LMFDB - Embedded newform 675.2.a.n.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [675,2,Mod(1,675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(675, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("675.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	675.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:	$5.38990213644$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.64575$ of defining polynomial
Character	$\chi$	$=$	675.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.64575 q^{2} +5.00000 q^{4} -3.00000 q^{7} +7.93725 q^{8} +O(q^{10})$	$q+2.64575 q^{2} +5.00000 q^{4} -3.00000 q^{7} +7.93725 q^{8} +5.29150 q^{11} +2.00000 q^{13} -7.93725 q^{14} +11.0000 q^{16} -5.29150 q^{17} +1.00000 q^{19} +14.0000 q^{22} +5.29150 q^{26} -15.0000 q^{28} -5.29150 q^{29} -3.00000 q^{31} +13.2288 q^{32} -14.0000 q^{34} +1.00000 q^{37} +2.64575 q^{38} -5.29150 q^{41} -1.00000 q^{43} +26.4575 q^{44} -5.29150 q^{47} +2.00000 q^{49} +10.0000 q^{52} -5.29150 q^{53} -23.8118 q^{56} -14.0000 q^{58} -10.5830 q^{59} +7.00000 q^{61} -7.93725 q^{62} +13.0000 q^{64} -12.0000 q^{67} -26.4575 q^{68} +10.5830 q^{71} +11.0000 q^{73} +2.64575 q^{74} +5.00000 q^{76} -15.8745 q^{77} +15.0000 q^{79} -14.0000 q^{82} -15.8745 q^{83} -2.64575 q^{86} +42.0000 q^{88} -6.00000 q^{91} -14.0000 q^{94} +7.00000 q^{97} +5.29150 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 10 q^{4} - 6 q^{7}+O(q^{10}) $ 2 * q + 10 * q^4 - 6 * q^7	$ 2 q + 10 q^{4} - 6 q^{7} + 4 q^{13} + 22 q^{16} + 2 q^{19} + 28 q^{22} - 30 q^{28} - 6 q^{31} - 28 q^{34} + 2 q^{37} - 2 q^{43} + 4 q^{49} + 20 q^{52} - 28 q^{58} + 14 q^{61} + 26 q^{64} - 24 q^{67} + 22 q^{73} + 10 q^{76} + 30 q^{79} - 28 q^{82} + 84 q^{88} - 12 q^{91} - 28 q^{94} + 14 q^{97}+O(q^{100}) $ 2 * q + 10 * q^4 - 6 * q^7 + 4 * q^13 + 22 * q^16 + 2 * q^19 + 28 * q^22 - 30 * q^28 - 6 * q^31 - 28 * q^34 + 2 * q^37 - 2 * q^43 + 4 * q^49 + 20 * q^52 - 28 * q^58 + 14 * q^61 + 26 * q^64 - 24 * q^67 + 22 * q^73 + 10 * q^76 + 30 * q^79 - 28 * q^82 + 84 * q^88 - 12 * q^91 - 28 * q^94 + 14 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	2.64575	1.87083	0.935414	−	0.353553i	$-0.115027\pi$
$2$	2.64575	1.87083	0.935414	+	0.353553i	$0.115027\pi$
$3$	0	0
$3$	0	0
$4$	5.00000	2.50000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	7.93725	2.80624
$9$	0	0
$10$	0	0
$11$	5.29150	1.59545	0.797724	−	0.603023i	$-0.206037\pi$
$11$	5.29150	1.59545	0.797724	+	0.603023i	$0.206037\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	−7.93725	−2.12132
$15$	0	0
$16$	11.0000	2.75000
$17$	−5.29150	−1.28338	−0.641689	−	0.766965i	$-0.721766\pi$
$17$	−5.29150	−1.28338	−0.641689	+	0.766965i	$0.721766\pi$
$18$	0	0
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	0	0
$21$	0	0
$22$	14.0000	2.98481
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	0	0
$26$	5.29150	1.03775
$27$	0	0
$28$	−15.0000	−2.83473
$29$	−5.29150	−0.982607	−0.491304	−	0.870988i	$-0.663479\pi$
$29$	−5.29150	−0.982607	−0.491304	+	0.870988i	$0.663479\pi$
$30$	0	0
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	13.2288	2.33854
$33$	0	0
$34$	−14.0000	−2.40098
$35$	0	0
$36$	0	0
$37$	1.00000	0.164399	0.0821995	−	0.996616i	$-0.473806\pi$
$37$	1.00000	0.164399	0.0821995	+	0.996616i	$0.473806\pi$
$38$	2.64575	0.429198
$39$	0	0
$40$	0	0
$41$	−5.29150	−0.826394	−0.413197	−	0.910642i	$-0.635588\pi$
$41$	−5.29150	−0.826394	−0.413197	+	0.910642i	$0.635588\pi$
$42$	0	0
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	26.4575	3.98862
$45$	0	0
$46$	0	0
$47$	−5.29150	−0.771845	−0.385922	−	0.922531i	$-0.626117\pi$
$47$	−5.29150	−0.771845	−0.385922	+	0.922531i	$0.626117\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	0	0
$51$	0	0
$52$	10.0000	1.38675
$53$	−5.29150	−0.726844	−0.363422	−	0.931625i	$-0.618392\pi$
$53$	−5.29150	−0.726844	−0.363422	+	0.931625i	$0.618392\pi$
$54$	0	0
$55$	0	0
$56$	−23.8118	−3.18198
$57$	0	0
$58$	−14.0000	−1.83829
$59$	−10.5830	−1.37779	−0.688895	−	0.724861i	$-0.741904\pi$
$59$	−10.5830	−1.37779	−0.688895	+	0.724861i	$0.741904\pi$
$60$	0	0
$61$	7.00000	0.896258	0.448129	−	0.893969i	$-0.352090\pi$
$61$	7.00000	0.896258	0.448129	+	0.893969i	$0.352090\pi$
$62$	−7.93725	−1.00803
$63$	0	0
$64$	13.0000	1.62500
$65$	0	0
$66$	0	0
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	−26.4575	−3.20844
$69$	0	0
$70$	0	0
$71$	10.5830	1.25597	0.627986	−	0.778225i	$-0.283880\pi$
$71$	10.5830	1.25597	0.627986	+	0.778225i	$0.283880\pi$
$72$	0	0
$73$	11.0000	1.28745	0.643726	−	0.765256i	$-0.277388\pi$
$73$	11.0000	1.28745	0.643726	+	0.765256i	$0.277388\pi$
$74$	2.64575	0.307562
$75$	0	0
$76$	5.00000	0.573539
$77$	−15.8745	−1.80907
$78$	0	0
$79$	15.0000	1.68763	0.843816	−	0.536633i	$-0.180304\pi$
$79$	15.0000	1.68763	0.843816	+	0.536633i	$0.180304\pi$
$80$	0	0
$81$	0	0
$82$	−14.0000	−1.54604
$83$	−15.8745	−1.74245	−0.871227	−	0.490881i	$-0.836675\pi$
$83$	−15.8745	−1.74245	−0.871227	+	0.490881i	$0.836675\pi$
$84$	0	0
$85$	0	0
$86$	−2.64575	−0.285299
$87$	0	0
$88$	42.0000	4.47722
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	−6.00000	−0.628971
$92$	0	0
$93$	0	0
$94$	−14.0000	−1.44399
$95$	0	0
$96$	0	0
$97$	7.00000	0.710742	0.355371	−	0.934725i	$-0.384354\pi$
$97$	7.00000	0.710742	0.355371	+	0.934725i	$0.384354\pi$
$98$	5.29150	0.534522
$99$	0	0
$100$	0	0
$101$	15.8745	1.57957	0.789786	−	0.613382i	$-0.210191\pi$
$101$	15.8745	1.57957	0.789786	+	0.613382i	$0.210191\pi$
$102$	0	0
$103$	7.00000	0.689730	0.344865	−	0.938652i	$-0.387925\pi$
$103$	7.00000	0.689730	0.344865	+	0.938652i	$0.387925\pi$
$104$	15.8745	1.55662
$105$	0	0
$106$	−14.0000	−1.35980
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	17.0000	1.62830	0.814152	−	0.580651i	$-0.197202\pi$
$109$	17.0000	1.62830	0.814152	+	0.580651i	$0.197202\pi$
$110$	0	0
$111$	0	0
$112$	−33.0000	−3.11821
$113$	10.5830	0.995565	0.497783	−	0.867302i	$-0.334148\pi$
$113$	10.5830	0.995565	0.497783	+	0.867302i	$0.334148\pi$
$114$	0	0
$115$	0	0
$116$	−26.4575	−2.45652
$117$	0	0
$118$	−28.0000	−2.57761
$119$	15.8745	1.45521
$120$	0	0
$121$	17.0000	1.54545
$122$	18.5203	1.67675
$123$	0	0
$124$	−15.0000	−1.34704
$125$	0	0
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	7.93725	0.701561
$129$	0	0
$130$	0	0
$131$	−15.8745	−1.38696	−0.693481	−	0.720475i	$-0.743924\pi$
$131$	−15.8745	−1.38696	−0.693481	+	0.720475i	$0.743924\pi$
$132$	0	0
$133$	−3.00000	−0.260133
$134$	−31.7490	−2.74270
$135$	0	0
$136$	−42.0000	−3.60147
$137$	15.8745	1.35625	0.678125	−	0.734946i	$-0.262793\pi$
$137$	15.8745	1.35625	0.678125	+	0.734946i	$0.262793\pi$
$138$	0	0
$139$	−7.00000	−0.593732	−0.296866	−	0.954919i	$-0.595942\pi$
$139$	−7.00000	−0.593732	−0.296866	+	0.954919i	$0.595942\pi$
$140$	0	0
$141$	0	0
$142$	28.0000	2.34971
$143$	10.5830	0.884995
$144$	0	0
$145$	0	0
$146$	29.1033	2.40860
$147$	0	0
$148$	5.00000	0.410997
$149$	−5.29150	−0.433497	−0.216748	−	0.976228i	$-0.569545\pi$
$149$	−5.29150	−0.433497	−0.216748	+	0.976228i	$0.569545\pi$
$150$	0	0
$151$	−11.0000	−0.895167	−0.447584	−	0.894242i	$-0.647715\pi$
$151$	−11.0000	−0.895167	−0.447584	+	0.894242i	$0.647715\pi$
$152$	7.93725	0.643796
$153$	0	0
$154$	−42.0000	−3.38446
$155$	0	0
$156$	0	0
$157$	−11.0000	−0.877896	−0.438948	−	0.898513i	$-0.644649\pi$
$157$	−11.0000	−0.877896	−0.438948	+	0.898513i	$0.644649\pi$
$158$	39.6863	3.15727
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	−26.4575	−2.06598
$165$	0	0
$166$	−42.0000	−3.25983
$167$	−15.8745	−1.22841	−0.614203	−	0.789148i	$-0.710522\pi$
$167$	−15.8745	−1.22841	−0.614203	+	0.789148i	$0.710522\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	−5.00000	−0.381246
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	58.2065	4.38748
$177$	0	0
$178$	0	0
$179$	−10.5830	−0.791011	−0.395505	−	0.918464i	$-0.629431\pi$
$179$	−10.5830	−0.791011	−0.395505	+	0.918464i	$0.629431\pi$
$180$	0	0
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	−15.8745	−1.17670
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−28.0000	−2.04756
$188$	−26.4575	−1.92961
$189$	0	0
$190$	0	0
$191$	−5.29150	−0.382880	−0.191440	−	0.981504i	$-0.561316\pi$
$191$	−5.29150	−0.382880	−0.191440	+	0.981504i	$0.561316\pi$
$192$	0	0
$193$	−23.0000	−1.65558	−0.827788	−	0.561041i	$-0.810401\pi$
$193$	−23.0000	−1.65558	−0.827788	+	0.561041i	$0.810401\pi$
$194$	18.5203	1.32968
$195$	0	0
$196$	10.0000	0.714286
$197$	15.8745	1.13101	0.565506	−	0.824744i	$-0.308681\pi$
$197$	15.8745	1.13101	0.565506	+	0.824744i	$0.308681\pi$
$198$	0	0
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	0	0
$201$	0	0
$202$	42.0000	2.95511
$203$	15.8745	1.11417
$204$	0	0
$205$	0	0
$206$	18.5203	1.29037
$207$	0	0
$208$	22.0000	1.52543
$209$	5.29150	0.366021
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	−26.4575	−1.81711
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	9.00000	0.610960
$218$	44.9778	3.04628
$219$	0	0
$220$	0	0
$221$	−10.5830	−0.711890
$222$	0	0
$223$	19.0000	1.27233	0.636167	−	0.771551i	$-0.280519\pi$
$223$	19.0000	1.27233	0.636167	+	0.771551i	$0.280519\pi$
$224$	−39.6863	−2.65165
$225$	0	0
$226$	28.0000	1.86253
$227$	21.1660	1.40484	0.702419	−	0.711764i	$-0.252103\pi$
$227$	21.1660	1.40484	0.702419	+	0.711764i	$0.252103\pi$
$228$	0	0
$229$	21.0000	1.38772	0.693860	−	0.720110i	$-0.255909\pi$
$229$	21.0000	1.38772	0.693860	+	0.720110i	$0.255909\pi$
$230$	0	0
$231$	0	0
$232$	−42.0000	−2.75744
$233$	−15.8745	−1.03997	−0.519987	−	0.854174i	$-0.674063\pi$
$233$	−15.8745	−1.03997	−0.519987	+	0.854174i	$0.674063\pi$
$234$	0	0
$235$	0	0
$236$	−52.9150	−3.44447
$237$	0	0
$238$	42.0000	2.72246
$239$	5.29150	0.342279	0.171139	−	0.985247i	$-0.445255\pi$
$239$	5.29150	0.342279	0.171139	+	0.985247i	$0.445255\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	44.9778	2.89128
$243$	0	0
$244$	35.0000	2.24065
$245$	0	0
$246$	0	0
$247$	2.00000	0.127257
$248$	−23.8118	−1.51205
$249$	0	0
$250$	0	0
$251$	−15.8745	−1.00199	−0.500995	−	0.865450i	$-0.667033\pi$
$251$	−15.8745	−1.00199	−0.500995	+	0.865450i	$0.667033\pi$
$252$	0	0
$253$	0	0
$254$	21.1660	1.32807
$255$	0	0
$256$	−5.00000	−0.312500
$257$	15.8745	0.990225	0.495112	−	0.868829i	$-0.335127\pi$
$257$	15.8745	0.990225	0.495112	+	0.868829i	$0.335127\pi$
$258$	0	0
$259$	−3.00000	−0.186411
$260$	0	0
$261$	0	0
$262$	−42.0000	−2.59477
$263$	5.29150	0.326288	0.163144	−	0.986602i	$-0.447836\pi$
$263$	5.29150	0.326288	0.163144	+	0.986602i	$0.447836\pi$
$264$	0	0
$265$	0	0
$266$	−7.93725	−0.486664
$267$	0	0
$268$	−60.0000	−3.66508
$269$	21.1660	1.29051	0.645257	−	0.763965i	$-0.276750\pi$
$269$	21.1660	1.29051	0.645257	+	0.763965i	$0.276750\pi$
$270$	0	0
$271$	−29.0000	−1.76162	−0.880812	−	0.473466i	$-0.843003\pi$
$271$	−29.0000	−1.76162	−0.880812	+	0.473466i	$0.843003\pi$
$272$	−58.2065	−3.52929
$273$	0	0
$274$	42.0000	2.53731
$275$	0	0
$276$	0	0
$277$	3.00000	0.180253	0.0901263	−	0.995930i	$-0.471273\pi$
$277$	3.00000	0.180253	0.0901263	+	0.995930i	$0.471273\pi$
$278$	−18.5203	−1.11077
$279$	0	0
$280$	0	0
$281$	15.8745	0.946994	0.473497	−	0.880795i	$-0.342992\pi$
$281$	15.8745	0.946994	0.473497	+	0.880795i	$0.342992\pi$
$282$	0	0
$283$	21.0000	1.24832	0.624160	−	0.781296i	$-0.285441\pi$
$283$	21.0000	1.24832	0.624160	+	0.781296i	$0.285441\pi$
$284$	52.9150	3.13993
$285$	0	0
$286$	28.0000	1.65567
$287$	15.8745	0.937043
$288$	0	0
$289$	11.0000	0.647059
$290$	0	0
$291$	0	0
$292$	55.0000	3.21863
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	0	0
$296$	7.93725	0.461344
$297$	0	0
$298$	−14.0000	−0.810998
$299$	0	0
$300$	0	0
$301$	3.00000	0.172917
$302$	−29.1033	−1.67470
$303$	0	0
$304$	11.0000	0.630893
$305$	0	0
$306$	0	0
$307$	5.00000	0.285365	0.142683	−	0.989769i	$-0.454427\pi$
$307$	5.00000	0.285365	0.142683	+	0.989769i	$0.454427\pi$
$308$	−79.3725	−4.52267
$309$	0	0
$310$	0	0
$311$	−15.8745	−0.900161	−0.450080	−	0.892988i	$-0.648605\pi$
$311$	−15.8745	−0.900161	−0.450080	+	0.892988i	$0.648605\pi$
$312$	0	0
$313$	−14.0000	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	−29.1033	−1.64239
$315$	0	0
$316$	75.0000	4.21908
$317$	5.29150	0.297200	0.148600	−	0.988897i	$-0.452523\pi$
$317$	5.29150	0.297200	0.148600	+	0.988897i	$0.452523\pi$
$318$	0	0
$319$	−28.0000	−1.56770
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−5.29150	−0.294427
$324$	0	0
$325$	0	0
$326$	10.5830	0.586138
$327$	0	0
$328$	−42.0000	−2.31906
$329$	15.8745	0.875190
$330$	0	0
$331$	−3.00000	−0.164895	−0.0824475	−	0.996595i	$-0.526274\pi$
$331$	−3.00000	−0.164895	−0.0824475	+	0.996595i	$0.526274\pi$
$332$	−79.3725	−4.35613
$333$	0	0
$334$	−42.0000	−2.29814
$335$	0	0
$336$	0	0
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	−23.8118	−1.29519
$339$	0	0
$340$	0	0
$341$	−15.8745	−0.859653
$342$	0	0
$343$	15.0000	0.809924
$344$	−7.93725	−0.427948
$345$	0	0
$346$	0	0
$347$	10.5830	0.568125	0.284063	−	0.958806i	$-0.408318\pi$
$347$	10.5830	0.568125	0.284063	+	0.958806i	$0.408318\pi$
$348$	0	0
$349$	−5.00000	−0.267644	−0.133822	−	0.991005i	$-0.542725\pi$
$349$	−5.00000	−0.267644	−0.133822	+	0.991005i	$0.542725\pi$
$350$	0	0
$351$	0	0
$352$	70.0000	3.73101
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	−28.0000	−1.47985
$359$	15.8745	0.837824	0.418912	−	0.908027i	$-0.362411\pi$
$359$	15.8745	0.837824	0.418912	+	0.908027i	$0.362411\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	37.0405	1.94681
$363$	0	0
$364$	−30.0000	−1.57243
$365$	0	0
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	15.8745	0.824163
$372$	0	0
$373$	−31.0000	−1.60512	−0.802560	−	0.596572i	$-0.796529\pi$
$373$	−31.0000	−1.60512	−0.802560	+	0.596572i	$0.796529\pi$
$374$	−74.0810	−3.83064
$375$	0	0
$376$	−42.0000	−2.16598
$377$	−10.5830	−0.545053
$378$	0	0
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	0	0
$382$	−14.0000	−0.716302
$383$	15.8745	0.811149	0.405575	−	0.914062i	$-0.367071\pi$
$383$	15.8745	0.811149	0.405575	+	0.914062i	$0.367071\pi$
$384$	0	0
$385$	0	0
$386$	−60.8523	−3.09730
$387$	0	0
$388$	35.0000	1.77686
$389$	15.8745	0.804869	0.402435	−	0.915449i	$-0.368164\pi$
$389$	15.8745	0.804869	0.402435	+	0.915449i	$0.368164\pi$
$390$	0	0
$391$	0	0
$392$	15.8745	0.801784
$393$	0	0
$394$	42.0000	2.11593
$395$	0	0
$396$	0	0
$397$	29.0000	1.45547	0.727734	−	0.685859i	$-0.240573\pi$
$397$	29.0000	1.45547	0.727734	+	0.685859i	$0.240573\pi$
$398$	10.5830	0.530478
$399$	0	0
$400$	0	0
$401$	−31.7490	−1.58547	−0.792735	−	0.609566i	$-0.791344\pi$
$401$	−31.7490	−1.58547	−0.792735	+	0.609566i	$0.791344\pi$
$402$	0	0
$403$	−6.00000	−0.298881
$404$	79.3725	3.94893
$405$	0	0
$406$	42.0000	2.08443
$407$	5.29150	0.262290
$408$	0	0
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	0	0
$411$	0	0
$412$	35.0000	1.72433
$413$	31.7490	1.56227
$414$	0	0
$415$	0	0
$416$	26.4575	1.29719
$417$	0	0
$418$	14.0000	0.684762
$419$	10.5830	0.517014	0.258507	−	0.966009i	$-0.416770\pi$
$419$	10.5830	0.517014	0.258507	+	0.966009i	$0.416770\pi$
$420$	0	0
$421$	13.0000	0.633581	0.316791	−	0.948495i	$-0.397395\pi$
$421$	13.0000	0.633581	0.316791	+	0.948495i	$0.397395\pi$
$422$	21.1660	1.03035
$423$	0	0
$424$	−42.0000	−2.03970
$425$	0	0
$426$	0	0
$427$	−21.0000	−1.01626
$428$	0	0
$429$	0	0
$430$	0	0
$431$	15.8745	0.764648	0.382324	−	0.924028i	$-0.375124\pi$
$431$	15.8745	0.764648	0.382324	+	0.924028i	$0.375124\pi$
$432$	0	0
$433$	−19.0000	−0.913082	−0.456541	−	0.889702i	$-0.650912\pi$
$433$	−19.0000	−0.913082	−0.456541	+	0.889702i	$0.650912\pi$
$434$	23.8118	1.14300
$435$	0	0
$436$	85.0000	4.07076
$437$	0	0
$438$	0	0
$439$	1.00000	0.0477274	0.0238637	−	0.999715i	$-0.492403\pi$
$439$	1.00000	0.0477274	0.0238637	+	0.999715i	$0.492403\pi$
$440$	0	0
$441$	0	0
$442$	−28.0000	−1.33182
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	0	0
$446$	50.2693	2.38032
$447$	0	0
$448$	−39.0000	−1.84258
$449$	−26.4575	−1.24861	−0.624304	−	0.781182i	$-0.714617\pi$
$449$	−26.4575	−1.24861	−0.624304	+	0.781182i	$0.714617\pi$
$450$	0	0
$451$	−28.0000	−1.31847
$452$	52.9150	2.48891
$453$	0	0
$454$	56.0000	2.62821
$455$	0	0
$456$	0	0
$457$	38.0000	1.77757	0.888783	−	0.458329i	$-0.151552\pi$
$457$	38.0000	1.77757	0.888783	+	0.458329i	$0.151552\pi$
$458$	55.5608	2.59618
$459$	0	0
$460$	0	0
$461$	−31.7490	−1.47870	−0.739350	−	0.673322i	$-0.764867\pi$
$461$	−31.7490	−1.47870	−0.739350	+	0.673322i	$0.764867\pi$
$462$	0	0
$463$	15.0000	0.697109	0.348555	−	0.937288i	$-0.386673\pi$
$463$	15.0000	0.697109	0.348555	+	0.937288i	$0.386673\pi$
$464$	−58.2065	−2.70217
$465$	0	0
$466$	−42.0000	−1.94561
$467$	−10.5830	−0.489723	−0.244862	−	0.969558i	$-0.578743\pi$
$467$	−10.5830	−0.489723	−0.244862	+	0.969558i	$0.578743\pi$
$468$	0	0
$469$	36.0000	1.66233
$470$	0	0
$471$	0	0
$472$	−84.0000	−3.86641
$473$	−5.29150	−0.243304
$474$	0	0
$475$	0	0
$476$	79.3725	3.63803
$477$	0	0
$478$	14.0000	0.640345
$479$	−31.7490	−1.45065	−0.725325	−	0.688407i	$-0.758310\pi$
$479$	−31.7490	−1.45065	−0.725325	+	0.688407i	$0.758310\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	26.4575	1.20511
$483$	0	0
$484$	85.0000	3.86364
$485$	0	0
$486$	0	0
$487$	4.00000	0.181257	0.0906287	−	0.995885i	$-0.471112\pi$
$487$	4.00000	0.181257	0.0906287	+	0.995885i	$0.471112\pi$
$488$	55.5608	2.51512
$489$	0	0
$490$	0	0
$491$	26.4575	1.19401	0.597005	−	0.802237i	$-0.296357\pi$
$491$	26.4575	1.19401	0.597005	+	0.802237i	$0.296357\pi$
$492$	0	0
$493$	28.0000	1.26106
$494$	5.29150	0.238076
$495$	0	0
$496$	−33.0000	−1.48174
$497$	−31.7490	−1.42414
$498$	0	0
$499$	−29.0000	−1.29822	−0.649109	−	0.760695i	$-0.724858\pi$
$499$	−29.0000	−1.29822	−0.649109	+	0.760695i	$0.724858\pi$
$500$	0	0
$501$	0	0
$502$	−42.0000	−1.87455
$503$	−26.4575	−1.17968	−0.589841	−	0.807519i	$-0.700810\pi$
$503$	−26.4575	−1.17968	−0.589841	+	0.807519i	$0.700810\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	40.0000	1.77471
$509$	−42.3320	−1.87633	−0.938167	−	0.346183i	$-0.887478\pi$
$509$	−42.3320	−1.87633	−0.938167	+	0.346183i	$0.887478\pi$
$510$	0	0
$511$	−33.0000	−1.45983
$512$	−29.1033	−1.28619
$513$	0	0
$514$	42.0000	1.85254
$515$	0	0
$516$	0	0
$517$	−28.0000	−1.23144
$518$	−7.93725	−0.348743
$519$	0	0
$520$	0	0
$521$	42.3320	1.85460	0.927300	−	0.374320i	$-0.122124\pi$
$521$	42.3320	1.85460	0.927300	+	0.374320i	$0.122124\pi$
$522$	0	0
$523$	−1.00000	−0.0437269	−0.0218635	−	0.999761i	$-0.506960\pi$
$523$	−1.00000	−0.0437269	−0.0218635	+	0.999761i	$0.506960\pi$
$524$	−79.3725	−3.46741
$525$	0	0
$526$	14.0000	0.610429
$527$	15.8745	0.691504
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	0	0
$532$	−15.0000	−0.650332
$533$	−10.5830	−0.458401
$534$	0	0
$535$	0	0
$536$	−95.2470	−4.11405
$537$	0	0
$538$	56.0000	2.41433
$539$	10.5830	0.455842
$540$	0	0
$541$	45.0000	1.93470	0.967351	−	0.253442i	$-0.0815627\pi$
$541$	45.0000	1.93470	0.967351	+	0.253442i	$0.0815627\pi$
$542$	−76.7268	−3.29570
$543$	0	0
$544$	−70.0000	−3.00123
$545$	0	0
$546$	0	0
$547$	−13.0000	−0.555840	−0.277920	−	0.960604i	$-0.589645\pi$
$547$	−13.0000	−0.555840	−0.277920	+	0.960604i	$0.589645\pi$
$548$	79.3725	3.39063
$549$	0	0
$550$	0	0
$551$	−5.29150	−0.225426
$552$	0	0
$553$	−45.0000	−1.91359
$554$	7.93725	0.337222
$555$	0	0
$556$	−35.0000	−1.48433
$557$	15.8745	0.672624	0.336312	−	0.941751i	$-0.390820\pi$
$557$	15.8745	0.672624	0.336312	+	0.941751i	$0.390820\pi$
$558$	0	0
$559$	−2.00000	−0.0845910
$560$	0	0
$561$	0	0
$562$	42.0000	1.77166
$563$	−31.7490	−1.33806	−0.669031	−	0.743235i	$-0.733291\pi$
$563$	−31.7490	−1.33806	−0.669031	+	0.743235i	$0.733291\pi$
$564$	0	0
$565$	0	0
$566$	55.5608	2.33539
$567$	0	0
$568$	84.0000	3.52456
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	5.00000	0.209243	0.104622	−	0.994512i	$-0.466637\pi$
$571$	5.00000	0.209243	0.104622	+	0.994512i	$0.466637\pi$
$572$	52.9150	2.21249
$573$	0	0
$574$	42.0000	1.75305
$575$	0	0
$576$	0	0
$577$	25.0000	1.04076	0.520382	−	0.853934i	$-0.325790\pi$
$577$	25.0000	1.04076	0.520382	+	0.853934i	$0.325790\pi$
$578$	29.1033	1.21054
$579$	0	0
$580$	0	0
$581$	47.6235	1.97576
$582$	0	0
$583$	−28.0000	−1.15964
$584$	87.3098	3.61290
$585$	0	0
$586$	0	0
$587$	−15.8745	−0.655211	−0.327606	−	0.944815i	$-0.606242\pi$
$587$	−15.8745	−0.655211	−0.327606	+	0.944815i	$0.606242\pi$
$588$	0	0
$589$	−3.00000	−0.123613
$590$	0	0
$591$	0	0
$592$	11.0000	0.452097
$593$	21.1660	0.869184	0.434592	−	0.900627i	$-0.356893\pi$
$593$	21.1660	0.869184	0.434592	+	0.900627i	$0.356893\pi$
$594$	0	0
$595$	0	0
$596$	−26.4575	−1.08374
$597$	0	0
$598$	0	0
$599$	10.5830	0.432410	0.216205	−	0.976348i	$-0.430632\pi$
$599$	10.5830	0.432410	0.216205	+	0.976348i	$0.430632\pi$
$600$	0	0
$601$	35.0000	1.42768	0.713840	−	0.700309i	$-0.246954\pi$
$601$	35.0000	1.42768	0.713840	+	0.700309i	$0.246954\pi$
$602$	7.93725	0.323498
$603$	0	0
$604$	−55.0000	−2.23792
$605$	0	0
$606$	0	0
$607$	−1.00000	−0.0405887	−0.0202944	−	0.999794i	$-0.506460\pi$
$607$	−1.00000	−0.0405887	−0.0202944	+	0.999794i	$0.506460\pi$
$608$	13.2288	0.536497
$609$	0	0
$610$	0	0
$611$	−10.5830	−0.428143
$612$	0	0
$613$	−19.0000	−0.767403	−0.383701	−	0.923457i	$-0.625351\pi$
$613$	−19.0000	−0.767403	−0.383701	+	0.923457i	$0.625351\pi$
$614$	13.2288	0.533869
$615$	0	0
$616$	−126.000	−5.07668
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	0	0
$619$	−31.0000	−1.24600	−0.622998	−	0.782224i	$-0.714085\pi$
$619$	−31.0000	−1.24600	−0.622998	+	0.782224i	$0.714085\pi$
$620$	0	0
$621$	0	0
$622$	−42.0000	−1.68405
$623$	0	0
$624$	0	0
$625$	0	0
$626$	−37.0405	−1.48044
$627$	0	0
$628$	−55.0000	−2.19474
$629$	−5.29150	−0.210986
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	119.059	4.73591
$633$	0	0
$634$	14.0000	0.556011
$635$	0	0
$636$	0	0
$637$	4.00000	0.158486
$638$	−74.0810	−2.93290
$639$	0	0
$640$	0	0
$641$	−31.7490	−1.25401	−0.627005	−	0.779015i	$-0.715720\pi$
$641$	−31.7490	−1.25401	−0.627005	+	0.779015i	$0.715720\pi$
$642$	0	0
$643$	12.0000	0.473234	0.236617	−	0.971603i	$-0.423961\pi$
$643$	12.0000	0.473234	0.236617	+	0.971603i	$0.423961\pi$
$644$	0	0
$645$	0	0
$646$	−14.0000	−0.550823
$647$	−21.1660	−0.832122	−0.416061	−	0.909337i	$-0.636590\pi$
$647$	−21.1660	−0.832122	−0.416061	+	0.909337i	$0.636590\pi$
$648$	0	0
$649$	−56.0000	−2.19819
$650$	0	0
$651$	0	0
$652$	20.0000	0.783260
$653$	−10.5830	−0.414145	−0.207072	−	0.978326i	$-0.566394\pi$
$653$	−10.5830	−0.414145	−0.207072	+	0.978326i	$0.566394\pi$
$654$	0	0
$655$	0	0
$656$	−58.2065	−2.27258
$657$	0	0
$658$	42.0000	1.63733
$659$	5.29150	0.206128	0.103064	−	0.994675i	$-0.467135\pi$
$659$	5.29150	0.206128	0.103064	+	0.994675i	$0.467135\pi$
$660$	0	0
$661$	−17.0000	−0.661223	−0.330612	−	0.943767i	$-0.607255\pi$
$661$	−17.0000	−0.661223	−0.330612	+	0.943767i	$0.607255\pi$
$662$	−7.93725	−0.308490
$663$	0	0
$664$	−126.000	−4.88975
$665$	0	0
$666$	0	0
$667$	0	0
$668$	−79.3725	−3.07102
$669$	0	0
$670$	0	0
$671$	37.0405	1.42993
$672$	0	0
$673$	−15.0000	−0.578208	−0.289104	−	0.957298i	$-0.593357\pi$
$673$	−15.0000	−0.578208	−0.289104	+	0.957298i	$0.593357\pi$
$674$	5.29150	0.203821
$675$	0	0
$676$	−45.0000	−1.73077
$677$	−15.8745	−0.610107	−0.305053	−	0.952335i	$-0.598674\pi$
$677$	−15.8745	−0.610107	−0.305053	+	0.952335i	$0.598674\pi$
$678$	0	0
$679$	−21.0000	−0.805906
$680$	0	0
$681$	0	0
$682$	−42.0000	−1.60826
$683$	15.8745	0.607421	0.303711	−	0.952764i	$-0.401774\pi$
$683$	15.8745	0.607421	0.303711	+	0.952764i	$0.401774\pi$
$684$	0	0
$685$	0	0
$686$	39.6863	1.51523
$687$	0	0
$688$	−11.0000	−0.419371
$689$	−10.5830	−0.403180
$690$	0	0
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	0	0
$693$	0	0
$694$	28.0000	1.06287
$695$	0	0
$696$	0	0
$697$	28.0000	1.06058
$698$	−13.2288	−0.500716
$699$	0	0
$700$	0	0
$701$	−21.1660	−0.799429	−0.399715	−	0.916640i	$-0.630891\pi$
$701$	−21.1660	−0.799429	−0.399715	+	0.916640i	$0.630891\pi$
$702$	0	0
$703$	1.00000	0.0377157
$704$	68.7895	2.59260
$705$	0	0
$706$	0	0
$707$	−47.6235	−1.79107
$708$	0	0
$709$	13.0000	0.488225	0.244113	−	0.969747i	$-0.421503\pi$
$709$	13.0000	0.488225	0.244113	+	0.969747i	$0.421503\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−52.9150	−1.97753
$717$	0	0
$718$	42.0000	1.56743
$719$	−15.8745	−0.592019	−0.296010	−	0.955185i	$-0.595656\pi$
$719$	−15.8745	−0.592019	−0.296010	+	0.955185i	$0.595656\pi$
$720$	0	0
$721$	−21.0000	−0.782081
$722$	−47.6235	−1.77236
$723$	0	0
$724$	70.0000	2.60153
$725$	0	0
$726$	0	0
$727$	7.00000	0.259616	0.129808	−	0.991539i	$-0.458564\pi$
$727$	7.00000	0.259616	0.129808	+	0.991539i	$0.458564\pi$
$728$	−47.6235	−1.76505
$729$	0	0
$730$	0	0
$731$	5.29150	0.195713
$732$	0	0
$733$	34.0000	1.25582	0.627909	−	0.778287i	$-0.283911\pi$
$733$	34.0000	1.25582	0.627909	+	0.778287i	$0.283911\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−63.4980	−2.33898
$738$	0	0
$739$	40.0000	1.47142	0.735712	−	0.677295i	$-0.236848\pi$
$739$	40.0000	1.47142	0.735712	+	0.677295i	$0.236848\pi$
$740$	0	0
$741$	0	0
$742$	42.0000	1.54187
$743$	−52.9150	−1.94126	−0.970632	−	0.240569i	$-0.922666\pi$
$743$	−52.9150	−1.94126	−0.970632	+	0.240569i	$0.922666\pi$
$744$	0	0
$745$	0	0
$746$	−82.0183	−3.00290
$747$	0	0
$748$	−140.000	−5.11891
$749$	0	0
$750$	0	0
$751$	−17.0000	−0.620339	−0.310169	−	0.950681i	$-0.600386\pi$
$751$	−17.0000	−0.620339	−0.310169	+	0.950681i	$0.600386\pi$
$752$	−58.2065	−2.12257
$753$	0	0
$754$	−28.0000	−1.01970
$755$	0	0
$756$	0	0
$757$	23.0000	0.835949	0.417975	−	0.908459i	$-0.362740\pi$
$757$	23.0000	0.835949	0.417975	+	0.908459i	$0.362740\pi$
$758$	−52.9150	−1.92196
$759$	0	0
$760$	0	0
$761$	47.6235	1.72635	0.863176	−	0.504904i	$-0.168472\pi$
$761$	47.6235	1.72635	0.863176	+	0.504904i	$0.168472\pi$
$762$	0	0
$763$	−51.0000	−1.84632
$764$	−26.4575	−0.957199
$765$	0	0
$766$	42.0000	1.51752
$767$	−21.1660	−0.764260
$768$	0	0
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	0	0
$772$	−115.000	−4.13894
$773$	−15.8745	−0.570966	−0.285483	−	0.958384i	$-0.592154\pi$
$773$	−15.8745	−0.570966	−0.285483	+	0.958384i	$0.592154\pi$
$774$	0	0
$775$	0	0
$776$	55.5608	1.99452
$777$	0	0
$778$	42.0000	1.50577
$779$	−5.29150	−0.189588
$780$	0	0
$781$	56.0000	2.00384
$782$	0	0
$783$	0	0
$784$	22.0000	0.785714
$785$	0	0
$786$	0	0
$787$	35.0000	1.24762	0.623808	−	0.781578i	$-0.285585\pi$
$787$	35.0000	1.24762	0.623808	+	0.781578i	$0.285585\pi$
$788$	79.3725	2.82753
$789$	0	0
$790$	0	0
$791$	−31.7490	−1.12887
$792$	0	0
$793$	14.0000	0.497155
$794$	76.7268	2.72293
$795$	0	0
$796$	20.0000	0.708881
$797$	−10.5830	−0.374869	−0.187435	−	0.982277i	$-0.560017\pi$
$797$	−10.5830	−0.374869	−0.187435	+	0.982277i	$0.560017\pi$
$798$	0	0
$799$	28.0000	0.990569
$800$	0	0
$801$	0	0
$802$	−84.0000	−2.96614
$803$	58.2065	2.05406
$804$	0	0
$805$	0	0
$806$	−15.8745	−0.559156
$807$	0	0
$808$	126.000	4.43266
$809$	−21.1660	−0.744157	−0.372079	−	0.928201i	$-0.621355\pi$
$809$	−21.1660	−0.744157	−0.372079	+	0.928201i	$0.621355\pi$
$810$	0	0
$811$	−33.0000	−1.15879	−0.579393	−	0.815048i	$-0.696710\pi$
$811$	−33.0000	−1.15879	−0.579393	+	0.815048i	$0.696710\pi$
$812$	79.3725	2.78543
$813$	0	0
$814$	14.0000	0.490700
$815$	0	0
$816$	0	0
$817$	−1.00000	−0.0349856
$818$	−26.4575	−0.925065
$819$	0	0
$820$	0	0
$821$	15.8745	0.554024	0.277012	−	0.960866i	$-0.410656\pi$
$821$	15.8745	0.554024	0.277012	+	0.960866i	$0.410656\pi$
$822$	0	0
$823$	16.0000	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$824$	55.5608	1.93555
$825$	0	0
$826$	84.0000	2.92273
$827$	−5.29150	−0.184004	−0.0920018	−	0.995759i	$-0.529327\pi$
$827$	−5.29150	−0.184004	−0.0920018	+	0.995759i	$0.529327\pi$
$828$	0	0
$829$	−3.00000	−0.104194	−0.0520972	−	0.998642i	$-0.516591\pi$
$829$	−3.00000	−0.104194	−0.0520972	+	0.998642i	$0.516591\pi$
$830$	0	0
$831$	0	0
$832$	26.0000	0.901388
$833$	−10.5830	−0.366679
$834$	0	0
$835$	0	0
$836$	26.4575	0.915052
$837$	0	0
$838$	28.0000	0.967244
$839$	−52.9150	−1.82683	−0.913415	−	0.407030i	$-0.866564\pi$
$839$	−52.9150	−1.82683	−0.913415	+	0.407030i	$0.866564\pi$
$840$	0	0
$841$	−1.00000	−0.0344828
$842$	34.3948	1.18532
$843$	0	0
$844$	40.0000	1.37686
$845$	0	0
$846$	0	0
$847$	−51.0000	−1.75238
$848$	−58.2065	−1.99882
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	42.0000	1.43805	0.719026	−	0.694983i	$-0.244588\pi$
$853$	42.0000	1.43805	0.719026	+	0.694983i	$0.244588\pi$
$854$	−55.5608	−1.90125
$855$	0	0
$856$	0	0
$857$	10.5830	0.361509	0.180754	−	0.983528i	$-0.442146\pi$
$857$	10.5830	0.361509	0.180754	+	0.983528i	$0.442146\pi$
$858$	0	0
$859$	13.0000	0.443554	0.221777	−	0.975097i	$-0.428814\pi$
$859$	13.0000	0.443554	0.221777	+	0.975097i	$0.428814\pi$
$860$	0	0
$861$	0	0
$862$	42.0000	1.43053
$863$	−26.4575	−0.900624	−0.450312	−	0.892871i	$-0.648687\pi$
$863$	−26.4575	−0.900624	−0.450312	+	0.892871i	$0.648687\pi$
$864$	0	0
$865$	0	0
$866$	−50.2693	−1.70822
$867$	0	0
$868$	45.0000	1.52740
$869$	79.3725	2.69253
$870$	0	0
$871$	−24.0000	−0.813209
$872$	134.933	4.56942
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−15.0000	−0.506514	−0.253257	−	0.967399i	$-0.581502\pi$
$877$	−15.0000	−0.506514	−0.253257	+	0.967399i	$0.581502\pi$
$878$	2.64575	0.0892898
$879$	0	0
$880$	0	0
$881$	5.29150	0.178275	0.0891376	−	0.996019i	$-0.471589\pi$
$881$	5.29150	0.178275	0.0891376	+	0.996019i	$0.471589\pi$
$882$	0	0
$883$	13.0000	0.437485	0.218742	−	0.975783i	$-0.429805\pi$
$883$	13.0000	0.437485	0.218742	+	0.975783i	$0.429805\pi$
$884$	−52.9150	−1.77972
$885$	0	0
$886$	0	0
$887$	47.6235	1.59904	0.799521	−	0.600639i	$-0.205087\pi$
$887$	47.6235	1.59904	0.799521	+	0.600639i	$0.205087\pi$
$888$	0	0
$889$	−24.0000	−0.804934
$890$	0	0
$891$	0	0
$892$	95.0000	3.18084
$893$	−5.29150	−0.177073
$894$	0	0
$895$	0	0
$896$	−23.8118	−0.795495
$897$	0	0
$898$	−70.0000	−2.33593
$899$	15.8745	0.529444
$900$	0	0
$901$	28.0000	0.932815
$902$	−74.0810	−2.46663
$903$	0	0
$904$	84.0000	2.79380
$905$	0	0
$906$	0	0
$907$	−3.00000	−0.0996134	−0.0498067	−	0.998759i	$-0.515861\pi$
$907$	−3.00000	−0.0996134	−0.0498067	+	0.998759i	$0.515861\pi$
$908$	105.830	3.51209
$909$	0	0
$910$	0	0
$911$	−26.4575	−0.876577	−0.438288	−	0.898834i	$-0.644415\pi$
$911$	−26.4575	−0.876577	−0.438288	+	0.898834i	$0.644415\pi$
$912$	0	0
$913$	−84.0000	−2.77999
$914$	100.539	3.32552
$915$	0	0
$916$	105.000	3.46930
$917$	47.6235	1.57267
$918$	0	0
$919$	−17.0000	−0.560778	−0.280389	−	0.959886i	$-0.590464\pi$
$919$	−17.0000	−0.560778	−0.280389	+	0.959886i	$0.590464\pi$
$920$	0	0
$921$	0	0
$922$	−84.0000	−2.76639
$923$	21.1660	0.696688
$924$	0	0
$925$	0	0
$926$	39.6863	1.30417
$927$	0	0
$928$	−70.0000	−2.29786
$929$	−21.1660	−0.694434	−0.347217	−	0.937785i	$-0.612873\pi$
$929$	−21.1660	−0.694434	−0.347217	+	0.937785i	$0.612873\pi$
$930$	0	0
$931$	2.00000	0.0655474
$932$	−79.3725	−2.59993
$933$	0	0
$934$	−28.0000	−0.916188
$935$	0	0
$936$	0	0
$937$	9.00000	0.294017	0.147009	−	0.989135i	$-0.453036\pi$
$937$	9.00000	0.294017	0.147009	+	0.989135i	$0.453036\pi$
$938$	95.2470	3.10993
$939$	0	0
$940$	0	0
$941$	26.4575	0.862490	0.431245	−	0.902235i	$-0.358074\pi$
$941$	26.4575	0.862490	0.431245	+	0.902235i	$0.358074\pi$
$942$	0	0
$943$	0	0
$944$	−116.413	−3.78892
$945$	0	0
$946$	−14.0000	−0.455179
$947$	−15.8745	−0.515852	−0.257926	−	0.966165i	$-0.583039\pi$
$947$	−15.8745	−0.515852	−0.257926	+	0.966165i	$0.583039\pi$
$948$	0	0
$949$	22.0000	0.714150
$950$	0	0
$951$	0	0
$952$	126.000	4.08368
$953$	−37.0405	−1.19986	−0.599930	−	0.800052i	$-0.704805\pi$
$953$	−37.0405	−1.19986	−0.599930	+	0.800052i	$0.704805\pi$
$954$	0	0
$955$	0	0
$956$	26.4575	0.855697
$957$	0	0
$958$	−84.0000	−2.71392
$959$	−47.6235	−1.53784
$960$	0	0
$961$	−22.0000	−0.709677
$962$	5.29150	0.170605
$963$	0	0
$964$	50.0000	1.61039
$965$	0	0
$966$	0	0
$967$	−23.0000	−0.739630	−0.369815	−	0.929105i	$-0.620579\pi$
$967$	−23.0000	−0.739630	−0.369815	+	0.929105i	$0.620579\pi$
$968$	134.933	4.33692
$969$	0	0
$970$	0	0
$971$	31.7490	1.01887	0.509437	−	0.860508i	$-0.329854\pi$
$971$	31.7490	1.01887	0.509437	+	0.860508i	$0.329854\pi$
$972$	0	0
$973$	21.0000	0.673229
$974$	10.5830	0.339101
$975$	0	0
$976$	77.0000	2.46471
$977$	42.3320	1.35432	0.677161	−	0.735835i	$-0.263210\pi$
$977$	42.3320	1.35432	0.677161	+	0.735835i	$0.263210\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	70.0000	2.23379
$983$	47.6235	1.51895	0.759477	−	0.650534i	$-0.225455\pi$
$983$	47.6235	1.51895	0.759477	+	0.650534i	$0.225455\pi$
$984$	0	0
$985$	0	0
$986$	74.0810	2.35922
$987$	0	0
$988$	10.0000	0.318142
$989$	0	0
$990$	0	0
$991$	23.0000	0.730619	0.365310	−	0.930886i	$-0.380963\pi$
$991$	23.0000	0.730619	0.365310	+	0.930886i	$0.380963\pi$
$992$	−39.6863	−1.26004
$993$	0	0
$994$	−84.0000	−2.66432
$995$	0	0
$996$	0	0
$997$	−18.0000	−0.570066	−0.285033	−	0.958518i	$-0.592005\pi$
$997$	−18.0000	−0.570066	−0.285033	+	0.958518i	$0.592005\pi$
$998$	−76.7268	−2.42874
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.2.a.n.1.2	yes	2
3.2	odd	2	inner	675.2.a.n.1.1	✓	2
5.2	odd	4		675.2.b.g.649.3		4
5.3	odd	4		675.2.b.g.649.2		4
5.4	even	2		675.2.a.o.1.1	yes	2
15.2	even	4		675.2.b.g.649.1		4
15.8	even	4		675.2.b.g.649.4		4
15.14	odd	2		675.2.a.o.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
675.2.a.n.1.1	✓	2	3.2	odd	2	inner
675.2.a.n.1.2	yes	2	1.1	even	1	trivial
675.2.a.o.1.1	yes	2	5.4	even	2
675.2.a.o.1.2	yes	2	15.14	odd	2
675.2.b.g.649.1		4	15.2	even	4
675.2.b.g.649.2		4	5.3	odd	4
675.2.b.g.649.3		4	5.2	odd	4
675.2.b.g.649.4		4	15.8	even	4

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:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	675.a (trivial)

\(f(q)\)	\(=\)	\(q+2.64575 q^{2} +5.00000 q^{4} -3.00000 q^{7} +7.93725 q^{8} +O(q^{10})\)	\(q+2.64575 q^{2} +5.00000 q^{4} -3.00000 q^{7} +7.93725 q^{8} +5.29150 q^{11} +2.00000 q^{13} -7.93725 q^{14} +11.0000 q^{16} -5.29150 q^{17} +1.00000 q^{19} +14.0000 q^{22} +5.29150 q^{26} -15.0000 q^{28} -5.29150 q^{29} -3.00000 q^{31} +13.2288 q^{32} -14.0000 q^{34} +1.00000 q^{37} +2.64575 q^{38} -5.29150 q^{41} -1.00000 q^{43} +26.4575 q^{44} -5.29150 q^{47} +2.00000 q^{49} +10.0000 q^{52} -5.29150 q^{53} -23.8118 q^{56} -14.0000 q^{58} -10.5830 q^{59} +7.00000 q^{61} -7.93725 q^{62} +13.0000 q^{64} -12.0000 q^{67} -26.4575 q^{68} +10.5830 q^{71} +11.0000 q^{73} +2.64575 q^{74} +5.00000 q^{76} -15.8745 q^{77} +15.0000 q^{79} -14.0000 q^{82} -15.8745 q^{83} -2.64575 q^{86} +42.0000 q^{88} -6.00000 q^{91} -14.0000 q^{94} +7.00000 q^{97} +5.29150 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 10 q^{4} - 6 q^{7}+O(q^{10}) \) 2 * q + 10 * q^4 - 6 * q^7	\( 2 q + 10 q^{4} - 6 q^{7} + 4 q^{13} + 22 q^{16} + 2 q^{19} + 28 q^{22} - 30 q^{28} - 6 q^{31} - 28 q^{34} + 2 q^{37} - 2 q^{43} + 4 q^{49} + 20 q^{52} - 28 q^{58} + 14 q^{61} + 26 q^{64} - 24 q^{67} + 22 q^{73} + 10 q^{76} + 30 q^{79} - 28 q^{82} + 84 q^{88} - 12 q^{91} - 28 q^{94} + 14 q^{97}+O(q^{100}) \) 2 * q + 10 * q^4 - 6 * q^7 + 4 * q^13 + 22 * q^16 + 2 * q^19 + 28 * q^22 - 30 * q^28 - 6 * q^31 - 28 * q^34 + 2 * q^37 - 2 * q^43 + 4 * q^49 + 20 * q^52 - 28 * q^58 + 14 * q^61 + 26 * q^64 - 24 * q^67 + 22 * q^73 + 10 * q^76 + 30 * q^79 - 28 * q^82 + 84 * q^88 - 12 * q^91 - 28 * q^94 + 14 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more