LMFDB - Embedded newform 6975.2.a.cj.1.10

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 = [11,3,0,15,0,0,-8,9,0,0,0,0,-14,14,0,27,12] 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:	$0$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 3x^{10} - 14x^{9} + 44x^{8} + 61x^{7} - 211x^{6} - 83x^{5} + 369x^{4} + 10x^{3} - 168x^{2} - 31x + 5 $ x^11 - 3x^10 - 14x^9 + 44x^8 + 61x^7 - 211x^6 - 83x^5 + 369x^4 + 10x^3 - 168x^2 - 31x + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 465)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$2.60996$ 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)

$f(q)$	$=$	$q+2.60996 q^{2} +4.81187 q^{4} -1.35174 q^{7} +7.33885 q^{8} -2.89708 q^{11} +4.64772 q^{13} -3.52797 q^{14} +9.53034 q^{16} +6.57693 q^{17} -4.94592 q^{19} -7.56126 q^{22} -3.70309 q^{23} +12.1303 q^{26} -6.50437 q^{28} +2.99394 q^{29} +1.00000 q^{31} +10.1961 q^{32} +17.1655 q^{34} +9.62684 q^{37} -12.9086 q^{38} +7.00839 q^{41} +2.99944 q^{43} -13.9404 q^{44} -9.66491 q^{46} +7.98377 q^{47} -5.17281 q^{49} +22.3642 q^{52} +13.0752 q^{53} -9.92018 q^{56} +7.81406 q^{58} -4.50584 q^{59} +7.87056 q^{61} +2.60996 q^{62} +7.55057 q^{64} +4.37424 q^{67} +31.6473 q^{68} +11.0706 q^{71} -4.81757 q^{73} +25.1256 q^{74} -23.7991 q^{76} +3.91609 q^{77} +5.87005 q^{79} +18.2916 q^{82} -0.458163 q^{83} +7.82841 q^{86} -21.2613 q^{88} -8.57129 q^{89} -6.28249 q^{91} -17.8188 q^{92} +20.8373 q^{94} +0.397729 q^{97} -13.5008 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 3 q^{2} + 15 q^{4} - 8 q^{7} + 9 q^{8} - 14 q^{13} + 14 q^{14} + 27 q^{16} + 12 q^{17} + 12 q^{19} - 10 q^{22} + 12 q^{23} + 6 q^{26} - 22 q^{28} + 8 q^{29} + 11 q^{31} + 21 q^{32} + 2 q^{34} - 16 q^{37}+ \cdots + 17 q^{98}+O(q^{100}) $ 11 * q + 3 * q^2 + 15 * q^4 - 8 * q^7 + 9 * q^8 - 14 * q^13 + 14 * q^14 + 27 * q^16 + 12 * q^17 + 12 * q^19 - 10 * q^22 + 12 * q^23 + 6 * q^26 - 22 * q^28 + 8 * q^29 + 11 * q^31 + 21 * q^32 + 2 * q^34 - 16 * q^37 + 26 * q^38 + 8 * q^41 - 28 * q^43 + 26 * q^44 + 8 * q^46 + 14 * q^47 + 23 * q^49 - 18 * q^52 + 26 * q^53 + 42 * q^56 - 6 * q^58 + 6 * q^59 + 14 * q^61 + 3 * q^62 + 29 * q^64 - 6 * q^67 + 54 * q^68 + 18 * q^71 - 26 * q^73 + 40 * q^74 + 10 * q^76 + 2 * q^77 + 8 * q^79 + 30 * q^82 + 48 * q^83 + 12 * q^86 - 40 * q^88 + 14 * q^89 - 4 * q^91 - 30 * q^92 - 4 * q^94 - 16 * q^97 + 17 * 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 $

$2$	2.60996	1.84552	0.922759	−	0.385378i	$-0.125929\pi$
$2$	2.60996	1.84552	0.922759	+	0.385378i	$0.125929\pi$
$3$	0	0
$3$	0	0
$4$	4.81187	2.40593
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.35174	−0.510908	−0.255454	−	0.966821i	$-0.582225\pi$
$7$	−1.35174	−0.510908	−0.255454	+	0.966821i	$0.582225\pi$
$8$	7.33885	2.59468
$9$	0	0
$10$	0	0
$11$	−2.89708	−0.873503	−0.436752	−	0.899582i	$-0.643871\pi$
$11$	−2.89708	−0.873503	−0.436752	+	0.899582i	$0.643871\pi$
$12$	0	0
$13$	4.64772	1.28905	0.644523	−	0.764585i	$-0.277056\pi$
$13$	4.64772	1.28905	0.644523	+	0.764585i	$0.277056\pi$
$14$	−3.52797	−0.942889
$15$	0	0
$16$	9.53034	2.38258
$17$	6.57693	1.59514	0.797570	−	0.603226i	$-0.206118\pi$
$17$	6.57693	1.59514	0.797570	+	0.603226i	$0.206118\pi$
$18$	0	0
$19$	−4.94592	−1.13467	−0.567335	−	0.823487i	$-0.692026\pi$
$19$	−4.94592	−1.13467	−0.567335	+	0.823487i	$0.692026\pi$
$20$	0	0
$21$	0	0
$22$	−7.56126	−1.61207
$23$	−3.70309	−0.772149	−0.386074	−	0.922468i	$-0.626169\pi$
$23$	−3.70309	−0.772149	−0.386074	+	0.922468i	$0.626169\pi$
$24$	0	0
$25$	0	0
$26$	12.1303	2.37896
$27$	0	0
$28$	−6.50437	−1.22921
$29$	2.99394	0.555961	0.277981	−	0.960587i	$-0.410335\pi$
$29$	2.99394	0.555961	0.277981	+	0.960587i	$0.410335\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	10.1961	1.80242
$33$	0	0
$34$	17.1655	2.94386
$35$	0	0
$36$	0	0
$37$	9.62684	1.58264	0.791321	−	0.611400i	$-0.209393\pi$
$37$	9.62684	1.58264	0.791321	+	0.611400i	$0.209393\pi$
$38$	−12.9086	−2.09405
$39$	0	0
$40$	0	0
$41$	7.00839	1.09453	0.547264	−	0.836960i	$-0.315670\pi$
$41$	7.00839	1.09453	0.547264	+	0.836960i	$0.315670\pi$
$42$	0	0
$43$	2.99944	0.457411	0.228705	−	0.973496i	$-0.426551\pi$
$43$	2.99944	0.457411	0.228705	+	0.973496i	$0.426551\pi$
$44$	−13.9404	−2.10159
$45$	0	0
$46$	−9.66491	−1.42501
$47$	7.98377	1.16455	0.582276	−	0.812991i	$-0.302162\pi$
$47$	7.98377	1.16455	0.582276	+	0.812991i	$0.302162\pi$
$48$	0	0
$49$	−5.17281	−0.738973
$50$	0	0
$51$	0	0
$52$	22.3642	3.10136
$53$	13.0752	1.79601	0.898006	−	0.439984i	$-0.145016\pi$
$53$	13.0752	1.79601	0.898006	+	0.439984i	$0.145016\pi$
$54$	0	0
$55$	0	0
$56$	−9.92018	−1.32564
$57$	0	0
$58$	7.81406	1.02604
$59$	−4.50584	−0.586610	−0.293305	−	0.956019i	$-0.594755\pi$
$59$	−4.50584	−0.586610	−0.293305	+	0.956019i	$0.594755\pi$
$60$	0	0
$61$	7.87056	1.00772	0.503861	−	0.863785i	$-0.331912\pi$
$61$	7.87056	1.00772	0.503861	+	0.863785i	$0.331912\pi$
$62$	2.60996	0.331465
$63$	0	0
$64$	7.55057	0.943821
$65$	0	0
$66$	0	0
$67$	4.37424	0.534398	0.267199	−	0.963641i	$-0.413902\pi$
$67$	4.37424	0.534398	0.267199	+	0.963641i	$0.413902\pi$
$68$	31.6473	3.83780
$69$	0	0
$70$	0	0
$71$	11.0706	1.31384	0.656918	−	0.753962i	$-0.271860\pi$
$71$	11.0706	1.31384	0.656918	+	0.753962i	$0.271860\pi$
$72$	0	0
$73$	−4.81757	−0.563854	−0.281927	−	0.959436i	$-0.590974\pi$
$73$	−4.81757	−0.563854	−0.281927	+	0.959436i	$0.590974\pi$
$74$	25.1256	2.92079
$75$	0	0
$76$	−23.7991	−2.72994
$77$	3.91609	0.446280
$78$	0	0
$79$	5.87005	0.660433	0.330216	−	0.943905i	$-0.392878\pi$
$79$	5.87005	0.660433	0.330216	+	0.943905i	$0.392878\pi$
$80$	0	0
$81$	0	0
$82$	18.2916	2.01997
$83$	−0.458163	−0.0502900	−0.0251450	−	0.999684i	$-0.508005\pi$
$83$	−0.458163	−0.0502900	−0.0251450	+	0.999684i	$0.508005\pi$
$84$	0	0
$85$	0	0
$86$	7.82841	0.844159
$87$	0	0
$88$	−21.2613	−2.26646
$89$	−8.57129	−0.908555	−0.454277	−	0.890860i	$-0.650102\pi$
$89$	−8.57129	−0.908555	−0.454277	+	0.890860i	$0.650102\pi$
$90$	0	0
$91$	−6.28249	−0.658584
$92$	−17.8188	−1.85774
$93$	0	0
$94$	20.8373	2.14920
$95$	0	0
$96$	0	0
$97$	0.397729	0.0403832	0.0201916	−	0.999796i	$-0.493572\pi$
$97$	0.397729	0.0403832	0.0201916	+	0.999796i	$0.493572\pi$
$98$	−13.5008	−1.36379
$99$	0	0
$100$	0	0
$101$	−11.8759	−1.18170	−0.590848	−	0.806783i	$-0.701207\pi$
$101$	−11.8759	−1.18170	−0.590848	+	0.806783i	$0.701207\pi$
$102$	0	0
$103$	−1.83276	−0.180587	−0.0902937	−	0.995915i	$-0.528781\pi$
$103$	−1.83276	−0.180587	−0.0902937	+	0.995915i	$0.528781\pi$
$104$	34.1089	3.34466
$105$	0	0
$106$	34.1256	3.31457
$107$	−5.21399	−0.504055	−0.252028	−	0.967720i	$-0.581097\pi$
$107$	−5.21399	−0.504055	−0.252028	+	0.967720i	$0.581097\pi$
$108$	0	0
$109$	−2.46632	−0.236231	−0.118115	−	0.993000i	$-0.537685\pi$
$109$	−2.46632	−0.236231	−0.118115	+	0.993000i	$0.537685\pi$
$110$	0	0
$111$	0	0
$112$	−12.8825	−1.21728
$113$	−19.3409	−1.81944	−0.909722	−	0.415219i	$-0.863705\pi$
$113$	−19.3409	−1.81944	−0.909722	+	0.415219i	$0.863705\pi$
$114$	0	0
$115$	0	0
$116$	14.4065	1.33761
$117$	0	0
$118$	−11.7600	−1.08260
$119$	−8.89027	−0.814970
$120$	0	0
$121$	−2.60691	−0.236992
$122$	20.5418	1.85977
$123$	0	0
$124$	4.81187	0.432118
$125$	0	0
$126$	0	0
$127$	−14.3830	−1.27629	−0.638144	−	0.769917i	$-0.720298\pi$
$127$	−14.3830	−1.27629	−0.638144	+	0.769917i	$0.720298\pi$
$128$	−0.685450	−0.0605858
$129$	0	0
$130$	0	0
$131$	12.9938	1.13527	0.567635	−	0.823280i	$-0.307859\pi$
$131$	12.9938	1.13527	0.567635	+	0.823280i	$0.307859\pi$
$132$	0	0
$133$	6.68557	0.579712
$134$	11.4166	0.986241
$135$	0	0
$136$	48.2671	4.13887
$137$	−14.3381	−1.22498	−0.612491	−	0.790477i	$-0.709833\pi$
$137$	−14.3381	−1.22498	−0.612491	+	0.790477i	$0.709833\pi$
$138$	0	0
$139$	−2.98385	−0.253087	−0.126544	−	0.991961i	$-0.540388\pi$
$139$	−2.98385	−0.253087	−0.126544	+	0.991961i	$0.540388\pi$
$140$	0	0
$141$	0	0
$142$	28.8937	2.42471
$143$	−13.4648	−1.12599
$144$	0	0
$145$	0	0
$146$	−12.5736	−1.04060
$147$	0	0
$148$	46.3231	3.80773
$149$	7.49650	0.614137	0.307069	−	0.951687i	$-0.400652\pi$
$149$	7.49650	0.614137	0.307069	+	0.951687i	$0.400652\pi$
$150$	0	0
$151$	−3.97530	−0.323505	−0.161753	−	0.986831i	$-0.551715\pi$
$151$	−3.97530	−0.323505	−0.161753	+	0.986831i	$0.551715\pi$
$152$	−36.2973	−2.94410
$153$	0	0
$154$	10.2208	0.823617
$155$	0	0
$156$	0	0
$157$	16.8595	1.34554	0.672768	−	0.739854i	$-0.265105\pi$
$157$	16.8595	1.34554	0.672768	+	0.739854i	$0.265105\pi$
$158$	15.3206	1.21884
$159$	0	0
$160$	0	0
$161$	5.00560	0.394497
$162$	0	0
$163$	−17.8435	−1.39761	−0.698805	−	0.715312i	$-0.746285\pi$
$163$	−17.8435	−1.39761	−0.698805	+	0.715312i	$0.746285\pi$
$164$	33.7235	2.63336
$165$	0	0
$166$	−1.19579	−0.0928110
$167$	1.16540	0.0901810	0.0450905	−	0.998983i	$-0.485642\pi$
$167$	1.16540	0.0901810	0.0450905	+	0.998983i	$0.485642\pi$
$168$	0	0
$169$	8.60132	0.661640
$170$	0	0
$171$	0	0
$172$	14.4329	1.10050
$173$	22.3609	1.70007	0.850035	−	0.526726i	$-0.176581\pi$
$173$	22.3609	1.70007	0.850035	+	0.526726i	$0.176581\pi$
$174$	0	0
$175$	0	0
$176$	−27.6102	−2.08119
$177$	0	0
$178$	−22.3707	−1.67675
$179$	−15.0243	−1.12297	−0.561483	−	0.827488i	$-0.689769\pi$
$179$	−15.0243	−1.12297	−0.561483	+	0.827488i	$0.689769\pi$
$180$	0	0
$181$	9.66473	0.718374	0.359187	−	0.933266i	$-0.383054\pi$
$181$	9.66473	0.718374	0.359187	+	0.933266i	$0.383054\pi$
$182$	−16.3970	−1.21543
$183$	0	0
$184$	−27.1765	−2.00347
$185$	0	0
$186$	0	0
$187$	−19.0539	−1.39336
$188$	38.4168	2.80184
$189$	0	0
$190$	0	0
$191$	25.1736	1.82150	0.910750	−	0.412959i	$-0.135505\pi$
$191$	25.1736	1.82150	0.910750	+	0.412959i	$0.135505\pi$
$192$	0	0
$193$	−20.3877	−1.46754	−0.733771	−	0.679397i	$-0.762242\pi$
$193$	−20.3877	−1.46754	−0.733771	+	0.679397i	$0.762242\pi$
$194$	1.03805	0.0745280
$195$	0	0
$196$	−24.8909	−1.77792
$197$	−11.0317	−0.785974	−0.392987	−	0.919544i	$-0.628558\pi$
$197$	−11.0317	−0.785974	−0.392987	+	0.919544i	$0.628558\pi$
$198$	0	0
$199$	−27.7798	−1.96926	−0.984628	−	0.174665i	$-0.944116\pi$
$199$	−27.7798	−1.96926	−0.984628	+	0.174665i	$0.944116\pi$
$200$	0	0
$201$	0	0
$202$	−30.9956	−2.18084
$203$	−4.04702	−0.284045
$204$	0	0
$205$	0	0
$206$	−4.78343	−0.333277
$207$	0	0
$208$	44.2943	3.07126
$209$	14.3287	0.991139
$210$	0	0
$211$	−6.73018	−0.463324	−0.231662	−	0.972796i	$-0.574416\pi$
$211$	−6.73018	−0.463324	−0.231662	+	0.972796i	$0.574416\pi$
$212$	62.9159	4.32108
$213$	0	0
$214$	−13.6083	−0.930243
$215$	0	0
$216$	0	0
$217$	−1.35174	−0.0917618
$218$	−6.43698	−0.435968
$219$	0	0
$220$	0	0
$221$	30.5677	2.05621
$222$	0	0
$223$	−20.2081	−1.35323	−0.676617	−	0.736335i	$-0.736555\pi$
$223$	−20.2081	−1.35323	−0.676617	+	0.736335i	$0.736555\pi$
$224$	−13.7824	−0.920873
$225$	0	0
$226$	−50.4790	−3.35781
$227$	11.4787	0.761871	0.380936	−	0.924602i	$-0.375602\pi$
$227$	11.4787	0.761871	0.380936	+	0.924602i	$0.375602\pi$
$228$	0	0
$229$	8.45308	0.558595	0.279298	−	0.960205i	$-0.409898\pi$
$229$	8.45308	0.558595	0.279298	+	0.960205i	$0.409898\pi$
$230$	0	0
$231$	0	0
$232$	21.9721	1.44254
$233$	6.39116	0.418699	0.209349	−	0.977841i	$-0.432865\pi$
$233$	6.39116	0.418699	0.209349	+	0.977841i	$0.432865\pi$
$234$	0	0
$235$	0	0
$236$	−21.6815	−1.41135
$237$	0	0
$238$	−23.2032	−1.50404
$239$	18.1098	1.17143	0.585714	−	0.810518i	$-0.300814\pi$
$239$	18.1098	1.17143	0.585714	+	0.810518i	$0.300814\pi$
$240$	0	0
$241$	6.62015	0.426442	0.213221	−	0.977004i	$-0.431605\pi$
$241$	6.62015	0.426442	0.213221	+	0.977004i	$0.431605\pi$
$242$	−6.80392	−0.437373
$243$	0	0
$244$	37.8721	2.42451
$245$	0	0
$246$	0	0
$247$	−22.9872	−1.46264
$248$	7.33885	0.466017
$249$	0	0
$250$	0	0
$251$	−20.4490	−1.29073	−0.645363	−	0.763876i	$-0.723294\pi$
$251$	−20.4490	−1.29073	−0.645363	+	0.763876i	$0.723294\pi$
$252$	0	0
$253$	10.7282	0.674474
$254$	−37.5391	−2.35541
$255$	0	0
$256$	−16.8901	−1.05563
$257$	16.8684	1.05222	0.526110	−	0.850417i	$-0.323650\pi$
$257$	16.8684	1.05222	0.526110	+	0.850417i	$0.323650\pi$
$258$	0	0
$259$	−13.0129	−0.808585
$260$	0	0
$261$	0	0
$262$	33.9131	2.09516
$263$	−19.6858	−1.21388	−0.606940	−	0.794748i	$-0.707603\pi$
$263$	−19.6858	−1.21388	−0.606940	+	0.794748i	$0.707603\pi$
$264$	0	0
$265$	0	0
$266$	17.4490	1.06987
$267$	0	0
$268$	21.0483	1.28573
$269$	23.0747	1.40689	0.703443	−	0.710751i	$-0.251645\pi$
$269$	23.0747	1.40689	0.703443	+	0.710751i	$0.251645\pi$
$270$	0	0
$271$	−23.0339	−1.39921	−0.699604	−	0.714530i	$-0.746640\pi$
$271$	−23.0339	−1.39921	−0.699604	+	0.714530i	$0.746640\pi$
$272$	62.6804	3.80055
$273$	0	0
$274$	−37.4217	−2.26073
$275$	0	0
$276$	0	0
$277$	1.34884	0.0810437	0.0405219	−	0.999179i	$-0.487098\pi$
$277$	1.34884	0.0810437	0.0405219	+	0.999179i	$0.487098\pi$
$278$	−7.78772	−0.467077
$279$	0	0
$280$	0	0
$281$	18.8851	1.12659	0.563296	−	0.826256i	$-0.309533\pi$
$281$	18.8851	1.12659	0.563296	+	0.826256i	$0.309533\pi$
$282$	0	0
$283$	−8.54033	−0.507670	−0.253835	−	0.967248i	$-0.581692\pi$
$283$	−8.54033	−0.507670	−0.253835	+	0.967248i	$0.581692\pi$
$284$	53.2701	3.16100
$285$	0	0
$286$	−35.1426	−2.07803
$287$	−9.47349	−0.559203
$288$	0	0
$289$	26.2560	1.54447
$290$	0	0
$291$	0	0
$292$	−23.1815	−1.35659
$293$	−1.42571	−0.0832907	−0.0416453	−	0.999132i	$-0.513260\pi$
$293$	−1.42571	−0.0832907	−0.0416453	+	0.999132i	$0.513260\pi$
$294$	0	0
$295$	0	0
$296$	70.6499	4.10644
$297$	0	0
$298$	19.5655	1.13340
$299$	−17.2110	−0.995335
$300$	0	0
$301$	−4.05445	−0.233695
$302$	−10.3754	−0.597035
$303$	0	0
$304$	−47.1362	−2.70345
$305$	0	0
$306$	0	0
$307$	−18.1183	−1.03406	−0.517032	−	0.855966i	$-0.672963\pi$
$307$	−18.1183	−1.03406	−0.517032	+	0.855966i	$0.672963\pi$
$308$	18.8437	1.07372
$309$	0	0
$310$	0	0
$311$	13.5982	0.771081	0.385540	−	0.922691i	$-0.374015\pi$
$311$	13.5982	0.771081	0.385540	+	0.922691i	$0.374015\pi$
$312$	0	0
$313$	−17.1840	−0.971298	−0.485649	−	0.874154i	$-0.661417\pi$
$313$	−17.1840	−0.971298	−0.485649	+	0.874154i	$0.661417\pi$
$314$	44.0026	2.48321
$315$	0	0
$316$	28.2459	1.58896
$317$	18.9517	1.06443	0.532217	−	0.846608i	$-0.321359\pi$
$317$	18.9517	1.06443	0.532217	+	0.846608i	$0.321359\pi$
$318$	0	0
$319$	−8.67370	−0.485634
$320$	0	0
$321$	0	0
$322$	13.0644	0.728051
$323$	−32.5289	−1.80996
$324$	0	0
$325$	0	0
$326$	−46.5707	−2.57931
$327$	0	0
$328$	51.4335	2.83994
$329$	−10.7919	−0.594979
$330$	0	0
$331$	22.5785	1.24103	0.620513	−	0.784196i	$-0.286924\pi$
$331$	22.5785	1.24103	0.620513	+	0.784196i	$0.286924\pi$
$332$	−2.20462	−0.120994
$333$	0	0
$334$	3.04163	0.166431
$335$	0	0
$336$	0	0
$337$	20.6429	1.12449	0.562245	−	0.826971i	$-0.309938\pi$
$337$	20.6429	1.12449	0.562245	+	0.826971i	$0.309938\pi$
$338$	22.4491	1.22107
$339$	0	0
$340$	0	0
$341$	−2.89708	−0.156886
$342$	0	0
$343$	16.4544	0.888455
$344$	22.0125	1.18683
$345$	0	0
$346$	58.3611	3.13751
$347$	17.8082	0.955992	0.477996	−	0.878362i	$-0.341363\pi$
$347$	17.8082	0.955992	0.477996	+	0.878362i	$0.341363\pi$
$348$	0	0
$349$	5.54713	0.296931	0.148466	−	0.988918i	$-0.452567\pi$
$349$	5.54713	0.296931	0.148466	+	0.988918i	$0.452567\pi$
$350$	0	0
$351$	0	0
$352$	−29.5388	−1.57442
$353$	0.894986	0.0476353	0.0238176	−	0.999716i	$-0.492418\pi$
$353$	0.894986	0.0476353	0.0238176	+	0.999716i	$0.492418\pi$
$354$	0	0
$355$	0	0
$356$	−41.2439	−2.18592
$357$	0	0
$358$	−39.2127	−2.07245
$359$	22.6591	1.19590	0.597950	−	0.801534i	$-0.295982\pi$
$359$	22.6591	1.19590	0.597950	+	0.801534i	$0.295982\pi$
$360$	0	0
$361$	5.46208	0.287478
$362$	25.2245	1.32577
$363$	0	0
$364$	−30.2305	−1.58451
$365$	0	0
$366$	0	0
$367$	−4.07528	−0.212728	−0.106364	−	0.994327i	$-0.533921\pi$
$367$	−4.07528	−0.212728	−0.106364	+	0.994327i	$0.533921\pi$
$368$	−35.2917	−1.83971
$369$	0	0
$370$	0	0
$371$	−17.6742	−0.917596
$372$	0	0
$373$	2.55182	0.132128	0.0660642	−	0.997815i	$-0.478956\pi$
$373$	2.55182	0.132128	0.0660642	+	0.997815i	$0.478956\pi$
$374$	−49.7299	−2.57147
$375$	0	0
$376$	58.5917	3.02163
$377$	13.9150	0.716660
$378$	0	0
$379$	9.03832	0.464267	0.232134	−	0.972684i	$-0.425429\pi$
$379$	9.03832	0.464267	0.232134	+	0.972684i	$0.425429\pi$
$380$	0	0
$381$	0	0
$382$	65.7020	3.36161
$383$	19.3181	0.987110	0.493555	−	0.869715i	$-0.335697\pi$
$383$	19.3181	0.987110	0.493555	+	0.869715i	$0.335697\pi$
$384$	0	0
$385$	0	0
$386$	−53.2111	−2.70837
$387$	0	0
$388$	1.91382	0.0971594
$389$	−16.5174	−0.837468	−0.418734	−	0.908109i	$-0.637526\pi$
$389$	−16.5174	−0.837468	−0.418734	+	0.908109i	$0.637526\pi$
$390$	0	0
$391$	−24.3550	−1.23169
$392$	−37.9625	−1.91740
$393$	0	0
$394$	−28.7922	−1.45053
$395$	0	0
$396$	0	0
$397$	−22.1351	−1.11093	−0.555464	−	0.831541i	$-0.687459\pi$
$397$	−22.1351	−1.11093	−0.555464	+	0.831541i	$0.687459\pi$
$398$	−72.5040	−3.63430
$399$	0	0
$400$	0	0
$401$	−24.6373	−1.23033	−0.615163	−	0.788400i	$-0.710910\pi$
$401$	−24.6373	−1.23033	−0.615163	+	0.788400i	$0.710910\pi$
$402$	0	0
$403$	4.64772	0.231520
$404$	−57.1453	−2.84308
$405$	0	0
$406$	−10.5625	−0.524210
$407$	−27.8898	−1.38244
$408$	0	0
$409$	−22.7649	−1.12565	−0.562825	−	0.826576i	$-0.690286\pi$
$409$	−22.7649	−1.12565	−0.562825	+	0.826576i	$0.690286\pi$
$410$	0	0
$411$	0	0
$412$	−8.81901	−0.434481
$413$	6.09070	0.299704
$414$	0	0
$415$	0	0
$416$	47.3884	2.32341
$417$	0	0
$418$	37.3973	1.82916
$419$	−29.0185	−1.41765	−0.708823	−	0.705387i	$-0.750773\pi$
$419$	−29.0185	−1.41765	−0.708823	+	0.705387i	$0.750773\pi$
$420$	0	0
$421$	−24.2411	−1.18144	−0.590720	−	0.806876i	$-0.701156\pi$
$421$	−24.2411	−1.18144	−0.590720	+	0.806876i	$0.701156\pi$
$422$	−17.5655	−0.855073
$423$	0	0
$424$	95.9566	4.66007
$425$	0	0
$426$	0	0
$427$	−10.6389	−0.514853
$428$	−25.0890	−1.21272
$429$	0	0
$430$	0	0
$431$	23.7355	1.14330	0.571649	−	0.820498i	$-0.306304\pi$
$431$	23.7355	1.14330	0.571649	+	0.820498i	$0.306304\pi$
$432$	0	0
$433$	−7.26850	−0.349302	−0.174651	−	0.984630i	$-0.555880\pi$
$433$	−7.26850	−0.349302	−0.174651	+	0.984630i	$0.555880\pi$
$434$	−3.52797	−0.169348
$435$	0	0
$436$	−11.8676	−0.568355
$437$	18.3152	0.876135
$438$	0	0
$439$	−24.3863	−1.16390	−0.581948	−	0.813226i	$-0.697709\pi$
$439$	−24.3863	−1.16390	−0.581948	+	0.813226i	$0.697709\pi$
$440$	0	0
$441$	0	0
$442$	79.7805	3.79477
$443$	12.6288	0.600012	0.300006	−	0.953937i	$-0.403011\pi$
$443$	12.6288	0.600012	0.300006	+	0.953937i	$0.403011\pi$
$444$	0	0
$445$	0	0
$446$	−52.7422	−2.49742
$447$	0	0
$448$	−10.2064	−0.482206
$449$	−35.7239	−1.68592	−0.842958	−	0.537979i	$-0.819188\pi$
$449$	−35.7239	−1.68592	−0.842958	+	0.537979i	$0.819188\pi$
$450$	0	0
$451$	−20.3039	−0.956073
$452$	−93.0661	−4.37746
$453$	0	0
$454$	29.9590	1.40605
$455$	0	0
$456$	0	0
$457$	−20.1012	−0.940296	−0.470148	−	0.882588i	$-0.655799\pi$
$457$	−20.1012	−0.940296	−0.470148	+	0.882588i	$0.655799\pi$
$458$	22.0622	1.03090
$459$	0	0
$460$	0	0
$461$	−17.4892	−0.814553	−0.407276	−	0.913305i	$-0.633521\pi$
$461$	−17.4892	−0.814553	−0.407276	+	0.913305i	$0.633521\pi$
$462$	0	0
$463$	20.5115	0.953250	0.476625	−	0.879107i	$-0.341860\pi$
$463$	20.5115	0.953250	0.476625	+	0.879107i	$0.341860\pi$
$464$	28.5333	1.32462
$465$	0	0
$466$	16.6806	0.772716
$467$	−3.72615	−0.172426	−0.0862129	−	0.996277i	$-0.527477\pi$
$467$	−3.72615	−0.172426	−0.0862129	+	0.996277i	$0.527477\pi$
$468$	0	0
$469$	−5.91281	−0.273028
$470$	0	0
$471$	0	0
$472$	−33.0677	−1.52206
$473$	−8.68963	−0.399550
$474$	0	0
$475$	0	0
$476$	−42.7788	−1.96076
$477$	0	0
$478$	47.2658	2.16189
$479$	10.5607	0.482529	0.241265	−	0.970459i	$-0.422438\pi$
$479$	10.5607	0.482529	0.241265	+	0.970459i	$0.422438\pi$
$480$	0	0
$481$	44.7429	2.04010
$482$	17.2783	0.787005
$483$	0	0
$484$	−12.5441	−0.570187
$485$	0	0
$486$	0	0
$487$	−32.7564	−1.48434	−0.742168	−	0.670214i	$-0.766202\pi$
$487$	−32.7564	−1.48434	−0.742168	+	0.670214i	$0.766202\pi$
$488$	57.7609	2.61471
$489$	0	0
$490$	0	0
$491$	−8.34967	−0.376815	−0.188408	−	0.982091i	$-0.560333\pi$
$491$	−8.34967	−0.376815	−0.188408	+	0.982091i	$0.560333\pi$
$492$	0	0
$493$	19.6910	0.886836
$494$	−59.9957	−2.69933
$495$	0	0
$496$	9.53034	0.427925
$497$	−14.9645	−0.671249
$498$	0	0
$499$	−9.06078	−0.405616	−0.202808	−	0.979218i	$-0.565007\pi$
$499$	−9.06078	−0.405616	−0.202808	+	0.979218i	$0.565007\pi$
$500$	0	0
$501$	0	0
$502$	−53.3709	−2.38206
$503$	−13.5436	−0.603879	−0.301939	−	0.953327i	$-0.597634\pi$
$503$	−13.5436	−0.603879	−0.301939	+	0.953327i	$0.597634\pi$
$504$	0	0
$505$	0	0
$506$	28.0001	1.24475
$507$	0	0
$508$	−69.2093	−3.07067
$509$	5.99518	0.265732	0.132866	−	0.991134i	$-0.457582\pi$
$509$	5.99518	0.265732	0.132866	+	0.991134i	$0.457582\pi$
$510$	0	0
$511$	6.51208	0.288077
$512$	−42.7116	−1.88760
$513$	0	0
$514$	44.0257	1.94189
$515$	0	0
$516$	0	0
$517$	−23.1296	−1.01724
$518$	−33.9632	−1.49226
$519$	0	0
$520$	0	0
$521$	−0.831312	−0.0364204	−0.0182102	−	0.999834i	$-0.505797\pi$
$521$	−0.831312	−0.0364204	−0.0182102	+	0.999834i	$0.505797\pi$
$522$	0	0
$523$	−34.3566	−1.50231	−0.751154	−	0.660127i	$-0.770502\pi$
$523$	−34.3566	−1.50231	−0.751154	+	0.660127i	$0.770502\pi$
$524$	62.5242	2.73138
$525$	0	0
$526$	−51.3791	−2.24024
$527$	6.57693	0.286496
$528$	0	0
$529$	−9.28709	−0.403786
$530$	0	0
$531$	0	0
$532$	32.1701	1.39475
$533$	32.5731	1.41090
$534$	0	0
$535$	0	0
$536$	32.1019	1.38659
$537$	0	0
$538$	60.2238	2.59643
$539$	14.9861	0.645495
$540$	0	0
$541$	30.0236	1.29082	0.645408	−	0.763838i	$-0.276687\pi$
$541$	30.0236	1.29082	0.645408	+	0.763838i	$0.276687\pi$
$542$	−60.1174	−2.58226
$543$	0	0
$544$	67.0587	2.87512
$545$	0	0
$546$	0	0
$547$	−32.3911	−1.38494	−0.692472	−	0.721445i	$-0.743478\pi$
$547$	−32.3911	−1.38494	−0.692472	+	0.721445i	$0.743478\pi$
$548$	−68.9928	−2.94723
$549$	0	0
$550$	0	0
$551$	−14.8078	−0.630833
$552$	0	0
$553$	−7.93476	−0.337420
$554$	3.52040	0.149568
$555$	0	0
$556$	−14.3579	−0.608911
$557$	−4.01126	−0.169963	−0.0849814	−	0.996383i	$-0.527083\pi$
$557$	−4.01126	−0.169963	−0.0849814	+	0.996383i	$0.527083\pi$
$558$	0	0
$559$	13.9406	0.589623
$560$	0	0
$561$	0	0
$562$	49.2893	2.07914
$563$	21.6277	0.911498	0.455749	−	0.890108i	$-0.349371\pi$
$563$	21.6277	0.911498	0.455749	+	0.890108i	$0.349371\pi$
$564$	0	0
$565$	0	0
$566$	−22.2899	−0.936914
$567$	0	0
$568$	81.2453	3.40898
$569$	−7.83046	−0.328270	−0.164135	−	0.986438i	$-0.552483\pi$
$569$	−7.83046	−0.328270	−0.164135	+	0.986438i	$0.552483\pi$
$570$	0	0
$571$	16.8246	0.704089	0.352045	−	0.935983i	$-0.385487\pi$
$571$	16.8246	0.704089	0.352045	+	0.935983i	$0.385487\pi$
$572$	−64.7910	−2.70905
$573$	0	0
$574$	−24.7254	−1.03202
$575$	0	0
$576$	0	0
$577$	11.5651	0.481463	0.240732	−	0.970592i	$-0.422613\pi$
$577$	11.5651	0.481463	0.240732	+	0.970592i	$0.422613\pi$
$578$	68.5270	2.85035
$579$	0	0
$580$	0	0
$581$	0.619316	0.0256935
$582$	0	0
$583$	−37.8798	−1.56882
$584$	−35.3554	−1.46302
$585$	0	0
$586$	−3.72103	−0.153714
$587$	4.42457	0.182622	0.0913108	−	0.995822i	$-0.470894\pi$
$587$	4.42457	0.182622	0.0913108	+	0.995822i	$0.470894\pi$
$588$	0	0
$589$	−4.94592	−0.203793
$590$	0	0
$591$	0	0
$592$	91.7470	3.77078
$593$	−22.0225	−0.904356	−0.452178	−	0.891928i	$-0.649353\pi$
$593$	−22.0225	−0.904356	−0.452178	+	0.891928i	$0.649353\pi$
$594$	0	0
$595$	0	0
$596$	36.0722	1.47757
$597$	0	0
$598$	−44.9198	−1.83691
$599$	36.0976	1.47491	0.737453	−	0.675398i	$-0.236028\pi$
$599$	36.0976	1.47491	0.737453	+	0.675398i	$0.236028\pi$
$600$	0	0
$601$	−38.4272	−1.56748	−0.783738	−	0.621091i	$-0.786690\pi$
$601$	−38.4272	−1.56748	−0.783738	+	0.621091i	$0.786690\pi$
$602$	−10.5819	−0.431288
$603$	0	0
$604$	−19.1286	−0.778332
$605$	0	0
$606$	0	0
$607$	4.75673	0.193070	0.0965349	−	0.995330i	$-0.469224\pi$
$607$	4.75673	0.193070	0.0965349	+	0.995330i	$0.469224\pi$
$608$	−50.4288	−2.04516
$609$	0	0
$610$	0	0
$611$	37.1063	1.50116
$612$	0	0
$613$	−35.7017	−1.44198	−0.720990	−	0.692946i	$-0.756313\pi$
$613$	−35.7017	−1.44198	−0.720990	+	0.692946i	$0.756313\pi$
$614$	−47.2879	−1.90838
$615$	0	0
$616$	28.7396	1.15795
$617$	−14.8874	−0.599345	−0.299673	−	0.954042i	$-0.596877\pi$
$617$	−14.8874	−0.599345	−0.299673	+	0.954042i	$0.596877\pi$
$618$	0	0
$619$	−2.66707	−0.107199	−0.0535993	−	0.998563i	$-0.517069\pi$
$619$	−2.66707	−0.107199	−0.0535993	+	0.998563i	$0.517069\pi$
$620$	0	0
$621$	0	0
$622$	35.4906	1.42304
$623$	11.5861	0.464188
$624$	0	0
$625$	0	0
$626$	−44.8495	−1.79255
$627$	0	0
$628$	81.1257	3.23727
$629$	63.3151	2.52454
$630$	0	0
$631$	33.4741	1.33258	0.666292	−	0.745691i	$-0.267880\pi$
$631$	33.4741	1.33258	0.666292	+	0.745691i	$0.267880\pi$
$632$	43.0794	1.71361
$633$	0	0
$634$	49.4631	1.96443
$635$	0	0
$636$	0	0
$637$	−24.0418	−0.952570
$638$	−22.6380	−0.896246
$639$	0	0
$640$	0	0
$641$	14.9709	0.591316	0.295658	−	0.955294i	$-0.404461\pi$
$641$	14.9709	0.591316	0.295658	+	0.955294i	$0.404461\pi$
$642$	0	0
$643$	12.3887	0.488563	0.244281	−	0.969704i	$-0.421448\pi$
$643$	12.3887	0.488563	0.244281	+	0.969704i	$0.421448\pi$
$644$	24.0863	0.949133
$645$	0	0
$646$	−84.8991	−3.34031
$647$	−22.7051	−0.892630	−0.446315	−	0.894876i	$-0.647264\pi$
$647$	−22.7051	−0.892630	−0.446315	+	0.894876i	$0.647264\pi$
$648$	0	0
$649$	13.0538	0.512406
$650$	0	0
$651$	0	0
$652$	−85.8605	−3.36256
$653$	−39.7149	−1.55416	−0.777082	−	0.629399i	$-0.783301\pi$
$653$	−39.7149	−1.55416	−0.777082	+	0.629399i	$0.783301\pi$
$654$	0	0
$655$	0	0
$656$	66.7923	2.60780
$657$	0	0
$658$	−28.1665	−1.09804
$659$	−46.4837	−1.81075	−0.905374	−	0.424615i	$-0.860409\pi$
$659$	−46.4837	−1.81075	−0.905374	+	0.424615i	$0.860409\pi$
$660$	0	0
$661$	−16.8797	−0.656545	−0.328272	−	0.944583i	$-0.606466\pi$
$661$	−16.8797	−0.656545	−0.328272	+	0.944583i	$0.606466\pi$
$662$	58.9289	2.29034
$663$	0	0
$664$	−3.36239	−0.130486
$665$	0	0
$666$	0	0
$667$	−11.0869	−0.429285
$668$	5.60773	0.216970
$669$	0	0
$670$	0	0
$671$	−22.8017	−0.880249
$672$	0	0
$673$	16.3739	0.631167	0.315583	−	0.948898i	$-0.397800\pi$
$673$	16.3739	0.631167	0.315583	+	0.948898i	$0.397800\pi$
$674$	53.8770	2.07527
$675$	0	0
$676$	41.3884	1.59186
$677$	−7.91943	−0.304368	−0.152184	−	0.988352i	$-0.548631\pi$
$677$	−7.91943	−0.304368	−0.152184	+	0.988352i	$0.548631\pi$
$678$	0	0
$679$	−0.537624	−0.0206321
$680$	0	0
$681$	0	0
$682$	−7.56126	−0.289535
$683$	0.937540	0.0358740	0.0179370	−	0.999839i	$-0.494290\pi$
$683$	0.937540	0.0358740	0.0179370	+	0.999839i	$0.494290\pi$
$684$	0	0
$685$	0	0
$686$	42.9453	1.63966
$687$	0	0
$688$	28.5857	1.08982
$689$	60.7697	2.31514
$690$	0	0
$691$	−23.9525	−0.911196	−0.455598	−	0.890186i	$-0.650575\pi$
$691$	−23.9525	−0.911196	−0.455598	+	0.890186i	$0.650575\pi$
$692$	107.598	4.09026
$693$	0	0
$694$	46.4785	1.76430
$695$	0	0
$696$	0	0
$697$	46.0937	1.74592
$698$	14.4778	0.547992
$699$	0	0
$700$	0	0
$701$	−33.9432	−1.28202	−0.641009	−	0.767534i	$-0.721484\pi$
$701$	−33.9432	−1.28202	−0.641009	+	0.767534i	$0.721484\pi$
$702$	0	0
$703$	−47.6135	−1.79578
$704$	−21.8746	−0.824431
$705$	0	0
$706$	2.33587	0.0879117
$707$	16.0531	0.603738
$708$	0	0
$709$	34.5150	1.29624	0.648118	−	0.761540i	$-0.275556\pi$
$709$	34.5150	1.29624	0.648118	+	0.761540i	$0.275556\pi$
$710$	0	0
$711$	0	0
$712$	−62.9034	−2.35740
$713$	−3.70309	−0.138682
$714$	0	0
$715$	0	0
$716$	−72.2948	−2.70178
$717$	0	0
$718$	59.1391	2.20705
$719$	−14.8414	−0.553489	−0.276745	−	0.960943i	$-0.589256\pi$
$719$	−14.8414	−0.553489	−0.276745	+	0.960943i	$0.589256\pi$
$720$	0	0
$721$	2.47741	0.0922636
$722$	14.2558	0.530545
$723$	0	0
$724$	46.5054	1.72836
$725$	0	0
$726$	0	0
$727$	27.3231	1.01336	0.506679	−	0.862135i	$-0.330873\pi$
$727$	27.3231	1.01336	0.506679	+	0.862135i	$0.330873\pi$
$728$	−46.1062	−1.70881
$729$	0	0
$730$	0	0
$731$	19.7271	0.729634
$732$	0	0
$733$	−41.6761	−1.53934	−0.769670	−	0.638441i	$-0.779579\pi$
$733$	−41.6761	−1.53934	−0.769670	+	0.638441i	$0.779579\pi$
$734$	−10.6363	−0.392593
$735$	0	0
$736$	−37.7569	−1.39174
$737$	−12.6725	−0.466799
$738$	0	0
$739$	3.72185	0.136910	0.0684552	−	0.997654i	$-0.478193\pi$
$739$	3.72185	0.136910	0.0684552	+	0.997654i	$0.478193\pi$
$740$	0	0
$741$	0	0
$742$	−46.1287	−1.69344
$743$	21.7070	0.796351	0.398175	−	0.917309i	$-0.369644\pi$
$743$	21.7070	0.796351	0.398175	+	0.917309i	$0.369644\pi$
$744$	0	0
$745$	0	0
$746$	6.66014	0.243845
$747$	0	0
$748$	−91.6849	−3.35233
$749$	7.04793	0.257526
$750$	0	0
$751$	24.8253	0.905887	0.452943	−	0.891539i	$-0.350374\pi$
$751$	24.8253	0.905887	0.452943	+	0.891539i	$0.350374\pi$
$752$	76.0880	2.77464
$753$	0	0
$754$	36.3176	1.32261
$755$	0	0
$756$	0	0
$757$	20.0128	0.727377	0.363688	−	0.931521i	$-0.381517\pi$
$757$	20.0128	0.727377	0.363688	+	0.931521i	$0.381517\pi$
$758$	23.5896	0.856814
$759$	0	0
$760$	0	0
$761$	1.88654	0.0683871	0.0341935	−	0.999415i	$-0.489114\pi$
$761$	1.88654	0.0683871	0.0341935	+	0.999415i	$0.489114\pi$
$762$	0	0
$763$	3.33381	0.120692
$764$	121.132	4.38241
$765$	0	0
$766$	50.4194	1.82173
$767$	−20.9419	−0.756168
$768$	0	0
$769$	53.3296	1.92311	0.961557	−	0.274604i	$-0.0885467\pi$
$769$	53.3296	1.92311	0.961557	+	0.274604i	$0.0885467\pi$
$770$	0	0
$771$	0	0
$772$	−98.1031	−3.53081
$773$	20.2423	0.728065	0.364033	−	0.931386i	$-0.381400\pi$
$773$	20.2423	0.728065	0.364033	+	0.931386i	$0.381400\pi$
$774$	0	0
$775$	0	0
$776$	2.91887	0.104781
$777$	0	0
$778$	−43.1098	−1.54556
$779$	−34.6629	−1.24193
$780$	0	0
$781$	−32.0724	−1.14764
$782$	−63.5655	−2.27310
$783$	0	0
$784$	−49.2986	−1.76067
$785$	0	0
$786$	0	0
$787$	−49.6194	−1.76874	−0.884370	−	0.466787i	$-0.845411\pi$
$787$	−49.6194	−1.76874	−0.884370	+	0.466787i	$0.845411\pi$
$788$	−53.0829	−1.89100
$789$	0	0
$790$	0	0
$791$	26.1438	0.929568
$792$	0	0
$793$	36.5802	1.29900
$794$	−57.7716	−2.05024
$795$	0	0
$796$	−133.673	−4.73790
$797$	−37.6997	−1.33539	−0.667695	−	0.744435i	$-0.732719\pi$
$797$	−37.6997	−1.33539	−0.667695	+	0.744435i	$0.732719\pi$
$798$	0	0
$799$	52.5087	1.85762
$800$	0	0
$801$	0	0
$802$	−64.3022	−2.27059
$803$	13.9569	0.492528
$804$	0	0
$805$	0	0
$806$	12.1303	0.427273
$807$	0	0
$808$	−87.1555	−3.06612
$809$	5.44384	0.191395	0.0956977	−	0.995410i	$-0.469492\pi$
$809$	5.44384	0.191395	0.0956977	+	0.995410i	$0.469492\pi$
$810$	0	0
$811$	29.4222	1.03315	0.516577	−	0.856241i	$-0.327206\pi$
$811$	29.4222	1.03315	0.516577	+	0.856241i	$0.327206\pi$
$812$	−19.4737	−0.683394
$813$	0	0
$814$	−72.7910	−2.55132
$815$	0	0
$816$	0	0
$817$	−14.8350	−0.519010
$818$	−59.4153	−2.07741
$819$	0	0
$820$	0	0
$821$	−3.90102	−0.136146	−0.0680732	−	0.997680i	$-0.521685\pi$
$821$	−3.90102	−0.136146	−0.0680732	+	0.997680i	$0.521685\pi$
$822$	0	0
$823$	45.1123	1.57252	0.786258	−	0.617898i	$-0.212016\pi$
$823$	45.1123	1.57252	0.786258	+	0.617898i	$0.212016\pi$
$824$	−13.4504	−0.468566
$825$	0	0
$826$	15.8965	0.553108
$827$	−9.38902	−0.326488	−0.163244	−	0.986586i	$-0.552196\pi$
$827$	−9.38902	−0.326488	−0.163244	+	0.986586i	$0.552196\pi$
$828$	0	0
$829$	−19.6667	−0.683054	−0.341527	−	0.939872i	$-0.610944\pi$
$829$	−19.6667	−0.683054	−0.341527	+	0.939872i	$0.610944\pi$
$830$	0	0
$831$	0	0
$832$	35.0930	1.21663
$833$	−34.0212	−1.17877
$834$	0	0
$835$	0	0
$836$	68.9479	2.38461
$837$	0	0
$838$	−75.7370	−2.61629
$839$	−12.8568	−0.443865	−0.221933	−	0.975062i	$-0.571237\pi$
$839$	−12.8568	−0.443865	−0.221933	+	0.975062i	$0.571237\pi$
$840$	0	0
$841$	−20.0363	−0.690907
$842$	−63.2683	−2.18037
$843$	0	0
$844$	−32.3847	−1.11473
$845$	0	0
$846$	0	0
$847$	3.52385	0.121081
$848$	124.611	4.27915
$849$	0	0
$850$	0	0
$851$	−35.6491	−1.22204
$852$	0	0
$853$	−10.3662	−0.354933	−0.177467	−	0.984127i	$-0.556790\pi$
$853$	−10.3662	−0.354933	−0.177467	+	0.984127i	$0.556790\pi$
$854$	−27.7671	−0.950171
$855$	0	0
$856$	−38.2647	−1.30786
$857$	−10.2071	−0.348667	−0.174333	−	0.984687i	$-0.555777\pi$
$857$	−10.2071	−0.348667	−0.174333	+	0.984687i	$0.555777\pi$
$858$	0	0
$859$	−3.34065	−0.113981	−0.0569907	−	0.998375i	$-0.518151\pi$
$859$	−3.34065	−0.113981	−0.0569907	+	0.998375i	$0.518151\pi$
$860$	0	0
$861$	0	0
$862$	61.9485	2.10998
$863$	29.7744	1.01353	0.506765	−	0.862084i	$-0.330841\pi$
$863$	29.7744	1.01353	0.506765	+	0.862084i	$0.330841\pi$
$864$	0	0
$865$	0	0
$866$	−18.9705	−0.644643
$867$	0	0
$868$	−6.50437	−0.220773
$869$	−17.0060	−0.576890
$870$	0	0
$871$	20.3302	0.688864
$872$	−18.0999	−0.612941
$873$	0	0
$874$	47.8018	1.61692
$875$	0	0
$876$	0	0
$877$	11.2457	0.379741	0.189870	−	0.981809i	$-0.439193\pi$
$877$	11.2457	0.379741	0.189870	+	0.981809i	$0.439193\pi$
$878$	−63.6473	−2.14799
$879$	0	0
$880$	0	0
$881$	−37.8238	−1.27432	−0.637158	−	0.770733i	$-0.719890\pi$
$881$	−37.8238	−1.27432	−0.637158	+	0.770733i	$0.719890\pi$
$882$	0	0
$883$	−4.37102	−0.147097	−0.0735483	−	0.997292i	$-0.523432\pi$
$883$	−4.37102	−0.147097	−0.0735483	+	0.997292i	$0.523432\pi$
$884$	147.088	4.94710
$885$	0	0
$886$	32.9606	1.10733
$887$	−17.0810	−0.573525	−0.286763	−	0.958002i	$-0.592579\pi$
$887$	−17.0810	−0.573525	−0.286763	+	0.958002i	$0.592579\pi$
$888$	0	0
$889$	19.4421	0.652066
$890$	0	0
$891$	0	0
$892$	−97.2386	−3.25579
$893$	−39.4870	−1.32138
$894$	0	0
$895$	0	0
$896$	0.926547	0.0309538
$897$	0	0
$898$	−93.2378	−3.11139
$899$	2.99394	0.0998536
$900$	0	0
$901$	85.9944	2.86489
$902$	−52.9923	−1.76445
$903$	0	0
$904$	−141.940	−4.72086
$905$	0	0
$906$	0	0
$907$	−4.72624	−0.156932	−0.0784661	−	0.996917i	$-0.525002\pi$
$907$	−4.72624	−0.156932	−0.0784661	+	0.996917i	$0.525002\pi$
$908$	55.2342	1.83301
$909$	0	0
$910$	0	0
$911$	−31.3184	−1.03762	−0.518812	−	0.854888i	$-0.673626\pi$
$911$	−31.3184	−1.03762	−0.518812	+	0.854888i	$0.673626\pi$
$912$	0	0
$913$	1.32734	0.0439285
$914$	−52.4633	−1.73533
$915$	0	0
$916$	40.6751	1.34394
$917$	−17.5641	−0.580018
$918$	0	0
$919$	6.50359	0.214534	0.107267	−	0.994230i	$-0.465790\pi$
$919$	6.50359	0.214534	0.107267	+	0.994230i	$0.465790\pi$
$920$	0	0
$921$	0	0
$922$	−45.6460	−1.50327
$923$	51.4529	1.69359
$924$	0	0
$925$	0	0
$926$	53.5341	1.75924
$927$	0	0
$928$	30.5264	1.00208
$929$	21.6418	0.710044	0.355022	−	0.934858i	$-0.384473\pi$
$929$	21.6418	0.710044	0.355022	+	0.934858i	$0.384473\pi$
$930$	0	0
$931$	25.5843	0.838491
$932$	30.7534	1.00736
$933$	0	0
$934$	−9.72509	−0.318215
$935$	0	0
$936$	0	0
$937$	−11.6829	−0.381663	−0.190831	−	0.981623i	$-0.561118\pi$
$937$	−11.6829	−0.381663	−0.190831	+	0.981623i	$0.561118\pi$
$938$	−15.4322	−0.503878
$939$	0	0
$940$	0	0
$941$	24.2911	0.791866	0.395933	−	0.918279i	$-0.370421\pi$
$941$	24.2911	0.791866	0.395933	+	0.918279i	$0.370421\pi$
$942$	0	0
$943$	−25.9527	−0.845138
$944$	−42.9421	−1.39765
$945$	0	0
$946$	−22.6795	−0.737376
$947$	55.8068	1.81348	0.906739	−	0.421692i	$-0.138564\pi$
$947$	55.8068	1.81348	0.906739	+	0.421692i	$0.138564\pi$
$948$	0	0
$949$	−22.3907	−0.726833
$950$	0	0
$951$	0	0
$952$	−65.2443	−2.11458
$953$	−27.5385	−0.892059	−0.446029	−	0.895018i	$-0.647162\pi$
$953$	−27.5385	−0.892059	−0.446029	+	0.895018i	$0.647162\pi$
$954$	0	0
$955$	0	0
$956$	87.1421	2.81838
$957$	0	0
$958$	27.5629	0.890516
$959$	19.3813	0.625853
$960$	0	0
$961$	1.00000	0.0322581
$962$	116.777	3.76504
$963$	0	0
$964$	31.8553	1.02599
$965$	0	0
$966$	0	0
$967$	40.2348	1.29386	0.646931	−	0.762548i	$-0.276052\pi$
$967$	40.2348	1.29386	0.646931	+	0.762548i	$0.276052\pi$
$968$	−19.1317	−0.614917
$969$	0	0
$970$	0	0
$971$	44.6382	1.43251	0.716254	−	0.697840i	$-0.245855\pi$
$971$	44.6382	1.43251	0.716254	+	0.697840i	$0.245855\pi$
$972$	0	0
$973$	4.03338	0.129304
$974$	−85.4928	−2.73937
$975$	0	0
$976$	75.0091	2.40098
$977$	59.0405	1.88887	0.944436	−	0.328696i	$-0.106609\pi$
$977$	59.0405	1.88887	0.944436	+	0.328696i	$0.106609\pi$
$978$	0	0
$979$	24.8317	0.793625
$980$	0	0
$981$	0	0
$982$	−21.7923	−0.695419
$983$	16.1265	0.514354	0.257177	−	0.966364i	$-0.417208\pi$
$983$	16.1265	0.514354	0.257177	+	0.966364i	$0.417208\pi$
$984$	0	0
$985$	0	0
$986$	51.3925	1.63667
$987$	0	0
$988$	−110.612	−3.51902
$989$	−11.1072	−0.353189
$990$	0	0
$991$	62.0192	1.97011	0.985053	−	0.172253i	$-0.0551048\pi$
$991$	62.0192	1.97011	0.985053	+	0.172253i	$0.0551048\pi$
$992$	10.1961	0.323725
$993$	0	0
$994$	−39.0566	−1.23880
$995$	0	0
$996$	0	0
$997$	−0.590602	−0.0187046	−0.00935228	−	0.999956i	$-0.502977\pi$
$997$	−0.590602	−0.0187046	−0.00935228	+	0.999956i	$0.502977\pi$
$998$	−23.6482	−0.748572
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6975.2.a.cj.1.10		11
3.2	odd	2		2325.2.a.bc.1.2		11
5.2	odd	4		1395.2.c.h.559.20		22
5.3	odd	4		1395.2.c.h.559.3		22
5.4	even	2		6975.2.a.ci.1.2		11
15.2	even	4		465.2.c.b.94.3	✓	22
15.8	even	4		465.2.c.b.94.20	yes	22
15.14	odd	2		2325.2.a.bd.1.10		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
465.2.c.b.94.3	✓	22	15.2	even	4
465.2.c.b.94.20	yes	22	15.8	even	4
1395.2.c.h.559.3		22	5.3	odd	4
1395.2.c.h.559.20		22	5.2	odd	4
2325.2.a.bc.1.2		11	3.2	odd	2
2325.2.a.bd.1.10		11	15.14	odd	2
6975.2.a.ci.1.2		11	5.4	even	2
6975.2.a.cj.1.10		11	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)

\(f(q)\)	\(=\)	\(q+2.60996 q^{2} +4.81187 q^{4} -1.35174 q^{7} +7.33885 q^{8} -2.89708 q^{11} +4.64772 q^{13} -3.52797 q^{14} +9.53034 q^{16} +6.57693 q^{17} -4.94592 q^{19} -7.56126 q^{22} -3.70309 q^{23} +12.1303 q^{26} -6.50437 q^{28} +2.99394 q^{29} +1.00000 q^{31} +10.1961 q^{32} +17.1655 q^{34} +9.62684 q^{37} -12.9086 q^{38} +7.00839 q^{41} +2.99944 q^{43} -13.9404 q^{44} -9.66491 q^{46} +7.98377 q^{47} -5.17281 q^{49} +22.3642 q^{52} +13.0752 q^{53} -9.92018 q^{56} +7.81406 q^{58} -4.50584 q^{59} +7.87056 q^{61} +2.60996 q^{62} +7.55057 q^{64} +4.37424 q^{67} +31.6473 q^{68} +11.0706 q^{71} -4.81757 q^{73} +25.1256 q^{74} -23.7991 q^{76} +3.91609 q^{77} +5.87005 q^{79} +18.2916 q^{82} -0.458163 q^{83} +7.82841 q^{86} -21.2613 q^{88} -8.57129 q^{89} -6.28249 q^{91} -17.8188 q^{92} +20.8373 q^{94} +0.397729 q^{97} -13.5008 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 3 q^{2} + 15 q^{4} - 8 q^{7} + 9 q^{8} - 14 q^{13} + 14 q^{14} + 27 q^{16} + 12 q^{17} + 12 q^{19} - 10 q^{22} + 12 q^{23} + 6 q^{26} - 22 q^{28} + 8 q^{29} + 11 q^{31} + 21 q^{32} + 2 q^{34} - 16 q^{37}+ \cdots + 17 q^{98}+O(q^{100}) \) 11 * q + 3 * q^2 + 15 * q^4 - 8 * q^7 + 9 * q^8 - 14 * q^13 + 14 * q^14 + 27 * q^16 + 12 * q^17 + 12 * q^19 - 10 * q^22 + 12 * q^23 + 6 * q^26 - 22 * q^28 + 8 * q^29 + 11 * q^31 + 21 * q^32 + 2 * q^34 - 16 * q^37 + 26 * q^38 + 8 * q^41 - 28 * q^43 + 26 * q^44 + 8 * q^46 + 14 * q^47 + 23 * q^49 - 18 * q^52 + 26 * q^53 + 42 * q^56 - 6 * q^58 + 6 * q^59 + 14 * q^61 + 3 * q^62 + 29 * q^64 - 6 * q^67 + 54 * q^68 + 18 * q^71 - 26 * q^73 + 40 * q^74 + 10 * q^76 + 2 * q^77 + 8 * q^79 + 30 * q^82 + 48 * q^83 + 12 * q^86 - 40 * q^88 + 14 * q^89 - 4 * q^91 - 30 * q^92 - 4 * q^94 - 16 * q^97 + 17 * q^98

Introduction

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