LMFDB - Embedded newform 207.4.a.b.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [207,4,Mod(1,207)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(207, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("207.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 207 = 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	207.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:	$12.2133953712$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 69)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	207.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+3.82843 q^{2} +6.65685 q^{4} -15.3137 q^{5} -16.8284 q^{7} -5.14214 q^{8} -58.6274 q^{10} +55.7990 q^{11} -46.9706 q^{13} -64.4264 q^{14} -72.9411 q^{16} -62.1421 q^{17} -141.882 q^{19} -101.941 q^{20} +213.622 q^{22} -23.0000 q^{23} +109.510 q^{25} -179.823 q^{26} -112.024 q^{28} +288.676 q^{29} +68.7351 q^{31} -238.113 q^{32} -237.907 q^{34} +257.706 q^{35} +179.622 q^{37} -543.186 q^{38} +78.7452 q^{40} +71.5391 q^{41} -159.088 q^{43} +371.446 q^{44} -88.0538 q^{46} +272.902 q^{47} -59.8040 q^{49} +419.250 q^{50} -312.676 q^{52} +12.2843 q^{53} -854.489 q^{55} +86.5341 q^{56} +1105.18 q^{58} -426.891 q^{59} -243.407 q^{61} +263.147 q^{62} -328.068 q^{64} +719.294 q^{65} -81.0496 q^{67} -413.671 q^{68} +986.607 q^{70} -696.930 q^{71} +568.392 q^{73} +687.671 q^{74} -944.489 q^{76} -939.009 q^{77} +719.123 q^{79} +1117.00 q^{80} +273.882 q^{82} -1336.80 q^{83} +951.627 q^{85} -609.058 q^{86} -286.926 q^{88} +337.935 q^{89} +790.441 q^{91} -153.108 q^{92} +1044.78 q^{94} +2172.74 q^{95} -1419.76 q^{97} -228.955 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 8 q^{5} - 28 q^{7} + 18 q^{8} - 72 q^{10} + 72 q^{11} - 60 q^{13} - 44 q^{14} - 78 q^{16} - 96 q^{17} - 148 q^{19} - 136 q^{20} + 184 q^{22} - 46 q^{23} + 38 q^{25} - 156 q^{26}+ \cdots + 170 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - 8 * q^5 - 28 * q^7 + 18 * q^8 - 72 * q^10 + 72 * q^11 - 60 * q^13 - 44 * q^14 - 78 * q^16 - 96 * q^17 - 148 * q^19 - 136 * q^20 + 184 * q^22 - 46 * q^23 + 38 * q^25 - 156 * q^26 - 60 * q^28 + 204 * q^29 - 168 * q^31 - 414 * q^32 - 176 * q^34 + 176 * q^35 + 116 * q^37 - 532 * q^38 + 248 * q^40 - 4 * q^41 - 420 * q^43 + 296 * q^44 - 46 * q^46 + 48 * q^47 - 278 * q^49 + 550 * q^50 - 252 * q^52 - 32 * q^53 - 736 * q^55 - 172 * q^56 + 1260 * q^58 + 40 * q^59 - 764 * q^61 + 696 * q^62 + 34 * q^64 + 624 * q^65 - 988 * q^67 - 256 * q^68 + 1136 * q^70 + 224 * q^71 + 820 * q^73 + 804 * q^74 - 916 * q^76 - 1120 * q^77 + 1772 * q^79 + 1080 * q^80 + 412 * q^82 - 1480 * q^83 + 704 * q^85 - 132 * q^86 + 88 * q^88 - 744 * q^89 + 936 * q^91 - 46 * q^92 + 1456 * q^94 + 2128 * q^95 - 260 * q^97 + 170 * q^98

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 $

$2$	3.82843	1.35355	0.676777	−	0.736188i	$-0.263376\pi$
$2$	3.82843	1.35355	0.676777	+	0.736188i	$0.263376\pi$
$3$	0	0
$3$	0	0
$4$	6.65685	0.832107
$5$	−15.3137	−1.36970	−0.684850	−	0.728684i	$-0.740132\pi$
$5$	−15.3137	−1.36970	−0.684850	+	0.728684i	$0.740132\pi$
$6$	0	0
$7$	−16.8284	−0.908650	−0.454325	−	0.890836i	$-0.650119\pi$
$7$	−16.8284	−0.908650	−0.454325	+	0.890836i	$0.650119\pi$
$8$	−5.14214	−0.227252
$9$	0	0
$10$	−58.6274	−1.85396
$11$	55.7990	1.52946	0.764729	−	0.644353i	$-0.222873\pi$
$11$	55.7990	1.52946	0.764729	+	0.644353i	$0.222873\pi$
$12$	0	0
$13$	−46.9706	−1.00210	−0.501050	−	0.865419i	$-0.667053\pi$
$13$	−46.9706	−1.00210	−0.501050	+	0.865419i	$0.667053\pi$
$14$	−64.4264	−1.22991
$15$	0	0
$16$	−72.9411	−1.13971
$17$	−62.1421	−0.886570	−0.443285	−	0.896381i	$-0.646187\pi$
$17$	−62.1421	−0.886570	−0.443285	+	0.896381i	$0.646187\pi$
$18$	0	0
$19$	−141.882	−1.71316	−0.856579	−	0.516015i	$-0.827415\pi$
$19$	−141.882	−1.71316	−0.856579	+	0.516015i	$0.827415\pi$
$20$	−101.941	−1.13974
$21$	0	0
$22$	213.622	2.07020
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	109.510	0.876077
$26$	−179.823	−1.35639
$27$	0	0
$28$	−112.024	−0.756094
$29$	288.676	1.84848	0.924238	−	0.381816i	$-0.124701\pi$
$29$	288.676	1.84848	0.924238	+	0.381816i	$0.124701\pi$
$30$	0	0
$31$	68.7351	0.398232	0.199116	−	0.979976i	$-0.436193\pi$
$31$	68.7351	0.398232	0.199116	+	0.979976i	$0.436193\pi$
$32$	−238.113	−1.31540
$33$	0	0
$34$	−237.907	−1.20002
$35$	257.706	1.24458
$36$	0	0
$37$	179.622	0.798101	0.399050	−	0.916929i	$-0.369340\pi$
$37$	179.622	0.798101	0.399050	+	0.916929i	$0.369340\pi$
$38$	−543.186	−2.31885
$39$	0	0
$40$	78.7452	0.311268
$41$	71.5391	0.272501	0.136250	−	0.990674i	$-0.456495\pi$
$41$	71.5391	0.272501	0.136250	+	0.990674i	$0.456495\pi$
$42$	0	0
$43$	−159.088	−0.564203	−0.282102	−	0.959385i	$-0.591032\pi$
$43$	−159.088	−0.564203	−0.282102	+	0.959385i	$0.591032\pi$
$44$	371.446	1.27267
$45$	0	0
$46$	−88.0538	−0.282235
$47$	272.902	0.846953	0.423476	−	0.905907i	$-0.360810\pi$
$47$	272.902	0.846953	0.423476	+	0.905907i	$0.360810\pi$
$48$	0	0
$49$	−59.8040	−0.174356
$50$	419.250	1.18582
$51$	0	0
$52$	−312.676	−0.833854
$53$	12.2843	0.0318373	0.0159186	−	0.999873i	$-0.494933\pi$
$53$	12.2843	0.0318373	0.0159186	+	0.999873i	$0.494933\pi$
$54$	0	0
$55$	−854.489	−2.09490
$56$	86.5341	0.206493
$57$	0	0
$58$	1105.18	2.50201
$59$	−426.891	−0.941975	−0.470988	−	0.882140i	$-0.656102\pi$
$59$	−426.891	−0.941975	−0.470988	+	0.882140i	$0.656102\pi$
$60$	0	0
$61$	−243.407	−0.510903	−0.255451	−	0.966822i	$-0.582224\pi$
$61$	−243.407	−0.510903	−0.255451	+	0.966822i	$0.582224\pi$
$62$	263.147	0.539028
$63$	0	0
$64$	−328.068	−0.640758
$65$	719.294	1.37258
$66$	0	0
$67$	−81.0496	−0.147788	−0.0738940	−	0.997266i	$-0.523543\pi$
$67$	−81.0496	−0.147788	−0.0738940	+	0.997266i	$0.523543\pi$
$68$	−413.671	−0.737721
$69$	0	0
$70$	986.607	1.68460
$71$	−696.930	−1.16494	−0.582468	−	0.812854i	$-0.697913\pi$
$71$	−696.930	−1.16494	−0.582468	+	0.812854i	$0.697913\pi$
$72$	0	0
$73$	568.392	0.911305	0.455652	−	0.890158i	$-0.349406\pi$
$73$	568.392	0.911305	0.455652	+	0.890158i	$0.349406\pi$
$74$	687.671	1.08027
$75$	0	0
$76$	−944.489	−1.42553
$77$	−939.009	−1.38974
$78$	0	0
$79$	719.123	1.02415	0.512074	−	0.858942i	$-0.328877\pi$
$79$	719.123	1.02415	0.512074	+	0.858942i	$0.328877\pi$
$80$	1117.00	1.56105
$81$	0	0
$82$	273.882	0.368844
$83$	−1336.80	−1.76786	−0.883932	−	0.467616i	$-0.845113\pi$
$83$	−1336.80	−1.76786	−0.883932	+	0.467616i	$0.845113\pi$
$84$	0	0
$85$	951.627	1.21433
$86$	−609.058	−0.763679
$87$	0	0
$88$	−286.926	−0.347573
$89$	337.935	0.402484	0.201242	−	0.979542i	$-0.435502\pi$
$89$	337.935	0.402484	0.201242	+	0.979542i	$0.435502\pi$
$90$	0	0
$91$	790.441	0.910557
$92$	−153.108	−0.173506
$93$	0	0
$94$	1044.78	1.14640
$95$	2172.74	2.34651
$96$	0	0
$97$	−1419.76	−1.48613	−0.743067	−	0.669217i	$-0.766630\pi$
$97$	−1419.76	−1.48613	−0.743067	+	0.669217i	$0.766630\pi$
$98$	−228.955	−0.236000
$99$	0	0
$100$	728.990	0.728990
$101$	−471.275	−0.464293	−0.232147	−	0.972681i	$-0.574575\pi$
$101$	−471.275	−0.464293	−0.232147	+	0.972681i	$0.574575\pi$
$102$	0	0
$103$	−1129.00	−1.08004	−0.540019	−	0.841653i	$-0.681583\pi$
$103$	−1129.00	−1.08004	−0.540019	+	0.841653i	$0.681583\pi$
$104$	241.529	0.227729
$105$	0	0
$106$	47.0294	0.0430934
$107$	−203.543	−0.183900	−0.0919499	−	0.995764i	$-0.529310\pi$
$107$	−203.543	−0.183900	−0.0919499	+	0.995764i	$0.529310\pi$
$108$	0	0
$109$	−1938.34	−1.70329	−0.851646	−	0.524117i	$-0.824395\pi$
$109$	−1938.34	−1.70329	−0.851646	+	0.524117i	$0.824395\pi$
$110$	−3271.35	−2.83555
$111$	0	0
$112$	1227.48	1.03559
$113$	−1663.79	−1.38510	−0.692548	−	0.721371i	$-0.743512\pi$
$113$	−1663.79	−1.38510	−0.692548	+	0.721371i	$0.743512\pi$
$114$	0	0
$115$	352.215	0.285602
$116$	1921.68	1.53813
$117$	0	0
$118$	−1634.32	−1.27501
$119$	1045.75	0.805581
$120$	0	0
$121$	1782.53	1.33924
$122$	−931.866	−0.691534
$123$	0	0
$124$	457.559	0.331371
$125$	237.214	0.169737
$126$	0	0
$127$	−734.676	−0.513323	−0.256661	−	0.966501i	$-0.582622\pi$
$127$	−734.676	−0.513323	−0.256661	+	0.966501i	$0.582622\pi$
$128$	648.917	0.448099
$129$	0	0
$130$	2753.76	1.85785
$131$	2855.44	1.90443	0.952217	−	0.305421i	$-0.0987973\pi$
$131$	2855.44	1.90443	0.952217	+	0.305421i	$0.0987973\pi$
$132$	0	0
$133$	2387.66	1.55666
$134$	−310.293	−0.200039
$135$	0	0
$136$	319.543	0.201475
$137$	−1430.56	−0.892127	−0.446063	−	0.895001i	$-0.647174\pi$
$137$	−1430.56	−0.892127	−0.446063	+	0.895001i	$0.647174\pi$
$138$	0	0
$139$	288.410	0.175990	0.0879951	−	0.996121i	$-0.471954\pi$
$139$	288.410	0.175990	0.0879951	+	0.996121i	$0.471954\pi$
$140$	1715.51	1.03562
$141$	0	0
$142$	−2668.15	−1.57680
$143$	−2620.91	−1.53267
$144$	0	0
$145$	−4420.70	−2.53186
$146$	2176.05	1.23350
$147$	0	0
$148$	1195.72	0.664105
$149$	1937.08	1.06504	0.532522	−	0.846416i	$-0.321244\pi$
$149$	1937.08	1.06504	0.532522	+	0.846416i	$0.321244\pi$
$150$	0	0
$151$	1847.98	0.995935	0.497967	−	0.867196i	$-0.334080\pi$
$151$	1847.98	0.995935	0.497967	+	0.867196i	$0.334080\pi$
$152$	729.578	0.389320
$153$	0	0
$154$	−3594.93	−1.88109
$155$	−1052.59	−0.545458
$156$	0	0
$157$	51.2489	0.0260516	0.0130258	−	0.999915i	$-0.495854\pi$
$157$	51.2489	0.0260516	0.0130258	+	0.999915i	$0.495854\pi$
$158$	2753.11	1.38624
$159$	0	0
$160$	3646.39	1.80170
$161$	387.054	0.189467
$162$	0	0
$163$	−607.960	−0.292142	−0.146071	−	0.989274i	$-0.546663\pi$
$163$	−607.960	−0.292142	−0.146071	+	0.989274i	$0.546663\pi$
$164$	476.225	0.226750
$165$	0	0
$166$	−5117.83	−2.39290
$167$	3253.06	1.50736	0.753682	−	0.657240i	$-0.228276\pi$
$167$	3253.06	1.50736	0.753682	+	0.657240i	$0.228276\pi$
$168$	0	0
$169$	9.23376	0.00420290
$170$	3643.23	1.64367
$171$	0	0
$172$	−1059.03	−0.469477
$173$	−1975.75	−0.868288	−0.434144	−	0.900843i	$-0.642949\pi$
$173$	−1975.75	−0.868288	−0.434144	+	0.900843i	$0.642949\pi$
$174$	0	0
$175$	−1842.88	−0.796047
$176$	−4070.04	−1.74313
$177$	0	0
$178$	1293.76	0.544783
$179$	−4261.58	−1.77947	−0.889737	−	0.456474i	$-0.849112\pi$
$179$	−4261.58	−1.77947	−0.889737	+	0.456474i	$0.849112\pi$
$180$	0	0
$181$	2043.29	0.839095	0.419548	−	0.907733i	$-0.362189\pi$
$181$	2043.29	0.839095	0.419548	+	0.907733i	$0.362189\pi$
$182$	3026.14	1.23249
$183$	0	0
$184$	118.269	0.0473854
$185$	−2750.68	−1.09316
$186$	0	0
$187$	−3467.47	−1.35597
$188$	1816.67	0.704755
$189$	0	0
$190$	8318.19	3.17613
$191$	73.1001	0.0276929	0.0138464	−	0.999904i	$-0.495592\pi$
$191$	73.1001	0.0276929	0.0138464	+	0.999904i	$0.495592\pi$
$192$	0	0
$193$	−227.901	−0.0849982	−0.0424991	−	0.999097i	$-0.513532\pi$
$193$	−227.901	−0.0849982	−0.0424991	+	0.999097i	$0.513532\pi$
$194$	−5435.46	−2.01156
$195$	0	0
$196$	−398.107	−0.145083
$197$	−577.781	−0.208960	−0.104480	−	0.994527i	$-0.533318\pi$
$197$	−577.781	−0.208960	−0.104480	+	0.994527i	$0.533318\pi$
$198$	0	0
$199$	−4569.01	−1.62758	−0.813790	−	0.581159i	$-0.802599\pi$
$199$	−4569.01	−1.62758	−0.813790	+	0.581159i	$0.802599\pi$
$200$	−563.114	−0.199091
$201$	0	0
$202$	−1804.24	−0.628446
$203$	−4857.97	−1.67962
$204$	0	0
$205$	−1095.53	−0.373244
$206$	−4322.31	−1.46189
$207$	0	0
$208$	3426.09	1.14210
$209$	−7916.89	−2.62020
$210$	0	0
$211$	−1816.71	−0.592736	−0.296368	−	0.955074i	$-0.595775\pi$
$211$	−1816.71	−0.592736	−0.296368	+	0.955074i	$0.595775\pi$
$212$	81.7746	0.0264920
$213$	0	0
$214$	−779.251	−0.248918
$215$	2436.23	0.772789
$216$	0	0
$217$	−1156.70	−0.361853
$218$	−7420.78	−2.30550
$219$	0	0
$220$	−5688.21	−1.74318
$221$	2918.85	0.888431
$222$	0	0
$223$	3494.47	1.04936	0.524679	−	0.851300i	$-0.324185\pi$
$223$	3494.47	1.04936	0.524679	+	0.851300i	$0.324185\pi$
$224$	4007.06	1.19524
$225$	0	0
$226$	−6369.69	−1.87480
$227$	−1044.51	−0.305402	−0.152701	−	0.988272i	$-0.548797\pi$
$227$	−1044.51	−0.305402	−0.152701	+	0.988272i	$0.548797\pi$
$228$	0	0
$229$	2141.54	0.617979	0.308989	−	0.951065i	$-0.400009\pi$
$229$	2141.54	0.617979	0.308989	+	0.951065i	$0.400009\pi$
$230$	1348.43	0.386578
$231$	0	0
$232$	−1484.41	−0.420071
$233$	−716.802	−0.201542	−0.100771	−	0.994910i	$-0.532131\pi$
$233$	−716.802	−0.201542	−0.100771	+	0.994910i	$0.532131\pi$
$234$	0	0
$235$	−4179.14	−1.16007
$236$	−2841.75	−0.783824
$237$	0	0
$238$	4003.59	1.09040
$239$	3515.96	0.951585	0.475792	−	0.879558i	$-0.342161\pi$
$239$	3515.96	0.951585	0.475792	+	0.879558i	$0.342161\pi$
$240$	0	0
$241$	5684.64	1.51942	0.759710	−	0.650262i	$-0.225341\pi$
$241$	5684.64	1.51942	0.759710	+	0.650262i	$0.225341\pi$
$242$	6824.28	1.81273
$243$	0	0
$244$	−1620.33	−0.425126
$245$	915.822	0.238815
$246$	0	0
$247$	6664.29	1.71676
$248$	−353.445	−0.0904991
$249$	0	0
$250$	908.158	0.229748
$251$	6946.90	1.74695	0.873475	−	0.486870i	$-0.161861\pi$
$251$	6946.90	1.74695	0.873475	+	0.486870i	$0.161861\pi$
$252$	0	0
$253$	−1283.38	−0.318914
$254$	−2812.65	−0.694810
$255$	0	0
$256$	5108.88	1.24728
$257$	1776.30	0.431139	0.215569	−	0.976489i	$-0.430839\pi$
$257$	1776.30	0.431139	0.215569	+	0.976489i	$0.430839\pi$
$258$	0	0
$259$	−3022.76	−0.725194
$260$	4788.23	1.14213
$261$	0	0
$262$	10931.8	2.57775
$263$	−112.405	−0.0263544	−0.0131772	−	0.999913i	$-0.504195\pi$
$263$	−112.405	−0.0263544	−0.0131772	+	0.999913i	$0.504195\pi$
$264$	0	0
$265$	−188.118	−0.0436075
$266$	9140.96	2.10702
$267$	0	0
$268$	−539.536	−0.122975
$269$	5916.43	1.34101	0.670504	−	0.741906i	$-0.266078\pi$
$269$	5916.43	1.34101	0.670504	+	0.741906i	$0.266078\pi$
$270$	0	0
$271$	−1011.96	−0.226836	−0.113418	−	0.993547i	$-0.536180\pi$
$271$	−1011.96	−0.226836	−0.113418	+	0.993547i	$0.536180\pi$
$272$	4532.72	1.01043
$273$	0	0
$274$	−5476.81	−1.20754
$275$	6110.53	1.33992
$276$	0	0
$277$	−3632.42	−0.787910	−0.393955	−	0.919130i	$-0.628893\pi$
$277$	−3632.42	−0.787910	−0.393955	+	0.919130i	$0.628893\pi$
$278$	1104.16	0.238212
$279$	0	0
$280$	−1325.16	−0.282833
$281$	−4382.47	−0.930378	−0.465189	−	0.885211i	$-0.654014\pi$
$281$	−4382.47	−0.930378	−0.465189	+	0.885211i	$0.654014\pi$
$282$	0	0
$283$	−2692.29	−0.565513	−0.282757	−	0.959192i	$-0.591249\pi$
$283$	−2692.29	−0.565513	−0.282757	+	0.959192i	$0.591249\pi$
$284$	−4639.36	−0.969350
$285$	0	0
$286$	−10034.0	−2.07455
$287$	−1203.89	−0.247608
$288$	0	0
$289$	−1051.35	−0.213995
$290$	−16924.3	−3.42700
$291$	0	0
$292$	3783.70	0.758303
$293$	−4023.60	−0.802256	−0.401128	−	0.916022i	$-0.631382\pi$
$293$	−4023.60	−0.802256	−0.401128	+	0.916022i	$0.631382\pi$
$294$	0	0
$295$	6537.29	1.29022
$296$	−923.643	−0.181370
$297$	0	0
$298$	7415.96	1.44159
$299$	1080.32	0.208952
$300$	0	0
$301$	2677.21	0.512663
$302$	7074.84	1.34805
$303$	0	0
$304$	10349.1	1.95250
$305$	3727.46	0.699784
$306$	0	0
$307$	5195.21	0.965818	0.482909	−	0.875670i	$-0.339580\pi$
$307$	5195.21	0.965818	0.482909	+	0.875670i	$0.339580\pi$
$308$	−6250.85	−1.15641
$309$	0	0
$310$	−4029.76	−0.738306
$311$	−5440.59	−0.991986	−0.495993	−	0.868327i	$-0.665196\pi$
$311$	−5440.59	−0.991986	−0.495993	+	0.868327i	$0.665196\pi$
$312$	0	0
$313$	−7390.51	−1.33462	−0.667310	−	0.744780i	$-0.732555\pi$
$313$	−7390.51	−1.33462	−0.667310	+	0.744780i	$0.732555\pi$
$314$	196.203	0.0352623
$315$	0	0
$316$	4787.10	0.852200
$317$	8785.02	1.55652	0.778258	−	0.627944i	$-0.216103\pi$
$317$	8785.02	1.55652	0.778258	+	0.627944i	$0.216103\pi$
$318$	0	0
$319$	16107.8	2.82717
$320$	5023.94	0.877646
$321$	0	0
$322$	1481.81	0.256453
$323$	8816.87	1.51883
$324$	0	0
$325$	−5143.73	−0.877916
$326$	−2327.53	−0.395429
$327$	0	0
$328$	−367.864	−0.0619265
$329$	−4592.50	−0.769583
$330$	0	0
$331$	−7998.03	−1.32813	−0.664066	−	0.747674i	$-0.731170\pi$
$331$	−7998.03	−1.32813	−0.664066	+	0.747674i	$0.731170\pi$
$332$	−8898.87	−1.47105
$333$	0	0
$334$	12454.1	2.04030
$335$	1241.17	0.202425
$336$	0	0
$337$	−5572.96	−0.900827	−0.450413	−	0.892820i	$-0.648723\pi$
$337$	−5572.96	−0.900827	−0.450413	+	0.892820i	$0.648723\pi$
$338$	35.3508	0.00568885
$339$	0	0
$340$	6334.84	1.01046
$341$	3835.35	0.609078
$342$	0	0
$343$	6778.56	1.06708
$344$	818.054	0.128217
$345$	0	0
$346$	−7564.03	−1.17527
$347$	8070.32	1.24852	0.624262	−	0.781215i	$-0.285400\pi$
$347$	8070.32	1.24852	0.624262	+	0.781215i	$0.285400\pi$
$348$	0	0
$349$	−10343.3	−1.58644	−0.793218	−	0.608937i	$-0.791596\pi$
$349$	−10343.3	−1.58644	−0.793218	+	0.608937i	$0.791596\pi$
$350$	−7055.31	−1.07749
$351$	0	0
$352$	−13286.4	−2.01185
$353$	2326.06	0.350718	0.175359	−	0.984505i	$-0.443891\pi$
$353$	2326.06	0.350718	0.175359	+	0.984505i	$0.443891\pi$
$354$	0	0
$355$	10672.6	1.59561
$356$	2249.59	0.334910
$357$	0	0
$358$	−16315.2	−2.40861
$359$	3333.39	0.490054	0.245027	−	0.969516i	$-0.421203\pi$
$359$	3333.39	0.490054	0.245027	+	0.969516i	$0.421203\pi$
$360$	0	0
$361$	13271.6	1.93491
$362$	7822.57	1.13576
$363$	0	0
$364$	5261.85	0.757681
$365$	−8704.19	−1.24821
$366$	0	0
$367$	6803.73	0.967716	0.483858	−	0.875147i	$-0.339235\pi$
$367$	6803.73	0.967716	0.483858	+	0.875147i	$0.339235\pi$
$368$	1677.65	0.237645
$369$	0	0
$370$	−10530.8	−1.47965
$371$	−206.725	−0.0289289
$372$	0	0
$373$	628.598	0.0872589	0.0436294	−	0.999048i	$-0.486108\pi$
$373$	628.598	0.0872589	0.0436294	+	0.999048i	$0.486108\pi$
$374$	−13275.0	−1.83538
$375$	0	0
$376$	−1403.30	−0.192472
$377$	−13559.3	−1.85236
$378$	0	0
$379$	3289.15	0.445785	0.222892	−	0.974843i	$-0.428450\pi$
$379$	3289.15	0.445785	0.222892	+	0.974843i	$0.428450\pi$
$380$	14463.6	1.95255
$381$	0	0
$382$	279.859	0.0374838
$383$	1889.81	0.252128	0.126064	−	0.992022i	$-0.459766\pi$
$383$	1889.81	0.252128	0.126064	+	0.992022i	$0.459766\pi$
$384$	0	0
$385$	14379.7	1.90353
$386$	−872.501	−0.115050
$387$	0	0
$388$	−9451.15	−1.23662
$389$	569.046	0.0741692	0.0370846	−	0.999312i	$-0.488193\pi$
$389$	569.046	0.0741692	0.0370846	+	0.999312i	$0.488193\pi$
$390$	0	0
$391$	1429.27	0.184863
$392$	307.520	0.0396228
$393$	0	0
$394$	−2211.99	−0.282839
$395$	−11012.4	−1.40277
$396$	0	0
$397$	4092.11	0.517323	0.258661	−	0.965968i	$-0.416719\pi$
$397$	4092.11	0.517323	0.258661	+	0.965968i	$0.416719\pi$
$398$	−17492.1	−2.20302
$399$	0	0
$400$	−7987.76	−0.998470
$401$	−3867.00	−0.481568	−0.240784	−	0.970579i	$-0.577405\pi$
$401$	−3867.00	−0.481568	−0.240784	+	0.970579i	$0.577405\pi$
$402$	0	0
$403$	−3228.52	−0.399068
$404$	−3137.21	−0.386342
$405$	0	0
$406$	−18598.4	−2.27345
$407$	10022.7	1.22066
$408$	0	0
$409$	−15631.6	−1.88981	−0.944906	−	0.327343i	$-0.893847\pi$
$409$	−15631.6	−1.88981	−0.944906	+	0.327343i	$0.893847\pi$
$410$	−4194.15	−0.505206
$411$	0	0
$412$	−7515.61	−0.898708
$413$	7183.91	0.855925
$414$	0	0
$415$	20471.3	2.42144
$416$	11184.3	1.31816
$417$	0	0
$418$	−30309.2	−3.54658
$419$	3562.17	0.415330	0.207665	−	0.978200i	$-0.433414\pi$
$419$	3562.17	0.415330	0.207665	+	0.978200i	$0.433414\pi$
$420$	0	0
$421$	10576.0	1.22433	0.612166	−	0.790729i	$-0.290298\pi$
$421$	10576.0	1.22433	0.612166	+	0.790729i	$0.290298\pi$
$422$	−6955.13	−0.802299
$423$	0	0
$424$	−63.1674	−0.00723509
$425$	−6805.16	−0.776703
$426$	0	0
$427$	4096.16	0.464232
$428$	−1354.96	−0.153024
$429$	0	0
$430$	9326.94	1.04601
$431$	−6965.77	−0.778490	−0.389245	−	0.921134i	$-0.627264\pi$
$431$	−6965.77	−0.778490	−0.389245	+	0.921134i	$0.627264\pi$
$432$	0	0
$433$	−16073.5	−1.78393	−0.891967	−	0.452100i	$-0.850675\pi$
$433$	−16073.5	−1.78393	−0.891967	+	0.452100i	$0.850675\pi$
$434$	−4428.35	−0.489787
$435$	0	0
$436$	−12903.2	−1.41732
$437$	3263.29	0.357218
$438$	0	0
$439$	−613.988	−0.0667518	−0.0333759	−	0.999443i	$-0.510626\pi$
$439$	−613.988	−0.0667518	−0.0333759	+	0.999443i	$0.510626\pi$
$440$	4393.90	0.476070
$441$	0	0
$442$	11174.6	1.20254
$443$	263.475	0.0282575	0.0141288	−	0.999900i	$-0.495503\pi$
$443$	263.475	0.0282575	0.0141288	+	0.999900i	$0.495503\pi$
$444$	0	0
$445$	−5175.04	−0.551282
$446$	13378.3	1.42036
$447$	0	0
$448$	5520.87	0.582225
$449$	−1644.47	−0.172845	−0.0864226	−	0.996259i	$-0.527544\pi$
$449$	−1644.47	−0.172845	−0.0864226	+	0.996259i	$0.527544\pi$
$450$	0	0
$451$	3991.81	0.416778
$452$	−11075.6	−1.15255
$453$	0	0
$454$	−3998.81	−0.413378
$455$	−12104.6	−1.24719
$456$	0	0
$457$	−16190.9	−1.65728	−0.828641	−	0.559781i	$-0.810885\pi$
$457$	−16190.9	−1.65728	−0.828641	+	0.559781i	$0.810885\pi$
$458$	8198.74	0.836467
$459$	0	0
$460$	2344.65	0.237651
$461$	−5783.08	−0.584262	−0.292131	−	0.956378i	$-0.594364\pi$
$461$	−5783.08	−0.584262	−0.292131	+	0.956378i	$0.594364\pi$
$462$	0	0
$463$	−3753.34	−0.376744	−0.188372	−	0.982098i	$-0.560321\pi$
$463$	−3753.34	−0.376744	−0.188372	+	0.982098i	$0.560321\pi$
$464$	−21056.4	−2.10672
$465$	0	0
$466$	−2744.23	−0.272798
$467$	8441.21	0.836430	0.418215	−	0.908348i	$-0.362656\pi$
$467$	8441.21	0.836430	0.418215	+	0.908348i	$0.362656\pi$
$468$	0	0
$469$	1363.94	0.134287
$470$	−15999.5	−1.57022
$471$	0	0
$472$	2195.13	0.214066
$473$	−8876.97	−0.862925
$474$	0	0
$475$	−15537.5	−1.50086
$476$	6961.43	0.670329
$477$	0	0
$478$	13460.6	1.28802
$479$	1001.54	0.0955359	0.0477680	−	0.998858i	$-0.484789\pi$
$479$	1001.54	0.0955359	0.0477680	+	0.998858i	$0.484789\pi$
$480$	0	0
$481$	−8436.96	−0.799776
$482$	21763.2	2.05662
$483$	0	0
$484$	11866.0	1.11439
$485$	21741.8	2.03556
$486$	0	0
$487$	13740.3	1.27851	0.639253	−	0.768997i	$-0.279244\pi$
$487$	13740.3	1.27851	0.639253	+	0.768997i	$0.279244\pi$
$488$	1251.63	0.116104
$489$	0	0
$490$	3506.16	0.323249
$491$	10708.1	0.984217	0.492109	−	0.870534i	$-0.336226\pi$
$491$	10708.1	0.984217	0.492109	+	0.870534i	$0.336226\pi$
$492$	0	0
$493$	−17939.0	−1.63880
$494$	25513.7	2.32372
$495$	0	0
$496$	−5013.61	−0.453867
$497$	11728.2	1.05852
$498$	0	0
$499$	9630.22	0.863944	0.431972	−	0.901887i	$-0.357818\pi$
$499$	9630.22	0.863944	0.431972	+	0.901887i	$0.357818\pi$
$500$	1579.10	0.141239
$501$	0	0
$502$	26595.7	2.36459
$503$	−18170.1	−1.61067	−0.805334	−	0.592822i	$-0.798014\pi$
$503$	−18170.1	−1.61067	−0.805334	+	0.592822i	$0.798014\pi$
$504$	0	0
$505$	7216.97	0.635942
$506$	−4913.31	−0.431667
$507$	0	0
$508$	−4890.63	−0.427139
$509$	−12525.5	−1.09073	−0.545365	−	0.838199i	$-0.683609\pi$
$509$	−12525.5	−1.09073	−0.545365	+	0.838199i	$0.683609\pi$
$510$	0	0
$511$	−9565.14	−0.828057
$512$	14367.6	1.24017
$513$	0	0
$514$	6800.45	0.583570
$515$	17289.2	1.47933
$516$	0	0
$517$	15227.6	1.29538
$518$	−11572.4	−0.981589
$519$	0	0
$520$	−3698.70	−0.311921
$521$	−3631.35	−0.305360	−0.152680	−	0.988276i	$-0.548790\pi$
$521$	−3631.35	−0.305360	−0.152680	+	0.988276i	$0.548790\pi$
$522$	0	0
$523$	12118.1	1.01317	0.506584	−	0.862191i	$-0.330908\pi$
$523$	12118.1	1.01317	0.506584	+	0.862191i	$0.330908\pi$
$524$	19008.2	1.58469
$525$	0	0
$526$	−430.336	−0.0356721
$527$	−4271.34	−0.353060
$528$	0	0
$529$	529.000	0.0434783
$530$	−720.195	−0.0590250
$531$	0	0
$532$	15894.3	1.29531
$533$	−3360.23	−0.273073
$534$	0	0
$535$	3117.00	0.251887
$536$	416.768	0.0335852
$537$	0	0
$538$	22650.6	1.81513
$539$	−3337.01	−0.266670
$540$	0	0
$541$	−24466.6	−1.94437	−0.972184	−	0.234220i	$-0.924746\pi$
$541$	−24466.6	−1.94437	−0.972184	+	0.234220i	$0.924746\pi$
$542$	−3874.23	−0.307034
$543$	0	0
$544$	14796.8	1.16619
$545$	29683.1	2.33300
$546$	0	0
$547$	−9781.99	−0.764621	−0.382311	−	0.924034i	$-0.624872\pi$
$547$	−9781.99	−0.764621	−0.382311	+	0.924034i	$0.624872\pi$
$548$	−9523.06	−0.742345
$549$	0	0
$550$	23393.7	1.81366
$551$	−40958.0	−3.16673
$552$	0	0
$553$	−12101.7	−0.930591
$554$	−13906.5	−1.06648
$555$	0	0
$556$	1919.91	0.146443
$557$	2198.74	0.167260	0.0836300	−	0.996497i	$-0.473349\pi$
$557$	2198.74	0.167260	0.0836300	+	0.996497i	$0.473349\pi$
$558$	0	0
$559$	7472.47	0.565388
$560$	−18797.3	−1.41845
$561$	0	0
$562$	−16778.0	−1.25932
$563$	−7927.38	−0.593427	−0.296714	−	0.954967i	$-0.595891\pi$
$563$	−7927.38	−0.593427	−0.296714	+	0.954967i	$0.595891\pi$
$564$	0	0
$565$	25478.8	1.89717
$566$	−10307.3	−0.765453
$567$	0	0
$568$	3583.71	0.264734
$569$	18860.7	1.38960	0.694799	−	0.719204i	$-0.255493\pi$
$569$	18860.7	1.38960	0.694799	+	0.719204i	$0.255493\pi$
$570$	0	0
$571$	−1044.79	−0.0765726	−0.0382863	−	0.999267i	$-0.512190\pi$
$571$	−1044.79	−0.0765726	−0.0382863	+	0.999267i	$0.512190\pi$
$572$	−17447.0	−1.27534
$573$	0	0
$574$	−4609.01	−0.335150
$575$	−2518.72	−0.182675
$576$	0	0
$577$	−3366.04	−0.242860	−0.121430	−	0.992600i	$-0.538748\pi$
$577$	−3366.04	−0.242860	−0.121430	+	0.992600i	$0.538748\pi$
$578$	−4025.04	−0.289653
$579$	0	0
$580$	−29428.0	−2.10678
$581$	22496.2	1.60637
$582$	0	0
$583$	685.450	0.0486937
$584$	−2922.75	−0.207096
$585$	0	0
$586$	−15404.1	−1.08590
$587$	17910.9	1.25939	0.629696	−	0.776842i	$-0.283180\pi$
$587$	17910.9	1.25939	0.629696	+	0.776842i	$0.283180\pi$
$588$	0	0
$589$	−9752.29	−0.682234
$590$	25027.5	1.74639
$591$	0	0
$592$	−13101.9	−0.909600
$593$	−3396.99	−0.235240	−0.117620	−	0.993059i	$-0.537527\pi$
$593$	−3396.99	−0.235240	−0.117620	+	0.993059i	$0.537527\pi$
$594$	0	0
$595$	−16014.4	−1.10340
$596$	12894.8	0.886230
$597$	0	0
$598$	4135.94	0.282828
$599$	139.199	0.00949505	0.00474752	−	0.999989i	$-0.498489\pi$
$599$	139.199	0.00949505	0.00474752	+	0.999989i	$0.498489\pi$
$600$	0	0
$601$	1384.91	0.0939959	0.0469979	−	0.998895i	$-0.485035\pi$
$601$	1384.91	0.0939959	0.0469979	+	0.998895i	$0.485035\pi$
$602$	10249.5	0.693917
$603$	0	0
$604$	12301.7	0.828724
$605$	−27297.1	−1.83436
$606$	0	0
$607$	1928.79	0.128974	0.0644871	−	0.997919i	$-0.479459\pi$
$607$	1928.79	0.128974	0.0644871	+	0.997919i	$0.479459\pi$
$608$	33784.0	2.25349
$609$	0	0
$610$	14270.3	0.947195
$611$	−12818.3	−0.848731
$612$	0	0
$613$	14251.4	0.939001	0.469500	−	0.882932i	$-0.344434\pi$
$613$	14251.4	0.939001	0.469500	+	0.882932i	$0.344434\pi$
$614$	19889.5	1.30729
$615$	0	0
$616$	4828.51	0.315822
$617$	6803.77	0.443938	0.221969	−	0.975054i	$-0.428752\pi$
$617$	6803.77	0.443938	0.221969	+	0.975054i	$0.428752\pi$
$618$	0	0
$619$	11856.9	0.769904	0.384952	−	0.922937i	$-0.374218\pi$
$619$	11856.9	0.769904	0.384952	+	0.922937i	$0.374218\pi$
$620$	−7006.93	−0.453879
$621$	0	0
$622$	−20828.9	−1.34271
$623$	−5686.92	−0.365717
$624$	0	0
$625$	−17321.3	−1.10857
$626$	−28294.0	−1.80648
$627$	0	0
$628$	341.157	0.0216778
$629$	−11162.1	−0.707572
$630$	0	0
$631$	−15491.7	−0.977362	−0.488681	−	0.872463i	$-0.662522\pi$
$631$	−15491.7	−0.977362	−0.488681	+	0.872463i	$0.662522\pi$
$632$	−3697.83	−0.232740
$633$	0	0
$634$	33632.8	2.10683
$635$	11250.6	0.703098
$636$	0	0
$637$	2809.03	0.174722
$638$	61667.7	3.82672
$639$	0	0
$640$	−9937.32	−0.613761
$641$	18427.5	1.13548	0.567740	−	0.823208i	$-0.307818\pi$
$641$	18427.5	1.13548	0.567740	+	0.823208i	$0.307818\pi$
$642$	0	0
$643$	4164.58	0.255420	0.127710	−	0.991812i	$-0.459237\pi$
$643$	4164.58	0.255420	0.127710	+	0.991812i	$0.459237\pi$
$644$	2576.56	0.157656
$645$	0	0
$646$	33754.7	2.05582
$647$	−24965.0	−1.51697	−0.758483	−	0.651693i	$-0.774059\pi$
$647$	−24965.0	−1.51697	−0.758483	+	0.651693i	$0.774059\pi$
$648$	0	0
$649$	−23820.1	−1.44071
$650$	−19692.4	−1.18831
$651$	0	0
$652$	−4047.10	−0.243093
$653$	19702.6	1.18074	0.590370	−	0.807133i	$-0.298982\pi$
$653$	19702.6	1.18074	0.590370	+	0.807133i	$0.298982\pi$
$654$	0	0
$655$	−43727.4	−2.60850
$656$	−5218.14	−0.310571
$657$	0	0
$658$	−17582.1	−1.04167
$659$	19243.1	1.13749	0.568744	−	0.822515i	$-0.307429\pi$
$659$	19243.1	1.13749	0.568744	+	0.822515i	$0.307429\pi$
$660$	0	0
$661$	11725.0	0.689939	0.344969	−	0.938614i	$-0.387889\pi$
$661$	11725.0	0.689939	0.344969	+	0.938614i	$0.387889\pi$
$662$	−30619.9	−1.79770
$663$	0	0
$664$	6874.00	0.401751
$665$	−36563.9	−2.13216
$666$	0	0
$667$	−6639.55	−0.385434
$668$	21655.2	1.25429
$669$	0	0
$670$	4751.73	0.273993
$671$	−13581.9	−0.781404
$672$	0	0
$673$	−27664.1	−1.58450	−0.792252	−	0.610194i	$-0.791091\pi$
$673$	−27664.1	−1.58450	−0.792252	+	0.610194i	$0.791091\pi$
$674$	−21335.7	−1.21932
$675$	0	0
$676$	61.4678	0.00349726
$677$	−16852.0	−0.956683	−0.478342	−	0.878174i	$-0.658762\pi$
$677$	−16852.0	−0.956683	−0.478342	+	0.878174i	$0.658762\pi$
$678$	0	0
$679$	23892.4	1.35038
$680$	−4893.39	−0.275960
$681$	0	0
$682$	14683.3	0.824420
$683$	−24478.8	−1.37138	−0.685692	−	0.727891i	$-0.740500\pi$
$683$	−24478.8	−1.37138	−0.685692	+	0.727891i	$0.740500\pi$
$684$	0	0
$685$	21907.2	1.22195
$686$	25951.2	1.44435
$687$	0	0
$688$	11604.1	0.643025
$689$	−576.999	−0.0319041
$690$	0	0
$691$	22681.5	1.24869	0.624346	−	0.781148i	$-0.285365\pi$
$691$	22681.5	1.24869	0.624346	+	0.781148i	$0.285365\pi$
$692$	−13152.3	−0.722508
$693$	0	0
$694$	30896.6	1.68994
$695$	−4416.63	−0.241054
$696$	0	0
$697$	−4445.59	−0.241591
$698$	−39598.7	−2.14733
$699$	0	0
$700$	−12267.8	−0.662396
$701$	−6898.78	−0.371703	−0.185851	−	0.982578i	$-0.559504\pi$
$701$	−6898.78	−0.371703	−0.185851	+	0.982578i	$0.559504\pi$
$702$	0	0
$703$	−25485.2	−1.36727
$704$	−18305.9	−0.980012
$705$	0	0
$706$	8905.15	0.474716
$707$	7930.82	0.421880
$708$	0	0
$709$	−1725.21	−0.0913845	−0.0456922	−	0.998956i	$-0.514549\pi$
$709$	−1725.21	−0.0913845	−0.0456922	+	0.998956i	$0.514549\pi$
$710$	40859.2	2.15975
$711$	0	0
$712$	−1737.71	−0.0914654
$713$	−1580.91	−0.0830370
$714$	0	0
$715$	40135.9	2.09929
$716$	−28368.7	−1.48071
$717$	0	0
$718$	12761.6	0.663315
$719$	9213.52	0.477895	0.238947	−	0.971033i	$-0.423198\pi$
$719$	9213.52	0.477895	0.238947	+	0.971033i	$0.423198\pi$
$720$	0	0
$721$	18999.4	0.981377
$722$	50809.3	2.61901
$723$	0	0
$724$	13601.9	0.698217
$725$	31612.8	1.61941
$726$	0	0
$727$	23768.6	1.21256	0.606278	−	0.795253i	$-0.292662\pi$
$727$	23768.6	1.21256	0.606278	+	0.795253i	$0.292662\pi$
$728$	−4064.55	−0.206926
$729$	0	0
$730$	−33323.4	−1.68952
$731$	9886.09	0.500205
$732$	0	0
$733$	10711.8	0.539767	0.269884	−	0.962893i	$-0.413015\pi$
$733$	10711.8	0.539767	0.269884	+	0.962893i	$0.413015\pi$
$734$	26047.6	1.30986
$735$	0	0
$736$	5476.59	0.274280
$737$	−4522.49	−0.226035
$738$	0	0
$739$	−25572.2	−1.27292	−0.636461	−	0.771309i	$-0.719602\pi$
$739$	−25572.2	−1.27292	−0.636461	+	0.771309i	$0.719602\pi$
$740$	−18310.9	−0.909625
$741$	0	0
$742$	−791.431	−0.0391568
$743$	3613.01	0.178397	0.0891983	−	0.996014i	$-0.471570\pi$
$743$	3613.01	0.178397	0.0891983	+	0.996014i	$0.471570\pi$
$744$	0	0
$745$	−29663.8	−1.45879
$746$	2406.54	0.118110
$747$	0	0
$748$	−23082.4	−1.12831
$749$	3425.31	0.167100
$750$	0	0
$751$	26247.1	1.27532	0.637662	−	0.770316i	$-0.279902\pi$
$751$	26247.1	1.27532	0.637662	+	0.770316i	$0.279902\pi$
$752$	−19905.7	−0.965277
$753$	0	0
$754$	−51910.7	−2.50726
$755$	−28299.4	−1.36413
$756$	0	0
$757$	−12700.0	−0.609761	−0.304880	−	0.952391i	$-0.598617\pi$
$757$	−12700.0	−0.609761	−0.304880	+	0.952391i	$0.598617\pi$
$758$	12592.3	0.603393
$759$	0	0
$760$	−11172.5	−0.533251
$761$	−22485.4	−1.07108	−0.535542	−	0.844508i	$-0.679893\pi$
$761$	−22485.4	−1.07108	−0.535542	+	0.844508i	$0.679893\pi$
$762$	0	0
$763$	32619.1	1.54770
$764$	486.617	0.0230434
$765$	0	0
$766$	7235.01	0.341268
$767$	20051.3	0.943953
$768$	0	0
$769$	−7141.34	−0.334881	−0.167440	−	0.985882i	$-0.553550\pi$
$769$	−7141.34	−0.334881	−0.167440	+	0.985882i	$0.553550\pi$
$770$	55051.7	2.57653
$771$	0	0
$772$	−1517.10	−0.0707276
$773$	−15819.1	−0.736060	−0.368030	−	0.929814i	$-0.619968\pi$
$773$	−15819.1	−0.736060	−0.368030	+	0.929814i	$0.619968\pi$
$774$	0	0
$775$	7527.15	0.348882
$776$	7300.61	0.337728
$777$	0	0
$778$	2178.55	0.100392
$779$	−10150.1	−0.466837
$780$	0	0
$781$	−38888.0	−1.78172
$782$	5471.85	0.250221
$783$	0	0
$784$	4362.17	0.198714
$785$	−784.811	−0.0356829
$786$	0	0
$787$	−2959.52	−0.134048	−0.0670239	−	0.997751i	$-0.521350\pi$
$787$	−2959.52	−0.134048	−0.0670239	+	0.997751i	$0.521350\pi$
$788$	−3846.21	−0.173877
$789$	0	0
$790$	−42160.3	−1.89873
$791$	27998.9	1.25857
$792$	0	0
$793$	11433.0	0.511975
$794$	15666.3	0.700224
$795$	0	0
$796$	−30415.2	−1.35432
$797$	−19932.4	−0.885876	−0.442938	−	0.896552i	$-0.646064\pi$
$797$	−19932.4	−0.885876	−0.442938	+	0.896552i	$0.646064\pi$
$798$	0	0
$799$	−16958.7	−0.750883
$800$	−26075.6	−1.15239
$801$	0	0
$802$	−14804.5	−0.651828
$803$	31715.7	1.39380
$804$	0	0
$805$	−5927.23	−0.259512
$806$	−12360.2	−0.540159
$807$	0	0
$808$	2423.36	0.105512
$809$	7334.79	0.318761	0.159380	−	0.987217i	$-0.449050\pi$
$809$	7334.79	0.318761	0.159380	+	0.987217i	$0.449050\pi$
$810$	0	0
$811$	−15908.5	−0.688807	−0.344404	−	0.938822i	$-0.611919\pi$
$811$	−15908.5	−0.688807	−0.344404	+	0.938822i	$0.611919\pi$
$812$	−32338.8	−1.39762
$813$	0	0
$814$	38371.4	1.65223
$815$	9310.12	0.400146
$816$	0	0
$817$	22571.8	0.966570
$818$	−59844.4	−2.55796
$819$	0	0
$820$	−7292.78	−0.310579
$821$	6167.56	0.262180	0.131090	−	0.991371i	$-0.458152\pi$
$821$	6167.56	0.262180	0.131090	+	0.991371i	$0.458152\pi$
$822$	0	0
$823$	−45695.7	−1.93542	−0.967710	−	0.252066i	$-0.918890\pi$
$823$	−45695.7	−1.93542	−0.967710	+	0.252066i	$0.918890\pi$
$824$	5805.49	0.245441
$825$	0	0
$826$	27503.1	1.15854
$827$	−10905.6	−0.458554	−0.229277	−	0.973361i	$-0.573636\pi$
$827$	−10905.6	−0.458554	−0.229277	+	0.973361i	$0.573636\pi$
$828$	0	0
$829$	−4256.81	−0.178341	−0.0891707	−	0.996016i	$-0.528422\pi$
$829$	−4256.81	−0.178341	−0.0891707	+	0.996016i	$0.528422\pi$
$830$	78373.0	3.27755
$831$	0	0
$832$	15409.5	0.642103
$833$	3716.35	0.154579
$834$	0	0
$835$	−49816.5	−2.06464
$836$	−52701.6	−2.18029
$837$	0	0
$838$	13637.5	0.562171
$839$	15166.7	0.624092	0.312046	−	0.950067i	$-0.398986\pi$
$839$	15166.7	0.624092	0.312046	+	0.950067i	$0.398986\pi$
$840$	0	0
$841$	58944.9	2.41687
$842$	40489.5	1.65720
$843$	0	0
$844$	−12093.6	−0.493219
$845$	−141.403	−0.00575671
$846$	0	0
$847$	−29997.1	−1.21690
$848$	−896.029	−0.0362851
$849$	0	0
$850$	−26053.1	−1.05131
$851$	−4131.31	−0.166416
$852$	0	0
$853$	−24050.4	−0.965379	−0.482690	−	0.875791i	$-0.660340\pi$
$853$	−24050.4	−0.965379	−0.482690	+	0.875791i	$0.660340\pi$
$854$	15681.8	0.628363
$855$	0	0
$856$	1046.65	0.0417917
$857$	−39207.3	−1.56277	−0.781387	−	0.624047i	$-0.785487\pi$
$857$	−39207.3	−1.56277	−0.781387	+	0.624047i	$0.785487\pi$
$858$	0	0
$859$	−10527.2	−0.418142	−0.209071	−	0.977900i	$-0.567044\pi$
$859$	−10527.2	−0.418142	−0.209071	+	0.977900i	$0.567044\pi$
$860$	16217.6	0.643043
$861$	0	0
$862$	−26667.9	−1.05373
$863$	11856.0	0.467651	0.233825	−	0.972279i	$-0.424876\pi$
$863$	11856.0	0.467651	0.233825	+	0.972279i	$0.424876\pi$
$864$	0	0
$865$	30256.1	1.18929
$866$	−61536.3	−2.41465
$867$	0	0
$868$	−7700.00	−0.301100
$869$	40126.3	1.56639
$870$	0	0
$871$	3806.95	0.148098
$872$	9967.18	0.387077
$873$	0	0
$874$	12493.3	0.483514
$875$	−3991.95	−0.154231
$876$	0	0
$877$	51435.9	1.98046	0.990232	−	0.139432i	$-0.0445278\pi$
$877$	51435.9	1.98046	0.990232	+	0.139432i	$0.0445278\pi$
$878$	−2350.61	−0.0903521
$879$	0	0
$880$	62327.4	2.38756
$881$	−8069.52	−0.308592	−0.154296	−	0.988025i	$-0.549311\pi$
$881$	−8069.52	−0.308592	−0.154296	+	0.988025i	$0.549311\pi$
$882$	0	0
$883$	−43901.7	−1.67317	−0.836586	−	0.547835i	$-0.815452\pi$
$883$	−43901.7	−1.67317	−0.836586	+	0.547835i	$0.815452\pi$
$884$	19430.4	0.739269
$885$	0	0
$886$	1008.70	0.0382481
$887$	5877.02	0.222470	0.111235	−	0.993794i	$-0.464519\pi$
$887$	5877.02	0.222470	0.111235	+	0.993794i	$0.464519\pi$
$888$	0	0
$889$	12363.4	0.466430
$890$	−19812.3	−0.746190
$891$	0	0
$892$	23262.1	0.873178
$893$	−38719.9	−1.45097
$894$	0	0
$895$	65260.7	2.43734
$896$	−10920.2	−0.407165
$897$	0	0
$898$	−6295.75	−0.233955
$899$	19842.2	0.736122
$900$	0	0
$901$	−763.371	−0.0282259
$902$	15282.4	0.564132
$903$	0	0
$904$	8555.42	0.314767
$905$	−31290.3	−1.14931
$906$	0	0
$907$	1967.03	0.0720110	0.0360055	−	0.999352i	$-0.488537\pi$
$907$	1967.03	0.0720110	0.0360055	+	0.999352i	$0.488537\pi$
$908$	−6953.12	−0.254127
$909$	0	0
$910$	−46341.5	−1.68814
$911$	−21073.2	−0.766396	−0.383198	−	0.923666i	$-0.625177\pi$
$911$	−21073.2	−0.766396	−0.383198	+	0.923666i	$0.625177\pi$
$912$	0	0
$913$	−74592.0	−2.70387
$914$	−61985.6	−2.24322
$915$	0	0
$916$	14255.9	0.514224
$917$	−48052.6	−1.73046
$918$	0	0
$919$	−7748.69	−0.278135	−0.139067	−	0.990283i	$-0.544410\pi$
$919$	−7748.69	−0.278135	−0.139067	+	0.990283i	$0.544410\pi$
$920$	−1811.14	−0.0649038
$921$	0	0
$922$	−22140.1	−0.790830
$923$	32735.2	1.16738
$924$	0	0
$925$	19670.4	0.699198
$926$	−14369.4	−0.509943
$927$	0	0
$928$	−68737.5	−2.43148
$929$	−35683.4	−1.26021	−0.630105	−	0.776510i	$-0.716988\pi$
$929$	−35683.4	−1.26021	−0.630105	+	0.776510i	$0.716988\pi$
$930$	0	0
$931$	8485.13	0.298699
$932$	−4771.65	−0.167704
$933$	0	0
$934$	32316.6	1.13215
$935$	53099.8	1.85727
$936$	0	0
$937$	46259.2	1.61283	0.806414	−	0.591351i	$-0.201405\pi$
$937$	46259.2	1.61283	0.806414	+	0.591351i	$0.201405\pi$
$938$	5221.74	0.181765
$939$	0	0
$940$	−27819.9	−0.965303
$941$	−23519.2	−0.814777	−0.407388	−	0.913255i	$-0.633560\pi$
$941$	−23519.2	−0.814777	−0.407388	+	0.913255i	$0.633560\pi$
$942$	0	0
$943$	−1645.40	−0.0568203
$944$	31137.9	1.07357
$945$	0	0
$946$	−33984.8	−1.16801
$947$	52061.1	1.78644	0.893220	−	0.449621i	$-0.148441\pi$
$947$	52061.1	1.78644	0.893220	+	0.449621i	$0.148441\pi$
$948$	0	0
$949$	−26697.7	−0.913218
$950$	−59484.1	−2.03149
$951$	0	0
$952$	−5377.41	−0.183070
$953$	15306.5	0.520278	0.260139	−	0.965571i	$-0.416232\pi$
$953$	15306.5	0.520278	0.260139	+	0.965571i	$0.416232\pi$
$954$	0	0
$955$	−1119.43	−0.0379309
$956$	23405.3	0.791820
$957$	0	0
$958$	3834.34	0.129313
$959$	24074.1	0.810631
$960$	0	0
$961$	−25066.5	−0.841412
$962$	−32300.3	−1.08254
$963$	0	0
$964$	37841.8	1.26432
$965$	3490.01	0.116422
$966$	0	0
$967$	19036.6	0.633068	0.316534	−	0.948581i	$-0.397481\pi$
$967$	19036.6	0.633068	0.316534	+	0.948581i	$0.397481\pi$
$968$	−9166.00	−0.304345
$969$	0	0
$970$	83237.0	2.75524
$971$	−9160.05	−0.302740	−0.151370	−	0.988477i	$-0.548368\pi$
$971$	−9160.05	−0.302740	−0.151370	+	0.988477i	$0.548368\pi$
$972$	0	0
$973$	−4853.49	−0.159914
$974$	52603.7	1.73053
$975$	0	0
$976$	17754.4	0.582279
$977$	−27640.7	−0.905122	−0.452561	−	0.891733i	$-0.649490\pi$
$977$	−27640.7	−0.905122	−0.452561	+	0.891733i	$0.649490\pi$
$978$	0	0
$979$	18856.4	0.615582
$980$	6096.49	0.198720
$981$	0	0
$982$	40995.2	1.33219
$983$	−40382.6	−1.31028	−0.655140	−	0.755508i	$-0.727390\pi$
$983$	−40382.6	−1.31028	−0.655140	+	0.755508i	$0.727390\pi$
$984$	0	0
$985$	8847.97	0.286213
$986$	−68678.0	−2.21821
$987$	0	0
$988$	44363.2	1.42852
$989$	3659.03	0.117645
$990$	0	0
$991$	17541.4	0.562283	0.281141	−	0.959666i	$-0.409287\pi$
$991$	17541.4	0.562283	0.281141	+	0.959666i	$0.409287\pi$
$992$	−16366.7	−0.523834
$993$	0	0
$994$	44900.7	1.43276
$995$	69968.5	2.22930
$996$	0	0
$997$	−42798.5	−1.35952	−0.679760	−	0.733435i	$-0.737916\pi$
$997$	−42798.5	−1.35952	−0.679760	+	0.733435i	$0.737916\pi$
$998$	36868.6	1.16939
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.4.a.b.1.2		2
3.2	odd	2		69.4.a.b.1.1	✓	2
12.11	even	2		1104.4.a.q.1.2		2
15.14	odd	2		1725.4.a.m.1.2		2
69.68	even	2		1587.4.a.c.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.4.a.b.1.1	✓	2	3.2	odd	2
207.4.a.b.1.2		2	1.1	even	1	trivial
1104.4.a.q.1.2		2	12.11	even	2
1587.4.a.c.1.1		2	69.68	even	2
1725.4.a.m.1.2		2	15.14	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 207 = 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	207.a (trivial)

\(f(q)\)	\(=\)	\(q+3.82843 q^{2} +6.65685 q^{4} -15.3137 q^{5} -16.8284 q^{7} -5.14214 q^{8} -58.6274 q^{10} +55.7990 q^{11} -46.9706 q^{13} -64.4264 q^{14} -72.9411 q^{16} -62.1421 q^{17} -141.882 q^{19} -101.941 q^{20} +213.622 q^{22} -23.0000 q^{23} +109.510 q^{25} -179.823 q^{26} -112.024 q^{28} +288.676 q^{29} +68.7351 q^{31} -238.113 q^{32} -237.907 q^{34} +257.706 q^{35} +179.622 q^{37} -543.186 q^{38} +78.7452 q^{40} +71.5391 q^{41} -159.088 q^{43} +371.446 q^{44} -88.0538 q^{46} +272.902 q^{47} -59.8040 q^{49} +419.250 q^{50} -312.676 q^{52} +12.2843 q^{53} -854.489 q^{55} +86.5341 q^{56} +1105.18 q^{58} -426.891 q^{59} -243.407 q^{61} +263.147 q^{62} -328.068 q^{64} +719.294 q^{65} -81.0496 q^{67} -413.671 q^{68} +986.607 q^{70} -696.930 q^{71} +568.392 q^{73} +687.671 q^{74} -944.489 q^{76} -939.009 q^{77} +719.123 q^{79} +1117.00 q^{80} +273.882 q^{82} -1336.80 q^{83} +951.627 q^{85} -609.058 q^{86} -286.926 q^{88} +337.935 q^{89} +790.441 q^{91} -153.108 q^{92} +1044.78 q^{94} +2172.74 q^{95} -1419.76 q^{97} -228.955 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 8 q^{5} - 28 q^{7} + 18 q^{8} - 72 q^{10} + 72 q^{11} - 60 q^{13} - 44 q^{14} - 78 q^{16} - 96 q^{17} - 148 q^{19} - 136 q^{20} + 184 q^{22} - 46 q^{23} + 38 q^{25} - 156 q^{26}+ \cdots + 170 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 8 * q^5 - 28 * q^7 + 18 * q^8 - 72 * q^10 + 72 * q^11 - 60 * q^13 - 44 * q^14 - 78 * q^16 - 96 * q^17 - 148 * q^19 - 136 * q^20 + 184 * q^22 - 46 * q^23 + 38 * q^25 - 156 * q^26 - 60 * q^28 + 204 * q^29 - 168 * q^31 - 414 * q^32 - 176 * q^34 + 176 * q^35 + 116 * q^37 - 532 * q^38 + 248 * q^40 - 4 * q^41 - 420 * q^43 + 296 * q^44 - 46 * q^46 + 48 * q^47 - 278 * q^49 + 550 * q^50 - 252 * q^52 - 32 * q^53 - 736 * q^55 - 172 * q^56 + 1260 * q^58 + 40 * q^59 - 764 * q^61 + 696 * q^62 + 34 * q^64 + 624 * q^65 - 988 * q^67 - 256 * q^68 + 1136 * q^70 + 224 * q^71 + 820 * q^73 + 804 * q^74 - 916 * q^76 - 1120 * q^77 + 1772 * q^79 + 1080 * q^80 + 412 * q^82 - 1480 * q^83 + 704 * q^85 - 132 * q^86 + 88 * q^88 - 744 * q^89 + 936 * q^91 - 46 * q^92 + 1456 * q^94 + 2128 * q^95 - 260 * q^97 + 170 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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