LMFDB - Embedded newform 8325.2.a.cm.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8325,2,Mod(1,8325)] 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("8325.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8325, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8325 = 3^{2} \cdot 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8325.a (trivial)

Newform invariants

comment:select newform

sage:traces = [7,-1,0,7,0,0,0,0,0,0,-16,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$66.4754596827$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 10x^{5} + 9x^{4} + 26x^{3} - 23x^{2} - 9x + 5 $ x^7 - x^6 - 10x^5 + 9x^4 + 26x^3 - 23x^2 - 9*x + 5
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 925)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.548343$ of defining polynomial
Character	$\chi$	$=$	8325.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.548343 q^{2} -1.69932 q^{4} -1.23269 q^{7} -2.02850 q^{8} +0.976048 q^{11} -6.96588 q^{13} -0.675937 q^{14} +2.28633 q^{16} +2.51901 q^{17} -2.92164 q^{19} +0.535210 q^{22} +8.99438 q^{23} -3.81970 q^{26} +2.09473 q^{28} +0.687767 q^{29} +6.60794 q^{31} +5.31069 q^{32} +1.38128 q^{34} +1.00000 q^{37} -1.60206 q^{38} +0.193259 q^{41} +9.21747 q^{43} -1.65862 q^{44} +4.93201 q^{46} -3.18723 q^{47} -5.48048 q^{49} +11.8373 q^{52} -1.28828 q^{53} +2.50051 q^{56} +0.377133 q^{58} -12.2588 q^{59} -3.67468 q^{61} +3.62342 q^{62} -1.66057 q^{64} +8.16762 q^{67} -4.28061 q^{68} +2.82331 q^{71} +0.837276 q^{73} +0.548343 q^{74} +4.96481 q^{76} -1.20316 q^{77} +3.44333 q^{79} +0.105972 q^{82} -4.66960 q^{83} +5.05434 q^{86} -1.97991 q^{88} +6.74431 q^{89} +8.58676 q^{91} -15.2843 q^{92} -1.74770 q^{94} +2.21791 q^{97} -3.00518 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - q^{2} + 7 q^{4} - 16 q^{11} - q^{13} + 3 q^{14} + 3 q^{16} - 4 q^{17} + 9 q^{19} + q^{22} + q^{23} - 24 q^{26} + 7 q^{28} - 3 q^{29} + 11 q^{31} - 21 q^{32} - 23 q^{34} + 7 q^{37} + 32 q^{38}+ \cdots - 6 q^{98}+O(q^{100}) $ 7 * q - q^2 + 7 * q^4 - 16 * q^11 - q^13 + 3 * q^14 + 3 * q^16 - 4 * q^17 + 9 * q^19 + q^22 + q^23 - 24 * q^26 + 7 * q^28 - 3 * q^29 + 11 * q^31 - 21 * q^32 - 23 * q^34 + 7 * q^37 + 32 * q^38 - 29 * q^41 + 3 * q^43 - 18 * q^44 + 7 * q^46 + 2 * q^47 + 5 * q^49 + 30 * q^52 + 6 * q^53 - 13 * q^56 - 28 * q^58 - 42 * q^59 - 11 * q^61 - 11 * q^62 - 16 * q^64 + 3 * q^67 + 4 * q^68 - 10 * q^71 - 14 * q^73 - q^74 + 11 * q^76 + 11 * q^77 - 9 * q^79 - 6 * q^82 + 6 * q^83 + 22 * q^86 - 32 * q^88 - 12 * q^89 - 2 * q^91 - 5 * q^92 + 9 * q^94 - q^97 - 6 * 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 $

$2$	0.548343	0.387737	0.193869	−	0.981027i	$-0.437896\pi$
$2$	0.548343	0.387737	0.193869	+	0.981027i	$0.437896\pi$
$3$	0	0
$3$	0	0
$4$	−1.69932	−0.849660
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.23269	−0.465912	−0.232956	−	0.972487i	$-0.574840\pi$
$7$	−1.23269	−0.465912	−0.232956	+	0.972487i	$0.574840\pi$
$8$	−2.02850	−0.717182
$9$	0	0
$10$	0	0
$11$	0.976048	0.294290	0.147145	−	0.989115i	$-0.452992\pi$
$11$	0.976048	0.294290	0.147145	+	0.989115i	$0.452992\pi$
$12$	0	0
$13$	−6.96588	−1.93199	−0.965994	−	0.258564i	$-0.916751\pi$
$13$	−6.96588	−1.93199	−0.965994	+	0.258564i	$0.916751\pi$
$14$	−0.675937	−0.180652
$15$	0	0
$16$	2.28633	0.571582
$17$	2.51901	0.610951	0.305475	−	0.952200i	$-0.401185\pi$
$17$	2.51901	0.610951	0.305475	+	0.952200i	$0.401185\pi$
$18$	0	0
$19$	−2.92164	−0.670271	−0.335136	−	0.942170i	$-0.608782\pi$
$19$	−2.92164	−0.670271	−0.335136	+	0.942170i	$0.608782\pi$
$20$	0	0
$21$	0	0
$22$	0.535210	0.114107
$23$	8.99438	1.87546	0.937729	−	0.347368i	$-0.112925\pi$
$23$	8.99438	1.87546	0.937729	+	0.347368i	$0.112925\pi$
$24$	0	0
$25$	0	0
$26$	−3.81970	−0.749104
$27$	0	0
$28$	2.09473	0.395867
$29$	0.687767	0.127715	0.0638576	−	0.997959i	$-0.479660\pi$
$29$	0.687767	0.127715	0.0638576	+	0.997959i	$0.479660\pi$
$30$	0	0
$31$	6.60794	1.18682	0.593411	−	0.804900i	$-0.297781\pi$
$31$	6.60794	1.18682	0.593411	+	0.804900i	$0.297781\pi$
$32$	5.31069	0.938806
$33$	0	0
$34$	1.38128	0.236888
$35$	0	0
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	−1.60206	−0.259889
$39$	0	0
$40$	0	0
$41$	0.193259	0.0301820	0.0150910	−	0.999886i	$-0.495196\pi$
$41$	0.193259	0.0301820	0.0150910	+	0.999886i	$0.495196\pi$
$42$	0	0
$43$	9.21747	1.40565	0.702826	−	0.711362i	$-0.251921\pi$
$43$	9.21747	1.40565	0.702826	+	0.711362i	$0.251921\pi$
$44$	−1.65862	−0.250046
$45$	0	0
$46$	4.93201	0.727185
$47$	−3.18723	−0.464905	−0.232453	−	0.972608i	$-0.574675\pi$
$47$	−3.18723	−0.464905	−0.232453	+	0.972608i	$0.574675\pi$
$48$	0	0
$49$	−5.48048	−0.782926
$50$	0	0
$51$	0	0
$52$	11.8373	1.64153
$53$	−1.28828	−0.176959	−0.0884795	−	0.996078i	$-0.528201\pi$
$53$	−1.28828	−0.176959	−0.0884795	+	0.996078i	$0.528201\pi$
$54$	0	0
$55$	0	0
$56$	2.50051	0.334144
$57$	0	0
$58$	0.377133	0.0495199
$59$	−12.2588	−1.59596	−0.797978	−	0.602686i	$-0.794097\pi$
$59$	−12.2588	−1.59596	−0.797978	+	0.602686i	$0.794097\pi$
$60$	0	0
$61$	−3.67468	−0.470495	−0.235247	−	0.971936i	$-0.575590\pi$
$61$	−3.67468	−0.470495	−0.235247	+	0.971936i	$0.575590\pi$
$62$	3.62342	0.460175
$63$	0	0
$64$	−1.66057	−0.207572
$65$	0	0
$66$	0	0
$67$	8.16762	0.997833	0.498917	−	0.866650i	$-0.333731\pi$
$67$	8.16762	0.997833	0.498917	+	0.866650i	$0.333731\pi$
$68$	−4.28061	−0.519100
$69$	0	0
$70$	0	0
$71$	2.82331	0.335066	0.167533	−	0.985866i	$-0.446420\pi$
$71$	2.82331	0.335066	0.167533	+	0.985866i	$0.446420\pi$
$72$	0	0
$73$	0.837276	0.0979958	0.0489979	−	0.998799i	$-0.484397\pi$
$73$	0.837276	0.0979958	0.0489979	+	0.998799i	$0.484397\pi$
$74$	0.548343	0.0637436
$75$	0	0
$76$	4.96481	0.569503
$77$	−1.20316	−0.137113
$78$	0	0
$79$	3.44333	0.387405	0.193703	−	0.981060i	$-0.437950\pi$
$79$	3.44333	0.387405	0.193703	+	0.981060i	$0.437950\pi$
$80$	0	0
$81$	0	0
$82$	0.105972	0.0117027
$83$	−4.66960	−0.512555	−0.256278	−	0.966603i	$-0.582496\pi$
$83$	−4.66960	−0.512555	−0.256278	+	0.966603i	$0.582496\pi$
$84$	0	0
$85$	0	0
$86$	5.05434	0.545023
$87$	0	0
$88$	−1.97991	−0.211059
$89$	6.74431	0.714895	0.357448	−	0.933933i	$-0.383647\pi$
$89$	6.74431	0.714895	0.357448	+	0.933933i	$0.383647\pi$
$90$	0	0
$91$	8.58676	0.900137
$92$	−15.2843	−1.59350
$93$	0	0
$94$	−1.74770	−0.180261
$95$	0	0
$96$	0	0
$97$	2.21791	0.225195	0.112597	−	0.993641i	$-0.464083\pi$
$97$	2.21791	0.225195	0.112597	+	0.993641i	$0.464083\pi$
$98$	−3.00518	−0.303569
$99$	0	0
$100$	0	0
$101$	−4.83925	−0.481523	−0.240762	−	0.970584i	$-0.577397\pi$
$101$	−4.83925	−0.481523	−0.240762	+	0.970584i	$0.577397\pi$
$102$	0	0
$103$	8.95741	0.882600	0.441300	−	0.897360i	$-0.354517\pi$
$103$	8.95741	0.882600	0.441300	+	0.897360i	$0.354517\pi$
$104$	14.1303	1.38559
$105$	0	0
$106$	−0.706420	−0.0686136
$107$	15.6463	1.51259	0.756293	−	0.654233i	$-0.227008\pi$
$107$	15.6463	1.51259	0.756293	+	0.654233i	$0.227008\pi$
$108$	0	0
$109$	−12.4852	−1.19586	−0.597931	−	0.801548i	$-0.704010\pi$
$109$	−12.4852	−1.19586	−0.597931	+	0.801548i	$0.704010\pi$
$110$	0	0
$111$	0	0
$112$	−2.81833	−0.266307
$113$	−19.4522	−1.82991	−0.914953	−	0.403561i	$-0.867772\pi$
$113$	−19.4522	−1.82991	−0.914953	+	0.403561i	$0.867772\pi$
$114$	0	0
$115$	0	0
$116$	−1.16874	−0.108514
$117$	0	0
$118$	−6.72202	−0.618812
$119$	−3.10516	−0.284650
$120$	0	0
$121$	−10.0473	−0.913394
$122$	−2.01499	−0.182428
$123$	0	0
$124$	−11.2290	−1.00839
$125$	0	0
$126$	0	0
$127$	−5.25778	−0.466553	−0.233276	−	0.972410i	$-0.574945\pi$
$127$	−5.25778	−0.466553	−0.233276	+	0.972410i	$0.574945\pi$
$128$	−11.5319	−1.01929
$129$	0	0
$130$	0	0
$131$	−15.2972	−1.33653	−0.668263	−	0.743926i	$-0.732962\pi$
$131$	−15.2972	−1.33653	−0.668263	+	0.743926i	$0.732962\pi$
$132$	0	0
$133$	3.60148	0.312288
$134$	4.47866	0.386897
$135$	0	0
$136$	−5.10981	−0.438163
$137$	3.28024	0.280249	0.140125	−	0.990134i	$-0.455250\pi$
$137$	3.28024	0.280249	0.140125	+	0.990134i	$0.455250\pi$
$138$	0	0
$139$	0.561230	0.0476029	0.0238015	−	0.999717i	$-0.492423\pi$
$139$	0.561230	0.0476029	0.0238015	+	0.999717i	$0.492423\pi$
$140$	0	0
$141$	0	0
$142$	1.54815	0.129918
$143$	−6.79904	−0.568564
$144$	0	0
$145$	0	0
$146$	0.459115	0.0379966
$147$	0	0
$148$	−1.69932	−0.139683
$149$	−12.7327	−1.04311	−0.521553	−	0.853219i	$-0.674647\pi$
$149$	−12.7327	−1.04311	−0.521553	+	0.853219i	$0.674647\pi$
$150$	0	0
$151$	−18.1594	−1.47779	−0.738894	−	0.673822i	$-0.764651\pi$
$151$	−18.1594	−1.47779	−0.738894	+	0.673822i	$0.764651\pi$
$152$	5.92655	0.480707
$153$	0	0
$154$	−0.659747	−0.0531639
$155$	0	0
$156$	0	0
$157$	−12.5146	−0.998777	−0.499388	−	0.866378i	$-0.666442\pi$
$157$	−12.5146	−0.998777	−0.499388	+	0.866378i	$0.666442\pi$
$158$	1.88813	0.150211
$159$	0	0
$160$	0	0
$161$	−11.0873	−0.873799
$162$	0	0
$163$	−8.40802	−0.658567	−0.329284	−	0.944231i	$-0.606807\pi$
$163$	−8.40802	−0.658567	−0.329284	+	0.944231i	$0.606807\pi$
$164$	−0.328409	−0.0256444
$165$	0	0
$166$	−2.56055	−0.198737
$167$	−11.9553	−0.925130	−0.462565	−	0.886585i	$-0.653071\pi$
$167$	−11.9553	−0.925130	−0.462565	+	0.886585i	$0.653071\pi$
$168$	0	0
$169$	35.5235	2.73258
$170$	0	0
$171$	0	0
$172$	−15.6634	−1.19433
$173$	−21.6980	−1.64967	−0.824834	−	0.565375i	$-0.808732\pi$
$173$	−21.6980	−1.64967	−0.824834	+	0.565375i	$0.808732\pi$
$174$	0	0
$175$	0	0
$176$	2.23156	0.168211
$177$	0	0
$178$	3.69820	0.277192
$179$	3.99480	0.298586	0.149293	−	0.988793i	$-0.452300\pi$
$179$	3.99480	0.298586	0.149293	+	0.988793i	$0.452300\pi$
$180$	0	0
$181$	17.2016	1.27859	0.639293	−	0.768964i	$-0.279227\pi$
$181$	17.2016	1.27859	0.639293	+	0.768964i	$0.279227\pi$
$182$	4.70849	0.349017
$183$	0	0
$184$	−18.2451	−1.34504
$185$	0	0
$186$	0	0
$187$	2.45868	0.179797
$188$	5.41612	0.395011
$189$	0	0
$190$	0	0
$191$	20.4730	1.48137	0.740686	−	0.671852i	$-0.234501\pi$
$191$	20.4730	1.48137	0.740686	+	0.671852i	$0.234501\pi$
$192$	0	0
$193$	−20.4667	−1.47322	−0.736611	−	0.676317i	$-0.763575\pi$
$193$	−20.4667	−1.47322	−0.736611	+	0.676317i	$0.763575\pi$
$194$	1.21618	0.0873164
$195$	0	0
$196$	9.31309	0.665220
$197$	17.1879	1.22459	0.612294	−	0.790630i	$-0.290247\pi$
$197$	17.1879	1.22459	0.612294	+	0.790630i	$0.290247\pi$
$198$	0	0
$199$	−12.8818	−0.913164	−0.456582	−	0.889681i	$-0.650926\pi$
$199$	−12.8818	−0.913164	−0.456582	+	0.889681i	$0.650926\pi$
$200$	0	0
$201$	0	0
$202$	−2.65357	−0.186704
$203$	−0.847803	−0.0595041
$204$	0	0
$205$	0	0
$206$	4.91174	0.342217
$207$	0	0
$208$	−15.9263	−1.10429
$209$	−2.85167	−0.197254
$210$	0	0
$211$	14.6270	1.00697	0.503484	−	0.864005i	$-0.332051\pi$
$211$	14.6270	1.00697	0.503484	+	0.864005i	$0.332051\pi$
$212$	2.18920	0.150355
$213$	0	0
$214$	8.57955	0.586486
$215$	0	0
$216$	0	0
$217$	−8.14554	−0.552955
$218$	−6.84615	−0.463680
$219$	0	0
$220$	0	0
$221$	−17.5472	−1.18035
$222$	0	0
$223$	3.59602	0.240807	0.120404	−	0.992725i	$-0.461581\pi$
$223$	3.59602	0.240807	0.120404	+	0.992725i	$0.461581\pi$
$224$	−6.54642	−0.437401
$225$	0	0
$226$	−10.6665	−0.709523
$227$	−21.7284	−1.44217	−0.721084	−	0.692848i	$-0.756356\pi$
$227$	−21.7284	−1.44217	−0.721084	+	0.692848i	$0.756356\pi$
$228$	0	0
$229$	−14.6078	−0.965314	−0.482657	−	0.875809i	$-0.660328\pi$
$229$	−14.6078	−0.965314	−0.482657	+	0.875809i	$0.660328\pi$
$230$	0	0
$231$	0	0
$232$	−1.39513	−0.0915950
$233$	−18.5161	−1.21303	−0.606517	−	0.795071i	$-0.707434\pi$
$233$	−18.5161	−1.21303	−0.606517	+	0.795071i	$0.707434\pi$
$234$	0	0
$235$	0	0
$236$	20.8316	1.35602
$237$	0	0
$238$	−1.70269	−0.110369
$239$	−18.0845	−1.16979	−0.584896	−	0.811109i	$-0.698864\pi$
$239$	−18.0845	−1.16979	−0.584896	+	0.811109i	$0.698864\pi$
$240$	0	0
$241$	−12.4643	−0.802899	−0.401449	−	0.915881i	$-0.631493\pi$
$241$	−12.4643	−0.802899	−0.401449	+	0.915881i	$0.631493\pi$
$242$	−5.50939	−0.354157
$243$	0	0
$244$	6.24446	0.399761
$245$	0	0
$246$	0	0
$247$	20.3518	1.29496
$248$	−13.4042	−0.851167
$249$	0	0
$250$	0	0
$251$	−4.32919	−0.273256	−0.136628	−	0.990622i	$-0.543627\pi$
$251$	−4.32919	−0.273256	−0.136628	+	0.990622i	$0.543627\pi$
$252$	0	0
$253$	8.77895	0.551928
$254$	−2.88307	−0.180900
$255$	0	0
$256$	−3.00231	−0.187645
$257$	16.4633	1.02695	0.513475	−	0.858105i	$-0.328358\pi$
$257$	16.4633	1.02695	0.513475	+	0.858105i	$0.328358\pi$
$258$	0	0
$259$	−1.23269	−0.0765955
$260$	0	0
$261$	0	0
$262$	−8.38813	−0.518221
$263$	19.3023	1.19023	0.595116	−	0.803640i	$-0.297106\pi$
$263$	19.3023	1.19023	0.595116	+	0.803640i	$0.297106\pi$
$264$	0	0
$265$	0	0
$266$	1.97485	0.121086
$267$	0	0
$268$	−13.8794	−0.847819
$269$	9.62970	0.587133	0.293567	−	0.955939i	$-0.405158\pi$
$269$	9.62970	0.587133	0.293567	+	0.955939i	$0.405158\pi$
$270$	0	0
$271$	−3.65132	−0.221802	−0.110901	−	0.993831i	$-0.535374\pi$
$271$	−3.65132	−0.221802	−0.110901	+	0.993831i	$0.535374\pi$
$272$	5.75929	0.349208
$273$	0	0
$274$	1.79870	0.108663
$275$	0	0
$276$	0	0
$277$	25.0654	1.50603	0.753017	−	0.658002i	$-0.228598\pi$
$277$	25.0654	1.50603	0.753017	+	0.658002i	$0.228598\pi$
$278$	0.307747	0.0184574
$279$	0	0
$280$	0	0
$281$	−11.5359	−0.688177	−0.344089	−	0.938937i	$-0.611812\pi$
$281$	−11.5359	−0.688177	−0.344089	+	0.938937i	$0.611812\pi$
$282$	0	0
$283$	30.7318	1.82682	0.913408	−	0.407045i	$-0.133441\pi$
$283$	30.7318	1.82682	0.913408	+	0.407045i	$0.133441\pi$
$284$	−4.79771	−0.284692
$285$	0	0
$286$	−3.72821	−0.220454
$287$	−0.238228	−0.0140622
$288$	0	0
$289$	−10.6546	−0.626739
$290$	0	0
$291$	0	0
$292$	−1.42280	−0.0832631
$293$	−16.2157	−0.947332	−0.473666	−	0.880705i	$-0.657070\pi$
$293$	−16.2157	−0.947332	−0.473666	+	0.880705i	$0.657070\pi$
$294$	0	0
$295$	0	0
$296$	−2.02850	−0.117904
$297$	0	0
$298$	−6.98190	−0.404451
$299$	−62.6538	−3.62336
$300$	0	0
$301$	−11.3623	−0.654910
$302$	−9.95756	−0.572993
$303$	0	0
$304$	−6.67983	−0.383115
$305$	0	0
$306$	0	0
$307$	0.434062	0.0247732	0.0123866	−	0.999923i	$-0.496057\pi$
$307$	0.434062	0.0247732	0.0123866	+	0.999923i	$0.496057\pi$
$308$	2.04456	0.116500
$309$	0	0
$310$	0	0
$311$	21.0126	1.19151	0.595756	−	0.803165i	$-0.296852\pi$
$311$	21.0126	1.19151	0.595756	+	0.803165i	$0.296852\pi$
$312$	0	0
$313$	2.16272	0.122244	0.0611222	−	0.998130i	$-0.480532\pi$
$313$	2.16272	0.122244	0.0611222	+	0.998130i	$0.480532\pi$
$314$	−6.86232	−0.387263
$315$	0	0
$316$	−5.85132	−0.329162
$317$	28.3991	1.59505	0.797526	−	0.603284i	$-0.206142\pi$
$317$	28.3991	1.59505	0.797526	+	0.603284i	$0.206142\pi$
$318$	0	0
$319$	0.671294	0.0375853
$320$	0	0
$321$	0	0
$322$	−6.07963	−0.338805
$323$	−7.35967	−0.409503
$324$	0	0
$325$	0	0
$326$	−4.61048	−0.255351
$327$	0	0
$328$	−0.392026	−0.0216460
$329$	3.92886	0.216605
$330$	0	0
$331$	16.2418	0.892728	0.446364	−	0.894851i	$-0.352719\pi$
$331$	16.2418	0.892728	0.446364	+	0.894851i	$0.352719\pi$
$332$	7.93515	0.435498
$333$	0	0
$334$	−6.55561	−0.358707
$335$	0	0
$336$	0	0
$337$	−11.0000	−0.599207	−0.299604	−	0.954064i	$-0.596854\pi$
$337$	−11.0000	−0.599207	−0.299604	+	0.954064i	$0.596854\pi$
$338$	19.4791	1.05952
$339$	0	0
$340$	0	0
$341$	6.44967	0.349269
$342$	0	0
$343$	15.3845	0.830687
$344$	−18.6976	−1.00811
$345$	0	0
$346$	−11.8980	−0.639638
$347$	−8.15833	−0.437962	−0.218981	−	0.975729i	$-0.570273\pi$
$347$	−8.15833	−0.437962	−0.218981	+	0.975729i	$0.570273\pi$
$348$	0	0
$349$	−28.2422	−1.51177	−0.755886	−	0.654704i	$-0.772793\pi$
$349$	−28.2422	−1.51177	−0.755886	+	0.654704i	$0.772793\pi$
$350$	0	0
$351$	0	0
$352$	5.18349	0.276281
$353$	−1.95167	−0.103877	−0.0519385	−	0.998650i	$-0.516540\pi$
$353$	−1.95167	−0.103877	−0.0519385	+	0.998650i	$0.516540\pi$
$354$	0	0
$355$	0	0
$356$	−11.4607	−0.607418
$357$	0	0
$358$	2.19052	0.115773
$359$	−25.2874	−1.33462	−0.667310	−	0.744780i	$-0.732554\pi$
$359$	−25.2874	−1.33462	−0.667310	+	0.744780i	$0.732554\pi$
$360$	0	0
$361$	−10.4640	−0.550736
$362$	9.43239	0.495755
$363$	0	0
$364$	−14.5917	−0.764811
$365$	0	0
$366$	0	0
$367$	7.03397	0.367170	0.183585	−	0.983004i	$-0.441230\pi$
$367$	7.03397	0.367170	0.183585	+	0.983004i	$0.441230\pi$
$368$	20.5641	1.07198
$369$	0	0
$370$	0	0
$371$	1.58805	0.0824474
$372$	0	0
$373$	25.0922	1.29923	0.649613	−	0.760265i	$-0.274931\pi$
$373$	25.0922	1.29923	0.649613	+	0.760265i	$0.274931\pi$
$374$	1.34820	0.0697138
$375$	0	0
$376$	6.46529	0.333422
$377$	−4.79091	−0.246744
$378$	0	0
$379$	−18.9929	−0.975601	−0.487801	−	0.872955i	$-0.662201\pi$
$379$	−18.9929	−0.975601	−0.487801	+	0.872955i	$0.662201\pi$
$380$	0	0
$381$	0	0
$382$	11.2262	0.574383
$383$	30.2394	1.54516	0.772580	−	0.634918i	$-0.218966\pi$
$383$	30.2394	1.54516	0.772580	+	0.634918i	$0.218966\pi$
$384$	0	0
$385$	0	0
$386$	−11.2228	−0.571223
$387$	0	0
$388$	−3.76894	−0.191339
$389$	20.8628	1.05779	0.528894	−	0.848688i	$-0.322607\pi$
$389$	20.8628	1.05779	0.528894	+	0.848688i	$0.322607\pi$
$390$	0	0
$391$	22.6570	1.14581
$392$	11.1171	0.561500
$393$	0	0
$394$	9.42488	0.474819
$395$	0	0
$396$	0	0
$397$	−28.4743	−1.42908	−0.714541	−	0.699593i	$-0.753365\pi$
$397$	−28.4743	−1.42908	−0.714541	+	0.699593i	$0.753365\pi$
$398$	−7.06363	−0.354068
$399$	0	0
$400$	0	0
$401$	26.5071	1.32370	0.661851	−	0.749636i	$-0.269771\pi$
$401$	26.5071	1.32370	0.661851	+	0.749636i	$0.269771\pi$
$402$	0	0
$403$	−46.0302	−2.29293
$404$	8.22343	0.409131
$405$	0	0
$406$	−0.464887	−0.0230720
$407$	0.976048	0.0483809
$408$	0	0
$409$	7.94050	0.392632	0.196316	−	0.980541i	$-0.437102\pi$
$409$	7.94050	0.392632	0.196316	+	0.980541i	$0.437102\pi$
$410$	0	0
$411$	0	0
$412$	−15.2215	−0.749910
$413$	15.1113	0.743576
$414$	0	0
$415$	0	0
$416$	−36.9936	−1.81376
$417$	0	0
$418$	−1.56369	−0.0764827
$419$	−31.9064	−1.55873	−0.779366	−	0.626569i	$-0.784459\pi$
$419$	−31.9064	−1.55873	−0.779366	+	0.626569i	$0.784459\pi$
$420$	0	0
$421$	11.0436	0.538232	0.269116	−	0.963108i	$-0.413268\pi$
$421$	11.0436	0.538232	0.269116	+	0.963108i	$0.413268\pi$
$422$	8.02064	0.390439
$423$	0	0
$424$	2.61327	0.126912
$425$	0	0
$426$	0	0
$427$	4.52974	0.219209
$428$	−26.5881	−1.28518
$429$	0	0
$430$	0	0
$431$	−40.0005	−1.92676	−0.963379	−	0.268143i	$-0.913590\pi$
$431$	−40.0005	−1.92676	−0.963379	+	0.268143i	$0.913590\pi$
$432$	0	0
$433$	−3.42091	−0.164398	−0.0821992	−	0.996616i	$-0.526194\pi$
$433$	−3.42091	−0.164398	−0.0821992	+	0.996616i	$0.526194\pi$
$434$	−4.46655	−0.214401
$435$	0	0
$436$	21.2163	1.01608
$437$	−26.2784	−1.25707
$438$	0	0
$439$	1.87199	0.0893451	0.0446725	−	0.999002i	$-0.485776\pi$
$439$	1.87199	0.0893451	0.0446725	+	0.999002i	$0.485776\pi$
$440$	0	0
$441$	0	0
$442$	−9.62187	−0.457666
$443$	39.0684	1.85620	0.928098	−	0.372335i	$-0.121443\pi$
$443$	39.0684	1.85620	0.928098	+	0.372335i	$0.121443\pi$
$444$	0	0
$445$	0	0
$446$	1.97185	0.0933700
$447$	0	0
$448$	2.04697	0.0967102
$449$	−27.7382	−1.30905	−0.654523	−	0.756042i	$-0.727130\pi$
$449$	−27.7382	−1.30905	−0.654523	+	0.756042i	$0.727130\pi$
$450$	0	0
$451$	0.188630	0.00888225
$452$	33.0554	1.55480
$453$	0	0
$454$	−11.9146	−0.559182
$455$	0	0
$456$	0	0
$457$	3.94335	0.184462	0.0922310	−	0.995738i	$-0.470600\pi$
$457$	3.94335	0.184462	0.0922310	+	0.995738i	$0.470600\pi$
$458$	−8.01012	−0.374288
$459$	0	0
$460$	0	0
$461$	−31.1870	−1.45252	−0.726261	−	0.687419i	$-0.758744\pi$
$461$	−31.1870	−1.45252	−0.726261	+	0.687419i	$0.758744\pi$
$462$	0	0
$463$	−29.8007	−1.38496	−0.692478	−	0.721439i	$-0.743481\pi$
$463$	−29.8007	−1.38496	−0.692478	+	0.721439i	$0.743481\pi$
$464$	1.57246	0.0729996
$465$	0	0
$466$	−10.1532	−0.470338
$467$	−17.4785	−0.808807	−0.404404	−	0.914581i	$-0.632521\pi$
$467$	−17.4785	−0.808807	−0.404404	+	0.914581i	$0.632521\pi$
$468$	0	0
$469$	−10.0681	−0.464903
$470$	0	0
$471$	0	0
$472$	24.8669	1.14459
$473$	8.99670	0.413669
$474$	0	0
$475$	0	0
$476$	5.27666	0.241855
$477$	0	0
$478$	−9.91653	−0.453572
$479$	4.30979	0.196919	0.0984597	−	0.995141i	$-0.468608\pi$
$479$	4.30979	0.196919	0.0984597	+	0.995141i	$0.468608\pi$
$480$	0	0
$481$	−6.96588	−0.317617
$482$	−6.83474	−0.311314
$483$	0	0
$484$	17.0736	0.776074
$485$	0	0
$486$	0	0
$487$	38.1250	1.72761	0.863804	−	0.503829i	$-0.168076\pi$
$487$	38.1250	1.72761	0.863804	+	0.503829i	$0.168076\pi$
$488$	7.45408	0.337430
$489$	0	0
$490$	0	0
$491$	−36.0217	−1.62564	−0.812818	−	0.582517i	$-0.802068\pi$
$491$	−36.0217	−1.62564	−0.812818	+	0.582517i	$0.802068\pi$
$492$	0	0
$493$	1.73250	0.0780277
$494$	11.1598	0.502103
$495$	0	0
$496$	15.1079	0.678366
$497$	−3.48027	−0.156111
$498$	0	0
$499$	20.1879	0.903733	0.451867	−	0.892085i	$-0.350758\pi$
$499$	20.1879	0.903733	0.451867	+	0.892085i	$0.350758\pi$
$500$	0	0
$501$	0	0
$502$	−2.37388	−0.105952
$503$	−3.01511	−0.134437	−0.0672185	−	0.997738i	$-0.521412\pi$
$503$	−3.01511	−0.134437	−0.0672185	+	0.997738i	$0.521412\pi$
$504$	0	0
$505$	0	0
$506$	4.81388	0.214003
$507$	0	0
$508$	8.93465	0.396411
$509$	−24.8656	−1.10215	−0.551073	−	0.834457i	$-0.685782\pi$
$509$	−24.8656	−1.10215	−0.551073	+	0.834457i	$0.685782\pi$
$510$	0	0
$511$	−1.03210	−0.0456575
$512$	21.4176	0.946532
$513$	0	0
$514$	9.02752	0.398187
$515$	0	0
$516$	0	0
$517$	−3.11089	−0.136817
$518$	−0.675937	−0.0296989
$519$	0	0
$520$	0	0
$521$	−38.3050	−1.67817	−0.839085	−	0.544000i	$-0.816909\pi$
$521$	−38.3050	−1.67817	−0.839085	+	0.544000i	$0.816909\pi$
$522$	0	0
$523$	3.02688	0.132356	0.0661781	−	0.997808i	$-0.478919\pi$
$523$	3.02688	0.132356	0.0661781	+	0.997808i	$0.478919\pi$
$524$	25.9949	1.13559
$525$	0	0
$526$	10.5843	0.461497
$527$	16.6455	0.725090
$528$	0	0
$529$	57.8989	2.51734
$530$	0	0
$531$	0	0
$532$	−6.12006	−0.265338
$533$	−1.34622	−0.0583113
$534$	0	0
$535$	0	0
$536$	−16.5680	−0.715628
$537$	0	0
$538$	5.28038	0.227653
$539$	−5.34921	−0.230407
$540$	0	0
$541$	−26.8271	−1.15339	−0.576693	−	0.816961i	$-0.695657\pi$
$541$	−26.8271	−1.15339	−0.576693	+	0.816961i	$0.695657\pi$
$542$	−2.00218	−0.0860008
$543$	0	0
$544$	13.3777	0.573564
$545$	0	0
$546$	0	0
$547$	−28.1420	−1.20326	−0.601632	−	0.798773i	$-0.705483\pi$
$547$	−28.1420	−1.20326	−0.601632	+	0.798773i	$0.705483\pi$
$548$	−5.57417	−0.238117
$549$	0	0
$550$	0	0
$551$	−2.00941	−0.0856038
$552$	0	0
$553$	−4.24455	−0.180497
$554$	13.7444	0.583945
$555$	0	0
$556$	−0.953710	−0.0404463
$557$	−16.3022	−0.690748	−0.345374	−	0.938465i	$-0.612248\pi$
$557$	−16.3022	−0.690748	−0.345374	+	0.938465i	$0.612248\pi$
$558$	0	0
$559$	−64.2078	−2.71570
$560$	0	0
$561$	0	0
$562$	−6.32566	−0.266832
$563$	−10.0364	−0.422985	−0.211492	−	0.977380i	$-0.567832\pi$
$563$	−10.0364	−0.422985	−0.211492	+	0.977380i	$0.567832\pi$
$564$	0	0
$565$	0	0
$566$	16.8516	0.708325
$567$	0	0
$568$	−5.72709	−0.240303
$569$	4.06296	0.170328	0.0851640	−	0.996367i	$-0.472859\pi$
$569$	4.06296	0.170328	0.0851640	+	0.996367i	$0.472859\pi$
$570$	0	0
$571$	−30.9715	−1.29612	−0.648058	−	0.761591i	$-0.724418\pi$
$571$	−30.9715	−1.29612	−0.648058	+	0.761591i	$0.724418\pi$
$572$	11.5537	0.483086
$573$	0	0
$574$	−0.130631	−0.00545243
$575$	0	0
$576$	0	0
$577$	−1.38260	−0.0575583	−0.0287792	−	0.999586i	$-0.509162\pi$
$577$	−1.38260	−0.0575583	−0.0287792	+	0.999586i	$0.509162\pi$
$578$	−5.84236	−0.243010
$579$	0	0
$580$	0	0
$581$	5.75616	0.238806
$582$	0	0
$583$	−1.25742	−0.0520772
$584$	−1.69841	−0.0702808
$585$	0	0
$586$	−8.89178	−0.367316
$587$	−20.3736	−0.840907	−0.420453	−	0.907314i	$-0.638129\pi$
$587$	−20.3736	−0.840907	−0.420453	+	0.907314i	$0.638129\pi$
$588$	0	0
$589$	−19.3061	−0.795493
$590$	0	0
$591$	0	0
$592$	2.28633	0.0939674
$593$	−28.4271	−1.16736	−0.583681	−	0.811983i	$-0.698388\pi$
$593$	−28.4271	−1.16736	−0.583681	+	0.811983i	$0.698388\pi$
$594$	0	0
$595$	0	0
$596$	21.6370	0.886284
$597$	0	0
$598$	−34.3558	−1.40491
$599$	−4.99991	−0.204291	−0.102145	−	0.994769i	$-0.532571\pi$
$599$	−4.99991	−0.204291	−0.102145	+	0.994769i	$0.532571\pi$
$600$	0	0
$601$	−26.3404	−1.07445	−0.537223	−	0.843440i	$-0.680527\pi$
$601$	−26.3404	−1.07445	−0.537223	+	0.843440i	$0.680527\pi$
$602$	−6.23043	−0.253933
$603$	0	0
$604$	30.8585	1.25562
$605$	0	0
$606$	0	0
$607$	21.0686	0.855147	0.427573	−	0.903981i	$-0.359369\pi$
$607$	21.0686	0.855147	0.427573	+	0.903981i	$0.359369\pi$
$608$	−15.5159	−0.629254
$609$	0	0
$610$	0	0
$611$	22.2019	0.898192
$612$	0	0
$613$	−2.19848	−0.0887959	−0.0443980	−	0.999014i	$-0.514137\pi$
$613$	−2.19848	−0.0887959	−0.0443980	+	0.999014i	$0.514137\pi$
$614$	0.238015	0.00960551
$615$	0	0
$616$	2.44061	0.0983351
$617$	−19.0587	−0.767275	−0.383638	−	0.923484i	$-0.625329\pi$
$617$	−19.0587	−0.767275	−0.383638	+	0.923484i	$0.625329\pi$
$618$	0	0
$619$	26.9627	1.08372	0.541862	−	0.840467i	$-0.317720\pi$
$619$	26.9627	1.08372	0.541862	+	0.840467i	$0.317720\pi$
$620$	0	0
$621$	0	0
$622$	11.5221	0.461994
$623$	−8.31363	−0.333079
$624$	0	0
$625$	0	0
$626$	1.18592	0.0473987
$627$	0	0
$628$	21.2664	0.848620
$629$	2.51901	0.100440
$630$	0	0
$631$	−32.6629	−1.30029	−0.650145	−	0.759810i	$-0.725292\pi$
$631$	−32.6629	−1.30029	−0.650145	+	0.759810i	$0.725292\pi$
$632$	−6.98479	−0.277840
$633$	0	0
$634$	15.5725	0.618461
$635$	0	0
$636$	0	0
$637$	38.1764	1.51260
$638$	0.368100	0.0145732
$639$	0	0
$640$	0	0
$641$	33.9051	1.33917	0.669586	−	0.742735i	$-0.266472\pi$
$641$	33.9051	1.33917	0.669586	+	0.742735i	$0.266472\pi$
$642$	0	0
$643$	−23.2105	−0.915331	−0.457666	−	0.889124i	$-0.651314\pi$
$643$	−23.2105	−0.915331	−0.457666	+	0.889124i	$0.651314\pi$
$644$	18.8408	0.742432
$645$	0	0
$646$	−4.03562	−0.158779
$647$	−23.4822	−0.923180	−0.461590	−	0.887093i	$-0.652721\pi$
$647$	−23.4822	−0.923180	−0.461590	+	0.887093i	$0.652721\pi$
$648$	0	0
$649$	−11.9652	−0.469674
$650$	0	0
$651$	0	0
$652$	14.2879	0.559558
$653$	−6.98761	−0.273446	−0.136723	−	0.990609i	$-0.543657\pi$
$653$	−6.98761	−0.273446	−0.136723	+	0.990609i	$0.543657\pi$
$654$	0	0
$655$	0	0
$656$	0.441853	0.0172515
$657$	0	0
$658$	2.15437	0.0839859
$659$	−13.4171	−0.522656	−0.261328	−	0.965250i	$-0.584160\pi$
$659$	−13.4171	−0.522656	−0.261328	+	0.965250i	$0.584160\pi$
$660$	0	0
$661$	4.09587	0.159311	0.0796555	−	0.996822i	$-0.474618\pi$
$661$	4.09587	0.159311	0.0796555	+	0.996822i	$0.474618\pi$
$662$	8.90606	0.346144
$663$	0	0
$664$	9.47227	0.367596
$665$	0	0
$666$	0	0
$667$	6.18604	0.239524
$668$	20.3159	0.786045
$669$	0	0
$670$	0	0
$671$	−3.58667	−0.138462
$672$	0	0
$673$	15.0709	0.580939	0.290469	−	0.956884i	$-0.406189\pi$
$673$	15.0709	0.580939	0.290469	+	0.956884i	$0.406189\pi$
$674$	−6.03177	−0.232335
$675$	0	0
$676$	−60.3658	−2.32176
$677$	7.36551	0.283080	0.141540	−	0.989933i	$-0.454795\pi$
$677$	7.36551	0.283080	0.141540	+	0.989933i	$0.454795\pi$
$678$	0	0
$679$	−2.73399	−0.104921
$680$	0	0
$681$	0	0
$682$	3.53664	0.135425
$683$	46.2794	1.77083	0.885415	−	0.464801i	$-0.153874\pi$
$683$	46.2794	1.77083	0.885415	+	0.464801i	$0.153874\pi$
$684$	0	0
$685$	0	0
$686$	8.43601	0.322088
$687$	0	0
$688$	21.0741	0.803444
$689$	8.97401	0.341883
$690$	0	0
$691$	5.06739	0.192773	0.0963863	−	0.995344i	$-0.469272\pi$
$691$	5.06739	0.192773	0.0963863	+	0.995344i	$0.469272\pi$
$692$	36.8718	1.40166
$693$	0	0
$694$	−4.47357	−0.169814
$695$	0	0
$696$	0	0
$697$	0.486823	0.0184397
$698$	−15.4864	−0.586170
$699$	0	0
$700$	0	0
$701$	−20.9220	−0.790213	−0.395107	−	0.918635i	$-0.629292\pi$
$701$	−20.9220	−0.790213	−0.395107	+	0.918635i	$0.629292\pi$
$702$	0	0
$703$	−2.92164	−0.110192
$704$	−1.62080	−0.0610862
$705$	0	0
$706$	−1.07019	−0.0402770
$707$	5.96528	0.224348
$708$	0	0
$709$	−11.5698	−0.434515	−0.217257	−	0.976114i	$-0.569711\pi$
$709$	−11.5698	−0.434515	−0.217257	+	0.976114i	$0.569711\pi$
$710$	0	0
$711$	0	0
$712$	−13.6808	−0.512710
$713$	59.4344	2.22583
$714$	0	0
$715$	0	0
$716$	−6.78845	−0.253696
$717$	0	0
$718$	−13.8662	−0.517482
$719$	−20.2226	−0.754176	−0.377088	−	0.926177i	$-0.623075\pi$
$719$	−20.2226	−0.754176	−0.377088	+	0.926177i	$0.623075\pi$
$720$	0	0
$721$	−11.0417	−0.411214
$722$	−5.73786	−0.213541
$723$	0	0
$724$	−29.2310	−1.08636
$725$	0	0
$726$	0	0
$727$	−33.7740	−1.25261	−0.626304	−	0.779579i	$-0.715433\pi$
$727$	−33.7740	−1.25261	−0.626304	+	0.779579i	$0.715433\pi$
$728$	−17.4182	−0.645562
$729$	0	0
$730$	0	0
$731$	23.2189	0.858784
$732$	0	0
$733$	−49.8177	−1.84006	−0.920030	−	0.391849i	$-0.871836\pi$
$733$	−49.8177	−1.84006	−0.920030	+	0.391849i	$0.871836\pi$
$734$	3.85703	0.142366
$735$	0	0
$736$	47.7663	1.76069
$737$	7.97199	0.293652
$738$	0	0
$739$	−6.25259	−0.230005	−0.115003	−	0.993365i	$-0.536688\pi$
$739$	−6.25259	−0.230005	−0.115003	+	0.993365i	$0.536688\pi$
$740$	0	0
$741$	0	0
$742$	0.870796	0.0319679
$743$	0.401960	0.0147465	0.00737324	−	0.999973i	$-0.497653\pi$
$743$	0.401960	0.0147465	0.00737324	+	0.999973i	$0.497653\pi$
$744$	0	0
$745$	0	0
$746$	13.7592	0.503758
$747$	0	0
$748$	−4.17808	−0.152766
$749$	−19.2870	−0.704733
$750$	0	0
$751$	48.4218	1.76694	0.883469	−	0.468490i	$-0.155202\pi$
$751$	48.4218	1.76694	0.883469	+	0.468490i	$0.155202\pi$
$752$	−7.28705	−0.265731
$753$	0	0
$754$	−2.62706	−0.0956719
$755$	0	0
$756$	0	0
$757$	16.3058	0.592643	0.296322	−	0.955088i	$-0.404240\pi$
$757$	16.3058	0.592643	0.296322	+	0.955088i	$0.404240\pi$
$758$	−10.4146	−0.378277
$759$	0	0
$760$	0	0
$761$	37.6100	1.36336	0.681681	−	0.731650i	$-0.261249\pi$
$761$	37.6100	1.36336	0.681681	+	0.731650i	$0.261249\pi$
$762$	0	0
$763$	15.3903	0.557167
$764$	−34.7901	−1.25866
$765$	0	0
$766$	16.5816	0.599116
$767$	85.3932	3.08337
$768$	0	0
$769$	35.9898	1.29783	0.648913	−	0.760863i	$-0.275224\pi$
$769$	35.9898	1.29783	0.648913	+	0.760863i	$0.275224\pi$
$770$	0	0
$771$	0	0
$772$	34.7794	1.25174
$773$	−11.1521	−0.401114	−0.200557	−	0.979682i	$-0.564275\pi$
$773$	−11.1521	−0.401114	−0.200557	+	0.979682i	$0.564275\pi$
$774$	0	0
$775$	0	0
$776$	−4.49903	−0.161506
$777$	0	0
$778$	11.4400	0.410144
$779$	−0.564634	−0.0202301
$780$	0	0
$781$	2.75569	0.0986064
$782$	12.4238	0.444274
$783$	0	0
$784$	−12.5302	−0.447506
$785$	0	0
$786$	0	0
$787$	−28.9665	−1.03254	−0.516271	−	0.856425i	$-0.672680\pi$
$787$	−28.9665	−1.03254	−0.516271	+	0.856425i	$0.672680\pi$
$788$	−29.2078	−1.04048
$789$	0	0
$790$	0	0
$791$	23.9785	0.852576
$792$	0	0
$793$	25.5974	0.908991
$794$	−15.6137	−0.554109
$795$	0	0
$796$	21.8902	0.775879
$797$	−54.2642	−1.92214	−0.961068	−	0.276312i	$-0.910888\pi$
$797$	−54.2642	−1.92214	−0.961068	+	0.276312i	$0.910888\pi$
$798$	0	0
$799$	−8.02868	−0.284034
$800$	0	0
$801$	0	0
$802$	14.5350	0.513249
$803$	0.817222	0.0288391
$804$	0	0
$805$	0	0
$806$	−25.2403	−0.889053
$807$	0	0
$808$	9.81640	0.345340
$809$	−16.1657	−0.568355	−0.284177	−	0.958772i	$-0.591720\pi$
$809$	−16.1657	−0.568355	−0.284177	+	0.958772i	$0.591720\pi$
$810$	0	0
$811$	−29.0693	−1.02076	−0.510381	−	0.859948i	$-0.670496\pi$
$811$	−29.0693	−1.02076	−0.510381	+	0.859948i	$0.670496\pi$
$812$	1.44069	0.0505582
$813$	0	0
$814$	0.535210	0.0187591
$815$	0	0
$816$	0	0
$817$	−26.9302	−0.942167
$818$	4.35412	0.152238
$819$	0	0
$820$	0	0
$821$	−30.7078	−1.07171	−0.535855	−	0.844310i	$-0.680010\pi$
$821$	−30.7078	−1.07171	−0.535855	+	0.844310i	$0.680010\pi$
$822$	0	0
$823$	−3.08648	−0.107588	−0.0537940	−	0.998552i	$-0.517131\pi$
$823$	−3.08648	−0.107588	−0.0537940	+	0.998552i	$0.517131\pi$
$824$	−18.1701	−0.632985
$825$	0	0
$826$	8.28615	0.288312
$827$	−37.0441	−1.28815	−0.644074	−	0.764963i	$-0.722757\pi$
$827$	−37.0441	−1.28815	−0.644074	+	0.764963i	$0.722757\pi$
$828$	0	0
$829$	−45.3315	−1.57443	−0.787215	−	0.616679i	$-0.788478\pi$
$829$	−45.3315	−1.57443	−0.787215	+	0.616679i	$0.788478\pi$
$830$	0	0
$831$	0	0
$832$	11.5674	0.401026
$833$	−13.8054	−0.478329
$834$	0	0
$835$	0	0
$836$	4.84589	0.167599
$837$	0	0
$838$	−17.4957	−0.604379
$839$	45.1775	1.55970	0.779850	−	0.625967i	$-0.215296\pi$
$839$	45.1775	1.55970	0.779850	+	0.625967i	$0.215296\pi$
$840$	0	0
$841$	−28.5270	−0.983689
$842$	6.05569	0.208693
$843$	0	0
$844$	−24.8560	−0.855579
$845$	0	0
$846$	0	0
$847$	12.3852	0.425561
$848$	−2.94543	−0.101147
$849$	0	0
$850$	0	0
$851$	8.99438	0.308323
$852$	0	0
$853$	21.6482	0.741220	0.370610	−	0.928789i	$-0.379149\pi$
$853$	21.6482	0.741220	0.370610	+	0.928789i	$0.379149\pi$
$854$	2.48385	0.0849957
$855$	0	0
$856$	−31.7385	−1.08480
$857$	−1.48050	−0.0505729	−0.0252865	−	0.999680i	$-0.508050\pi$
$857$	−1.48050	−0.0505729	−0.0252865	+	0.999680i	$0.508050\pi$
$858$	0	0
$859$	20.1546	0.687666	0.343833	−	0.939031i	$-0.388275\pi$
$859$	20.1546	0.687666	0.343833	+	0.939031i	$0.388275\pi$
$860$	0	0
$861$	0	0
$862$	−21.9340	−0.747076
$863$	−9.07168	−0.308803	−0.154402	−	0.988008i	$-0.549345\pi$
$863$	−9.07168	−0.308803	−0.154402	+	0.988008i	$0.549345\pi$
$864$	0	0
$865$	0	0
$866$	−1.87583	−0.0637434
$867$	0	0
$868$	13.8419	0.469824
$869$	3.36086	0.114009
$870$	0	0
$871$	−56.8946	−1.92780
$872$	25.3261	0.857650
$873$	0	0
$874$	−14.4096	−0.487411
$875$	0	0
$876$	0	0
$877$	−3.52847	−0.119148	−0.0595739	−	0.998224i	$-0.518974\pi$
$877$	−3.52847	−0.119148	−0.0595739	+	0.998224i	$0.518974\pi$
$878$	1.02649	0.0346424
$879$	0	0
$880$	0	0
$881$	32.2604	1.08688	0.543441	−	0.839448i	$-0.317121\pi$
$881$	32.2604	1.08688	0.543441	+	0.839448i	$0.317121\pi$
$882$	0	0
$883$	31.5621	1.06215	0.531074	−	0.847326i	$-0.321789\pi$
$883$	31.5621	1.06215	0.531074	+	0.847326i	$0.321789\pi$
$884$	29.8182	1.00290
$885$	0	0
$886$	21.4229	0.719717
$887$	34.8507	1.17017	0.585087	−	0.810971i	$-0.301061\pi$
$887$	34.8507	1.17017	0.585087	+	0.810971i	$0.301061\pi$
$888$	0	0
$889$	6.48121	0.217373
$890$	0	0
$891$	0	0
$892$	−6.11079	−0.204604
$893$	9.31195	0.311613
$894$	0	0
$895$	0	0
$896$	14.2153	0.474899
$897$	0	0
$898$	−15.2100	−0.507566
$899$	4.54473	0.151575
$900$	0	0
$901$	−3.24520	−0.108113
$902$	0.103434	0.00344398
$903$	0	0
$904$	39.4587	1.31238
$905$	0	0
$906$	0	0
$907$	−36.6888	−1.21823	−0.609116	−	0.793081i	$-0.708476\pi$
$907$	−36.6888	−1.21823	−0.609116	+	0.793081i	$0.708476\pi$
$908$	36.9236	1.22535
$909$	0	0
$910$	0	0
$911$	−27.6843	−0.917221	−0.458610	−	0.888637i	$-0.651653\pi$
$911$	−27.6843	−0.917221	−0.458610	+	0.888637i	$0.651653\pi$
$912$	0	0
$913$	−4.55776	−0.150840
$914$	2.16231	0.0715228
$915$	0	0
$916$	24.8234	0.820188
$917$	18.8567	0.622704
$918$	0	0
$919$	34.3200	1.13211	0.566057	−	0.824366i	$-0.308468\pi$
$919$	34.3200	1.13211	0.566057	+	0.824366i	$0.308468\pi$
$920$	0	0
$921$	0	0
$922$	−17.1012	−0.563197
$923$	−19.6669	−0.647343
$924$	0	0
$925$	0	0
$926$	−16.3410	−0.536999
$927$	0	0
$928$	3.65252	0.119900
$929$	22.9783	0.753894	0.376947	−	0.926235i	$-0.376974\pi$
$929$	22.9783	0.753894	0.376947	+	0.926235i	$0.376974\pi$
$930$	0	0
$931$	16.0120	0.524772
$932$	31.4649	1.03067
$933$	0	0
$934$	−9.58421	−0.313605
$935$	0	0
$936$	0	0
$937$	45.7823	1.49564	0.747822	−	0.663899i	$-0.231100\pi$
$937$	45.7823	1.49564	0.747822	+	0.663899i	$0.231100\pi$
$938$	−5.52079	−0.180260
$939$	0	0
$940$	0	0
$941$	16.9929	0.553952	0.276976	−	0.960877i	$-0.410668\pi$
$941$	16.9929	0.553952	0.276976	+	0.960877i	$0.410668\pi$
$942$	0	0
$943$	1.73825	0.0566051
$944$	−28.0276	−0.912219
$945$	0	0
$946$	4.93328	0.160395
$947$	23.4730	0.762769	0.381384	−	0.924417i	$-0.375447\pi$
$947$	23.4730	0.762769	0.381384	+	0.924417i	$0.375447\pi$
$948$	0	0
$949$	−5.83237	−0.189327
$950$	0	0
$951$	0	0
$952$	6.29881	0.204146
$953$	24.6541	0.798623	0.399312	−	0.916815i	$-0.369249\pi$
$953$	24.6541	0.798623	0.399312	+	0.916815i	$0.369249\pi$
$954$	0	0
$955$	0	0
$956$	30.7314	0.993924
$957$	0	0
$958$	2.36325	0.0763530
$959$	−4.04351	−0.130572
$960$	0	0
$961$	12.6649	0.408546
$962$	−3.81970	−0.123152
$963$	0	0
$964$	21.1809	0.682191
$965$	0	0
$966$	0	0
$967$	11.3662	0.365512	0.182756	−	0.983158i	$-0.441498\pi$
$967$	11.3662	0.365512	0.182756	+	0.983158i	$0.441498\pi$
$968$	20.3810	0.655070
$969$	0	0
$970$	0	0
$971$	1.93826	0.0622017	0.0311008	−	0.999516i	$-0.490099\pi$
$971$	1.93826	0.0622017	0.0311008	+	0.999516i	$0.490099\pi$
$972$	0	0
$973$	−0.691822	−0.0221788
$974$	20.9056	0.669858
$975$	0	0
$976$	−8.40152	−0.268926
$977$	12.9186	0.413303	0.206651	−	0.978415i	$-0.433743\pi$
$977$	12.9186	0.413303	0.206651	+	0.978415i	$0.433743\pi$
$978$	0	0
$979$	6.58277	0.210386
$980$	0	0
$981$	0	0
$982$	−19.7523	−0.630320
$983$	−42.8843	−1.36780	−0.683898	−	0.729578i	$-0.739717\pi$
$983$	−42.8843	−1.36780	−0.683898	+	0.729578i	$0.739717\pi$
$984$	0	0
$985$	0	0
$986$	0.950002	0.0302542
$987$	0	0
$988$	−34.5843	−1.10027
$989$	82.9054	2.63624
$990$	0	0
$991$	−30.3017	−0.962564	−0.481282	−	0.876566i	$-0.659829\pi$
$991$	−30.3017	−0.962564	−0.481282	+	0.876566i	$0.659829\pi$
$992$	35.0927	1.11419
$993$	0	0
$994$	−1.90838	−0.0605302
$995$	0	0
$996$	0	0
$997$	−9.76058	−0.309121	−0.154560	−	0.987983i	$-0.549396\pi$
$997$	−9.76058	−0.309121	−0.154560	+	0.987983i	$0.549396\pi$
$998$	11.0699	0.350411
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8325.2.a.cm.1.5		7
3.2	odd	2		925.2.a.k.1.3	yes	7
5.4	even	2		8325.2.a.cn.1.3		7
15.2	even	4		925.2.b.i.149.6		14
15.8	even	4		925.2.b.i.149.9		14
15.14	odd	2		925.2.a.j.1.5	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
925.2.a.j.1.5	✓	7	15.14	odd	2
925.2.a.k.1.3	yes	7	3.2	odd	2
925.2.b.i.149.6		14	15.2	even	4
925.2.b.i.149.9		14	15.8	even	4
8325.2.a.cm.1.5		7	1.1	even	1	trivial
8325.2.a.cn.1.3		7	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8325 = 3^{2} \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8325.a (trivial)

\(f(q)\)	\(=\)	\(q+0.548343 q^{2} -1.69932 q^{4} -1.23269 q^{7} -2.02850 q^{8} +0.976048 q^{11} -6.96588 q^{13} -0.675937 q^{14} +2.28633 q^{16} +2.51901 q^{17} -2.92164 q^{19} +0.535210 q^{22} +8.99438 q^{23} -3.81970 q^{26} +2.09473 q^{28} +0.687767 q^{29} +6.60794 q^{31} +5.31069 q^{32} +1.38128 q^{34} +1.00000 q^{37} -1.60206 q^{38} +0.193259 q^{41} +9.21747 q^{43} -1.65862 q^{44} +4.93201 q^{46} -3.18723 q^{47} -5.48048 q^{49} +11.8373 q^{52} -1.28828 q^{53} +2.50051 q^{56} +0.377133 q^{58} -12.2588 q^{59} -3.67468 q^{61} +3.62342 q^{62} -1.66057 q^{64} +8.16762 q^{67} -4.28061 q^{68} +2.82331 q^{71} +0.837276 q^{73} +0.548343 q^{74} +4.96481 q^{76} -1.20316 q^{77} +3.44333 q^{79} +0.105972 q^{82} -4.66960 q^{83} +5.05434 q^{86} -1.97991 q^{88} +6.74431 q^{89} +8.58676 q^{91} -15.2843 q^{92} -1.74770 q^{94} +2.21791 q^{97} -3.00518 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - q^{2} + 7 q^{4} - 16 q^{11} - q^{13} + 3 q^{14} + 3 q^{16} - 4 q^{17} + 9 q^{19} + q^{22} + q^{23} - 24 q^{26} + 7 q^{28} - 3 q^{29} + 11 q^{31} - 21 q^{32} - 23 q^{34} + 7 q^{37} + 32 q^{38}+ \cdots - 6 q^{98}+O(q^{100}) \) 7 * q - q^2 + 7 * q^4 - 16 * q^11 - q^13 + 3 * q^14 + 3 * q^16 - 4 * q^17 + 9 * q^19 + q^22 + q^23 - 24 * q^26 + 7 * q^28 - 3 * q^29 + 11 * q^31 - 21 * q^32 - 23 * q^34 + 7 * q^37 + 32 * q^38 - 29 * q^41 + 3 * q^43 - 18 * q^44 + 7 * q^46 + 2 * q^47 + 5 * q^49 + 30 * q^52 + 6 * q^53 - 13 * q^56 - 28 * q^58 - 42 * q^59 - 11 * q^61 - 11 * q^62 - 16 * q^64 + 3 * q^67 + 4 * q^68 - 10 * q^71 - 14 * q^73 - q^74 + 11 * q^76 + 11 * q^77 - 9 * q^79 - 6 * q^82 + 6 * q^83 + 22 * q^86 - 32 * q^88 - 12 * q^89 - 2 * q^91 - 5 * q^92 + 9 * q^94 - q^97 - 6 * q^98

Introduction

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