LMFDB - Embedded newform 925.2.a.k.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [925,2,Mod(1,925)] 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("925.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(925, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 925 = 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	925.a (trivial)

Newform invariants

comment:select newform

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

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

Self dual:	yes
Analytic conductor:	$7.38616218697$
Analytic rank:	$0$
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:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.548343$ of defining polynomial
Character	$\chi$	$=$	925.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.548343 q^{2} -2.30011 q^{3} -1.69932 q^{4} +1.26125 q^{6} -1.23269 q^{7} +2.02850 q^{8} +2.29051 q^{9} -0.976048 q^{11} +3.90863 q^{12} -6.96588 q^{13} +0.675937 q^{14} +2.28633 q^{16} -2.51901 q^{17} -1.25599 q^{18} -2.92164 q^{19} +2.83532 q^{21} +0.535210 q^{22} -8.99438 q^{23} -4.66577 q^{24} +3.81970 q^{26} +1.63190 q^{27} +2.09473 q^{28} -0.687767 q^{29} +6.60794 q^{31} -5.31069 q^{32} +2.24502 q^{33} +1.38128 q^{34} -3.89232 q^{36} +1.00000 q^{37} +1.60206 q^{38} +16.0223 q^{39} -0.193259 q^{41} -1.55473 q^{42} +9.21747 q^{43} +1.65862 q^{44} +4.93201 q^{46} +3.18723 q^{47} -5.25881 q^{48} -5.48048 q^{49} +5.79402 q^{51} +11.8373 q^{52} +1.28828 q^{53} -0.894839 q^{54} -2.50051 q^{56} +6.72011 q^{57} +0.377133 q^{58} +12.2588 q^{59} -3.67468 q^{61} -3.62342 q^{62} -2.82349 q^{63} -1.66057 q^{64} -1.23104 q^{66} +8.16762 q^{67} +4.28061 q^{68} +20.6881 q^{69} -2.82331 q^{71} +4.64630 q^{72} +0.837276 q^{73} -0.548343 q^{74} +4.96481 q^{76} +1.20316 q^{77} -8.78573 q^{78} +3.44333 q^{79} -10.6251 q^{81} +0.105972 q^{82} +4.66960 q^{83} -4.81812 q^{84} -5.05434 q^{86} +1.58194 q^{87} -1.97991 q^{88} -6.74431 q^{89} +8.58676 q^{91} +15.2843 q^{92} -15.1990 q^{93} -1.74770 q^{94} +12.2152 q^{96} +2.21791 q^{97} +3.00518 q^{98} -2.23565 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + q^{2} - q^{3} + 7 q^{4} + 6 q^{6} + 10 q^{9} + 16 q^{11} + 11 q^{12} - q^{13} - 3 q^{14} + 3 q^{16} + 4 q^{17} - 23 q^{18} + 9 q^{19} + 2 q^{21} + q^{22} - q^{23} + 24 q^{26} - q^{27} + 7 q^{28}+ \cdots + 39 q^{99}+O(q^{100}) $ 7 * q + q^2 - q^3 + 7 * q^4 + 6 * q^6 + 10 * q^9 + 16 * q^11 + 11 * q^12 - q^13 - 3 * q^14 + 3 * q^16 + 4 * q^17 - 23 * q^18 + 9 * q^19 + 2 * q^21 + q^22 - q^23 + 24 * q^26 - q^27 + 7 * q^28 + 3 * q^29 + 11 * q^31 + 21 * q^32 - 15 * q^33 - 23 * q^34 + 19 * q^36 + 7 * q^37 - 32 * q^38 + 2 * q^39 + 29 * q^41 + 44 * q^42 + 3 * q^43 + 18 * q^44 + 7 * q^46 - 2 * q^47 - 4 * q^48 + 5 * q^49 + 5 * q^51 + 30 * q^52 - 6 * q^53 + 8 * q^54 + 13 * q^56 + q^57 - 28 * q^58 + 42 * q^59 - 11 * q^61 + 11 * q^62 - 2 * q^63 - 16 * q^64 + 29 * q^66 + 3 * q^67 - 4 * q^68 + 2 * q^69 + 10 * q^71 - 29 * q^72 - 14 * q^73 + q^74 + 11 * q^76 - 11 * q^77 + 3 * q^78 - 9 * q^79 - 13 * q^81 - 6 * q^82 - 6 * q^83 - 41 * q^84 - 22 * q^86 + 11 * q^87 - 32 * q^88 + 12 * q^89 - 2 * q^91 + 5 * q^92 + 28 * q^93 + 9 * q^94 - 5 * q^96 - q^97 + 6 * q^98 + 39 * q^99

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	$-0.562104\pi$
$2$	−0.548343	−0.387737	−0.193869	+	0.981027i	$0.562104\pi$
$3$	−2.30011	−1.32797	−0.663985	−	0.747746i	$-0.731136\pi$
$3$	−2.30011	−1.32797	−0.663985	+	0.747746i	$0.731136\pi$
$4$	−1.69932	−0.849660
$5$	0	0
$5$	0	0
$6$	1.26125	0.514904
$7$	−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
$9$	2.29051	0.763505
$10$	0	0
$11$	−0.976048	−0.294290	−0.147145	−	0.989115i	$-0.547008\pi$
$11$	−0.976048	−0.294290	−0.147145	+	0.989115i	$0.547008\pi$
$12$	3.90863	1.12832
$13$	−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	$-0.598815\pi$
$17$	−2.51901	−0.610951	−0.305475	+	0.952200i	$0.598815\pi$
$18$	−1.25599	−0.296039
$19$	−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
$21$	2.83532	0.618718
$22$	0.535210	0.114107
$23$	−8.99438	−1.87546	−0.937729	−	0.347368i	$-0.887075\pi$
$23$	−8.99438	−1.87546	−0.937729	+	0.347368i	$0.887075\pi$
$24$	−4.66577	−0.952396
$25$	0	0
$26$	3.81970	0.749104
$27$	1.63190	0.314059
$28$	2.09473	0.395867
$29$	−0.687767	−0.127715	−0.0638576	−	0.997959i	$-0.520340\pi$
$29$	−0.687767	−0.127715	−0.0638576	+	0.997959i	$0.520340\pi$
$30$	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
$33$	2.24502	0.390808
$34$	1.38128	0.236888
$35$	0	0
$36$	−3.89232	−0.648719
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	1.60206	0.259889
$39$	16.0223	2.56562
$40$	0	0
$41$	−0.193259	−0.0301820	−0.0150910	−	0.999886i	$-0.504804\pi$
$41$	−0.193259	−0.0301820	−0.0150910	+	0.999886i	$0.504804\pi$
$42$	−1.55473	−0.239900
$43$	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	$-0.425325\pi$
$47$	3.18723	0.464905	0.232453	+	0.972608i	$0.425325\pi$
$48$	−5.25881	−0.759043
$49$	−5.48048	−0.782926
$50$	0	0
$51$	5.79402	0.811324
$52$	11.8373	1.64153
$53$	1.28828	0.176959	0.0884795	−	0.996078i	$-0.471799\pi$
$53$	1.28828	0.176959	0.0884795	+	0.996078i	$0.471799\pi$
$54$	−0.894839	−0.121772
$55$	0	0
$56$	−2.50051	−0.334144
$57$	6.72011	0.890100
$58$	0.377133	0.0495199
$59$	12.2588	1.59596	0.797978	−	0.602686i	$-0.205903\pi$
$59$	12.2588	1.59596	0.797978	+	0.602686i	$0.205903\pi$
$60$	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
$63$	−2.82349	−0.355726
$64$	−1.66057	−0.207572
$65$	0	0
$66$	−1.23104	−0.151531
$67$	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
$69$	20.6881	2.49055
$70$	0	0
$71$	−2.82331	−0.335066	−0.167533	−	0.985866i	$-0.553580\pi$
$71$	−2.82331	−0.335066	−0.167533	+	0.985866i	$0.553580\pi$
$72$	4.64630	0.547572
$73$	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
$78$	−8.78573	−0.994788
$79$	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
$81$	−10.6251	−1.18057
$82$	0.105972	0.0117027
$83$	4.66960	0.512555	0.256278	−	0.966603i	$-0.417504\pi$
$83$	4.66960	0.512555	0.256278	+	0.966603i	$0.417504\pi$
$84$	−4.81812	−0.525700
$85$	0	0
$86$	−5.05434	−0.545023
$87$	1.58194	0.169602
$88$	−1.97991	−0.211059
$89$	−6.74431	−0.714895	−0.357448	−	0.933933i	$-0.616353\pi$
$89$	−6.74431	−0.714895	−0.357448	+	0.933933i	$0.616353\pi$
$90$	0	0
$91$	8.58676	0.900137
$92$	15.2843	1.59350
$93$	−15.1990	−1.57606
$94$	−1.74770	−0.180261
$95$	0	0
$96$	12.2152	1.24671
$97$	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
$99$	−2.23565	−0.224692
$100$	0	0
$101$	4.83925	0.481523	0.240762	−	0.970584i	$-0.422603\pi$
$101$	4.83925	0.481523	0.240762	+	0.970584i	$0.422603\pi$
$102$	−3.17711	−0.314581
$103$	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	$-0.772992\pi$
$107$	−15.6463	−1.51259	−0.756293	+	0.654233i	$0.772992\pi$
$108$	−2.77311	−0.266843
$109$	−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
$111$	−2.30011	−0.218317
$112$	−2.81833	−0.266307
$113$	19.4522	1.82991	0.914953	−	0.403561i	$-0.132228\pi$
$113$	19.4522	1.82991	0.914953	+	0.403561i	$0.132228\pi$
$114$	−3.68493	−0.345125
$115$	0	0
$116$	1.16874	0.108514
$117$	−15.9555	−1.47508
$118$	−6.72202	−0.618812
$119$	3.10516	0.284650
$120$	0	0
$121$	−10.0473	−0.913394
$122$	2.01499	0.182428
$123$	0.444518	0.0400808
$124$	−11.2290	−1.00839
$125$	0	0
$126$	1.54824	0.137928
$127$	−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
$129$	−21.2012	−1.86666
$130$	0	0
$131$	15.2972	1.33653	0.668263	−	0.743926i	$-0.267038\pi$
$131$	15.2972	1.33653	0.668263	+	0.743926i	$0.267038\pi$
$132$	−3.81501	−0.332054
$133$	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	$-0.544750\pi$
$137$	−3.28024	−0.280249	−0.140125	+	0.990134i	$0.544750\pi$
$138$	−11.3442	−0.965680
$139$	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
$141$	−7.33099	−0.617380
$142$	1.54815	0.129918
$143$	6.79904	0.568564
$144$	5.23686	0.436405
$145$	0	0
$146$	−0.459115	−0.0379966
$147$	12.6057	1.03970
$148$	−1.69932	−0.139683
$149$	12.7327	1.04311	0.521553	−	0.853219i	$-0.325353\pi$
$149$	12.7327	1.04311	0.521553	+	0.853219i	$0.325353\pi$
$150$	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
$153$	−5.76984	−0.466464
$154$	−0.659747	−0.0531639
$155$	0	0
$156$	−27.2270	−2.17991
$157$	−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
$159$	−2.96319	−0.234996
$160$	0	0
$161$	11.0873	0.873799
$162$	5.82620	0.457749
$163$	−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	$-0.346929\pi$
$167$	11.9553	0.925130	0.462565	+	0.886585i	$0.346929\pi$
$168$	5.75144	0.443733
$169$	35.5235	2.73258
$170$	0	0
$171$	−6.69207	−0.511755
$172$	−15.6634	−1.19433
$173$	21.6980	1.64967	0.824834	−	0.565375i	$-0.191268\pi$
$173$	21.6980	1.64967	0.824834	+	0.565375i	$0.191268\pi$
$174$	−0.867447	−0.0657610
$175$	0	0
$176$	−2.23156	−0.168211
$177$	−28.1966	−2.11938
$178$	3.69820	0.277192
$179$	−3.99480	−0.298586	−0.149293	−	0.988793i	$-0.547700\pi$
$179$	−3.99480	−0.298586	−0.149293	+	0.988793i	$0.547700\pi$
$180$	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
$183$	8.45218	0.624803
$184$	−18.2451	−1.34504
$185$	0	0
$186$	8.33428	0.611099
$187$	2.45868	0.179797
$188$	−5.41612	−0.395011
$189$	−2.01162	−0.146324
$190$	0	0
$191$	−20.4730	−1.48137	−0.740686	−	0.671852i	$-0.765499\pi$
$191$	−20.4730	−1.48137	−0.740686	+	0.671852i	$0.765499\pi$
$192$	3.81950	0.275649
$193$	−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	$-0.709753\pi$
$197$	−17.1879	−1.22459	−0.612294	+	0.790630i	$0.709753\pi$
$198$	1.22591	0.0871213
$199$	−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
$201$	−18.7864	−1.32509
$202$	−2.65357	−0.186704
$203$	0.847803	0.0595041
$204$	−9.84588	−0.689350
$205$	0	0
$206$	−4.91174	−0.342217
$207$	−20.6018	−1.43192
$208$	−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
$213$	6.49394	0.444957
$214$	8.57955	0.586486
$215$	0	0
$216$	3.31030	0.225237
$217$	−8.14554	−0.552955
$218$	6.84615	0.463680
$219$	−1.92583	−0.130135
$220$	0	0
$221$	17.5472	1.18035
$222$	1.26125	0.0846496
$223$	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	$-0.243644\pi$
$227$	21.7284	1.44217	0.721084	+	0.692848i	$0.243644\pi$
$228$	−11.4196	−0.756282
$229$	−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
$231$	−2.76741	−0.182082
$232$	−1.39513	−0.0915950
$233$	18.5161	1.21303	0.606517	−	0.795071i	$-0.292566\pi$
$233$	18.5161	1.21303	0.606517	+	0.795071i	$0.292566\pi$
$234$	8.74907	0.571944
$235$	0	0
$236$	−20.8316	−1.35602
$237$	−7.92005	−0.514462
$238$	−1.70269	−0.110369
$239$	18.0845	1.16979	0.584896	−	0.811109i	$-0.301136\pi$
$239$	18.0845	1.16979	0.584896	+	0.811109i	$0.301136\pi$
$240$	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
$243$	19.5432	1.25370
$244$	6.24446	0.399761
$245$	0	0
$246$	−0.243748	−0.0155408
$247$	20.3518	1.29496
$248$	13.4042	0.851167
$249$	−10.7406	−0.680658
$250$	0	0
$251$	4.32919	0.273256	0.136628	−	0.990622i	$-0.456373\pi$
$251$	4.32919	0.273256	0.136628	+	0.990622i	$0.456373\pi$
$252$	4.79801	0.302246
$253$	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	$-0.671642\pi$
$257$	−16.4633	−1.02695	−0.513475	+	0.858105i	$0.671642\pi$
$258$	11.6255	0.723775
$259$	−1.23269	−0.0765955
$260$	0	0
$261$	−1.57534	−0.0975111
$262$	−8.38813	−0.518221
$263$	−19.3023	−1.19023	−0.595116	−	0.803640i	$-0.702894\pi$
$263$	−19.3023	−1.19023	−0.595116	+	0.803640i	$0.702894\pi$
$264$	4.55402	0.280280
$265$	0	0
$266$	−1.97485	−0.121086
$267$	15.5127	0.949359
$268$	−13.8794	−0.847819
$269$	−9.62970	−0.587133	−0.293567	−	0.955939i	$-0.594842\pi$
$269$	−9.62970	−0.587133	−0.293567	+	0.955939i	$0.594842\pi$
$270$	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
$273$	−19.7505	−1.19536
$274$	1.79870	0.108663
$275$	0	0
$276$	−35.1557	−2.11612
$277$	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
$279$	15.1356	0.906144
$280$	0	0
$281$	11.5359	0.688177	0.344089	−	0.938937i	$-0.388188\pi$
$281$	11.5359	0.688177	0.344089	+	0.938937i	$0.388188\pi$
$282$	4.01990	0.239381
$283$	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
$288$	−12.1642	−0.716783
$289$	−10.6546	−0.626739
$290$	0	0
$291$	−5.10144	−0.299052
$292$	−1.42280	−0.0832631
$293$	16.2157	0.947332	0.473666	−	0.880705i	$-0.342930\pi$
$293$	16.2157	0.947332	0.473666	+	0.880705i	$0.342930\pi$
$294$	−6.91226	−0.403131
$295$	0	0
$296$	2.02850	0.117904
$297$	−1.59281	−0.0924242
$298$	−6.98190	−0.404451
$299$	62.6538	3.62336
$300$	0	0
$301$	−11.3623	−0.654910
$302$	9.95756	0.572993
$303$	−11.1308	−0.639448
$304$	−6.67983	−0.383115
$305$	0	0
$306$	3.16385	0.180865
$307$	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
$309$	−20.6030	−1.17207
$310$	0	0
$311$	−21.0126	−1.19151	−0.595756	−	0.803165i	$-0.703148\pi$
$311$	−21.0126	−1.19151	−0.595756	+	0.803165i	$0.703148\pi$
$312$	32.5012	1.84002
$313$	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	$-0.793858\pi$
$317$	−28.3991	−1.59505	−0.797526	+	0.603284i	$0.793858\pi$
$318$	1.62485	0.0911168
$319$	0.671294	0.0375853
$320$	0	0
$321$	35.9883	2.00867
$322$	−6.07963	−0.338805
$323$	7.35967	0.409503
$324$	18.0554	1.00308
$325$	0	0
$326$	4.61048	0.255351
$327$	28.7173	1.58807
$328$	−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
$333$	2.29051	0.125519
$334$	−6.55561	−0.358707
$335$	0	0
$336$	6.48247	0.353648
$337$	−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
$339$	−44.7422	−2.43006
$340$	0	0
$341$	−6.44967	−0.349269
$342$	3.66955	0.198427
$343$	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	$-0.429727\pi$
$347$	8.15833	0.437962	0.218981	+	0.975729i	$0.429727\pi$
$348$	−2.68822	−0.144104
$349$	−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
$351$	−11.3676	−0.606757
$352$	5.18349	0.276281
$353$	1.95167	0.103877	0.0519385	−	0.998650i	$-0.483460\pi$
$353$	1.95167	0.103877	0.0519385	+	0.998650i	$0.483460\pi$
$354$	15.4614	0.821764
$355$	0	0
$356$	11.4607	0.607418
$357$	−7.14222	−0.378006
$358$	2.19052	0.115773
$359$	25.2874	1.33462	0.667310	−	0.744780i	$-0.267446\pi$
$359$	25.2874	1.33462	0.667310	+	0.744780i	$0.267446\pi$
$360$	0	0
$361$	−10.4640	−0.550736
$362$	−9.43239	−0.495755
$363$	23.1100	1.21296
$364$	−14.5917	−0.764811
$365$	0	0
$366$	−4.63470	−0.242259
$367$	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
$369$	−0.442663	−0.0230441
$370$	0	0
$371$	−1.58805	−0.0824474
$372$	25.8280	1.33912
$373$	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
$378$	1.10306	0.0567352
$379$	−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
$381$	12.0935	0.619568
$382$	11.2262	0.574383
$383$	−30.2394	−1.54516	−0.772580	−	0.634918i	$-0.781034\pi$
$383$	−30.2394	−1.54516	−0.772580	+	0.634918i	$0.781034\pi$
$384$	−26.5247	−1.35359
$385$	0	0
$386$	11.2228	0.571223
$387$	21.1127	1.07322
$388$	−3.76894	−0.191339
$389$	−20.8628	−1.05779	−0.528894	−	0.848688i	$-0.677393\pi$
$389$	−20.8628	−1.05779	−0.528894	+	0.848688i	$0.677393\pi$
$390$	0	0
$391$	22.6570	1.14581
$392$	−11.1171	−0.561500
$393$	−35.1853	−1.77487
$394$	9.42488	0.474819
$395$	0	0
$396$	3.79909	0.190911
$397$	−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
$399$	−8.28380	−0.414709
$400$	0	0
$401$	−26.5071	−1.32370	−0.661851	−	0.749636i	$-0.730229\pi$
$401$	−26.5071	−1.32370	−0.661851	+	0.749636i	$0.730229\pi$
$402$	10.3014	0.513788
$403$	−46.0302	−2.29293
$404$	−8.22343	−0.409131
$405$	0	0
$406$	−0.464887	−0.0230720
$407$	−0.976048	−0.0483809
$408$	11.7531	0.581867
$409$	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
$411$	7.54491	0.372163
$412$	−15.2215	−0.749910
$413$	−15.1113	−0.743576
$414$	11.2968	0.555209
$415$	0	0
$416$	36.9936	1.81376
$417$	−1.29089	−0.0632153
$418$	−1.56369	−0.0764827
$419$	31.9064	1.55873	0.779366	−	0.626569i	$-0.215541\pi$
$419$	31.9064	1.55873	0.779366	+	0.626569i	$0.215541\pi$
$420$	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
$423$	7.30040	0.354957
$424$	2.61327	0.126912
$425$	0	0
$426$	−3.56091	−0.172527
$427$	4.52974	0.219209
$428$	26.5881	1.28518
$429$	−15.6385	−0.755036
$430$	0	0
$431$	40.0005	1.92676	0.963379	−	0.268143i	$-0.0864100\pi$
$431$	40.0005	1.92676	0.963379	+	0.268143i	$0.0864100\pi$
$432$	3.73105	0.179510
$433$	−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
$438$	1.05602	0.0504584
$439$	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
$441$	−12.5531	−0.597767
$442$	−9.62187	−0.457666
$443$	−39.0684	−1.85620	−0.928098	−	0.372335i	$-0.878557\pi$
$443$	−39.0684	−1.85620	−0.928098	+	0.372335i	$0.878557\pi$
$444$	3.90863	0.185495
$445$	0	0
$446$	−1.97185	−0.0933700
$447$	−29.2867	−1.38521
$448$	2.04697	0.0967102
$449$	27.7382	1.30905	0.654523	−	0.756042i	$-0.272870\pi$
$449$	27.7382	1.30905	0.654523	+	0.756042i	$0.272870\pi$
$450$	0	0
$451$	0.188630	0.00888225
$452$	−33.0554	−1.55480
$453$	41.7685	1.96246
$454$	−11.9146	−0.559182
$455$	0	0
$456$	13.6317	0.638364
$457$	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
$459$	−4.11077	−0.191874
$460$	0	0
$461$	31.1870	1.45252	0.726261	−	0.687419i	$-0.241256\pi$
$461$	31.1870	1.45252	0.726261	+	0.687419i	$0.241256\pi$
$462$	1.51749	0.0706001
$463$	−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	$-0.367479\pi$
$467$	17.4785	0.808807	0.404404	+	0.914581i	$0.367479\pi$
$468$	27.1134	1.25332
$469$	−10.0681	−0.464903
$470$	0	0
$471$	28.7851	1.32635
$472$	24.8669	1.14459
$473$	−8.99670	−0.413669
$474$	4.34291	0.199476
$475$	0	0
$476$	−5.27666	−0.241855
$477$	2.95083	0.135109
$478$	−9.91653	−0.453572
$479$	−4.30979	−0.196919	−0.0984597	−	0.995141i	$-0.531392\pi$
$479$	−4.30979	−0.196919	−0.0984597	+	0.995141i	$0.531392\pi$
$480$	0	0
$481$	−6.96588	−0.317617
$482$	6.83474	0.311314
$483$	−25.5020	−1.16038
$484$	17.0736	0.776074
$485$	0	0
$486$	−10.7164	−0.486105
$487$	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
$489$	19.3394	0.874558
$490$	0	0
$491$	36.0217	1.62564	0.812818	−	0.582517i	$-0.197932\pi$
$491$	36.0217	1.62564	0.812818	+	0.582517i	$0.197932\pi$
$492$	−0.755377	−0.0340550
$493$	1.73250	0.0780277
$494$	−11.1598	−0.502103
$495$	0	0
$496$	15.1079	0.678366
$497$	3.48027	0.156111
$498$	5.88954	0.263917
$499$	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
$501$	−27.4985	−1.22854
$502$	−2.37388	−0.105952
$503$	3.01511	0.134437	0.0672185	−	0.997738i	$-0.478588\pi$
$503$	3.01511	0.134437	0.0672185	+	0.997738i	$0.478588\pi$
$504$	−5.72744	−0.255121
$505$	0	0
$506$	−4.81388	−0.214003
$507$	−81.7081	−3.62878
$508$	8.93465	0.396411
$509$	24.8656	1.10215	0.551073	−	0.834457i	$-0.314218\pi$
$509$	24.8656	1.10215	0.551073	+	0.834457i	$0.314218\pi$
$510$	0	0
$511$	−1.03210	−0.0456575
$512$	−21.4176	−0.946532
$513$	−4.76782	−0.210504
$514$	9.02752	0.398187
$515$	0	0
$516$	36.0276	1.58603
$517$	−3.11089	−0.136817
$518$	0.675937	0.0296989
$519$	−49.9078	−2.19071
$520$	0	0
$521$	38.3050	1.67817	0.839085	−	0.544000i	$-0.183091\pi$
$521$	38.3050	1.67817	0.839085	+	0.544000i	$0.183091\pi$
$522$	0.863828	0.0378087
$523$	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
$528$	5.13285	0.223379
$529$	57.8989	2.51734
$530$	0	0
$531$	28.0789	1.21852
$532$	−6.12006	−0.265338
$533$	1.34622	0.0583113
$534$	−8.50626	−0.368102
$535$	0	0
$536$	16.5680	0.715628
$537$	9.18850	0.396513
$538$	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
$543$	−39.5656	−1.69792
$544$	13.3777	0.573564
$545$	0	0
$546$	10.8301	0.463484
$547$	−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
$549$	−8.41691	−0.359225
$550$	0	0
$551$	2.00941	0.0856038
$552$	41.9657	1.78618
$553$	−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	$-0.387752\pi$
$557$	16.3022	0.690748	0.345374	+	0.938465i	$0.387752\pi$
$558$	−8.29950	−0.351346
$559$	−64.2078	−2.71570
$560$	0	0
$561$	−5.65524	−0.238764
$562$	−6.32566	−0.266832
$563$	10.0364	0.422985	0.211492	−	0.977380i	$-0.432168\pi$
$563$	10.0364	0.422985	0.211492	+	0.977380i	$0.432168\pi$
$564$	12.4577	0.524563
$565$	0	0
$566$	−16.8516	−0.708325
$567$	13.0974	0.550040
$568$	−5.72709	−0.240303
$569$	−4.06296	−0.170328	−0.0851640	−	0.996367i	$-0.527141\pi$
$569$	−4.06296	−0.170328	−0.0851640	+	0.996367i	$0.527141\pi$
$570$	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
$573$	47.0901	1.96722
$574$	−0.130631	−0.00545243
$575$	0	0
$576$	−3.80357	−0.158482
$577$	−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
$579$	47.0756	1.95639
$580$	0	0
$581$	−5.75616	−0.238806
$582$	2.79734	0.115954
$583$	−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	$-0.361871\pi$
$587$	20.3736	0.840907	0.420453	+	0.907314i	$0.361871\pi$
$588$	−21.4211	−0.883393
$589$	−19.3061	−0.795493
$590$	0	0
$591$	39.5341	1.62622
$592$	2.28633	0.0939674
$593$	28.4271	1.16736	0.583681	−	0.811983i	$-0.301612\pi$
$593$	28.4271	1.16736	0.583681	+	0.811983i	$0.301612\pi$
$594$	0.873407	0.0358363
$595$	0	0
$596$	−21.6370	−0.886284
$597$	29.6295	1.21265
$598$	−34.3558	−1.40491
$599$	4.99991	0.204291	0.102145	−	0.994769i	$-0.467429\pi$
$599$	4.99991	0.204291	0.102145	+	0.994769i	$0.467429\pi$
$600$	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
$603$	18.7080	0.761850
$604$	30.8585	1.25562
$605$	0	0
$606$	6.10350	0.247938
$607$	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
$609$	−1.95004	−0.0790197
$610$	0	0
$611$	−22.2019	−0.898192
$612$	9.80480	0.396336
$613$	−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	$-0.374671\pi$
$617$	19.0587	0.767275	0.383638	+	0.923484i	$0.374671\pi$
$618$	11.2975	0.454454
$619$	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
$621$	−14.6779	−0.589004
$622$	11.5221	0.461994
$623$	8.31363	0.333079
$624$	36.6322	1.46646
$625$	0	0
$626$	−1.18592	−0.0473987
$627$	−6.55915	−0.261947
$628$	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
$633$	−33.6438	−1.33722
$634$	15.5725	0.618461
$635$	0	0
$636$	5.03541	0.199667
$637$	38.1764	1.51260
$638$	−0.368100	−0.0145732
$639$	−6.46684	−0.255824
$640$	0	0
$641$	−33.9051	−1.33917	−0.669586	−	0.742735i	$-0.733528\pi$
$641$	−33.9051	−1.33917	−0.669586	+	0.742735i	$0.733528\pi$
$642$	−19.7339	−0.778836
$643$	−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	$-0.347279\pi$
$647$	23.4822	0.923180	0.461590	+	0.887093i	$0.347279\pi$
$648$	−21.5530	−0.846680
$649$	−11.9652	−0.469674
$650$	0	0
$651$	18.7356	0.734308
$652$	14.2879	0.559558
$653$	6.98761	0.273446	0.136723	−	0.990609i	$-0.456343\pi$
$653$	6.98761	0.273446	0.136723	+	0.990609i	$0.456343\pi$
$654$	−15.7469	−0.615753
$655$	0	0
$656$	−0.441853	−0.0172515
$657$	1.91779	0.0748203
$658$	2.15437	0.0839859
$659$	13.4171	0.522656	0.261328	−	0.965250i	$-0.415840\pi$
$659$	13.4171	0.522656	0.261328	+	0.965250i	$0.415840\pi$
$660$	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
$663$	−40.3604	−1.56747
$664$	9.47227	0.367596
$665$	0	0
$666$	−1.25599	−0.0486686
$667$	6.18604	0.239524
$668$	−20.3159	−0.786045
$669$	−8.27125	−0.319785
$670$	0	0
$671$	3.58667	0.138462
$672$	−15.0575	−0.580856
$673$	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	$-0.545205\pi$
$677$	−7.36551	−0.283080	−0.141540	+	0.989933i	$0.545205\pi$
$678$	24.5341	0.942225
$679$	−2.73399	−0.104921
$680$	0	0
$681$	−49.9778	−1.91515
$682$	3.53664	0.135425
$683$	−46.2794	−1.77083	−0.885415	−	0.464801i	$-0.846126\pi$
$683$	−46.2794	−1.77083	−0.885415	+	0.464801i	$0.846126\pi$
$684$	11.3720	0.434818
$685$	0	0
$686$	−8.43601	−0.322088
$687$	33.5997	1.28191
$688$	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
$693$	2.75586	0.104687
$694$	−4.47357	−0.169814
$695$	0	0
$696$	3.20896	0.121635
$697$	0.486823	0.0184397
$698$	15.4864	0.586170
$699$	−42.5892	−1.61087
$700$	0	0
$701$	20.9220	0.790213	0.395107	−	0.918635i	$-0.370708\pi$
$701$	20.9220	0.790213	0.395107	+	0.918635i	$0.370708\pi$
$702$	6.23335	0.235262
$703$	−2.92164	−0.110192
$704$	1.62080	0.0610862
$705$	0	0
$706$	−1.07019	−0.0402770
$707$	−5.96528	−0.224348
$708$	47.9150	1.80075
$709$	−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
$711$	7.88700	0.295786
$712$	−13.6808	−0.512710
$713$	−59.4344	−2.22583
$714$	3.91639	0.146567
$715$	0	0
$716$	6.78845	0.253696
$717$	−41.5964	−1.55345
$718$	−13.8662	−0.517482
$719$	20.2226	0.754176	0.377088	−	0.926177i	$-0.376925\pi$
$719$	20.2226	0.754176	0.377088	+	0.926177i	$0.376925\pi$
$720$	0	0
$721$	−11.0417	−0.411214
$722$	5.73786	0.213541
$723$	28.6694	1.06623
$724$	−29.2310	−1.08636
$725$	0	0
$726$	−12.6722	−0.470310
$727$	−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
$729$	−13.0763	−0.484307
$730$	0	0
$731$	−23.2189	−0.858784
$732$	−14.3630	−0.530870
$733$	−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
$738$	0.242731	0.00893506
$739$	−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
$741$	−46.8115	−1.71966
$742$	0.870796	0.0319679
$743$	−0.401960	−0.0147465	−0.00737324	−	0.999973i	$-0.502347\pi$
$743$	−0.401960	−0.0147465	−0.00737324	+	0.999973i	$0.502347\pi$
$744$	−30.8312	−1.13032
$745$	0	0
$746$	−13.7592	−0.503758
$747$	10.6958	0.391338
$748$	−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
$753$	−9.95763	−0.362876
$754$	−2.62706	−0.0956719
$755$	0	0
$756$	3.41838	0.124325
$757$	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
$759$	−20.1926	−0.732944
$760$	0	0
$761$	−37.6100	−1.36336	−0.681681	−	0.731650i	$-0.738751\pi$
$761$	−37.6100	−1.36336	−0.681681	+	0.731650i	$0.738751\pi$
$762$	−6.63138	−0.240230
$763$	15.3903	0.557167
$764$	34.7901	1.25866
$765$	0	0
$766$	16.5816	0.599116
$767$	−85.3932	−3.08337
$768$	6.90566	0.249187
$769$	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
$771$	37.8673	1.36376
$772$	34.7794	1.25174
$773$	11.1521	0.401114	0.200557	−	0.979682i	$-0.435725\pi$
$773$	11.1521	0.401114	0.200557	+	0.979682i	$0.435725\pi$
$774$	−11.5770	−0.416128
$775$	0	0
$776$	4.49903	0.161506
$777$	2.83532	0.101717
$778$	11.4400	0.410144
$779$	0.564634	0.0202301
$780$	0	0
$781$	2.75569	0.0986064
$782$	−12.4238	−0.444274
$783$	−1.12236	−0.0401100
$784$	−12.5302	−0.447506
$785$	0	0
$786$	19.2936	0.688181
$787$	−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
$789$	44.3975	1.58059
$790$	0	0
$791$	−23.9785	−0.852576
$792$	−4.53502	−0.161145
$793$	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	$-0.0891122\pi$
$797$	54.2642	1.92214	0.961068	+	0.276312i	$0.0891122\pi$
$798$	4.54237	0.160798
$799$	−8.02868	−0.284034
$800$	0	0
$801$	−15.4479	−0.545826
$802$	14.5350	0.513249
$803$	−0.817222	−0.0288391
$804$	31.9241	1.12588
$805$	0	0
$806$	25.2403	0.889053
$807$	22.1494	0.779695
$808$	9.81640	0.345340
$809$	16.1657	0.568355	0.284177	−	0.958772i	$-0.408280\pi$
$809$	16.1657	0.568355	0.284177	+	0.958772i	$0.408280\pi$
$810$	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
$813$	8.39844	0.294546
$814$	0.535210	0.0187591
$815$	0	0
$816$	13.2470	0.463738
$817$	−26.9302	−0.942167
$818$	−4.35412	−0.152238
$819$	19.6681	0.687259
$820$	0	0
$821$	30.7078	1.07171	0.535855	−	0.844310i	$-0.319990\pi$
$821$	30.7078	1.07171	0.535855	+	0.844310i	$0.319990\pi$
$822$	−4.13720	−0.144301
$823$	−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	$-0.277243\pi$
$827$	37.0441	1.28815	0.644074	+	0.764963i	$0.277243\pi$
$828$	35.0090	1.21665
$829$	−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
$831$	−57.6532	−1.99997
$832$	11.5674	0.401026
$833$	13.8054	0.478329
$834$	0.707853	0.0245109
$835$	0	0
$836$	−4.84589	−0.167599
$837$	10.7835	0.372732
$838$	−17.4957	−0.604379
$839$	−45.1775	−1.55970	−0.779850	−	0.625967i	$-0.784704\pi$
$839$	−45.1775	−1.55970	−0.779850	+	0.625967i	$0.784704\pi$
$840$	0	0
$841$	−28.5270	−0.983689
$842$	−6.05569	−0.208693
$843$	−26.5340	−0.913879
$844$	−24.8560	−0.855579
$845$	0	0
$846$	−4.00312	−0.137630
$847$	12.3852	0.425561
$848$	2.94543	0.101147
$849$	−70.6866	−2.42596
$850$	0	0
$851$	−8.99438	−0.308323
$852$	−11.0353	−0.378062
$853$	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	$-0.491950\pi$
$857$	1.48050	0.0505729	0.0252865	+	0.999680i	$0.491950\pi$
$858$	8.57529	0.292756
$859$	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
$861$	−0.547952	−0.0186741
$862$	−21.9340	−0.747076
$863$	9.07168	0.308803	0.154402	−	0.988008i	$-0.450655\pi$
$863$	9.07168	0.308803	0.154402	+	0.988008i	$0.450655\pi$
$864$	−8.66649	−0.294840
$865$	0	0
$866$	1.87583	0.0637434
$867$	24.5067	0.832291
$868$	13.8419	0.469824
$869$	−3.36086	−0.114009
$870$	0	0
$871$	−56.8946	−1.92780
$872$	−25.3261	−0.857650
$873$	5.08016	0.171937
$874$	−14.4096	−0.487411
$875$	0	0
$876$	3.27260	0.110571
$877$	−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
$879$	−37.2980	−1.25803
$880$	0	0
$881$	−32.2604	−1.08688	−0.543441	−	0.839448i	$-0.682879\pi$
$881$	−32.2604	−1.08688	−0.543441	+	0.839448i	$0.682879\pi$
$882$	6.88342	0.231777
$883$	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	$-0.698939\pi$
$887$	−34.8507	−1.17017	−0.585087	+	0.810971i	$0.698939\pi$
$888$	−4.66577	−0.156573
$889$	6.48121	0.217373
$890$	0	0
$891$	10.3706	0.347428
$892$	−6.11079	−0.204604
$893$	−9.31195	−0.311613
$894$	16.0592	0.537098
$895$	0	0
$896$	−14.2153	−0.474899
$897$	−144.111	−4.81172
$898$	−15.2100	−0.507566
$899$	−4.54473	−0.151575
$900$	0	0
$901$	−3.24520	−0.108113
$902$	−0.103434	−0.00344398
$903$	26.1345	0.869701
$904$	39.4587	1.31238
$905$	0	0
$906$	−22.9035	−0.760918
$907$	−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
$909$	11.0844	0.367645
$910$	0	0
$911$	27.6843	0.917221	0.458610	−	0.888637i	$-0.348347\pi$
$911$	27.6843	0.917221	0.458610	+	0.888637i	$0.348347\pi$
$912$	15.3644	0.508765
$913$	−4.55776	−0.150840
$914$	−2.16231	−0.0715228
$915$	0	0
$916$	24.8234	0.820188
$917$	−18.8567	−0.622704
$918$	2.25411	0.0743968
$919$	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
$921$	−0.998391	−0.0328981
$922$	−17.1012	−0.563197
$923$	19.6669	0.647343
$924$	4.70272	0.154708
$925$	0	0
$926$	16.3410	0.536999
$927$	20.5171	0.673869
$928$	3.65252	0.119900
$929$	−22.9783	−0.753894	−0.376947	−	0.926235i	$-0.623026\pi$
$929$	−22.9783	−0.753894	−0.376947	+	0.926235i	$0.623026\pi$
$930$	0	0
$931$	16.0120	0.524772
$932$	−31.4649	−1.03067
$933$	48.3312	1.58229
$934$	−9.58421	−0.313605
$935$	0	0
$936$	−32.3656	−1.05790
$937$	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
$939$	−4.97451	−0.162337
$940$	0	0
$941$	−16.9929	−0.553952	−0.276976	−	0.960877i	$-0.589332\pi$
$941$	−16.9929	−0.553952	−0.276976	+	0.960877i	$0.589332\pi$
$942$	−15.7841	−0.514274
$943$	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	$-0.624553\pi$
$947$	−23.4730	−0.762769	−0.381384	+	0.924417i	$0.624553\pi$
$948$	13.4587	0.437118
$949$	−5.83237	−0.189327
$950$	0	0
$951$	65.3211	2.11818
$952$	6.29881	0.204146
$953$	−24.6541	−0.798623	−0.399312	−	0.916815i	$-0.630751\pi$
$953$	−24.6541	−0.798623	−0.399312	+	0.916815i	$0.630751\pi$
$954$	−1.61807	−0.0523868
$955$	0	0
$956$	−30.7314	−0.993924
$957$	−1.54405	−0.0499121
$958$	2.36325	0.0763530
$959$	4.04351	0.130572
$960$	0	0
$961$	12.6649	0.408546
$962$	3.81970	0.123152
$963$	−35.8381	−1.15487
$964$	21.1809	0.682191
$965$	0	0
$966$	13.9838	0.449922
$967$	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
$969$	−16.9281	−0.543807
$970$	0	0
$971$	−1.93826	−0.0622017	−0.0311008	−	0.999516i	$-0.509901\pi$
$971$	−1.93826	−0.0622017	−0.0311008	+	0.999516i	$0.509901\pi$
$972$	−33.2101	−1.06522
$973$	−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	$-0.566257\pi$
$977$	−12.9186	−0.413303	−0.206651	+	0.978415i	$0.566257\pi$
$978$	−10.6046	−0.339099
$979$	6.58277	0.210386
$980$	0	0
$981$	−28.5974	−0.913046
$982$	−19.7523	−0.630320
$983$	42.8843	1.36780	0.683898	−	0.729578i	$-0.260283\pi$
$983$	42.8843	1.36780	0.683898	+	0.729578i	$0.260283\pi$
$984$	0.901703	0.0287452
$985$	0	0
$986$	−0.950002	−0.0302542
$987$	9.03682	0.287645
$988$	−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
$993$	−37.3579	−1.18552
$994$	−1.90838	−0.0605302
$995$	0	0
$996$	18.2517	0.578328
$997$	−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
$999$	1.63190	0.0516309

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
925.2.a.j.1.5	✓	7	5.4	even	2
925.2.a.k.1.3	yes	7	1.1	even	1	trivial
925.2.b.i.149.6		14	5.2	odd	4
925.2.b.i.149.9		14	5.3	odd	4
8325.2.a.cm.1.5		7	3.2	odd	2
8325.2.a.cn.1.3		7	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 925 = 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	925.a (trivial)

\(f(q)\)	\(=\)	\(q-0.548343 q^{2} -2.30011 q^{3} -1.69932 q^{4} +1.26125 q^{6} -1.23269 q^{7} +2.02850 q^{8} +2.29051 q^{9} -0.976048 q^{11} +3.90863 q^{12} -6.96588 q^{13} +0.675937 q^{14} +2.28633 q^{16} -2.51901 q^{17} -1.25599 q^{18} -2.92164 q^{19} +2.83532 q^{21} +0.535210 q^{22} -8.99438 q^{23} -4.66577 q^{24} +3.81970 q^{26} +1.63190 q^{27} +2.09473 q^{28} -0.687767 q^{29} +6.60794 q^{31} -5.31069 q^{32} +2.24502 q^{33} +1.38128 q^{34} -3.89232 q^{36} +1.00000 q^{37} +1.60206 q^{38} +16.0223 q^{39} -0.193259 q^{41} -1.55473 q^{42} +9.21747 q^{43} +1.65862 q^{44} +4.93201 q^{46} +3.18723 q^{47} -5.25881 q^{48} -5.48048 q^{49} +5.79402 q^{51} +11.8373 q^{52} +1.28828 q^{53} -0.894839 q^{54} -2.50051 q^{56} +6.72011 q^{57} +0.377133 q^{58} +12.2588 q^{59} -3.67468 q^{61} -3.62342 q^{62} -2.82349 q^{63} -1.66057 q^{64} -1.23104 q^{66} +8.16762 q^{67} +4.28061 q^{68} +20.6881 q^{69} -2.82331 q^{71} +4.64630 q^{72} +0.837276 q^{73} -0.548343 q^{74} +4.96481 q^{76} +1.20316 q^{77} -8.78573 q^{78} +3.44333 q^{79} -10.6251 q^{81} +0.105972 q^{82} +4.66960 q^{83} -4.81812 q^{84} -5.05434 q^{86} +1.58194 q^{87} -1.97991 q^{88} -6.74431 q^{89} +8.58676 q^{91} +15.2843 q^{92} -15.1990 q^{93} -1.74770 q^{94} +12.2152 q^{96} +2.21791 q^{97} +3.00518 q^{98} -2.23565 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + q^{2} - q^{3} + 7 q^{4} + 6 q^{6} + 10 q^{9} + 16 q^{11} + 11 q^{12} - q^{13} - 3 q^{14} + 3 q^{16} + 4 q^{17} - 23 q^{18} + 9 q^{19} + 2 q^{21} + q^{22} - q^{23} + 24 q^{26} - q^{27} + 7 q^{28}+ \cdots + 39 q^{99}+O(q^{100}) \) 7 * q + q^2 - q^3 + 7 * q^4 + 6 * q^6 + 10 * q^9 + 16 * q^11 + 11 * q^12 - q^13 - 3 * q^14 + 3 * q^16 + 4 * q^17 - 23 * q^18 + 9 * q^19 + 2 * q^21 + q^22 - q^23 + 24 * q^26 - q^27 + 7 * q^28 + 3 * q^29 + 11 * q^31 + 21 * q^32 - 15 * q^33 - 23 * q^34 + 19 * q^36 + 7 * q^37 - 32 * q^38 + 2 * q^39 + 29 * q^41 + 44 * q^42 + 3 * q^43 + 18 * q^44 + 7 * q^46 - 2 * q^47 - 4 * q^48 + 5 * q^49 + 5 * q^51 + 30 * q^52 - 6 * q^53 + 8 * q^54 + 13 * q^56 + q^57 - 28 * q^58 + 42 * q^59 - 11 * q^61 + 11 * q^62 - 2 * q^63 - 16 * q^64 + 29 * q^66 + 3 * q^67 - 4 * q^68 + 2 * q^69 + 10 * q^71 - 29 * q^72 - 14 * q^73 + q^74 + 11 * q^76 - 11 * q^77 + 3 * q^78 - 9 * q^79 - 13 * q^81 - 6 * q^82 - 6 * q^83 - 41 * q^84 - 22 * q^86 + 11 * q^87 - 32 * q^88 + 12 * q^89 - 2 * q^91 + 5 * q^92 + 28 * q^93 + 9 * q^94 - 5 * q^96 - q^97 + 6 * q^98 + 39 * q^99

Introduction

L-functions

Modular forms

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Database

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