0.269778i	−0.990861	+	0.134889i	$0.956932\pi$
$48$	3.34596i	0.482948i
$49$	0.251772	0.0359674
$50$	0	0
$51$	2.50354	0.350566
$52$	6.69193i	0.928003i
$53$	− 10.4472i	− 1.43504i	−0.696538	−	0.717520i	$-0.745277\pi$
$53$	− 10.4472i	− 1.43504i	0.696538	−	0.717520i	$-0.254723\pi$
$54$	17.3839	2.36564
$55$	0	0
$56$	−2.59774	−0.347137
$57$	11.1955i	1.48288i
$58$	− 3.94370i	− 0.517833i
$59$	−5.84951	−0.761541	−0.380770	−	0.924670i	$-0.624341\pi$
$59$	−5.84951	−0.761541	−0.380770	+	0.924670i	$0.624341\pi$
$60$	0	0
$61$	7.94370	1.01709	0.508543	−	0.861036i	$-0.330184\pi$
$61$	7.94370	1.01709	0.508543	+	0.861036i	$0.330184\pi$
$62$	7.79321i	0.989738i
$63$	21.2897i	2.68225i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−15.8874	−1.95560
$67$	1.84951i	0.225953i	0.993598	+	0.112977i	$0.0360385\pi$
$67$	1.84951i	0.225953i	−0.993598	+	0.112977i	$0.963961\pi$
$68$	0.748228i	0.0907360i
$69$	5.00709	0.602783
$70$	0	0
$71$	−3.88740	−0.461349	−0.230675	−	0.973031i	$-0.574093\pi$
$71$	−3.88740	−0.461349	−0.230675	+	0.973031i	$0.574093\pi$
$72$	8.19547i	0.965845i
$73$	7.49646i	0.877394i	0.898635	+	0.438697i	$0.144560\pi$
$73$	7.49646i	0.877394i	−0.898635	+	0.438697i	$0.855440\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	−3.34596	−0.383808
$77$	− 12.3346i	− 1.40566i
$78$	22.3909i	2.53527i
$79$	16.5414	1.86106	0.930528	−	0.366220i	$-0.119348\pi$
$79$	16.5414	1.86106	0.930528	+	0.366220i	$0.119348\pi$
$80$	0	0
$81$	33.5793	3.73104
$82$	− 6.44724i	− 0.711979i
$83$	− 15.2334i	− 1.67208i	−0.548670	−	0.836039i	$-0.684866\pi$
$83$	− 15.2334i	− 1.67208i	0.548670	−	0.836039i	$-0.315134\pi$
$84$	−8.69193	−0.948367
$85$	0	0
$86$	1.94370	0.209594
$87$	− 13.1955i	− 1.41470i
$88$	− 4.74823i	− 0.506163i
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	−17.3839	−1.82232
$92$	1.49646i	0.156016i
$93$	26.0758i	2.70393i
$94$	−1.84951	−0.190762
$95$	0	0
$96$	−3.34596	−0.341496
$97$	− 10.4472i	− 1.06076i	−0.847761	−	0.530378i	$-0.822050\pi$
$97$	− 10.4472i	− 1.06076i	0.847761	−	0.530378i	$-0.177950\pi$
$98$	0.251772i	0.0254328i
$99$	−38.9140	−3.91100
$100$	0	0
$101$	−12.1884	−1.21279	−0.606395	−	0.795164i	$-0.707385\pi$
$101$	−12.1884	−1.21279	−0.606395	+	0.795164i	$0.707385\pi$
$102$	2.50354i	0.247888i
$103$	1.30807i	0.128888i	0.997921	+	0.0644442i	$0.0205274\pi$
$103$	1.30807i	0.128888i	−0.997921	+	0.0644442i	$0.979473\pi$
$104$	−6.69193	−0.656197
$105$	0	0
$106$	10.4472	1.01473
$107$	− 3.04498i	− 0.294369i	−0.989109	−	0.147185i	$-0.952979\pi$
$107$	− 3.04498i	− 0.294369i	0.989109	−	0.147185i	$-0.0470211\pi$
$108$	17.3839i	1.67276i
$109$	1.44015	0.137942	0.0689709	−	0.997619i	$-0.478028\pi$
$109$	1.44015	0.137942	0.0689709	+	0.997619i	$0.478028\pi$
$110$	0	0
$111$	3.34596	0.317585
$112$	− 2.59774i	− 0.245463i
$113$	− 11.1392i	− 1.04788i	−0.851754	−	0.523942i	$-0.824461\pi$
$113$	− 11.1392i	− 1.04788i	0.851754	−	0.523942i	$-0.175539\pi$
$114$	−11.1955	−1.04855
$115$	0	0
$116$	3.94370	0.366163
$117$	54.8435i	5.07028i
$118$	− 5.84951i	− 0.538491i
$119$	−1.94370	−0.178179
$120$	0	0
$121$	11.5457	1.04961
$122$	7.94370i	0.719189i
$123$	− 21.5722i	− 1.94510i
$124$	−7.79321	−0.699851
$125$	0	0
$126$	−21.2897	−1.89663
$127$	− 8.84242i	− 0.784638i	−0.919829	−	0.392319i	$-0.871673\pi$
$127$	− 8.84242i	− 0.784638i	0.919829	−	0.392319i	$-0.128327\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	6.50354	0.572605
$130$	0	0
$131$	6.15049	0.537371	0.268686	−	0.963228i	$-0.413411\pi$
$131$	6.15049	0.537371	0.268686	+	0.963228i	$0.413411\pi$
$132$	− 15.8874i	− 1.38282i
$133$	− 8.69193i	− 0.753686i
$134$	−1.84951	−0.159773
$135$	0	0
$136$	−0.748228	−0.0641600
$137$	10.8945i	0.930779i	0.885106	+	0.465389i	$0.154086\pi$
$137$	10.8945i	0.930779i	−0.885106	+	0.465389i	$0.845914\pi$
$138$	5.00709i	0.426232i
$139$	−1.04921	−0.0889932	−0.0444966	−	0.999010i	$-0.514168\pi$
$139$	−1.04921	−0.0889932	−0.0444966	+	0.999010i	$0.514168\pi$
$140$	0	0
$141$	−6.18838	−0.521156
$142$	− 3.88740i	− 0.326223i
$143$	− 31.7748i	− 2.65714i
$144$	−8.19547	−0.682956
$145$	0	0
$146$	−7.49646	−0.620411
$147$	0.842420i	0.0694816i
$148$	1.00000i	0.0821995i
$149$	4.18838	0.343126	0.171563	−	0.985173i	$-0.445118\pi$
$149$	4.18838	0.343126	0.171563	+	0.985173i	$0.445118\pi$
$150$	0	0
$151$	−6.69193	−0.544581	−0.272291	−	0.962215i	$-0.587781\pi$
$151$	−6.69193	−0.544581	−0.272291	+	0.962215i	$0.587781\pi$
$152$	− 3.34596i	− 0.271393i
$153$	6.13208i	0.495749i
$154$	12.3346	0.993954
$155$	0	0
$156$	−22.3909	−1.79271
$157$	− 3.94370i	− 0.314741i	−0.987540	−	0.157371i	$-0.949698\pi$
$157$	− 3.94370i	− 0.314741i	0.987540	−	0.157371i	$-0.0503018\pi$
$158$	16.5414i	1.31597i
$159$	34.9561	2.77220
$160$	0	0
$161$	−3.88740	−0.306370
$162$	33.5793i	2.63824i
$163$	12.1463i	0.951368i	0.879616	+	0.475684i	$0.157799\pi$
$163$	12.1463i	0.951368i	−0.879616	+	0.475684i	$0.842201\pi$
$164$	6.44724	0.503445
$165$	0	0
$166$	15.2334	1.18234
$167$	17.0829i	1.32191i	0.750425	+	0.660956i	$0.229849\pi$
$167$	17.0829i	1.32191i	−0.750425	+	0.660956i	$0.770151\pi$
$168$	− 8.69193i	− 0.670597i
$169$	−31.7819	−2.44476
$170$	0	0
$171$	−27.4217	−2.09699
$172$	1.94370i	0.148206i
$173$	15.3417i	1.16641i	0.812325	+	0.583205i	$0.198202\pi$
$173$	15.3417i	1.16641i	−0.812325	+	0.583205i	$0.801798\pi$
$174$	13.1955	1.00035
$175$	0	0
$176$	4.74823	0.357911
$177$	− 19.5722i	− 1.47114i
$178$	− 6.00000i	− 0.449719i
$179$	−2.45148	−0.183232	−0.0916161	−	0.995794i	$-0.529203\pi$
$179$	−2.45148	−0.183232	−0.0916161	+	0.995794i	$0.529203\pi$
$180$	0	0
$181$	13.1955	0.980812	0.490406	−	0.871494i	$-0.336849\pi$
$181$	13.1955	0.980812	0.490406	+	0.871494i	$0.336849\pi$
$182$	− 17.3839i	− 1.28858i
$183$	26.5793i	1.96480i
$184$	−1.49646	−0.110320
$185$	0	0
$186$	−26.0758	−1.91197
$187$	− 3.55276i	− 0.259803i
$188$	− 1.84951i	− 0.134889i
$189$	−45.1586	−3.28481
$190$	0	0
$191$	−21.7790	−1.57588	−0.787938	−	0.615755i	$-0.788851\pi$
$191$	−21.7790	−1.57588	−0.787938	+	0.615755i	$0.788851\pi$
$192$	− 3.34596i	− 0.241474i
$193$	12.9929i	0.935250i	0.883927	+	0.467625i	$0.154890\pi$
$193$	12.9929i	0.935250i	−0.883927	+	0.467625i	$0.845110\pi$
$194$	10.4472	0.750068
$195$	0	0
$196$	−0.251772	−0.0179837
$197$	0.616147i	0.0438986i	0.999759	+	0.0219493i	$0.00698725\pi$
$197$	0.616147i	0.0438986i	−0.999759	+	0.0219493i	$0.993013\pi$
$198$	− 38.9140i	− 2.76549i
$199$	1.54852	0.109772	0.0548859	−	0.998493i	$-0.482520\pi$
$199$	1.54852	0.109772	0.0548859	+	0.998493i	$0.482520\pi$
$200$	0	0
$201$	−6.18838	−0.436495
$202$	− 12.1884i	− 0.857572i
$203$	10.2447i	0.719036i
$204$	−2.50354	−0.175283
$205$	0	0
$206$	−1.30807	−0.0911378
$207$	12.2642i	0.852418i
$208$	− 6.69193i	− 0.464002i
$209$	15.8874	1.09895
$210$	0	0
$211$	20.9366	1.44134	0.720668	−	0.693280i	$-0.243835\pi$
$211$	20.9366	1.44134	0.720668	+	0.693280i	$0.243835\pi$
$212$	10.4472i	0.717520i
$213$	− 13.0071i	− 0.891231i
$214$	3.04498	0.208150
$215$	0	0
$216$	−17.3839	−1.18282
$217$	− 20.2447i	− 1.37430i
$218$	1.44015i	0.0975396i
$219$	−25.0829	−1.69494
$220$	0	0
$221$	−5.00709	−0.336813
$222$	3.34596i	0.224566i
$223$	2.71034i	0.181498i	0.995874	+	0.0907488i	$0.0289260\pi$
$223$	2.71034i	0.181498i	−0.995874	+	0.0907488i	$0.971074\pi$
$224$	2.59774	0.173568
$225$	0	0
$226$	11.1392	0.740966
$227$	26.1321i	1.73445i	0.497919	+	0.867224i	$0.334098\pi$
$227$	26.1321i	1.73445i	−0.497919	+	0.867224i	$0.665902\pi$
$228$	− 11.1955i	− 0.741438i
$229$	24.7677	1.63670	0.818348	−	0.574723i	$-0.194890\pi$
$229$	24.7677	1.63670	0.818348	+	0.574723i	$0.194890\pi$
$230$	0	0
$231$	41.2713	2.71545
$232$	3.94370i	0.258916i
$233$	12.2783i	0.804381i	0.915556	+	0.402190i	$0.131751\pi$
$233$	12.2783i	0.804381i	−0.915556	+	0.402190i	$0.868249\pi$
$234$	−54.8435	−3.58523
$235$	0	0
$236$	5.84951	0.380770
$237$	55.3470i	3.59518i
$238$	− 1.94370i	− 0.125991i
$239$	17.1771	1.11109	0.555546	−	0.831486i	$-0.312509\pi$
$239$	17.1771	1.11109	0.555546	+	0.831486i	$0.312509\pi$
$240$	0	0
$241$	−15.2713	−0.983708	−0.491854	−	0.870678i	$-0.663681\pi$
$241$	−15.2713	−0.983708	−0.491854	+	0.870678i	$0.663681\pi$
$242$	11.5457i	0.742184i
$243$	60.2036i	3.86206i
$244$	−7.94370	−0.508543
$245$	0	0
$246$	21.5722	1.37540
$247$	− 22.3909i	− 1.42470i
$248$	− 7.79321i	− 0.494869i
$249$	50.9703	3.23011
$250$	0	0
$251$	10.0379	0.633586	0.316793	−	0.948495i	$-0.397394\pi$
$251$	10.0379	0.633586	0.316793	+	0.948495i	$0.397394\pi$
$252$	− 21.2897i	− 1.34112i
$253$	− 7.10552i	− 0.446720i
$254$	8.84242	0.554823
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 20.9703i	− 1.30809i	−0.756456	−	0.654045i	$-0.773071\pi$
$257$	− 20.9703i	− 1.30809i	0.756456	−	0.654045i	$-0.226929\pi$
$258$	6.50354i	0.404893i
$259$	−2.59774	−0.161415
$260$	0	0
$261$	32.3205	2.00059
$262$	6.15049i	0.379979i
$263$	− 22.3725i	− 1.37955i	−0.724024	−	0.689775i	$-0.757710\pi$
$263$	− 22.3725i	− 1.37955i	0.724024	−	0.689775i	$-0.242290\pi$
$264$	15.8874	0.977802
$265$	0	0
$266$	8.69193	0.532936
$267$	− 20.0758i	− 1.22862i
$268$	− 1.84951i	− 0.112977i
$269$	4.18838	0.255370	0.127685	−	0.991815i	$-0.459245\pi$
$269$	4.18838	0.255370	0.127685	+	0.991815i	$0.459245\pi$
$270$	0	0
$271$	−12.1126	−0.735788	−0.367894	−	0.929868i	$-0.619921\pi$
$271$	−12.1126	−0.735788	−0.367894	+	0.929868i	$0.619921\pi$
$272$	− 0.748228i	− 0.0453680i
$273$	− 58.1657i	− 3.52035i
$274$	−10.8945	−0.658160
$275$	0	0
$276$	−5.00709	−0.301391
$277$	− 9.30807i	− 0.559268i	−0.960107	−	0.279634i	$-0.909787\pi$
$277$	− 9.30807i	− 0.559268i	0.960107	−	0.279634i	$-0.0902131\pi$
$278$	− 1.04921i	− 0.0629277i
$279$	−63.8690	−3.82374
$280$	0	0
$281$	19.2713	1.14963	0.574813	−	0.818285i	$-0.305075\pi$
$281$	19.2713	1.14963	0.574813	+	0.818285i	$0.305075\pi$
$282$	− 6.18838i	− 0.368513i
$283$	13.6848i	0.813479i	0.913544	+	0.406740i	$0.133334\pi$
$283$	13.6848i	0.813479i	−0.913544	+	0.406740i	$0.866666\pi$
$284$	3.88740	0.230675
$285$	0	0
$286$	31.7748	1.87888
$287$	16.7482i	0.988617i
$288$	− 8.19547i	− 0.482923i
$289$	16.4402	0.967068
$290$	0	0
$291$	34.9561	2.04916
$292$	− 7.49646i	− 0.438697i
$293$	10.4472i	0.610334i	0.952299	+	0.305167i	$0.0987124\pi$
$293$	10.4472i	0.610334i	−0.952299	+	0.305167i	$0.901288\pi$
$294$	−0.842420	−0.0491309
$295$	0	0
$296$	−1.00000	−0.0581238
$297$	− 82.5425i	− 4.78960i
$298$	4.18838i	0.242627i
$299$	−10.0142	−0.579135
$300$	0	0
$301$	−5.04921	−0.291032
$302$	− 6.69193i	− 0.385077i
$303$	− 40.7819i	− 2.34286i
$304$	3.34596	0.191904
$305$	0	0
$306$	−6.13208	−0.350548
$307$	− 4.95502i	− 0.282798i	−0.989953	−	0.141399i	$-0.954840\pi$
$307$	− 4.95502i	− 0.282798i	0.989953	−	0.141399i	$-0.0451601\pi$
$308$	12.3346i	0.702831i
$309$	−4.37677	−0.248985
$310$	0	0
$311$	3.68060	0.208708	0.104354	−	0.994540i	$-0.466723\pi$
$311$	3.68060	0.208708	0.104354	+	0.994540i	$0.466723\pi$
$312$	− 22.3909i	− 1.26764i
$313$	0.992912i	0.0561227i	0.999606	+	0.0280614i	$0.00893338\pi$
$313$	0.992912i	0.0561227i	−0.999606	+	0.0280614i	$0.991067\pi$
$314$	3.94370	0.222556
$315$	0	0
$316$	−16.5414	−0.930528
$317$	− 6.55985i	− 0.368438i	−0.982885	−	0.184219i	$-0.941024\pi$
$317$	− 6.55985i	− 0.368438i	0.982885	−	0.184219i	$-0.0589755\pi$
$318$	34.9561i	1.96024i
$319$	−18.7256	−1.04843
$320$	0	0
$321$	10.1884	0.568660
$322$	− 3.88740i	− 0.216636i
$323$	− 2.50354i	− 0.139301i
$324$	−33.5793	−1.86552
$325$	0	0
$326$	−12.1463	−0.672719
$327$	4.81870i	0.266475i
$328$	6.44724i	0.355989i
$329$	4.80453	0.264882
$330$	0	0
$331$	−18.0379	−0.991452	−0.495726	−	0.868479i	$-0.665098\pi$
$331$	−18.0379	−0.991452	−0.495726	+	0.868479i	$0.665098\pi$
$332$	15.2334i	0.836039i
$333$	8.19547i	0.449109i
$334$	−17.0829	−0.934733
$335$	0	0
$336$	8.69193	0.474183
$337$	− 14.8945i	− 0.811354i	−0.914016	−	0.405677i	$-0.867036\pi$
$337$	− 14.8945i	− 0.811354i	0.914016	−	0.405677i	$-0.132964\pi$
$338$	− 31.7819i	− 1.72871i
$339$	37.2713	2.02430
$340$	0	0
$341$	37.0039	2.00387
$342$	− 27.4217i	− 1.48280i
$343$	− 18.8382i	− 1.01717i
$344$	−1.94370	−0.104797
$345$	0	0
$346$	−15.3417	−0.824776
$347$	− 9.38385i	− 0.503752i	−0.967760	−	0.251876i	$-0.918953\pi$
$347$	− 9.38385i	− 0.503752i	0.967760	−	0.251876i	$-0.0810475\pi$
$348$	13.1955i	0.707351i
$349$	32.9703	1.76486	0.882429	−	0.470446i	$-0.155907\pi$
$349$	32.9703	1.76486	0.882429	+	0.470446i	$0.155907\pi$
$350$	0	0
$351$	−116.331	−6.20931
$352$	4.74823i	0.253081i
$353$	− 27.4260i	− 1.45974i	−0.683587	−	0.729869i	$-0.739581\pi$
$353$	− 27.4260i	− 1.45974i	0.683587	−	0.729869i	$-0.260419\pi$
$354$	19.5722	1.04025
$355$	0	0
$356$	6.00000	0.317999
$357$	− 6.50354i	− 0.344204i
$358$	− 2.45148i	− 0.129565i
$359$	25.0829	1.32382	0.661912	−	0.749582i	$-0.269745\pi$
$359$	25.0829	1.32382	0.661912	+	0.749582i	$0.269745\pi$
$360$	0	0
$361$	−7.80453	−0.410765
$362$	13.1955i	0.693539i
$363$	38.6314i	2.02762i
$364$	17.3839	0.911161
$365$	0	0
$366$	−26.5793	−1.38932
$367$	2.48513i	0.129723i	0.997894	+	0.0648614i	$0.0206605\pi$
$367$	2.48513i	0.129723i	−0.997894	+	0.0648614i	$0.979339\pi$
$368$	− 1.49646i	− 0.0780082i
$369$	52.8382	2.75065
$370$	0	0
$371$	−27.1392	−1.40900
$372$	− 26.0758i	− 1.35197i
$373$	35.1586i	1.82045i	0.414119	+	0.910223i	$0.364090\pi$
$373$	35.1586i	1.82045i	−0.414119	+	0.910223i	$0.635910\pi$
$374$	3.55276	0.183709
$375$	0	0
$376$	1.84951	0.0953810
$377$	26.3909i	1.35920i
$378$	− 45.1586i	− 2.32271i
$379$	−24.6778	−1.26761	−0.633805	−	0.773492i	$-0.718508\pi$
$379$	−24.6778	−1.26761	−0.633805	+	0.773492i	$0.718508\pi$
$380$	0	0
$381$	29.5864	1.51576
$382$	− 21.7790i	− 1.11431i
$383$	13.3839i	0.683883i	0.939721	+	0.341941i	$0.111084\pi$
$383$	13.3839i	0.683883i	−0.939721	+	0.341941i	$0.888916\pi$
$384$	3.34596	0.170748
$385$	0	0
$386$	−12.9929	−0.661322
$387$	15.9295i	0.809743i
$388$	10.4472i	0.530378i
$389$	−18.2220	−0.923894	−0.461947	−	0.886908i	$-0.652849\pi$
$389$	−18.2220	−0.923894	−0.461947	+	0.886908i	$0.652849\pi$
$390$	0	0
$391$	−1.11969	−0.0566252
$392$	− 0.251772i	− 0.0127164i
$393$	20.5793i	1.03809i
$394$	−0.616147	−0.0310410
$395$	0	0
$396$	38.9140	1.95550
$397$	− 2.22521i	− 0.111680i	−0.998440	−	0.0558399i	$-0.982216\pi$
$397$	− 2.22521i	− 0.111680i	0.998440	−	0.0558399i	$-0.0177837\pi$
$398$	1.54852i	0.0776204i
$399$	29.0829	1.45596
$400$	0	0
$401$	12.3909	0.618774	0.309387	−	0.950936i	$-0.399876\pi$
$401$	12.3909	0.618774	0.309387	+	0.950936i	$0.399876\pi$
$402$	− 6.18838i	− 0.308648i
$403$	− 52.1516i	− 2.59785i
$404$	12.1884	0.606395
$405$	0	0
$406$	−10.2447	−0.508435
$407$	− 4.74823i	− 0.235361i
$408$	− 2.50354i	− 0.123944i
$409$	−7.27125	−0.359540	−0.179770	−	0.983709i	$-0.557535\pi$
$409$	−7.27125	−0.359540	−0.179770	+	0.983709i	$0.557535\pi$
$410$	0	0
$411$	−36.4525	−1.79807
$412$	− 1.30807i	− 0.0644442i
$413$	15.1955i	0.747720i
$414$	−12.2642	−0.602751
$415$	0	0
$416$	6.69193	0.328099
$417$	− 3.51063i	− 0.171916i
$418$	15.8874i	0.777078i
$419$	−28.4809	−1.39138	−0.695691	−	0.718341i	$-0.744902\pi$
$419$	−28.4809	−1.39138	−0.695691	+	0.718341i	$0.744902\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	20.9366i	1.01918i
$423$	− 15.1576i	− 0.736987i
$424$	−10.4472	−0.507363
$425$	0	0
$426$	13.0071	0.630195
$427$	− 20.6356i	− 0.998628i
$428$	3.04498i	0.147185i
$429$	106.317	5.13305
$430$	0	0
$431$	1.51487	0.0729686	0.0364843	−	0.999334i	$-0.488384\pi$
$431$	1.51487	0.0729686	0.0364843	+	0.999334i	$0.488384\pi$
$432$	− 17.3839i	− 0.836381i
$433$	− 24.9929i	− 1.20108i	−0.799594	−	0.600541i	$-0.794952\pi$
$433$	− 24.9929i	− 1.20108i	0.799594	−	0.600541i	$-0.205048\pi$
$434$	20.2447	0.971776
$435$	0	0
$436$	−1.44015	−0.0689709
$437$	− 5.00709i	− 0.239521i
$438$	− 25.0829i	− 1.19851i
$439$	5.10128	0.243471	0.121735	−	0.992563i	$-0.461154\pi$
$439$	5.10128	0.243471	0.121735	+	0.992563i	$0.461154\pi$
$440$	0	0
$441$	−2.06339	−0.0982566
$442$	− 5.00709i	− 0.238163i
$443$	− 25.7369i	− 1.22280i	−0.791323	−	0.611399i	$-0.790607\pi$
$443$	− 25.7369i	− 1.22280i	0.791323	−	0.611399i	$-0.209393\pi$
$444$	−3.34596	−0.158792
$445$	0	0
$446$	−2.71034	−0.128338
$447$	14.0142i	0.662848i
$448$	2.59774i	0.122731i
$449$	17.7748	0.838844	0.419422	−	0.907791i	$-0.362233\pi$
$449$	17.7748	0.838844	0.419422	+	0.907791i	$0.362233\pi$
$450$	0	0
$451$	−30.6130	−1.44151
$452$	11.1392i	0.523942i
$453$	− 22.3909i	− 1.05202i
$454$	−26.1321	−1.22644
$455$	0	0
$456$	11.1955	0.524276
$457$	34.3346i	1.60611i	0.595907	+	0.803053i	$0.296793\pi$
$457$	34.3346i	1.60611i	−0.595907	+	0.803053i	$0.703207\pi$
$458$	24.7677i	1.15732i
$459$	−13.0071	−0.607119
$460$	0	0
$461$	−16.9508	−0.789477	−0.394738	−	0.918794i	$-0.629165\pi$
$461$	−16.9508	−0.789477	−0.394738	+	0.918794i	$0.629165\pi$
$462$	41.2713i	1.92011i
$463$	− 10.9929i	− 0.510884i	−0.966824	−	0.255442i	$-0.917779\pi$
$463$	− 10.9929i	− 0.510884i	0.966824	−	0.255442i	$-0.0822210\pi$
$464$	−3.94370	−0.183082
$465$	0	0
$466$	−12.2783	−0.568783
$467$	37.4175i	1.73148i	0.500498	+	0.865738i	$0.333150\pi$
$467$	37.4175i	1.73148i	−0.500498	+	0.865738i	$0.666850\pi$
$468$	− 54.8435i	− 2.53514i
$469$	4.80453	0.221853
$470$	0	0
$471$	13.1955	0.608015
$472$	5.84951i	0.269245i
$473$	− 9.22912i	− 0.424356i
$474$	−55.3470	−2.54217
$475$	0	0
$476$	1.94370	0.0890893
$477$	85.6201i	3.92027i
$478$	17.1771i	0.785660i
$479$	0.465654	0.0212763	0.0106381	−	0.999943i	$-0.496614\pi$
$479$	0.465654	0.0212763	0.0106381	+	0.999943i	$0.496614\pi$
$480$	0	0
$481$	−6.69193	−0.305126
$482$	− 15.2713i	− 0.695586i
$483$	− 13.0071i	− 0.591843i
$484$	−11.5457	−0.524803
$485$	0	0
$486$	−60.2036	−2.73089
$487$	34.9561i	1.58401i	0.610514	+	0.792006i	$0.290963\pi$
$487$	34.9561i	1.58401i	−0.610514	+	0.792006i	$0.709037\pi$
$488$	− 7.94370i	− 0.359594i
$489$	−40.6409	−1.83785
$490$	0	0
$491$	−24.7819	−1.11839	−0.559195	−	0.829036i	$-0.688890\pi$
$491$	−24.7819	−1.11839	−0.559195	+	0.829036i	$0.688890\pi$
$492$	21.5722i	0.972552i
$493$	2.95079i	0.132897i
$494$	22.3909	1.00742
$495$	0	0
$496$	7.79321	0.349925
$497$	10.0984i	0.452976i
$498$	50.9703i	2.28403i
$499$	2.33888	0.104702	0.0523512	−	0.998629i	$-0.483328\pi$
$499$	2.33888	0.104702	0.0523512	+	0.998629i	$0.483328\pi$
$500$	0	0
$501$	−57.1586	−2.55366
$502$	10.0379i	0.448013i
$503$	18.6161i	0.830053i	0.909809	+	0.415026i	$0.136228\pi$
$503$	18.6161i	0.830053i	−0.909809	+	0.415026i	$0.863772\pi$
$504$	21.2897	0.948317
$505$	0	0
$506$	7.10552	0.315879
$507$	− 106.341i	− 4.72277i
$508$	8.84242i	0.392319i
$509$	−6.89448	−0.305593	−0.152796	−	0.988258i	$-0.548828\pi$
$509$	−6.89448	−0.305593	−0.152796	+	0.988258i	$0.548828\pi$
$510$	0	0
$511$	19.4738	0.861471
$512$	1.00000i	0.0441942i
$513$	− 58.1657i	− 2.56808i
$514$	20.9703	0.924959
$515$	0	0
$516$	−6.50354	−0.286303
$517$	8.78188i	0.386227i
$518$	− 2.59774i	− 0.114138i
$519$	−51.3329	−2.25326
$520$	0	0
$521$	4.86083	0.212957	0.106478	−	0.994315i	$-0.466042\pi$
$521$	4.86083	0.212957	0.106478	+	0.994315i	$0.466042\pi$
$522$	32.3205i	1.41463i
$523$	− 22.7677i	− 0.995562i	−0.867303	−	0.497781i	$-0.834148\pi$
$523$	− 22.7677i	− 0.995562i	0.867303	−	0.497781i	$-0.165852\pi$
$524$	−6.15049	−0.268686
$525$	0	0
$526$	22.3725	0.975489
$527$	− 5.83110i	− 0.254007i
$528$	15.8874i	0.691410i
$529$	20.7606	0.902636
$530$	0	0
$531$	47.9395	2.08040
$532$	8.69193i	0.376843i
$533$	43.1445i	1.86879i
$534$	20.0758	0.868764
$535$	0	0
$536$	1.84951	0.0798865
$537$	− 8.20256i	− 0.353967i
$538$	4.18838i	0.180574i
$539$	1.19547	0.0514926
$540$	0	0
$541$	8.99291	0.386636	0.193318	−	0.981136i	$-0.438075\pi$
$541$	8.99291	0.386636	0.193318	+	0.981136i	$0.438075\pi$
$542$	− 12.1126i	− 0.520281i
$543$	44.1516i	1.89472i
$544$	0.748228	0.0320800
$545$	0	0
$546$	58.1657	2.48926
$547$	− 18.8382i	− 0.805463i	−0.915318	−	0.402731i	$-0.868061\pi$
$547$	− 18.8382i	− 0.805463i	0.915318	−	0.402731i	$-0.131939\pi$
$548$	− 10.8945i	− 0.465389i
$549$	−65.1023	−2.77850
$550$	0	0
$551$	−13.1955	−0.562146
$552$	− 5.00709i	− 0.213116i
$553$	− 42.9703i	− 1.82728i
$554$	9.30807	0.395462
$555$	0	0
$556$	1.04921	0.0444966
$557$	3.69901i	0.156732i	0.996925	+	0.0783661i	$0.0249703\pi$
$557$	3.69901i	0.156732i	−0.996925	+	0.0783661i	$0.975030\pi$
$558$	− 63.8690i	− 2.70379i
$559$	−13.0071	−0.550141
$560$	0	0
$561$	11.8874	0.501886
$562$	19.2713i	0.812909i
$563$	14.3488i	0.604730i	0.953192	+	0.302365i	$0.0977762\pi$
$563$	14.3488i	0.604730i	−0.953192	+	0.302365i	$0.902224\pi$
$564$	6.18838	0.260578
$565$	0	0
$566$	−13.6848	−0.575217
$567$	− 87.2302i	− 3.66332i
$568$	3.88740i	0.163112i
$569$	−15.9858	−0.670161	−0.335080	−	0.942190i	$-0.608763\pi$
$569$	−15.9858	−0.670161	−0.335080	+	0.942190i	$0.608763\pi$
$570$	0	0
$571$	−30.2079	−1.26416	−0.632080	−	0.774903i	$-0.717799\pi$
$571$	−30.2079	−1.26416	−0.632080	+	0.774903i	$0.717799\pi$
$572$	31.7748i	1.32857i
$573$	− 72.8718i	− 3.04426i
$574$	−16.7482	−0.699058
$575$	0	0
$576$	8.19547	0.341478
$577$	− 19.1813i	− 0.798528i	−0.916836	−	0.399264i	$-0.869266\pi$
$577$	− 19.1813i	− 0.798528i	0.916836	−	0.399264i	$-0.130734\pi$
$578$	16.4402i	0.683820i
$579$	−43.4738	−1.80671
$580$	0	0
$581$	−39.5722	−1.64173
$582$	34.9561i	1.44898i
$583$	− 49.6059i	− 2.05447i
$584$	7.49646	0.310206
$585$	0	0
$586$	−10.4472	−0.431572
$587$	− 26.6130i	− 1.09844i	−0.835679	−	0.549218i	$-0.814926\pi$
$587$	− 26.6130i	− 1.09844i	0.835679	−	0.549218i	$-0.185074\pi$
$588$	− 0.842420i	− 0.0347408i
$589$	26.0758	1.07443
$590$	0	0
$591$	−2.06160	−0.0848031
$592$	− 1.00000i	− 0.0410997i
$593$	− 26.2642i	− 1.07854i	−0.842133	−	0.539270i	$-0.818700\pi$
$593$	− 26.2642i	− 1.07854i	0.842133	−	0.539270i	$-0.181300\pi$
$594$	82.5425	3.38676
$595$	0	0
$596$	−4.18838	−0.171563
$597$	5.18130i	0.212056i
$598$	− 10.0142i	− 0.409510i
$599$	38.2415	1.56251	0.781253	−	0.624215i	$-0.214581\pi$
$599$	38.2415	1.56251	0.781253	+	0.624215i	$0.214581\pi$
$600$	0	0
$601$	30.2447	1.23371	0.616853	−	0.787078i	$-0.288407\pi$
$601$	30.2447	1.23371	0.616853	+	0.787078i	$0.288407\pi$
$602$	− 5.04921i	− 0.205791i
$603$	− 15.1576i	− 0.617264i
$604$	6.69193	0.272291
$605$	0	0
$606$	40.7819	1.65665
$607$	− 5.90157i	− 0.239537i	−0.992802	−	0.119769i	$-0.961785\pi$
$607$	− 5.90157i	− 0.239537i	0.992802	−	0.119769i	$-0.0382153\pi$
$608$	3.34596i	0.135697i
$609$	−34.2783	−1.38903
$610$	0	0
$611$	12.3768	0.500710
$612$	− 6.13208i	− 0.247875i
$613$	− 6.15473i	− 0.248587i	−0.992245	−	0.124294i	$-0.960334\pi$
$613$	− 6.15473i	− 0.248587i	0.992245	−	0.124294i	$-0.0396665\pi$
$614$	4.95502	0.199968
$615$	0	0
$616$	−12.3346	−0.496977
$617$	16.5035i	0.664408i	0.943208	+	0.332204i	$0.107792\pi$
$617$	16.5035i	0.664408i	−0.943208	+	0.332204i	$0.892208\pi$
$618$	− 4.37677i	− 0.176059i
$619$	−9.04921	−0.363719	−0.181859	−	0.983325i	$-0.558212\pi$
$619$	−9.04921	−0.363719	−0.181859	+	0.983325i	$0.558212\pi$
$620$	0	0
$621$	−26.0142	−1.04391
$622$	3.68060i	0.147579i
$623$	15.5864i	0.624456i
$624$	22.3909	0.896355
$625$	0	0
$626$	−0.992912	−0.0396848
$627$	53.1586i	2.12295i
$628$	3.94370i	0.157371i
$629$	−0.748228	−0.0298338
$630$	0	0
$631$	−0.507780	−0.0202144	−0.0101072	−	0.999949i	$-0.503217\pi$
$631$	−0.507780	−0.0202144	−0.0101072	+	0.999949i	$0.503217\pi$
$632$	− 16.5414i	− 0.657983i
$633$	70.0531i	2.78436i
$634$	6.55985	0.260525
$635$	0	0
$636$	−34.9561	−1.38610
$637$	− 1.68484i	− 0.0667558i
$638$	− 18.7256i	− 0.741353i
$639$	31.8590	1.26032
$640$	0	0
$641$	−40.6356	−1.60501	−0.802505	−	0.596645i	$-0.796500\pi$
$641$	−40.6356	−1.60501	−0.802505	+	0.596645i	$0.796500\pi$
$642$	10.1884i	0.402103i
$643$	9.53011i	0.375831i	0.982185	+	0.187915i	$0.0601731\pi$
$643$	9.53011i	0.375831i	−0.982185	+	0.187915i	$0.939827\pi$
$644$	3.88740	0.153185
$645$	0	0
$646$	2.50354	0.0985006
$647$	− 21.0071i	− 0.825874i	−0.910760	−	0.412937i	$-0.864503\pi$
$647$	− 21.0071i	− 0.825874i	0.910760	−	0.412937i	$-0.135497\pi$
$648$	− 33.5793i	− 1.31912i
$649$	−27.7748	−1.09026
$650$	0	0
$651$	67.7380	2.65486
$652$	− 12.1463i	− 0.475684i
$653$	45.5496i	1.78249i	0.453519	+	0.891247i	$0.350168\pi$
$653$	45.5496i	1.78249i	−0.453519	+	0.891247i	$0.649832\pi$
$654$	−4.81870	−0.188426
$655$	0	0
$656$	−6.44724	−0.251723
$657$	− 61.4370i	− 2.39689i
$658$	4.80453i	0.187300i
$659$	−9.38385	−0.365543	−0.182772	−	0.983155i	$-0.558507\pi$
$659$	−9.38385	−0.365543	−0.182772	+	0.983155i	$0.558507\pi$
$660$	0	0
$661$	4.05630	0.157772	0.0788859	−	0.996884i	$-0.474864\pi$
$661$	4.05630	0.157772	0.0788859	+	0.996884i	$0.474864\pi$
$662$	− 18.0379i	− 0.701062i
$663$	− 16.7535i	− 0.650653i
$664$	−15.2334	−0.591169
$665$	0	0
$666$	−8.19547	−0.317568
$667$	5.90157i	0.228510i
$668$	− 17.0829i	− 0.660956i
$669$	−9.06869	−0.350616
$670$	0	0
$671$	37.7185	1.45611
$672$	8.69193i	0.335298i
$673$	− 7.60906i	− 0.293308i	−0.989188	−	0.146654i	$-0.953150\pi$
$673$	− 7.60906i	− 0.293308i	0.989188	−	0.146654i	$-0.0468503\pi$
$674$	14.8945	0.573714
$675$	0	0
$676$	31.7819	1.22238
$677$	28.7677i	1.10563i	0.833303	+	0.552816i	$0.186447\pi$
$677$	28.7677i	1.10563i	−0.833303	+	0.552816i	$0.813553\pi$
$678$	37.2713i	1.43139i
$679$	−27.1392	−1.04151
$680$	0	0
$681$	−87.4370	−3.35059
$682$	37.0039i	1.41695i
$683$	− 0.0704767i	− 0.00269671i	−0.999999	−	0.00134836i	$-0.999571\pi$
$683$	− 0.0704767i	− 0.00269671i	0.999999	−	0.00134836i	$-0.000429196\pi$
$684$	27.4217	1.04850
$685$	0	0
$686$	18.8382	0.719245
$687$	82.8718i	3.16176i
$688$	− 1.94370i	− 0.0741028i
$689$	−69.9122	−2.66344
$690$	0	0
$691$	10.6498	0.405137	0.202569	−	0.979268i	$-0.435071\pi$
$691$	10.6498	0.405137	0.202569	+	0.979268i	$0.435071\pi$
$692$	− 15.3417i	− 0.583205i
$693$	101.088i	3.84002i
$694$	9.38385	0.356206
$695$	0	0
$696$	−13.1955	−0.500173
$697$	4.82401i	0.182722i
$698$	32.9703i	1.24794i
$699$	−41.0829	−1.55390
$700$	0	0
$701$	−7.98582	−0.301620	−0.150810	−	0.988563i	$-0.548188\pi$
$701$	−7.98582	−0.301620	−0.150810	+	0.988563i	$0.548188\pi$
$702$	− 116.331i	− 4.39065i
$703$	− 3.34596i	− 0.126195i
$704$	−4.74823	−0.178956
$705$	0	0
$706$	27.4260	1.03219
$707$	31.6622i	1.19078i
$708$	19.5722i	0.735570i
$709$	37.2149	1.39764	0.698818	−	0.715299i	$-0.253710\pi$
$709$	37.2149	1.39764	0.698818	+	0.715299i	$0.253710\pi$
$710$	0	0
$711$	−135.565	−5.08408
$712$	6.00000i	0.224860i
$713$	− 11.6622i	− 0.436752i
$714$	6.50354	0.243389
$715$	0	0
$716$	2.45148	0.0916161
$717$	57.4738i	2.14640i
$718$	25.0829i	0.936084i
$719$	−30.1742	−1.12531	−0.562654	−	0.826692i	$-0.690220\pi$
$719$	−30.1742	−1.12531	−0.562654	+	0.826692i	$0.690220\pi$
$720$	0	0
$721$	3.39803	0.126549
$722$	− 7.80453i	− 0.290455i
$723$	− 51.0970i	− 1.90032i
$724$	−13.1955	−0.490406
$725$	0	0
$726$	−38.6314	−1.43375
$727$	− 18.7677i	− 0.696056i	−0.937484	−	0.348028i	$-0.886851\pi$
$727$	− 18.7677i	− 0.696056i	0.937484	−	0.348028i	$-0.113149\pi$
$728$	17.3839i	0.644288i
$729$	−100.701	−3.72967
$730$	0	0
$731$	−1.45433	−0.0537903
$732$	− 26.5793i	− 0.982400i
$733$	− 26.1094i	− 0.964374i	−0.876068	−	0.482187i	$-0.839843\pi$
$733$	− 26.1094i	− 0.964374i	0.876068	−	0.482187i	$-0.160157\pi$
$734$	−2.48513	−0.0917279
$735$	0	0
$736$	1.49646	0.0551601
$737$	8.78188i	0.323485i
$738$	52.8382i	1.94500i
$739$	−42.6130	−1.56754	−0.783772	−	0.621049i	$-0.786707\pi$
$739$	−42.6130	−1.56754	−0.783772	+	0.621049i	$0.786707\pi$
$740$	0	0
$741$	74.9193	2.75223
$742$	− 27.1392i	− 0.996310i
$743$	35.4922i	1.30208i	0.759042	+	0.651042i	$0.225668\pi$
$743$	35.4922i	1.30208i	−0.759042	+	0.651042i	$0.774332\pi$
$744$	26.0758	0.955984
$745$	0	0
$746$	−35.1586	−1.28725
$747$	124.845i	4.56782i
$748$	3.55276i	0.129902i
$749$	−7.91005	−0.289027
$750$	0	0
$751$	21.5722	0.787182	0.393591	−	0.919286i	$-0.371233\pi$
$751$	21.5722	0.787182	0.393591	+	0.919286i	$0.371233\pi$
$752$	1.84951i	0.0674446i
$753$	33.5864i	1.22396i
$754$	−26.3909	−0.961101
$755$	0	0
$756$	45.1586	1.64240
$757$	− 23.7890i	− 0.864625i	−0.901724	−	0.432312i	$-0.857698\pi$
$757$	− 23.7890i	− 0.864625i	0.901724	−	0.432312i	$-0.142302\pi$
$758$	− 24.6778i	− 0.896336i
$759$	23.7748	0.862970
$760$	0	0
$761$	−18.6724	−0.676876	−0.338438	−	0.940989i	$-0.609898\pi$
$761$	−18.6724	−0.676876	−0.338438	+	0.940989i	$0.609898\pi$
$762$	29.5864i	1.07180i
$763$	− 3.74114i	− 0.135438i
$764$	21.7790	0.787938
$765$	0	0
$766$	−13.3839	−0.483578
$767$	39.1445i	1.41342i
$768$	3.34596i	0.120737i
$769$	2.48937	0.0897689	0.0448845	−	0.998992i	$-0.485708\pi$
$769$	2.48937	0.0897689	0.0448845	+	0.998992i	$0.485708\pi$
$770$	0	0
$771$	70.1657	2.52696
$772$	− 12.9929i	− 0.467625i
$773$	− 0.950786i	− 0.0341974i	−0.999854	−	0.0170987i	$-0.994557\pi$
$773$	− 0.950786i	− 0.0341974i	0.999854	−	0.0170987i	$-0.00544295\pi$
$774$	−15.9295	−0.572575
$775$	0	0
$776$	−10.4472	−0.375034
$777$	− 8.69193i	− 0.311821i
$778$	− 18.2220i	− 0.653292i
$779$	−21.5722	−0.772906
$780$	0	0
$781$	−18.4582	−0.660488
$782$	− 1.11969i	− 0.0400401i
$783$	68.5567i	2.45002i
$784$	0.251772	0.00899185
$785$	0	0
$786$	−20.5793	−0.734040
$787$	− 29.4359i	− 1.04928i	−0.851325	−	0.524639i	$-0.824200\pi$
$787$	− 29.4359i	− 1.04928i	0.851325	−	0.524639i	$-0.175800\pi$
$788$	− 0.616147i	− 0.0219493i
$789$	74.8577	2.66500
$790$	0	0
$791$	−28.9366	−1.02887
$792$	38.9140i	1.38275i
$793$	− 53.1586i	− 1.88772i
$794$	2.22521	0.0789696
$795$	0	0
$796$	−1.54852	−0.0548859
$797$	− 14.1742i	− 0.502076i	−0.967977	−	0.251038i	$-0.919228\pi$
$797$	− 14.1742i	− 0.502076i	0.967977	−	0.251038i	$-0.0807719\pi$
$798$	29.0829i	1.02952i
$799$	1.38385	0.0489572
$800$	0	0
$801$	49.1728	1.73744
$802$	12.3909i	0.437539i
$803$	35.5949i	1.25612i
$804$	6.18838	0.218247
$805$	0	0
$806$	52.1516	1.83696
$807$	14.0142i	0.493322i
$808$	12.1884i	0.428786i
$809$	32.9929	1.15997	0.579985	−	0.814627i	$-0.303059\pi$
$809$	32.9929	1.15997	0.579985	+	0.814627i	$0.303059\pi$
$810$	0	0
$811$	40.5567	1.42414	0.712069	−	0.702110i	$-0.247758\pi$
$811$	40.5567	1.42414	0.712069	+	0.702110i	$0.247758\pi$
$812$	− 10.2447i	− 0.359518i
$813$	− 40.5283i	− 1.42139i
$814$	4.74823	0.166425
$815$	0	0
$816$	2.50354	0.0876416
$817$	− 6.50354i	− 0.227530i
$818$	− 7.27125i	− 0.254233i
$819$	142.469	4.97826
$820$	0	0
$821$	−38.8661	−1.35644	−0.678219	−	0.734860i	$-0.737248\pi$
$821$	−38.8661	−1.35644	−0.678219	+	0.734860i	$0.737248\pi$
$822$	− 36.4525i	− 1.27143i
$823$	49.6243i	1.72979i	0.501949	+	0.864897i	$0.332616\pi$
$823$	49.6243i	1.72979i	−0.501949	+	0.864897i	$0.667384\pi$
$824$	1.30807	0.0455689
$825$	0	0
$826$	−15.1955	−0.528718
$827$	− 30.0195i	− 1.04388i	−0.852982	−	0.521940i	$-0.825209\pi$
$827$	− 30.0195i	− 1.04388i	0.852982	−	0.521940i	$-0.174791\pi$
$828$	− 12.2642i	− 0.426209i
$829$	−10.5598	−0.366759	−0.183379	−	0.983042i	$-0.558704\pi$
$829$	−10.5598	−0.366759	−0.183379	+	0.983042i	$0.558704\pi$
$830$	0	0
$831$	31.1445	1.08039
$832$	6.69193i	0.232001i
$833$	− 0.188383i	− 0.00652708i
$834$	3.51063	0.121563
$835$	0	0
$836$	−15.8874	−0.549477
$837$	− 135.476i	− 4.68273i
$838$	− 28.4809i	− 0.983856i
$839$	−29.3839	−1.01444	−0.507222	−	0.861816i	$-0.669327\pi$
$839$	−29.3839	−1.01444	−0.507222	+	0.861816i	$0.669327\pi$
$840$	0	0
$841$	−13.4472	−0.463698
$842$	22.0000i	0.758170i
$843$	64.4809i	2.22084i
$844$	−20.9366	−0.720668
$845$	0	0
$846$	15.1576	0.521128
$847$	− 29.9926i	− 1.03056i
$848$	− 10.4472i	− 0.358760i
$849$	−45.7890	−1.57147
$850$	0	0
$851$	−1.49646	−0.0512979
$852$	13.0071i	0.445615i
$853$	− 6.22521i	− 0.213147i	−0.994305	−	0.106573i	$-0.966012\pi$
$853$	− 6.22521i	− 0.213147i	0.994305	−	0.106573i	$-0.0339879\pi$
$854$	20.6356	0.706137
$855$	0	0
$856$	−3.04498	−0.104075
$857$	− 3.65119i	− 0.124722i	−0.998054	−	0.0623611i	$-0.980137\pi$
$857$	− 3.65119i	− 0.124722i	0.998054	−	0.0623611i	$-0.0198630\pi$
$858$	106.317i	3.62961i
$859$	36.3162	1.23909	0.619547	−	0.784960i	$-0.287316\pi$
$859$	36.3162	1.23909	0.619547	+	0.784960i	$0.287316\pi$
$860$	0	0
$861$	−56.0390	−1.90980
$862$	1.51487i	0.0515966i
$863$	− 24.5751i	− 0.836546i	−0.908321	−	0.418273i	$-0.862636\pi$
$863$	− 24.5751i	− 0.836546i	0.908321	−	0.418273i	$-0.137364\pi$
$864$	17.3839	0.591411
$865$	0	0
$866$	24.9929	0.849294
$867$	55.0082i	1.86817i
$868$	20.2447i	0.687149i
$869$	78.5425	2.66437
$870$	0	0
$871$	12.3768	0.419371
$872$	− 1.44015i	− 0.0487698i
$873$	85.6201i	2.89780i
$874$	5.00709	0.169367
$875$	0	0
$876$	25.0829	0.847472
$877$	20.0563i	0.677253i	0.940921	+	0.338627i	$0.109962\pi$
$877$	20.0563i	0.677253i	−0.940921	+	0.338627i	$0.890038\pi$
$878$	5.10128i	0.172160i
$879$	−34.9561	−1.17904
$880$	0	0
$881$	−15.0266	−0.506258	−0.253129	−	0.967433i	$-0.581460\pi$
$881$	−15.0266	−0.506258	−0.253129	+	0.967433i	$0.581460\pi$
$882$	− 2.06339i	− 0.0694779i
$883$	6.62145i	0.222830i	0.993774	+	0.111415i	$0.0355382\pi$
$883$	6.62145i	0.222830i	−0.993774	+	0.111415i	$0.964462\pi$
$884$	5.00709	0.168407
$885$	0	0
$886$	25.7369	0.864648
$887$	− 37.5538i	− 1.26093i	−0.776216	−	0.630467i	$-0.782863\pi$
$887$	− 37.5538i	− 1.26093i	0.776216	−	0.630467i	$-0.217137\pi$
$888$	− 3.34596i	− 0.112283i
$889$	−22.9703	−0.770398
$890$	0	0
$891$	159.442	5.34152
$892$	− 2.71034i	− 0.0907488i
$893$	6.18838i	0.207086i
$894$	−14.0142	−0.468704
$895$	0	0
$896$	−2.59774	−0.0867842
$897$	− 33.5071i	− 1.11877i
$898$	17.7748i	0.593153i
$899$	−30.7341	−1.02504
$900$	0	0
$901$	−7.81692	−0.260419
$902$	− 30.6130i	− 1.01930i
$903$	− 16.8945i	− 0.562213i
$904$	−11.1392	−0.370483
$905$	0	0
$906$	22.3909	0.743889
$907$	50.4667i	1.67572i	0.545885	+	0.837860i	$0.316193\pi$
$907$	50.4667i	1.67572i	−0.545885	+	0.837860i	$0.683807\pi$
$908$	− 26.1321i	− 0.867224i
$909$	99.8895	3.31313
$910$	0	0
$911$	15.9395	0.528098	0.264049	−	0.964509i	$-0.414942\pi$
$911$	15.9395	0.528098	0.264049	+	0.964509i	$0.414942\pi$
$912$	11.1955i	0.370719i
$913$	− 72.3315i	− 2.39382i
$914$	−34.3346	−1.13569
$915$	0	0
$916$	−24.7677	−0.818348
$917$	− 15.9774i	− 0.527619i
$918$	− 13.0071i	− 0.429298i
$919$	−32.5414	−1.07344	−0.536721	−	0.843760i	$-0.680337\pi$
$919$	−32.5414	−1.07344	−0.536721	+	0.843760i	$0.680337\pi$
$920$	0	0
$921$	16.5793	0.546307
$922$	− 16.9508i	− 0.558244i
$923$	26.0142i	0.856267i
$924$	−41.2713	−1.35772
$925$	0	0
$926$	10.9929	0.361250
$927$	− 10.7203i	− 0.352100i
$928$	− 3.94370i	− 0.129458i
$929$	34.1094	1.11909	0.559547	−	0.828799i	$-0.310975\pi$
$929$	34.1094	1.11909	0.559547	+	0.828799i	$0.310975\pi$
$930$	0	0
$931$	0.842420	0.0276092
$932$	− 12.2783i	− 0.402190i
$933$	12.3152i	0.403180i
$934$	−37.4175	−1.22434
$935$	0	0
$936$	54.8435	1.79262
$937$	0.0141751i	0 0.000463082i	1.00000		0.000231541i	$7.37017e-5\pi$
$937$	0.0141751i	0 0.000463082i	−1.00000		0.000231541i	$0.999926\pi$
$938$	4.80453i	0.156873i
$939$	−3.32225	−0.108417
$940$	0	0
$941$	11.1813	0.364500	0.182250	−	0.983252i	$-0.441662\pi$
$941$	11.1813	0.364500	0.182250	+	0.983252i	$0.441662\pi$
$942$	13.1955i	0.429932i
$943$	9.64802i	0.314183i
$944$	−5.84951	−0.190385
$945$	0	0
$946$	9.22912	0.300065
$947$	− 53.6427i	− 1.74315i	−0.490259	−	0.871577i	$-0.663098\pi$
$947$	− 53.6427i	− 1.74315i	0.490259	−	0.871577i	$-0.336902\pi$
$948$	− 55.3470i	− 1.79759i
$949$	50.1657	1.62845
$950$	0	0
$951$	21.9490	0.711745
$952$	1.94370i	0.0629956i
$953$	9.88740i	0.320284i	0.987094	+	0.160142i	$0.0511952\pi$
$953$	9.88740i	0.320284i	−0.987094	+	0.160142i	$0.948805\pi$
$954$	−85.6201	−2.77205
$955$	0	0
$956$	−17.1771	−0.555546
$957$	− 62.6551i	− 2.02535i
$958$	0.465654i	0.0150446i
$959$	28.3010	0.913886
$960$	0	0
$961$	29.7341	0.959163
$962$	− 6.69193i	− 0.215756i
$963$	24.9550i	0.804164i
$964$	15.2713	0.491854
$965$	0	0
$966$	13.0071	0.418496
$967$	51.6254i	1.66016i	0.557644	+	0.830080i	$0.311705\pi$
$967$	51.6254i	1.66016i	−0.557644	+	0.830080i	$0.688295\pi$
$968$	− 11.5457i	− 0.371092i
$969$	8.37677	0.269100
$970$	0	0
$971$	20.5598	0.659797	0.329898	−	0.944016i	$-0.392986\pi$
$971$	20.5598	0.659797	0.329898	+	0.944016i	$0.392986\pi$
$972$	− 60.2036i	− 1.93103i
$973$	2.72558i	0.0873781i
$974$	−34.9561	−1.12007
$975$	0	0
$976$	7.94370	0.254272
$977$	38.4472i	1.23004i	0.788513	+	0.615018i	$0.210851\pi$
$977$	38.4472i	1.23004i	−0.788513	+	0.615018i	$0.789149\pi$
$978$	− 40.6409i	− 1.29955i
$979$	−28.4894	−0.910524
$980$	0	0
$981$	−11.8027	−0.376833
$982$	− 24.7819i	− 0.790822i
$983$	− 11.8973i	− 0.379466i	−0.981836	−	0.189733i	$-0.939238\pi$
$983$	− 11.8973i	− 0.379466i	0.981836	−	0.189733i	$-0.0607622\pi$
$984$	−21.5722	−0.687698
$985$	0	0
$986$	−2.95079	−0.0939722
$987$	16.0758i	0.511698i
$988$	22.3909i	0.712351i
$989$	−2.90866	−0.0924900
$990$	0	0
$991$	−19.7932	−0.628752	−0.314376	−	0.949299i	$-0.601795\pi$
$991$	−19.7932	−0.628752	−0.314376	+	0.949299i	$0.601795\pi$
$992$	7.79321i	0.247435i
$993$	− 60.3541i	− 1.91528i
$994$	−10.0984	−0.320303
$995$	0	0
$996$	−50.9703	−1.61505
$997$	5.88740i	0.186456i	0.995645	+	0.0932279i	$0.0297185\pi$
$997$	5.88740i	0.186456i	−0.995645	+	0.0932279i	$0.970281\pi$
$998$	2.33888i	0.0740358i
$999$	−17.3839	−0.550001

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1850.2.b.o.149.6		6
5.2	odd	4		1850.2.a.z.1.3		3
5.3	odd	4		370.2.a.g.1.1	✓	3
5.4	even	2	inner	1850.2.b.o.149.1		6
15.8	even	4		3330.2.a.bg.1.1		3
20.3	even	4		2960.2.a.u.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
370.2.a.g.1.1	✓	3	5.3	odd	4
1850.2.a.z.1.3		3	5.2	odd	4
1850.2.b.o.149.1		6	5.4	even	2	inner
1850.2.b.o.149.6		6	1.1	even	1	trivial
2960.2.a.u.1.3		3	20.3	even	4
3330.2.a.bg.1.1		3	15.8	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:	\( N \)	\(=\)	\( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1850.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +3.34596i q^{3} -1.00000 q^{4} -3.34596 q^{6} -2.59774i q^{7} -1.00000i q^{8} -8.19547 q^{9} +4.74823 q^{11} -3.34596i q^{12} -6.69193i q^{13} +2.59774 q^{14} +1.00000 q^{16} -0.748228i q^{17} -8.19547i q^{18} +3.34596 q^{19} +8.69193 q^{21} +4.74823i q^{22} -1.49646i q^{23} +3.34596 q^{24} +6.69193 q^{26} -17.3839i q^{27} +2.59774i q^{28} -3.94370 q^{29} +7.79321 q^{31} +1.00000i q^{32} +15.8874i q^{33} +0.748228 q^{34} +8.19547 q^{36} -1.00000i q^{37} +3.34596i q^{38} +22.3909 q^{39} -6.44724 q^{41} +8.69193i q^{42} -1.94370i q^{43} -4.74823 q^{44} +1.49646 q^{46} +1.84951i q^{47} +3.34596i q^{48} +0.251772 q^{49} +2.50354 q^{51} +6.69193i q^{52} -10.4472i q^{53} +17.3839 q^{54} -2.59774 q^{56} +11.1955i q^{57} -3.94370i q^{58} -5.84951 q^{59} +7.94370 q^{61} +7.79321i q^{62} +21.2897i q^{63} -1.00000 q^{64} -15.8874 q^{66} +1.84951i q^{67} +0.748228i q^{68} +5.00709 q^{69} -3.88740 q^{71} +8.19547i q^{72} +7.49646i q^{73} +1.00000 q^{74} -3.34596 q^{76} -12.3346i q^{77} +22.3909i q^{78} +16.5414 q^{79} +33.5793 q^{81} -6.44724i q^{82} -15.2334i q^{83} -8.69193 q^{84} +1.94370 q^{86} -13.1955i q^{87} -4.74823i q^{88} -6.00000 q^{89} -17.3839 q^{91} +1.49646i q^{92} +26.0758i q^{93} -1.84951 q^{94} -3.34596 q^{96} -10.4472i q^{97} +0.251772i q^{98} -38.9140 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{4} - 22 q^{9} + 22 q^{11} + 2 q^{14} + 6 q^{16} + 12 q^{21} + 10 q^{29} + 6 q^{31} - 2 q^{34} + 22 q^{36} + 80 q^{39} - 18 q^{41} - 22 q^{44} - 4 q^{46} + 8 q^{49} + 28 q^{51} + 24 q^{54} - 2 q^{56}+ \cdots - 38 q^{99}+O(q^{100}) \) 6 * q - 6 * q^4 - 22 * q^9 + 22 * q^11 + 2 * q^14 + 6 * q^16 + 12 * q^21 + 10 * q^29 + 6 * q^31 - 2 * q^34 + 22 * q^36 + 80 * q^39 - 18 * q^41 - 22 * q^44 - 4 * q^46 + 8 * q^49 + 28 * q^51 + 24 * q^54 - 2 * q^56 - 28 * q^59 + 14 * q^61 - 6 * q^64 - 28 * q^66 + 56 * q^69 + 44 * q^71 + 6 * q^74 + 52 * q^79 + 94 * q^81 - 12 * q^84 - 22 * q^86 - 36 * q^89 - 24 * q^91 - 4 * q^94 - 38 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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