LMFDB - Embedded newform 7623.2.a.z.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7623,2,Mod(1,7623)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7623, 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("7623.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7623 = 3^{2} \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7623.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:	$60.8699614608$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	7623.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.61803 q^{2} +0.618034 q^{4} -0.618034 q^{5} +1.00000 q^{7} +2.23607 q^{8} +O(q^{10})$	$q-1.61803 q^{2} +0.618034 q^{4} -0.618034 q^{5} +1.00000 q^{7} +2.23607 q^{8} +1.00000 q^{10} +0.236068 q^{13} -1.61803 q^{14} -4.85410 q^{16} +5.47214 q^{17} +2.47214 q^{19} -0.381966 q^{20} +7.70820 q^{23} -4.61803 q^{25} -0.381966 q^{26} +0.618034 q^{28} +4.23607 q^{29} +3.00000 q^{31} +3.38197 q^{32} -8.85410 q^{34} -0.618034 q^{35} +6.00000 q^{37} -4.00000 q^{38} -1.38197 q^{40} -3.14590 q^{41} +2.52786 q^{43} -12.4721 q^{46} +11.8541 q^{47} +1.00000 q^{49} +7.47214 q^{50} +0.145898 q^{52} -7.56231 q^{53} +2.23607 q^{56} -6.85410 q^{58} +2.38197 q^{59} -9.94427 q^{61} -4.85410 q^{62} +4.23607 q^{64} -0.145898 q^{65} +8.76393 q^{67} +3.38197 q^{68} +1.00000 q^{70} +13.4721 q^{71} -9.61803 q^{73} -9.70820 q^{74} +1.52786 q^{76} -5.09017 q^{79} +3.00000 q^{80} +5.09017 q^{82} -0.472136 q^{83} -3.38197 q^{85} -4.09017 q^{86} +14.3262 q^{89} +0.236068 q^{91} +4.76393 q^{92} -19.1803 q^{94} -1.52786 q^{95} -18.7082 q^{97} -1.61803 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - q^{4} + q^{5} + 2 q^{7}+O(q^{10}) $ 2 * q - q^2 - q^4 + q^5 + 2 * q^7	$ 2 q - q^{2} - q^{4} + q^{5} + 2 q^{7} + 2 q^{10} - 4 q^{13} - q^{14} - 3 q^{16} + 2 q^{17} - 4 q^{19} - 3 q^{20} + 2 q^{23} - 7 q^{25} - 3 q^{26} - q^{28} + 4 q^{29} + 6 q^{31} + 9 q^{32} - 11 q^{34} + q^{35} + 12 q^{37} - 8 q^{38} - 5 q^{40} - 13 q^{41} + 14 q^{43} - 16 q^{46} + 17 q^{47} + 2 q^{49} + 6 q^{50} + 7 q^{52} + 5 q^{53} - 7 q^{58} + 7 q^{59} - 2 q^{61} - 3 q^{62} + 4 q^{64} - 7 q^{65} + 22 q^{67} + 9 q^{68} + 2 q^{70} + 18 q^{71} - 17 q^{73} - 6 q^{74} + 12 q^{76} + q^{79} + 6 q^{80} - q^{82} + 8 q^{83} - 9 q^{85} + 3 q^{86} + 13 q^{89} - 4 q^{91} + 14 q^{92} - 16 q^{94} - 12 q^{95} - 24 q^{97} - q^{98}+O(q^{100}) $ 2 * q - q^2 - q^4 + q^5 + 2 * q^7 + 2 * q^10 - 4 * q^13 - q^14 - 3 * q^16 + 2 * q^17 - 4 * q^19 - 3 * q^20 + 2 * q^23 - 7 * q^25 - 3 * q^26 - q^28 + 4 * q^29 + 6 * q^31 + 9 * q^32 - 11 * q^34 + q^35 + 12 * q^37 - 8 * q^38 - 5 * q^40 - 13 * q^41 + 14 * q^43 - 16 * q^46 + 17 * q^47 + 2 * q^49 + 6 * q^50 + 7 * q^52 + 5 * q^53 - 7 * q^58 + 7 * q^59 - 2 * q^61 - 3 * q^62 + 4 * q^64 - 7 * q^65 + 22 * q^67 + 9 * q^68 + 2 * q^70 + 18 * q^71 - 17 * q^73 - 6 * q^74 + 12 * q^76 + q^79 + 6 * q^80 - q^82 + 8 * q^83 - 9 * q^85 + 3 * q^86 + 13 * q^89 - 4 * q^91 + 14 * q^92 - 16 * q^94 - 12 * q^95 - 24 * q^97 - q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.61803	−1.14412	−0.572061	−	0.820211i	$-0.693856\pi$
$2$	−1.61803	−1.14412	−0.572061	+	0.820211i	$0.693856\pi$
$3$	0	0
$3$	0	0
$4$	0.618034	0.309017
$5$	−0.618034	−0.276393	−0.138197	−	0.990405i	$-0.544131\pi$
$5$	−0.618034	−0.276393	−0.138197	+	0.990405i	$0.544131\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.23607	0.790569
$9$	0	0
$10$	1.00000	0.316228
$11$	0	0
$11$	0	0
$12$	0	0
$13$	0.236068	0.0654735	0.0327367	−	0.999464i	$-0.489578\pi$
$13$	0.236068	0.0654735	0.0327367	+	0.999464i	$0.489578\pi$
$14$	−1.61803	−0.432438
$15$	0	0
$16$	−4.85410	−1.21353
$17$	5.47214	1.32719	0.663594	−	0.748093i	$-0.269030\pi$
$17$	5.47214	1.32719	0.663594	+	0.748093i	$0.269030\pi$
$18$	0	0
$19$	2.47214	0.567147	0.283573	−	0.958951i	$-0.408480\pi$
$19$	2.47214	0.567147	0.283573	+	0.958951i	$0.408480\pi$
$20$	−0.381966	−0.0854102
$21$	0	0
$22$	0	0
$23$	7.70820	1.60727	0.803636	−	0.595121i	$-0.202896\pi$
$23$	7.70820	1.60727	0.803636	+	0.595121i	$0.202896\pi$
$24$	0	0
$25$	−4.61803	−0.923607
$26$	−0.381966	−0.0749097
$27$	0	0
$28$	0.618034	0.116797
$29$	4.23607	0.786618	0.393309	−	0.919406i	$-0.371330\pi$
$29$	4.23607	0.786618	0.393309	+	0.919406i	$0.371330\pi$
$30$	0	0
$31$	3.00000	0.538816	0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000	0.538816	0.269408	+	0.963026i	$0.413172\pi$
$32$	3.38197	0.597853
$33$	0	0
$34$	−8.85410	−1.51847
$35$	−0.618034	−0.104467
$36$	0	0
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	−4.00000	−0.648886
$39$	0	0
$40$	−1.38197	−0.218508
$41$	−3.14590	−0.491307	−0.245653	−	0.969358i	$-0.579002\pi$
$41$	−3.14590	−0.491307	−0.245653	+	0.969358i	$0.579002\pi$
$42$	0	0
$43$	2.52786	0.385496	0.192748	−	0.981248i	$-0.438260\pi$
$43$	2.52786	0.385496	0.192748	+	0.981248i	$0.438260\pi$
$44$	0	0
$45$	0	0
$46$	−12.4721	−1.83892
$47$	11.8541	1.72910	0.864549	−	0.502548i	$-0.167604\pi$
$47$	11.8541	1.72910	0.864549	+	0.502548i	$0.167604\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	7.47214	1.05672
$51$	0	0
$52$	0.145898	0.0202324
$53$	−7.56231	−1.03876	−0.519381	−	0.854543i	$-0.673838\pi$
$53$	−7.56231	−1.03876	−0.519381	+	0.854543i	$0.673838\pi$
$54$	0	0
$55$	0	0
$56$	2.23607	0.298807
$57$	0	0
$58$	−6.85410	−0.899988
$59$	2.38197	0.310106	0.155053	−	0.987906i	$-0.450445\pi$
$59$	2.38197	0.310106	0.155053	+	0.987906i	$0.450445\pi$
$60$	0	0
$61$	−9.94427	−1.27323	−0.636617	−	0.771180i	$-0.719667\pi$
$61$	−9.94427	−1.27323	−0.636617	+	0.771180i	$0.719667\pi$
$62$	−4.85410	−0.616472
$63$	0	0
$64$	4.23607	0.529508
$65$	−0.145898	−0.0180964
$66$	0	0
$67$	8.76393	1.07068	0.535342	−	0.844635i	$-0.320183\pi$
$67$	8.76393	1.07068	0.535342	+	0.844635i	$0.320183\pi$
$68$	3.38197	0.410124
$69$	0	0
$70$	1.00000	0.119523
$71$	13.4721	1.59885	0.799424	−	0.600767i	$-0.205138\pi$
$71$	13.4721	1.59885	0.799424	+	0.600767i	$0.205138\pi$
$72$	0	0
$73$	−9.61803	−1.12571	−0.562853	−	0.826557i	$-0.690296\pi$
$73$	−9.61803	−1.12571	−0.562853	+	0.826557i	$0.690296\pi$
$74$	−9.70820	−1.12856
$75$	0	0
$76$	1.52786	0.175258
$77$	0	0
$78$	0	0
$79$	−5.09017	−0.572689	−0.286344	−	0.958127i	$-0.592440\pi$
$79$	−5.09017	−0.572689	−0.286344	+	0.958127i	$0.592440\pi$
$80$	3.00000	0.335410
$81$	0	0
$82$	5.09017	0.562115
$83$	−0.472136	−0.0518237	−0.0259118	−	0.999664i	$-0.508249\pi$
$83$	−0.472136	−0.0518237	−0.0259118	+	0.999664i	$0.508249\pi$
$84$	0	0
$85$	−3.38197	−0.366826
$86$	−4.09017	−0.441054
$87$	0	0
$88$	0	0
$89$	14.3262	1.51858	0.759289	−	0.650753i	$-0.225547\pi$
$89$	14.3262	1.51858	0.759289	+	0.650753i	$0.225547\pi$
$90$	0	0
$91$	0.236068	0.0247466
$92$	4.76393	0.496674
$93$	0	0
$94$	−19.1803	−1.97830
$95$	−1.52786	−0.156756
$96$	0	0
$97$	−18.7082	−1.89953	−0.949765	−	0.312963i	$-0.898678\pi$
$97$	−18.7082	−1.89953	−0.949765	+	0.312963i	$0.898678\pi$
$98$	−1.61803	−0.163446
$99$	0	0
$100$	−2.85410	−0.285410
$101$	6.76393	0.673036	0.336518	−	0.941677i	$-0.390751\pi$
$101$	6.76393	0.673036	0.336518	+	0.941677i	$0.390751\pi$
$102$	0	0
$103$	−6.52786	−0.643210	−0.321605	−	0.946874i	$-0.604222\pi$
$103$	−6.52786	−0.643210	−0.321605	+	0.946874i	$0.604222\pi$
$104$	0.527864	0.0517613
$105$	0	0
$106$	12.2361	1.18847
$107$	10.2361	0.989558	0.494779	−	0.869019i	$-0.335249\pi$
$107$	10.2361	0.989558	0.494779	+	0.869019i	$0.335249\pi$
$108$	0	0
$109$	10.8541	1.03963	0.519817	−	0.854278i	$-0.326000\pi$
$109$	10.8541	1.03963	0.519817	+	0.854278i	$0.326000\pi$
$110$	0	0
$111$	0	0
$112$	−4.85410	−0.458670
$113$	−11.1803	−1.05176	−0.525879	−	0.850559i	$-0.676264\pi$
$113$	−11.1803	−1.05176	−0.525879	+	0.850559i	$0.676264\pi$
$114$	0	0
$115$	−4.76393	−0.444239
$116$	2.61803	0.243078
$117$	0	0
$118$	−3.85410	−0.354799
$119$	5.47214	0.501630
$120$	0	0
$121$	0	0
$122$	16.0902	1.45674
$123$	0	0
$124$	1.85410	0.166503
$125$	5.94427	0.531672
$126$	0	0
$127$	−8.32624	−0.738834	−0.369417	−	0.929264i	$-0.620443\pi$
$127$	−8.32624	−0.738834	−0.369417	+	0.929264i	$0.620443\pi$
$128$	−13.6180	−1.20368
$129$	0	0
$130$	0.236068	0.0207045
$131$	0.145898	0.0127472	0.00637359	−	0.999980i	$-0.497971\pi$
$131$	0.145898	0.0127472	0.00637359	+	0.999980i	$0.497971\pi$
$132$	0	0
$133$	2.47214	0.214361
$134$	−14.1803	−1.22499
$135$	0	0
$136$	12.2361	1.04923
$137$	1.52786	0.130534	0.0652671	−	0.997868i	$-0.479210\pi$
$137$	1.52786	0.130534	0.0652671	+	0.997868i	$0.479210\pi$
$138$	0	0
$139$	−0.381966	−0.0323979	−0.0161990	−	0.999869i	$-0.505157\pi$
$139$	−0.381966	−0.0323979	−0.0161990	+	0.999869i	$0.505157\pi$
$140$	−0.381966	−0.0322820
$141$	0	0
$142$	−21.7984	−1.82928
$143$	0	0
$144$	0	0
$145$	−2.61803	−0.217416
$146$	15.5623	1.28795
$147$	0	0
$148$	3.70820	0.304812
$149$	−3.61803	−0.296401	−0.148200	−	0.988957i	$-0.547348\pi$
$149$	−3.61803	−0.296401	−0.148200	+	0.988957i	$0.547348\pi$
$150$	0	0
$151$	9.32624	0.758958	0.379479	−	0.925200i	$-0.376103\pi$
$151$	9.32624	0.758958	0.379479	+	0.925200i	$0.376103\pi$
$152$	5.52786	0.448369
$153$	0	0
$154$	0	0
$155$	−1.85410	−0.148925
$156$	0	0
$157$	17.7984	1.42046	0.710232	−	0.703967i	$-0.248590\pi$
$157$	17.7984	1.42046	0.710232	+	0.703967i	$0.248590\pi$
$158$	8.23607	0.655226
$159$	0	0
$160$	−2.09017	−0.165242
$161$	7.70820	0.607492
$162$	0	0
$163$	−2.67376	−0.209425	−0.104713	−	0.994503i	$-0.533392\pi$
$163$	−2.67376	−0.209425	−0.104713	+	0.994503i	$0.533392\pi$
$164$	−1.94427	−0.151822
$165$	0	0
$166$	0.763932	0.0592926
$167$	−8.85410	−0.685151	−0.342575	−	0.939490i	$-0.611299\pi$
$167$	−8.85410	−0.685151	−0.342575	+	0.939490i	$0.611299\pi$
$168$	0	0
$169$	−12.9443	−0.995713
$170$	5.47214	0.419694
$171$	0	0
$172$	1.56231	0.119125
$173$	−13.7984	−1.04907	−0.524535	−	0.851389i	$-0.675761\pi$
$173$	−13.7984	−1.04907	−0.524535	+	0.851389i	$0.675761\pi$
$174$	0	0
$175$	−4.61803	−0.349091
$176$	0	0
$177$	0	0
$178$	−23.1803	−1.73744
$179$	5.29180	0.395527	0.197764	−	0.980250i	$-0.436632\pi$
$179$	5.29180	0.395527	0.197764	+	0.980250i	$0.436632\pi$
$180$	0	0
$181$	−9.47214	−0.704058	−0.352029	−	0.935989i	$-0.614508\pi$
$181$	−9.47214	−0.704058	−0.352029	+	0.935989i	$0.614508\pi$
$182$	−0.381966	−0.0283132
$183$	0	0
$184$	17.2361	1.27066
$185$	−3.70820	−0.272633
$186$	0	0
$187$	0	0
$188$	7.32624	0.534321
$189$	0	0
$190$	2.47214	0.179348
$191$	11.7639	0.851208	0.425604	−	0.904909i	$-0.360062\pi$
$191$	11.7639	0.851208	0.425604	+	0.904909i	$0.360062\pi$
$192$	0	0
$193$	9.41641	0.677808	0.338904	−	0.940821i	$-0.389944\pi$
$193$	9.41641	0.677808	0.338904	+	0.940821i	$0.389944\pi$
$194$	30.2705	2.17330
$195$	0	0
$196$	0.618034	0.0441453
$197$	−26.6525	−1.89891	−0.949455	−	0.313903i	$-0.898363\pi$
$197$	−26.6525	−1.89891	−0.949455	+	0.313903i	$0.898363\pi$
$198$	0	0
$199$	25.2148	1.78743	0.893714	−	0.448637i	$-0.148090\pi$
$199$	25.2148	1.78743	0.893714	+	0.448637i	$0.148090\pi$
$200$	−10.3262	−0.730175
$201$	0	0
$202$	−10.9443	−0.770036
$203$	4.23607	0.297314
$204$	0	0
$205$	1.94427	0.135794
$206$	10.5623	0.735911
$207$	0	0
$208$	−1.14590	−0.0794537
$209$	0	0
$210$	0	0
$211$	5.29180	0.364302	0.182151	−	0.983271i	$-0.441694\pi$
$211$	5.29180	0.364302	0.182151	+	0.983271i	$0.441694\pi$
$212$	−4.67376	−0.320995
$213$	0	0
$214$	−16.5623	−1.13218
$215$	−1.56231	−0.106548
$216$	0	0
$217$	3.00000	0.203653
$218$	−17.5623	−1.18947
$219$	0	0
$220$	0	0
$221$	1.29180	0.0868956
$222$	0	0
$223$	2.05573	0.137662	0.0688309	−	0.997628i	$-0.478073\pi$
$223$	2.05573	0.137662	0.0688309	+	0.997628i	$0.478073\pi$
$224$	3.38197	0.225967
$225$	0	0
$226$	18.0902	1.20334
$227$	−12.0902	−0.802453	−0.401226	−	0.915979i	$-0.631416\pi$
$227$	−12.0902	−0.802453	−0.401226	+	0.915979i	$0.631416\pi$
$228$	0	0
$229$	−25.1246	−1.66028	−0.830141	−	0.557554i	$-0.811740\pi$
$229$	−25.1246	−1.66028	−0.830141	+	0.557554i	$0.811740\pi$
$230$	7.70820	0.508264
$231$	0	0
$232$	9.47214	0.621876
$233$	−20.2361	−1.32571	−0.662854	−	0.748748i	$-0.730655\pi$
$233$	−20.2361	−1.32571	−0.662854	+	0.748748i	$0.730655\pi$
$234$	0	0
$235$	−7.32624	−0.477911
$236$	1.47214	0.0958279
$237$	0	0
$238$	−8.85410	−0.573926
$239$	−19.3820	−1.25372	−0.626858	−	0.779134i	$-0.715659\pi$
$239$	−19.3820	−1.25372	−0.626858	+	0.779134i	$0.715659\pi$
$240$	0	0
$241$	16.4164	1.05747	0.528737	−	0.848786i	$-0.322666\pi$
$241$	16.4164	1.05747	0.528737	+	0.848786i	$0.322666\pi$
$242$	0	0
$243$	0	0
$244$	−6.14590	−0.393451
$245$	−0.618034	−0.0394847
$246$	0	0
$247$	0.583592	0.0371331
$248$	6.70820	0.425971
$249$	0	0
$250$	−9.61803	−0.608298
$251$	−3.94427	−0.248960	−0.124480	−	0.992222i	$-0.539726\pi$
$251$	−3.94427	−0.248960	−0.124480	+	0.992222i	$0.539726\pi$
$252$	0	0
$253$	0	0
$254$	13.4721	0.845317
$255$	0	0
$256$	13.5623	0.847644
$257$	11.8885	0.741587	0.370793	−	0.928715i	$-0.379086\pi$
$257$	11.8885	0.741587	0.370793	+	0.928715i	$0.379086\pi$
$258$	0	0
$259$	6.00000	0.372822
$260$	−0.0901699	−0.00559210
$261$	0	0
$262$	−0.236068	−0.0145843
$263$	−22.7082	−1.40025	−0.700124	−	0.714021i	$-0.746872\pi$
$263$	−22.7082	−1.40025	−0.700124	+	0.714021i	$0.746872\pi$
$264$	0	0
$265$	4.67376	0.287107
$266$	−4.00000	−0.245256
$267$	0	0
$268$	5.41641	0.330860
$269$	6.81966	0.415802	0.207901	−	0.978150i	$-0.433337\pi$
$269$	6.81966	0.415802	0.207901	+	0.978150i	$0.433337\pi$
$270$	0	0
$271$	−18.1246	−1.10099	−0.550496	−	0.834838i	$-0.685561\pi$
$271$	−18.1246	−1.10099	−0.550496	+	0.834838i	$0.685561\pi$
$272$	−26.5623	−1.61058
$273$	0	0
$274$	−2.47214	−0.149347
$275$	0	0
$276$	0	0
$277$	0.472136	0.0283679	0.0141840	−	0.999899i	$-0.495485\pi$
$277$	0.472136	0.0283679	0.0141840	+	0.999899i	$0.495485\pi$
$278$	0.618034	0.0370672
$279$	0	0
$280$	−1.38197	−0.0825883
$281$	4.14590	0.247324	0.123662	−	0.992324i	$-0.460536\pi$
$281$	4.14590	0.247324	0.123662	+	0.992324i	$0.460536\pi$
$282$	0	0
$283$	22.8885	1.36058	0.680291	−	0.732942i	$-0.261853\pi$
$283$	22.8885	1.36058	0.680291	+	0.732942i	$0.261853\pi$
$284$	8.32624	0.494071
$285$	0	0
$286$	0	0
$287$	−3.14590	−0.185696
$288$	0	0
$289$	12.9443	0.761428
$290$	4.23607	0.248750
$291$	0	0
$292$	−5.94427	−0.347862
$293$	5.70820	0.333477	0.166738	−	0.986001i	$-0.446676\pi$
$293$	5.70820	0.333477	0.166738	+	0.986001i	$0.446676\pi$
$294$	0	0
$295$	−1.47214	−0.0857111
$296$	13.4164	0.779813
$297$	0	0
$298$	5.85410	0.339119
$299$	1.81966	0.105234
$300$	0	0
$301$	2.52786	0.145704
$302$	−15.0902	−0.868342
$303$	0	0
$304$	−12.0000	−0.688247
$305$	6.14590	0.351913
$306$	0	0
$307$	−25.9443	−1.48072	−0.740359	−	0.672212i	$-0.765344\pi$
$307$	−25.9443	−1.48072	−0.740359	+	0.672212i	$0.765344\pi$
$308$	0	0
$309$	0	0
$310$	3.00000	0.170389
$311$	17.7984	1.00925	0.504627	−	0.863338i	$-0.331630\pi$
$311$	17.7984	1.00925	0.504627	+	0.863338i	$0.331630\pi$
$312$	0	0
$313$	20.9098	1.18189	0.590947	−	0.806711i	$-0.298754\pi$
$313$	20.9098	1.18189	0.590947	+	0.806711i	$0.298754\pi$
$314$	−28.7984	−1.62519
$315$	0	0
$316$	−3.14590	−0.176971
$317$	−0.888544	−0.0499056	−0.0249528	−	0.999689i	$-0.507944\pi$
$317$	−0.888544	−0.0499056	−0.0249528	+	0.999689i	$0.507944\pi$
$318$	0	0
$319$	0	0
$320$	−2.61803	−0.146353
$321$	0	0
$322$	−12.4721	−0.695045
$323$	13.5279	0.752710
$324$	0	0
$325$	−1.09017	−0.0604717
$326$	4.32624	0.239608
$327$	0	0
$328$	−7.03444	−0.388412
$329$	11.8541	0.653538
$330$	0	0
$331$	−32.7082	−1.79781	−0.898903	−	0.438148i	$-0.855635\pi$
$331$	−32.7082	−1.79781	−0.898903	+	0.438148i	$0.855635\pi$
$332$	−0.291796	−0.0160144
$333$	0	0
$334$	14.3262	0.783897
$335$	−5.41641	−0.295930
$336$	0	0
$337$	−0.236068	−0.0128594	−0.00642972	−	0.999979i	$-0.502047\pi$
$337$	−0.236068	−0.0128594	−0.00642972	+	0.999979i	$0.502047\pi$
$338$	20.9443	1.13922
$339$	0	0
$340$	−2.09017	−0.113355
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	5.65248	0.304761
$345$	0	0
$346$	22.3262	1.20027
$347$	8.47214	0.454808	0.227404	−	0.973800i	$-0.426976\pi$
$347$	8.47214	0.454808	0.227404	+	0.973800i	$0.426976\pi$
$348$	0	0
$349$	1.67376	0.0895944	0.0447972	−	0.998996i	$-0.485736\pi$
$349$	1.67376	0.0895944	0.0447972	+	0.998996i	$0.485736\pi$
$350$	7.47214	0.399402
$351$	0	0
$352$	0	0
$353$	−36.1246	−1.92272	−0.961360	−	0.275296i	$-0.911224\pi$
$353$	−36.1246	−1.92272	−0.961360	+	0.275296i	$0.911224\pi$
$354$	0	0
$355$	−8.32624	−0.441911
$356$	8.85410	0.469266
$357$	0	0
$358$	−8.56231	−0.452532
$359$	12.6180	0.665954	0.332977	−	0.942935i	$-0.391947\pi$
$359$	12.6180	0.665954	0.332977	+	0.942935i	$0.391947\pi$
$360$	0	0
$361$	−12.8885	−0.678344
$362$	15.3262	0.805529
$363$	0	0
$364$	0.145898	0.00764713
$365$	5.94427	0.311137
$366$	0	0
$367$	−5.00000	−0.260998	−0.130499	−	0.991448i	$-0.541658\pi$
$367$	−5.00000	−0.260998	−0.130499	+	0.991448i	$0.541658\pi$
$368$	−37.4164	−1.95047
$369$	0	0
$370$	6.00000	0.311925
$371$	−7.56231	−0.392615
$372$	0	0
$373$	33.7082	1.74534	0.872672	−	0.488306i	$-0.162385\pi$
$373$	33.7082	1.74534	0.872672	+	0.488306i	$0.162385\pi$
$374$	0	0
$375$	0	0
$376$	26.5066	1.36697
$377$	1.00000	0.0515026
$378$	0	0
$379$	−13.0902	−0.672397	−0.336198	−	0.941791i	$-0.609141\pi$
$379$	−13.0902	−0.672397	−0.336198	+	0.941791i	$0.609141\pi$
$380$	−0.944272	−0.0484401
$381$	0	0
$382$	−19.0344	−0.973887
$383$	18.3262	0.936427	0.468214	−	0.883615i	$-0.344898\pi$
$383$	18.3262	0.936427	0.468214	+	0.883615i	$0.344898\pi$
$384$	0	0
$385$	0	0
$386$	−15.2361	−0.775495
$387$	0	0
$388$	−11.5623	−0.586987
$389$	12.7639	0.647157	0.323579	−	0.946201i	$-0.395114\pi$
$389$	12.7639	0.647157	0.323579	+	0.946201i	$0.395114\pi$
$390$	0	0
$391$	42.1803	2.13315
$392$	2.23607	0.112938
$393$	0	0
$394$	43.1246	2.17259
$395$	3.14590	0.158287
$396$	0	0
$397$	9.56231	0.479918	0.239959	−	0.970783i	$-0.422866\pi$
$397$	9.56231	0.479918	0.239959	+	0.970783i	$0.422866\pi$
$398$	−40.7984	−2.04504
$399$	0	0
$400$	22.4164	1.12082
$401$	14.7426	0.736213	0.368106	−	0.929784i	$-0.380006\pi$
$401$	14.7426	0.736213	0.368106	+	0.929784i	$0.380006\pi$
$402$	0	0
$403$	0.708204	0.0352782
$404$	4.18034	0.207980
$405$	0	0
$406$	−6.85410	−0.340163
$407$	0	0
$408$	0	0
$409$	−32.7984	−1.62178	−0.810888	−	0.585202i	$-0.801015\pi$
$409$	−32.7984	−1.62178	−0.810888	+	0.585202i	$0.801015\pi$
$410$	−3.14590	−0.155365
$411$	0	0
$412$	−4.03444	−0.198763
$413$	2.38197	0.117209
$414$	0	0
$415$	0.291796	0.0143237
$416$	0.798374	0.0391435
$417$	0	0
$418$	0	0
$419$	33.3820	1.63082	0.815408	−	0.578887i	$-0.196513\pi$
$419$	33.3820	1.63082	0.815408	+	0.578887i	$0.196513\pi$
$420$	0	0
$421$	24.8541	1.21131	0.605657	−	0.795726i	$-0.292910\pi$
$421$	24.8541	1.21131	0.605657	+	0.795726i	$0.292910\pi$
$422$	−8.56231	−0.416807
$423$	0	0
$424$	−16.9098	−0.821214
$425$	−25.2705	−1.22580
$426$	0	0
$427$	−9.94427	−0.481237
$428$	6.32624	0.305790
$429$	0	0
$430$	2.52786	0.121904
$431$	−13.8541	−0.667329	−0.333664	−	0.942692i	$-0.608285\pi$
$431$	−13.8541	−0.667329	−0.333664	+	0.942692i	$0.608285\pi$
$432$	0	0
$433$	10.9443	0.525948	0.262974	−	0.964803i	$-0.415297\pi$
$433$	10.9443	0.525948	0.262974	+	0.964803i	$0.415297\pi$
$434$	−4.85410	−0.233004
$435$	0	0
$436$	6.70820	0.321265
$437$	19.0557	0.911559
$438$	0	0
$439$	−4.34752	−0.207496	−0.103748	−	0.994604i	$-0.533084\pi$
$439$	−4.34752	−0.207496	−0.103748	+	0.994604i	$0.533084\pi$
$440$	0	0
$441$	0	0
$442$	−2.09017	−0.0994192
$443$	6.65248	0.316069	0.158034	−	0.987434i	$-0.449484\pi$
$443$	6.65248	0.316069	0.158034	+	0.987434i	$0.449484\pi$
$444$	0	0
$445$	−8.85410	−0.419725
$446$	−3.32624	−0.157502
$447$	0	0
$448$	4.23607	0.200135
$449$	−28.5967	−1.34956	−0.674782	−	0.738017i	$-0.735762\pi$
$449$	−28.5967	−1.34956	−0.674782	+	0.738017i	$0.735762\pi$
$450$	0	0
$451$	0	0
$452$	−6.90983	−0.325011
$453$	0	0
$454$	19.5623	0.918105
$455$	−0.145898	−0.00683981
$456$	0	0
$457$	23.7426	1.11063	0.555317	−	0.831639i	$-0.312597\pi$
$457$	23.7426	1.11063	0.555317	+	0.831639i	$0.312597\pi$
$458$	40.6525	1.89957
$459$	0	0
$460$	−2.94427	−0.137277
$461$	−34.3050	−1.59774	−0.798870	−	0.601503i	$-0.794569\pi$
$461$	−34.3050	−1.59774	−0.798870	+	0.601503i	$0.794569\pi$
$462$	0	0
$463$	31.9787	1.48618	0.743088	−	0.669193i	$-0.233360\pi$
$463$	31.9787	1.48618	0.743088	+	0.669193i	$0.233360\pi$
$464$	−20.5623	−0.954581
$465$	0	0
$466$	32.7426	1.51677
$467$	13.2918	0.615071	0.307535	−	0.951537i	$-0.400496\pi$
$467$	13.2918	0.615071	0.307535	+	0.951537i	$0.400496\pi$
$468$	0	0
$469$	8.76393	0.404681
$470$	11.8541	0.546789
$471$	0	0
$472$	5.32624	0.245160
$473$	0	0
$474$	0	0
$475$	−11.4164	−0.523821
$476$	3.38197	0.155012
$477$	0	0
$478$	31.3607	1.43440
$479$	32.6180	1.49036	0.745178	−	0.666866i	$-0.232364\pi$
$479$	32.6180	1.49036	0.745178	+	0.666866i	$0.232364\pi$
$480$	0	0
$481$	1.41641	0.0645826
$482$	−26.5623	−1.20988
$483$	0	0
$484$	0	0
$485$	11.5623	0.525017
$486$	0	0
$487$	−17.6180	−0.798349	−0.399175	−	0.916875i	$-0.630703\pi$
$487$	−17.6180	−0.798349	−0.399175	+	0.916875i	$0.630703\pi$
$488$	−22.2361	−1.00658
$489$	0	0
$490$	1.00000	0.0451754
$491$	−21.7082	−0.979678	−0.489839	−	0.871813i	$-0.662944\pi$
$491$	−21.7082	−0.979678	−0.489839	+	0.871813i	$0.662944\pi$
$492$	0	0
$493$	23.1803	1.04399
$494$	−0.944272	−0.0424848
$495$	0	0
$496$	−14.5623	−0.653867
$497$	13.4721	0.604308
$498$	0	0
$499$	7.00000	0.313363	0.156682	−	0.987649i	$-0.449920\pi$
$499$	7.00000	0.313363	0.156682	+	0.987649i	$0.449920\pi$
$500$	3.67376	0.164296
$501$	0	0
$502$	6.38197	0.284841
$503$	34.0132	1.51657	0.758286	−	0.651922i	$-0.226037\pi$
$503$	34.0132	1.51657	0.758286	+	0.651922i	$0.226037\pi$
$504$	0	0
$505$	−4.18034	−0.186023
$506$	0	0
$507$	0	0
$508$	−5.14590	−0.228312
$509$	28.1246	1.24660	0.623301	−	0.781982i	$-0.285791\pi$
$509$	28.1246	1.24660	0.623301	+	0.781982i	$0.285791\pi$
$510$	0	0
$511$	−9.61803	−0.425477
$512$	5.29180	0.233867
$513$	0	0
$514$	−19.2361	−0.848467
$515$	4.03444	0.177779
$516$	0	0
$517$	0	0
$518$	−9.70820	−0.426554
$519$	0	0
$520$	−0.326238	−0.0143065
$521$	18.4164	0.806837	0.403419	−	0.915015i	$-0.367822\pi$
$521$	18.4164	0.806837	0.403419	+	0.915015i	$0.367822\pi$
$522$	0	0
$523$	37.5410	1.64155	0.820777	−	0.571249i	$-0.193541\pi$
$523$	37.5410	1.64155	0.820777	+	0.571249i	$0.193541\pi$
$524$	0.0901699	0.00393909
$525$	0	0
$526$	36.7426	1.60206
$527$	16.4164	0.715110
$528$	0	0
$529$	36.4164	1.58332
$530$	−7.56231	−0.328486
$531$	0	0
$532$	1.52786	0.0662413
$533$	−0.742646	−0.0321676
$534$	0	0
$535$	−6.32624	−0.273507
$536$	19.5967	0.846451
$537$	0	0
$538$	−11.0344	−0.475729
$539$	0	0
$540$	0	0
$541$	40.0344	1.72122	0.860608	−	0.509269i	$-0.170084\pi$
$541$	40.0344	1.72122	0.860608	+	0.509269i	$0.170084\pi$
$542$	29.3262	1.25967
$543$	0	0
$544$	18.5066	0.793463
$545$	−6.70820	−0.287348
$546$	0	0
$547$	−39.0689	−1.67046	−0.835232	−	0.549897i	$-0.814667\pi$
$547$	−39.0689	−1.67046	−0.835232	+	0.549897i	$0.814667\pi$
$548$	0.944272	0.0403373
$549$	0	0
$550$	0	0
$551$	10.4721	0.446128
$552$	0	0
$553$	−5.09017	−0.216456
$554$	−0.763932	−0.0324564
$555$	0	0
$556$	−0.236068	−0.0100115
$557$	46.5410	1.97201	0.986003	−	0.166727i	$-0.0533198\pi$
$557$	46.5410	1.97201	0.986003	+	0.166727i	$0.0533198\pi$
$558$	0	0
$559$	0.596748	0.0252397
$560$	3.00000	0.126773
$561$	0	0
$562$	−6.70820	−0.282969
$563$	−10.0557	−0.423798	−0.211899	−	0.977292i	$-0.567965\pi$
$563$	−10.0557	−0.423798	−0.211899	+	0.977292i	$0.567965\pi$
$564$	0	0
$565$	6.90983	0.290699
$566$	−37.0344	−1.55667
$567$	0	0
$568$	30.1246	1.26400
$569$	−25.2918	−1.06029	−0.530144	−	0.847908i	$-0.677862\pi$
$569$	−25.2918	−1.06029	−0.530144	+	0.847908i	$0.677862\pi$
$570$	0	0
$571$	−18.0557	−0.755609	−0.377804	−	0.925885i	$-0.623321\pi$
$571$	−18.0557	−0.755609	−0.377804	+	0.925885i	$0.623321\pi$
$572$	0	0
$573$	0	0
$574$	5.09017	0.212460
$575$	−35.5967	−1.48449
$576$	0	0
$577$	43.8541	1.82567	0.912835	−	0.408328i	$-0.133888\pi$
$577$	43.8541	1.82567	0.912835	+	0.408328i	$0.133888\pi$
$578$	−20.9443	−0.871167
$579$	0	0
$580$	−1.61803	−0.0671852
$581$	−0.472136	−0.0195875
$582$	0	0
$583$	0	0
$584$	−21.5066	−0.889949
$585$	0	0
$586$	−9.23607	−0.381538
$587$	−48.2492	−1.99146	−0.995729	−	0.0923211i	$-0.970571\pi$
$587$	−48.2492	−1.99146	−0.995729	+	0.0923211i	$0.970571\pi$
$588$	0	0
$589$	7.41641	0.305588
$590$	2.38197	0.0980640
$591$	0	0
$592$	−29.1246	−1.19701
$593$	10.0000	0.410651	0.205325	−	0.978694i	$-0.434175\pi$
$593$	10.0000	0.410651	0.205325	+	0.978694i	$0.434175\pi$
$594$	0	0
$595$	−3.38197	−0.138647
$596$	−2.23607	−0.0915929
$597$	0	0
$598$	−2.94427	−0.120400
$599$	−5.56231	−0.227270	−0.113635	−	0.993523i	$-0.536249\pi$
$599$	−5.56231	−0.227270	−0.113635	+	0.993523i	$0.536249\pi$
$600$	0	0
$601$	31.4164	1.28150	0.640751	−	0.767749i	$-0.278623\pi$
$601$	31.4164	1.28150	0.640751	+	0.767749i	$0.278623\pi$
$602$	−4.09017	−0.166703
$603$	0	0
$604$	5.76393	0.234531
$605$	0	0
$606$	0	0
$607$	−46.1591	−1.87354	−0.936769	−	0.349948i	$-0.886199\pi$
$607$	−46.1591	−1.87354	−0.936769	+	0.349948i	$0.886199\pi$
$608$	8.36068	0.339070
$609$	0	0
$610$	−9.94427	−0.402632
$611$	2.79837	0.113210
$612$	0	0
$613$	−4.43769	−0.179237	−0.0896184	−	0.995976i	$-0.528565\pi$
$613$	−4.43769	−0.179237	−0.0896184	+	0.995976i	$0.528565\pi$
$614$	41.9787	1.69412
$615$	0	0
$616$	0	0
$617$	−14.3262	−0.576753	−0.288376	−	0.957517i	$-0.593115\pi$
$617$	−14.3262	−0.576753	−0.288376	+	0.957517i	$0.593115\pi$
$618$	0	0
$619$	0.180340	0.00724847	0.00362424	−	0.999993i	$-0.498846\pi$
$619$	0.180340	0.00724847	0.00362424	+	0.999993i	$0.498846\pi$
$620$	−1.14590	−0.0460204
$621$	0	0
$622$	−28.7984	−1.15471
$623$	14.3262	0.573969
$624$	0	0
$625$	19.4164	0.776656
$626$	−33.8328	−1.35223
$627$	0	0
$628$	11.0000	0.438948
$629$	32.8328	1.30913
$630$	0	0
$631$	−12.7082	−0.505906	−0.252953	−	0.967479i	$-0.581402\pi$
$631$	−12.7082	−0.505906	−0.252953	+	0.967479i	$0.581402\pi$
$632$	−11.3820	−0.452750
$633$	0	0
$634$	1.43769	0.0570981
$635$	5.14590	0.204209
$636$	0	0
$637$	0.236068	0.00935335
$638$	0	0
$639$	0	0
$640$	8.41641	0.332688
$641$	6.32624	0.249871	0.124936	−	0.992165i	$-0.460128\pi$
$641$	6.32624	0.249871	0.124936	+	0.992165i	$0.460128\pi$
$642$	0	0
$643$	−29.8885	−1.17869	−0.589345	−	0.807882i	$-0.700614\pi$
$643$	−29.8885	−1.17869	−0.589345	+	0.807882i	$0.700614\pi$
$644$	4.76393	0.187725
$645$	0	0
$646$	−21.8885	−0.861193
$647$	20.8885	0.821213	0.410607	−	0.911813i	$-0.365317\pi$
$647$	20.8885	0.821213	0.410607	+	0.911813i	$0.365317\pi$
$648$	0	0
$649$	0	0
$650$	1.76393	0.0691871
$651$	0	0
$652$	−1.65248	−0.0647159
$653$	8.81966	0.345140	0.172570	−	0.984997i	$-0.444793\pi$
$653$	8.81966	0.345140	0.172570	+	0.984997i	$0.444793\pi$
$654$	0	0
$655$	−0.0901699	−0.00352323
$656$	15.2705	0.596213
$657$	0	0
$658$	−19.1803	−0.747727
$659$	14.8885	0.579975	0.289988	−	0.957030i	$-0.406349\pi$
$659$	14.8885	0.579975	0.289988	+	0.957030i	$0.406349\pi$
$660$	0	0
$661$	−30.5967	−1.19008	−0.595038	−	0.803698i	$-0.702863\pi$
$661$	−30.5967	−1.19008	−0.595038	+	0.803698i	$0.702863\pi$
$662$	52.9230	2.05691
$663$	0	0
$664$	−1.05573	−0.0409702
$665$	−1.52786	−0.0592480
$666$	0	0
$667$	32.6525	1.26431
$668$	−5.47214	−0.211723
$669$	0	0
$670$	8.76393	0.338580
$671$	0	0
$672$	0	0
$673$	35.9443	1.38555	0.692775	−	0.721154i	$-0.256388\pi$
$673$	35.9443	1.38555	0.692775	+	0.721154i	$0.256388\pi$
$674$	0.381966	0.0147128
$675$	0	0
$676$	−8.00000	−0.307692
$677$	29.3262	1.12710	0.563549	−	0.826082i	$-0.309435\pi$
$677$	29.3262	1.12710	0.563549	+	0.826082i	$0.309435\pi$
$678$	0	0
$679$	−18.7082	−0.717955
$680$	−7.56231	−0.290001
$681$	0	0
$682$	0	0
$683$	6.52786	0.249782	0.124891	−	0.992170i	$-0.460142\pi$
$683$	6.52786	0.249782	0.124891	+	0.992170i	$0.460142\pi$
$684$	0	0
$685$	−0.944272	−0.0360788
$686$	−1.61803	−0.0617768
$687$	0	0
$688$	−12.2705	−0.467809
$689$	−1.78522	−0.0680114
$690$	0	0
$691$	−30.0344	−1.14256	−0.571282	−	0.820754i	$-0.693554\pi$
$691$	−30.0344	−1.14256	−0.571282	+	0.820754i	$0.693554\pi$
$692$	−8.52786	−0.324181
$693$	0	0
$694$	−13.7082	−0.520356
$695$	0.236068	0.00895457
$696$	0	0
$697$	−17.2148	−0.652056
$698$	−2.70820	−0.102507
$699$	0	0
$700$	−2.85410	−0.107875
$701$	16.0557	0.606416	0.303208	−	0.952924i	$-0.401942\pi$
$701$	16.0557	0.606416	0.303208	+	0.952924i	$0.401942\pi$
$702$	0	0
$703$	14.8328	0.559430
$704$	0	0
$705$	0	0
$706$	58.4508	2.19983
$707$	6.76393	0.254384
$708$	0	0
$709$	7.70820	0.289488	0.144744	−	0.989469i	$-0.453764\pi$
$709$	7.70820	0.289488	0.144744	+	0.989469i	$0.453764\pi$
$710$	13.4721	0.505600
$711$	0	0
$712$	32.0344	1.20054
$713$	23.1246	0.866024
$714$	0	0
$715$	0	0
$716$	3.27051	0.122225
$717$	0	0
$718$	−20.4164	−0.761934
$719$	−22.4508	−0.837275	−0.418638	−	0.908153i	$-0.637492\pi$
$719$	−22.4508	−0.837275	−0.418638	+	0.908153i	$0.637492\pi$
$720$	0	0
$721$	−6.52786	−0.243110
$722$	20.8541	0.776109
$723$	0	0
$724$	−5.85410	−0.217566
$725$	−19.5623	−0.726526
$726$	0	0
$727$	27.0344	1.00265	0.501326	−	0.865258i	$-0.332846\pi$
$727$	27.0344	1.00265	0.501326	+	0.865258i	$0.332846\pi$
$728$	0.527864	0.0195639
$729$	0	0
$730$	−9.61803	−0.355979
$731$	13.8328	0.511625
$732$	0	0
$733$	−44.4164	−1.64056	−0.820279	−	0.571964i	$-0.806182\pi$
$733$	−44.4164	−1.64056	−0.820279	+	0.571964i	$0.806182\pi$
$734$	8.09017	0.298614
$735$	0	0
$736$	26.0689	0.960912
$737$	0	0
$738$	0	0
$739$	46.8541	1.72356	0.861778	−	0.507286i	$-0.169351\pi$
$739$	46.8541	1.72356	0.861778	+	0.507286i	$0.169351\pi$
$740$	−2.29180	−0.0842481
$741$	0	0
$742$	12.2361	0.449200
$743$	26.9098	0.987226	0.493613	−	0.869682i	$-0.335676\pi$
$743$	26.9098	0.987226	0.493613	+	0.869682i	$0.335676\pi$
$744$	0	0
$745$	2.23607	0.0819232
$746$	−54.5410	−1.99689
$747$	0	0
$748$	0	0
$749$	10.2361	0.374018
$750$	0	0
$751$	30.4377	1.11069	0.555344	−	0.831621i	$-0.312587\pi$
$751$	30.4377	1.11069	0.555344	+	0.831621i	$0.312587\pi$
$752$	−57.5410	−2.09831
$753$	0	0
$754$	−1.61803	−0.0589253
$755$	−5.76393	−0.209771
$756$	0	0
$757$	10.8754	0.395273	0.197636	−	0.980275i	$-0.436673\pi$
$757$	10.8754	0.395273	0.197636	+	0.980275i	$0.436673\pi$
$758$	21.1803	0.769305
$759$	0	0
$760$	−3.41641	−0.123926
$761$	24.1459	0.875288	0.437644	−	0.899148i	$-0.355813\pi$
$761$	24.1459	0.875288	0.437644	+	0.899148i	$0.355813\pi$
$762$	0	0
$763$	10.8541	0.392945
$764$	7.27051	0.263038
$765$	0	0
$766$	−29.6525	−1.07139
$767$	0.562306	0.0203037
$768$	0	0
$769$	−17.7771	−0.641058	−0.320529	−	0.947239i	$-0.603861\pi$
$769$	−17.7771	−0.641058	−0.320529	+	0.947239i	$0.603861\pi$
$770$	0	0
$771$	0	0
$772$	5.81966	0.209454
$773$	13.0557	0.469582	0.234791	−	0.972046i	$-0.424559\pi$
$773$	13.0557	0.469582	0.234791	+	0.972046i	$0.424559\pi$
$774$	0	0
$775$	−13.8541	−0.497654
$776$	−41.8328	−1.50171
$777$	0	0
$778$	−20.6525	−0.740427
$779$	−7.77709	−0.278643
$780$	0	0
$781$	0	0
$782$	−68.2492	−2.44059
$783$	0	0
$784$	−4.85410	−0.173361
$785$	−11.0000	−0.392607
$786$	0	0
$787$	19.5623	0.697321	0.348660	−	0.937249i	$-0.386637\pi$
$787$	19.5623	0.697321	0.348660	+	0.937249i	$0.386637\pi$
$788$	−16.4721	−0.586796
$789$	0	0
$790$	−5.09017	−0.181100
$791$	−11.1803	−0.397527
$792$	0	0
$793$	−2.34752	−0.0833630
$794$	−15.4721	−0.549086
$795$	0	0
$796$	15.5836	0.552346
$797$	−43.8541	−1.55339	−0.776696	−	0.629876i	$-0.783106\pi$
$797$	−43.8541	−1.55339	−0.776696	+	0.629876i	$0.783106\pi$
$798$	0	0
$799$	64.8673	2.29484
$800$	−15.6180	−0.552181
$801$	0	0
$802$	−23.8541	−0.842318
$803$	0	0
$804$	0	0
$805$	−4.76393	−0.167907
$806$	−1.14590	−0.0403625
$807$	0	0
$808$	15.1246	0.532082
$809$	19.3951	0.681896	0.340948	−	0.940082i	$-0.389252\pi$
$809$	19.3951	0.681896	0.340948	+	0.940082i	$0.389252\pi$
$810$	0	0
$811$	−34.2918	−1.20415	−0.602074	−	0.798440i	$-0.705659\pi$
$811$	−34.2918	−1.20415	−0.602074	+	0.798440i	$0.705659\pi$
$812$	2.61803	0.0918750
$813$	0	0
$814$	0	0
$815$	1.65248	0.0578837
$816$	0	0
$817$	6.24922	0.218633
$818$	53.0689	1.85551
$819$	0	0
$820$	1.20163	0.0419626
$821$	−36.1803	−1.26270	−0.631351	−	0.775497i	$-0.717499\pi$
$821$	−36.1803	−1.26270	−0.631351	+	0.775497i	$0.717499\pi$
$822$	0	0
$823$	13.5279	0.471552	0.235776	−	0.971807i	$-0.424237\pi$
$823$	13.5279	0.471552	0.235776	+	0.971807i	$0.424237\pi$
$824$	−14.5967	−0.508502
$825$	0	0
$826$	−3.85410	−0.134101
$827$	24.0557	0.836500	0.418250	−	0.908332i	$-0.362644\pi$
$827$	24.0557	0.836500	0.418250	+	0.908332i	$0.362644\pi$
$828$	0	0
$829$	−16.3262	−0.567034	−0.283517	−	0.958967i	$-0.591501\pi$
$829$	−16.3262	−0.567034	−0.283517	+	0.958967i	$0.591501\pi$
$830$	−0.472136	−0.0163881
$831$	0	0
$832$	1.00000	0.0346688
$833$	5.47214	0.189598
$834$	0	0
$835$	5.47214	0.189371
$836$	0	0
$837$	0	0
$838$	−54.0132	−1.86585
$839$	23.4721	0.810348	0.405174	−	0.914240i	$-0.367211\pi$
$839$	23.4721	0.810348	0.405174	+	0.914240i	$0.367211\pi$
$840$	0	0
$841$	−11.0557	−0.381232
$842$	−40.2148	−1.38589
$843$	0	0
$844$	3.27051	0.112576
$845$	8.00000	0.275208
$846$	0	0
$847$	0	0
$848$	36.7082	1.26056
$849$	0	0
$850$	40.8885	1.40247
$851$	46.2492	1.58540
$852$	0	0
$853$	−0.201626	−0.00690355	−0.00345177	−	0.999994i	$-0.501099\pi$
$853$	−0.201626	−0.00690355	−0.00345177	+	0.999994i	$0.501099\pi$
$854$	16.0902	0.550594
$855$	0	0
$856$	22.8885	0.782314
$857$	−14.0557	−0.480135	−0.240067	−	0.970756i	$-0.577170\pi$
$857$	−14.0557	−0.480135	−0.240067	+	0.970756i	$0.577170\pi$
$858$	0	0
$859$	−29.6525	−1.01173	−0.505865	−	0.862613i	$-0.668827\pi$
$859$	−29.6525	−1.01173	−0.505865	+	0.862613i	$0.668827\pi$
$860$	−0.965558	−0.0329253
$861$	0	0
$862$	22.4164	0.763506
$863$	49.2492	1.67646	0.838232	−	0.545314i	$-0.183590\pi$
$863$	49.2492	1.67646	0.838232	+	0.545314i	$0.183590\pi$
$864$	0	0
$865$	8.52786	0.289956
$866$	−17.7082	−0.601749
$867$	0	0
$868$	1.85410	0.0629323
$869$	0	0
$870$	0	0
$871$	2.06888	0.0701015
$872$	24.2705	0.821903
$873$	0	0
$874$	−30.8328	−1.04294
$875$	5.94427	0.200953
$876$	0	0
$877$	18.8541	0.636658	0.318329	−	0.947980i	$-0.396878\pi$
$877$	18.8541	0.636658	0.318329	+	0.947980i	$0.396878\pi$
$878$	7.03444	0.237401
$879$	0	0
$880$	0	0
$881$	18.7082	0.630295	0.315148	−	0.949043i	$-0.397946\pi$
$881$	18.7082	0.630295	0.315148	+	0.949043i	$0.397946\pi$
$882$	0	0
$883$	−32.9098	−1.10750	−0.553752	−	0.832682i	$-0.686804\pi$
$883$	−32.9098	−1.10750	−0.553752	+	0.832682i	$0.686804\pi$
$884$	0.798374	0.0268522
$885$	0	0
$886$	−10.7639	−0.361621
$887$	25.2918	0.849215	0.424608	−	0.905377i	$-0.360412\pi$
$887$	25.2918	0.849215	0.424608	+	0.905377i	$0.360412\pi$
$888$	0	0
$889$	−8.32624	−0.279253
$890$	14.3262	0.480217
$891$	0	0
$892$	1.27051	0.0425398
$893$	29.3050	0.980653
$894$	0	0
$895$	−3.27051	−0.109321
$896$	−13.6180	−0.454947
$897$	0	0
$898$	46.2705	1.54407
$899$	12.7082	0.423842
$900$	0	0
$901$	−41.3820	−1.37863
$902$	0	0
$903$	0	0
$904$	−25.0000	−0.831488
$905$	5.85410	0.194597
$906$	0	0
$907$	−49.5066	−1.64384	−0.821919	−	0.569604i	$-0.807097\pi$
$907$	−49.5066	−1.64384	−0.821919	+	0.569604i	$0.807097\pi$
$908$	−7.47214	−0.247972
$909$	0	0
$910$	0.236068	0.00782558
$911$	43.0902	1.42764	0.713821	−	0.700329i	$-0.246963\pi$
$911$	43.0902	1.42764	0.713821	+	0.700329i	$0.246963\pi$
$912$	0	0
$913$	0	0
$914$	−38.4164	−1.27070
$915$	0	0
$916$	−15.5279	−0.513055
$917$	0.145898	0.00481798
$918$	0	0
$919$	4.00000	0.131948	0.0659739	−	0.997821i	$-0.478985\pi$
$919$	4.00000	0.131948	0.0659739	+	0.997821i	$0.478985\pi$
$920$	−10.6525	−0.351202
$921$	0	0
$922$	55.5066	1.82801
$923$	3.18034	0.104682
$924$	0	0
$925$	−27.7082	−0.911040
$926$	−51.7426	−1.70037
$927$	0	0
$928$	14.3262	0.470282
$929$	−25.8885	−0.849376	−0.424688	−	0.905340i	$-0.639616\pi$
$929$	−25.8885	−0.849376	−0.424688	+	0.905340i	$0.639616\pi$
$930$	0	0
$931$	2.47214	0.0810210
$932$	−12.5066	−0.409667
$933$	0	0
$934$	−21.5066	−0.703717
$935$	0	0
$936$	0	0
$937$	−18.1246	−0.592105	−0.296053	−	0.955172i	$-0.595670\pi$
$937$	−18.1246	−0.592105	−0.296053	+	0.955172i	$0.595670\pi$
$938$	−14.1803	−0.463005
$939$	0	0
$940$	−4.52786	−0.147683
$941$	−51.5967	−1.68201	−0.841003	−	0.541031i	$-0.818034\pi$
$941$	−51.5967	−1.68201	−0.841003	+	0.541031i	$0.818034\pi$
$942$	0	0
$943$	−24.2492	−0.789663
$944$	−11.5623	−0.376321
$945$	0	0
$946$	0	0
$947$	41.9787	1.36412	0.682062	−	0.731294i	$-0.261083\pi$
$947$	41.9787	1.36412	0.682062	+	0.731294i	$0.261083\pi$
$948$	0	0
$949$	−2.27051	−0.0737039
$950$	18.4721	0.599315
$951$	0	0
$952$	12.2361	0.396573
$953$	19.0344	0.616586	0.308293	−	0.951291i	$-0.400242\pi$
$953$	19.0344	0.616586	0.308293	+	0.951291i	$0.400242\pi$
$954$	0	0
$955$	−7.27051	−0.235268
$956$	−11.9787	−0.387419
$957$	0	0
$958$	−52.7771	−1.70515
$959$	1.52786	0.0493373
$960$	0	0
$961$	−22.0000	−0.709677
$962$	−2.29180	−0.0738905
$963$	0	0
$964$	10.1459	0.326777
$965$	−5.81966	−0.187341
$966$	0	0
$967$	17.3475	0.557859	0.278929	−	0.960312i	$-0.410020\pi$
$967$	17.3475	0.557859	0.278929	+	0.960312i	$0.410020\pi$
$968$	0	0
$969$	0	0
$970$	−18.7082	−0.600684
$971$	−16.4164	−0.526828	−0.263414	−	0.964683i	$-0.584848\pi$
$971$	−16.4164	−0.526828	−0.263414	+	0.964683i	$0.584848\pi$
$972$	0	0
$973$	−0.381966	−0.0122453
$974$	28.5066	0.913410
$975$	0	0
$976$	48.2705	1.54510
$977$	28.7771	0.920661	0.460330	−	0.887748i	$-0.347731\pi$
$977$	28.7771	0.920661	0.460330	+	0.887748i	$0.347731\pi$
$978$	0	0
$979$	0	0
$980$	−0.381966	−0.0122015
$981$	0	0
$982$	35.1246	1.12087
$983$	−1.40325	−0.0447568	−0.0223784	−	0.999750i	$-0.507124\pi$
$983$	−1.40325	−0.0447568	−0.0223784	+	0.999750i	$0.507124\pi$
$984$	0	0
$985$	16.4721	0.524846
$986$	−37.5066	−1.19445
$987$	0	0
$988$	0.360680	0.0114748
$989$	19.4853	0.619596
$990$	0	0
$991$	−26.3820	−0.838051	−0.419025	−	0.907975i	$-0.637628\pi$
$991$	−26.3820	−0.838051	−0.419025	+	0.907975i	$0.637628\pi$
$992$	10.1459	0.322133
$993$	0	0
$994$	−21.7984	−0.691402
$995$	−15.5836	−0.494033
$996$	0	0
$997$	−57.1935	−1.81134	−0.905668	−	0.423987i	$-0.860630\pi$
$997$	−57.1935	−1.81134	−0.905668	+	0.423987i	$0.860630\pi$
$998$	−11.3262	−0.358526
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.z.1.1		2
3.2	odd	2		2541.2.a.x.1.2		2
11.7	odd	10		693.2.m.d.379.1		4
11.8	odd	10		693.2.m.d.64.1		4
11.10	odd	2		7623.2.a.bo.1.2		2
33.8	even	10		231.2.j.b.64.1	✓	4
33.29	even	10		231.2.j.b.148.1	yes	4
33.32	even	2		2541.2.a.p.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.j.b.64.1	✓	4	33.8	even	10
231.2.j.b.148.1	yes	4	33.29	even	10
693.2.m.d.64.1		4	11.8	odd	10
693.2.m.d.379.1		4	11.7	odd	10
2541.2.a.p.1.1		2	33.32	even	2
2541.2.a.x.1.2		2	3.2	odd	2
7623.2.a.z.1.1		2	1.1	even	1	trivial
7623.2.a.bo.1.2		2	11.10	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:	\( N \)	\(=\)	\( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7623.a (trivial)

\(f(q)\)	\(=\)	\(q-1.61803 q^{2} +0.618034 q^{4} -0.618034 q^{5} +1.00000 q^{7} +2.23607 q^{8} +O(q^{10})\)	\(q-1.61803 q^{2} +0.618034 q^{4} -0.618034 q^{5} +1.00000 q^{7} +2.23607 q^{8} +1.00000 q^{10} +0.236068 q^{13} -1.61803 q^{14} -4.85410 q^{16} +5.47214 q^{17} +2.47214 q^{19} -0.381966 q^{20} +7.70820 q^{23} -4.61803 q^{25} -0.381966 q^{26} +0.618034 q^{28} +4.23607 q^{29} +3.00000 q^{31} +3.38197 q^{32} -8.85410 q^{34} -0.618034 q^{35} +6.00000 q^{37} -4.00000 q^{38} -1.38197 q^{40} -3.14590 q^{41} +2.52786 q^{43} -12.4721 q^{46} +11.8541 q^{47} +1.00000 q^{49} +7.47214 q^{50} +0.145898 q^{52} -7.56231 q^{53} +2.23607 q^{56} -6.85410 q^{58} +2.38197 q^{59} -9.94427 q^{61} -4.85410 q^{62} +4.23607 q^{64} -0.145898 q^{65} +8.76393 q^{67} +3.38197 q^{68} +1.00000 q^{70} +13.4721 q^{71} -9.61803 q^{73} -9.70820 q^{74} +1.52786 q^{76} -5.09017 q^{79} +3.00000 q^{80} +5.09017 q^{82} -0.472136 q^{83} -3.38197 q^{85} -4.09017 q^{86} +14.3262 q^{89} +0.236068 q^{91} +4.76393 q^{92} -19.1803 q^{94} -1.52786 q^{95} -18.7082 q^{97} -1.61803 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - q^{4} + q^{5} + 2 q^{7}+O(q^{10}) \) 2 * q - q^2 - q^4 + q^5 + 2 * q^7	\( 2 q - q^{2} - q^{4} + q^{5} + 2 q^{7} + 2 q^{10} - 4 q^{13} - q^{14} - 3 q^{16} + 2 q^{17} - 4 q^{19} - 3 q^{20} + 2 q^{23} - 7 q^{25} - 3 q^{26} - q^{28} + 4 q^{29} + 6 q^{31} + 9 q^{32} - 11 q^{34} + q^{35} + 12 q^{37} - 8 q^{38} - 5 q^{40} - 13 q^{41} + 14 q^{43} - 16 q^{46} + 17 q^{47} + 2 q^{49} + 6 q^{50} + 7 q^{52} + 5 q^{53} - 7 q^{58} + 7 q^{59} - 2 q^{61} - 3 q^{62} + 4 q^{64} - 7 q^{65} + 22 q^{67} + 9 q^{68} + 2 q^{70} + 18 q^{71} - 17 q^{73} - 6 q^{74} + 12 q^{76} + q^{79} + 6 q^{80} - q^{82} + 8 q^{83} - 9 q^{85} + 3 q^{86} + 13 q^{89} - 4 q^{91} + 14 q^{92} - 16 q^{94} - 12 q^{95} - 24 q^{97} - q^{98}+O(q^{100}) \) 2 * q - q^2 - q^4 + q^5 + 2 * q^7 + 2 * q^10 - 4 * q^13 - q^14 - 3 * q^16 + 2 * q^17 - 4 * q^19 - 3 * q^20 + 2 * q^23 - 7 * q^25 - 3 * q^26 - q^28 + 4 * q^29 + 6 * q^31 + 9 * q^32 - 11 * q^34 + q^35 + 12 * q^37 - 8 * q^38 - 5 * q^40 - 13 * q^41 + 14 * q^43 - 16 * q^46 + 17 * q^47 + 2 * q^49 + 6 * q^50 + 7 * q^52 + 5 * q^53 - 7 * q^58 + 7 * q^59 - 2 * q^61 - 3 * q^62 + 4 * q^64 - 7 * q^65 + 22 * q^67 + 9 * q^68 + 2 * q^70 + 18 * q^71 - 17 * q^73 - 6 * q^74 + 12 * q^76 + q^79 + 6 * q^80 - q^82 + 8 * q^83 - 9 * q^85 + 3 * q^86 + 13 * q^89 - 4 * q^91 + 14 * q^92 - 16 * q^94 - 12 * q^95 - 24 * q^97 - q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more