LMFDB - Embedded newform 7500.2.a.m.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7500,2,Mod(1,7500)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7500, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("7500.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7500 = 2^{2} \cdot 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7500.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$59.8878015160$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 6 x^{11} - 11 x^{10} + 94 x^{9} + 27 x^{8} - 460 x^{7} + 55 x^{6} + 812 x^{5} - 127 x^{4} + \cdots - 4 $ x^12 - 6x^11 - 11x^10 + 94x^9 + 27x^8 - 460x^7 + 55x^6 + 812x^5 - 127x^4 - 512x^3 + 40x^2 + 72*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 5^{3} $
Twist minimal:	no (minimal twist has level 300)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$-2.34221$ of defining polynomial
Character	$\chi$	$=$	7500.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +3.54704 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +3.54704 q^{7} +1.00000 q^{9} -2.20616 q^{11} +7.17771 q^{13} -6.36532 q^{17} -2.31428 q^{19} -3.54704 q^{21} -2.17594 q^{23} -1.00000 q^{27} +0.847877 q^{29} -4.30090 q^{31} +2.20616 q^{33} +7.22998 q^{37} -7.17771 q^{39} -1.34457 q^{41} -8.18973 q^{43} -6.05453 q^{47} +5.58150 q^{49} +6.36532 q^{51} -11.9583 q^{53} +2.31428 q^{57} -12.4714 q^{59} -6.92041 q^{61} +3.54704 q^{63} -4.73409 q^{67} +2.17594 q^{69} +13.7791 q^{71} -1.08917 q^{73} -7.82536 q^{77} -5.84245 q^{79} +1.00000 q^{81} -12.5277 q^{83} -0.847877 q^{87} -7.02611 q^{89} +25.4596 q^{91} +4.30090 q^{93} +18.1445 q^{97} -2.20616 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{3} - 8 q^{7} + 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^3 - 8 * q^7 + 12 * q^9	$ 12 q - 12 q^{3} - 8 q^{7} + 12 q^{9} + 2 q^{11} - 8 q^{17} + 10 q^{19} + 8 q^{21} - 18 q^{23} - 12 q^{27} + 8 q^{29} - 2 q^{31} - 2 q^{33} - 4 q^{37} + 10 q^{41} - 28 q^{43} - 22 q^{47} + 28 q^{49} + 8 q^{51} - 16 q^{53} - 10 q^{57} - 2 q^{59} + 34 q^{61} - 8 q^{63} - 32 q^{67} + 18 q^{69} - 24 q^{73} - 18 q^{77} + 6 q^{79} + 12 q^{81} - 28 q^{83} - 8 q^{87} + 10 q^{89} + 20 q^{91} + 2 q^{93} - 16 q^{97} + 2 q^{99}+O(q^{100}) $ 12 * q - 12 * q^3 - 8 * q^7 + 12 * q^9 + 2 * q^11 - 8 * q^17 + 10 * q^19 + 8 * q^21 - 18 * q^23 - 12 * q^27 + 8 * q^29 - 2 * q^31 - 2 * q^33 - 4 * q^37 + 10 * q^41 - 28 * q^43 - 22 * q^47 + 28 * q^49 + 8 * q^51 - 16 * q^53 - 10 * q^57 - 2 * q^59 + 34 * q^61 - 8 * q^63 - 32 * q^67 + 18 * q^69 - 24 * q^73 - 18 * q^77 + 6 * q^79 + 12 * q^81 - 28 * q^83 - 8 * q^87 + 10 * q^89 + 20 * q^91 + 2 * q^93 - 16 * q^97 + 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.54704	1.34066	0.670328	−	0.742065i	$-0.266154\pi$
$7$	3.54704	1.34066	0.670328	+	0.742065i	$0.266154\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.20616	−0.665184	−0.332592	−	0.943071i	$-0.607923\pi$
$11$	−2.20616	−0.665184	−0.332592	+	0.943071i	$0.607923\pi$
$12$	0	0
$13$	7.17771	1.99074	0.995370	−	0.0961203i	$-0.0306434\pi$
$13$	7.17771	1.99074	0.995370	+	0.0961203i	$0.0306434\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.36532	−1.54382	−0.771909	−	0.635733i	$-0.780698\pi$
$17$	−6.36532	−1.54382	−0.771909	+	0.635733i	$0.780698\pi$
$18$	0	0
$19$	−2.31428	−0.530931	−0.265466	−	0.964120i	$-0.585526\pi$
$19$	−2.31428	−0.530931	−0.265466	+	0.964120i	$0.585526\pi$
$20$	0	0
$21$	−3.54704	−0.774028
$22$	0	0
$23$	−2.17594	−0.453716	−0.226858	−	0.973928i	$-0.572845\pi$
$23$	−2.17594	−0.453716	−0.226858	+	0.973928i	$0.572845\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	0.847877	0.157447	0.0787234	−	0.996896i	$-0.474916\pi$
$29$	0.847877	0.157447	0.0787234	+	0.996896i	$0.474916\pi$
$30$	0	0
$31$	−4.30090	−0.772465	−0.386232	−	0.922402i	$-0.626224\pi$
$31$	−4.30090	−0.772465	−0.386232	+	0.922402i	$0.626224\pi$
$32$	0	0
$33$	2.20616	0.384044
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.22998	1.18860	0.594301	−	0.804243i	$-0.297429\pi$
$37$	7.22998	1.18860	0.594301	+	0.804243i	$0.297429\pi$
$38$	0	0
$39$	−7.17771	−1.14935
$40$	0	0
$41$	−1.34457	−0.209986	−0.104993	−	0.994473i	$-0.533482\pi$
$41$	−1.34457	−0.209986	−0.104993	+	0.994473i	$0.533482\pi$
$42$	0	0
$43$	−8.18973	−1.24892	−0.624461	−	0.781056i	$-0.714681\pi$
$43$	−8.18973	−1.24892	−0.624461	+	0.781056i	$0.714681\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.05453	−0.883144	−0.441572	−	0.897226i	$-0.645579\pi$
$47$	−6.05453	−0.883144	−0.441572	+	0.897226i	$0.645579\pi$
$48$	0	0
$49$	5.58150	0.797357
$50$	0	0
$51$	6.36532	0.891323
$52$	0	0
$53$	−11.9583	−1.64259	−0.821296	−	0.570502i	$-0.806749\pi$
$53$	−11.9583	−1.64259	−0.821296	+	0.570502i	$0.806749\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.31428	0.306533
$58$	0	0
$59$	−12.4714	−1.62364	−0.811819	−	0.583910i	$-0.801522\pi$
$59$	−12.4714	−1.62364	−0.811819	+	0.583910i	$0.801522\pi$
$60$	0	0
$61$	−6.92041	−0.886068	−0.443034	−	0.896505i	$-0.646098\pi$
$61$	−6.92041	−0.886068	−0.443034	+	0.896505i	$0.646098\pi$
$62$	0	0
$63$	3.54704	0.446885
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−4.73409	−0.578361	−0.289181	−	0.957275i	$-0.593383\pi$
$67$	−4.73409	−0.578361	−0.289181	+	0.957275i	$0.593383\pi$
$68$	0	0
$69$	2.17594	0.261953
$70$	0	0
$71$	13.7791	1.63528	0.817642	−	0.575727i	$-0.195281\pi$
$71$	13.7791	1.63528	0.817642	+	0.575727i	$0.195281\pi$
$72$	0	0
$73$	−1.08917	−0.127477	−0.0637387	−	0.997967i	$-0.520302\pi$
$73$	−1.08917	−0.127477	−0.0637387	+	0.997967i	$0.520302\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−7.82536	−0.891782
$78$	0	0
$79$	−5.84245	−0.657327	−0.328664	−	0.944447i	$-0.606598\pi$
$79$	−5.84245	−0.657327	−0.328664	+	0.944447i	$0.606598\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−12.5277	−1.37509	−0.687546	−	0.726141i	$-0.741312\pi$
$83$	−12.5277	−1.37509	−0.687546	+	0.726141i	$0.741312\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−0.847877	−0.0909020
$88$	0	0
$89$	−7.02611	−0.744766	−0.372383	−	0.928079i	$-0.621459\pi$
$89$	−7.02611	−0.744766	−0.372383	+	0.928079i	$0.621459\pi$
$90$	0	0
$91$	25.4596	2.66890
$92$	0	0
$93$	4.30090	0.445983
$94$	0	0
$95$	0	0
$96$	0	0
$97$	18.1445	1.84229	0.921146	−	0.389218i	$-0.127255\pi$
$97$	18.1445	1.84229	0.921146	+	0.389218i	$0.127255\pi$
$98$	0	0
$99$	−2.20616	−0.221728
$100$	0	0
$101$	5.97473	0.594508	0.297254	−	0.954798i	$-0.403929\pi$
$101$	5.97473	0.594508	0.297254	+	0.954798i	$0.403929\pi$
$102$	0	0
$103$	0.459568	0.0452826	0.0226413	−	0.999744i	$-0.492792\pi$
$103$	0.459568	0.0452826	0.0226413	+	0.999744i	$0.492792\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.47862	−0.239617	−0.119809	−	0.992797i	$-0.538228\pi$
$107$	−2.47862	−0.239617	−0.119809	+	0.992797i	$0.538228\pi$
$108$	0	0
$109$	−3.34126	−0.320035	−0.160017	−	0.987114i	$-0.551155\pi$
$109$	−3.34126	−0.320035	−0.160017	+	0.987114i	$0.551155\pi$
$110$	0	0
$111$	−7.22998	−0.686240
$112$	0	0
$113$	−17.5979	−1.65547	−0.827734	−	0.561121i	$-0.810370\pi$
$113$	−17.5979	−1.65547	−0.827734	+	0.561121i	$0.810370\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	7.17771	0.663580
$118$	0	0
$119$	−22.5781	−2.06973
$120$	0	0
$121$	−6.13284	−0.557531
$122$	0	0
$123$	1.34457	0.121236
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−12.4119	−1.10138	−0.550691	−	0.834709i	$-0.685636\pi$
$127$	−12.4119	−1.10138	−0.550691	+	0.834709i	$0.685636\pi$
$128$	0	0
$129$	8.18973	0.721066
$130$	0	0
$131$	15.8232	1.38248	0.691239	−	0.722626i	$-0.257065\pi$
$131$	15.8232	1.38248	0.691239	+	0.722626i	$0.257065\pi$
$132$	0	0
$133$	−8.20883	−0.711796
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−7.39493	−0.631792	−0.315896	−	0.948794i	$-0.602305\pi$
$137$	−7.39493	−0.631792	−0.315896	+	0.948794i	$0.602305\pi$
$138$	0	0
$139$	3.93894	0.334096	0.167048	−	0.985949i	$-0.446576\pi$
$139$	3.93894	0.334096	0.167048	+	0.985949i	$0.446576\pi$
$140$	0	0
$141$	6.05453	0.509883
$142$	0	0
$143$	−15.8352	−1.32421
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−5.58150	−0.460354
$148$	0	0
$149$	3.92892	0.321870	0.160935	−	0.986965i	$-0.448549\pi$
$149$	3.92892	0.321870	0.160935	+	0.986965i	$0.448549\pi$
$150$	0	0
$151$	7.93418	0.645674	0.322837	−	0.946455i	$-0.395363\pi$
$151$	7.93418	0.645674	0.322837	+	0.946455i	$0.395363\pi$
$152$	0	0
$153$	−6.36532	−0.514606
$154$	0	0
$155$	0	0
$156$	0	0
$157$	6.09738	0.486624	0.243312	−	0.969948i	$-0.421766\pi$
$157$	6.09738	0.486624	0.243312	+	0.969948i	$0.421766\pi$
$158$	0	0
$159$	11.9583	0.948351
$160$	0	0
$161$	−7.71816	−0.608277
$162$	0	0
$163$	0.00255632	0.000200227 0	0.000100113	−	1.00000i	$-0.499968\pi$
$163$	0.00255632	0.000200227 0	0.000100113		1.00000i	$0.499968\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	10.1007	0.781613	0.390806	−	0.920473i	$-0.372196\pi$
$167$	10.1007	0.781613	0.390806	+	0.920473i	$0.372196\pi$
$168$	0	0
$169$	38.5196	2.96304
$170$	0	0
$171$	−2.31428	−0.176977
$172$	0	0
$173$	6.17988	0.469847	0.234924	−	0.972014i	$-0.424516\pi$
$173$	6.17988	0.469847	0.234924	+	0.972014i	$0.424516\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	12.4714	0.937407
$178$	0	0
$179$	14.4326	1.07874	0.539370	−	0.842069i	$-0.318662\pi$
$179$	14.4326	1.07874	0.539370	+	0.842069i	$0.318662\pi$
$180$	0	0
$181$	6.74285	0.501192	0.250596	−	0.968092i	$-0.419373\pi$
$181$	6.74285	0.501192	0.250596	+	0.968092i	$0.419373\pi$
$182$	0	0
$183$	6.92041	0.511572
$184$	0	0
$185$	0	0
$186$	0	0
$187$	14.0430	1.02692
$188$	0	0
$189$	−3.54704	−0.258009
$190$	0	0
$191$	15.3905	1.11362	0.556810	−	0.830640i	$-0.312025\pi$
$191$	15.3905	1.11362	0.556810	+	0.830640i	$0.312025\pi$
$192$	0	0
$193$	−16.3875	−1.17960	−0.589799	−	0.807550i	$-0.700793\pi$
$193$	−16.3875	−1.17960	−0.589799	+	0.807550i	$0.700793\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−9.85035	−0.701808	−0.350904	−	0.936411i	$-0.614126\pi$
$197$	−9.85035	−0.701808	−0.350904	+	0.936411i	$0.614126\pi$
$198$	0	0
$199$	−4.96275	−0.351800	−0.175900	−	0.984408i	$-0.556284\pi$
$199$	−4.96275	−0.351800	−0.175900	+	0.984408i	$0.556284\pi$
$200$	0	0
$201$	4.73409	0.333917
$202$	0	0
$203$	3.00746	0.211082
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−2.17594	−0.151239
$208$	0	0
$209$	5.10568	0.353167
$210$	0	0
$211$	3.49980	0.240936	0.120468	−	0.992717i	$-0.461560\pi$
$211$	3.49980	0.240936	0.120468	+	0.992717i	$0.461560\pi$
$212$	0	0
$213$	−13.7791	−0.944132
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−15.2555	−1.03561
$218$	0	0
$219$	1.08917	0.0735991
$220$	0	0
$221$	−45.6885	−3.07334
$222$	0	0
$223$	−23.0786	−1.54546	−0.772729	−	0.634736i	$-0.781109\pi$
$223$	−23.0786	−1.54546	−0.772729	+	0.634736i	$0.781109\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5.85059	0.388317	0.194159	−	0.980970i	$-0.437802\pi$
$227$	5.85059	0.388317	0.194159	+	0.980970i	$0.437802\pi$
$228$	0	0
$229$	−6.39071	−0.422310	−0.211155	−	0.977453i	$-0.567723\pi$
$229$	−6.39071	−0.422310	−0.211155	+	0.977453i	$0.567723\pi$
$230$	0	0
$231$	7.82536	0.514871
$232$	0	0
$233$	−4.54311	−0.297629	−0.148815	−	0.988865i	$-0.547546\pi$
$233$	−4.54311	−0.297629	−0.148815	+	0.988865i	$0.547546\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	5.84245	0.379508
$238$	0	0
$239$	23.0411	1.49041	0.745204	−	0.666837i	$-0.232352\pi$
$239$	23.0411	1.49041	0.745204	+	0.666837i	$0.232352\pi$
$240$	0	0
$241$	16.4476	1.05948	0.529742	−	0.848159i	$-0.322289\pi$
$241$	16.4476	1.05948	0.529742	+	0.848159i	$0.322289\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−16.6112	−1.05695
$248$	0	0
$249$	12.5277	0.793910
$250$	0	0
$251$	4.66327	0.294343	0.147171	−	0.989111i	$-0.452983\pi$
$251$	4.66327	0.294343	0.147171	+	0.989111i	$0.452983\pi$
$252$	0	0
$253$	4.80049	0.301804
$254$	0	0
$255$	0	0
$256$	0	0
$257$	5.78734	0.361004	0.180502	−	0.983575i	$-0.442228\pi$
$257$	5.78734	0.361004	0.180502	+	0.983575i	$0.442228\pi$
$258$	0	0
$259$	25.6450	1.59351
$260$	0	0
$261$	0.847877	0.0524823
$262$	0	0
$263$	−22.9498	−1.41515	−0.707574	−	0.706639i	$-0.750210\pi$
$263$	−22.9498	−1.41515	−0.707574	+	0.706639i	$0.750210\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	7.02611	0.429991
$268$	0	0
$269$	−4.50997	−0.274978	−0.137489	−	0.990503i	$-0.543903\pi$
$269$	−4.50997	−0.274978	−0.137489	+	0.990503i	$0.543903\pi$
$270$	0	0
$271$	−9.93132	−0.603285	−0.301642	−	0.953421i	$-0.597535\pi$
$271$	−9.93132	−0.603285	−0.301642	+	0.953421i	$0.597535\pi$
$272$	0	0
$273$	−25.4596	−1.54089
$274$	0	0
$275$	0	0
$276$	0	0
$277$	23.0153	1.38285	0.691427	−	0.722447i	$-0.256983\pi$
$277$	23.0153	1.38285	0.691427	+	0.722447i	$0.256983\pi$
$278$	0	0
$279$	−4.30090	−0.257488
$280$	0	0
$281$	0.734004	0.0437870	0.0218935	−	0.999760i	$-0.493031\pi$
$281$	0.734004	0.0437870	0.0218935	+	0.999760i	$0.493031\pi$
$282$	0	0
$283$	−21.4921	−1.27758	−0.638788	−	0.769383i	$-0.720564\pi$
$283$	−21.4921	−1.27758	−0.638788	+	0.769383i	$0.720564\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−4.76924	−0.281519
$288$	0	0
$289$	23.5173	1.38337
$290$	0	0
$291$	−18.1445	−1.06365
$292$	0	0
$293$	26.5961	1.55376	0.776880	−	0.629649i	$-0.216801\pi$
$293$	26.5961	1.55376	0.776880	+	0.629649i	$0.216801\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	2.20616	0.128015
$298$	0	0
$299$	−15.6183	−0.903230
$300$	0	0
$301$	−29.0493	−1.67437
$302$	0	0
$303$	−5.97473	−0.343239
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−12.1736	−0.694785	−0.347393	−	0.937720i	$-0.612933\pi$
$307$	−12.1736	−0.694785	−0.347393	+	0.937720i	$0.612933\pi$
$308$	0	0
$309$	−0.459568	−0.0261439
$310$	0	0
$311$	−24.7621	−1.40413	−0.702065	−	0.712113i	$-0.747739\pi$
$311$	−24.7621	−1.40413	−0.702065	+	0.712113i	$0.747739\pi$
$312$	0	0
$313$	−14.8454	−0.839110	−0.419555	−	0.907730i	$-0.637814\pi$
$313$	−14.8454	−0.839110	−0.419555	+	0.907730i	$0.637814\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1.94231	−0.109091	−0.0545455	−	0.998511i	$-0.517371\pi$
$317$	−1.94231	−0.109091	−0.0545455	+	0.998511i	$0.517371\pi$
$318$	0	0
$319$	−1.87056	−0.104731
$320$	0	0
$321$	2.47862	0.138343
$322$	0	0
$323$	14.7311	0.819661
$324$	0	0
$325$	0	0
$326$	0	0
$327$	3.34126	0.184772
$328$	0	0
$329$	−21.4757	−1.18399
$330$	0	0
$331$	−27.7307	−1.52421	−0.762107	−	0.647451i	$-0.775835\pi$
$331$	−27.7307	−1.52421	−0.762107	+	0.647451i	$0.775835\pi$
$332$	0	0
$333$	7.22998	0.396201
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−10.8904	−0.593235	−0.296618	−	0.954996i	$-0.595859\pi$
$337$	−10.8904	−0.593235	−0.296618	+	0.954996i	$0.595859\pi$
$338$	0	0
$339$	17.5979	0.955784
$340$	0	0
$341$	9.48850	0.513831
$342$	0	0
$343$	−5.03148	−0.271675
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−22.8539	−1.22686	−0.613430	−	0.789749i	$-0.710211\pi$
$347$	−22.8539	−1.22686	−0.613430	+	0.789749i	$0.710211\pi$
$348$	0	0
$349$	−28.8539	−1.54451	−0.772256	−	0.635311i	$-0.780872\pi$
$349$	−28.8539	−1.54451	−0.772256	+	0.635311i	$0.780872\pi$
$350$	0	0
$351$	−7.17771	−0.383118
$352$	0	0
$353$	27.2412	1.44990	0.724951	−	0.688800i	$-0.241862\pi$
$353$	27.2412	1.44990	0.724951	+	0.688800i	$0.241862\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	22.5781	1.19496
$358$	0	0
$359$	−23.9362	−1.26331	−0.631653	−	0.775251i	$-0.717623\pi$
$359$	−23.9362	−1.26331	−0.631653	+	0.775251i	$0.717623\pi$
$360$	0	0
$361$	−13.6441	−0.718112
$362$	0	0
$363$	6.13284	0.321890
$364$	0	0
$365$	0	0
$366$	0	0
$367$	6.71322	0.350427	0.175214	−	0.984530i	$-0.443938\pi$
$367$	6.71322	0.350427	0.175214	+	0.984530i	$0.443938\pi$
$368$	0	0
$369$	−1.34457	−0.0699954
$370$	0	0
$371$	−42.4164	−2.20215
$372$	0	0
$373$	−13.6091	−0.704652	−0.352326	−	0.935877i	$-0.614609\pi$
$373$	−13.6091	−0.704652	−0.352326	+	0.935877i	$0.614609\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6.08582	0.313436
$378$	0	0
$379$	0.444222	0.0228182	0.0114091	−	0.999935i	$-0.496368\pi$
$379$	0.444222	0.0228182	0.0114091	+	0.999935i	$0.496368\pi$
$380$	0	0
$381$	12.4119	0.635883
$382$	0	0
$383$	−10.2305	−0.522755	−0.261378	−	0.965237i	$-0.584177\pi$
$383$	−10.2305	−0.522755	−0.261378	+	0.965237i	$0.584177\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−8.18973	−0.416307
$388$	0	0
$389$	−24.2064	−1.22732	−0.613658	−	0.789572i	$-0.710303\pi$
$389$	−24.2064	−1.22732	−0.613658	+	0.789572i	$0.710303\pi$
$390$	0	0
$391$	13.8506	0.700454
$392$	0	0
$393$	−15.8232	−0.798174
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−2.72194	−0.136610	−0.0683050	−	0.997664i	$-0.521759\pi$
$397$	−2.72194	−0.136610	−0.0683050	+	0.997664i	$0.521759\pi$
$398$	0	0
$399$	8.20883	0.410956
$400$	0	0
$401$	31.3538	1.56573	0.782867	−	0.622189i	$-0.213756\pi$
$401$	31.3538	1.56573	0.782867	+	0.622189i	$0.213756\pi$
$402$	0	0
$403$	−30.8706	−1.53778
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−15.9505	−0.790639
$408$	0	0
$409$	3.32261	0.164293	0.0821463	−	0.996620i	$-0.473823\pi$
$409$	3.32261	0.164293	0.0821463	+	0.996620i	$0.473823\pi$
$410$	0	0
$411$	7.39493	0.364765
$412$	0	0
$413$	−44.2365	−2.17674
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−3.93894	−0.192891
$418$	0	0
$419$	−11.0462	−0.539643	−0.269822	−	0.962910i	$-0.586965\pi$
$419$	−11.0462	−0.539643	−0.269822	+	0.962910i	$0.586965\pi$
$420$	0	0
$421$	−24.4661	−1.19240	−0.596202	−	0.802834i	$-0.703324\pi$
$421$	−24.4661	−1.19240	−0.596202	+	0.802834i	$0.703324\pi$
$422$	0	0
$423$	−6.05453	−0.294381
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−24.5470	−1.18791
$428$	0	0
$429$	15.8352	0.764532
$430$	0	0
$431$	1.14758	0.0552769	0.0276385	−	0.999618i	$-0.491201\pi$
$431$	1.14758	0.0552769	0.0276385	+	0.999618i	$0.491201\pi$
$432$	0	0
$433$	−5.85457	−0.281352	−0.140676	−	0.990056i	$-0.544928\pi$
$433$	−5.85457	−0.281352	−0.140676	+	0.990056i	$0.544928\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.03574	0.240892
$438$	0	0
$439$	−20.3300	−0.970299	−0.485150	−	0.874431i	$-0.661235\pi$
$439$	−20.3300	−0.970299	−0.485150	+	0.874431i	$0.661235\pi$
$440$	0	0
$441$	5.58150	0.265786
$442$	0	0
$443$	17.5912	0.835783	0.417891	−	0.908497i	$-0.362769\pi$
$443$	17.5912	0.835783	0.417891	+	0.908497i	$0.362769\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−3.92892	−0.185831
$448$	0	0
$449$	−2.83956	−0.134007	−0.0670037	−	0.997753i	$-0.521344\pi$
$449$	−2.83956	−0.134007	−0.0670037	+	0.997753i	$0.521344\pi$
$450$	0	0
$451$	2.96634	0.139679
$452$	0	0
$453$	−7.93418	−0.372780
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−8.12004	−0.379840	−0.189920	−	0.981800i	$-0.560823\pi$
$457$	−8.12004	−0.379840	−0.189920	+	0.981800i	$0.560823\pi$
$458$	0	0
$459$	6.36532	0.297108
$460$	0	0
$461$	18.0007	0.838377	0.419188	−	0.907899i	$-0.362315\pi$
$461$	18.0007	0.838377	0.419188	+	0.907899i	$0.362315\pi$
$462$	0	0
$463$	28.6718	1.33249	0.666246	−	0.745732i	$-0.267900\pi$
$463$	28.6718	1.33249	0.666246	+	0.745732i	$0.267900\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	20.0449	0.927567	0.463784	−	0.885948i	$-0.346491\pi$
$467$	20.0449	0.927567	0.463784	+	0.885948i	$0.346491\pi$
$468$	0	0
$469$	−16.7920	−0.775383
$470$	0	0
$471$	−6.09738	−0.280953
$472$	0	0
$473$	18.0679	0.830763
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−11.9583	−0.547531
$478$	0	0
$479$	−15.4759	−0.707113	−0.353557	−	0.935413i	$-0.615028\pi$
$479$	−15.4759	−0.707113	−0.353557	+	0.935413i	$0.615028\pi$
$480$	0	0
$481$	51.8947	2.36620
$482$	0	0
$483$	7.71816	0.351189
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−37.4665	−1.69777	−0.848884	−	0.528580i	$-0.822725\pi$
$487$	−37.4665	−1.69777	−0.848884	+	0.528580i	$0.822725\pi$
$488$	0	0
$489$	−0.00255632	−0.000115601 0
$490$	0	0
$491$	39.9889	1.80467	0.902336	−	0.431033i	$-0.141851\pi$
$491$	39.9889	1.80467	0.902336	+	0.431033i	$0.141851\pi$
$492$	0	0
$493$	−5.39701	−0.243069
$494$	0	0
$495$	0	0
$496$	0	0
$497$	48.8752	2.19235
$498$	0	0
$499$	28.3040	1.26706	0.633530	−	0.773718i	$-0.281605\pi$
$499$	28.3040	1.26706	0.633530	+	0.773718i	$0.281605\pi$
$500$	0	0
$501$	−10.1007	−0.451264
$502$	0	0
$503$	−32.6836	−1.45729	−0.728644	−	0.684893i	$-0.759849\pi$
$503$	−32.6836	−1.45729	−0.728644	+	0.684893i	$0.759849\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−38.5196	−1.71071
$508$	0	0
$509$	17.1344	0.759467	0.379734	−	0.925096i	$-0.376016\pi$
$509$	17.1344	0.759467	0.379734	+	0.925096i	$0.376016\pi$
$510$	0	0
$511$	−3.86332	−0.170903
$512$	0	0
$513$	2.31428	0.102178
$514$	0	0
$515$	0	0
$516$	0	0
$517$	13.3573	0.587453
$518$	0	0
$519$	−6.17988	−0.271267
$520$	0	0
$521$	20.1367	0.882207	0.441103	−	0.897456i	$-0.354587\pi$
$521$	20.1367	0.882207	0.441103	+	0.897456i	$0.354587\pi$
$522$	0	0
$523$	−27.9075	−1.22031	−0.610156	−	0.792282i	$-0.708893\pi$
$523$	−27.9075	−1.22031	−0.610156	+	0.792282i	$0.708893\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	27.3766	1.19254
$528$	0	0
$529$	−18.2653	−0.794142
$530$	0	0
$531$	−12.4714	−0.541212
$532$	0	0
$533$	−9.65092	−0.418028
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−14.4326	−0.622811
$538$	0	0
$539$	−12.3137	−0.530389
$540$	0	0
$541$	−11.1257	−0.478332	−0.239166	−	0.970979i	$-0.576874\pi$
$541$	−11.1257	−0.478332	−0.239166	+	0.970979i	$0.576874\pi$
$542$	0	0
$543$	−6.74285	−0.289363
$544$	0	0
$545$	0	0
$546$	0	0
$547$	27.8305	1.18995	0.594973	−	0.803746i	$-0.297163\pi$
$547$	27.8305	1.18995	0.594973	+	0.803746i	$0.297163\pi$
$548$	0	0
$549$	−6.92041	−0.295356
$550$	0	0
$551$	−1.96222	−0.0835935
$552$	0	0
$553$	−20.7234	−0.881250
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−6.36580	−0.269728	−0.134864	−	0.990864i	$-0.543060\pi$
$557$	−6.36580	−0.269728	−0.134864	+	0.990864i	$0.543060\pi$
$558$	0	0
$559$	−58.7835	−2.48628
$560$	0	0
$561$	−14.0430	−0.592894
$562$	0	0
$563$	17.1281	0.721864	0.360932	−	0.932592i	$-0.382459\pi$
$563$	17.1281	0.721864	0.360932	+	0.932592i	$0.382459\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	3.54704	0.148962
$568$	0	0
$569$	15.3862	0.645025	0.322512	−	0.946565i	$-0.395473\pi$
$569$	15.3862	0.645025	0.322512	+	0.946565i	$0.395473\pi$
$570$	0	0
$571$	8.53546	0.357198	0.178599	−	0.983922i	$-0.442844\pi$
$571$	8.53546	0.357198	0.178599	+	0.983922i	$0.442844\pi$
$572$	0	0
$573$	−15.3905	−0.642949
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−2.30516	−0.0959650	−0.0479825	−	0.998848i	$-0.515279\pi$
$577$	−2.30516	−0.0959650	−0.0479825	+	0.998848i	$0.515279\pi$
$578$	0	0
$579$	16.3875	0.681042
$580$	0	0
$581$	−44.4362	−1.84352
$582$	0	0
$583$	26.3819	1.09263
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0.919218	0.0379402	0.0189701	−	0.999820i	$-0.493961\pi$
$587$	0.919218	0.0379402	0.0189701	+	0.999820i	$0.493961\pi$
$588$	0	0
$589$	9.95347	0.410126
$590$	0	0
$591$	9.85035	0.405189
$592$	0	0
$593$	−25.5925	−1.05096	−0.525478	−	0.850807i	$-0.676114\pi$
$593$	−25.5925	−1.05096	−0.525478	+	0.850807i	$0.676114\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	4.96275	0.203112
$598$	0	0
$599$	−37.0204	−1.51261	−0.756307	−	0.654217i	$-0.772998\pi$
$599$	−37.0204	−1.51261	−0.756307	+	0.654217i	$0.772998\pi$
$600$	0	0
$601$	33.4191	1.36319	0.681597	−	0.731728i	$-0.261286\pi$
$601$	33.4191	1.36319	0.681597	+	0.731728i	$0.261286\pi$
$602$	0	0
$603$	−4.73409	−0.192787
$604$	0	0
$605$	0	0
$606$	0	0
$607$	4.34502	0.176359	0.0881795	−	0.996105i	$-0.471895\pi$
$607$	4.34502	0.176359	0.0881795	+	0.996105i	$0.471895\pi$
$608$	0	0
$609$	−3.00746	−0.121868
$610$	0	0
$611$	−43.4577	−1.75811
$612$	0	0
$613$	3.95996	0.159941	0.0799706	−	0.996797i	$-0.474517\pi$
$613$	3.95996	0.159941	0.0799706	+	0.996797i	$0.474517\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	20.2639	0.815794	0.407897	−	0.913028i	$-0.366262\pi$
$617$	20.2639	0.815794	0.407897	+	0.913028i	$0.366262\pi$
$618$	0	0
$619$	−15.0778	−0.606028	−0.303014	−	0.952986i	$-0.597993\pi$
$619$	−15.0778	−0.606028	−0.303014	+	0.952986i	$0.597993\pi$
$620$	0	0
$621$	2.17594	0.0873176
$622$	0	0
$623$	−24.9219	−0.998474
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−5.10568	−0.203901
$628$	0	0
$629$	−46.0212	−1.83498
$630$	0	0
$631$	−2.44569	−0.0973615	−0.0486808	−	0.998814i	$-0.515502\pi$
$631$	−2.44569	−0.0973615	−0.0486808	+	0.998814i	$0.515502\pi$
$632$	0	0
$633$	−3.49980	−0.139105
$634$	0	0
$635$	0	0
$636$	0	0
$637$	40.0624	1.58733
$638$	0	0
$639$	13.7791	0.545095
$640$	0	0
$641$	13.3931	0.528997	0.264498	−	0.964386i	$-0.414794\pi$
$641$	13.3931	0.528997	0.264498	+	0.964386i	$0.414794\pi$
$642$	0	0
$643$	35.4836	1.39934	0.699669	−	0.714467i	$-0.253331\pi$
$643$	35.4836	1.39934	0.699669	+	0.714467i	$0.253331\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	22.3841	0.880010	0.440005	−	0.897995i	$-0.354977\pi$
$647$	22.3841	0.880010	0.440005	+	0.897995i	$0.354977\pi$
$648$	0	0
$649$	27.5140	1.08002
$650$	0	0
$651$	15.2555	0.597909
$652$	0	0
$653$	18.9285	0.740729	0.370365	−	0.928886i	$-0.379233\pi$
$653$	18.9285	0.740729	0.370365	+	0.928886i	$0.379233\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−1.08917	−0.0424925
$658$	0	0
$659$	5.66578	0.220707	0.110354	−	0.993892i	$-0.464802\pi$
$659$	5.66578	0.220707	0.110354	+	0.993892i	$0.464802\pi$
$660$	0	0
$661$	−25.2436	−0.981861	−0.490930	−	0.871199i	$-0.663343\pi$
$661$	−25.2436	−0.981861	−0.490930	+	0.871199i	$0.663343\pi$
$662$	0	0
$663$	45.6885	1.77439
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.84493	−0.0714361
$668$	0	0
$669$	23.0786	0.892270
$670$	0	0
$671$	15.2676	0.589398
$672$	0	0
$673$	−13.1688	−0.507621	−0.253810	−	0.967254i	$-0.581684\pi$
$673$	−13.1688	−0.507621	−0.253810	+	0.967254i	$0.581684\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−26.9634	−1.03629	−0.518145	−	0.855293i	$-0.673377\pi$
$677$	−26.9634	−1.03629	−0.518145	+	0.855293i	$0.673377\pi$
$678$	0	0
$679$	64.3592	2.46988
$680$	0	0
$681$	−5.85059	−0.224195
$682$	0	0
$683$	−51.7797	−1.98129	−0.990647	−	0.136449i	$-0.956431\pi$
$683$	−51.7797	−1.98129	−0.990647	+	0.136449i	$0.956431\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	6.39071	0.243821
$688$	0	0
$689$	−85.8329	−3.26997
$690$	0	0
$691$	−3.71620	−0.141371	−0.0706854	−	0.997499i	$-0.522519\pi$
$691$	−3.71620	−0.141371	−0.0706854	+	0.997499i	$0.522519\pi$
$692$	0	0
$693$	−7.82536	−0.297261
$694$	0	0
$695$	0	0
$696$	0	0
$697$	8.55861	0.324180
$698$	0	0
$699$	4.54311	0.171836
$700$	0	0
$701$	−28.4299	−1.07378	−0.536890	−	0.843652i	$-0.680401\pi$
$701$	−28.4299	−1.07378	−0.536890	+	0.843652i	$0.680401\pi$
$702$	0	0
$703$	−16.7322	−0.631066
$704$	0	0
$705$	0	0
$706$	0	0
$707$	21.1926	0.797030
$708$	0	0
$709$	40.9295	1.53714	0.768569	−	0.639767i	$-0.220969\pi$
$709$	40.9295	1.53714	0.768569	+	0.639767i	$0.220969\pi$
$710$	0	0
$711$	−5.84245	−0.219109
$712$	0	0
$713$	9.35852	0.350479
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−23.0411	−0.860487
$718$	0	0
$719$	4.73657	0.176644	0.0883222	−	0.996092i	$-0.471849\pi$
$719$	4.73657	0.176644	0.0883222	+	0.996092i	$0.471849\pi$
$720$	0	0
$721$	1.63011	0.0607084
$722$	0	0
$723$	−16.4476	−0.611693
$724$	0	0
$725$	0	0
$726$	0	0
$727$	5.57086	0.206612	0.103306	−	0.994650i	$-0.467058\pi$
$727$	5.57086	0.206612	0.103306	+	0.994650i	$0.467058\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	52.1303	1.92811
$732$	0	0
$733$	30.3597	1.12136	0.560681	−	0.828032i	$-0.310540\pi$
$733$	30.3597	1.12136	0.560681	+	0.828032i	$0.310540\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	10.4442	0.384716
$738$	0	0
$739$	48.9640	1.80117	0.900584	−	0.434681i	$-0.143139\pi$
$739$	48.9640	1.80117	0.900584	+	0.434681i	$0.143139\pi$
$740$	0	0
$741$	16.6112	0.610228
$742$	0	0
$743$	−45.5953	−1.67273	−0.836364	−	0.548174i	$-0.815323\pi$
$743$	−45.5953	−1.67273	−0.836364	+	0.548174i	$0.815323\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−12.5277	−0.458364
$748$	0	0
$749$	−8.79176	−0.321244
$750$	0	0
$751$	−27.9100	−1.01845	−0.509225	−	0.860633i	$-0.670068\pi$
$751$	−27.9100	−1.01845	−0.509225	+	0.860633i	$0.670068\pi$
$752$	0	0
$753$	−4.66327	−0.169939
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−27.6680	−1.00561	−0.502804	−	0.864400i	$-0.667698\pi$
$757$	−27.6680	−1.00561	−0.502804	+	0.864400i	$0.667698\pi$
$758$	0	0
$759$	−4.80049	−0.174247
$760$	0	0
$761$	−12.1179	−0.439275	−0.219638	−	0.975582i	$-0.570487\pi$
$761$	−12.1179	−0.439275	−0.219638	+	0.975582i	$0.570487\pi$
$762$	0	0
$763$	−11.8516	−0.429056
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−89.5161	−3.23224
$768$	0	0
$769$	−10.1418	−0.365722	−0.182861	−	0.983139i	$-0.558536\pi$
$769$	−10.1418	−0.365722	−0.182861	+	0.983139i	$0.558536\pi$
$770$	0	0
$771$	−5.78734	−0.208426
$772$	0	0
$773$	50.1727	1.80459	0.902293	−	0.431122i	$-0.141882\pi$
$773$	50.1727	1.80459	0.902293	+	0.431122i	$0.141882\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−25.6450	−0.920011
$778$	0	0
$779$	3.11170	0.111488
$780$	0	0
$781$	−30.3991	−1.08776
$782$	0	0
$783$	−0.847877	−0.0303007
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−15.4581	−0.551023	−0.275512	−	0.961298i	$-0.588847\pi$
$787$	−15.4581	−0.551023	−0.275512	+	0.961298i	$0.588847\pi$
$788$	0	0
$789$	22.9498	0.817036
$790$	0	0
$791$	−62.4203	−2.21941
$792$	0	0
$793$	−49.6727	−1.76393
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−5.73139	−0.203016	−0.101508	−	0.994835i	$-0.532367\pi$
$797$	−5.73139	−0.203016	−0.101508	+	0.994835i	$0.532367\pi$
$798$	0	0
$799$	38.5390	1.36341
$800$	0	0
$801$	−7.02611	−0.248255
$802$	0	0
$803$	2.40288	0.0847959
$804$	0	0
$805$	0	0
$806$	0	0
$807$	4.50997	0.158758
$808$	0	0
$809$	−28.4181	−0.999128	−0.499564	−	0.866277i	$-0.666506\pi$
$809$	−28.4181	−0.999128	−0.499564	+	0.866277i	$0.666506\pi$
$810$	0	0
$811$	−12.1882	−0.427986	−0.213993	−	0.976835i	$-0.568647\pi$
$811$	−12.1882	−0.427986	−0.213993	+	0.976835i	$0.568647\pi$
$812$	0	0
$813$	9.93132	0.348307
$814$	0	0
$815$	0	0
$816$	0	0
$817$	18.9533	0.663092
$818$	0	0
$819$	25.4596	0.889632
$820$	0	0
$821$	14.6223	0.510322	0.255161	−	0.966899i	$-0.417872\pi$
$821$	14.6223	0.510322	0.255161	+	0.966899i	$0.417872\pi$
$822$	0	0
$823$	21.6407	0.754347	0.377174	−	0.926143i	$-0.376896\pi$
$823$	21.6407	0.754347	0.377174	+	0.926143i	$0.376896\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	11.1622	0.388146	0.194073	−	0.980987i	$-0.437830\pi$
$827$	11.1622	0.388146	0.194073	+	0.980987i	$0.437830\pi$
$828$	0	0
$829$	27.8484	0.967215	0.483607	−	0.875285i	$-0.339326\pi$
$829$	27.8484	0.967215	0.483607	+	0.875285i	$0.339326\pi$
$830$	0	0
$831$	−23.0153	−0.798391
$832$	0	0
$833$	−35.5280	−1.23097
$834$	0	0
$835$	0	0
$836$	0	0
$837$	4.30090	0.148661
$838$	0	0
$839$	−18.9805	−0.655281	−0.327640	−	0.944803i	$-0.606253\pi$
$839$	−18.9805	−0.655281	−0.327640	+	0.944803i	$0.606253\pi$
$840$	0	0
$841$	−28.2811	−0.975210
$842$	0	0
$843$	−0.734004	−0.0252804
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−21.7534	−0.747456
$848$	0	0
$849$	21.4921	0.737609
$850$	0	0
$851$	−15.7320	−0.539287
$852$	0	0
$853$	−16.5066	−0.565175	−0.282587	−	0.959242i	$-0.591193\pi$
$853$	−16.5066	−0.565175	−0.282587	+	0.959242i	$0.591193\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	28.7615	0.982475	0.491237	−	0.871026i	$-0.336545\pi$
$857$	28.7615	0.982475	0.491237	+	0.871026i	$0.336545\pi$
$858$	0	0
$859$	−15.1352	−0.516405	−0.258202	−	0.966091i	$-0.583130\pi$
$859$	−15.1352	−0.516405	−0.258202	+	0.966091i	$0.583130\pi$
$860$	0	0
$861$	4.76924	0.162535
$862$	0	0
$863$	−43.0471	−1.46534	−0.732670	−	0.680584i	$-0.761726\pi$
$863$	−43.0471	−1.46534	−0.732670	+	0.680584i	$0.761726\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−23.5173	−0.798690
$868$	0	0
$869$	12.8894	0.437244
$870$	0	0
$871$	−33.9799	−1.15137
$872$	0	0
$873$	18.1445	0.614097
$874$	0	0
$875$	0	0
$876$	0	0
$877$	50.6953	1.71186	0.855930	−	0.517092i	$-0.172985\pi$
$877$	50.6953	1.71186	0.855930	+	0.517092i	$0.172985\pi$
$878$	0	0
$879$	−26.5961	−0.897064
$880$	0	0
$881$	−2.65023	−0.0892885	−0.0446442	−	0.999003i	$-0.514215\pi$
$881$	−2.65023	−0.0892885	−0.0446442	+	0.999003i	$0.514215\pi$
$882$	0	0
$883$	33.7718	1.13651	0.568256	−	0.822852i	$-0.307618\pi$
$883$	33.7718	1.13651	0.568256	+	0.822852i	$0.307618\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	4.32390	0.145183	0.0725913	−	0.997362i	$-0.476873\pi$
$887$	4.32390	0.145183	0.0725913	+	0.997362i	$0.476873\pi$
$888$	0	0
$889$	−44.0256	−1.47657
$890$	0	0
$891$	−2.20616	−0.0739093
$892$	0	0
$893$	14.0119	0.468889
$894$	0	0
$895$	0	0
$896$	0	0
$897$	15.6183	0.521480
$898$	0	0
$899$	−3.64664	−0.121622
$900$	0	0
$901$	76.1182	2.53586
$902$	0	0
$903$	29.0493	0.966701
$904$	0	0
$905$	0	0
$906$	0	0
$907$	47.2507	1.56893	0.784467	−	0.620171i	$-0.212937\pi$
$907$	47.2507	1.56893	0.784467	+	0.620171i	$0.212937\pi$
$908$	0	0
$909$	5.97473	0.198169
$910$	0	0
$911$	50.8749	1.68556	0.842780	−	0.538259i	$-0.180918\pi$
$911$	50.8749	1.68556	0.842780	+	0.538259i	$0.180918\pi$
$912$	0	0
$913$	27.6381	0.914689
$914$	0	0
$915$	0	0
$916$	0	0
$917$	56.1255	1.85343
$918$	0	0
$919$	−36.6958	−1.21048	−0.605242	−	0.796042i	$-0.706924\pi$
$919$	−36.6958	−1.21048	−0.605242	+	0.796042i	$0.706924\pi$
$920$	0	0
$921$	12.1736	0.401135
$922$	0	0
$923$	98.9028	3.25542
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0.459568	0.0150942
$928$	0	0
$929$	48.3043	1.58481	0.792407	−	0.609993i	$-0.208828\pi$
$929$	48.3043	1.58481	0.792407	+	0.609993i	$0.208828\pi$
$930$	0	0
$931$	−12.9171	−0.423342
$932$	0	0
$933$	24.7621	0.810675
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−9.48340	−0.309809	−0.154905	−	0.987929i	$-0.549507\pi$
$937$	−9.48340	−0.309809	−0.154905	+	0.987929i	$0.549507\pi$
$938$	0	0
$939$	14.8454	0.484460
$940$	0	0
$941$	−21.2940	−0.694164	−0.347082	−	0.937835i	$-0.612827\pi$
$941$	−21.2940	−0.694164	−0.347082	+	0.937835i	$0.612827\pi$
$942$	0	0
$943$	2.92570	0.0952740
$944$	0	0
$945$	0	0
$946$	0	0
$947$	31.5118	1.02400	0.511998	−	0.858987i	$-0.328906\pi$
$947$	31.5118	1.02400	0.511998	+	0.858987i	$0.328906\pi$
$948$	0	0
$949$	−7.81773	−0.253774
$950$	0	0
$951$	1.94231	0.0629837
$952$	0	0
$953$	11.2134	0.363238	0.181619	−	0.983369i	$-0.441866\pi$
$953$	11.2134	0.363238	0.181619	+	0.983369i	$0.441866\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	1.87056	0.0604665
$958$	0	0
$959$	−26.2301	−0.847015
$960$	0	0
$961$	−12.5023	−0.403298
$962$	0	0
$963$	−2.47862	−0.0798723
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−35.6217	−1.14552	−0.572758	−	0.819724i	$-0.694127\pi$
$967$	−35.6217	−1.14552	−0.572758	+	0.819724i	$0.694127\pi$
$968$	0	0
$969$	−14.7311	−0.473232
$970$	0	0
$971$	42.7896	1.37318	0.686592	−	0.727043i	$-0.259106\pi$
$971$	42.7896	1.37318	0.686592	+	0.727043i	$0.259106\pi$
$972$	0	0
$973$	13.9716	0.447908
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−40.6813	−1.30151	−0.650755	−	0.759288i	$-0.725548\pi$
$977$	−40.6813	−1.30151	−0.650755	+	0.759288i	$0.725548\pi$
$978$	0	0
$979$	15.5007	0.495406
$980$	0	0
$981$	−3.34126	−0.106678
$982$	0	0
$983$	−48.9680	−1.56184	−0.780918	−	0.624634i	$-0.785248\pi$
$983$	−48.9680	−1.56184	−0.780918	+	0.624634i	$0.785248\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	21.4757	0.683578
$988$	0	0
$989$	17.8204	0.566656
$990$	0	0
$991$	−2.46390	−0.0782685	−0.0391343	−	0.999234i	$-0.512460\pi$
$991$	−2.46390	−0.0782685	−0.0391343	+	0.999234i	$0.512460\pi$
$992$	0	0
$993$	27.7307	0.880006
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−54.2528	−1.71820	−0.859102	−	0.511804i	$-0.828977\pi$
$997$	−54.2528	−1.71820	−0.859102	+	0.511804i	$0.828977\pi$
$998$	0	0
$999$	−7.22998	−0.228747

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7500.2.a.m.1.11		12
5.2	odd	4		7500.2.d.g.1249.23		24
5.3	odd	4		7500.2.d.g.1249.2		24
5.4	even	2		7500.2.a.n.1.2		12
25.3	odd	20		300.2.o.a.109.4	✓	24
25.4	even	10		1500.2.m.c.1201.1		24
25.6	even	5		1500.2.m.d.301.6		24
25.8	odd	20		1500.2.o.c.949.3		24
25.17	odd	20		300.2.o.a.289.4	yes	24
25.19	even	10		1500.2.m.c.301.1		24
25.21	even	5		1500.2.m.d.1201.6		24
25.22	odd	20		1500.2.o.c.49.3		24
75.17	even	20		900.2.w.c.289.5		24
75.53	even	20		900.2.w.c.109.5		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.2.o.a.109.4	✓	24	25.3	odd	20
300.2.o.a.289.4	yes	24	25.17	odd	20
900.2.w.c.109.5		24	75.53	even	20
900.2.w.c.289.5		24	75.17	even	20
1500.2.m.c.301.1		24	25.19	even	10
1500.2.m.c.1201.1		24	25.4	even	10
1500.2.m.d.301.6		24	25.6	even	5
1500.2.m.d.1201.6		24	25.21	even	5
1500.2.o.c.49.3		24	25.22	odd	20
1500.2.o.c.949.3		24	25.8	odd	20
7500.2.a.m.1.11		12	1.1	even	1	trivial
7500.2.a.n.1.2		12	5.4	even	2
7500.2.d.g.1249.2		24	5.3	odd	4
7500.2.d.g.1249.23		24	5.2	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7500 = 2^{2} \cdot 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7500.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.54704 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +3.54704 q^{7} +1.00000 q^{9} -2.20616 q^{11} +7.17771 q^{13} -6.36532 q^{17} -2.31428 q^{19} -3.54704 q^{21} -2.17594 q^{23} -1.00000 q^{27} +0.847877 q^{29} -4.30090 q^{31} +2.20616 q^{33} +7.22998 q^{37} -7.17771 q^{39} -1.34457 q^{41} -8.18973 q^{43} -6.05453 q^{47} +5.58150 q^{49} +6.36532 q^{51} -11.9583 q^{53} +2.31428 q^{57} -12.4714 q^{59} -6.92041 q^{61} +3.54704 q^{63} -4.73409 q^{67} +2.17594 q^{69} +13.7791 q^{71} -1.08917 q^{73} -7.82536 q^{77} -5.84245 q^{79} +1.00000 q^{81} -12.5277 q^{83} -0.847877 q^{87} -7.02611 q^{89} +25.4596 q^{91} +4.30090 q^{93} +18.1445 q^{97} -2.20616 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{3} - 8 q^{7} + 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^3 - 8 * q^7 + 12 * q^9	\( 12 q - 12 q^{3} - 8 q^{7} + 12 q^{9} + 2 q^{11} - 8 q^{17} + 10 q^{19} + 8 q^{21} - 18 q^{23} - 12 q^{27} + 8 q^{29} - 2 q^{31} - 2 q^{33} - 4 q^{37} + 10 q^{41} - 28 q^{43} - 22 q^{47} + 28 q^{49} + 8 q^{51} - 16 q^{53} - 10 q^{57} - 2 q^{59} + 34 q^{61} - 8 q^{63} - 32 q^{67} + 18 q^{69} - 24 q^{73} - 18 q^{77} + 6 q^{79} + 12 q^{81} - 28 q^{83} - 8 q^{87} + 10 q^{89} + 20 q^{91} + 2 q^{93} - 16 q^{97} + 2 q^{99}+O(q^{100}) \) 12 * q - 12 * q^3 - 8 * q^7 + 12 * q^9 + 2 * q^11 - 8 * q^17 + 10 * q^19 + 8 * q^21 - 18 * q^23 - 12 * q^27 + 8 * q^29 - 2 * q^31 - 2 * q^33 - 4 * q^37 + 10 * q^41 - 28 * q^43 - 22 * q^47 + 28 * q^49 + 8 * q^51 - 16 * q^53 - 10 * q^57 - 2 * q^59 + 34 * q^61 - 8 * q^63 - 32 * q^67 + 18 * q^69 - 24 * q^73 - 18 * q^77 + 6 * q^79 + 12 * q^81 - 28 * q^83 - 8 * q^87 + 10 * q^89 + 20 * q^91 + 2 * q^93 - 16 * q^97 + 2 * q^99

Introduction

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