LMFDB - Embedded newform 5950.2.a.bm.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5950,2,Mod(1,5950)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5950, 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("5950.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5950 = 2 \cdot 5^{2} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5950.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:	$47.5109892027$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.2597.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 9x + 8 $ x^3 - x^2 - 9*x + 8
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1190)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.07912$ of defining polynomial
Character	$\chi$	$=$	5950.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +3.07912 q^{3} +1.00000 q^{4} +3.07912 q^{6} +1.00000 q^{7} +1.00000 q^{8} +6.48097 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +3.07912 q^{3} +1.00000 q^{4} +3.07912 q^{6} +1.00000 q^{7} +1.00000 q^{8} +6.48097 q^{9} -3.48097 q^{11} +3.07912 q^{12} +1.00000 q^{14} +1.00000 q^{16} +1.00000 q^{17} +6.48097 q^{18} +7.48097 q^{19} +3.07912 q^{21} -3.48097 q^{22} +3.07912 q^{23} +3.07912 q^{24} +10.7183 q^{27} +1.00000 q^{28} -3.07912 q^{29} -7.63921 q^{31} +1.00000 q^{32} -10.7183 q^{33} +1.00000 q^{34} +6.48097 q^{36} +2.15824 q^{37} +7.48097 q^{38} +0.519027 q^{41} +3.07912 q^{42} +7.07912 q^{43} -3.48097 q^{44} +3.07912 q^{46} +3.07912 q^{47} +3.07912 q^{48} +1.00000 q^{49} +3.07912 q^{51} -5.07912 q^{53} +10.7183 q^{54} +1.00000 q^{56} +23.0348 q^{57} -3.07912 q^{58} -1.07912 q^{59} +14.3165 q^{61} -7.63921 q^{62} +6.48097 q^{63} +1.00000 q^{64} -10.7183 q^{66} -13.7974 q^{67} +1.00000 q^{68} +9.48097 q^{69} +13.1202 q^{71} +6.48097 q^{72} -6.00000 q^{73} +2.15824 q^{74} +7.48097 q^{76} -3.48097 q^{77} -0.803708 q^{79} +13.5601 q^{81} +0.519027 q^{82} -6.31648 q^{83} +3.07912 q^{84} +7.07912 q^{86} -9.48097 q^{87} -3.48097 q^{88} -2.80371 q^{89} +3.07912 q^{92} -23.5220 q^{93} +3.07912 q^{94} +3.07912 q^{96} -14.3165 q^{97} +1.00000 q^{98} -22.5601 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + q^{3} + 3 q^{4} + q^{6} + 3 q^{7} + 3 q^{8} + 10 q^{9}+O(q^{10}) $ 3 * q + 3 * q^2 + q^3 + 3 * q^4 + q^6 + 3 * q^7 + 3 * q^8 + 10 * q^9	$ 3 q + 3 q^{2} + q^{3} + 3 q^{4} + q^{6} + 3 q^{7} + 3 q^{8} + 10 q^{9} - q^{11} + q^{12} + 3 q^{14} + 3 q^{16} + 3 q^{17} + 10 q^{18} + 13 q^{19} + q^{21} - q^{22} + q^{23} + q^{24} - 2 q^{27} + 3 q^{28} - q^{29} + 3 q^{31} + 3 q^{32} + 2 q^{33} + 3 q^{34} + 10 q^{36} - 10 q^{37} + 13 q^{38} + 11 q^{41} + q^{42} + 13 q^{43} - q^{44} + q^{46} + q^{47} + q^{48} + 3 q^{49} + q^{51} - 7 q^{53} - 2 q^{54} + 3 q^{56} + 2 q^{57} - q^{58} + 5 q^{59} + 10 q^{61} + 3 q^{62} + 10 q^{63} + 3 q^{64} + 2 q^{66} + q^{67} + 3 q^{68} + 19 q^{69} + 4 q^{71} + 10 q^{72} - 18 q^{73} - 10 q^{74} + 13 q^{76} - q^{77} + 23 q^{81} + 11 q^{82} + 14 q^{83} + q^{84} + 13 q^{86} - 19 q^{87} - q^{88} - 6 q^{89} + q^{92} - 34 q^{93} + q^{94} + q^{96} - 10 q^{97} + 3 q^{98} - 50 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 + q^3 + 3 * q^4 + q^6 + 3 * q^7 + 3 * q^8 + 10 * q^9 - q^11 + q^12 + 3 * q^14 + 3 * q^16 + 3 * q^17 + 10 * q^18 + 13 * q^19 + q^21 - q^22 + q^23 + q^24 - 2 * q^27 + 3 * q^28 - q^29 + 3 * q^31 + 3 * q^32 + 2 * q^33 + 3 * q^34 + 10 * q^36 - 10 * q^37 + 13 * q^38 + 11 * q^41 + q^42 + 13 * q^43 - q^44 + q^46 + q^47 + q^48 + 3 * q^49 + q^51 - 7 * q^53 - 2 * q^54 + 3 * q^56 + 2 * q^57 - q^58 + 5 * q^59 + 10 * q^61 + 3 * q^62 + 10 * q^63 + 3 * q^64 + 2 * q^66 + q^67 + 3 * q^68 + 19 * q^69 + 4 * q^71 + 10 * q^72 - 18 * q^73 - 10 * q^74 + 13 * q^76 - q^77 + 23 * q^81 + 11 * q^82 + 14 * q^83 + q^84 + 13 * q^86 - 19 * q^87 - q^88 - 6 * q^89 + q^92 - 34 * q^93 + q^94 + q^96 - 10 * q^97 + 3 * q^98 - 50 * q^99

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
$3$	3.07912	1.77773	0.888865	−	0.458169i	$-0.151495\pi$
$3$	3.07912	1.77773	0.888865	+	0.458169i	$0.151495\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	3.07912	1.25705
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	6.48097	2.16032
$10$	0	0
$11$	−3.48097	−1.04955	−0.524776	−	0.851240i	$-0.675851\pi$
$11$	−3.48097	−1.04955	−0.524776	+	0.851240i	$0.675851\pi$
$12$	3.07912	0.888865
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	6.48097	1.52758
$19$	7.48097	1.71625	0.858126	−	0.513438i	$-0.171629\pi$
$19$	7.48097	1.71625	0.858126	+	0.513438i	$0.171629\pi$
$20$	0	0
$21$	3.07912	0.671919
$22$	−3.48097	−0.742146
$23$	3.07912	0.642041	0.321020	−	0.947072i	$-0.395974\pi$
$23$	3.07912	0.642041	0.321020	+	0.947072i	$0.395974\pi$
$24$	3.07912	0.628523
$25$	0	0
$26$	0	0
$27$	10.7183	2.06274
$28$	1.00000	0.188982
$29$	−3.07912	−0.571778	−0.285889	−	0.958263i	$-0.592289\pi$
$29$	−3.07912	−0.571778	−0.285889	+	0.958263i	$0.592289\pi$
$30$	0	0
$31$	−7.63921	−1.37204	−0.686021	−	0.727581i	$-0.740644\pi$
$31$	−7.63921	−1.37204	−0.686021	+	0.727581i	$0.740644\pi$
$32$	1.00000	0.176777
$33$	−10.7183	−1.86582
$34$	1.00000	0.171499
$35$	0	0
$36$	6.48097	1.08016
$37$	2.15824	0.354812	0.177406	−	0.984138i	$-0.443229\pi$
$37$	2.15824	0.354812	0.177406	+	0.984138i	$0.443229\pi$
$38$	7.48097	1.21357
$39$	0	0
$40$	0	0
$41$	0.519027	0.0810584	0.0405292	−	0.999178i	$-0.487096\pi$
$41$	0.519027	0.0810584	0.0405292	+	0.999178i	$0.487096\pi$
$42$	3.07912	0.475118
$43$	7.07912	1.07956	0.539778	−	0.841808i	$-0.318508\pi$
$43$	7.07912	1.07956	0.539778	+	0.841808i	$0.318508\pi$
$44$	−3.48097	−0.524776
$45$	0	0
$46$	3.07912	0.453991
$47$	3.07912	0.449136	0.224568	−	0.974458i	$-0.427903\pi$
$47$	3.07912	0.449136	0.224568	+	0.974458i	$0.427903\pi$
$48$	3.07912	0.444433
$49$	1.00000	0.142857
$50$	0	0
$51$	3.07912	0.431163
$52$	0	0
$53$	−5.07912	−0.697671	−0.348835	−	0.937184i	$-0.613423\pi$
$53$	−5.07912	−0.697671	−0.348835	+	0.937184i	$0.613423\pi$
$54$	10.7183	1.45858
$55$	0	0
$56$	1.00000	0.133631
$57$	23.0348	3.05103
$58$	−3.07912	−0.404308
$59$	−1.07912	−0.140489	−0.0702447	−	0.997530i	$-0.522378\pi$
$59$	−1.07912	−0.140489	−0.0702447	+	0.997530i	$0.522378\pi$
$60$	0	0
$61$	14.3165	1.83304	0.916518	−	0.399992i	$-0.130987\pi$
$61$	14.3165	1.83304	0.916518	+	0.399992i	$0.130987\pi$
$62$	−7.63921	−0.970181
$63$	6.48097	0.816526
$64$	1.00000	0.125000
$65$	0	0
$66$	−10.7183	−1.31934
$67$	−13.7974	−1.68563	−0.842813	−	0.538206i	$-0.819102\pi$
$67$	−13.7974	−1.68563	−0.842813	+	0.538206i	$0.819102\pi$
$68$	1.00000	0.121268
$69$	9.48097	1.14138
$70$	0	0
$71$	13.1202	1.55708	0.778540	−	0.627595i	$-0.215961\pi$
$71$	13.1202	1.55708	0.778540	+	0.627595i	$0.215961\pi$
$72$	6.48097	0.763790
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	2.15824	0.250890
$75$	0	0
$76$	7.48097	0.858126
$77$	−3.48097	−0.396694
$78$	0	0
$79$	−0.803708	−0.0904242	−0.0452121	−	0.998977i	$-0.514396\pi$
$79$	−0.803708	−0.0904242	−0.0452121	+	0.998977i	$0.514396\pi$
$80$	0	0
$81$	13.5601	1.50668
$82$	0.519027	0.0573169
$83$	−6.31648	−0.693323	−0.346662	−	0.937990i	$-0.612685\pi$
$83$	−6.31648	−0.693323	−0.346662	+	0.937990i	$0.612685\pi$
$84$	3.07912	0.335959
$85$	0	0
$86$	7.07912	0.763361
$87$	−9.48097	−1.01647
$88$	−3.48097	−0.371073
$89$	−2.80371	−0.297192	−0.148596	−	0.988898i	$-0.547475\pi$
$89$	−2.80371	−0.297192	−0.148596	+	0.988898i	$0.547475\pi$
$90$	0	0
$91$	0	0
$92$	3.07912	0.321020
$93$	−23.5220	−2.43912
$94$	3.07912	0.317587
$95$	0	0
$96$	3.07912	0.314261
$97$	−14.3165	−1.45362	−0.726809	−	0.686840i	$-0.758997\pi$
$97$	−14.3165	−1.45362	−0.726809	+	0.686840i	$0.758997\pi$
$98$	1.00000	0.101015
$99$	−22.5601	−2.26737
$100$	0	0
$101$	19.6392	1.95417	0.977087	−	0.212839i	$-0.0682710\pi$
$101$	19.6392	1.95417	0.977087	+	0.212839i	$0.0682710\pi$
$102$	3.07912	0.304878
$103$	−19.3956	−1.91110	−0.955552	−	0.294821i	$-0.904740\pi$
$103$	−19.3956	−1.91110	−0.955552	+	0.294821i	$0.904740\pi$
$104$	0	0
$105$	0	0
$106$	−5.07912	−0.493328
$107$	6.80371	0.657739	0.328870	−	0.944375i	$-0.393332\pi$
$107$	6.80371	0.657739	0.328870	+	0.944375i	$0.393332\pi$
$108$	10.7183	1.03137
$109$	−10.6012	−1.01541	−0.507703	−	0.861532i	$-0.669505\pi$
$109$	−10.6012	−1.01541	−0.507703	+	0.861532i	$0.669505\pi$
$110$	0	0
$111$	6.64547	0.630760
$112$	1.00000	0.0944911
$113$	1.32274	0.124432	0.0622162	−	0.998063i	$-0.480183\pi$
$113$	1.32274	0.124432	0.0622162	+	0.998063i	$0.480183\pi$
$114$	23.0348	2.15741
$115$	0	0
$116$	−3.07912	−0.285889
$117$	0	0
$118$	−1.07912	−0.0993409
$119$	1.00000	0.0916698
$120$	0	0
$121$	1.11717	0.101561
$122$	14.3165	1.29615
$123$	1.59815	0.144100
$124$	−7.63921	−0.686021
$125$	0	0
$126$	6.48097	0.577371
$127$	15.2784	1.35574	0.677870	−	0.735181i	$-0.262903\pi$
$127$	15.2784	1.35574	0.677870	+	0.735181i	$0.262903\pi$
$128$	1.00000	0.0883883
$129$	21.7974	1.91916
$130$	0	0
$131$	−10.9619	−0.957750	−0.478875	−	0.877883i	$-0.658955\pi$
$131$	−10.9619	−0.957750	−0.478875	+	0.877883i	$0.658955\pi$
$132$	−10.7183	−0.932911
$133$	7.48097	0.648683
$134$	−13.7974	−1.19192
$135$	0	0
$136$	1.00000	0.0857493
$137$	9.27842	0.792709	0.396355	−	0.918098i	$-0.370275\pi$
$137$	9.27842	0.792709	0.396355	+	0.918098i	$0.370275\pi$
$138$	9.48097	0.807074
$139$	6.96195	0.590505	0.295252	−	0.955419i	$-0.404596\pi$
$139$	6.96195	0.590505	0.295252	+	0.955419i	$0.404596\pi$
$140$	0	0
$141$	9.48097	0.798442
$142$	13.1202	1.10102
$143$	0	0
$144$	6.48097	0.540081
$145$	0	0
$146$	−6.00000	−0.496564
$147$	3.07912	0.253961
$148$	2.15824	0.177406
$149$	21.2784	1.74320	0.871598	−	0.490221i	$-0.163084\pi$
$149$	21.2784	1.74320	0.871598	+	0.490221i	$0.163084\pi$
$150$	0	0
$151$	−7.63921	−0.621670	−0.310835	−	0.950464i	$-0.600609\pi$
$151$	−7.63921	−0.621670	−0.310835	+	0.950464i	$0.600609\pi$
$152$	7.48097	0.606787
$153$	6.48097	0.523956
$154$	−3.48097	−0.280505
$155$	0	0
$156$	0	0
$157$	−12.0000	−0.957704	−0.478852	−	0.877896i	$-0.658947\pi$
$157$	−12.0000	−0.957704	−0.478852	+	0.877896i	$0.658947\pi$
$158$	−0.803708	−0.0639396
$159$	−15.6392	−1.24027
$160$	0	0
$161$	3.07912	0.242669
$162$	13.5601	1.06538
$163$	−20.9619	−1.64187	−0.820933	−	0.571024i	$-0.806546\pi$
$163$	−20.9619	−1.64187	−0.820933	+	0.571024i	$0.806546\pi$
$164$	0.519027	0.0405292
$165$	0	0
$166$	−6.31648	−0.490254
$167$	−1.12018	−0.0866824	−0.0433412	−	0.999060i	$-0.513800\pi$
$167$	−1.12018	−0.0866824	−0.0433412	+	0.999060i	$0.513800\pi$
$168$	3.07912	0.237559
$169$	−13.0000	−1.00000
$170$	0	0
$171$	48.4840	3.70766
$172$	7.07912	0.539778
$173$	−13.9557	−1.06103	−0.530516	−	0.847675i	$-0.678002\pi$
$173$	−13.9557	−1.06103	−0.530516	+	0.847675i	$0.678002\pi$
$174$	−9.48097	−0.718751
$175$	0	0
$176$	−3.48097	−0.262388
$177$	−3.32274	−0.249752
$178$	−2.80371	−0.210147
$179$	−1.03805	−0.0775878	−0.0387939	−	0.999247i	$-0.512352\pi$
$179$	−1.03805	−0.0775878	−0.0387939	+	0.999247i	$0.512352\pi$
$180$	0	0
$181$	20.1582	1.49835	0.749175	−	0.662372i	$-0.230450\pi$
$181$	20.1582	1.49835	0.749175	+	0.662372i	$0.230450\pi$
$182$	0	0
$183$	44.0821	3.25864
$184$	3.07912	0.226996
$185$	0	0
$186$	−23.5220	−1.72472
$187$	−3.48097	−0.254554
$188$	3.07912	0.224568
$189$	10.7183	0.779644
$190$	0	0
$191$	6.28468	0.454744	0.227372	−	0.973808i	$-0.426987\pi$
$191$	6.28468	0.454744	0.227372	+	0.973808i	$0.426987\pi$
$192$	3.07912	0.222216
$193$	−16.8448	−1.21251	−0.606257	−	0.795269i	$-0.707330\pi$
$193$	−16.8448	−1.21251	−0.606257	+	0.795269i	$0.707330\pi$
$194$	−14.3165	−1.02786
$195$	0	0
$196$	1.00000	0.0714286
$197$	14.1582	1.00873	0.504366	−	0.863490i	$-0.331726\pi$
$197$	14.1582	1.00873	0.504366	+	0.863490i	$0.331726\pi$
$198$	−22.5601	−1.60328
$199$	0.117173	0.00830617	0.00415308	−	0.999991i	$-0.498678\pi$
$199$	0.117173	0.00830617	0.00415308	+	0.999991i	$0.498678\pi$
$200$	0	0
$201$	−42.4840	−2.99659
$202$	19.6392	1.38181
$203$	−3.07912	−0.216112
$204$	3.07912	0.215581
$205$	0	0
$206$	−19.3956	−1.35136
$207$	19.9557	1.38702
$208$	0	0
$209$	−26.0411	−1.80130
$210$	0	0
$211$	6.80371	0.468387	0.234193	−	0.972190i	$-0.424755\pi$
$211$	6.80371	0.468387	0.234193	+	0.972190i	$0.424755\pi$
$212$	−5.07912	−0.348835
$213$	40.3986	2.76807
$214$	6.80371	0.465092
$215$	0	0
$216$	10.7183	0.729290
$217$	−7.63921	−0.518583
$218$	−10.6012	−0.718001
$219$	−18.4747	−1.24841
$220$	0	0
$221$	0	0
$222$	6.64547	0.446015
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	1.32274	0.0879870
$227$	−8.75939	−0.581381	−0.290691	−	0.956817i	$-0.593885\pi$
$227$	−8.75939	−0.581381	−0.290691	+	0.956817i	$0.593885\pi$
$228$	23.0348	1.52552
$229$	−11.3956	−0.753042	−0.376521	−	0.926408i	$-0.622880\pi$
$229$	−11.3956	−0.753042	−0.376521	+	0.926408i	$0.622880\pi$
$230$	0	0
$231$	−10.7183	−0.705214
$232$	−3.07912	−0.202154
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	0	0
$236$	−1.07912	−0.0702447
$237$	−2.47471	−0.160750
$238$	1.00000	0.0648204
$239$	9.24663	0.598115	0.299057	−	0.954235i	$-0.403328\pi$
$239$	9.24663	0.598115	0.299057	+	0.954235i	$0.403328\pi$
$240$	0	0
$241$	−0.0410648	−0.00264522	−0.00132261	−	0.999999i	$-0.500421\pi$
$241$	−0.0410648	−0.00264522	−0.00132261	+	0.999999i	$0.500421\pi$
$242$	1.11717	0.0718146
$243$	9.59815	0.615721
$244$	14.3165	0.916518
$245$	0	0
$246$	1.59815	0.101894
$247$	0	0
$248$	−7.63921	−0.485090
$249$	−19.4492	−1.23254
$250$	0	0
$251$	−4.83550	−0.305214	−0.152607	−	0.988287i	$-0.548767\pi$
$251$	−4.83550	−0.305214	−0.152607	+	0.988287i	$0.548767\pi$
$252$	6.48097	0.408263
$253$	−10.7183	−0.673856
$254$	15.2784	0.958653
$255$	0	0
$256$	1.00000	0.0625000
$257$	4.76264	0.297086	0.148543	−	0.988906i	$-0.452542\pi$
$257$	4.76264	0.297086	0.148543	+	0.988906i	$0.452542\pi$
$258$	21.7974	1.35705
$259$	2.15824	0.134106
$260$	0	0
$261$	−19.9557	−1.23523
$262$	−10.9619	−0.677231
$263$	−4.31648	−0.266165	−0.133083	−	0.991105i	$-0.542488\pi$
$263$	−4.31648	−0.266165	−0.133083	+	0.991105i	$0.542488\pi$
$264$	−10.7183	−0.659668
$265$	0	0
$266$	7.48097	0.458688
$267$	−8.63295	−0.528328
$268$	−13.7974	−0.842813
$269$	−30.6330	−1.86772	−0.933862	−	0.357634i	$-0.883584\pi$
$269$	−30.6330	−1.86772	−0.933862	+	0.357634i	$0.883584\pi$
$270$	0	0
$271$	31.5949	1.91925	0.959627	−	0.281277i	$-0.0907579\pi$
$271$	31.5949	1.91925	0.959627	+	0.281277i	$0.0907579\pi$
$272$	1.00000	0.0606339
$273$	0	0
$274$	9.27842	0.560530
$275$	0	0
$276$	9.48097	0.570688
$277$	−25.4367	−1.52834	−0.764170	−	0.645014i	$-0.776851\pi$
$277$	−25.4367	−1.52834	−0.764170	+	0.645014i	$0.776851\pi$
$278$	6.96195	0.417550
$279$	−49.5095	−2.96406
$280$	0	0
$281$	−28.3575	−1.69167	−0.845835	−	0.533445i	$-0.820897\pi$
$281$	−28.3575	−1.69167	−0.845835	+	0.533445i	$0.820897\pi$
$282$	9.48097	0.564584
$283$	−28.6330	−1.70205	−0.851026	−	0.525123i	$-0.824019\pi$
$283$	−28.6330	−1.70205	−0.851026	+	0.525123i	$0.824019\pi$
$284$	13.1202	0.778540
$285$	0	0
$286$	0	0
$287$	0.519027	0.0306372
$288$	6.48097	0.381895
$289$	1.00000	0.0588235
$290$	0	0
$291$	−44.0821	−2.58414
$292$	−6.00000	−0.351123
$293$	24.8858	1.45385	0.726923	−	0.686719i	$-0.240950\pi$
$293$	24.8858	1.45385	0.726923	+	0.686719i	$0.240950\pi$
$294$	3.07912	0.179578
$295$	0	0
$296$	2.15824	0.125445
$297$	−37.3102	−2.16496
$298$	21.2784	1.23263
$299$	0	0
$300$	0	0
$301$	7.07912	0.408034
$302$	−7.63921	−0.439587
$303$	60.4715	3.47399
$304$	7.48097	0.429063
$305$	0	0
$306$	6.48097	0.370493
$307$	−12.1582	−0.693907	−0.346954	−	0.937882i	$-0.612784\pi$
$307$	−12.1582	−0.693907	−0.346954	+	0.937882i	$0.612784\pi$
$308$	−3.48097	−0.198347
$309$	−59.7213	−3.39743
$310$	0	0
$311$	8.11717	0.460283	0.230141	−	0.973157i	$-0.426081\pi$
$311$	8.11717	0.460283	0.230141	+	0.973157i	$0.426081\pi$
$312$	0	0
$313$	5.76565	0.325894	0.162947	−	0.986635i	$-0.447900\pi$
$313$	5.76565	0.325894	0.162947	+	0.986635i	$0.447900\pi$
$314$	−12.0000	−0.677199
$315$	0	0
$316$	−0.803708	−0.0452121
$317$	−10.9619	−0.615684	−0.307842	−	0.951437i	$-0.599607\pi$
$317$	−10.9619	−0.615684	−0.307842	+	0.951437i	$0.599607\pi$
$318$	−15.6392	−0.877003
$319$	10.7183	0.600111
$320$	0	0
$321$	20.9494	1.16928
$322$	3.07912	0.171593
$323$	7.48097	0.416252
$324$	13.5601	0.753338
$325$	0	0
$326$	−20.9619	−1.16097
$327$	−32.6422	−1.80512
$328$	0.519027	0.0286585
$329$	3.07912	0.169757
$330$	0	0
$331$	−25.9239	−1.42491	−0.712453	−	0.701720i	$-0.752416\pi$
$331$	−25.9239	−1.42491	−0.712453	+	0.701720i	$0.752416\pi$
$332$	−6.31648	−0.346662
$333$	13.9875	0.766509
$334$	−1.12018	−0.0612937
$335$	0	0
$336$	3.07912	0.167980
$337$	1.31346	0.0715490	0.0357745	−	0.999360i	$-0.488610\pi$
$337$	1.31346	0.0715490	0.0357745	+	0.999360i	$0.488610\pi$
$338$	−13.0000	−0.707107
$339$	4.07286	0.221207
$340$	0	0
$341$	26.5919	1.44003
$342$	48.4840	2.62171
$343$	1.00000	0.0539949
$344$	7.07912	0.381681
$345$	0	0
$346$	−13.9557	−0.750262
$347$	−12.4747	−0.669678	−0.334839	−	0.942275i	$-0.608682\pi$
$347$	−12.4747	−0.669678	−0.334839	+	0.942275i	$0.608682\pi$
$348$	−9.48097	−0.508233
$349$	0.487233	0.0260810	0.0130405	−	0.999915i	$-0.495849\pi$
$349$	0.487233	0.0260810	0.0130405	+	0.999915i	$0.495849\pi$
$350$	0	0
$351$	0	0
$352$	−3.48097	−0.185536
$353$	24.6012	1.30939	0.654694	−	0.755894i	$-0.272798\pi$
$353$	24.6012	1.30939	0.654694	+	0.755894i	$0.272798\pi$
$354$	−3.32274	−0.176601
$355$	0	0
$356$	−2.80371	−0.148596
$357$	3.07912	0.162964
$358$	−1.03805	−0.0548629
$359$	20.1993	1.06608	0.533039	−	0.846091i	$-0.321050\pi$
$359$	20.1993	1.06608	0.533039	+	0.846091i	$0.321050\pi$
$360$	0	0
$361$	36.9650	1.94552
$362$	20.1582	1.05949
$363$	3.43991	0.180548
$364$	0	0
$365$	0	0
$366$	44.0821	2.30421
$367$	−2.07611	−0.108372	−0.0541860	−	0.998531i	$-0.517256\pi$
$367$	−2.07611	−0.108372	−0.0541860	+	0.998531i	$0.517256\pi$
$368$	3.07912	0.160510
$369$	3.36380	0.175112
$370$	0	0
$371$	−5.07912	−0.263695
$372$	−23.5220	−1.21956
$373$	−21.4777	−1.11207	−0.556037	−	0.831157i	$-0.687679\pi$
$373$	−21.4777	−1.11207	−0.556037	+	0.831157i	$0.687679\pi$
$374$	−3.48097	−0.179997
$375$	0	0
$376$	3.07912	0.158793
$377$	0	0
$378$	10.7183	0.551291
$379$	7.00301	0.359721	0.179860	−	0.983692i	$-0.442435\pi$
$379$	7.00301	0.359721	0.179860	+	0.983692i	$0.442435\pi$
$380$	0	0
$381$	47.0441	2.41014
$382$	6.28468	0.321552
$383$	−8.44292	−0.431413	−0.215707	−	0.976458i	$-0.569205\pi$
$383$	−8.44292	−0.431413	−0.215707	+	0.976458i	$0.569205\pi$
$384$	3.07912	0.157131
$385$	0	0
$386$	−16.8448	−0.857376
$387$	45.8796	2.33219
$388$	−14.3165	−0.726809
$389$	−22.3986	−1.13565	−0.567827	−	0.823148i	$-0.692216\pi$
$389$	−22.3986	−1.13565	−0.567827	+	0.823148i	$0.692216\pi$
$390$	0	0
$391$	3.07912	0.155718
$392$	1.00000	0.0505076
$393$	−33.7531	−1.70262
$394$	14.1582	0.713282
$395$	0	0
$396$	−22.5601	−1.13369
$397$	−10.6865	−0.536342	−0.268171	−	0.963371i	$-0.586419\pi$
$397$	−10.6865	−0.536342	−0.268171	+	0.963371i	$0.586419\pi$
$398$	0.117173	0.00587335
$399$	23.0348	1.15318
$400$	0	0
$401$	−20.9619	−1.04679	−0.523395	−	0.852090i	$-0.675335\pi$
$401$	−20.9619	−1.04679	−0.523395	+	0.852090i	$0.675335\pi$
$402$	−42.4840	−2.11891
$403$	0	0
$404$	19.6392	0.977087
$405$	0	0
$406$	−3.07912	−0.152814
$407$	−7.51277	−0.372394
$408$	3.07912	0.152439
$409$	−29.5949	−1.46337	−0.731687	−	0.681641i	$-0.761267\pi$
$409$	−29.5949	−1.46337	−0.731687	+	0.681641i	$0.761267\pi$
$410$	0	0
$411$	28.5694	1.40922
$412$	−19.3956	−0.955552
$413$	−1.07912	−0.0531000
$414$	19.9557	0.980768
$415$	0	0
$416$	0	0
$417$	21.4367	1.04976
$418$	−26.0411	−1.27371
$419$	35.0441	1.71202	0.856008	−	0.516963i	$-0.172938\pi$
$419$	35.0441	1.71202	0.856008	+	0.516963i	$0.172938\pi$
$420$	0	0
$421$	−4.15824	−0.202660	−0.101330	−	0.994853i	$-0.532310\pi$
$421$	−4.15824	−0.202660	−0.101330	+	0.994853i	$0.532310\pi$
$422$	6.80371	0.331199
$423$	19.9557	0.970279
$424$	−5.07912	−0.246664
$425$	0	0
$426$	40.3986	1.95732
$427$	14.3165	0.692823
$428$	6.80371	0.328870
$429$	0	0
$430$	0	0
$431$	−32.6330	−1.57187	−0.785937	−	0.618307i	$-0.787819\pi$
$431$	−32.6330	−1.57187	−0.785937	+	0.618307i	$0.787819\pi$
$432$	10.7183	0.515686
$433$	−9.68352	−0.465360	−0.232680	−	0.972553i	$-0.574750\pi$
$433$	−9.68352	−0.465360	−0.232680	+	0.972553i	$0.574750\pi$
$434$	−7.63921	−0.366694
$435$	0	0
$436$	−10.6012	−0.507703
$437$	23.0348	1.10190
$438$	−18.4747	−0.882756
$439$	−30.9084	−1.47518	−0.737588	−	0.675251i	$-0.764035\pi$
$439$	−30.9084	−1.47518	−0.737588	+	0.675251i	$0.764035\pi$
$440$	0	0
$441$	6.48097	0.308618
$442$	0	0
$443$	9.00301	0.427746	0.213873	−	0.976861i	$-0.431392\pi$
$443$	9.00301	0.427746	0.213873	+	0.976861i	$0.431392\pi$
$444$	6.64547	0.315380
$445$	0	0
$446$	4.00000	0.189405
$447$	65.5188	3.09893
$448$	1.00000	0.0472456
$449$	21.5313	1.01613	0.508063	−	0.861320i	$-0.330362\pi$
$449$	21.5313	1.01613	0.508063	+	0.861320i	$0.330362\pi$
$450$	0	0
$451$	−1.80672	−0.0850751
$452$	1.32274	0.0622162
$453$	−23.5220	−1.10516
$454$	−8.75939	−0.411099
$455$	0	0
$456$	23.0348	1.07870
$457$	19.3545	0.905367	0.452683	−	0.891671i	$-0.350467\pi$
$457$	19.3545	0.905367	0.452683	+	0.891671i	$0.350467\pi$
$458$	−11.3956	−0.532481
$459$	10.7183	0.500289
$460$	0	0
$461$	10.6012	0.493745	0.246873	−	0.969048i	$-0.420597\pi$
$461$	10.6012	0.493745	0.246873	+	0.969048i	$0.420597\pi$
$462$	−10.7183	−0.498662
$463$	−23.2784	−1.08184	−0.540920	−	0.841074i	$-0.681924\pi$
$463$	−23.2784	−1.08184	−0.540920	+	0.841074i	$0.681924\pi$
$464$	−3.07912	−0.142945
$465$	0	0
$466$	6.00000	0.277945
$467$	12.7276	0.588963	0.294482	−	0.955657i	$-0.404853\pi$
$467$	12.7276	0.588963	0.294482	+	0.955657i	$0.404853\pi$
$468$	0	0
$469$	−13.7974	−0.637107
$470$	0	0
$471$	−36.9494	−1.70254
$472$	−1.07912	−0.0496705
$473$	−24.6422	−1.13305
$474$	−2.47471	−0.113667
$475$	0	0
$476$	1.00000	0.0458349
$477$	−32.9176	−1.50719
$478$	9.24663	0.422931
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	0	0
$481$	0	0
$482$	−0.0410648	−0.00187045
$483$	9.48097	0.431399
$484$	1.11717	0.0507806
$485$	0	0
$486$	9.59815	0.435381
$487$	27.7213	1.25617	0.628087	−	0.778143i	$-0.283838\pi$
$487$	27.7213	1.25617	0.628087	+	0.778143i	$0.283838\pi$
$488$	14.3165	0.648076
$489$	−64.5443	−2.91880
$490$	0	0
$491$	20.6330	0.931152	0.465576	−	0.885008i	$-0.345847\pi$
$491$	20.6330	0.931152	0.465576	+	0.885008i	$0.345847\pi$
$492$	1.59815	0.0720500
$493$	−3.07912	−0.138677
$494$	0	0
$495$	0	0
$496$	−7.63921	−0.343011
$497$	13.1202	0.588521
$498$	−19.4492	−0.871539
$499$	3.22809	0.144509	0.0722545	−	0.997386i	$-0.476981\pi$
$499$	3.22809	0.144509	0.0722545	+	0.997386i	$0.476981\pi$
$500$	0	0
$501$	−3.44918	−0.154098
$502$	−4.83550	−0.215819
$503$	6.64547	0.296307	0.148153	−	0.988964i	$-0.452667\pi$
$503$	6.64547	0.296307	0.148153	+	0.988964i	$0.452667\pi$
$504$	6.48097	0.288685
$505$	0	0
$506$	−10.7183	−0.476488
$507$	−40.0285	−1.77773
$508$	15.2784	0.677870
$509$	−25.3195	−1.12227	−0.561133	−	0.827725i	$-0.689634\pi$
$509$	−25.3195	−1.12227	−0.561133	+	0.827725i	$0.689634\pi$
$510$	0	0
$511$	−6.00000	−0.265424
$512$	1.00000	0.0441942
$513$	80.1835	3.54019
$514$	4.76264	0.210071
$515$	0	0
$516$	21.7974	0.959579
$517$	−10.7183	−0.471392
$518$	2.15824	0.0948275
$519$	−42.9712	−1.88623
$520$	0	0
$521$	34.8323	1.52603	0.763014	−	0.646382i	$-0.223719\pi$
$521$	34.8323	1.52603	0.763014	+	0.646382i	$0.223719\pi$
$522$	−19.9557	−0.873437
$523$	−16.1582	−0.706550	−0.353275	−	0.935519i	$-0.614932\pi$
$523$	−16.1582	−0.706550	−0.353275	+	0.935519i	$0.614932\pi$
$524$	−10.9619	−0.478875
$525$	0	0
$526$	−4.31648	−0.188207
$527$	−7.63921	−0.332769
$528$	−10.7183	−0.466455
$529$	−13.5190	−0.587784
$530$	0	0
$531$	−6.99374	−0.303502
$532$	7.48097	0.324341
$533$	0	0
$534$	−8.63295	−0.373584
$535$	0	0
$536$	−13.7974	−0.595959
$537$	−3.19629	−0.137930
$538$	−30.6330	−1.32068
$539$	−3.48097	−0.149936
$540$	0	0
$541$	17.8796	0.768703	0.384352	−	0.923187i	$-0.374425\pi$
$541$	17.8796	0.768703	0.384352	+	0.923187i	$0.374425\pi$
$542$	31.5949	1.35712
$543$	62.0696	2.66366
$544$	1.00000	0.0428746
$545$	0	0
$546$	0	0
$547$	1.44918	0.0619624	0.0309812	−	0.999520i	$-0.490137\pi$
$547$	1.44918	0.0619624	0.0309812	+	0.999520i	$0.490137\pi$
$548$	9.27842	0.396355
$549$	92.7847	3.95995
$550$	0	0
$551$	−23.0348	−0.981316
$552$	9.48097	0.403537
$553$	−0.803708	−0.0341772
$554$	−25.4367	−1.08070
$555$	0	0
$556$	6.96195	0.295252
$557$	−21.3227	−0.903473	−0.451737	−	0.892151i	$-0.649195\pi$
$557$	−21.3227	−0.903473	−0.451737	+	0.892151i	$0.649195\pi$
$558$	−49.5095	−2.09591
$559$	0	0
$560$	0	0
$561$	−10.7183	−0.452528
$562$	−28.3575	−1.19619
$563$	34.6330	1.45960	0.729802	−	0.683658i	$-0.239612\pi$
$563$	34.6330	1.45960	0.729802	+	0.683658i	$0.239612\pi$
$564$	9.48097	0.399221
$565$	0	0
$566$	−28.6330	−1.20353
$567$	13.5601	0.569470
$568$	13.1202	0.550511
$569$	34.4429	1.44392	0.721961	−	0.691934i	$-0.243241\pi$
$569$	34.4429	1.44392	0.721961	+	0.691934i	$0.243241\pi$
$570$	0	0
$571$	−13.1520	−0.550393	−0.275197	−	0.961388i	$-0.588743\pi$
$571$	−13.1520	−0.550393	−0.275197	+	0.961388i	$0.588743\pi$
$572$	0	0
$573$	19.3513	0.808411
$574$	0.519027	0.0216638
$575$	0	0
$576$	6.48097	0.270041
$577$	20.9176	0.870812	0.435406	−	0.900234i	$-0.356605\pi$
$577$	20.9176	0.870812	0.435406	+	0.900234i	$0.356605\pi$
$578$	1.00000	0.0415945
$579$	−51.8671	−2.15552
$580$	0	0
$581$	−6.31648	−0.262052
$582$	−44.0821	−1.82726
$583$	17.6803	0.732242
$584$	−6.00000	−0.248282
$585$	0	0
$586$	24.8858	1.02802
$587$	−33.2784	−1.37355	−0.686774	−	0.726871i	$-0.740974\pi$
$587$	−33.2784	−1.37355	−0.686774	+	0.726871i	$0.740974\pi$
$588$	3.07912	0.126981
$589$	−57.1487	−2.35477
$590$	0	0
$591$	43.5949	1.79325
$592$	2.15824	0.0887030
$593$	−28.8448	−1.18451	−0.592256	−	0.805750i	$-0.701763\pi$
$593$	−28.8448	−1.18451	−0.592256	+	0.805750i	$0.701763\pi$
$594$	−37.3102	−1.53086
$595$	0	0
$596$	21.2784	0.871598
$597$	0.360789	0.0147661
$598$	0	0
$599$	24.7501	1.01126	0.505631	−	0.862750i	$-0.331260\pi$
$599$	24.7501	1.01126	0.505631	+	0.862750i	$0.331260\pi$
$600$	0	0
$601$	−33.7213	−1.37552	−0.687761	−	0.725937i	$-0.741406\pi$
$601$	−33.7213	−1.37552	−0.687761	+	0.725937i	$0.741406\pi$
$602$	7.07912	0.288523
$603$	−89.4209	−3.64150
$604$	−7.63921	−0.310835
$605$	0	0
$606$	60.4715	2.45649
$607$	15.5128	0.629644	0.314822	−	0.949151i	$-0.398055\pi$
$607$	15.5128	0.629644	0.314822	+	0.949151i	$0.398055\pi$
$608$	7.48097	0.303394
$609$	−9.48097	−0.384188
$610$	0	0
$611$	0	0
$612$	6.48097	0.261978
$613$	−25.0255	−1.01077	−0.505386	−	0.862893i	$-0.668650\pi$
$613$	−25.0255	−1.01077	−0.505386	+	0.862893i	$0.668650\pi$
$614$	−12.1582	−0.490667
$615$	0	0
$616$	−3.48097	−0.140252
$617$	15.6360	0.629480	0.314740	−	0.949178i	$-0.398083\pi$
$617$	15.6360	0.629480	0.314740	+	0.949178i	$0.398083\pi$
$618$	−59.7213	−2.40234
$619$	−5.52529	−0.222080	−0.111040	−	0.993816i	$-0.535418\pi$
$619$	−5.52529	−0.222080	−0.111040	+	0.993816i	$0.535418\pi$
$620$	0	0
$621$	33.0030	1.32437
$622$	8.11717	0.325469
$623$	−2.80371	−0.112328
$624$	0	0
$625$	0	0
$626$	5.76565	0.230442
$627$	−80.1835	−3.20222
$628$	−12.0000	−0.478852
$629$	2.15824	0.0860546
$630$	0	0
$631$	41.8796	1.66720	0.833600	−	0.552369i	$-0.186276\pi$
$631$	41.8796	1.66720	0.833600	+	0.552369i	$0.186276\pi$
$632$	−0.803708	−0.0319698
$633$	20.9494	0.832665
$634$	−10.9619	−0.435354
$635$	0	0
$636$	−15.6392	−0.620135
$637$	0	0
$638$	10.7183	0.424343
$639$	85.0316	3.36380
$640$	0	0
$641$	−5.43064	−0.214497	−0.107249	−	0.994232i	$-0.534204\pi$
$641$	−5.43064	−0.214497	−0.107249	+	0.994232i	$0.534204\pi$
$642$	20.9494	0.826808
$643$	−43.8478	−1.72919	−0.864594	−	0.502471i	$-0.832424\pi$
$643$	−43.8478	−1.72919	−0.864594	+	0.502471i	$0.832424\pi$
$644$	3.07912	0.121334
$645$	0	0
$646$	7.48097	0.294335
$647$	−21.9239	−0.861917	−0.430959	−	0.902372i	$-0.641824\pi$
$647$	−21.9239	−0.861917	−0.430959	+	0.902372i	$0.641824\pi$
$648$	13.5601	0.532691
$649$	3.75638	0.147451
$650$	0	0
$651$	−23.5220	−0.921901
$652$	−20.9619	−0.820933
$653$	−20.5508	−0.804216	−0.402108	−	0.915592i	$-0.631722\pi$
$653$	−20.5508	−0.804216	−0.402108	+	0.915592i	$0.631722\pi$
$654$	−32.6422	−1.27641
$655$	0	0
$656$	0.519027	0.0202646
$657$	−38.8858	−1.51708
$658$	3.07912	0.120037
$659$	−40.8858	−1.59269	−0.796343	−	0.604845i	$-0.793235\pi$
$659$	−40.8858	−1.59269	−0.796343	+	0.604845i	$0.793235\pi$
$660$	0	0
$661$	−20.2722	−0.788496	−0.394248	−	0.919004i	$-0.628995\pi$
$661$	−20.2722	−0.788496	−0.394248	+	0.919004i	$0.628995\pi$
$662$	−25.9239	−1.00756
$663$	0	0
$664$	−6.31648	−0.245127
$665$	0	0
$666$	13.9875	0.542004
$667$	−9.48097	−0.367105
$668$	−1.12018	−0.0433412
$669$	12.3165	0.476182
$670$	0	0
$671$	−49.8353	−1.92387
$672$	3.07912	0.118780
$673$	−12.2847	−0.473540	−0.236770	−	0.971566i	$-0.576089\pi$
$673$	−12.2847	−0.473540	−0.236770	+	0.971566i	$0.576089\pi$
$674$	1.31346	0.0505928
$675$	0	0
$676$	−13.0000	−0.500000
$677$	1.39559	0.0536370	0.0268185	−	0.999640i	$-0.491462\pi$
$677$	1.39559	0.0536370	0.0268185	+	0.999640i	$0.491462\pi$
$678$	4.07286	0.156417
$679$	−14.3165	−0.549416
$680$	0	0
$681$	−26.9712	−1.03354
$682$	26.5919	1.01826
$683$	−8.55684	−0.327418	−0.163709	−	0.986509i	$-0.552346\pi$
$683$	−8.55684	−0.327418	−0.163709	+	0.986509i	$0.552346\pi$
$684$	48.4840	1.85383
$685$	0	0
$686$	1.00000	0.0381802
$687$	−35.0884	−1.33871
$688$	7.07912	0.269889
$689$	0	0
$690$	0	0
$691$	−27.7657	−1.05626	−0.528128	−	0.849165i	$-0.677106\pi$
$691$	−27.7657	−1.05626	−0.528128	+	0.849165i	$0.677106\pi$
$692$	−13.9557	−0.530516
$693$	−22.5601	−0.856987
$694$	−12.4747	−0.473534
$695$	0	0
$696$	−9.48097	−0.359375
$697$	0.519027	0.0196596
$698$	0.487233	0.0184420
$699$	18.4747	0.698778
$700$	0	0
$701$	9.19629	0.347339	0.173670	−	0.984804i	$-0.444438\pi$
$701$	9.19629	0.347339	0.173670	+	0.984804i	$0.444438\pi$
$702$	0	0
$703$	16.1457	0.608947
$704$	−3.48097	−0.131194
$705$	0	0
$706$	24.6012	0.925877
$707$	19.6392	0.738609
$708$	−3.32274	−0.124876
$709$	41.8703	1.57247	0.786236	−	0.617926i	$-0.212027\pi$
$709$	41.8703	1.57247	0.786236	+	0.617926i	$0.212027\pi$
$710$	0	0
$711$	−5.20881	−0.195346
$712$	−2.80371	−0.105073
$713$	−23.5220	−0.880907
$714$	3.07912	0.115233
$715$	0	0
$716$	−1.03805	−0.0387939
$717$	28.4715	1.06329
$718$	20.1993	0.753831
$719$	−8.99374	−0.335410	−0.167705	−	0.985837i	$-0.553636\pi$
$719$	−8.99374	−0.335410	−0.167705	+	0.985837i	$0.553636\pi$
$720$	0	0
$721$	−19.3956	−0.722330
$722$	36.9650	1.37569
$723$	−0.126443	−0.00470248
$724$	20.1582	0.749175
$725$	0	0
$726$	3.43991	0.127667
$727$	0.838751	0.0311076	0.0155538	−	0.999879i	$-0.495049\pi$
$727$	0.838751	0.0311076	0.0155538	+	0.999879i	$0.495049\pi$
$728$	0	0
$729$	−11.1264	−0.412090
$730$	0	0
$731$	7.07912	0.261831
$732$	44.0821	1.62932
$733$	12.8858	0.475949	0.237975	−	0.971271i	$-0.423516\pi$
$733$	12.8858	0.475949	0.237975	+	0.971271i	$0.423516\pi$
$734$	−2.07611	−0.0766305
$735$	0	0
$736$	3.07912	0.113498
$737$	48.0285	1.76915
$738$	3.36380	0.123823
$739$	17.1202	0.629776	0.314888	−	0.949129i	$-0.398033\pi$
$739$	17.1202	0.629776	0.314888	+	0.949129i	$0.398033\pi$
$740$	0	0
$741$	0	0
$742$	−5.07912	−0.186460
$743$	−22.2404	−0.815920	−0.407960	−	0.913000i	$-0.633760\pi$
$743$	−22.2404	−0.815920	−0.407960	+	0.913000i	$0.633760\pi$
$744$	−23.5220	−0.862360
$745$	0	0
$746$	−21.4777	−0.786355
$747$	−40.9369	−1.49780
$748$	−3.48097	−0.127277
$749$	6.80371	0.248602
$750$	0	0
$751$	32.1643	1.17369	0.586845	−	0.809699i	$-0.300370\pi$
$751$	32.1643	1.17369	0.586845	+	0.809699i	$0.300370\pi$
$752$	3.07912	0.112284
$753$	−14.8891	−0.542588
$754$	0	0
$755$	0	0
$756$	10.7183	0.389822
$757$	41.8478	1.52098	0.760492	−	0.649348i	$-0.224958\pi$
$757$	41.8478	1.52098	0.760492	+	0.649348i	$0.224958\pi$
$758$	7.00301	0.254361
$759$	−33.0030	−1.19793
$760$	0	0
$761$	52.5568	1.90518	0.952592	−	0.304251i	$-0.0984060\pi$
$761$	52.5568	1.90518	0.952592	+	0.304251i	$0.0984060\pi$
$762$	47.0441	1.70423
$763$	−10.6012	−0.383788
$764$	6.28468	0.227372
$765$	0	0
$766$	−8.44292	−0.305055
$767$	0	0
$768$	3.07912	0.111108
$769$	47.9875	1.73047	0.865236	−	0.501364i	$-0.167168\pi$
$769$	47.9875	1.73047	0.865236	+	0.501364i	$0.167168\pi$
$770$	0	0
$771$	14.6647	0.528138
$772$	−16.8448	−0.606257
$773$	49.2659	1.77197	0.885986	−	0.463713i	$-0.153483\pi$
$773$	49.2659	1.77197	0.885986	+	0.463713i	$0.153483\pi$
$774$	45.8796	1.64911
$775$	0	0
$776$	−14.3165	−0.513932
$777$	6.64547	0.238405
$778$	−22.3986	−0.803029
$779$	3.88283	0.139117
$780$	0	0
$781$	−45.6710	−1.63424
$782$	3.07912	0.110109
$783$	−33.0030	−1.17943
$784$	1.00000	0.0357143
$785$	0	0
$786$	−33.7531	−1.20393
$787$	−18.2026	−0.648851	−0.324425	−	0.945911i	$-0.605171\pi$
$787$	−18.2026	−0.648851	−0.324425	+	0.945911i	$0.605171\pi$
$788$	14.1582	0.504366
$789$	−13.2909	−0.473170
$790$	0	0
$791$	1.32274	0.0470311
$792$	−22.5601	−0.801638
$793$	0	0
$794$	−10.6865	−0.379251
$795$	0	0
$796$	0.117173	0.00415308
$797$	20.3165	0.719646	0.359823	−	0.933020i	$-0.382837\pi$
$797$	20.3165	0.719646	0.359823	+	0.933020i	$0.382837\pi$
$798$	23.0348	0.815423
$799$	3.07912	0.108931
$800$	0	0
$801$	−18.1708	−0.642032
$802$	−20.9619	−0.740192
$803$	20.8858	0.737045
$804$	−42.4840	−1.49829
$805$	0	0
$806$	0	0
$807$	−94.3225	−3.32031
$808$	19.6392	0.690905
$809$	−3.12018	−0.109700	−0.0548499	−	0.998495i	$-0.517468\pi$
$809$	−3.12018	−0.109700	−0.0548499	+	0.998495i	$0.517468\pi$
$810$	0	0
$811$	−31.3606	−1.10122	−0.550609	−	0.834763i	$-0.685604\pi$
$811$	−31.3606	−1.10122	−0.550609	+	0.834763i	$0.685604\pi$
$812$	−3.07912	−0.108056
$813$	97.2844	3.41191
$814$	−7.51277	−0.263322
$815$	0	0
$816$	3.07912	0.107791
$817$	52.9587	1.85279
$818$	−29.5949	−1.03476
$819$	0	0
$820$	0	0
$821$	32.1457	1.12189	0.560947	−	0.827852i	$-0.310437\pi$
$821$	32.1457	1.12189	0.560947	+	0.827852i	$0.310437\pi$
$822$	28.5694	0.996471
$823$	7.08839	0.247086	0.123543	−	0.992339i	$-0.460574\pi$
$823$	7.08839	0.247086	0.123543	+	0.992339i	$0.460574\pi$
$824$	−19.3956	−0.675678
$825$	0	0
$826$	−1.07912	−0.0375473
$827$	−16.6455	−0.578820	−0.289410	−	0.957205i	$-0.593459\pi$
$827$	−16.6455	−0.578820	−0.289410	+	0.957205i	$0.593459\pi$
$828$	19.9557	0.693508
$829$	42.1325	1.46332	0.731660	−	0.681669i	$-0.238746\pi$
$829$	42.1325	1.46332	0.731660	+	0.681669i	$0.238746\pi$
$830$	0	0
$831$	−78.3225	−2.71698
$832$	0	0
$833$	1.00000	0.0346479
$834$	21.4367	0.742291
$835$	0	0
$836$	−26.0411	−0.900649
$837$	−81.8796	−2.83017
$838$	35.0441	1.21058
$839$	−7.64848	−0.264055	−0.132027	−	0.991246i	$-0.542149\pi$
$839$	−7.64848	−0.264055	−0.132027	+	0.991246i	$0.542149\pi$
$840$	0	0
$841$	−19.5190	−0.673070
$842$	−4.15824	−0.143302
$843$	−87.3162	−3.00733
$844$	6.80371	0.234193
$845$	0	0
$846$	19.9557	0.686091
$847$	1.11717	0.0383865
$848$	−5.07912	−0.174418
$849$	−88.1643	−3.02579
$850$	0	0
$851$	6.64547	0.227804
$852$	40.3986	1.38403
$853$	−13.3135	−0.455844	−0.227922	−	0.973679i	$-0.573193\pi$
$853$	−13.3135	−0.455844	−0.227922	+	0.973679i	$0.573193\pi$
$854$	14.3165	0.489900
$855$	0	0
$856$	6.80371	0.232546
$857$	20.6455	0.705236	0.352618	−	0.935767i	$-0.385292\pi$
$857$	20.6455	0.705236	0.352618	+	0.935767i	$0.385292\pi$
$858$	0	0
$859$	57.3606	1.95712	0.978558	−	0.205970i	$-0.0660348\pi$
$859$	57.3606	1.95712	0.978558	+	0.205970i	$0.0660348\pi$
$860$	0	0
$861$	1.59815	0.0544647
$862$	−32.6330	−1.11148
$863$	−20.7151	−0.705150	−0.352575	−	0.935784i	$-0.614694\pi$
$863$	−20.7151	−0.705150	−0.352575	+	0.935784i	$0.614694\pi$
$864$	10.7183	0.364645
$865$	0	0
$866$	−9.68352	−0.329060
$867$	3.07912	0.104572
$868$	−7.63921	−0.259292
$869$	2.79769	0.0949050
$870$	0	0
$871$	0	0
$872$	−10.6012	−0.359000
$873$	−92.7847	−3.14029
$874$	23.0348	0.779164
$875$	0	0
$876$	−18.4747	−0.624203
$877$	9.35453	0.315880	0.157940	−	0.987449i	$-0.449515\pi$
$877$	9.35453	0.315880	0.157940	+	0.987449i	$0.449515\pi$
$878$	−30.9084	−1.04311
$879$	76.6265	2.58455
$880$	0	0
$881$	−49.9650	−1.68336	−0.841681	−	0.539975i	$-0.818434\pi$
$881$	−49.9650	−1.68336	−0.841681	+	0.539975i	$0.818434\pi$
$882$	6.48097	0.218226
$883$	5.29094	0.178054	0.0890272	−	0.996029i	$-0.471624\pi$
$883$	5.29094	0.178054	0.0890272	+	0.996029i	$0.471624\pi$
$884$	0	0
$885$	0	0
$886$	9.00301	0.302462
$887$	9.75313	0.327478	0.163739	−	0.986504i	$-0.447644\pi$
$887$	9.75313	0.327478	0.163739	+	0.986504i	$0.447644\pi$
$888$	6.64547	0.223007
$889$	15.2784	0.512422
$890$	0	0
$891$	−47.2023	−1.58134
$892$	4.00000	0.133930
$893$	23.0348	0.770830
$894$	65.5188	2.19128
$895$	0	0
$896$	1.00000	0.0334077
$897$	0	0
$898$	21.5313	0.718509
$899$	23.5220	0.784504
$900$	0	0
$901$	−5.07912	−0.169210
$902$	−1.80672	−0.0601572
$903$	21.7974	0.725374
$904$	1.32274	0.0439935
$905$	0	0
$906$	−23.5220	−0.781467
$907$	41.5949	1.38114	0.690568	−	0.723268i	$-0.257361\pi$
$907$	41.5949	1.38114	0.690568	+	0.723268i	$0.257361\pi$
$908$	−8.75939	−0.290691
$909$	127.281	4.22165
$910$	0	0
$911$	−23.8292	−0.789498	−0.394749	−	0.918789i	$-0.629168\pi$
$911$	−23.8292	−0.789498	−0.394749	+	0.918789i	$0.629168\pi$
$912$	23.0348	0.762759
$913$	21.9875	0.727679
$914$	19.3545	0.640191
$915$	0	0
$916$	−11.3956	−0.376521
$917$	−10.9619	−0.361995
$918$	10.7183	0.353758
$919$	−44.5979	−1.47115	−0.735575	−	0.677444i	$-0.763088\pi$
$919$	−44.5979	−1.47115	−0.735575	+	0.677444i	$0.763088\pi$
$920$	0	0
$921$	−37.4367	−1.23358
$922$	10.6012	0.349131
$923$	0	0
$924$	−10.7183	−0.352607
$925$	0	0
$926$	−23.2784	−0.764976
$927$	−125.702	−4.12861
$928$	−3.07912	−0.101077
$929$	−17.0255	−0.558590	−0.279295	−	0.960205i	$-0.590101\pi$
$929$	−17.0255	−0.558590	−0.279295	+	0.960205i	$0.590101\pi$
$930$	0	0
$931$	7.48097	0.245179
$932$	6.00000	0.196537
$933$	24.9937	0.818258
$934$	12.7276	0.416460
$935$	0	0
$936$	0	0
$937$	35.5538	1.16149	0.580747	−	0.814084i	$-0.302761\pi$
$937$	35.5538	1.16149	0.580747	+	0.814084i	$0.302761\pi$
$938$	−13.7974	−0.450503
$939$	17.7531	0.579352
$940$	0	0
$941$	−30.0000	−0.977972	−0.488986	−	0.872292i	$-0.662633\pi$
$941$	−30.0000	−0.977972	−0.488986	+	0.872292i	$0.662633\pi$
$942$	−36.9494	−1.20388
$943$	1.59815	0.0520428
$944$	−1.07912	−0.0351223
$945$	0	0
$946$	−24.6422	−0.801188
$947$	43.4367	1.41150	0.705751	−	0.708460i	$-0.250610\pi$
$947$	43.4367	1.41150	0.705751	+	0.708460i	$0.250610\pi$
$948$	−2.47471	−0.0803749
$949$	0	0
$950$	0	0
$951$	−33.7531	−1.09452
$952$	1.00000	0.0324102
$953$	−51.7531	−1.67645	−0.838224	−	0.545326i	$-0.816406\pi$
$953$	−51.7531	−1.67645	−0.838224	+	0.545326i	$0.816406\pi$
$954$	−32.9176	−1.06575
$955$	0	0
$956$	9.24663	0.299057
$957$	33.0030	1.06684
$958$	−16.0000	−0.516937
$959$	9.27842	0.299616
$960$	0	0
$961$	27.3575	0.882501
$962$	0	0
$963$	44.0946	1.42093
$964$	−0.0410648	−0.00132261
$965$	0	0
$966$	9.48097	0.305045
$967$	−16.8673	−0.542416	−0.271208	−	0.962521i	$-0.587423\pi$
$967$	−16.8673	−0.542416	−0.271208	+	0.962521i	$0.587423\pi$
$968$	1.11717	0.0359073
$969$	23.0348	0.739985
$970$	0	0
$971$	50.6708	1.62610	0.813051	−	0.582192i	$-0.197805\pi$
$971$	50.6708	1.62610	0.813051	+	0.582192i	$0.197805\pi$
$972$	9.59815	0.307861
$973$	6.96195	0.223190
$974$	27.7213	0.888249
$975$	0	0
$976$	14.3165	0.458259
$977$	−58.8858	−1.88392	−0.941962	−	0.335718i	$-0.891021\pi$
$977$	−58.8858	−1.88392	−0.941962	+	0.335718i	$0.891021\pi$
$978$	−64.5443	−2.06390
$979$	9.75963	0.311919
$980$	0	0
$981$	−68.7058	−2.19361
$982$	20.6330	0.658424
$983$	16.2343	0.517795	0.258898	−	0.965905i	$-0.416641\pi$
$983$	16.2343	0.517795	0.258898	+	0.965905i	$0.416641\pi$
$984$	1.59815	0.0509470
$985$	0	0
$986$	−3.07912	−0.0980591
$987$	9.48097	0.301783
$988$	0	0
$989$	21.7974	0.693119
$990$	0	0
$991$	−20.6330	−0.655427	−0.327714	−	0.944777i	$-0.606278\pi$
$991$	−20.6330	−0.655427	−0.327714	+	0.944777i	$0.606278\pi$
$992$	−7.63921	−0.242545
$993$	−79.8227	−2.53310
$994$	13.1202	0.416147
$995$	0	0
$996$	−19.4492	−0.616271
$997$	60.5568	1.91785	0.958927	−	0.283652i	$-0.0915461\pi$
$997$	60.5568	1.91785	0.958927	+	0.283652i	$0.0915461\pi$
$998$	3.22809	0.102183
$999$	23.1327	0.731886

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5950.2.a.bm.1.3		3
5.4	even	2		1190.2.a.j.1.1	✓	3
20.19	odd	2		9520.2.a.z.1.3		3
35.34	odd	2		8330.2.a.bu.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1190.2.a.j.1.1	✓	3	5.4	even	2
5950.2.a.bm.1.3		3	1.1	even	1	trivial
8330.2.a.bu.1.3		3	35.34	odd	2
9520.2.a.z.1.3		3	20.19	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:	\( N \)	\(=\)	\( 5950 = 2 \cdot 5^{2} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5950.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +3.07912 q^{3} +1.00000 q^{4} +3.07912 q^{6} +1.00000 q^{7} +1.00000 q^{8} +6.48097 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +3.07912 q^{3} +1.00000 q^{4} +3.07912 q^{6} +1.00000 q^{7} +1.00000 q^{8} +6.48097 q^{9} -3.48097 q^{11} +3.07912 q^{12} +1.00000 q^{14} +1.00000 q^{16} +1.00000 q^{17} +6.48097 q^{18} +7.48097 q^{19} +3.07912 q^{21} -3.48097 q^{22} +3.07912 q^{23} +3.07912 q^{24} +10.7183 q^{27} +1.00000 q^{28} -3.07912 q^{29} -7.63921 q^{31} +1.00000 q^{32} -10.7183 q^{33} +1.00000 q^{34} +6.48097 q^{36} +2.15824 q^{37} +7.48097 q^{38} +0.519027 q^{41} +3.07912 q^{42} +7.07912 q^{43} -3.48097 q^{44} +3.07912 q^{46} +3.07912 q^{47} +3.07912 q^{48} +1.00000 q^{49} +3.07912 q^{51} -5.07912 q^{53} +10.7183 q^{54} +1.00000 q^{56} +23.0348 q^{57} -3.07912 q^{58} -1.07912 q^{59} +14.3165 q^{61} -7.63921 q^{62} +6.48097 q^{63} +1.00000 q^{64} -10.7183 q^{66} -13.7974 q^{67} +1.00000 q^{68} +9.48097 q^{69} +13.1202 q^{71} +6.48097 q^{72} -6.00000 q^{73} +2.15824 q^{74} +7.48097 q^{76} -3.48097 q^{77} -0.803708 q^{79} +13.5601 q^{81} +0.519027 q^{82} -6.31648 q^{83} +3.07912 q^{84} +7.07912 q^{86} -9.48097 q^{87} -3.48097 q^{88} -2.80371 q^{89} +3.07912 q^{92} -23.5220 q^{93} +3.07912 q^{94} +3.07912 q^{96} -14.3165 q^{97} +1.00000 q^{98} -22.5601 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + q^{3} + 3 q^{4} + q^{6} + 3 q^{7} + 3 q^{8} + 10 q^{9}+O(q^{10}) \) 3 * q + 3 * q^2 + q^3 + 3 * q^4 + q^6 + 3 * q^7 + 3 * q^8 + 10 * q^9	\( 3 q + 3 q^{2} + q^{3} + 3 q^{4} + q^{6} + 3 q^{7} + 3 q^{8} + 10 q^{9} - q^{11} + q^{12} + 3 q^{14} + 3 q^{16} + 3 q^{17} + 10 q^{18} + 13 q^{19} + q^{21} - q^{22} + q^{23} + q^{24} - 2 q^{27} + 3 q^{28} - q^{29} + 3 q^{31} + 3 q^{32} + 2 q^{33} + 3 q^{34} + 10 q^{36} - 10 q^{37} + 13 q^{38} + 11 q^{41} + q^{42} + 13 q^{43} - q^{44} + q^{46} + q^{47} + q^{48} + 3 q^{49} + q^{51} - 7 q^{53} - 2 q^{54} + 3 q^{56} + 2 q^{57} - q^{58} + 5 q^{59} + 10 q^{61} + 3 q^{62} + 10 q^{63} + 3 q^{64} + 2 q^{66} + q^{67} + 3 q^{68} + 19 q^{69} + 4 q^{71} + 10 q^{72} - 18 q^{73} - 10 q^{74} + 13 q^{76} - q^{77} + 23 q^{81} + 11 q^{82} + 14 q^{83} + q^{84} + 13 q^{86} - 19 q^{87} - q^{88} - 6 q^{89} + q^{92} - 34 q^{93} + q^{94} + q^{96} - 10 q^{97} + 3 q^{98} - 50 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 + q^3 + 3 * q^4 + q^6 + 3 * q^7 + 3 * q^8 + 10 * q^9 - q^11 + q^12 + 3 * q^14 + 3 * q^16 + 3 * q^17 + 10 * q^18 + 13 * q^19 + q^21 - q^22 + q^23 + q^24 - 2 * q^27 + 3 * q^28 - q^29 + 3 * q^31 + 3 * q^32 + 2 * q^33 + 3 * q^34 + 10 * q^36 - 10 * q^37 + 13 * q^38 + 11 * q^41 + q^42 + 13 * q^43 - q^44 + q^46 + q^47 + q^48 + 3 * q^49 + q^51 - 7 * q^53 - 2 * q^54 + 3 * q^56 + 2 * q^57 - q^58 + 5 * q^59 + 10 * q^61 + 3 * q^62 + 10 * q^63 + 3 * q^64 + 2 * q^66 + q^67 + 3 * q^68 + 19 * q^69 + 4 * q^71 + 10 * q^72 - 18 * q^73 - 10 * q^74 + 13 * q^76 - q^77 + 23 * q^81 + 11 * q^82 + 14 * q^83 + q^84 + 13 * q^86 - 19 * q^87 - q^88 - 6 * q^89 + q^92 - 34 * q^93 + q^94 + q^96 - 10 * q^97 + 3 * q^98 - 50 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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