LMFDB - Embedded newform 6975.2.a.cc.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6975,2,Mod(1,6975)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6975.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6975, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6975 = 3^{2} \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6975.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,1,0,7,0,0,-2,-3,0,0,-7,0,-4,-10,0,17,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$55.6956554098$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.75968016.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 9x^{4} + 9x^{3} + 14x^{2} - 6x - 6 $ x^6 - x^5 - 9x^4 + 9x^3 + 14x^2 - 6x - 6
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2325)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$2.46251$ of defining polynomial
Character	$\chi$	$=$	6975.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.46251 q^{2} +4.06395 q^{4} -3.91086 q^{7} +5.08250 q^{8} -3.77308 q^{11} +4.83704 q^{13} -9.63054 q^{14} +4.38781 q^{16} -5.89630 q^{17} +5.60730 q^{19} -9.29125 q^{22} -5.19547 q^{23} +11.9112 q^{26} -15.8936 q^{28} +6.33880 q^{29} -1.00000 q^{31} +0.640010 q^{32} -14.5197 q^{34} -11.2913 q^{37} +13.8080 q^{38} -0.663143 q^{41} +3.39308 q^{43} -15.3336 q^{44} -12.7939 q^{46} -1.31684 q^{47} +8.29486 q^{49} +19.6575 q^{52} -10.4443 q^{53} -19.8770 q^{56} +15.6093 q^{58} -7.58816 q^{59} -10.3208 q^{61} -2.46251 q^{62} -7.19958 q^{64} +10.2874 q^{67} -23.9623 q^{68} +3.68083 q^{71} -12.3909 q^{73} -27.8048 q^{74} +22.7878 q^{76} +14.7560 q^{77} -9.23657 q^{79} -1.63300 q^{82} -0.590907 q^{83} +8.35548 q^{86} -19.1767 q^{88} -15.8673 q^{89} -18.9170 q^{91} -21.1141 q^{92} -3.24273 q^{94} -3.98585 q^{97} +20.4262 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + q^{2} + 7 q^{4} - 2 q^{7} - 3 q^{8} - 7 q^{11} - 4 q^{13} - 10 q^{14} + 17 q^{16} + 17 q^{19} - 2 q^{22} - q^{23} - 2 q^{26} - 22 q^{28} + 8 q^{29} - 6 q^{31} - 35 q^{32} - 13 q^{34} - 14 q^{37}+ \cdots + 37 q^{98}+O(q^{100}) $ 6 * q + q^2 + 7 * q^4 - 2 * q^7 - 3 * q^8 - 7 * q^11 - 4 * q^13 - 10 * q^14 + 17 * q^16 + 17 * q^19 - 2 * q^22 - q^23 - 2 * q^26 - 22 * q^28 + 8 * q^29 - 6 * q^31 - 35 * q^32 - 13 * q^34 - 14 * q^37 + 10 * q^38 - 18 * q^41 - 15 * q^43 - 4 * q^44 - 10 * q^46 - 13 * q^47 + 18 * q^49 + 38 * q^52 - 13 * q^53 - 32 * q^56 - 23 * q^58 - 32 * q^59 - q^62 + 29 * q^64 - 9 * q^67 + 9 * q^68 - q^71 - 14 * q^73 - 20 * q^74 + 42 * q^76 - 14 * q^77 + 13 * q^79 + 32 * q^82 + 9 * q^83 + 11 * q^86 - 36 * q^88 - 34 * q^89 - 2 * q^91 - 46 * q^92 + 19 * q^94 - 30 * q^97 + 37 * q^98

Coefficient data

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

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

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

$2$	2.46251	1.74126	0.870629	−	0.491941i	$-0.163712\pi$
$2$	2.46251	1.74126	0.870629	+	0.491941i	$0.163712\pi$
$3$	0	0
$3$	0	0
$4$	4.06395	2.03198
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.91086	−1.47817	−0.739084	−	0.673613i	$-0.764741\pi$
$7$	−3.91086	−1.47817	−0.739084	+	0.673613i	$0.764741\pi$
$8$	5.08250	1.79694
$9$	0	0
$10$	0	0
$11$	−3.77308	−1.13763	−0.568814	−	0.822466i	$-0.692597\pi$
$11$	−3.77308	−1.13763	−0.568814	+	0.822466i	$0.692597\pi$
$12$	0	0
$13$	4.83704	1.34155	0.670776	−	0.741660i	$-0.265961\pi$
$13$	4.83704	1.34155	0.670776	+	0.741660i	$0.265961\pi$
$14$	−9.63054	−2.57387
$15$	0	0
$16$	4.38781	1.09695
$17$	−5.89630	−1.43006	−0.715032	−	0.699092i	$-0.753588\pi$
$17$	−5.89630	−1.43006	−0.715032	+	0.699092i	$0.753588\pi$
$18$	0	0
$19$	5.60730	1.28640	0.643202	−	0.765697i	$-0.277606\pi$
$19$	5.60730	1.28640	0.643202	+	0.765697i	$0.277606\pi$
$20$	0	0
$21$	0	0
$22$	−9.29125	−1.98090
$23$	−5.19547	−1.08333	−0.541665	−	0.840595i	$-0.682206\pi$
$23$	−5.19547	−1.08333	−0.541665	+	0.840595i	$0.682206\pi$
$24$	0	0
$25$	0	0
$26$	11.9112	2.33599
$27$	0	0
$28$	−15.8936	−3.00360
$29$	6.33880	1.17709	0.588543	−	0.808466i	$-0.299702\pi$
$29$	6.33880	1.17709	0.588543	+	0.808466i	$0.299702\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0.640010	0.113139
$33$	0	0
$34$	−14.5197	−2.49011
$35$	0	0
$36$	0	0
$37$	−11.2913	−1.85627	−0.928135	−	0.372243i	$-0.878589\pi$
$37$	−11.2913	−1.85627	−0.928135	+	0.372243i	$0.878589\pi$
$38$	13.8080	2.23996
$39$	0	0
$40$	0	0
$41$	−0.663143	−0.103566	−0.0517828	−	0.998658i	$-0.516490\pi$
$41$	−0.663143	−0.103566	−0.0517828	+	0.998658i	$0.516490\pi$
$42$	0	0
$43$	3.39308	0.517439	0.258720	−	0.965952i	$-0.416699\pi$
$43$	3.39308	0.517439	0.258720	+	0.965952i	$0.416699\pi$
$44$	−15.3336	−2.31163
$45$	0	0
$46$	−12.7939	−1.88636
$47$	−1.31684	−0.192081	−0.0960405	−	0.995377i	$-0.530618\pi$
$47$	−1.31684	−0.192081	−0.0960405	+	0.995377i	$0.530618\pi$
$48$	0	0
$49$	8.29486	1.18498
$50$	0	0
$51$	0	0
$52$	19.6575	2.72600
$53$	−10.4443	−1.43464	−0.717320	−	0.696744i	$-0.754632\pi$
$53$	−10.4443	−1.43464	−0.717320	+	0.696744i	$0.754632\pi$
$54$	0	0
$55$	0	0
$56$	−19.8770	−2.65617
$57$	0	0
$58$	15.6093	2.04961
$59$	−7.58816	−0.987895	−0.493947	−	0.869492i	$-0.664446\pi$
$59$	−7.58816	−0.987895	−0.493947	+	0.869492i	$0.664446\pi$
$60$	0	0
$61$	−10.3208	−1.32145	−0.660724	−	0.750629i	$-0.729751\pi$
$61$	−10.3208	−1.32145	−0.660724	+	0.750629i	$0.729751\pi$
$62$	−2.46251	−0.312739
$63$	0	0
$64$	−7.19958	−0.899948
$65$	0	0
$66$	0	0
$67$	10.2874	1.25680	0.628400	−	0.777890i	$-0.283710\pi$
$67$	10.2874	1.25680	0.628400	+	0.777890i	$0.283710\pi$
$68$	−23.9623	−2.90586
$69$	0	0
$70$	0	0
$71$	3.68083	0.436834	0.218417	−	0.975856i	$-0.429911\pi$
$71$	3.68083	0.436834	0.218417	+	0.975856i	$0.429911\pi$
$72$	0	0
$73$	−12.3909	−1.45025	−0.725125	−	0.688618i	$-0.758218\pi$
$73$	−12.3909	−1.45025	−0.725125	+	0.688618i	$0.758218\pi$
$74$	−27.8048	−3.23224
$75$	0	0
$76$	22.7878	2.61394
$77$	14.7560	1.68160
$78$	0	0
$79$	−9.23657	−1.03919	−0.519597	−	0.854411i	$-0.673918\pi$
$79$	−9.23657	−1.03919	−0.519597	+	0.854411i	$0.673918\pi$
$80$	0	0
$81$	0	0
$82$	−1.63300	−0.180334
$83$	−0.590907	−0.0648605	−0.0324302	−	0.999474i	$-0.510325\pi$
$83$	−0.590907	−0.0648605	−0.0324302	+	0.999474i	$0.510325\pi$
$84$	0	0
$85$	0	0
$86$	8.35548	0.900995
$87$	0	0
$88$	−19.1767	−2.04424
$89$	−15.8673	−1.68193	−0.840964	−	0.541090i	$-0.818012\pi$
$89$	−15.8673	−1.68193	−0.840964	+	0.541090i	$0.818012\pi$
$90$	0	0
$91$	−18.9170	−1.98304
$92$	−21.1141	−2.20130
$93$	0	0
$94$	−3.24273	−0.334462
$95$	0	0
$96$	0	0
$97$	−3.98585	−0.404701	−0.202351	−	0.979313i	$-0.564858\pi$
$97$	−3.98585	−0.404701	−0.202351	+	0.979313i	$0.564858\pi$
$98$	20.4262	2.06335
$99$	0	0
$100$	0	0
$101$	4.48372	0.446147	0.223073	−	0.974802i	$-0.428391\pi$
$101$	4.48372	0.446147	0.223073	+	0.974802i	$0.428391\pi$
$102$	0	0
$103$	−14.9600	−1.47405	−0.737025	−	0.675865i	$-0.763770\pi$
$103$	−14.9600	−1.47405	−0.737025	+	0.675865i	$0.763770\pi$
$104$	24.5842	2.41068
$105$	0	0
$106$	−25.7193	−2.49808
$107$	1.47421	0.142517	0.0712585	−	0.997458i	$-0.477298\pi$
$107$	1.47421	0.142517	0.0712585	+	0.997458i	$0.477298\pi$
$108$	0	0
$109$	12.7771	1.22382	0.611911	−	0.790927i	$-0.290401\pi$
$109$	12.7771	1.22382	0.611911	+	0.790927i	$0.290401\pi$
$110$	0	0
$111$	0	0
$112$	−17.1601	−1.62148
$113$	−4.71980	−0.444002	−0.222001	−	0.975046i	$-0.571259\pi$
$113$	−4.71980	−0.444002	−0.222001	+	0.975046i	$0.571259\pi$
$114$	0	0
$115$	0	0
$116$	25.7606	2.39181
$117$	0	0
$118$	−18.6859	−1.72018
$119$	23.0596	2.11387
$120$	0	0
$121$	3.23615	0.294196
$122$	−25.4152	−2.30098
$123$	0	0
$124$	−4.06395	−0.364954
$125$	0	0
$126$	0	0
$127$	14.7204	1.30622	0.653111	−	0.757262i	$-0.273464\pi$
$127$	14.7204	1.30622	0.653111	+	0.757262i	$0.273464\pi$
$128$	−19.0091	−1.68018
$129$	0	0
$130$	0	0
$131$	−10.9970	−0.960810	−0.480405	−	0.877047i	$-0.659510\pi$
$131$	−10.9970	−0.960810	−0.480405	+	0.877047i	$0.659510\pi$
$132$	0	0
$133$	−21.9294	−1.90152
$134$	25.3327	2.18841
$135$	0	0
$136$	−29.9680	−2.56973
$137$	3.29038	0.281116	0.140558	−	0.990072i	$-0.455110\pi$
$137$	3.29038	0.281116	0.140558	+	0.990072i	$0.455110\pi$
$138$	0	0
$139$	11.7607	0.997527	0.498764	−	0.866738i	$-0.333788\pi$
$139$	11.7607	0.997527	0.498764	+	0.866738i	$0.333788\pi$
$140$	0	0
$141$	0	0
$142$	9.06408	0.760641
$143$	−18.2505	−1.52619
$144$	0	0
$145$	0	0
$146$	−30.5128	−2.52526
$147$	0	0
$148$	−45.8871	−3.77190
$149$	0.0294614	0.00241357	0.00120679	−	0.999999i	$-0.499616\pi$
$149$	0.0294614	0.00241357	0.00120679	+	0.999999i	$0.499616\pi$
$150$	0	0
$151$	12.0018	0.976689	0.488345	−	0.872651i	$-0.337601\pi$
$151$	12.0018	0.976689	0.488345	+	0.872651i	$0.337601\pi$
$152$	28.4991	2.31159
$153$	0	0
$154$	36.3368	2.92810
$155$	0	0
$156$	0	0
$157$	−7.34191	−0.585948	−0.292974	−	0.956120i	$-0.594645\pi$
$157$	−7.34191	−0.585948	−0.292974	+	0.956120i	$0.594645\pi$
$158$	−22.7451	−1.80951
$159$	0	0
$160$	0	0
$161$	20.3188	1.60134
$162$	0	0
$163$	11.3447	0.888582	0.444291	−	0.895882i	$-0.353456\pi$
$163$	11.3447	0.888582	0.444291	+	0.895882i	$0.353456\pi$
$164$	−2.69498	−0.210443
$165$	0	0
$166$	−1.45511	−0.112939
$167$	−7.42413	−0.574496	−0.287248	−	0.957856i	$-0.592740\pi$
$167$	−7.42413	−0.574496	−0.287248	+	0.957856i	$0.592740\pi$
$168$	0	0
$169$	10.3969	0.799762
$170$	0	0
$171$	0	0
$172$	13.7893	1.05142
$173$	21.4708	1.63239	0.816196	−	0.577775i	$-0.196079\pi$
$173$	21.4708	1.63239	0.816196	+	0.577775i	$0.196079\pi$
$174$	0	0
$175$	0	0
$176$	−16.5556	−1.24792
$177$	0	0
$178$	−39.0733	−2.92867
$179$	−9.85343	−0.736480	−0.368240	−	0.929731i	$-0.620040\pi$
$179$	−9.85343	−0.736480	−0.368240	+	0.929731i	$0.620040\pi$
$180$	0	0
$181$	−3.95811	−0.294204	−0.147102	−	0.989121i	$-0.546995\pi$
$181$	−3.95811	−0.294204	−0.147102	+	0.989121i	$0.546995\pi$
$182$	−46.5833	−3.45298
$183$	0	0
$184$	−26.4060	−1.94667
$185$	0	0
$186$	0	0
$187$	22.2472	1.62688
$188$	−5.35158	−0.390304
$189$	0	0
$190$	0	0
$191$	16.6250	1.20294	0.601470	−	0.798896i	$-0.294582\pi$
$191$	16.6250	1.20294	0.601470	+	0.798896i	$0.294582\pi$
$192$	0	0
$193$	−4.17565	−0.300570	−0.150285	−	0.988643i	$-0.548019\pi$
$193$	−4.17565	−0.300570	−0.150285	+	0.988643i	$0.548019\pi$
$194$	−9.81518	−0.704689
$195$	0	0
$196$	33.7099	2.40785
$197$	10.0370	0.715108	0.357554	−	0.933892i	$-0.383611\pi$
$197$	10.0370	0.715108	0.357554	+	0.933892i	$0.383611\pi$
$198$	0	0
$199$	11.7746	0.834679	0.417339	−	0.908751i	$-0.362963\pi$
$199$	11.7746	0.834679	0.417339	+	0.908751i	$0.362963\pi$
$200$	0	0
$201$	0	0
$202$	11.0412	0.776856
$203$	−24.7902	−1.73993
$204$	0	0
$205$	0	0
$206$	−36.8391	−2.56670
$207$	0	0
$208$	21.2240	1.47162
$209$	−21.1568	−1.46345
$210$	0	0
$211$	−19.9525	−1.37359	−0.686795	−	0.726851i	$-0.740983\pi$
$211$	−19.9525	−1.37359	−0.686795	+	0.726851i	$0.740983\pi$
$212$	−42.4453	−2.91516
$213$	0	0
$214$	3.63025	0.248159
$215$	0	0
$216$	0	0
$217$	3.91086	0.265487
$218$	31.4637	2.13099
$219$	0	0
$220$	0	0
$221$	−28.5206	−1.91851
$222$	0	0
$223$	17.5032	1.17210	0.586051	−	0.810274i	$-0.300682\pi$
$223$	17.5032	1.17210	0.586051	+	0.810274i	$0.300682\pi$
$224$	−2.50299	−0.167238
$225$	0	0
$226$	−11.6226	−0.773121
$227$	8.55366	0.567727	0.283863	−	0.958865i	$-0.408384\pi$
$227$	8.55366	0.567727	0.283863	+	0.958865i	$0.408384\pi$
$228$	0	0
$229$	−18.2864	−1.20840	−0.604198	−	0.796834i	$-0.706507\pi$
$229$	−18.2864	−1.20840	−0.604198	+	0.796834i	$0.706507\pi$
$230$	0	0
$231$	0	0
$232$	32.2170	2.11515
$233$	−17.7085	−1.16012	−0.580061	−	0.814573i	$-0.696971\pi$
$233$	−17.7085	−1.16012	−0.580061	+	0.814573i	$0.696971\pi$
$234$	0	0
$235$	0	0
$236$	−30.8379	−2.00738
$237$	0	0
$238$	56.7846	3.68080
$239$	−1.18501	−0.0766517	−0.0383258	−	0.999265i	$-0.512202\pi$
$239$	−1.18501	−0.0766517	−0.0383258	+	0.999265i	$0.512202\pi$
$240$	0	0
$241$	18.0403	1.16208	0.581039	−	0.813876i	$-0.302646\pi$
$241$	18.0403	1.16208	0.581039	+	0.813876i	$0.302646\pi$
$242$	7.96906	0.512270
$243$	0	0
$244$	−41.9434	−2.68515
$245$	0	0
$246$	0	0
$247$	27.1227	1.72578
$248$	−5.08250	−0.322739
$249$	0	0
$250$	0	0
$251$	2.86868	0.181070	0.0905348	−	0.995893i	$-0.471142\pi$
$251$	2.86868	0.181070	0.0905348	+	0.995893i	$0.471142\pi$
$252$	0	0
$253$	19.6029	1.23243
$254$	36.2491	2.27447
$255$	0	0
$256$	−32.4108	−2.02568
$257$	10.5692	0.659292	0.329646	−	0.944105i	$-0.393071\pi$
$257$	10.5692	0.659292	0.329646	+	0.944105i	$0.393071\pi$
$258$	0	0
$259$	44.1586	2.74388
$260$	0	0
$261$	0	0
$262$	−27.0802	−1.67302
$263$	−27.8993	−1.72034	−0.860171	−	0.510005i	$-0.829643\pi$
$263$	−27.8993	−1.72034	−0.860171	+	0.510005i	$0.829643\pi$
$264$	0	0
$265$	0	0
$266$	−54.0014	−3.31104
$267$	0	0
$268$	41.8073	2.55379
$269$	24.3497	1.48463	0.742313	−	0.670053i	$-0.233729\pi$
$269$	24.3497	1.48463	0.742313	+	0.670053i	$0.233729\pi$
$270$	0	0
$271$	26.2165	1.59254	0.796268	−	0.604943i	$-0.206804\pi$
$271$	26.2165	1.59254	0.796268	+	0.604943i	$0.206804\pi$
$272$	−25.8718	−1.56871
$273$	0	0
$274$	8.10259	0.489495
$275$	0	0
$276$	0	0
$277$	−5.34031	−0.320868	−0.160434	−	0.987047i	$-0.551289\pi$
$277$	−5.34031	−0.320868	−0.160434	+	0.987047i	$0.551289\pi$
$278$	28.9608	1.73695
$279$	0	0
$280$	0	0
$281$	−0.305796	−0.0182422	−0.00912112	−	0.999958i	$-0.502903\pi$
$281$	−0.305796	−0.0182422	−0.00912112	+	0.999958i	$0.502903\pi$
$282$	0	0
$283$	17.0008	1.01059	0.505296	−	0.862946i	$-0.331383\pi$
$283$	17.0008	1.01059	0.505296	+	0.862946i	$0.331383\pi$
$284$	14.9587	0.887637
$285$	0	0
$286$	−44.9421	−2.65748
$287$	2.59346	0.153087
$288$	0	0
$289$	17.7664	1.04508
$290$	0	0
$291$	0	0
$292$	−50.3562	−2.94687
$293$	−23.8670	−1.39433	−0.697164	−	0.716912i	$-0.745555\pi$
$293$	−23.8670	−1.39433	−0.697164	+	0.716912i	$0.745555\pi$
$294$	0	0
$295$	0	0
$296$	−57.3878	−3.33560
$297$	0	0
$298$	0.0725490	0.00420265
$299$	−25.1307	−1.45334
$300$	0	0
$301$	−13.2699	−0.764862
$302$	29.5545	1.70067
$303$	0	0
$304$	24.6038	1.41112
$305$	0	0
$306$	0	0
$307$	10.0221	0.571992	0.285996	−	0.958231i	$-0.407676\pi$
$307$	10.0221	0.571992	0.285996	+	0.958231i	$0.407676\pi$
$308$	59.9677	3.41698
$309$	0	0
$310$	0	0
$311$	3.00341	0.170307	0.0851537	−	0.996368i	$-0.472862\pi$
$311$	3.00341	0.170307	0.0851537	+	0.996368i	$0.472862\pi$
$312$	0	0
$313$	10.2325	0.578374	0.289187	−	0.957273i	$-0.406615\pi$
$313$	10.2325	0.578374	0.289187	+	0.957273i	$0.406615\pi$
$314$	−18.0795	−1.02029
$315$	0	0
$316$	−37.5370	−2.11162
$317$	−23.4147	−1.31510	−0.657550	−	0.753411i	$-0.728407\pi$
$317$	−23.4147	−1.31510	−0.657550	+	0.753411i	$0.728407\pi$
$318$	0	0
$319$	−23.9168	−1.33908
$320$	0	0
$321$	0	0
$322$	50.0352	2.78835
$323$	−33.0624	−1.83964
$324$	0	0
$325$	0	0
$326$	27.9363	1.54725
$327$	0	0
$328$	−3.37043	−0.186101
$329$	5.14999	0.283928
$330$	0	0
$331$	20.2889	1.11518	0.557590	−	0.830116i	$-0.311726\pi$
$331$	20.2889	1.11518	0.557590	+	0.830116i	$0.311726\pi$
$332$	−2.40142	−0.131795
$333$	0	0
$334$	−18.2820	−1.00035
$335$	0	0
$336$	0	0
$337$	15.9577	0.869274	0.434637	−	0.900606i	$-0.356877\pi$
$337$	15.9577	0.869274	0.434637	+	0.900606i	$0.356877\pi$
$338$	25.6025	1.39259
$339$	0	0
$340$	0	0
$341$	3.77308	0.204324
$342$	0	0
$343$	−5.06402	−0.273431
$344$	17.2453	0.929806
$345$	0	0
$346$	52.8720	2.84242
$347$	6.93796	0.372449	0.186224	−	0.982507i	$-0.440375\pi$
$347$	6.93796	0.372449	0.186224	+	0.982507i	$0.440375\pi$
$348$	0	0
$349$	−9.56079	−0.511778	−0.255889	−	0.966706i	$-0.582368\pi$
$349$	−9.56079	−0.511778	−0.255889	+	0.966706i	$0.582368\pi$
$350$	0	0
$351$	0	0
$352$	−2.41481	−0.128710
$353$	12.0956	0.643783	0.321892	−	0.946776i	$-0.395681\pi$
$353$	12.0956	0.643783	0.321892	+	0.946776i	$0.395681\pi$
$354$	0	0
$355$	0	0
$356$	−64.4839	−3.41764
$357$	0	0
$358$	−24.2642	−1.28240
$359$	28.8573	1.52303	0.761514	−	0.648148i	$-0.224456\pi$
$359$	28.8573	1.52303	0.761514	+	0.648148i	$0.224456\pi$
$360$	0	0
$361$	12.4419	0.654835
$362$	−9.74688	−0.512284
$363$	0	0
$364$	−76.8778	−4.02949
$365$	0	0
$366$	0	0
$367$	−23.3643	−1.21961	−0.609804	−	0.792552i	$-0.708752\pi$
$367$	−23.3643	−1.21961	−0.609804	+	0.792552i	$0.708752\pi$
$368$	−22.7967	−1.18836
$369$	0	0
$370$	0	0
$371$	40.8464	2.12064
$372$	0	0
$373$	16.9679	0.878563	0.439281	−	0.898350i	$-0.355233\pi$
$373$	16.9679	0.878563	0.439281	+	0.898350i	$0.355233\pi$
$374$	54.7840	2.83282
$375$	0	0
$376$	−6.69285	−0.345157
$377$	30.6610	1.57912
$378$	0	0
$379$	−23.3454	−1.19917	−0.599585	−	0.800311i	$-0.704668\pi$
$379$	−23.3454	−1.19917	−0.599585	+	0.800311i	$0.704668\pi$
$380$	0	0
$381$	0	0
$382$	40.9391	2.09463
$383$	−25.3541	−1.29554	−0.647768	−	0.761838i	$-0.724297\pi$
$383$	−25.3541	−1.29554	−0.647768	+	0.761838i	$0.724297\pi$
$384$	0	0
$385$	0	0
$386$	−10.2826	−0.523370
$387$	0	0
$388$	−16.1983	−0.822343
$389$	12.6350	0.640622	0.320311	−	0.947312i	$-0.396213\pi$
$389$	12.6350	0.640622	0.320311	+	0.947312i	$0.396213\pi$
$390$	0	0
$391$	30.6340	1.54923
$392$	42.1587	2.12933
$393$	0	0
$394$	24.7163	1.24519
$395$	0	0
$396$	0	0
$397$	−25.5883	−1.28424	−0.642119	−	0.766605i	$-0.721945\pi$
$397$	−25.5883	−1.28424	−0.642119	+	0.766605i	$0.721945\pi$
$398$	28.9950	1.45339
$399$	0	0
$400$	0	0
$401$	−37.1198	−1.85368	−0.926838	−	0.375461i	$-0.877484\pi$
$401$	−37.1198	−1.85368	−0.926838	+	0.375461i	$0.877484\pi$
$402$	0	0
$403$	−4.83704	−0.240950
$404$	18.2216	0.906560
$405$	0	0
$406$	−61.0460	−3.02966
$407$	42.6028	2.11174
$408$	0	0
$409$	11.2361	0.555589	0.277794	−	0.960641i	$-0.410397\pi$
$409$	11.2361	0.555589	0.277794	+	0.960641i	$0.410397\pi$
$410$	0	0
$411$	0	0
$412$	−60.7966	−2.99524
$413$	29.6763	1.46027
$414$	0	0
$415$	0	0
$416$	3.09575	0.151782
$417$	0	0
$418$	−52.0989	−2.54824
$419$	26.5740	1.29823	0.649113	−	0.760692i	$-0.275140\pi$
$419$	26.5740	1.29823	0.649113	+	0.760692i	$0.275140\pi$
$420$	0	0
$421$	21.5334	1.04948	0.524738	−	0.851264i	$-0.324163\pi$
$421$	21.5334	1.04948	0.524738	+	0.851264i	$0.324163\pi$
$422$	−49.1333	−2.39177
$423$	0	0
$424$	−53.0834	−2.57796
$425$	0	0
$426$	0	0
$427$	40.3634	1.95332
$428$	5.99111	0.289591
$429$	0	0
$430$	0	0
$431$	−7.20557	−0.347080	−0.173540	−	0.984827i	$-0.555521\pi$
$431$	−7.20557	−0.347080	−0.173540	+	0.984827i	$0.555521\pi$
$432$	0	0
$433$	12.1236	0.582624	0.291312	−	0.956628i	$-0.405908\pi$
$433$	12.1236	0.582624	0.291312	+	0.956628i	$0.405908\pi$
$434$	9.63054	0.462281
$435$	0	0
$436$	51.9254	2.48678
$437$	−29.1326	−1.39360
$438$	0	0
$439$	38.1847	1.82245	0.911227	−	0.411905i	$-0.135136\pi$
$439$	38.1847	1.82245	0.911227	+	0.411905i	$0.135136\pi$
$440$	0	0
$441$	0	0
$442$	−70.2323	−3.34061
$443$	24.9128	1.18364	0.591822	−	0.806069i	$-0.298409\pi$
$443$	24.9128	1.18364	0.591822	+	0.806069i	$0.298409\pi$
$444$	0	0
$445$	0	0
$446$	43.1018	2.04093
$447$	0	0
$448$	28.1566	1.33027
$449$	14.0489	0.663007	0.331504	−	0.943454i	$-0.392444\pi$
$449$	14.0489	0.663007	0.331504	+	0.943454i	$0.392444\pi$
$450$	0	0
$451$	2.50209	0.117819
$452$	−19.1811	−0.902201
$453$	0	0
$454$	21.0635	0.988558
$455$	0	0
$456$	0	0
$457$	−40.4110	−1.89034	−0.945172	−	0.326572i	$-0.894106\pi$
$457$	−40.4110	−1.89034	−0.945172	+	0.326572i	$0.894106\pi$
$458$	−45.0303	−2.10413
$459$	0	0
$460$	0	0
$461$	−13.7596	−0.640850	−0.320425	−	0.947274i	$-0.603826\pi$
$461$	−13.7596	−0.640850	−0.320425	+	0.947274i	$0.603826\pi$
$462$	0	0
$463$	−23.9445	−1.11280	−0.556398	−	0.830916i	$-0.687817\pi$
$463$	−23.9445	−1.11280	−0.556398	+	0.830916i	$0.687817\pi$
$464$	27.8134	1.29121
$465$	0	0
$466$	−43.6073	−2.02007
$467$	−7.84527	−0.363036	−0.181518	−	0.983388i	$-0.558101\pi$
$467$	−7.84527	−0.363036	−0.181518	+	0.983388i	$0.558101\pi$
$468$	0	0
$469$	−40.2324	−1.85776
$470$	0	0
$471$	0	0
$472$	−38.5669	−1.77518
$473$	−12.8024	−0.588653
$474$	0	0
$475$	0	0
$476$	93.7133	4.29534
$477$	0	0
$478$	−2.91809	−0.133470
$479$	−36.2715	−1.65729	−0.828644	−	0.559776i	$-0.810887\pi$
$479$	−36.2715	−1.65729	−0.828644	+	0.559776i	$0.810887\pi$
$480$	0	0
$481$	−54.6162	−2.49028
$482$	44.4244	2.02348
$483$	0	0
$484$	13.1516	0.597799
$485$	0	0
$486$	0	0
$487$	5.83307	0.264322	0.132161	−	0.991228i	$-0.457808\pi$
$487$	5.83307	0.264322	0.132161	+	0.991228i	$0.457808\pi$
$488$	−52.4557	−2.37456
$489$	0	0
$490$	0	0
$491$	−3.76466	−0.169897	−0.0849484	−	0.996385i	$-0.527073\pi$
$491$	−3.76466	−0.169897	−0.0849484	+	0.996385i	$0.527073\pi$
$492$	0	0
$493$	−37.3755	−1.68331
$494$	66.7900	3.00502
$495$	0	0
$496$	−4.38781	−0.197018
$497$	−14.3952	−0.645714
$498$	0	0
$499$	11.2661	0.504339	0.252169	−	0.967683i	$-0.418856\pi$
$499$	11.2661	0.504339	0.252169	+	0.967683i	$0.418856\pi$
$500$	0	0
$501$	0	0
$502$	7.06416	0.315289
$503$	36.7533	1.63875	0.819374	−	0.573259i	$-0.194321\pi$
$503$	36.7533	1.63875	0.819374	+	0.573259i	$0.194321\pi$
$504$	0	0
$505$	0	0
$506$	48.2724	2.14597
$507$	0	0
$508$	59.8229	2.65421
$509$	−36.8000	−1.63113	−0.815565	−	0.578666i	$-0.803573\pi$
$509$	−36.8000	−1.63113	−0.815565	+	0.578666i	$0.803573\pi$
$510$	0	0
$511$	48.4593	2.14371
$512$	−41.7939	−1.84704
$513$	0	0
$514$	26.0269	1.14800
$515$	0	0
$516$	0	0
$517$	4.96855	0.218517
$518$	108.741	4.77780
$519$	0	0
$520$	0	0
$521$	8.57486	0.375672	0.187836	−	0.982200i	$-0.439853\pi$
$521$	8.57486	0.375672	0.187836	+	0.982200i	$0.439853\pi$
$522$	0	0
$523$	−6.23174	−0.272495	−0.136247	−	0.990675i	$-0.543504\pi$
$523$	−6.23174	−0.272495	−0.136247	+	0.990675i	$0.543504\pi$
$524$	−44.6912	−1.95234
$525$	0	0
$526$	−68.7022	−2.99556
$527$	5.89630	0.256847
$528$	0	0
$529$	3.99288	0.173603
$530$	0	0
$531$	0	0
$532$	−89.1201	−3.86385
$533$	−3.20765	−0.138939
$534$	0	0
$535$	0	0
$536$	52.2855	2.25839
$537$	0	0
$538$	59.9613	2.58512
$539$	−31.2972	−1.34807
$540$	0	0
$541$	11.9562	0.514037	0.257018	−	0.966407i	$-0.417260\pi$
$541$	11.9562	0.514037	0.257018	+	0.966407i	$0.417260\pi$
$542$	64.5583	2.77302
$543$	0	0
$544$	−3.77369	−0.161796
$545$	0	0
$546$	0	0
$547$	−36.2669	−1.55066	−0.775330	−	0.631557i	$-0.782416\pi$
$547$	−36.2669	−1.55066	−0.775330	+	0.631557i	$0.782416\pi$
$548$	13.3719	0.571221
$549$	0	0
$550$	0	0
$551$	35.5436	1.51421
$552$	0	0
$553$	36.1230	1.53610
$554$	−13.1506	−0.558714
$555$	0	0
$556$	47.7948	2.02695
$557$	36.6763	1.55403	0.777013	−	0.629484i	$-0.216734\pi$
$557$	36.6763	1.55403	0.777013	+	0.629484i	$0.216734\pi$
$558$	0	0
$559$	16.4124	0.694172
$560$	0	0
$561$	0	0
$562$	−0.753025	−0.0317644
$563$	11.7391	0.494742	0.247371	−	0.968921i	$-0.420433\pi$
$563$	11.7391	0.494742	0.247371	+	0.968921i	$0.420433\pi$
$564$	0	0
$565$	0	0
$566$	41.8646	1.75970
$567$	0	0
$568$	18.7078	0.784963
$569$	−0.948286	−0.0397542	−0.0198771	−	0.999802i	$-0.506327\pi$
$569$	−0.948286	−0.0397542	−0.0198771	+	0.999802i	$0.506327\pi$
$570$	0	0
$571$	41.3896	1.73210	0.866051	−	0.499956i	$-0.166651\pi$
$571$	41.3896	1.73210	0.866051	+	0.499956i	$0.166651\pi$
$572$	−74.1693	−3.10117
$573$	0	0
$574$	6.38643	0.266564
$575$	0	0
$576$	0	0
$577$	−11.6673	−0.485716	−0.242858	−	0.970062i	$-0.578085\pi$
$577$	−11.6673	−0.485716	−0.242858	+	0.970062i	$0.578085\pi$
$578$	43.7499	1.81976
$579$	0	0
$580$	0	0
$581$	2.31096	0.0958747
$582$	0	0
$583$	39.4074	1.63209
$584$	−62.9770	−2.60601
$585$	0	0
$586$	−58.7728	−2.42788
$587$	−22.5741	−0.931735	−0.465867	−	0.884855i	$-0.654258\pi$
$587$	−22.5741	−0.931735	−0.465867	+	0.884855i	$0.654258\pi$
$588$	0	0
$589$	−5.60730	−0.231045
$590$	0	0
$591$	0	0
$592$	−49.5438	−2.03624
$593$	−1.50814	−0.0619317	−0.0309659	−	0.999520i	$-0.509858\pi$
$593$	−1.50814	−0.0619317	−0.0309659	+	0.999520i	$0.509858\pi$
$594$	0	0
$595$	0	0
$596$	0.119730	0.00490433
$597$	0	0
$598$	−61.8845	−2.53064
$599$	−2.77147	−0.113239	−0.0566196	−	0.998396i	$-0.518032\pi$
$599$	−2.77147	−0.113239	−0.0566196	+	0.998396i	$0.518032\pi$
$600$	0	0
$601$	−18.0452	−0.736078	−0.368039	−	0.929810i	$-0.619971\pi$
$601$	−18.0452	−0.736078	−0.368039	+	0.929810i	$0.619971\pi$
$602$	−32.6772	−1.33182
$603$	0	0
$604$	48.7746	1.98461
$605$	0	0
$606$	0	0
$607$	11.3139	0.459219	0.229609	−	0.973283i	$-0.426255\pi$
$607$	11.3139	0.459219	0.229609	+	0.973283i	$0.426255\pi$
$608$	3.58873	0.145542
$609$	0	0
$610$	0	0
$611$	−6.36961	−0.257687
$612$	0	0
$613$	−1.32044	−0.0533319	−0.0266659	−	0.999644i	$-0.508489\pi$
$613$	−1.32044	−0.0533319	−0.0266659	+	0.999644i	$0.508489\pi$
$614$	24.6795	0.995985
$615$	0	0
$616$	74.9975	3.02174
$617$	−31.4084	−1.26446	−0.632228	−	0.774783i	$-0.717859\pi$
$617$	−31.4084	−1.26446	−0.632228	+	0.774783i	$0.717859\pi$
$618$	0	0
$619$	−24.2173	−0.973377	−0.486689	−	0.873576i	$-0.661795\pi$
$619$	−24.2173	−0.973377	−0.486689	+	0.873576i	$0.661795\pi$
$620$	0	0
$621$	0	0
$622$	7.39591	0.296549
$623$	62.0548	2.48617
$624$	0	0
$625$	0	0
$626$	25.1976	1.00710
$627$	0	0
$628$	−29.8372	−1.19063
$629$	66.5766	2.65458
$630$	0	0
$631$	−16.7012	−0.664863	−0.332432	−	0.943127i	$-0.607869\pi$
$631$	−16.7012	−0.664863	−0.332432	+	0.943127i	$0.607869\pi$
$632$	−46.9449	−1.86737
$633$	0	0
$634$	−57.6589	−2.28993
$635$	0	0
$636$	0	0
$637$	40.1225	1.58971
$638$	−58.8954	−2.33169
$639$	0	0
$640$	0	0
$641$	−45.6730	−1.80398	−0.901988	−	0.431762i	$-0.857892\pi$
$641$	−45.6730	−1.80398	−0.901988	+	0.431762i	$0.857892\pi$
$642$	0	0
$643$	−36.4701	−1.43824	−0.719120	−	0.694886i	$-0.755455\pi$
$643$	−36.4701	−1.43824	−0.719120	+	0.694886i	$0.755455\pi$
$644$	82.5745	3.25389
$645$	0	0
$646$	−81.4164	−3.20329
$647$	−3.16329	−0.124362	−0.0621808	−	0.998065i	$-0.519806\pi$
$647$	−3.16329	−0.124362	−0.0621808	+	0.998065i	$0.519806\pi$
$648$	0	0
$649$	28.6308	1.12386
$650$	0	0
$651$	0	0
$652$	46.1042	1.80558
$653$	−34.2753	−1.34130	−0.670648	−	0.741776i	$-0.733984\pi$
$653$	−34.2753	−1.34130	−0.670648	+	0.741776i	$0.733984\pi$
$654$	0	0
$655$	0	0
$656$	−2.90974	−0.113606
$657$	0	0
$658$	12.6819	0.494392
$659$	−22.2146	−0.865357	−0.432679	−	0.901548i	$-0.642432\pi$
$659$	−22.2146	−0.865357	−0.432679	+	0.901548i	$0.642432\pi$
$660$	0	0
$661$	−22.4547	−0.873387	−0.436693	−	0.899610i	$-0.643851\pi$
$661$	−22.4547	−0.873387	−0.436693	+	0.899610i	$0.643851\pi$
$662$	49.9617	1.94182
$663$	0	0
$664$	−3.00329	−0.116550
$665$	0	0
$666$	0	0
$667$	−32.9330	−1.27517
$668$	−30.1713	−1.16736
$669$	0	0
$670$	0	0
$671$	38.9414	1.50332
$672$	0	0
$673$	9.90870	0.381952	0.190976	−	0.981595i	$-0.438835\pi$
$673$	9.90870	0.381952	0.190976	+	0.981595i	$0.438835\pi$
$674$	39.2961	1.51363
$675$	0	0
$676$	42.2526	1.62510
$677$	−16.1570	−0.620963	−0.310482	−	0.950579i	$-0.600490\pi$
$677$	−16.1570	−0.620963	−0.310482	+	0.950579i	$0.600490\pi$
$678$	0	0
$679$	15.5881	0.598216
$680$	0	0
$681$	0	0
$682$	9.29125	0.355780
$683$	−14.6765	−0.561583	−0.280791	−	0.959769i	$-0.590597\pi$
$683$	−14.6765	−0.561583	−0.280791	+	0.959769i	$0.590597\pi$
$684$	0	0
$685$	0	0
$686$	−12.4702	−0.476114
$687$	0	0
$688$	14.8882	0.567606
$689$	−50.5196	−1.92465
$690$	0	0
$691$	−4.78243	−0.181932	−0.0909662	−	0.995854i	$-0.528996\pi$
$691$	−4.78243	−0.181932	−0.0909662	+	0.995854i	$0.528996\pi$
$692$	87.2562	3.31698
$693$	0	0
$694$	17.0848	0.648529
$695$	0	0
$696$	0	0
$697$	3.91009	0.148105
$698$	−23.5435	−0.891136
$699$	0	0
$700$	0	0
$701$	−14.9186	−0.563467	−0.281734	−	0.959493i	$-0.590909\pi$
$701$	−14.9186	−0.563467	−0.281734	+	0.959493i	$0.590909\pi$
$702$	0	0
$703$	−63.3135	−2.38791
$704$	27.1646	1.02381
$705$	0	0
$706$	29.7855	1.12099
$707$	−17.5352	−0.659480
$708$	0	0
$709$	31.0264	1.16522	0.582611	−	0.812751i	$-0.302031\pi$
$709$	31.0264	1.16522	0.582611	+	0.812751i	$0.302031\pi$
$710$	0	0
$711$	0	0
$712$	−80.6455	−3.02232
$713$	5.19547	0.194572
$714$	0	0
$715$	0	0
$716$	−40.0439	−1.49651
$717$	0	0
$718$	71.0613	2.65198
$719$	−22.5881	−0.842394	−0.421197	−	0.906969i	$-0.638390\pi$
$719$	−22.5881	−0.842394	−0.421197	+	0.906969i	$0.638390\pi$
$720$	0	0
$721$	58.5064	2.17889
$722$	30.6382	1.14024
$723$	0	0
$724$	−16.0856	−0.597815
$725$	0	0
$726$	0	0
$727$	30.6620	1.13719	0.568596	−	0.822617i	$-0.307487\pi$
$727$	30.6620	1.13719	0.568596	+	0.822617i	$0.307487\pi$
$728$	−96.1457	−3.56340
$729$	0	0
$730$	0	0
$731$	−20.0066	−0.739971
$732$	0	0
$733$	17.6648	0.652463	0.326232	−	0.945290i	$-0.394221\pi$
$733$	17.6648	0.652463	0.326232	+	0.945290i	$0.394221\pi$
$734$	−57.5349	−2.12365
$735$	0	0
$736$	−3.32515	−0.122567
$737$	−38.8150	−1.42977
$738$	0	0
$739$	34.0146	1.25125	0.625623	−	0.780126i	$-0.284845\pi$
$739$	34.0146	1.25125	0.625623	+	0.780126i	$0.284845\pi$
$740$	0	0
$741$	0	0
$742$	100.585	3.69258
$743$	1.10230	0.0404394	0.0202197	−	0.999796i	$-0.493563\pi$
$743$	1.10230	0.0404394	0.0202197	+	0.999796i	$0.493563\pi$
$744$	0	0
$745$	0	0
$746$	41.7835	1.52980
$747$	0	0
$748$	90.4117	3.30578
$749$	−5.76543	−0.210664
$750$	0	0
$751$	29.9534	1.09302	0.546508	−	0.837454i	$-0.315957\pi$
$751$	29.9534	1.09302	0.546508	+	0.837454i	$0.315957\pi$
$752$	−5.77804	−0.210704
$753$	0	0
$754$	75.5030	2.74966
$755$	0	0
$756$	0	0
$757$	−20.3970	−0.741340	−0.370670	−	0.928765i	$-0.620872\pi$
$757$	−20.3970	−0.741340	−0.370670	+	0.928765i	$0.620872\pi$
$758$	−57.4882	−2.08806
$759$	0	0
$760$	0	0
$761$	42.3063	1.53360	0.766802	−	0.641884i	$-0.221847\pi$
$761$	42.3063	1.53360	0.766802	+	0.641884i	$0.221847\pi$
$762$	0	0
$763$	−49.9694	−1.80901
$764$	67.5630	2.44435
$765$	0	0
$766$	−62.4348	−2.25586
$767$	−36.7042	−1.32531
$768$	0	0
$769$	−2.54358	−0.0917239	−0.0458620	−	0.998948i	$-0.514603\pi$
$769$	−2.54358	−0.0917239	−0.0458620	+	0.998948i	$0.514603\pi$
$770$	0	0
$771$	0	0
$772$	−16.9697	−0.610751
$773$	−43.4158	−1.56156	−0.780778	−	0.624808i	$-0.785177\pi$
$773$	−43.4158	−1.56156	−0.780778	+	0.624808i	$0.785177\pi$
$774$	0	0
$775$	0	0
$776$	−20.2581	−0.727222
$777$	0	0
$778$	31.1139	1.11549
$779$	−3.71845	−0.133227
$780$	0	0
$781$	−13.8881	−0.496955
$782$	75.4366	2.69761
$783$	0	0
$784$	36.3962	1.29987
$785$	0	0
$786$	0	0
$787$	11.5860	0.412997	0.206498	−	0.978447i	$-0.433793\pi$
$787$	11.5860	0.412997	0.206498	+	0.978447i	$0.433793\pi$
$788$	40.7900	1.45308
$789$	0	0
$790$	0	0
$791$	18.4585	0.656309
$792$	0	0
$793$	−49.9223	−1.77279
$794$	−63.0114	−2.23619
$795$	0	0
$796$	47.8514	1.69605
$797$	37.0975	1.31406	0.657030	−	0.753864i	$-0.271812\pi$
$797$	37.0975	1.31406	0.657030	+	0.753864i	$0.271812\pi$
$798$	0	0
$799$	7.76449	0.274688
$800$	0	0
$801$	0	0
$802$	−91.4079	−3.22773
$803$	46.7520	1.64984
$804$	0	0
$805$	0	0
$806$	−11.9112	−0.419556
$807$	0	0
$808$	22.7885	0.801697
$809$	22.2275	0.781479	0.390739	−	0.920501i	$-0.372219\pi$
$809$	22.2275	0.781479	0.390739	+	0.920501i	$0.372219\pi$
$810$	0	0
$811$	4.80625	0.168770	0.0843852	−	0.996433i	$-0.473107\pi$
$811$	4.80625	0.168770	0.0843852	+	0.996433i	$0.473107\pi$
$812$	−100.746	−3.53550
$813$	0	0
$814$	104.910	3.67709
$815$	0	0
$816$	0	0
$817$	19.0260	0.665636
$818$	27.6690	0.967423
$819$	0	0
$820$	0	0
$821$	−54.7778	−1.91176	−0.955880	−	0.293758i	$-0.905094\pi$
$821$	−54.7778	−1.91176	−0.955880	+	0.293758i	$0.905094\pi$
$822$	0	0
$823$	−40.1760	−1.40045	−0.700224	−	0.713924i	$-0.746916\pi$
$823$	−40.1760	−1.40045	−0.700224	+	0.713924i	$0.746916\pi$
$824$	−76.0341	−2.64877
$825$	0	0
$826$	73.0781	2.54271
$827$	−1.84956	−0.0643154	−0.0321577	−	0.999483i	$-0.510238\pi$
$827$	−1.84956	−0.0643154	−0.0321577	+	0.999483i	$0.510238\pi$
$828$	0	0
$829$	12.8593	0.446622	0.223311	−	0.974747i	$-0.428313\pi$
$829$	12.8593	0.446622	0.223311	+	0.974747i	$0.428313\pi$
$830$	0	0
$831$	0	0
$832$	−34.8246	−1.20733
$833$	−48.9090	−1.69460
$834$	0	0
$835$	0	0
$836$	−85.9803	−2.97369
$837$	0	0
$838$	65.4388	2.26054
$839$	49.4500	1.70720	0.853601	−	0.520927i	$-0.174414\pi$
$839$	49.4500	1.70720	0.853601	+	0.520927i	$0.174414\pi$
$840$	0	0
$841$	11.1803	0.385529
$842$	53.0263	1.82741
$843$	0	0
$844$	−81.0862	−2.79110
$845$	0	0
$846$	0	0
$847$	−12.6562	−0.434871
$848$	−45.8278	−1.57373
$849$	0	0
$850$	0	0
$851$	58.6633	2.01095
$852$	0	0
$853$	−17.9548	−0.614762	−0.307381	−	0.951587i	$-0.599453\pi$
$853$	−17.9548	−0.614762	−0.307381	+	0.951587i	$0.599453\pi$
$854$	99.3953	3.40124
$855$	0	0
$856$	7.49267	0.256094
$857$	28.9951	0.990452	0.495226	−	0.868764i	$-0.335085\pi$
$857$	28.9951	0.990452	0.495226	+	0.868764i	$0.335085\pi$
$858$	0	0
$859$	−39.3399	−1.34226	−0.671130	−	0.741339i	$-0.734191\pi$
$859$	−39.3399	−1.34226	−0.671130	+	0.741339i	$0.734191\pi$
$860$	0	0
$861$	0	0
$862$	−17.7438	−0.604356
$863$	25.2944	0.861032	0.430516	−	0.902583i	$-0.358332\pi$
$863$	25.2944	0.861032	0.430516	+	0.902583i	$0.358332\pi$
$864$	0	0
$865$	0	0
$866$	29.8545	1.01450
$867$	0	0
$868$	15.8936	0.539463
$869$	34.8503	1.18222
$870$	0	0
$871$	49.7603	1.68606
$872$	64.9395	2.19913
$873$	0	0
$874$	−71.7392	−2.42662
$875$	0	0
$876$	0	0
$877$	−45.7315	−1.54424	−0.772121	−	0.635476i	$-0.780804\pi$
$877$	−45.7315	−1.54424	−0.772121	+	0.635476i	$0.780804\pi$
$878$	94.0301	3.17336
$879$	0	0
$880$	0	0
$881$	7.69529	0.259261	0.129630	−	0.991562i	$-0.458621\pi$
$881$	7.69529	0.259261	0.129630	+	0.991562i	$0.458621\pi$
$882$	0	0
$883$	5.44217	0.183144	0.0915719	−	0.995798i	$-0.470811\pi$
$883$	5.44217	0.183144	0.0915719	+	0.995798i	$0.470811\pi$
$884$	−115.906	−3.89836
$885$	0	0
$886$	61.3481	2.06103
$887$	45.0909	1.51400	0.757002	−	0.653412i	$-0.226663\pi$
$887$	45.0909	1.51400	0.757002	+	0.653412i	$0.226663\pi$
$888$	0	0
$889$	−57.5694	−1.93081
$890$	0	0
$891$	0	0
$892$	71.1322	2.38168
$893$	−7.38393	−0.247094
$894$	0	0
$895$	0	0
$896$	74.3419	2.48359
$897$	0	0
$898$	34.5955	1.15447
$899$	−6.33880	−0.211411
$900$	0	0
$901$	61.5830	2.05163
$902$	6.16143	0.205153
$903$	0	0
$904$	−23.9884	−0.797843
$905$	0	0
$906$	0	0
$907$	−25.3635	−0.842180	−0.421090	−	0.907019i	$-0.638352\pi$
$907$	−25.3635	−0.842180	−0.421090	+	0.907019i	$0.638352\pi$
$908$	34.7617	1.15361
$909$	0	0
$910$	0	0
$911$	2.49514	0.0826676	0.0413338	−	0.999145i	$-0.486839\pi$
$911$	2.49514	0.0826676	0.0413338	+	0.999145i	$0.486839\pi$
$912$	0	0
$913$	2.22954	0.0737871
$914$	−99.5124	−3.29158
$915$	0	0
$916$	−74.3149	−2.45543
$917$	43.0077	1.42024
$918$	0	0
$919$	17.7435	0.585306	0.292653	−	0.956219i	$-0.405462\pi$
$919$	17.7435	0.585306	0.292653	+	0.956219i	$0.405462\pi$
$920$	0	0
$921$	0	0
$922$	−33.8832	−1.11588
$923$	17.8043	0.586036
$924$	0	0
$925$	0	0
$926$	−58.9636	−1.93766
$927$	0	0
$928$	4.05689	0.133174
$929$	−28.5183	−0.935654	−0.467827	−	0.883820i	$-0.654963\pi$
$929$	−28.5183	−0.935654	−0.467827	+	0.883820i	$0.654963\pi$
$930$	0	0
$931$	46.5118	1.52436
$932$	−71.9665	−2.35734
$933$	0	0
$934$	−19.3191	−0.632139
$935$	0	0
$936$	0	0
$937$	7.18881	0.234848	0.117424	−	0.993082i	$-0.462536\pi$
$937$	7.18881	0.234848	0.117424	+	0.993082i	$0.462536\pi$
$938$	−99.0728	−3.23484
$939$	0	0
$940$	0	0
$941$	4.41233	0.143838	0.0719189	−	0.997410i	$-0.477088\pi$
$941$	4.41233	0.143838	0.0719189	+	0.997410i	$0.477088\pi$
$942$	0	0
$943$	3.44534	0.112196
$944$	−33.2954	−1.08367
$945$	0	0
$946$	−31.5259	−1.02500
$947$	−46.3158	−1.50506	−0.752530	−	0.658558i	$-0.771167\pi$
$947$	−46.3158	−1.50506	−0.752530	+	0.658558i	$0.771167\pi$
$948$	0	0
$949$	−59.9354	−1.94558
$950$	0	0
$951$	0	0
$952$	117.201	3.79850
$953$	−56.1358	−1.81842	−0.909209	−	0.416341i	$-0.863312\pi$
$953$	−56.1358	−1.81842	−0.909209	+	0.416341i	$0.863312\pi$
$954$	0	0
$955$	0	0
$956$	−4.81581	−0.155754
$957$	0	0
$958$	−89.3189	−2.88576
$959$	−12.8682	−0.415537
$960$	0	0
$961$	1.00000	0.0322581
$962$	−134.493	−4.33622
$963$	0	0
$964$	73.3149	2.36132
$965$	0	0
$966$	0	0
$967$	21.4773	0.690664	0.345332	−	0.938481i	$-0.387766\pi$
$967$	21.4773	0.690664	0.345332	+	0.938481i	$0.387766\pi$
$968$	16.4478	0.528651
$969$	0	0
$970$	0	0
$971$	−23.1016	−0.741366	−0.370683	−	0.928759i	$-0.620876\pi$
$971$	−23.1016	−0.741366	−0.370683	+	0.928759i	$0.620876\pi$
$972$	0	0
$973$	−45.9944	−1.47451
$974$	14.3640	0.460252
$975$	0	0
$976$	−45.2859	−1.44957
$977$	14.3103	0.457827	0.228913	−	0.973447i	$-0.426483\pi$
$977$	14.3103	0.457827	0.228913	+	0.973447i	$0.426483\pi$
$978$	0	0
$979$	59.8686	1.91341
$980$	0	0
$981$	0	0
$982$	−9.27052	−0.295834
$983$	−32.1775	−1.02630	−0.513152	−	0.858298i	$-0.671522\pi$
$983$	−32.1775	−1.02630	−0.513152	+	0.858298i	$0.671522\pi$
$984$	0	0
$985$	0	0
$986$	−92.0374	−2.93107
$987$	0	0
$988$	110.226	3.50674
$989$	−17.6286	−0.560557
$990$	0	0
$991$	−50.8207	−1.61437	−0.807186	−	0.590298i	$-0.799010\pi$
$991$	−50.8207	−1.61437	−0.807186	+	0.590298i	$0.799010\pi$
$992$	−0.640010	−0.0203203
$993$	0	0
$994$	−35.4484	−1.12435
$995$	0	0
$996$	0	0
$997$	39.1814	1.24089	0.620444	−	0.784251i	$-0.286952\pi$
$997$	39.1814	1.24089	0.620444	+	0.784251i	$0.286952\pi$
$998$	27.7428	0.878183
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6975.2.a.cc.1.6		6
3.2	odd	2		2325.2.a.y.1.1	✓	6
5.4	even	2		6975.2.a.ca.1.1		6
15.2	even	4		2325.2.c.r.1024.2		12
15.8	even	4		2325.2.c.r.1024.11		12
15.14	odd	2		2325.2.a.bb.1.6	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2325.2.a.y.1.1	✓	6	3.2	odd	2
2325.2.a.bb.1.6	yes	6	15.14	odd	2
2325.2.c.r.1024.2		12	15.2	even	4
2325.2.c.r.1024.11		12	15.8	even	4
6975.2.a.ca.1.1		6	5.4	even	2
6975.2.a.cc.1.6		6	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6975 = 3^{2} \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6975.a (trivial)

\(f(q)\)	\(=\)	\(q+2.46251 q^{2} +4.06395 q^{4} -3.91086 q^{7} +5.08250 q^{8} -3.77308 q^{11} +4.83704 q^{13} -9.63054 q^{14} +4.38781 q^{16} -5.89630 q^{17} +5.60730 q^{19} -9.29125 q^{22} -5.19547 q^{23} +11.9112 q^{26} -15.8936 q^{28} +6.33880 q^{29} -1.00000 q^{31} +0.640010 q^{32} -14.5197 q^{34} -11.2913 q^{37} +13.8080 q^{38} -0.663143 q^{41} +3.39308 q^{43} -15.3336 q^{44} -12.7939 q^{46} -1.31684 q^{47} +8.29486 q^{49} +19.6575 q^{52} -10.4443 q^{53} -19.8770 q^{56} +15.6093 q^{58} -7.58816 q^{59} -10.3208 q^{61} -2.46251 q^{62} -7.19958 q^{64} +10.2874 q^{67} -23.9623 q^{68} +3.68083 q^{71} -12.3909 q^{73} -27.8048 q^{74} +22.7878 q^{76} +14.7560 q^{77} -9.23657 q^{79} -1.63300 q^{82} -0.590907 q^{83} +8.35548 q^{86} -19.1767 q^{88} -15.8673 q^{89} -18.9170 q^{91} -21.1141 q^{92} -3.24273 q^{94} -3.98585 q^{97} +20.4262 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + q^{2} + 7 q^{4} - 2 q^{7} - 3 q^{8} - 7 q^{11} - 4 q^{13} - 10 q^{14} + 17 q^{16} + 17 q^{19} - 2 q^{22} - q^{23} - 2 q^{26} - 22 q^{28} + 8 q^{29} - 6 q^{31} - 35 q^{32} - 13 q^{34} - 14 q^{37}+ \cdots + 37 q^{98}+O(q^{100}) \) 6 * q + q^2 + 7 * q^4 - 2 * q^7 - 3 * q^8 - 7 * q^11 - 4 * q^13 - 10 * q^14 + 17 * q^16 + 17 * q^19 - 2 * q^22 - q^23 - 2 * q^26 - 22 * q^28 + 8 * q^29 - 6 * q^31 - 35 * q^32 - 13 * q^34 - 14 * q^37 + 10 * q^38 - 18 * q^41 - 15 * q^43 - 4 * q^44 - 10 * q^46 - 13 * q^47 + 18 * q^49 + 38 * q^52 - 13 * q^53 - 32 * q^56 - 23 * q^58 - 32 * q^59 - q^62 + 29 * q^64 - 9 * q^67 + 9 * q^68 - q^71 - 14 * q^73 - 20 * q^74 + 42 * q^76 - 14 * q^77 + 13 * q^79 + 32 * q^82 + 9 * q^83 + 11 * q^86 - 36 * q^88 - 34 * q^89 - 2 * q^91 - 46 * q^92 + 19 * q^94 - 30 * q^97 + 37 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more