LMFDB - Embedded newform 2368.2.a.bc.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2368,2,Mod(1,2368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2368, 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("2368.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2368 = 2^{6} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2368.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:	$18.9085751986$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.621.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 3 $ x^3 - 6*x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1184)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.14510$ of defining polynomial
Character	$\chi$	$=$	2368.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.14510 q^{3} +3.14510 q^{5} +1.74657 q^{7} +1.60147 q^{9} +O(q^{10})$	$q+2.14510 q^{3} +3.14510 q^{5} +1.74657 q^{7} +1.60147 q^{9} +2.89167 q^{11} +2.39853 q^{13} +6.74657 q^{15} -5.49314 q^{17} -2.00000 q^{19} +3.74657 q^{21} -1.60147 q^{23} +4.89167 q^{25} -3.00000 q^{27} +5.89167 q^{29} +3.94217 q^{31} +6.20293 q^{33} +5.49314 q^{35} +1.00000 q^{37} +5.14510 q^{39} +6.14510 q^{41} -5.20293 q^{43} +5.03677 q^{45} +0.253432 q^{47} -3.94950 q^{49} -11.7833 q^{51} -0.543637 q^{53} +9.09460 q^{55} -4.29021 q^{57} -10.6961 q^{59} -6.63824 q^{61} +2.79707 q^{63} +7.54364 q^{65} -7.14510 q^{67} -3.43531 q^{69} -4.03677 q^{71} +3.18188 q^{73} +10.4931 q^{75} +5.05050 q^{77} +9.89167 q^{79} -11.2397 q^{81} -6.03677 q^{83} -17.2765 q^{85} +12.6382 q^{87} +4.50686 q^{89} +4.18920 q^{91} +8.45636 q^{93} -6.29021 q^{95} +12.9863 q^{97} +4.63091 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^5 - 3 * q^7 + 3 * q^9	$ 3 q + 3 q^{5} - 3 q^{7} + 3 q^{9} - 6 q^{11} + 9 q^{13} + 12 q^{15} - 6 q^{19} + 3 q^{21} - 3 q^{23} - 9 q^{27} + 3 q^{29} + 9 q^{31} + 15 q^{33} + 3 q^{37} + 9 q^{39} + 12 q^{41} - 12 q^{43} - 6 q^{45} + 9 q^{47} - 6 q^{51} + 3 q^{53} + 9 q^{55} - 12 q^{59} + 3 q^{61} + 12 q^{63} + 18 q^{65} - 15 q^{67} + 9 q^{69} + 9 q^{71} - 18 q^{73} + 15 q^{75} + 27 q^{77} + 15 q^{79} - 9 q^{81} + 3 q^{83} - 6 q^{85} + 15 q^{87} + 30 q^{89} - 24 q^{91} + 30 q^{93} - 6 q^{95} + 6 q^{97}+O(q^{100}) $ 3 * q + 3 * q^5 - 3 * q^7 + 3 * q^9 - 6 * q^11 + 9 * q^13 + 12 * q^15 - 6 * q^19 + 3 * q^21 - 3 * q^23 - 9 * q^27 + 3 * q^29 + 9 * q^31 + 15 * q^33 + 3 * q^37 + 9 * q^39 + 12 * q^41 - 12 * q^43 - 6 * q^45 + 9 * q^47 - 6 * q^51 + 3 * q^53 + 9 * q^55 - 12 * q^59 + 3 * q^61 + 12 * q^63 + 18 * q^65 - 15 * q^67 + 9 * q^69 + 9 * q^71 - 18 * q^73 + 15 * q^75 + 27 * q^77 + 15 * q^79 - 9 * q^81 + 3 * q^83 - 6 * q^85 + 15 * q^87 + 30 * q^89 - 24 * q^91 + 30 * q^93 - 6 * q^95 + 6 * q^97

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 $

$2$	0	0
$2$	0	0
$3$	2.14510	1.23848	0.619238	−	0.785204i	$-0.287442\pi$
$3$	2.14510	1.23848	0.619238	+	0.785204i	$0.287442\pi$
$4$	0	0
$5$	3.14510	1.40653	0.703266	−	0.710926i	$-0.251724\pi$
$5$	3.14510	1.40653	0.703266	+	0.710926i	$0.251724\pi$
$6$	0	0
$7$	1.74657	0.660141	0.330070	−	0.943956i	$-0.392928\pi$
$7$	1.74657	0.660141	0.330070	+	0.943956i	$0.392928\pi$
$8$	0	0
$9$	1.60147	0.533822
$10$	0	0
$11$	2.89167	0.871872	0.435936	−	0.899978i	$-0.356417\pi$
$11$	2.89167	0.871872	0.435936	+	0.899978i	$0.356417\pi$
$12$	0	0
$13$	2.39853	0.665234	0.332617	−	0.943062i	$-0.392068\pi$
$13$	2.39853	0.665234	0.332617	+	0.943062i	$0.392068\pi$
$14$	0	0
$15$	6.74657	1.74196
$16$	0	0
$17$	−5.49314	−1.33228	−0.666141	−	0.745826i	$-0.732055\pi$
$17$	−5.49314	−1.33228	−0.666141	+	0.745826i	$0.732055\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	3.74657	0.817568
$22$	0	0
$23$	−1.60147	−0.333929	−0.166964	−	0.985963i	$-0.553396\pi$
$23$	−1.60147	−0.333929	−0.166964	+	0.985963i	$0.553396\pi$
$24$	0	0
$25$	4.89167	0.978334
$26$	0	0
$27$	−3.00000	−0.577350
$28$	0	0
$29$	5.89167	1.09406	0.547028	−	0.837114i	$-0.315759\pi$
$29$	5.89167	1.09406	0.547028	+	0.837114i	$0.315759\pi$
$30$	0	0
$31$	3.94217	0.708035	0.354017	−	0.935239i	$-0.384815\pi$
$31$	3.94217	0.708035	0.354017	+	0.935239i	$0.384815\pi$
$32$	0	0
$33$	6.20293	1.07979
$34$	0	0
$35$	5.49314	0.928510
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	5.14510	0.823876
$40$	0	0
$41$	6.14510	0.959704	0.479852	−	0.877350i	$-0.340690\pi$
$41$	6.14510	0.959704	0.479852	+	0.877350i	$0.340690\pi$
$42$	0	0
$43$	−5.20293	−0.793440	−0.396720	−	0.917940i	$-0.629852\pi$
$43$	−5.20293	−0.793440	−0.396720	+	0.917940i	$0.629852\pi$
$44$	0	0
$45$	5.03677	0.750838
$46$	0	0
$47$	0.253432	0.0369668	0.0184834	−	0.999829i	$-0.494116\pi$
$47$	0.253432	0.0369668	0.0184834	+	0.999829i	$0.494116\pi$
$48$	0	0
$49$	−3.94950	−0.564214
$50$	0	0
$51$	−11.7833	−1.65000
$52$	0	0
$53$	−0.543637	−0.0746743	−0.0373372	−	0.999303i	$-0.511888\pi$
$53$	−0.543637	−0.0746743	−0.0373372	+	0.999303i	$0.511888\pi$
$54$	0	0
$55$	9.09460	1.22632
$56$	0	0
$57$	−4.29021	−0.568252
$58$	0	0
$59$	−10.6961	−1.39251	−0.696255	−	0.717795i	$-0.745152\pi$
$59$	−10.6961	−1.39251	−0.696255	+	0.717795i	$0.745152\pi$
$60$	0	0
$61$	−6.63824	−0.849939	−0.424970	−	0.905208i	$-0.639715\pi$
$61$	−6.63824	−0.849939	−0.424970	+	0.905208i	$0.639715\pi$
$62$	0	0
$63$	2.79707	0.352398
$64$	0	0
$65$	7.54364	0.935673
$66$	0	0
$67$	−7.14510	−0.872913	−0.436457	−	0.899725i	$-0.643767\pi$
$67$	−7.14510	−0.872913	−0.436457	+	0.899725i	$0.643767\pi$
$68$	0	0
$69$	−3.43531	−0.413562
$70$	0	0
$71$	−4.03677	−0.479077	−0.239538	−	0.970887i	$-0.576996\pi$
$71$	−4.03677	−0.479077	−0.239538	+	0.970887i	$0.576996\pi$
$72$	0	0
$73$	3.18188	0.372410	0.186205	−	0.982511i	$-0.440381\pi$
$73$	3.18188	0.372410	0.186205	+	0.982511i	$0.440381\pi$
$74$	0	0
$75$	10.4931	1.21164
$76$	0	0
$77$	5.05050	0.575558
$78$	0	0
$79$	9.89167	1.11290	0.556450	−	0.830881i	$-0.312163\pi$
$79$	9.89167	1.11290	0.556450	+	0.830881i	$0.312163\pi$
$80$	0	0
$81$	−11.2397	−1.24886
$82$	0	0
$83$	−6.03677	−0.662622	−0.331311	−	0.943522i	$-0.607491\pi$
$83$	−6.03677	−0.662622	−0.331311	+	0.943522i	$0.607491\pi$
$84$	0	0
$85$	−17.2765	−1.87390
$86$	0	0
$87$	12.6382	1.35496
$88$	0	0
$89$	4.50686	0.477727	0.238863	−	0.971053i	$-0.423225\pi$
$89$	4.50686	0.477727	0.238863	+	0.971053i	$0.423225\pi$
$90$	0	0
$91$	4.18920	0.439148
$92$	0	0
$93$	8.45636	0.876884
$94$	0	0
$95$	−6.29021	−0.645361
$96$	0	0
$97$	12.9863	1.31856	0.659278	−	0.751899i	$-0.270862\pi$
$97$	12.9863	1.31856	0.659278	+	0.751899i	$0.270862\pi$
$98$	0	0
$99$	4.63091	0.465424
$100$	0	0
$101$	−16.6172	−1.65347	−0.826736	−	0.562590i	$-0.809805\pi$
$101$	−16.6172	−1.65347	−0.826736	+	0.562590i	$0.809805\pi$
$102$	0	0
$103$	−8.13138	−0.801208	−0.400604	−	0.916251i	$-0.631200\pi$
$103$	−8.13138	−0.801208	−0.400604	+	0.916251i	$0.631200\pi$
$104$	0	0
$105$	11.7833	1.14994
$106$	0	0
$107$	19.3848	1.87400	0.937000	−	0.349329i	$-0.113591\pi$
$107$	19.3848	1.87400	0.937000	+	0.349329i	$0.113591\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	2.14510	0.203604
$112$	0	0
$113$	20.4794	1.92654	0.963270	−	0.268533i	$-0.0865389\pi$
$113$	20.4794	1.92654	0.963270	+	0.268533i	$0.0865389\pi$
$114$	0	0
$115$	−5.03677	−0.469682
$116$	0	0
$117$	3.84117	0.355116
$118$	0	0
$119$	−9.59414	−0.879493
$120$	0	0
$121$	−2.63824	−0.239840
$122$	0	0
$123$	13.1819	1.18857
$124$	0	0
$125$	−0.340706	−0.0304737
$126$	0	0
$127$	−18.4426	−1.63652	−0.818260	−	0.574849i	$-0.805061\pi$
$127$	−18.4426	−1.63652	−0.818260	+	0.574849i	$0.805061\pi$
$128$	0	0
$129$	−11.1608	−0.982656
$130$	0	0
$131$	−17.3775	−1.51828	−0.759139	−	0.650929i	$-0.774380\pi$
$131$	−17.3775	−1.51828	−0.759139	+	0.650929i	$0.774380\pi$
$132$	0	0
$133$	−3.49314	−0.302893
$134$	0	0
$135$	−9.43531	−0.812062
$136$	0	0
$137$	−9.84117	−0.840788	−0.420394	−	0.907342i	$-0.638108\pi$
$137$	−9.84117	−0.840788	−0.420394	+	0.907342i	$0.638108\pi$
$138$	0	0
$139$	−6.80440	−0.577141	−0.288571	−	0.957459i	$-0.593180\pi$
$139$	−6.80440	−0.577141	−0.288571	+	0.957459i	$0.593180\pi$
$140$	0	0
$141$	0.543637	0.0457825
$142$	0	0
$143$	6.93577	0.579998
$144$	0	0
$145$	18.5299	1.53883
$146$	0	0
$147$	−8.47208	−0.698766
$148$	0	0
$149$	13.8201	1.13219	0.566094	−	0.824341i	$-0.308454\pi$
$149$	13.8201	1.13219	0.566094	+	0.824341i	$0.308454\pi$
$150$	0	0
$151$	−10.8706	−0.884638	−0.442319	−	0.896858i	$-0.645844\pi$
$151$	−10.8706	−0.884638	−0.442319	+	0.896858i	$0.645844\pi$
$152$	0	0
$153$	−8.79707	−0.711201
$154$	0	0
$155$	12.3985	0.995874
$156$	0	0
$157$	22.5162	1.79699	0.898494	−	0.438987i	$-0.144663\pi$
$157$	22.5162	1.79699	0.898494	+	0.438987i	$0.144663\pi$
$158$	0	0
$159$	−1.16616	−0.0924823
$160$	0	0
$161$	−2.79707	−0.220440
$162$	0	0
$163$	−14.0735	−1.10233	−0.551163	−	0.834398i	$-0.685816\pi$
$163$	−14.0735	−1.10233	−0.551163	+	0.834398i	$0.685816\pi$
$164$	0	0
$165$	19.5089	1.51876
$166$	0	0
$167$	−17.7255	−1.37164	−0.685821	−	0.727771i	$-0.740557\pi$
$167$	−17.7255	−1.37164	−0.685821	+	0.727771i	$0.740557\pi$
$168$	0	0
$169$	−7.24703	−0.557464
$170$	0	0
$171$	−3.20293	−0.244934
$172$	0	0
$173$	15.4564	1.17513	0.587563	−	0.809179i	$-0.300088\pi$
$173$	15.4564	1.17513	0.587563	+	0.809179i	$0.300088\pi$
$174$	0	0
$175$	8.54364	0.645838
$176$	0	0
$177$	−22.9442	−1.72459
$178$	0	0
$179$	4.69607	0.351001	0.175500	−	0.984479i	$-0.443846\pi$
$179$	4.69607	0.351001	0.175500	+	0.984479i	$0.443846\pi$
$180$	0	0
$181$	−3.05050	−0.226742	−0.113371	−	0.993553i	$-0.536165\pi$
$181$	−3.05050	−0.226742	−0.113371	+	0.993553i	$0.536165\pi$
$182$	0	0
$183$	−14.2397	−1.05263
$184$	0	0
$185$	3.14510	0.231233
$186$	0	0
$187$	−15.8843	−1.16158
$188$	0	0
$189$	−5.23970	−0.381132
$190$	0	0
$191$	−2.73924	−0.198204	−0.0991022	−	0.995077i	$-0.531597\pi$
$191$	−2.73924	−0.198204	−0.0991022	+	0.995077i	$0.531597\pi$
$192$	0	0
$193$	16.0000	1.15171	0.575853	−	0.817554i	$-0.304670\pi$
$193$	16.0000	1.15171	0.575853	+	0.817554i	$0.304670\pi$
$194$	0	0
$195$	16.1819	1.15881
$196$	0	0
$197$	−23.0230	−1.64032	−0.820162	−	0.572131i	$-0.806117\pi$
$197$	−23.0230	−1.64032	−0.820162	+	0.572131i	$0.806117\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	−15.3270	−1.08108
$202$	0	0
$203$	10.2902	0.722231
$204$	0	0
$205$	19.3270	1.34985
$206$	0	0
$207$	−2.56469	−0.178258
$208$	0	0
$209$	−5.78334	−0.400042
$210$	0	0
$211$	−14.9652	−1.03025	−0.515124	−	0.857116i	$-0.672254\pi$
$211$	−14.9652	−1.03025	−0.515124	+	0.857116i	$0.672254\pi$
$212$	0	0
$213$	−8.65929	−0.593325
$214$	0	0
$215$	−16.3638	−1.11600
$216$	0	0
$217$	6.88527	0.467403
$218$	0	0
$219$	6.82545	0.461221
$220$	0	0
$221$	−13.1755	−0.886278
$222$	0	0
$223$	22.2260	1.48836	0.744181	−	0.667978i	$-0.232840\pi$
$223$	22.2260	1.48836	0.744181	+	0.667978i	$0.232840\pi$
$224$	0	0
$225$	7.83384	0.522256
$226$	0	0
$227$	24.6961	1.63914	0.819568	−	0.572982i	$-0.194214\pi$
$227$	24.6961	1.63914	0.819568	+	0.572982i	$0.194214\pi$
$228$	0	0
$229$	1.92112	0.126951	0.0634755	−	0.997983i	$-0.479782\pi$
$229$	1.92112	0.126951	0.0634755	+	0.997983i	$0.479782\pi$
$230$	0	0
$231$	10.8338	0.712815
$232$	0	0
$233$	16.4721	1.07912	0.539561	−	0.841947i	$-0.318590\pi$
$233$	16.4721	1.07912	0.539561	+	0.841947i	$0.318590\pi$
$234$	0	0
$235$	0.797069	0.0519950
$236$	0	0
$237$	21.2186	1.37830
$238$	0	0
$239$	8.52258	0.551280	0.275640	−	0.961261i	$-0.411110\pi$
$239$	8.52258	0.551280	0.275640	+	0.961261i	$0.411110\pi$
$240$	0	0
$241$	6.17455	0.397738	0.198869	−	0.980026i	$-0.436273\pi$
$241$	6.17455	0.397738	0.198869	+	0.980026i	$0.436273\pi$
$242$	0	0
$243$	−15.1103	−0.969328
$244$	0	0
$245$	−12.4216	−0.793586
$246$	0	0
$247$	−4.79707	−0.305230
$248$	0	0
$249$	−12.9495	−0.820641
$250$	0	0
$251$	−25.9725	−1.63937	−0.819686	−	0.572813i	$-0.805852\pi$
$251$	−25.9725	−1.63937	−0.819686	+	0.572813i	$0.805852\pi$
$252$	0	0
$253$	−4.63091	−0.291143
$254$	0	0
$255$	−37.0598	−2.32078
$256$	0	0
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	1.74657	0.108526
$260$	0	0
$261$	9.43531	0.584031
$262$	0	0
$263$	−18.1524	−1.11933	−0.559663	−	0.828720i	$-0.689070\pi$
$263$	−18.1524	−1.11933	−0.559663	+	0.828720i	$0.689070\pi$
$264$	0	0
$265$	−1.70979	−0.105032
$266$	0	0
$267$	9.66769	0.591653
$268$	0	0
$269$	17.6677	1.07722	0.538609	−	0.842556i	$-0.318950\pi$
$269$	17.6677	1.07722	0.538609	+	0.842556i	$0.318950\pi$
$270$	0	0
$271$	16.4426	0.998819	0.499409	−	0.866366i	$-0.333550\pi$
$271$	16.4426	0.998819	0.499409	+	0.866366i	$0.333550\pi$
$272$	0	0
$273$	8.98627	0.543874
$274$	0	0
$275$	14.1451	0.852982
$276$	0	0
$277$	20.3059	1.22006	0.610032	−	0.792376i	$-0.291156\pi$
$277$	20.3059	1.22006	0.610032	+	0.792376i	$0.291156\pi$
$278$	0	0
$279$	6.31325	0.377965
$280$	0	0
$281$	15.5667	0.928630	0.464315	−	0.885670i	$-0.346300\pi$
$281$	15.5667	0.928630	0.464315	+	0.885670i	$0.346300\pi$
$282$	0	0
$283$	21.7833	1.29489	0.647443	−	0.762114i	$-0.275839\pi$
$283$	21.7833	1.29489	0.647443	+	0.762114i	$0.275839\pi$
$284$	0	0
$285$	−13.4931	−0.799264
$286$	0	0
$287$	10.7328	0.633540
$288$	0	0
$289$	13.1745	0.774973
$290$	0	0
$291$	27.8569	1.63300
$292$	0	0
$293$	7.60879	0.444510	0.222255	−	0.974989i	$-0.428658\pi$
$293$	7.60879	0.444510	0.222255	+	0.974989i	$0.428658\pi$
$294$	0	0
$295$	−33.6402	−1.95861
$296$	0	0
$297$	−8.67501	−0.503375
$298$	0	0
$299$	−3.84117	−0.222141
$300$	0	0
$301$	−9.08727	−0.523782
$302$	0	0
$303$	−35.6456	−2.04778
$304$	0	0
$305$	−20.8779	−1.19547
$306$	0	0
$307$	−11.7392	−0.669994	−0.334997	−	0.942219i	$-0.608735\pi$
$307$	−11.7392	−0.669994	−0.334997	+	0.942219i	$0.608735\pi$
$308$	0	0
$309$	−17.4426	−0.992277
$310$	0	0
$311$	12.5731	0.712954	0.356477	−	0.934304i	$-0.383978\pi$
$311$	12.5731	0.712954	0.356477	+	0.934304i	$0.383978\pi$
$312$	0	0
$313$	−25.4510	−1.43858	−0.719289	−	0.694711i	$-0.755532\pi$
$313$	−25.4510	−1.43858	−0.719289	+	0.694711i	$0.755532\pi$
$314$	0	0
$315$	8.79707	0.495659
$316$	0	0
$317$	−16.0735	−0.902780	−0.451390	−	0.892327i	$-0.649072\pi$
$317$	−16.0735	−0.902780	−0.451390	+	0.892327i	$0.649072\pi$
$318$	0	0
$319$	17.0368	0.953876
$320$	0	0
$321$	41.5824	2.32090
$322$	0	0
$323$	10.9863	0.611293
$324$	0	0
$325$	11.7328	0.650821
$326$	0	0
$327$	21.4510	1.18624
$328$	0	0
$329$	0.442636	0.0244033
$330$	0	0
$331$	9.08727	0.499482	0.249741	−	0.968313i	$-0.419655\pi$
$331$	9.08727	0.499482	0.249741	+	0.968313i	$0.419655\pi$
$332$	0	0
$333$	1.60147	0.0877598
$334$	0	0
$335$	−22.4721	−1.22778
$336$	0	0
$337$	−5.97895	−0.325694	−0.162847	−	0.986651i	$-0.552068\pi$
$337$	−5.97895	−0.325694	−0.162847	+	0.986651i	$0.552068\pi$
$338$	0	0
$339$	43.9304	2.38597
$340$	0	0
$341$	11.3995	0.617316
$342$	0	0
$343$	−19.1240	−1.03260
$344$	0	0
$345$	−10.8044	−0.581689
$346$	0	0
$347$	−22.8706	−1.22776	−0.613880	−	0.789400i	$-0.710392\pi$
$347$	−22.8706	−1.22776	−0.613880	+	0.789400i	$0.710392\pi$
$348$	0	0
$349$	−8.39121	−0.449171	−0.224585	−	0.974454i	$-0.572103\pi$
$349$	−8.39121	−0.449171	−0.224585	+	0.974454i	$0.572103\pi$
$350$	0	0
$351$	−7.19560	−0.384073
$352$	0	0
$353$	−17.8569	−0.950426	−0.475213	−	0.879871i	$-0.657629\pi$
$353$	−17.8569	−0.950426	−0.475213	+	0.879871i	$0.657629\pi$
$354$	0	0
$355$	−12.6961	−0.673837
$356$	0	0
$357$	−20.5804	−1.08923
$358$	0	0
$359$	−8.03677	−0.424165	−0.212082	−	0.977252i	$-0.568025\pi$
$359$	−8.03677	−0.424165	−0.212082	+	0.977252i	$0.568025\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	−5.65929	−0.297036
$364$	0	0
$365$	10.0073	0.523807
$366$	0	0
$367$	17.6088	0.919172	0.459586	−	0.888133i	$-0.347998\pi$
$367$	17.6088	0.919172	0.459586	+	0.888133i	$0.347998\pi$
$368$	0	0
$369$	9.84117	0.512311
$370$	0	0
$371$	−0.949499	−0.0492956
$372$	0	0
$373$	15.0084	0.777105	0.388553	−	0.921427i	$-0.372975\pi$
$373$	15.0084	0.777105	0.388553	+	0.921427i	$0.372975\pi$
$374$	0	0
$375$	−0.730849	−0.0377409
$376$	0	0
$377$	14.1314	0.727803
$378$	0	0
$379$	−5.56469	−0.285839	−0.142920	−	0.989734i	$-0.545649\pi$
$379$	−5.56469	−0.285839	−0.142920	+	0.989734i	$0.545649\pi$
$380$	0	0
$381$	−39.5613	−2.02679
$382$	0	0
$383$	26.1745	1.33746	0.668728	−	0.743507i	$-0.266839\pi$
$383$	26.1745	1.33746	0.668728	+	0.743507i	$0.266839\pi$
$384$	0	0
$385$	15.8843	0.809541
$386$	0	0
$387$	−8.33231	−0.423555
$388$	0	0
$389$	−27.5005	−1.39433	−0.697164	−	0.716911i	$-0.745555\pi$
$389$	−27.5005	−1.39433	−0.697164	+	0.716911i	$0.745555\pi$
$390$	0	0
$391$	8.79707	0.444887
$392$	0	0
$393$	−37.2765	−1.88035
$394$	0	0
$395$	31.1103	1.56533
$396$	0	0
$397$	−23.8201	−1.19550	−0.597749	−	0.801684i	$-0.703938\pi$
$397$	−23.8201	−1.19550	−0.597749	+	0.801684i	$0.703938\pi$
$398$	0	0
$399$	−7.49314	−0.375126
$400$	0	0
$401$	24.4373	1.22034	0.610170	−	0.792270i	$-0.291101\pi$
$401$	24.4373	1.22034	0.610170	+	0.792270i	$0.291101\pi$
$402$	0	0
$403$	9.45543	0.471009
$404$	0	0
$405$	−35.3500	−1.75656
$406$	0	0
$407$	2.89167	0.143335
$408$	0	0
$409$	−8.62252	−0.426356	−0.213178	−	0.977013i	$-0.568381\pi$
$409$	−8.62252	−0.426356	−0.213178	+	0.977013i	$0.568381\pi$
$410$	0	0
$411$	−21.1103	−1.04130
$412$	0	0
$413$	−18.6814	−0.919252
$414$	0	0
$415$	−18.9863	−0.931999
$416$	0	0
$417$	−14.5961	−0.714776
$418$	0	0
$419$	10.2186	0.499214	0.249607	−	0.968347i	$-0.419699\pi$
$419$	10.2186	0.499214	0.249607	+	0.968347i	$0.419699\pi$
$420$	0	0
$421$	−29.7339	−1.44914	−0.724571	−	0.689200i	$-0.757962\pi$
$421$	−29.7339	−1.44914	−0.724571	+	0.689200i	$0.757962\pi$
$422$	0	0
$423$	0.405862	0.0197337
$424$	0	0
$425$	−26.8706	−1.30342
$426$	0	0
$427$	−11.5941	−0.561080
$428$	0	0
$429$	14.8779	0.718314
$430$	0	0
$431$	16.6540	0.802193	0.401096	−	0.916036i	$-0.368629\pi$
$431$	16.6540	0.802193	0.401096	+	0.916036i	$0.368629\pi$
$432$	0	0
$433$	18.0020	0.865121	0.432560	−	0.901605i	$-0.357610\pi$
$433$	18.0020	0.865121	0.432560	+	0.901605i	$0.357610\pi$
$434$	0	0
$435$	39.7486	1.90580
$436$	0	0
$437$	3.20293	0.153217
$438$	0	0
$439$	−0.587739	−0.0280512	−0.0140256	−	0.999902i	$-0.504465\pi$
$439$	−0.587739	−0.0280512	−0.0140256	+	0.999902i	$0.504465\pi$
$440$	0	0
$441$	−6.32499	−0.301190
$442$	0	0
$443$	−15.0388	−0.714513	−0.357257	−	0.934006i	$-0.616288\pi$
$443$	−15.0388	−0.714513	−0.357257	+	0.934006i	$0.616288\pi$
$444$	0	0
$445$	14.1745	0.671938
$446$	0	0
$447$	29.6456	1.40219
$448$	0	0
$449$	−11.1608	−0.526712	−0.263356	−	0.964699i	$-0.584829\pi$
$449$	−11.1608	−0.526712	−0.263356	+	0.964699i	$0.584829\pi$
$450$	0	0
$451$	17.7696	0.836738
$452$	0	0
$453$	−23.3186	−1.09560
$454$	0	0
$455$	13.1755	0.617676
$456$	0	0
$457$	−25.9579	−1.21426	−0.607129	−	0.794603i	$-0.707679\pi$
$457$	−25.9579	−1.21426	−0.607129	+	0.794603i	$0.707679\pi$
$458$	0	0
$459$	16.4794	0.769193
$460$	0	0
$461$	4.17455	0.194428	0.0972141	−	0.995263i	$-0.469007\pi$
$461$	4.17455	0.194428	0.0972141	+	0.995263i	$0.469007\pi$
$462$	0	0
$463$	−2.10833	−0.0979824	−0.0489912	−	0.998799i	$-0.515601\pi$
$463$	−2.10833	−0.0979824	−0.0489912	+	0.998799i	$0.515601\pi$
$464$	0	0
$465$	26.5961	1.23337
$466$	0	0
$467$	−33.6677	−1.55795	−0.778977	−	0.627052i	$-0.784261\pi$
$467$	−33.6677	−1.55795	−0.778977	+	0.627052i	$0.784261\pi$
$468$	0	0
$469$	−12.4794	−0.576246
$470$	0	0
$471$	48.2995	2.22552
$472$	0	0
$473$	−15.0452	−0.691777
$474$	0	0
$475$	−9.78334	−0.448891
$476$	0	0
$477$	−0.870616	−0.0398628
$478$	0	0
$479$	2.10833	0.0963320	0.0481660	−	0.998839i	$-0.484662\pi$
$479$	2.10833	0.0963320	0.0481660	+	0.998839i	$0.484662\pi$
$480$	0	0
$481$	2.39853	0.109364
$482$	0	0
$483$	−6.00000	−0.273009
$484$	0	0
$485$	40.8432	1.85459
$486$	0	0
$487$	−25.7412	−1.16645	−0.583223	−	0.812312i	$-0.698209\pi$
$487$	−25.7412	−1.16645	−0.583223	+	0.812312i	$0.698209\pi$
$488$	0	0
$489$	−30.1892	−1.36520
$490$	0	0
$491$	18.5877	0.838853	0.419426	−	0.907789i	$-0.362231\pi$
$491$	18.5877	0.838853	0.419426	+	0.907789i	$0.362231\pi$
$492$	0	0
$493$	−32.3638	−1.45759
$494$	0	0
$495$	14.5647	0.654634
$496$	0	0
$497$	−7.05050	−0.316258
$498$	0	0
$499$	−22.9716	−1.02835	−0.514175	−	0.857685i	$-0.671902\pi$
$499$	−22.9716	−1.02835	−0.514175	+	0.857685i	$0.671902\pi$
$500$	0	0
$501$	−38.0230	−1.69874
$502$	0	0
$503$	31.9001	1.42235	0.711177	−	0.703013i	$-0.248163\pi$
$503$	31.9001	1.42235	0.711177	+	0.703013i	$0.248163\pi$
$504$	0	0
$505$	−52.2628	−2.32566
$506$	0	0
$507$	−15.5456	−0.690406
$508$	0	0
$509$	22.3270	0.989626	0.494813	−	0.868999i	$-0.335237\pi$
$509$	22.3270	0.989626	0.494813	+	0.868999i	$0.335237\pi$
$510$	0	0
$511$	5.55736	0.245843
$512$	0	0
$513$	6.00000	0.264906
$514$	0	0
$515$	−25.5740	−1.12693
$516$	0	0
$517$	0.732841	0.0322303
$518$	0	0
$519$	33.1555	1.45536
$520$	0	0
$521$	−0.110321	−0.00483325	−0.00241662	−	0.999997i	$-0.500769\pi$
$521$	−0.110321	−0.00483325	−0.00241662	+	0.999997i	$0.500769\pi$
$522$	0	0
$523$	20.9442	0.915824	0.457912	−	0.888997i	$-0.348597\pi$
$523$	20.9442	0.915824	0.457912	+	0.888997i	$0.348597\pi$
$524$	0	0
$525$	18.3270	0.799855
$526$	0	0
$527$	−21.6549	−0.943302
$528$	0	0
$529$	−20.4353	−0.888492
$530$	0	0
$531$	−17.1294	−0.743352
$532$	0	0
$533$	14.7392	0.638427
$534$	0	0
$535$	60.9672	2.63584
$536$	0	0
$537$	10.0735	0.434706
$538$	0	0
$539$	−11.4207	−0.491922
$540$	0	0
$541$	−25.5510	−1.09852	−0.549261	−	0.835651i	$-0.685091\pi$
$541$	−25.5510	−1.09852	−0.549261	+	0.835651i	$0.685091\pi$
$542$	0	0
$543$	−6.54364	−0.280814
$544$	0	0
$545$	31.4510	1.34721
$546$	0	0
$547$	−11.6823	−0.499501	−0.249750	−	0.968310i	$-0.580349\pi$
$547$	−11.6823	−0.499501	−0.249750	+	0.968310i	$0.580349\pi$
$548$	0	0
$549$	−10.6309	−0.453716
$550$	0	0
$551$	−11.7833	−0.501987
$552$	0	0
$553$	17.2765	0.734671
$554$	0	0
$555$	6.74657	0.286376
$556$	0	0
$557$	16.6971	0.707480	0.353740	−	0.935344i	$-0.384910\pi$
$557$	16.6971	0.707480	0.353740	+	0.935344i	$0.384910\pi$
$558$	0	0
$559$	−12.4794	−0.527823
$560$	0	0
$561$	−34.0735	−1.43859
$562$	0	0
$563$	34.9588	1.47334	0.736669	−	0.676253i	$-0.236397\pi$
$563$	34.9588	1.47334	0.736669	+	0.676253i	$0.236397\pi$
$564$	0	0
$565$	64.4098	2.70974
$566$	0	0
$567$	−19.6309	−0.824421
$568$	0	0
$569$	19.4931	0.817195	0.408597	−	0.912715i	$-0.366018\pi$
$569$	19.4931	0.817195	0.408597	+	0.912715i	$0.366018\pi$
$570$	0	0
$571$	−30.9422	−1.29489	−0.647445	−	0.762112i	$-0.724162\pi$
$571$	−30.9422	−1.29489	−0.647445	+	0.762112i	$0.724162\pi$
$572$	0	0
$573$	−5.87595	−0.245471
$574$	0	0
$575$	−7.83384	−0.326694
$576$	0	0
$577$	−8.87062	−0.369289	−0.184644	−	0.982805i	$-0.559113\pi$
$577$	−8.87062	−0.369289	−0.184644	+	0.982805i	$0.559113\pi$
$578$	0	0
$579$	34.3216	1.42636
$580$	0	0
$581$	−10.5436	−0.437424
$582$	0	0
$583$	−1.57202	−0.0651064
$584$	0	0
$585$	12.0809	0.499483
$586$	0	0
$587$	−20.1745	−0.832693	−0.416346	−	0.909206i	$-0.636690\pi$
$587$	−20.1745	−0.832693	−0.416346	+	0.909206i	$0.636690\pi$
$588$	0	0
$589$	−7.88434	−0.324869
$590$	0	0
$591$	−49.3868	−2.03150
$592$	0	0
$593$	−32.5319	−1.33593	−0.667963	−	0.744195i	$-0.732833\pi$
$593$	−32.5319	−1.33593	−0.667963	+	0.744195i	$0.732833\pi$
$594$	0	0
$595$	−30.1745	−1.23704
$596$	0	0
$597$	34.3216	1.40469
$598$	0	0
$599$	13.3554	0.545685	0.272843	−	0.962059i	$-0.412036\pi$
$599$	13.3554	0.545685	0.272843	+	0.962059i	$0.412036\pi$
$600$	0	0
$601$	−25.6099	−1.04465	−0.522324	−	0.852747i	$-0.674935\pi$
$601$	−25.6099	−1.04465	−0.522324	+	0.852747i	$0.674935\pi$
$602$	0	0
$603$	−11.4426	−0.465980
$604$	0	0
$605$	−8.29753	−0.337343
$606$	0	0
$607$	37.6015	1.52620	0.763098	−	0.646283i	$-0.223677\pi$
$607$	37.6015	1.52620	0.763098	+	0.646283i	$0.223677\pi$
$608$	0	0
$609$	22.0735	0.894465
$610$	0	0
$611$	0.607865	0.0245916
$612$	0	0
$613$	−31.8201	−1.28520	−0.642601	−	0.766201i	$-0.722145\pi$
$613$	−31.8201	−1.28520	−0.642601	+	0.766201i	$0.722145\pi$
$614$	0	0
$615$	41.4584	1.67176
$616$	0	0
$617$	43.8128	1.76384	0.881918	−	0.471402i	$-0.156252\pi$
$617$	43.8128	1.76384	0.881918	+	0.471402i	$0.156252\pi$
$618$	0	0
$619$	−28.9063	−1.16184	−0.580922	−	0.813959i	$-0.697308\pi$
$619$	−28.9063	−1.16184	−0.580922	+	0.813959i	$0.697308\pi$
$620$	0	0
$621$	4.80440	0.192794
$622$	0	0
$623$	7.87154	0.315367
$624$	0	0
$625$	−25.5299	−1.02120
$626$	0	0
$627$	−12.4059	−0.495442
$628$	0	0
$629$	−5.49314	−0.219026
$630$	0	0
$631$	35.7907	1.42480	0.712402	−	0.701772i	$-0.247607\pi$
$631$	35.7907	1.42480	0.712402	+	0.701772i	$0.247607\pi$
$632$	0	0
$633$	−32.1019	−1.27594
$634$	0	0
$635$	−58.0040	−2.30182
$636$	0	0
$637$	−9.47301	−0.375334
$638$	0	0
$639$	−6.46475	−0.255742
$640$	0	0
$641$	−12.8412	−0.507196	−0.253598	−	0.967310i	$-0.581614\pi$
$641$	−12.8412	−0.507196	−0.253598	+	0.967310i	$0.581614\pi$
$642$	0	0
$643$	−21.9579	−0.865935	−0.432967	−	0.901410i	$-0.642533\pi$
$643$	−21.9579	−0.865935	−0.432967	+	0.901410i	$0.642533\pi$
$644$	0	0
$645$	−35.1019	−1.38214
$646$	0	0
$647$	−15.4858	−0.608810	−0.304405	−	0.952543i	$-0.598458\pi$
$647$	−15.4858	−0.608810	−0.304405	+	0.952543i	$0.598458\pi$
$648$	0	0
$649$	−30.9295	−1.21409
$650$	0	0
$651$	14.7696	0.578867
$652$	0	0
$653$	23.0525	0.902114	0.451057	−	0.892495i	$-0.351047\pi$
$653$	23.0525	0.902114	0.451057	+	0.892495i	$0.351047\pi$
$654$	0	0
$655$	−54.6540	−2.13551
$656$	0	0
$657$	5.09567	0.198801
$658$	0	0
$659$	−5.05783	−0.197025	−0.0985125	−	0.995136i	$-0.531408\pi$
$659$	−5.05783	−0.197025	−0.0985125	+	0.995136i	$0.531408\pi$
$660$	0	0
$661$	−34.9789	−1.36052	−0.680262	−	0.732969i	$-0.738134\pi$
$661$	−34.9789	−1.36052	−0.680262	+	0.732969i	$0.738134\pi$
$662$	0	0
$663$	−28.2628	−1.09763
$664$	0	0
$665$	−10.9863	−0.426029
$666$	0	0
$667$	−9.43531	−0.365337
$668$	0	0
$669$	47.6770	1.84330
$670$	0	0
$671$	−19.1956	−0.741038
$672$	0	0
$673$	7.73924	0.298326	0.149163	−	0.988813i	$-0.452342\pi$
$673$	7.73924	0.298326	0.149163	+	0.988813i	$0.452342\pi$
$674$	0	0
$675$	−14.6750	−0.564842
$676$	0	0
$677$	37.5446	1.44295	0.721477	−	0.692438i	$-0.243463\pi$
$677$	37.5446	1.44295	0.721477	+	0.692438i	$0.243463\pi$
$678$	0	0
$679$	22.6814	0.870433
$680$	0	0
$681$	52.9756	2.03003
$682$	0	0
$683$	4.85596	0.185808	0.0929041	−	0.995675i	$-0.470385\pi$
$683$	4.85596	0.185808	0.0929041	+	0.995675i	$0.470385\pi$
$684$	0	0
$685$	−30.9515	−1.18260
$686$	0	0
$687$	4.12099	0.157226
$688$	0	0
$689$	−1.30393	−0.0496759
$690$	0	0
$691$	24.1471	0.918599	0.459299	−	0.888282i	$-0.348101\pi$
$691$	24.1471	0.918599	0.459299	+	0.888282i	$0.348101\pi$
$692$	0	0
$693$	8.08820	0.307245
$694$	0	0
$695$	−21.4005	−0.811768
$696$	0	0
$697$	−33.7559	−1.27860
$698$	0	0
$699$	35.3343	1.33647
$700$	0	0
$701$	13.0441	0.492669	0.246334	−	0.969185i	$-0.420774\pi$
$701$	13.0441	0.492669	0.246334	+	0.969185i	$0.420774\pi$
$702$	0	0
$703$	−2.00000	−0.0754314
$704$	0	0
$705$	1.70979	0.0643946
$706$	0	0
$707$	−29.0230	−1.09152
$708$	0	0
$709$	−42.5098	−1.59649	−0.798244	−	0.602334i	$-0.794238\pi$
$709$	−42.5098	−1.59649	−0.798244	+	0.602334i	$0.794238\pi$
$710$	0	0
$711$	15.8412	0.594090
$712$	0	0
$713$	−6.31325	−0.236433
$714$	0	0
$715$	21.8137	0.815787
$716$	0	0
$717$	18.2818	0.682747
$718$	0	0
$719$	26.9220	1.00402	0.502011	−	0.864861i	$-0.332594\pi$
$719$	26.9220	1.00402	0.502011	+	0.864861i	$0.332594\pi$
$720$	0	0
$721$	−14.2020	−0.528910
$722$	0	0
$723$	13.2450	0.492588
$724$	0	0
$725$	28.8201	1.07035
$726$	0	0
$727$	−41.4814	−1.53846	−0.769230	−	0.638972i	$-0.779360\pi$
$727$	−41.4814	−1.53846	−0.769230	+	0.638972i	$0.779360\pi$
$728$	0	0
$729$	1.30592	0.0483676
$730$	0	0
$731$	28.5804	1.05708
$732$	0	0
$733$	23.5299	0.869097	0.434549	−	0.900648i	$-0.356908\pi$
$733$	23.5299	0.869097	0.434549	+	0.900648i	$0.356908\pi$
$734$	0	0
$735$	−26.6456	−0.982837
$736$	0	0
$737$	−20.6613	−0.761068
$738$	0	0
$739$	11.3250	0.416596	0.208298	−	0.978065i	$-0.433208\pi$
$739$	11.3250	0.416596	0.208298	+	0.978065i	$0.433208\pi$
$740$	0	0
$741$	−10.2902	−0.378020
$742$	0	0
$743$	−46.3731	−1.70126	−0.850632	−	0.525762i	$-0.823780\pi$
$743$	−46.3731	−1.70126	−0.850632	+	0.525762i	$0.823780\pi$
$744$	0	0
$745$	43.4657	1.59246
$746$	0	0
$747$	−9.66769	−0.353722
$748$	0	0
$749$	33.8569	1.23710
$750$	0	0
$751$	19.9632	0.728468	0.364234	−	0.931307i	$-0.381331\pi$
$751$	19.9632	0.728468	0.364234	+	0.931307i	$0.381331\pi$
$752$	0	0
$753$	−55.7138	−2.03032
$754$	0	0
$755$	−34.1892	−1.24427
$756$	0	0
$757$	11.2922	0.410422	0.205211	−	0.978718i	$-0.434212\pi$
$757$	11.2922	0.410422	0.205211	+	0.978718i	$0.434212\pi$
$758$	0	0
$759$	−9.93378	−0.360573
$760$	0	0
$761$	48.5549	1.76012	0.880058	−	0.474867i	$-0.157504\pi$
$761$	48.5549	1.76012	0.880058	+	0.474867i	$0.157504\pi$
$762$	0	0
$763$	17.4657	0.632300
$764$	0	0
$765$	−27.6677	−1.00033
$766$	0	0
$767$	−25.6549	−0.926344
$768$	0	0
$769$	7.31859	0.263915	0.131958	−	0.991255i	$-0.457874\pi$
$769$	7.31859	0.263915	0.131958	+	0.991255i	$0.457874\pi$
$770$	0	0
$771$	12.8706	0.463524
$772$	0	0
$773$	−35.2711	−1.26861	−0.634307	−	0.773081i	$-0.718715\pi$
$773$	−35.2711	−1.26861	−0.634307	+	0.773081i	$0.718715\pi$
$774$	0	0
$775$	19.2838	0.692695
$776$	0	0
$777$	3.74657	0.134407
$778$	0	0
$779$	−12.2902	−0.440342
$780$	0	0
$781$	−11.6730	−0.417694
$782$	0	0
$783$	−17.6750	−0.631653
$784$	0	0
$785$	70.8157	2.52752
$786$	0	0
$787$	−22.9220	−0.817083	−0.408541	−	0.912740i	$-0.633962\pi$
$787$	−22.9220	−0.817083	−0.408541	+	0.912740i	$0.633962\pi$
$788$	0	0
$789$	−38.9388	−1.38626
$790$	0	0
$791$	35.7687	1.27179
$792$	0	0
$793$	−15.9220	−0.565408
$794$	0	0
$795$	−3.66769	−0.130079
$796$	0	0
$797$	22.1819	0.785722	0.392861	−	0.919598i	$-0.371485\pi$
$797$	22.1819	0.785722	0.392861	+	0.919598i	$0.371485\pi$
$798$	0	0
$799$	−1.39214	−0.0492502
$800$	0	0
$801$	7.21759	0.255021
$802$	0	0
$803$	9.20094	0.324694
$804$	0	0
$805$	−8.79707	−0.310056
$806$	0	0
$807$	37.8990	1.33411
$808$	0	0
$809$	27.7098	0.974224	0.487112	−	0.873339i	$-0.338050\pi$
$809$	27.7098	0.974224	0.487112	+	0.873339i	$0.338050\pi$
$810$	0	0
$811$	−40.4437	−1.42017	−0.710085	−	0.704116i	$-0.751344\pi$
$811$	−40.4437	−1.42017	−0.710085	+	0.704116i	$0.751344\pi$
$812$	0	0
$813$	35.2711	1.23701
$814$	0	0
$815$	−44.2628	−1.55046
$816$	0	0
$817$	10.4059	0.364055
$818$	0	0
$819$	6.70887	0.234427
$820$	0	0
$821$	−1.93577	−0.0675589	−0.0337795	−	0.999429i	$-0.510754\pi$
$821$	−1.93577	−0.0675589	−0.0337795	+	0.999429i	$0.510754\pi$
$822$	0	0
$823$	28.3784	0.989209	0.494605	−	0.869118i	$-0.335313\pi$
$823$	28.3784	0.989209	0.494605	+	0.869118i	$0.335313\pi$
$824$	0	0
$825$	30.3427	1.05640
$826$	0	0
$827$	−16.2902	−0.566466	−0.283233	−	0.959051i	$-0.591407\pi$
$827$	−16.2902	−0.566466	−0.283233	+	0.959051i	$0.591407\pi$
$828$	0	0
$829$	14.6152	0.507607	0.253803	−	0.967256i	$-0.418318\pi$
$829$	14.6152	0.507607	0.253803	+	0.967256i	$0.418318\pi$
$830$	0	0
$831$	43.5583	1.51102
$832$	0	0
$833$	21.6951	0.751692
$834$	0	0
$835$	−55.7486	−1.92926
$836$	0	0
$837$	−11.8265	−0.408784
$838$	0	0
$839$	−22.5215	−0.777529	−0.388765	−	0.921337i	$-0.627098\pi$
$839$	−22.5215	−0.777529	−0.388765	+	0.921337i	$0.627098\pi$
$840$	0	0
$841$	5.71179	0.196958
$842$	0	0
$843$	33.3921	1.15009
$844$	0	0
$845$	−22.7927	−0.784091
$846$	0	0
$847$	−4.60786	−0.158328
$848$	0	0
$849$	46.7275	1.60368
$850$	0	0
$851$	−1.60147	−0.0548975
$852$	0	0
$853$	22.0578	0.755246	0.377623	−	0.925960i	$-0.376742\pi$
$853$	22.0578	0.755246	0.377623	+	0.925960i	$0.376742\pi$
$854$	0	0
$855$	−10.0735	−0.344508
$856$	0	0
$857$	9.26182	0.316378	0.158189	−	0.987409i	$-0.449434\pi$
$857$	9.26182	0.316378	0.158189	+	0.987409i	$0.449434\pi$
$858$	0	0
$859$	−43.8148	−1.49494	−0.747470	−	0.664295i	$-0.768732\pi$
$859$	−43.8148	−1.49494	−0.747470	+	0.664295i	$0.768732\pi$
$860$	0	0
$861$	23.0230	0.784623
$862$	0	0
$863$	−0.231314	−0.00787401	−0.00393700	−	0.999992i	$-0.501253\pi$
$863$	−0.231314	−0.00787401	−0.00393700	+	0.999992i	$0.501253\pi$
$864$	0	0
$865$	48.6118	1.65285
$866$	0	0
$867$	28.2608	0.959786
$868$	0	0
$869$	28.6035	0.970306
$870$	0	0
$871$	−17.1378	−0.580691
$872$	0	0
$873$	20.7971	0.703874
$874$	0	0
$875$	−0.595066	−0.0201169
$876$	0	0
$877$	−34.1471	−1.15307	−0.576533	−	0.817074i	$-0.695595\pi$
$877$	−34.1471	−1.15307	−0.576533	+	0.817074i	$0.695595\pi$
$878$	0	0
$879$	16.3216	0.550515
$880$	0	0
$881$	−40.7118	−1.37161	−0.685807	−	0.727783i	$-0.740551\pi$
$881$	−40.7118	−1.37161	−0.685807	+	0.727783i	$0.740551\pi$
$882$	0	0
$883$	36.2206	1.21892	0.609461	−	0.792816i	$-0.291386\pi$
$883$	36.2206	1.21892	0.609461	+	0.792816i	$0.291386\pi$
$884$	0	0
$885$	−72.1618	−2.42569
$886$	0	0
$887$	2.36909	0.0795462	0.0397731	−	0.999209i	$-0.487336\pi$
$887$	2.36909	0.0795462	0.0397731	+	0.999209i	$0.487336\pi$
$888$	0	0
$889$	−32.2113	−1.08033
$890$	0	0
$891$	−32.5015	−1.08884
$892$	0	0
$893$	−0.506864	−0.0169615
$894$	0	0
$895$	14.7696	0.493694
$896$	0	0
$897$	−8.23970	−0.275116
$898$	0	0
$899$	23.2260	0.774630
$900$	0	0
$901$	2.98627	0.0994872
$902$	0	0
$903$	−19.4931	−0.648691
$904$	0	0
$905$	−9.59414	−0.318920
$906$	0	0
$907$	33.5667	1.11456	0.557282	−	0.830323i	$-0.311844\pi$
$907$	33.5667	1.11456	0.557282	+	0.830323i	$0.311844\pi$
$908$	0	0
$909$	−26.6118	−0.882659
$910$	0	0
$911$	7.92645	0.262615	0.131308	−	0.991342i	$-0.458082\pi$
$911$	7.92645	0.262615	0.131308	+	0.991342i	$0.458082\pi$
$912$	0	0
$913$	−17.4564	−0.577721
$914$	0	0
$915$	−44.7853	−1.48056
$916$	0	0
$917$	−30.3510	−1.00228
$918$	0	0
$919$	−4.00000	−0.131948	−0.0659739	−	0.997821i	$-0.521015\pi$
$919$	−4.00000	−0.131948	−0.0659739	+	0.997821i	$0.521015\pi$
$920$	0	0
$921$	−25.1819	−0.829771
$922$	0	0
$923$	−9.68234	−0.318698
$924$	0	0
$925$	4.89167	0.160837
$926$	0	0
$927$	−13.0221	−0.427702
$928$	0	0
$929$	4.68035	0.153557	0.0767786	−	0.997048i	$-0.475537\pi$
$929$	4.68035	0.153557	0.0767786	+	0.997048i	$0.475537\pi$
$930$	0	0
$931$	7.89900	0.258879
$932$	0	0
$933$	26.9706	0.882976
$934$	0	0
$935$	−49.9579	−1.63380
$936$	0	0
$937$	28.1912	0.920966	0.460483	−	0.887668i	$-0.347676\pi$
$937$	28.1912	0.920966	0.460483	+	0.887668i	$0.347676\pi$
$938$	0	0
$939$	−54.5951	−1.78164
$940$	0	0
$941$	1.05584	0.0344193	0.0172096	−	0.999852i	$-0.494522\pi$
$941$	1.05584	0.0344193	0.0172096	+	0.999852i	$0.494522\pi$
$942$	0	0
$943$	−9.84117	−0.320473
$944$	0	0
$945$	−16.4794	−0.536075
$946$	0	0
$947$	46.4794	1.51038	0.755189	−	0.655507i	$-0.227545\pi$
$947$	46.4794	1.51038	0.755189	+	0.655507i	$0.227545\pi$
$948$	0	0
$949$	7.63184	0.247740
$950$	0	0
$951$	−34.4794	−1.11807
$952$	0	0
$953$	11.8358	0.383400	0.191700	−	0.981454i	$-0.438600\pi$
$953$	11.8358	0.383400	0.191700	+	0.981454i	$0.438600\pi$
$954$	0	0
$955$	−8.61519	−0.278781
$956$	0	0
$957$	36.5456	1.18135
$958$	0	0
$959$	−17.1883	−0.555038
$960$	0	0
$961$	−15.4593	−0.498687
$962$	0	0
$963$	31.0441	1.00038
$964$	0	0
$965$	50.3216	1.61991
$966$	0	0
$967$	55.2226	1.77584	0.887920	−	0.459998i	$-0.152150\pi$
$967$	55.2226	1.77584	0.887920	+	0.459998i	$0.152150\pi$
$968$	0	0
$969$	23.5667	0.757071
$970$	0	0
$971$	−57.4353	−1.84319	−0.921593	−	0.388157i	$-0.873112\pi$
$971$	−57.4353	−1.84319	−0.921593	+	0.388157i	$0.873112\pi$
$972$	0	0
$973$	−11.8843	−0.380995
$974$	0	0
$975$	25.1681	0.806026
$976$	0	0
$977$	−5.26182	−0.168341	−0.0841703	−	0.996451i	$-0.526824\pi$
$977$	−5.26182	−0.168341	−0.0841703	+	0.996451i	$0.526824\pi$
$978$	0	0
$979$	13.0324	0.416516
$980$	0	0
$981$	16.0147	0.511309
$982$	0	0
$983$	−57.6770	−1.83961	−0.919805	−	0.392375i	$-0.871654\pi$
$983$	−57.6770	−1.83961	−0.919805	+	0.392375i	$0.871654\pi$
$984$	0	0
$985$	−72.4098	−2.30717
$986$	0	0
$987$	0.949499	0.0302229
$988$	0	0
$989$	8.33231	0.264952
$990$	0	0
$991$	29.0250	0.922011	0.461005	−	0.887397i	$-0.347489\pi$
$991$	29.0250	0.922011	0.461005	+	0.887397i	$0.347489\pi$
$992$	0	0
$993$	19.4931	0.618596
$994$	0	0
$995$	50.3216	1.59530
$996$	0	0
$997$	19.0873	0.604500	0.302250	−	0.953229i	$-0.402262\pi$
$997$	19.0873	0.604500	0.302250	+	0.953229i	$0.402262\pi$
$998$	0	0
$999$	−3.00000	−0.0949158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2368.2.a.bc.1.3		3
4.3	odd	2		2368.2.a.bd.1.1		3
8.3	odd	2		1184.2.a.m.1.3	yes	3
8.5	even	2		1184.2.a.l.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1184.2.a.l.1.1	✓	3	8.5	even	2
1184.2.a.m.1.3	yes	3	8.3	odd	2
2368.2.a.bc.1.3		3	1.1	even	1	trivial
2368.2.a.bd.1.1		3	4.3	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:	\( N \)	\(=\)	\( 2368 = 2^{6} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2368.a (trivial)

\(f(q)\)	\(=\)	\(q+2.14510 q^{3} +3.14510 q^{5} +1.74657 q^{7} +1.60147 q^{9} +O(q^{10})\)	\(q+2.14510 q^{3} +3.14510 q^{5} +1.74657 q^{7} +1.60147 q^{9} +2.89167 q^{11} +2.39853 q^{13} +6.74657 q^{15} -5.49314 q^{17} -2.00000 q^{19} +3.74657 q^{21} -1.60147 q^{23} +4.89167 q^{25} -3.00000 q^{27} +5.89167 q^{29} +3.94217 q^{31} +6.20293 q^{33} +5.49314 q^{35} +1.00000 q^{37} +5.14510 q^{39} +6.14510 q^{41} -5.20293 q^{43} +5.03677 q^{45} +0.253432 q^{47} -3.94950 q^{49} -11.7833 q^{51} -0.543637 q^{53} +9.09460 q^{55} -4.29021 q^{57} -10.6961 q^{59} -6.63824 q^{61} +2.79707 q^{63} +7.54364 q^{65} -7.14510 q^{67} -3.43531 q^{69} -4.03677 q^{71} +3.18188 q^{73} +10.4931 q^{75} +5.05050 q^{77} +9.89167 q^{79} -11.2397 q^{81} -6.03677 q^{83} -17.2765 q^{85} +12.6382 q^{87} +4.50686 q^{89} +4.18920 q^{91} +8.45636 q^{93} -6.29021 q^{95} +12.9863 q^{97} +4.63091 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^5 - 3 * q^7 + 3 * q^9	\( 3 q + 3 q^{5} - 3 q^{7} + 3 q^{9} - 6 q^{11} + 9 q^{13} + 12 q^{15} - 6 q^{19} + 3 q^{21} - 3 q^{23} - 9 q^{27} + 3 q^{29} + 9 q^{31} + 15 q^{33} + 3 q^{37} + 9 q^{39} + 12 q^{41} - 12 q^{43} - 6 q^{45} + 9 q^{47} - 6 q^{51} + 3 q^{53} + 9 q^{55} - 12 q^{59} + 3 q^{61} + 12 q^{63} + 18 q^{65} - 15 q^{67} + 9 q^{69} + 9 q^{71} - 18 q^{73} + 15 q^{75} + 27 q^{77} + 15 q^{79} - 9 q^{81} + 3 q^{83} - 6 q^{85} + 15 q^{87} + 30 q^{89} - 24 q^{91} + 30 q^{93} - 6 q^{95} + 6 q^{97}+O(q^{100}) \) 3 * q + 3 * q^5 - 3 * q^7 + 3 * q^9 - 6 * q^11 + 9 * q^13 + 12 * q^15 - 6 * q^19 + 3 * q^21 - 3 * q^23 - 9 * q^27 + 3 * q^29 + 9 * q^31 + 15 * q^33 + 3 * q^37 + 9 * q^39 + 12 * q^41 - 12 * q^43 - 6 * q^45 + 9 * q^47 - 6 * q^51 + 3 * q^53 + 9 * q^55 - 12 * q^59 + 3 * q^61 + 12 * q^63 + 18 * q^65 - 15 * q^67 + 9 * q^69 + 9 * q^71 - 18 * q^73 + 15 * q^75 + 27 * q^77 + 15 * q^79 - 9 * q^81 + 3 * q^83 - 6 * q^85 + 15 * q^87 + 30 * q^89 - 24 * q^91 + 30 * q^93 - 6 * q^95 + 6 * q^97

Introduction

L-functions

Modular forms

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Database

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