LMFDB - Embedded newform 7400.2.a.r.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7400,2,Mod(1,7400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7400, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7400.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7400 = 2^{3} \cdot 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7400.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:	$59.0892974957$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.583504.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 8x^{3} - 2x^{2} + 15x + 8 $ x^5 - 8x^3 - 2x^2 + 15*x + 8
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1480)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.34794$ of defining polynomial
Character	$\chi$	$=$	7400.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.34794 q^{3} +2.75056 q^{7} -1.18306 q^{9} +O(q^{10})$	$q-1.34794 q^{3} +2.75056 q^{7} -1.18306 q^{9} -4.31429 q^{11} +1.72307 q^{13} -5.15688 q^{17} -7.66223 q^{19} -3.70759 q^{21} -8.12970 q^{23} +5.63851 q^{27} -8.12970 q^{29} -6.73508 q^{31} +5.81541 q^{33} +1.00000 q^{37} -2.32259 q^{39} +4.31429 q^{41} +4.32977 q^{43} +4.42079 q^{47} +0.565572 q^{49} +6.95117 q^{51} +9.82650 q^{53} +10.3282 q^{57} +1.54032 q^{59} +4.61748 q^{61} -3.25406 q^{63} +3.17935 q^{67} +10.9584 q^{69} +7.39492 q^{71} -5.08857 q^{73} -11.8667 q^{77} +5.14325 q^{79} -4.05121 q^{81} -8.99108 q^{83} +10.9584 q^{87} -1.83089 q^{89} +4.73940 q^{91} +9.07849 q^{93} +14.3922 q^{97} +5.10405 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{3} + 5 q^{7} + 6 q^{9}+O(q^{10}) $ 5 * q + 5 * q^3 + 5 * q^7 + 6 * q^9	$ 5 q + 5 q^{3} + 5 q^{7} + 6 q^{9} - 7 q^{11} + 6 q^{13} - 12 q^{19} - q^{21} + 6 q^{23} + 17 q^{27} + 6 q^{29} - 10 q^{31} - 3 q^{33} + 5 q^{37} + 4 q^{39} + 7 q^{41} + 22 q^{43} + 13 q^{47} + 14 q^{49} - 10 q^{51} + 11 q^{53} + 8 q^{57} - 10 q^{59} + 10 q^{63} + 24 q^{67} + 26 q^{69} - 13 q^{71} + 13 q^{73} + 19 q^{77} + 2 q^{79} + q^{81} + 13 q^{83} + 26 q^{87} + 8 q^{89} - 8 q^{91} + 20 q^{93} + 20 q^{97} + 2 q^{99}+O(q^{100}) $ 5 * q + 5 * q^3 + 5 * q^7 + 6 * q^9 - 7 * q^11 + 6 * q^13 - 12 * q^19 - q^21 + 6 * q^23 + 17 * q^27 + 6 * q^29 - 10 * q^31 - 3 * q^33 + 5 * q^37 + 4 * q^39 + 7 * q^41 + 22 * q^43 + 13 * q^47 + 14 * q^49 - 10 * q^51 + 11 * q^53 + 8 * q^57 - 10 * q^59 + 10 * q^63 + 24 * q^67 + 26 * q^69 - 13 * q^71 + 13 * q^73 + 19 * q^77 + 2 * q^79 + q^81 + 13 * q^83 + 26 * q^87 + 8 * q^89 - 8 * q^91 + 20 * q^93 + 20 * q^97 + 2 * 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 $

$2$	0	0
$2$	0	0
$3$	−1.34794	−0.778234	−0.389117	−	0.921188i	$-0.627220\pi$
$3$	−1.34794	−0.778234	−0.389117	+	0.921188i	$0.627220\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.75056	1.03961	0.519807	−	0.854284i	$-0.326004\pi$
$7$	2.75056	1.03961	0.519807	+	0.854284i	$0.326004\pi$
$8$	0	0
$9$	−1.18306	−0.394352
$10$	0	0
$11$	−4.31429	−1.30081	−0.650404	−	0.759589i	$-0.725400\pi$
$11$	−4.31429	−1.30081	−0.650404	+	0.759589i	$0.725400\pi$
$12$	0	0
$13$	1.72307	0.477893	0.238947	−	0.971033i	$-0.423198\pi$
$13$	1.72307	0.477893	0.238947	+	0.971033i	$0.423198\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.15688	−1.25073	−0.625364	−	0.780333i	$-0.715050\pi$
$17$	−5.15688	−1.25073	−0.625364	+	0.780333i	$0.715050\pi$
$18$	0	0
$19$	−7.66223	−1.75784	−0.878918	−	0.476973i	$-0.841734\pi$
$19$	−7.66223	−1.75784	−0.878918	+	0.476973i	$0.841734\pi$
$20$	0	0
$21$	−3.70759	−0.809062
$22$	0	0
$23$	−8.12970	−1.69516	−0.847580	−	0.530668i	$-0.821941\pi$
$23$	−8.12970	−1.69516	−0.847580	+	0.530668i	$0.821941\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.63851	1.08513
$28$	0	0
$29$	−8.12970	−1.50965	−0.754823	−	0.655928i	$-0.772277\pi$
$29$	−8.12970	−1.50965	−0.754823	+	0.655928i	$0.772277\pi$
$30$	0	0
$31$	−6.73508	−1.20966	−0.604828	−	0.796356i	$-0.706758\pi$
$31$	−6.73508	−1.20966	−0.604828	+	0.796356i	$0.706758\pi$
$32$	0	0
$33$	5.81541	1.01233
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	−2.32259	−0.371913
$40$	0	0
$41$	4.31429	0.673779	0.336889	−	0.941544i	$-0.390625\pi$
$41$	4.31429	0.673779	0.336889	+	0.941544i	$0.390625\pi$
$42$	0	0
$43$	4.32977	0.660284	0.330142	−	0.943931i	$-0.392903\pi$
$43$	4.32977	0.660284	0.330142	+	0.943931i	$0.392903\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.42079	0.644838	0.322419	−	0.946597i	$-0.395504\pi$
$47$	4.42079	0.644838	0.322419	+	0.946597i	$0.395504\pi$
$48$	0	0
$49$	0.565572	0.0807960
$50$	0	0
$51$	6.95117	0.973359
$52$	0	0
$53$	9.82650	1.34977	0.674887	−	0.737921i	$-0.264192\pi$
$53$	9.82650	1.34977	0.674887	+	0.737921i	$0.264192\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	10.3282	1.36801
$58$	0	0
$59$	1.54032	0.200532	0.100266	−	0.994961i	$-0.468031\pi$
$59$	1.54032	0.200532	0.100266	+	0.994961i	$0.468031\pi$
$60$	0	0
$61$	4.61748	0.591208	0.295604	−	0.955311i	$-0.404479\pi$
$61$	4.61748	0.591208	0.295604	+	0.955311i	$0.404479\pi$
$62$	0	0
$63$	−3.25406	−0.409974
$64$	0	0
$65$	0	0
$66$	0	0
$67$	3.17935	0.388419	0.194210	−	0.980960i	$-0.437786\pi$
$67$	3.17935	0.388419	0.194210	+	0.980960i	$0.437786\pi$
$68$	0	0
$69$	10.9584	1.31923
$70$	0	0
$71$	7.39492	0.877616	0.438808	−	0.898581i	$-0.355401\pi$
$71$	7.39492	0.877616	0.438808	+	0.898581i	$0.355401\pi$
$72$	0	0
$73$	−5.08857	−0.595572	−0.297786	−	0.954633i	$-0.596248\pi$
$73$	−5.08857	−0.595572	−0.297786	+	0.954633i	$0.596248\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−11.8667	−1.35234
$78$	0	0
$79$	5.14325	0.578660	0.289330	−	0.957229i	$-0.406567\pi$
$79$	5.14325	0.578660	0.289330	+	0.957229i	$0.406567\pi$
$80$	0	0
$81$	−4.05121	−0.450134
$82$	0	0
$83$	−8.99108	−0.986899	−0.493449	−	0.869775i	$-0.664264\pi$
$83$	−8.99108	−0.986899	−0.493449	+	0.869775i	$0.664264\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	10.9584	1.17486
$88$	0	0
$89$	−1.83089	−0.194074	−0.0970368	−	0.995281i	$-0.530936\pi$
$89$	−1.83089	−0.194074	−0.0970368	+	0.995281i	$0.530936\pi$
$90$	0	0
$91$	4.73940	0.496824
$92$	0	0
$93$	9.07849	0.941395
$94$	0	0
$95$	0	0
$96$	0	0
$97$	14.3922	1.46130	0.730652	−	0.682751i	$-0.239216\pi$
$97$	14.3922	1.46130	0.730652	+	0.682751i	$0.239216\pi$
$98$	0	0
$99$	5.10405	0.512976
$100$	0	0
$101$	14.8629	1.47892	0.739459	−	0.673202i	$-0.235081\pi$
$101$	14.8629	1.47892	0.739459	+	0.673202i	$0.235081\pi$
$102$	0	0
$103$	8.86763	0.873754	0.436877	−	0.899521i	$-0.356085\pi$
$103$	8.86763	0.873754	0.436877	+	0.899521i	$0.356085\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	19.9559	1.92921	0.964605	−	0.263700i	$-0.0849429\pi$
$107$	19.9559	1.92921	0.964605	+	0.263700i	$0.0849429\pi$
$108$	0	0
$109$	6.36611	0.609763	0.304881	−	0.952390i	$-0.401383\pi$
$109$	6.36611	0.609763	0.304881	+	0.952390i	$0.401383\pi$
$110$	0	0
$111$	−1.34794	−0.127941
$112$	0	0
$113$	−9.14457	−0.860248	−0.430124	−	0.902770i	$-0.641530\pi$
$113$	−9.14457	−0.860248	−0.430124	+	0.902770i	$0.641530\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−2.03849	−0.188458
$118$	0	0
$119$	−14.1843	−1.30027
$120$	0	0
$121$	7.61310	0.692100
$122$	0	0
$123$	−5.81541	−0.524358
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−11.8185	−1.04872	−0.524361	−	0.851496i	$-0.675696\pi$
$127$	−11.8185	−1.04872	−0.524361	+	0.851496i	$0.675696\pi$
$128$	0	0
$129$	−5.83627	−0.513855
$130$	0	0
$131$	−15.0053	−1.31102	−0.655511	−	0.755186i	$-0.727547\pi$
$131$	−15.0053	−1.31102	−0.655511	+	0.755186i	$0.727547\pi$
$132$	0	0
$133$	−21.0754	−1.82747
$134$	0	0
$135$	0	0
$136$	0	0
$137$	7.13500	0.609585	0.304792	−	0.952419i	$-0.401413\pi$
$137$	7.13500	0.609585	0.304792	+	0.952419i	$0.401413\pi$
$138$	0	0
$139$	−2.12706	−0.180415	−0.0902074	−	0.995923i	$-0.528753\pi$
$139$	−2.12706	−0.180415	−0.0902074	+	0.995923i	$0.528753\pi$
$140$	0	0
$141$	−5.95896	−0.501835
$142$	0	0
$143$	−7.43382	−0.621647
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−0.762357	−0.0628782
$148$	0	0
$149$	14.4035	1.17998	0.589989	−	0.807411i	$-0.299132\pi$
$149$	14.4035	1.17998	0.589989	+	0.807411i	$0.299132\pi$
$150$	0	0
$151$	12.3811	1.00756	0.503779	−	0.863833i	$-0.331943\pi$
$151$	12.3811	1.00756	0.503779	+	0.863833i	$0.331943\pi$
$152$	0	0
$153$	6.10088	0.493227
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−1.88271	−0.150256	−0.0751282	−	0.997174i	$-0.523937\pi$
$157$	−1.88271	−0.150256	−0.0751282	+	0.997174i	$0.523937\pi$
$158$	0	0
$159$	−13.2455	−1.05044
$160$	0	0
$161$	−22.3612	−1.76231
$162$	0	0
$163$	−4.84900	−0.379803	−0.189901	−	0.981803i	$-0.560817\pi$
$163$	−4.84900	−0.379803	−0.189901	+	0.981803i	$0.560817\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−9.89328	−0.765565	−0.382783	−	0.923838i	$-0.625034\pi$
$167$	−9.89328	−0.765565	−0.382783	+	0.923838i	$0.625034\pi$
$168$	0	0
$169$	−10.0310	−0.771618
$170$	0	0
$171$	9.06485	0.693206
$172$	0	0
$173$	−6.57676	−0.500022	−0.250011	−	0.968243i	$-0.580434\pi$
$173$	−6.57676	−0.500022	−0.250011	+	0.968243i	$0.580434\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−2.07625	−0.156061
$178$	0	0
$179$	−24.2526	−1.81273	−0.906363	−	0.422500i	$-0.861153\pi$
$179$	−24.2526	−1.81273	−0.906363	+	0.422500i	$0.861153\pi$
$180$	0	0
$181$	14.7696	1.09781	0.548907	−	0.835883i	$-0.315044\pi$
$181$	14.7696	1.09781	0.548907	+	0.835883i	$0.315044\pi$
$182$	0	0
$183$	−6.22409	−0.460098
$184$	0	0
$185$	0	0
$186$	0	0
$187$	22.2483	1.62696
$188$	0	0
$189$	15.5091	1.12812
$190$	0	0
$191$	−15.4859	−1.12052	−0.560262	−	0.828316i	$-0.689299\pi$
$191$	−15.4859	−1.12052	−0.560262	+	0.828316i	$0.689299\pi$
$192$	0	0
$193$	−3.61047	−0.259887	−0.129944	−	0.991521i	$-0.541480\pi$
$193$	−3.61047	−0.259887	−0.129944	+	0.991521i	$0.541480\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	18.1142	1.29058	0.645292	−	0.763936i	$-0.276736\pi$
$197$	18.1142	1.29058	0.645292	+	0.763936i	$0.276736\pi$
$198$	0	0
$199$	8.86932	0.628729	0.314365	−	0.949302i	$-0.398209\pi$
$199$	8.86932	0.628729	0.314365	+	0.949302i	$0.398209\pi$
$200$	0	0
$201$	−4.28557	−0.302281
$202$	0	0
$203$	−22.3612	−1.56945
$204$	0	0
$205$	0	0
$206$	0	0
$207$	9.61789	0.668489
$208$	0	0
$209$	33.0571	2.28661
$210$	0	0
$211$	−3.43820	−0.236695	−0.118348	−	0.992972i	$-0.537760\pi$
$211$	−3.43820	−0.236695	−0.118348	+	0.992972i	$0.537760\pi$
$212$	0	0
$213$	−9.96792	−0.682990
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−18.5252	−1.25757
$218$	0	0
$219$	6.85909	0.463494
$220$	0	0
$221$	−8.88566	−0.597714
$222$	0	0
$223$	27.4184	1.83607	0.918035	−	0.396500i	$-0.129775\pi$
$223$	27.4184	1.83607	0.918035	+	0.396500i	$0.129775\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	18.1660	1.20572	0.602861	−	0.797846i	$-0.294027\pi$
$227$	18.1660	1.20572	0.602861	+	0.797846i	$0.294027\pi$
$228$	0	0
$229$	25.6592	1.69561	0.847804	−	0.530309i	$-0.177924\pi$
$229$	25.6592	1.69561	0.847804	+	0.530309i	$0.177924\pi$
$230$	0	0
$231$	15.9956	1.05243
$232$	0	0
$233$	−10.0129	−0.655969	−0.327984	−	0.944683i	$-0.606369\pi$
$233$	−10.0129	−0.655969	−0.327984	+	0.944683i	$0.606369\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−6.93279	−0.450333
$238$	0	0
$239$	13.9610	0.903065	0.451532	−	0.892255i	$-0.350878\pi$
$239$	13.9610	0.903065	0.451532	+	0.892255i	$0.350878\pi$
$240$	0	0
$241$	0.864995	0.0557192	0.0278596	−	0.999612i	$-0.491131\pi$
$241$	0.864995	0.0557192	0.0278596	+	0.999612i	$0.491131\pi$
$242$	0	0
$243$	−11.4547	−0.734822
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−13.2025	−0.840058
$248$	0	0
$249$	12.1194	0.768038
$250$	0	0
$251$	−10.6608	−0.672903	−0.336451	−	0.941701i	$-0.609227\pi$
$251$	−10.6608	−0.672903	−0.336451	+	0.941701i	$0.609227\pi$
$252$	0	0
$253$	35.0739	2.20508
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−16.1158	−1.00528	−0.502639	−	0.864496i	$-0.667638\pi$
$257$	−16.1158	−1.00528	−0.502639	+	0.864496i	$0.667638\pi$
$258$	0	0
$259$	2.75056	0.170911
$260$	0	0
$261$	9.61789	0.595332
$262$	0	0
$263$	−21.0572	−1.29844	−0.649220	−	0.760601i	$-0.724904\pi$
$263$	−21.0572	−1.29844	−0.649220	+	0.760601i	$0.724904\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2.46793	0.151035
$268$	0	0
$269$	−12.0096	−0.732236	−0.366118	−	0.930568i	$-0.619313\pi$
$269$	−12.0096	−0.732236	−0.366118	+	0.930568i	$0.619313\pi$
$270$	0	0
$271$	−27.8450	−1.69146	−0.845732	−	0.533609i	$-0.820836\pi$
$271$	−27.8450	−1.69146	−0.845732	+	0.533609i	$0.820836\pi$
$272$	0	0
$273$	−6.38843	−0.386645
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−6.94879	−0.417512	−0.208756	−	0.977968i	$-0.566941\pi$
$277$	−6.94879	−0.417512	−0.208756	+	0.977968i	$0.566941\pi$
$278$	0	0
$279$	7.96798	0.477030
$280$	0	0
$281$	−6.52862	−0.389465	−0.194732	−	0.980856i	$-0.562384\pi$
$281$	−6.52862	−0.389465	−0.194732	+	0.980856i	$0.562384\pi$
$282$	0	0
$283$	14.8724	0.884073	0.442037	−	0.896997i	$-0.354256\pi$
$283$	14.8724	0.884073	0.442037	+	0.896997i	$0.354256\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	11.8667	0.700470
$288$	0	0
$289$	9.59346	0.564321
$290$	0	0
$291$	−19.3998	−1.13724
$292$	0	0
$293$	2.85654	0.166881	0.0834403	−	0.996513i	$-0.473409\pi$
$293$	2.85654	0.166881	0.0834403	+	0.996513i	$0.473409\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−24.3262	−1.41155
$298$	0	0
$299$	−14.0080	−0.810105
$300$	0	0
$301$	11.9093	0.686440
$302$	0	0
$303$	−20.0344	−1.15094
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−3.93300	−0.224468	−0.112234	−	0.993682i	$-0.535801\pi$
$307$	−3.93300	−0.224468	−0.112234	+	0.993682i	$0.535801\pi$
$308$	0	0
$309$	−11.9530	−0.679985
$310$	0	0
$311$	4.14548	0.235069	0.117534	−	0.993069i	$-0.462501\pi$
$311$	4.14548	0.235069	0.117534	+	0.993069i	$0.462501\pi$
$312$	0	0
$313$	−1.94502	−0.109939	−0.0549695	−	0.998488i	$-0.517506\pi$
$313$	−1.94502	−0.109939	−0.0549695	+	0.998488i	$0.517506\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	27.9543	1.57007	0.785034	−	0.619453i	$-0.212645\pi$
$317$	27.9543	1.57007	0.785034	+	0.619453i	$0.212645\pi$
$318$	0	0
$319$	35.0739	1.96376
$320$	0	0
$321$	−26.8994	−1.50138
$322$	0	0
$323$	39.5132	2.19858
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−8.58114	−0.474538
$328$	0	0
$329$	12.1596	0.670383
$330$	0	0
$331$	−3.64991	−0.200617	−0.100309	−	0.994956i	$-0.531983\pi$
$331$	−3.64991	−0.200617	−0.100309	+	0.994956i	$0.531983\pi$
$332$	0	0
$333$	−1.18306	−0.0648311
$334$	0	0
$335$	0	0
$336$	0	0
$337$	15.0764	0.821265	0.410633	−	0.911801i	$-0.365308\pi$
$337$	15.0764	0.821265	0.410633	+	0.911801i	$0.365308\pi$
$338$	0	0
$339$	12.3263	0.669474
$340$	0	0
$341$	29.0571	1.57353
$342$	0	0
$343$	−17.6983	−0.955617
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−32.4761	−1.74341	−0.871705	−	0.490031i	$-0.836985\pi$
$347$	−32.4761	−1.74341	−0.871705	+	0.490031i	$0.836985\pi$
$348$	0	0
$349$	36.8006	1.96989	0.984946	−	0.172861i	$-0.0553012\pi$
$349$	36.8006	1.96989	0.984946	+	0.172861i	$0.0553012\pi$
$350$	0	0
$351$	9.71554	0.518577
$352$	0	0
$353$	6.10364	0.324864	0.162432	−	0.986720i	$-0.448066\pi$
$353$	6.10364	0.324864	0.162432	+	0.986720i	$0.448066\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	19.1196	1.01192
$358$	0	0
$359$	21.1491	1.11621	0.558104	−	0.829771i	$-0.311529\pi$
$359$	21.1491	1.11621	0.558104	+	0.829771i	$0.311529\pi$
$360$	0	0
$361$	39.7098	2.08999
$362$	0	0
$363$	−10.2620	−0.538616
$364$	0	0
$365$	0	0
$366$	0	0
$367$	10.8689	0.567351	0.283675	−	0.958920i	$-0.408446\pi$
$367$	10.8689	0.567351	0.283675	+	0.958920i	$0.408446\pi$
$368$	0	0
$369$	−5.10405	−0.265706
$370$	0	0
$371$	27.0284	1.40324
$372$	0	0
$373$	18.6046	0.963312	0.481656	−	0.876360i	$-0.340035\pi$
$373$	18.6046	0.963312	0.481656	+	0.876360i	$0.340035\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−14.0080	−0.721450
$378$	0	0
$379$	−27.4846	−1.41179	−0.705894	−	0.708317i	$-0.749455\pi$
$379$	−27.4846	−1.41179	−0.705894	+	0.708317i	$0.749455\pi$
$380$	0	0
$381$	15.9306	0.816152
$382$	0	0
$383$	−17.9963	−0.919569	−0.459785	−	0.888031i	$-0.652073\pi$
$383$	−17.9963	−0.919569	−0.459785	+	0.888031i	$0.652073\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−5.12236	−0.260384
$388$	0	0
$389$	7.62327	0.386515	0.193258	−	0.981148i	$-0.438095\pi$
$389$	7.62327	0.386515	0.193258	+	0.981148i	$0.438095\pi$
$390$	0	0
$391$	41.9239	2.12018
$392$	0	0
$393$	20.2263	1.02028
$394$	0	0
$395$	0	0
$396$	0	0
$397$	24.3599	1.22259	0.611294	−	0.791404i	$-0.290649\pi$
$397$	24.3599	1.22259	0.611294	+	0.791404i	$0.290649\pi$
$398$	0	0
$399$	28.4084	1.42220
$400$	0	0
$401$	−12.3427	−0.616365	−0.308182	−	0.951327i	$-0.599721\pi$
$401$	−12.3427	−0.616365	−0.308182	+	0.951327i	$0.599721\pi$
$402$	0	0
$403$	−11.6050	−0.578086
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−4.31429	−0.213851
$408$	0	0
$409$	−28.7263	−1.42042	−0.710212	−	0.703988i	$-0.751401\pi$
$409$	−28.7263	−1.42042	−0.710212	+	0.703988i	$0.751401\pi$
$410$	0	0
$411$	−9.61756	−0.474399
$412$	0	0
$413$	4.23673	0.208476
$414$	0	0
$415$	0	0
$416$	0	0
$417$	2.86715	0.140405
$418$	0	0
$419$	−12.4021	−0.605880	−0.302940	−	0.953010i	$-0.597968\pi$
$419$	−12.4021	−0.605880	−0.302940	+	0.953010i	$0.597968\pi$
$420$	0	0
$421$	15.1603	0.738865	0.369433	−	0.929257i	$-0.379552\pi$
$421$	15.1603	0.738865	0.369433	+	0.929257i	$0.379552\pi$
$422$	0	0
$423$	−5.23004	−0.254293
$424$	0	0
$425$	0	0
$426$	0	0
$427$	12.7007	0.614628
$428$	0	0
$429$	10.0203	0.483787
$430$	0	0
$431$	−37.2804	−1.79573	−0.897866	−	0.440269i	$-0.854883\pi$
$431$	−37.2804	−1.79573	−0.897866	+	0.440269i	$0.854883\pi$
$432$	0	0
$433$	−2.57676	−0.123831	−0.0619156	−	0.998081i	$-0.519721\pi$
$433$	−2.57676	−0.123831	−0.0619156	+	0.998081i	$0.519721\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	62.2916	2.97981
$438$	0	0
$439$	8.03202	0.383348	0.191674	−	0.981459i	$-0.438608\pi$
$439$	8.03202	0.383348	0.191674	+	0.981459i	$0.438608\pi$
$440$	0	0
$441$	−0.669103	−0.0318621
$442$	0	0
$443$	−34.3158	−1.63039	−0.815196	−	0.579185i	$-0.803371\pi$
$443$	−34.3158	−1.63039	−0.815196	+	0.579185i	$0.803371\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−19.4150	−0.918299
$448$	0	0
$449$	−21.9062	−1.03382	−0.516909	−	0.856040i	$-0.672917\pi$
$449$	−21.9062	−1.03382	−0.516909	+	0.856040i	$0.672917\pi$
$450$	0	0
$451$	−18.6131	−0.876457
$452$	0	0
$453$	−16.6889	−0.784115
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−4.01851	−0.187978	−0.0939891	−	0.995573i	$-0.529962\pi$
$457$	−4.01851	−0.187978	−0.0939891	+	0.995573i	$0.529962\pi$
$458$	0	0
$459$	−29.0772	−1.35721
$460$	0	0
$461$	−18.0129	−0.838946	−0.419473	−	0.907768i	$-0.637785\pi$
$461$	−18.0129	−0.838946	−0.419473	+	0.907768i	$0.637785\pi$
$462$	0	0
$463$	11.9141	0.553693	0.276847	−	0.960914i	$-0.410711\pi$
$463$	11.9141	0.553693	0.276847	+	0.960914i	$0.410711\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	7.63266	0.353198	0.176599	−	0.984283i	$-0.443490\pi$
$467$	7.63266	0.353198	0.176599	+	0.984283i	$0.443490\pi$
$468$	0	0
$469$	8.74498	0.403806
$470$	0	0
$471$	2.53778	0.116935
$472$	0	0
$473$	−18.6799	−0.858902
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−11.6253	−0.532286
$478$	0	0
$479$	−8.90659	−0.406953	−0.203476	−	0.979080i	$-0.565224\pi$
$479$	−8.90659	−0.406953	−0.203476	+	0.979080i	$0.565224\pi$
$480$	0	0
$481$	1.72307	0.0785651
$482$	0	0
$483$	30.1416	1.37149
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−1.44411	−0.0654386	−0.0327193	−	0.999465i	$-0.510417\pi$
$487$	−1.44411	−0.0654386	−0.0327193	+	0.999465i	$0.510417\pi$
$488$	0	0
$489$	6.53616	0.295575
$490$	0	0
$491$	39.7679	1.79470	0.897351	−	0.441318i	$-0.145489\pi$
$491$	39.7679	1.79470	0.897351	+	0.441318i	$0.145489\pi$
$492$	0	0
$493$	41.9239	1.88816
$494$	0	0
$495$	0	0
$496$	0	0
$497$	20.3402	0.912381
$498$	0	0
$499$	21.7811	0.975055	0.487527	−	0.873108i	$-0.337899\pi$
$499$	21.7811	0.975055	0.487527	+	0.873108i	$0.337899\pi$
$500$	0	0
$501$	13.3356	0.595789
$502$	0	0
$503$	−1.05275	−0.0469397	−0.0234698	−	0.999725i	$-0.507471\pi$
$503$	−1.05275	−0.0469397	−0.0234698	+	0.999725i	$0.507471\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	13.5212	0.600499
$508$	0	0
$509$	−27.7721	−1.23098	−0.615489	−	0.788145i	$-0.711041\pi$
$509$	−27.7721	−1.23098	−0.615489	+	0.788145i	$0.711041\pi$
$510$	0	0
$511$	−13.9964	−0.619165
$512$	0	0
$513$	−43.2036	−1.90748
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−19.0726	−0.838811
$518$	0	0
$519$	8.86508	0.389134
$520$	0	0
$521$	−17.4835	−0.765966	−0.382983	−	0.923755i	$-0.625103\pi$
$521$	−17.4835	−0.765966	−0.382983	+	0.923755i	$0.625103\pi$
$522$	0	0
$523$	4.74840	0.207633	0.103817	−	0.994596i	$-0.466894\pi$
$523$	4.74840	0.207633	0.103817	+	0.994596i	$0.466894\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	34.7320	1.51295
$528$	0	0
$529$	43.0920	1.87356
$530$	0	0
$531$	−1.82228	−0.0790802
$532$	0	0
$533$	7.43382	0.321994
$534$	0	0
$535$	0	0
$536$	0	0
$537$	32.6911	1.41072
$538$	0	0
$539$	−2.44004	−0.105100
$540$	0	0
$541$	39.8878	1.71491	0.857454	−	0.514560i	$-0.172045\pi$
$541$	39.8878	1.71491	0.857454	+	0.514560i	$0.172045\pi$
$542$	0	0
$543$	−19.9085	−0.854356
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−37.5673	−1.60626	−0.803130	−	0.595804i	$-0.796833\pi$
$547$	−37.5673	−1.60626	−0.803130	+	0.595804i	$0.796833\pi$
$548$	0	0
$549$	−5.46274	−0.233144
$550$	0	0
$551$	62.2916	2.65371
$552$	0	0
$553$	14.1468	0.601583
$554$	0	0
$555$	0	0
$556$	0	0
$557$	38.3652	1.62558	0.812792	−	0.582553i	$-0.197946\pi$
$557$	38.3652	1.62558	0.812792	+	0.582553i	$0.197946\pi$
$558$	0	0
$559$	7.46049	0.315545
$560$	0	0
$561$	−29.9894	−1.26615
$562$	0	0
$563$	−45.9437	−1.93629	−0.968147	−	0.250382i	$-0.919444\pi$
$563$	−45.9437	−1.93629	−0.968147	+	0.250382i	$0.919444\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−11.1431	−0.467966
$568$	0	0
$569$	13.4312	0.563064	0.281532	−	0.959552i	$-0.409157\pi$
$569$	13.4312	0.563064	0.281532	+	0.959552i	$0.409157\pi$
$570$	0	0
$571$	17.7210	0.741601	0.370801	−	0.928712i	$-0.379083\pi$
$571$	17.7210	0.741601	0.370801	+	0.928712i	$0.379083\pi$
$572$	0	0
$573$	20.8741	0.872029
$574$	0	0
$575$	0	0
$576$	0	0
$577$	1.17860	0.0490656	0.0245328	−	0.999699i	$-0.492190\pi$
$577$	1.17860	0.0490656	0.0245328	+	0.999699i	$0.492190\pi$
$578$	0	0
$579$	4.86670	0.202253
$580$	0	0
$581$	−24.7305	−1.02599
$582$	0	0
$583$	−42.3944	−1.75580
$584$	0	0
$585$	0	0
$586$	0	0
$587$	23.8110	0.982785	0.491393	−	0.870938i	$-0.336488\pi$
$587$	23.8110	0.982785	0.491393	+	0.870938i	$0.336488\pi$
$588$	0	0
$589$	51.6057	2.12638
$590$	0	0
$591$	−24.4169	−1.00438
$592$	0	0
$593$	47.3367	1.94389	0.971944	−	0.235214i	$-0.0755792\pi$
$593$	47.3367	1.94389	0.971944	+	0.235214i	$0.0755792\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−11.9553	−0.489298
$598$	0	0
$599$	−20.1889	−0.824898	−0.412449	−	0.910981i	$-0.635326\pi$
$599$	−20.1889	−0.824898	−0.412449	+	0.910981i	$0.635326\pi$
$600$	0	0
$601$	−18.6771	−0.761856	−0.380928	−	0.924605i	$-0.624395\pi$
$601$	−18.6771	−0.761856	−0.380928	+	0.924605i	$0.624395\pi$
$602$	0	0
$603$	−3.76135	−0.153174
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−17.2797	−0.701360	−0.350680	−	0.936495i	$-0.614049\pi$
$607$	−17.2797	−0.701360	−0.350680	+	0.936495i	$0.614049\pi$
$608$	0	0
$609$	30.1416	1.22140
$610$	0	0
$611$	7.61732	0.308164
$612$	0	0
$613$	−43.6926	−1.76473	−0.882363	−	0.470569i	$-0.844049\pi$
$613$	−43.6926	−1.76473	−0.882363	+	0.470569i	$0.844049\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	38.0451	1.53164	0.765818	−	0.643057i	$-0.222334\pi$
$617$	38.0451	1.53164	0.765818	+	0.643057i	$0.222334\pi$
$618$	0	0
$619$	−6.23864	−0.250752	−0.125376	−	0.992109i	$-0.540014\pi$
$619$	−6.23864	−0.250752	−0.125376	+	0.992109i	$0.540014\pi$
$620$	0	0
$621$	−45.8394	−1.83947
$622$	0	0
$623$	−5.03596	−0.201761
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−44.5590	−1.77951
$628$	0	0
$629$	−5.15688	−0.205618
$630$	0	0
$631$	−3.71843	−0.148029	−0.0740143	−	0.997257i	$-0.523581\pi$
$631$	−3.71843	−0.148029	−0.0740143	+	0.997257i	$0.523581\pi$
$632$	0	0
$633$	4.63449	0.184204
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0.974519	0.0386118
$638$	0	0
$639$	−8.74861	−0.346090
$640$	0	0
$641$	−29.3123	−1.15777	−0.578884	−	0.815410i	$-0.696512\pi$
$641$	−29.3123	−1.15777	−0.578884	+	0.815410i	$0.696512\pi$
$642$	0	0
$643$	30.1939	1.19073	0.595366	−	0.803455i	$-0.297007\pi$
$643$	30.1939	1.19073	0.595366	+	0.803455i	$0.297007\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−4.13074	−0.162396	−0.0811981	−	0.996698i	$-0.525875\pi$
$647$	−4.13074	−0.162396	−0.0811981	+	0.996698i	$0.525875\pi$
$648$	0	0
$649$	−6.64537	−0.260854
$650$	0	0
$651$	24.9709	0.978687
$652$	0	0
$653$	25.4367	0.995417	0.497708	−	0.867344i	$-0.334175\pi$
$653$	25.4367	0.995417	0.497708	+	0.867344i	$0.334175\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	6.02006	0.234865
$658$	0	0
$659$	−17.3373	−0.675365	−0.337683	−	0.941260i	$-0.609643\pi$
$659$	−17.3373	−0.675365	−0.337683	+	0.941260i	$0.609643\pi$
$660$	0	0
$661$	−51.2968	−1.99521	−0.997607	−	0.0691355i	$-0.977976\pi$
$661$	−51.2968	−1.99521	−0.997607	+	0.0691355i	$0.977976\pi$
$662$	0	0
$663$	11.9773	0.465162
$664$	0	0
$665$	0	0
$666$	0	0
$667$	66.0920	2.55909
$668$	0	0
$669$	−36.9583	−1.42889
$670$	0	0
$671$	−19.9212	−0.769048
$672$	0	0
$673$	26.9495	1.03883	0.519413	−	0.854523i	$-0.326151\pi$
$673$	26.9495	1.03883	0.519413	+	0.854523i	$0.326151\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	45.3779	1.74401	0.872007	−	0.489494i	$-0.162818\pi$
$677$	45.3779	1.74401	0.872007	+	0.489494i	$0.162818\pi$
$678$	0	0
$679$	39.5865	1.51919
$680$	0	0
$681$	−24.4867	−0.938334
$682$	0	0
$683$	5.88140	0.225045	0.112523	−	0.993649i	$-0.464107\pi$
$683$	5.88140	0.225045	0.112523	+	0.993649i	$0.464107\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−34.5871	−1.31958
$688$	0	0
$689$	16.9317	0.645048
$690$	0	0
$691$	30.6836	1.16726	0.583629	−	0.812021i	$-0.301632\pi$
$691$	30.6836	1.16726	0.583629	+	0.812021i	$0.301632\pi$
$692$	0	0
$693$	14.0390	0.533297
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−22.2483	−0.842714
$698$	0	0
$699$	13.4968	0.510497
$700$	0	0
$701$	12.1117	0.457451	0.228726	−	0.973491i	$-0.426544\pi$
$701$	12.1117	0.457451	0.228726	+	0.973491i	$0.426544\pi$
$702$	0	0
$703$	−7.66223	−0.288987
$704$	0	0
$705$	0	0
$706$	0	0
$707$	40.8814	1.53750
$708$	0	0
$709$	33.2904	1.25025	0.625124	−	0.780525i	$-0.285048\pi$
$709$	33.2904	1.25025	0.625124	+	0.780525i	$0.285048\pi$
$710$	0	0
$711$	−6.08475	−0.228196
$712$	0	0
$713$	54.7542	2.05056
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−18.8187	−0.702796
$718$	0	0
$719$	−10.7261	−0.400017	−0.200008	−	0.979794i	$-0.564097\pi$
$719$	−10.7261	−0.400017	−0.200008	+	0.979794i	$0.564097\pi$
$720$	0	0
$721$	24.3909	0.908366
$722$	0	0
$723$	−1.16596	−0.0433626
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−12.8826	−0.477789	−0.238894	−	0.971046i	$-0.576785\pi$
$727$	−12.8826	−0.477789	−0.238894	+	0.971046i	$0.576785\pi$
$728$	0	0
$729$	27.5939	1.02200
$730$	0	0
$731$	−22.3281	−0.825835
$732$	0	0
$733$	−37.9729	−1.40256	−0.701280	−	0.712886i	$-0.747388\pi$
$733$	−37.9729	−1.40256	−0.701280	+	0.712886i	$0.747388\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−13.7166	−0.505259
$738$	0	0
$739$	−42.3682	−1.55854	−0.779270	−	0.626688i	$-0.784410\pi$
$739$	−42.3682	−1.55854	−0.779270	+	0.626688i	$0.784410\pi$
$740$	0	0
$741$	17.7962	0.653762
$742$	0	0
$743$	26.3869	0.968043	0.484021	−	0.875056i	$-0.339176\pi$
$743$	26.3869	0.968043	0.484021	+	0.875056i	$0.339176\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	10.6369	0.389186
$748$	0	0
$749$	54.8899	2.00563
$750$	0	0
$751$	22.2028	0.810192	0.405096	−	0.914274i	$-0.367238\pi$
$751$	22.2028	0.810192	0.405096	+	0.914274i	$0.367238\pi$
$752$	0	0
$753$	14.3701	0.523676
$754$	0	0
$755$	0	0
$756$	0	0
$757$	25.1775	0.915093	0.457546	−	0.889186i	$-0.348728\pi$
$757$	25.1775	0.915093	0.457546	+	0.889186i	$0.348728\pi$
$758$	0	0
$759$	−47.2775	−1.71606
$760$	0	0
$761$	−46.5088	−1.68594	−0.842972	−	0.537958i	$-0.819196\pi$
$761$	−46.5088	−1.68594	−0.842972	+	0.537958i	$0.819196\pi$
$762$	0	0
$763$	17.5104	0.633918
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.65407	0.0958329
$768$	0	0
$769$	26.1946	0.944600	0.472300	−	0.881438i	$-0.343424\pi$
$769$	26.1946	0.944600	0.472300	+	0.881438i	$0.343424\pi$
$770$	0	0
$771$	21.7232	0.782342
$772$	0	0
$773$	18.2957	0.658049	0.329024	−	0.944321i	$-0.393280\pi$
$773$	18.2957	0.658049	0.329024	+	0.944321i	$0.393280\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−3.70759	−0.133009
$778$	0	0
$779$	−33.0571	−1.18439
$780$	0	0
$781$	−31.9038	−1.14161
$782$	0	0
$783$	−45.8394	−1.63817
$784$	0	0
$785$	0	0
$786$	0	0
$787$	5.40992	0.192843	0.0964213	−	0.995341i	$-0.469260\pi$
$787$	5.40992	0.192843	0.0964213	+	0.995341i	$0.469260\pi$
$788$	0	0
$789$	28.3838	1.01049
$790$	0	0
$791$	−25.1527	−0.894326
$792$	0	0
$793$	7.95624	0.282534
$794$	0	0
$795$	0	0
$796$	0	0
$797$	8.16219	0.289120	0.144560	−	0.989496i	$-0.453823\pi$
$797$	8.16219	0.289120	0.144560	+	0.989496i	$0.453823\pi$
$798$	0	0
$799$	−22.7975	−0.806518
$800$	0	0
$801$	2.16604	0.0765333
$802$	0	0
$803$	21.9536	0.774724
$804$	0	0
$805$	0	0
$806$	0	0
$807$	16.1882	0.569851
$808$	0	0
$809$	24.2261	0.851745	0.425872	−	0.904783i	$-0.359967\pi$
$809$	24.2261	0.851745	0.425872	+	0.904783i	$0.359967\pi$
$810$	0	0
$811$	2.36927	0.0831964	0.0415982	−	0.999134i	$-0.486755\pi$
$811$	2.36927	0.0831964	0.0415982	+	0.999134i	$0.486755\pi$
$812$	0	0
$813$	37.5334	1.31635
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−33.1757	−1.16067
$818$	0	0
$819$	−5.60698	−0.195924
$820$	0	0
$821$	−11.7721	−0.410850	−0.205425	−	0.978673i	$-0.565858\pi$
$821$	−11.7721	−0.410850	−0.205425	+	0.978673i	$0.565858\pi$
$822$	0	0
$823$	−14.6452	−0.510499	−0.255249	−	0.966875i	$-0.582158\pi$
$823$	−14.6452	−0.510499	−0.255249	+	0.966875i	$0.582158\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	45.2646	1.57400	0.787001	−	0.616951i	$-0.211632\pi$
$827$	45.2646	1.57400	0.787001	+	0.616951i	$0.211632\pi$
$828$	0	0
$829$	−34.0248	−1.18173	−0.590865	−	0.806770i	$-0.701214\pi$
$829$	−34.0248	−1.18173	−0.590865	+	0.806770i	$0.701214\pi$
$830$	0	0
$831$	9.36656	0.324922
$832$	0	0
$833$	−2.91659	−0.101054
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−37.9758	−1.31264
$838$	0	0
$839$	−37.8847	−1.30793	−0.653963	−	0.756527i	$-0.726895\pi$
$839$	−37.8847	−1.30793	−0.653963	+	0.756527i	$0.726895\pi$
$840$	0	0
$841$	37.0920	1.27903
$842$	0	0
$843$	8.80019	0.303095
$844$	0	0
$845$	0	0
$846$	0	0
$847$	20.9403	0.719517
$848$	0	0
$849$	−20.0471	−0.688016
$850$	0	0
$851$	−8.12970	−0.278682
$852$	0	0
$853$	6.38585	0.218647	0.109324	−	0.994006i	$-0.465132\pi$
$853$	6.38585	0.218647	0.109324	+	0.994006i	$0.465132\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−22.5675	−0.770890	−0.385445	−	0.922731i	$-0.625952\pi$
$857$	−22.5675	−0.770890	−0.385445	+	0.922731i	$0.625952\pi$
$858$	0	0
$859$	−12.9293	−0.441143	−0.220572	−	0.975371i	$-0.570792\pi$
$859$	−12.9293	−0.441143	−0.220572	+	0.975371i	$0.570792\pi$
$860$	0	0
$861$	−15.9956	−0.545129
$862$	0	0
$863$	56.6075	1.92694	0.963471	−	0.267813i	$-0.0863008\pi$
$863$	56.6075	1.92694	0.963471	+	0.267813i	$0.0863008\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−12.9314	−0.439174
$868$	0	0
$869$	−22.1895	−0.752726
$870$	0	0
$871$	5.47823	0.185623
$872$	0	0
$873$	−17.0267	−0.576268
$874$	0	0
$875$	0	0
$876$	0	0
$877$	45.8870	1.54949	0.774747	−	0.632271i	$-0.217877\pi$
$877$	45.8870	1.54949	0.774747	+	0.632271i	$0.217877\pi$
$878$	0	0
$879$	−3.85044	−0.129872
$880$	0	0
$881$	48.7211	1.64145	0.820727	−	0.571320i	$-0.193569\pi$
$881$	48.7211	1.64145	0.820727	+	0.571320i	$0.193569\pi$
$882$	0	0
$883$	19.5541	0.658049	0.329025	−	0.944321i	$-0.393280\pi$
$883$	19.5541	0.658049	0.329025	+	0.944321i	$0.393280\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1.99614	−0.0670238	−0.0335119	−	0.999438i	$-0.510669\pi$
$887$	−1.99614	−0.0670238	−0.0335119	+	0.999438i	$0.510669\pi$
$888$	0	0
$889$	−32.5075	−1.09027
$890$	0	0
$891$	17.4781	0.585538
$892$	0	0
$893$	−33.8731	−1.13352
$894$	0	0
$895$	0	0
$896$	0	0
$897$	18.8820	0.630451
$898$	0	0
$899$	54.7542	1.82615
$900$	0	0
$901$	−50.6742	−1.68820
$902$	0	0
$903$	−16.0530	−0.534211
$904$	0	0
$905$	0	0
$906$	0	0
$907$	38.3162	1.27227	0.636134	−	0.771578i	$-0.280532\pi$
$907$	38.3162	1.27227	0.636134	+	0.771578i	$0.280532\pi$
$908$	0	0
$909$	−17.5837	−0.583214
$910$	0	0
$911$	32.6766	1.08262	0.541312	−	0.840822i	$-0.317928\pi$
$911$	32.6766	1.08262	0.541312	+	0.840822i	$0.317928\pi$
$912$	0	0
$913$	38.7901	1.28377
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−41.2730	−1.36296
$918$	0	0
$919$	−45.4614	−1.49963	−0.749816	−	0.661646i	$-0.769858\pi$
$919$	−45.4614	−1.49963	−0.749816	+	0.661646i	$0.769858\pi$
$920$	0	0
$921$	5.30145	0.174689
$922$	0	0
$923$	12.7420	0.419407
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−10.4909	−0.344567
$928$	0	0
$929$	−45.1627	−1.48174	−0.740871	−	0.671648i	$-0.765587\pi$
$929$	−45.1627	−1.48174	−0.740871	+	0.671648i	$0.765587\pi$
$930$	0	0
$931$	−4.33354	−0.142026
$932$	0	0
$933$	−5.58786	−0.182938
$934$	0	0
$935$	0	0
$936$	0	0
$937$	26.8750	0.877967	0.438983	−	0.898495i	$-0.355339\pi$
$937$	26.8750	0.877967	0.438983	+	0.898495i	$0.355339\pi$
$938$	0	0
$939$	2.62177	0.0855583
$940$	0	0
$941$	−12.1286	−0.395381	−0.197690	−	0.980264i	$-0.563344\pi$
$941$	−12.1286	−0.395381	−0.197690	+	0.980264i	$0.563344\pi$
$942$	0	0
$943$	−35.0739	−1.14216
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−39.4063	−1.28053	−0.640266	−	0.768153i	$-0.721176\pi$
$947$	−39.4063	−1.28053	−0.640266	+	0.768153i	$0.721176\pi$
$948$	0	0
$949$	−8.76795	−0.284620
$950$	0	0
$951$	−37.6807	−1.22188
$952$	0	0
$953$	−4.36823	−0.141501	−0.0707504	−	0.997494i	$-0.522539\pi$
$953$	−4.36823	−0.141501	−0.0707504	+	0.997494i	$0.522539\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−47.2775	−1.52826
$958$	0	0
$959$	19.6252	0.633732
$960$	0	0
$961$	14.3613	0.463268
$962$	0	0
$963$	−23.6089	−0.760788
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−20.1859	−0.649135	−0.324567	−	0.945863i	$-0.605219\pi$
$967$	−20.1859	−0.649135	−0.324567	+	0.945863i	$0.605219\pi$
$968$	0	0
$969$	−53.2615	−1.71101
$970$	0	0
$971$	41.9791	1.34717	0.673587	−	0.739108i	$-0.264753\pi$
$971$	41.9791	1.34717	0.673587	+	0.739108i	$0.264753\pi$
$972$	0	0
$973$	−5.85060	−0.187562
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−28.0938	−0.898801	−0.449400	−	0.893330i	$-0.648362\pi$
$977$	−28.0938	−0.898801	−0.449400	+	0.893330i	$0.648362\pi$
$978$	0	0
$979$	7.89897	0.252452
$980$	0	0
$981$	−7.53147	−0.240461
$982$	0	0
$983$	43.8210	1.39767	0.698837	−	0.715281i	$-0.253701\pi$
$983$	43.8210	1.39767	0.698837	+	0.715281i	$0.253701\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−16.3905	−0.521714
$988$	0	0
$989$	−35.1997	−1.11929
$990$	0	0
$991$	6.04515	0.192031	0.0960153	−	0.995380i	$-0.469390\pi$
$991$	6.04515	0.192031	0.0960153	+	0.995380i	$0.469390\pi$
$992$	0	0
$993$	4.91986	0.156127
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−30.4559	−0.964548	−0.482274	−	0.876020i	$-0.660189\pi$
$997$	−30.4559	−0.964548	−0.482274	+	0.876020i	$0.660189\pi$
$998$	0	0
$999$	5.63851	0.178395

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7400.2.a.r.1.1		5
5.4	even	2		1480.2.a.g.1.5	✓	5
20.19	odd	2		2960.2.a.bb.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1480.2.a.g.1.5	✓	5	5.4	even	2
2960.2.a.bb.1.1		5	20.19	odd	2
7400.2.a.r.1.1		5	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7400 = 2^{3} \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7400.a (trivial)

\(f(q)\)	\(=\)	\(q-1.34794 q^{3} +2.75056 q^{7} -1.18306 q^{9} +O(q^{10})\)	\(q-1.34794 q^{3} +2.75056 q^{7} -1.18306 q^{9} -4.31429 q^{11} +1.72307 q^{13} -5.15688 q^{17} -7.66223 q^{19} -3.70759 q^{21} -8.12970 q^{23} +5.63851 q^{27} -8.12970 q^{29} -6.73508 q^{31} +5.81541 q^{33} +1.00000 q^{37} -2.32259 q^{39} +4.31429 q^{41} +4.32977 q^{43} +4.42079 q^{47} +0.565572 q^{49} +6.95117 q^{51} +9.82650 q^{53} +10.3282 q^{57} +1.54032 q^{59} +4.61748 q^{61} -3.25406 q^{63} +3.17935 q^{67} +10.9584 q^{69} +7.39492 q^{71} -5.08857 q^{73} -11.8667 q^{77} +5.14325 q^{79} -4.05121 q^{81} -8.99108 q^{83} +10.9584 q^{87} -1.83089 q^{89} +4.73940 q^{91} +9.07849 q^{93} +14.3922 q^{97} +5.10405 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{3} + 5 q^{7} + 6 q^{9}+O(q^{10}) \) 5 * q + 5 * q^3 + 5 * q^7 + 6 * q^9	\( 5 q + 5 q^{3} + 5 q^{7} + 6 q^{9} - 7 q^{11} + 6 q^{13} - 12 q^{19} - q^{21} + 6 q^{23} + 17 q^{27} + 6 q^{29} - 10 q^{31} - 3 q^{33} + 5 q^{37} + 4 q^{39} + 7 q^{41} + 22 q^{43} + 13 q^{47} + 14 q^{49} - 10 q^{51} + 11 q^{53} + 8 q^{57} - 10 q^{59} + 10 q^{63} + 24 q^{67} + 26 q^{69} - 13 q^{71} + 13 q^{73} + 19 q^{77} + 2 q^{79} + q^{81} + 13 q^{83} + 26 q^{87} + 8 q^{89} - 8 q^{91} + 20 q^{93} + 20 q^{97} + 2 q^{99}+O(q^{100}) \) 5 * q + 5 * q^3 + 5 * q^7 + 6 * q^9 - 7 * q^11 + 6 * q^13 - 12 * q^19 - q^21 + 6 * q^23 + 17 * q^27 + 6 * q^29 - 10 * q^31 - 3 * q^33 + 5 * q^37 + 4 * q^39 + 7 * q^41 + 22 * q^43 + 13 * q^47 + 14 * q^49 - 10 * q^51 + 11 * q^53 + 8 * q^57 - 10 * q^59 + 10 * q^63 + 24 * q^67 + 26 * q^69 - 13 * q^71 + 13 * q^73 + 19 * q^77 + 2 * q^79 + q^81 + 13 * q^83 + 26 * q^87 + 8 * q^89 - 8 * q^91 + 20 * q^93 + 20 * q^97 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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