LMFDB - Embedded newform 7500.2.a.n.1.1

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:	$0$
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			1.1
Root			$4.54905$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -4.13266 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -4.13266 q^{7} +1.00000 q^{9} -3.76632 q^{11} -0.698665 q^{13} -5.29251 q^{17} +5.73494 q^{19} -4.13266 q^{21} +5.46649 q^{23} +1.00000 q^{27} -7.02438 q^{29} -10.0920 q^{31} -3.76632 q^{33} +5.78488 q^{37} -0.698665 q^{39} -6.59552 q^{41} +4.79668 q^{43} +9.67379 q^{47} +10.0789 q^{49} -5.29251 q^{51} -3.39638 q^{53} +5.73494 q^{57} +0.745297 q^{59} +11.8079 q^{61} -4.13266 q^{63} -4.79759 q^{67} +5.46649 q^{69} -3.14256 q^{71} +10.7473 q^{73} +15.5649 q^{77} -9.12492 q^{79} +1.00000 q^{81} +8.44098 q^{83} -7.02438 q^{87} -3.38783 q^{89} +2.88735 q^{91} -10.0920 q^{93} +10.4351 q^{97} -3.76632 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	−4.13266	−1.56200	−0.780999	−	0.624532i	$-0.785290\pi$
$7$	−4.13266	−1.56200	−0.780999	+	0.624532i	$0.785290\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.76632	−1.13559	−0.567794	−	0.823171i	$-0.692203\pi$
$11$	−3.76632	−1.13559	−0.567794	+	0.823171i	$0.692203\pi$
$12$	0	0
$13$	−0.698665	−0.193775	−0.0968875	−	0.995295i	$-0.530889\pi$
$13$	−0.698665	−0.193775	−0.0968875	+	0.995295i	$0.530889\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.29251	−1.28362	−0.641811	−	0.766863i	$-0.721816\pi$
$17$	−5.29251	−1.28362	−0.641811	+	0.766863i	$0.721816\pi$
$18$	0	0
$19$	5.73494	1.31568	0.657842	−	0.753156i	$-0.271469\pi$
$19$	5.73494	1.31568	0.657842	+	0.753156i	$0.271469\pi$
$20$	0	0
$21$	−4.13266	−0.901820
$22$	0	0
$23$	5.46649	1.13984	0.569921	−	0.821699i	$-0.306974\pi$
$23$	5.46649	1.13984	0.569921	+	0.821699i	$0.306974\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−7.02438	−1.30439	−0.652197	−	0.758049i	$-0.726153\pi$
$29$	−7.02438	−1.30439	−0.652197	+	0.758049i	$0.726153\pi$
$30$	0	0
$31$	−10.0920	−1.81258	−0.906288	−	0.422660i	$-0.861096\pi$
$31$	−10.0920	−1.81258	−0.906288	+	0.422660i	$0.861096\pi$
$32$	0	0
$33$	−3.76632	−0.655632
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.78488	0.951029	0.475515	−	0.879708i	$-0.342262\pi$
$37$	5.78488	0.951029	0.475515	+	0.879708i	$0.342262\pi$
$38$	0	0
$39$	−0.698665	−0.111876
$40$	0	0
$41$	−6.59552	−1.03005	−0.515024	−	0.857176i	$-0.672217\pi$
$41$	−6.59552	−1.03005	−0.515024	+	0.857176i	$0.672217\pi$
$42$	0	0
$43$	4.79668	0.731488	0.365744	−	0.930716i	$-0.380815\pi$
$43$	4.79668	0.731488	0.365744	+	0.930716i	$0.380815\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.67379	1.41107	0.705534	−	0.708676i	$-0.250707\pi$
$47$	9.67379	1.41107	0.705534	+	0.708676i	$0.250707\pi$
$48$	0	0
$49$	10.0789	1.43984
$50$	0	0
$51$	−5.29251	−0.741099
$52$	0	0
$53$	−3.39638	−0.466528	−0.233264	−	0.972413i	$-0.574941\pi$
$53$	−3.39638	−0.466528	−0.233264	+	0.972413i	$0.574941\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	5.73494	0.759611
$58$	0	0
$59$	0.745297	0.0970294	0.0485147	−	0.998822i	$-0.484551\pi$
$59$	0.745297	0.0970294	0.0485147	+	0.998822i	$0.484551\pi$
$60$	0	0
$61$	11.8079	1.51185	0.755926	−	0.654657i	$-0.227187\pi$
$61$	11.8079	1.51185	0.755926	+	0.654657i	$0.227187\pi$
$62$	0	0
$63$	−4.13266	−0.520666
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−4.79759	−0.586119	−0.293059	−	0.956094i	$-0.594673\pi$
$67$	−4.79759	−0.586119	−0.293059	+	0.956094i	$0.594673\pi$
$68$	0	0
$69$	5.46649	0.658088
$70$	0	0
$71$	−3.14256	−0.372953	−0.186477	−	0.982459i	$-0.559707\pi$
$71$	−3.14256	−0.372953	−0.186477	+	0.982459i	$0.559707\pi$
$72$	0	0
$73$	10.7473	1.25788	0.628939	−	0.777455i	$-0.283490\pi$
$73$	10.7473	1.25788	0.628939	+	0.777455i	$0.283490\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	15.5649	1.77379
$78$	0	0
$79$	−9.12492	−1.02663	−0.513317	−	0.858199i	$-0.671583\pi$
$79$	−9.12492	−1.02663	−0.513317	+	0.858199i	$0.671583\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	8.44098	0.926518	0.463259	−	0.886223i	$-0.346680\pi$
$83$	8.44098	0.926518	0.463259	+	0.886223i	$0.346680\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−7.02438	−0.753093
$88$	0	0
$89$	−3.38783	−0.359109	−0.179555	−	0.983748i	$-0.557466\pi$
$89$	−3.38783	−0.359109	−0.179555	+	0.983748i	$0.557466\pi$
$90$	0	0
$91$	2.88735	0.302676
$92$	0	0
$93$	−10.0920	−1.04649
$94$	0	0
$95$	0	0
$96$	0	0
$97$	10.4351	1.05952	0.529760	−	0.848147i	$-0.322282\pi$
$97$	10.4351	1.05952	0.529760	+	0.848147i	$0.322282\pi$
$98$	0	0
$99$	−3.76632	−0.378529
$100$	0	0
$101$	7.10799	0.707272	0.353636	−	0.935383i	$-0.384945\pi$
$101$	7.10799	0.707272	0.353636	+	0.935383i	$0.384945\pi$
$102$	0	0
$103$	−12.2091	−1.20299	−0.601497	−	0.798875i	$-0.705429\pi$
$103$	−12.2091	−1.20299	−0.601497	+	0.798875i	$0.705429\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.04303	0.390855	0.195427	−	0.980718i	$-0.437391\pi$
$107$	4.04303	0.390855	0.195427	+	0.980718i	$0.437391\pi$
$108$	0	0
$109$	−0.773815	−0.0741180	−0.0370590	−	0.999313i	$-0.511799\pi$
$109$	−0.773815	−0.0741180	−0.0370590	+	0.999313i	$0.511799\pi$
$110$	0	0
$111$	5.78488	0.549077
$112$	0	0
$113$	1.13943	0.107189	0.0535944	−	0.998563i	$-0.482932\pi$
$113$	1.13943	0.107189	0.0535944	+	0.998563i	$0.482932\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−0.698665	−0.0645916
$118$	0	0
$119$	21.8721	2.00501
$120$	0	0
$121$	3.18514	0.289559
$122$	0	0
$123$	−6.59552	−0.594698
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−0.537868	−0.0477281	−0.0238640	−	0.999715i	$-0.507597\pi$
$127$	−0.537868	−0.0477281	−0.0238640	+	0.999715i	$0.507597\pi$
$128$	0	0
$129$	4.79668	0.422325
$130$	0	0
$131$	−6.72242	−0.587341	−0.293670	−	0.955907i	$-0.594877\pi$
$131$	−6.72242	−0.587341	−0.293670	+	0.955907i	$0.594877\pi$
$132$	0	0
$133$	−23.7005	−2.05510
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−0.183379	−0.0156671	−0.00783355	−	0.999969i	$-0.502494\pi$
$137$	−0.183379	−0.0156671	−0.00783355	+	0.999969i	$0.502494\pi$
$138$	0	0
$139$	8.22006	0.697217	0.348608	−	0.937268i	$-0.386654\pi$
$139$	8.22006	0.697217	0.348608	+	0.937268i	$0.386654\pi$
$140$	0	0
$141$	9.67379	0.814680
$142$	0	0
$143$	2.63140	0.220048
$144$	0	0
$145$	0	0
$146$	0	0
$147$	10.0789	0.831292
$148$	0	0
$149$	5.53790	0.453682	0.226841	−	0.973932i	$-0.427160\pi$
$149$	5.53790	0.453682	0.226841	+	0.973932i	$0.427160\pi$
$150$	0	0
$151$	12.8736	1.04764	0.523821	−	0.851828i	$-0.324506\pi$
$151$	12.8736	1.04764	0.523821	+	0.851828i	$0.324506\pi$
$152$	0	0
$153$	−5.29251	−0.427874
$154$	0	0
$155$	0	0
$156$	0	0
$157$	14.6395	1.16836	0.584179	−	0.811625i	$-0.301417\pi$
$157$	14.6395	1.16836	0.584179	+	0.811625i	$0.301417\pi$
$158$	0	0
$159$	−3.39638	−0.269350
$160$	0	0
$161$	−22.5912	−1.78043
$162$	0	0
$163$	4.44424	0.348099	0.174050	−	0.984737i	$-0.444315\pi$
$163$	4.44424	0.348099	0.174050	+	0.984737i	$0.444315\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	23.7655	1.83903	0.919514	−	0.393057i	$-0.128583\pi$
$167$	23.7655	1.83903	0.919514	+	0.393057i	$0.128583\pi$
$168$	0	0
$169$	−12.5119	−0.962451
$170$	0	0
$171$	5.73494	0.438562
$172$	0	0
$173$	15.0868	1.14703	0.573513	−	0.819196i	$-0.305580\pi$
$173$	15.0868	1.14703	0.573513	+	0.819196i	$0.305580\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0.745297	0.0560200
$178$	0	0
$179$	16.7163	1.24944	0.624718	−	0.780850i	$-0.285214\pi$
$179$	16.7163	1.24944	0.624718	+	0.780850i	$0.285214\pi$
$180$	0	0
$181$	10.3923	0.772454	0.386227	−	0.922404i	$-0.373778\pi$
$181$	10.3923	0.772454	0.386227	+	0.922404i	$0.373778\pi$
$182$	0	0
$183$	11.8079	0.872869
$184$	0	0
$185$	0	0
$186$	0	0
$187$	19.9333	1.45766
$188$	0	0
$189$	−4.13266	−0.300607
$190$	0	0
$191$	−19.7557	−1.42947	−0.714737	−	0.699393i	$-0.753454\pi$
$191$	−19.7557	−1.42947	−0.714737	+	0.699393i	$0.753454\pi$
$192$	0	0
$193$	2.23549	0.160914	0.0804571	−	0.996758i	$-0.474362\pi$
$193$	2.23549	0.160914	0.0804571	+	0.996758i	$0.474362\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−22.7568	−1.62135	−0.810677	−	0.585494i	$-0.800901\pi$
$197$	−22.7568	−1.62135	−0.810677	+	0.585494i	$0.800901\pi$
$198$	0	0
$199$	22.8171	1.61746	0.808731	−	0.588179i	$-0.200155\pi$
$199$	22.8171	1.61746	0.808731	+	0.588179i	$0.200155\pi$
$200$	0	0
$201$	−4.79759	−0.338396
$202$	0	0
$203$	29.0294	2.03746
$204$	0	0
$205$	0	0
$206$	0	0
$207$	5.46649	0.379947
$208$	0	0
$209$	−21.5996	−1.49408
$210$	0	0
$211$	20.0966	1.38351	0.691753	−	0.722134i	$-0.256839\pi$
$211$	20.0966	1.38351	0.691753	+	0.722134i	$0.256839\pi$
$212$	0	0
$213$	−3.14256	−0.215325
$214$	0	0
$215$	0	0
$216$	0	0
$217$	41.7068	2.83124
$218$	0	0
$219$	10.7473	0.726236
$220$	0	0
$221$	3.69769	0.248734
$222$	0	0
$223$	−13.4188	−0.898593	−0.449296	−	0.893383i	$-0.648325\pi$
$223$	−13.4188	−0.898593	−0.449296	+	0.893383i	$0.648325\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	18.4913	1.22731	0.613654	−	0.789575i	$-0.289699\pi$
$227$	18.4913	1.22731	0.613654	+	0.789575i	$0.289699\pi$
$228$	0	0
$229$	−19.4125	−1.28281	−0.641406	−	0.767202i	$-0.721648\pi$
$229$	−19.4125	−1.28281	−0.641406	+	0.767202i	$0.721648\pi$
$230$	0	0
$231$	15.5649	1.02410
$232$	0	0
$233$	12.1833	0.798151	0.399076	−	0.916918i	$-0.369331\pi$
$233$	12.1833	0.798151	0.399076	+	0.916918i	$0.369331\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−9.12492	−0.592727
$238$	0	0
$239$	4.52476	0.292682	0.146341	−	0.989234i	$-0.453250\pi$
$239$	4.52476	0.292682	0.146341	+	0.989234i	$0.453250\pi$
$240$	0	0
$241$	9.01177	0.580499	0.290250	−	0.956951i	$-0.406262\pi$
$241$	9.01177	0.580499	0.290250	+	0.956951i	$0.406262\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−4.00680	−0.254947
$248$	0	0
$249$	8.44098	0.534925
$250$	0	0
$251$	3.16965	0.200066	0.100033	−	0.994984i	$-0.468105\pi$
$251$	3.16965	0.200066	0.100033	+	0.994984i	$0.468105\pi$
$252$	0	0
$253$	−20.5885	−1.29439
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3.25241	−0.202880	−0.101440	−	0.994842i	$-0.532345\pi$
$257$	−3.25241	−0.202880	−0.101440	+	0.994842i	$0.532345\pi$
$258$	0	0
$259$	−23.9070	−1.48551
$260$	0	0
$261$	−7.02438	−0.434798
$262$	0	0
$263$	11.0600	0.681991	0.340995	−	0.940065i	$-0.389236\pi$
$263$	11.0600	0.681991	0.340995	+	0.940065i	$0.389236\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−3.38783	−0.207332
$268$	0	0
$269$	24.7558	1.50939	0.754695	−	0.656076i	$-0.227785\pi$
$269$	24.7558	1.50939	0.754695	+	0.656076i	$0.227785\pi$
$270$	0	0
$271$	0.250205	0.0151989	0.00759945	−	0.999971i	$-0.497581\pi$
$271$	0.250205	0.0151989	0.00759945	+	0.999971i	$0.497581\pi$
$272$	0	0
$273$	2.88735	0.174750
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−8.87074	−0.532991	−0.266496	−	0.963836i	$-0.585866\pi$
$277$	−8.87074	−0.532991	−0.266496	+	0.963836i	$0.585866\pi$
$278$	0	0
$279$	−10.0920	−0.604192
$280$	0	0
$281$	19.7379	1.17746	0.588732	−	0.808328i	$-0.299627\pi$
$281$	19.7379	1.17746	0.588732	+	0.808328i	$0.299627\pi$
$282$	0	0
$283$	26.4418	1.57180	0.785901	−	0.618352i	$-0.212199\pi$
$283$	26.4418	1.57180	0.785901	+	0.618352i	$0.212199\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	27.2570	1.60893
$288$	0	0
$289$	11.0106	0.647683
$290$	0	0
$291$	10.4351	0.611714
$292$	0	0
$293$	−19.1882	−1.12099	−0.560494	−	0.828158i	$-0.689389\pi$
$293$	−19.1882	−1.12099	−0.560494	+	0.828158i	$0.689389\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−3.76632	−0.218544
$298$	0	0
$299$	−3.81925	−0.220873
$300$	0	0
$301$	−19.8231	−1.14258
$302$	0	0
$303$	7.10799	0.408344
$304$	0	0
$305$	0	0
$306$	0	0
$307$	9.26979	0.529055	0.264527	−	0.964378i	$-0.414784\pi$
$307$	9.26979	0.529055	0.264527	+	0.964378i	$0.414784\pi$
$308$	0	0
$309$	−12.2091	−0.694549
$310$	0	0
$311$	−24.8936	−1.41159	−0.705793	−	0.708418i	$-0.749409\pi$
$311$	−24.8936	−1.41159	−0.705793	+	0.708418i	$0.749409\pi$
$312$	0	0
$313$	−6.04712	−0.341803	−0.170902	−	0.985288i	$-0.554668\pi$
$313$	−6.04712	−0.341803	−0.170902	+	0.985288i	$0.554668\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−14.3864	−0.808023	−0.404012	−	0.914754i	$-0.632384\pi$
$317$	−14.3864	−0.808023	−0.404012	+	0.914754i	$0.632384\pi$
$318$	0	0
$319$	26.4560	1.48125
$320$	0	0
$321$	4.04303	0.225660
$322$	0	0
$323$	−30.3522	−1.68884
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−0.773815	−0.0427921
$328$	0	0
$329$	−39.9785	−2.20409
$330$	0	0
$331$	1.06688	0.0586411	0.0293206	−	0.999570i	$-0.490666\pi$
$331$	1.06688	0.0586411	0.0293206	+	0.999570i	$0.490666\pi$
$332$	0	0
$333$	5.78488	0.317010
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−14.6020	−0.795421	−0.397710	−	0.917511i	$-0.630195\pi$
$337$	−14.6020	−0.795421	−0.397710	+	0.917511i	$0.630195\pi$
$338$	0	0
$339$	1.13943	0.0618854
$340$	0	0
$341$	38.0097	2.05834
$342$	0	0
$343$	−12.7239	−0.687029
$344$	0	0
$345$	0	0
$346$	0	0
$347$	16.1453	0.866723	0.433361	−	0.901220i	$-0.357327\pi$
$347$	16.1453	0.866723	0.433361	+	0.901220i	$0.357327\pi$
$348$	0	0
$349$	−22.2622	−1.19167	−0.595834	−	0.803108i	$-0.703178\pi$
$349$	−22.2622	−1.19167	−0.595834	+	0.803108i	$0.703178\pi$
$350$	0	0
$351$	−0.698665	−0.0372920
$352$	0	0
$353$	−9.52997	−0.507229	−0.253615	−	0.967305i	$-0.581619\pi$
$353$	−9.52997	−0.507229	−0.253615	+	0.967305i	$0.581619\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	21.8721	1.15760
$358$	0	0
$359$	−16.7872	−0.885995	−0.442998	−	0.896523i	$-0.646085\pi$
$359$	−16.7872	−0.885995	−0.442998	+	0.896523i	$0.646085\pi$
$360$	0	0
$361$	13.8895	0.731027
$362$	0	0
$363$	3.18514	0.167177
$364$	0	0
$365$	0	0
$366$	0	0
$367$	32.4766	1.69527	0.847633	−	0.530583i	$-0.178027\pi$
$367$	32.4766	1.69527	0.847633	+	0.530583i	$0.178027\pi$
$368$	0	0
$369$	−6.59552	−0.343349
$370$	0	0
$371$	14.0361	0.728717
$372$	0	0
$373$	−14.2912	−0.739972	−0.369986	−	0.929037i	$-0.620637\pi$
$373$	−14.2912	−0.739972	−0.369986	+	0.929037i	$0.620637\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.90769	0.252759
$378$	0	0
$379$	−3.21569	−0.165179	−0.0825895	−	0.996584i	$-0.526319\pi$
$379$	−3.21569	−0.165179	−0.0825895	+	0.996584i	$0.526319\pi$
$380$	0	0
$381$	−0.537868	−0.0275558
$382$	0	0
$383$	−12.3598	−0.631556	−0.315778	−	0.948833i	$-0.602266\pi$
$383$	−12.3598	−0.631556	−0.315778	+	0.948833i	$0.602266\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4.79668	0.243829
$388$	0	0
$389$	23.1248	1.17247	0.586236	−	0.810140i	$-0.300609\pi$
$389$	23.1248	1.17247	0.586236	+	0.810140i	$0.300609\pi$
$390$	0	0
$391$	−28.9314	−1.46313
$392$	0	0
$393$	−6.72242	−0.339101
$394$	0	0
$395$	0	0
$396$	0	0
$397$	2.12824	0.106814	0.0534068	−	0.998573i	$-0.482992\pi$
$397$	2.12824	0.106814	0.0534068	+	0.998573i	$0.482992\pi$
$398$	0	0
$399$	−23.7005	−1.18651
$400$	0	0
$401$	−33.8250	−1.68914	−0.844569	−	0.535447i	$-0.820143\pi$
$401$	−33.8250	−1.68914	−0.844569	+	0.535447i	$0.820143\pi$
$402$	0	0
$403$	7.05093	0.351232
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−21.7877	−1.07998
$408$	0	0
$409$	−34.7663	−1.71908	−0.859542	−	0.511065i	$-0.829251\pi$
$409$	−34.7663	−1.71908	−0.859542	+	0.511065i	$0.829251\pi$
$410$	0	0
$411$	−0.183379	−0.00904541
$412$	0	0
$413$	−3.08006	−0.151560
$414$	0	0
$415$	0	0
$416$	0	0
$417$	8.22006	0.402538
$418$	0	0
$419$	−6.79744	−0.332077	−0.166038	−	0.986119i	$-0.553098\pi$
$419$	−6.79744	−0.332077	−0.166038	+	0.986119i	$0.553098\pi$
$420$	0	0
$421$	−10.8246	−0.527560	−0.263780	−	0.964583i	$-0.584969\pi$
$421$	−10.8246	−0.527560	−0.263780	+	0.964583i	$0.584969\pi$
$422$	0	0
$423$	9.67379	0.470356
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−48.7982	−2.36151
$428$	0	0
$429$	2.63140	0.127045
$430$	0	0
$431$	−10.3869	−0.500319	−0.250159	−	0.968205i	$-0.580483\pi$
$431$	−10.3869	−0.500319	−0.250159	+	0.968205i	$0.580483\pi$
$432$	0	0
$433$	−15.6627	−0.752701	−0.376350	−	0.926477i	$-0.622821\pi$
$433$	−15.6627	−0.752701	−0.376350	+	0.926477i	$0.622821\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	31.3500	1.49967
$438$	0	0
$439$	22.0075	1.05036	0.525181	−	0.850990i	$-0.323997\pi$
$439$	22.0075	1.05036	0.525181	+	0.850990i	$0.323997\pi$
$440$	0	0
$441$	10.0789	0.479946
$442$	0	0
$443$	29.4447	1.39896	0.699480	−	0.714652i	$-0.253415\pi$
$443$	29.4447	1.39896	0.699480	+	0.714652i	$0.253415\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	5.53790	0.261933
$448$	0	0
$449$	27.3445	1.29047	0.645234	−	0.763985i	$-0.276760\pi$
$449$	27.3445	1.29047	0.645234	+	0.763985i	$0.276760\pi$
$450$	0	0
$451$	24.8408	1.16971
$452$	0	0
$453$	12.8736	0.604856
$454$	0	0
$455$	0	0
$456$	0	0
$457$	33.6394	1.57359	0.786793	−	0.617217i	$-0.211740\pi$
$457$	33.6394	1.57359	0.786793	+	0.617217i	$0.211740\pi$
$458$	0	0
$459$	−5.29251	−0.247033
$460$	0	0
$461$	25.4419	1.18495	0.592474	−	0.805589i	$-0.298151\pi$
$461$	25.4419	1.18495	0.592474	+	0.805589i	$0.298151\pi$
$462$	0	0
$463$	7.84127	0.364415	0.182207	−	0.983260i	$-0.441676\pi$
$463$	7.84127	0.364415	0.182207	+	0.983260i	$0.441676\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−10.8007	−0.499795	−0.249897	−	0.968272i	$-0.580397\pi$
$467$	−10.8007	−0.499795	−0.249897	+	0.968272i	$0.580397\pi$
$468$	0	0
$469$	19.8268	0.915517
$470$	0	0
$471$	14.6395	0.674551
$472$	0	0
$473$	−18.0658	−0.830668
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−3.39638	−0.155509
$478$	0	0
$479$	22.6753	1.03606	0.518030	−	0.855362i	$-0.326665\pi$
$479$	22.6753	1.03606	0.518030	+	0.855362i	$0.326665\pi$
$480$	0	0
$481$	−4.04170	−0.184286
$482$	0	0
$483$	−22.5912	−1.02793
$484$	0	0
$485$	0	0
$486$	0	0
$487$	4.98587	0.225931	0.112966	−	0.993599i	$-0.463965\pi$
$487$	4.98587	0.225931	0.112966	+	0.993599i	$0.463965\pi$
$488$	0	0
$489$	4.44424	0.200975
$490$	0	0
$491$	−26.4530	−1.19381	−0.596904	−	0.802313i	$-0.703603\pi$
$491$	−26.4530	−1.19381	−0.596904	+	0.802313i	$0.703603\pi$
$492$	0	0
$493$	37.1766	1.67435
$494$	0	0
$495$	0	0
$496$	0	0
$497$	12.9871	0.582553
$498$	0	0
$499$	38.0875	1.70503	0.852516	−	0.522702i	$-0.175076\pi$
$499$	38.0875	1.70503	0.852516	+	0.522702i	$0.175076\pi$
$500$	0	0
$501$	23.7655	1.06176
$502$	0	0
$503$	−2.35598	−0.105048	−0.0525240	−	0.998620i	$-0.516727\pi$
$503$	−2.35598	−0.105048	−0.0525240	+	0.998620i	$0.516727\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−12.5119	−0.555672
$508$	0	0
$509$	−29.1985	−1.29420	−0.647099	−	0.762406i	$-0.724018\pi$
$509$	−29.1985	−1.29420	−0.647099	+	0.762406i	$0.724018\pi$
$510$	0	0
$511$	−44.4150	−1.96480
$512$	0	0
$513$	5.73494	0.253204
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−36.4346	−1.60239
$518$	0	0
$519$	15.0868	0.662236
$520$	0	0
$521$	−42.6181	−1.86713	−0.933565	−	0.358407i	$-0.883320\pi$
$521$	−42.6181	−1.86713	−0.933565	+	0.358407i	$0.883320\pi$
$522$	0	0
$523$	20.6750	0.904055	0.452028	−	0.892004i	$-0.350701\pi$
$523$	20.6750	0.904055	0.452028	+	0.892004i	$0.350701\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	53.4120	2.32666
$528$	0	0
$529$	6.88254	0.299241
$530$	0	0
$531$	0.745297	0.0323431
$532$	0	0
$533$	4.60806	0.199597
$534$	0	0
$535$	0	0
$536$	0	0
$537$	16.7163	0.721363
$538$	0	0
$539$	−37.9602	−1.63506
$540$	0	0
$541$	11.9549	0.513980	0.256990	−	0.966414i	$-0.417269\pi$
$541$	11.9549	0.513980	0.256990	+	0.966414i	$0.417269\pi$
$542$	0	0
$543$	10.3923	0.445976
$544$	0	0
$545$	0	0
$546$	0	0
$547$	26.6027	1.13745	0.568724	−	0.822529i	$-0.307437\pi$
$547$	26.6027	1.13745	0.568724	+	0.822529i	$0.307437\pi$
$548$	0	0
$549$	11.8079	0.503951
$550$	0	0
$551$	−40.2844	−1.71617
$552$	0	0
$553$	37.7102	1.60360
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−28.3420	−1.20089	−0.600444	−	0.799667i	$-0.705010\pi$
$557$	−28.3420	−1.20089	−0.600444	+	0.799667i	$0.705010\pi$
$558$	0	0
$559$	−3.35128	−0.141744
$560$	0	0
$561$	19.9333	0.841583
$562$	0	0
$563$	4.05616	0.170947	0.0854735	−	0.996340i	$-0.472760\pi$
$563$	4.05616	0.170947	0.0854735	+	0.996340i	$0.472760\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−4.13266	−0.173555
$568$	0	0
$569$	1.96688	0.0824558	0.0412279	−	0.999150i	$-0.486873\pi$
$569$	1.96688	0.0824558	0.0412279	+	0.999150i	$0.486873\pi$
$570$	0	0
$571$	5.15778	0.215846	0.107923	−	0.994159i	$-0.465580\pi$
$571$	5.15778	0.215846	0.107923	+	0.994159i	$0.465580\pi$
$572$	0	0
$573$	−19.7557	−0.825307
$574$	0	0
$575$	0	0
$576$	0	0
$577$	38.6187	1.60772	0.803859	−	0.594819i	$-0.202776\pi$
$577$	38.6187	1.60772	0.803859	+	0.594819i	$0.202776\pi$
$578$	0	0
$579$	2.23549	0.0929039
$580$	0	0
$581$	−34.8837	−1.44722
$582$	0	0
$583$	12.7918	0.529784
$584$	0	0
$585$	0	0
$586$	0	0
$587$	27.1719	1.12151	0.560753	−	0.827983i	$-0.310512\pi$
$587$	27.1719	1.12151	0.560753	+	0.827983i	$0.310512\pi$
$588$	0	0
$589$	−57.8770	−2.38478
$590$	0	0
$591$	−22.7568	−0.936089
$592$	0	0
$593$	−30.3486	−1.24627	−0.623135	−	0.782114i	$-0.714141\pi$
$593$	−30.3486	−1.24627	−0.623135	+	0.782114i	$0.714141\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	22.8171	0.933842
$598$	0	0
$599$	−26.9205	−1.09994	−0.549971	−	0.835184i	$-0.685361\pi$
$599$	−26.9205	−1.09994	−0.549971	+	0.835184i	$0.685361\pi$
$600$	0	0
$601$	−16.9133	−0.689910	−0.344955	−	0.938619i	$-0.612106\pi$
$601$	−16.9133	−0.689910	−0.344955	+	0.938619i	$0.612106\pi$
$602$	0	0
$603$	−4.79759	−0.195373
$604$	0	0
$605$	0	0
$606$	0	0
$607$	18.9715	0.770028	0.385014	−	0.922911i	$-0.374197\pi$
$607$	18.9715	0.770028	0.385014	+	0.922911i	$0.374197\pi$
$608$	0	0
$609$	29.0294	1.17633
$610$	0	0
$611$	−6.75874	−0.273430
$612$	0	0
$613$	5.45816	0.220453	0.110227	−	0.993906i	$-0.464842\pi$
$613$	5.45816	0.220453	0.110227	+	0.993906i	$0.464842\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	11.8970	0.478954	0.239477	−	0.970902i	$-0.423024\pi$
$617$	11.8970	0.478954	0.239477	+	0.970902i	$0.423024\pi$
$618$	0	0
$619$	−13.2121	−0.531038	−0.265519	−	0.964106i	$-0.585543\pi$
$619$	−13.2121	−0.531038	−0.265519	+	0.964106i	$0.585543\pi$
$620$	0	0
$621$	5.46649	0.219363
$622$	0	0
$623$	14.0008	0.560928
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−21.5996	−0.862605
$628$	0	0
$629$	−30.6165	−1.22076
$630$	0	0
$631$	−11.6182	−0.462513	−0.231257	−	0.972893i	$-0.574284\pi$
$631$	−11.6182	−0.462513	−0.231257	+	0.972893i	$0.574284\pi$
$632$	0	0
$633$	20.0966	0.798768
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−7.04176	−0.279005
$638$	0	0
$639$	−3.14256	−0.124318
$640$	0	0
$641$	−34.8796	−1.37766	−0.688831	−	0.724922i	$-0.741876\pi$
$641$	−34.8796	−1.37766	−0.688831	+	0.724922i	$0.741876\pi$
$642$	0	0
$643$	−2.90629	−0.114613	−0.0573065	−	0.998357i	$-0.518251\pi$
$643$	−2.90629	−0.114613	−0.0573065	+	0.998357i	$0.518251\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−12.1402	−0.477280	−0.238640	−	0.971108i	$-0.576702\pi$
$647$	−12.1402	−0.477280	−0.238640	+	0.971108i	$0.576702\pi$
$648$	0	0
$649$	−2.80702	−0.110185
$650$	0	0
$651$	41.7068	1.63462
$652$	0	0
$653$	−26.0987	−1.02132	−0.510661	−	0.859782i	$-0.670599\pi$
$653$	−26.0987	−1.02132	−0.510661	+	0.859782i	$0.670599\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	10.7473	0.419293
$658$	0	0
$659$	21.3268	0.830775	0.415388	−	0.909644i	$-0.363646\pi$
$659$	21.3268	0.830775	0.415388	+	0.909644i	$0.363646\pi$
$660$	0	0
$661$	−31.6863	−1.23246	−0.616228	−	0.787568i	$-0.711340\pi$
$661$	−31.6863	−1.23246	−0.616228	+	0.787568i	$0.711340\pi$
$662$	0	0
$663$	3.69769	0.143606
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−38.3987	−1.48680
$668$	0	0
$669$	−13.4188	−0.518803
$670$	0	0
$671$	−44.4725	−1.71684
$672$	0	0
$673$	−1.98577	−0.0765456	−0.0382728	−	0.999267i	$-0.512186\pi$
$673$	−1.98577	−0.0765456	−0.0382728	+	0.999267i	$0.512186\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	39.4359	1.51564	0.757822	−	0.652462i	$-0.226264\pi$
$677$	39.4359	1.51564	0.757822	+	0.652462i	$0.226264\pi$
$678$	0	0
$679$	−43.1246	−1.65497
$680$	0	0
$681$	18.4913	0.708587
$682$	0	0
$683$	−12.2965	−0.470511	−0.235256	−	0.971934i	$-0.575593\pi$
$683$	−12.2965	−0.470511	−0.235256	+	0.971934i	$0.575593\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−19.4125	−0.740631
$688$	0	0
$689$	2.37293	0.0904015
$690$	0	0
$691$	15.7058	0.597475	0.298737	−	0.954335i	$-0.403435\pi$
$691$	15.7058	0.597475	0.298737	+	0.954335i	$0.403435\pi$
$692$	0	0
$693$	15.5649	0.591262
$694$	0	0
$695$	0	0
$696$	0	0
$697$	34.9068	1.32219
$698$	0	0
$699$	12.1833	0.460813
$700$	0	0
$701$	−26.4222	−0.997953	−0.498976	−	0.866616i	$-0.666291\pi$
$701$	−26.4222	−0.997953	−0.498976	+	0.866616i	$0.666291\pi$
$702$	0	0
$703$	33.1759	1.25125
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−29.3749	−1.10476
$708$	0	0
$709$	48.7475	1.83075	0.915376	−	0.402601i	$-0.131894\pi$
$709$	48.7475	1.83075	0.915376	+	0.402601i	$0.131894\pi$
$710$	0	0
$711$	−9.12492	−0.342211
$712$	0	0
$713$	−55.1678	−2.06605
$714$	0	0
$715$	0	0
$716$	0	0
$717$	4.52476	0.168980
$718$	0	0
$719$	−34.2535	−1.27744	−0.638721	−	0.769439i	$-0.720536\pi$
$719$	−34.2535	−1.27744	−0.638721	+	0.769439i	$0.720536\pi$
$720$	0	0
$721$	50.4559	1.87907
$722$	0	0
$723$	9.01177	0.335151
$724$	0	0
$725$	0	0
$726$	0	0
$727$	21.3624	0.792287	0.396144	−	0.918189i	$-0.370348\pi$
$727$	21.3624	0.792287	0.396144	+	0.918189i	$0.370348\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−25.3865	−0.938953
$732$	0	0
$733$	−31.2827	−1.15545	−0.577727	−	0.816230i	$-0.696060\pi$
$733$	−31.2827	−1.15545	−0.577727	+	0.816230i	$0.696060\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	18.0692	0.665589
$738$	0	0
$739$	1.31491	0.0483699	0.0241849	−	0.999708i	$-0.492301\pi$
$739$	1.31491	0.0483699	0.0241849	+	0.999708i	$0.492301\pi$
$740$	0	0
$741$	−4.00680	−0.147194
$742$	0	0
$743$	−8.76431	−0.321531	−0.160766	−	0.986993i	$-0.551396\pi$
$743$	−8.76431	−0.321531	−0.160766	+	0.986993i	$0.551396\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	8.44098	0.308839
$748$	0	0
$749$	−16.7085	−0.610515
$750$	0	0
$751$	44.6570	1.62956	0.814779	−	0.579772i	$-0.196858\pi$
$751$	44.6570	1.62956	0.814779	+	0.579772i	$0.196858\pi$
$752$	0	0
$753$	3.16965	0.115508
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−7.70729	−0.280126	−0.140063	−	0.990143i	$-0.544731\pi$
$757$	−7.70729	−0.280126	−0.140063	+	0.990143i	$0.544731\pi$
$758$	0	0
$759$	−20.5885	−0.747317
$760$	0	0
$761$	39.5914	1.43519	0.717594	−	0.696461i	$-0.245243\pi$
$761$	39.5914	1.43519	0.717594	+	0.696461i	$0.245243\pi$
$762$	0	0
$763$	3.19791	0.115772
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−0.520713	−0.0188019
$768$	0	0
$769$	29.3363	1.05790	0.528948	−	0.848655i	$-0.322587\pi$
$769$	29.3363	1.05790	0.528948	+	0.848655i	$0.322587\pi$
$770$	0	0
$771$	−3.25241	−0.117133
$772$	0	0
$773$	10.9101	0.392411	0.196205	−	0.980563i	$-0.437138\pi$
$773$	10.9101	0.392411	0.196205	+	0.980563i	$0.437138\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−23.9070	−0.857657
$778$	0	0
$779$	−37.8249	−1.35522
$780$	0	0
$781$	11.8359	0.423521
$782$	0	0
$783$	−7.02438	−0.251031
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−37.7075	−1.34413	−0.672063	−	0.740494i	$-0.734591\pi$
$787$	−37.7075	−1.34413	−0.672063	+	0.740494i	$0.734591\pi$
$788$	0	0
$789$	11.0600	0.393747
$790$	0	0
$791$	−4.70888	−0.167429
$792$	0	0
$793$	−8.24980	−0.292959
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−49.2412	−1.74421	−0.872106	−	0.489317i	$-0.837246\pi$
$797$	−49.2412	−1.74421	−0.872106	+	0.489317i	$0.837246\pi$
$798$	0	0
$799$	−51.1986	−1.81128
$800$	0	0
$801$	−3.38783	−0.119703
$802$	0	0
$803$	−40.4778	−1.42843
$804$	0	0
$805$	0	0
$806$	0	0
$807$	24.7558	0.871447
$808$	0	0
$809$	26.9783	0.948507	0.474253	−	0.880388i	$-0.342718\pi$
$809$	26.9783	0.948507	0.474253	+	0.880388i	$0.342718\pi$
$810$	0	0
$811$	8.17256	0.286977	0.143489	−	0.989652i	$-0.454168\pi$
$811$	8.17256	0.286977	0.143489	+	0.989652i	$0.454168\pi$
$812$	0	0
$813$	0.250205	0.00877509
$814$	0	0
$815$	0	0
$816$	0	0
$817$	27.5087	0.962407
$818$	0	0
$819$	2.88735	0.100892
$820$	0	0
$821$	−43.7884	−1.52823	−0.764113	−	0.645082i	$-0.776823\pi$
$821$	−43.7884	−1.52823	−0.764113	+	0.645082i	$0.776823\pi$
$822$	0	0
$823$	16.4567	0.573644	0.286822	−	0.957984i	$-0.407401\pi$
$823$	16.4567	0.573644	0.286822	+	0.957984i	$0.407401\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−4.01834	−0.139731	−0.0698657	−	0.997556i	$-0.522257\pi$
$827$	−4.01834	−0.139731	−0.0698657	+	0.997556i	$0.522257\pi$
$828$	0	0
$829$	37.4656	1.30124	0.650618	−	0.759405i	$-0.274510\pi$
$829$	37.4656	1.30124	0.650618	+	0.759405i	$0.274510\pi$
$830$	0	0
$831$	−8.87074	−0.307723
$832$	0	0
$833$	−53.3425	−1.84821
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−10.0920	−0.348831
$838$	0	0
$839$	−11.5960	−0.400337	−0.200169	−	0.979761i	$-0.564149\pi$
$839$	−11.5960	−0.400337	−0.200169	+	0.979761i	$0.564149\pi$
$840$	0	0
$841$	20.3419	0.701445
$842$	0	0
$843$	19.7379	0.679809
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−13.1631	−0.452290
$848$	0	0
$849$	26.4418	0.907481
$850$	0	0
$851$	31.6230	1.08402
$852$	0	0
$853$	1.32197	0.0452635	0.0226317	−	0.999744i	$-0.492795\pi$
$853$	1.32197	0.0452635	0.0226317	+	0.999744i	$0.492795\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−15.5623	−0.531598	−0.265799	−	0.964028i	$-0.585636\pi$
$857$	−15.5623	−0.531598	−0.265799	+	0.964028i	$0.585636\pi$
$858$	0	0
$859$	−27.3826	−0.934283	−0.467141	−	0.884183i	$-0.654716\pi$
$859$	−27.3826	−0.934283	−0.467141	+	0.884183i	$0.654716\pi$
$860$	0	0
$861$	27.2570	0.928917
$862$	0	0
$863$	−32.0293	−1.09029	−0.545145	−	0.838342i	$-0.683525\pi$
$863$	−32.0293	−1.09029	−0.545145	+	0.838342i	$0.683525\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	11.0106	0.373940
$868$	0	0
$869$	34.3673	1.16583
$870$	0	0
$871$	3.35191	0.113575
$872$	0	0
$873$	10.4351	0.353173
$874$	0	0
$875$	0	0
$876$	0	0
$877$	4.29121	0.144904	0.0724520	−	0.997372i	$-0.476918\pi$
$877$	4.29121	0.144904	0.0724520	+	0.997372i	$0.476918\pi$
$878$	0	0
$879$	−19.1882	−0.647203
$880$	0	0
$881$	−54.0804	−1.82202	−0.911008	−	0.412388i	$-0.864695\pi$
$881$	−54.0804	−1.82202	−0.911008	+	0.412388i	$0.864695\pi$
$882$	0	0
$883$	31.7020	1.06686	0.533428	−	0.845845i	$-0.320903\pi$
$883$	31.7020	1.06686	0.533428	+	0.845845i	$0.320903\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	5.38001	0.180643	0.0903216	−	0.995913i	$-0.471211\pi$
$887$	5.38001	0.180643	0.0903216	+	0.995913i	$0.471211\pi$
$888$	0	0
$889$	2.22283	0.0745512
$890$	0	0
$891$	−3.76632	−0.126176
$892$	0	0
$893$	55.4786	1.85652
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−3.81925	−0.127521
$898$	0	0
$899$	70.8900	2.36432
$900$	0	0
$901$	17.9753	0.598846
$902$	0	0
$903$	−19.8231	−0.659670
$904$	0	0
$905$	0	0
$906$	0	0
$907$	45.7593	1.51941	0.759706	−	0.650266i	$-0.225343\pi$
$907$	45.7593	1.51941	0.759706	+	0.650266i	$0.225343\pi$
$908$	0	0
$909$	7.10799	0.235757
$910$	0	0
$911$	2.16774	0.0718206	0.0359103	−	0.999355i	$-0.488567\pi$
$911$	2.16774	0.0718206	0.0359103	+	0.999355i	$0.488567\pi$
$912$	0	0
$913$	−31.7914	−1.05214
$914$	0	0
$915$	0	0
$916$	0	0
$917$	27.7815	0.917426
$918$	0	0
$919$	−19.6600	−0.648523	−0.324262	−	0.945967i	$-0.605116\pi$
$919$	−19.6600	−0.648523	−0.324262	+	0.945967i	$0.605116\pi$
$920$	0	0
$921$	9.26979	0.305450
$922$	0	0
$923$	2.19560	0.0722690
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−12.2091	−0.400998
$928$	0	0
$929$	43.5995	1.43045	0.715227	−	0.698892i	$-0.246323\pi$
$929$	43.5995	1.43045	0.715227	+	0.698892i	$0.246323\pi$
$930$	0	0
$931$	57.8017	1.89438
$932$	0	0
$933$	−24.8936	−0.814980
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−42.0735	−1.37448	−0.687241	−	0.726429i	$-0.741178\pi$
$937$	−42.0735	−1.37448	−0.687241	+	0.726429i	$0.741178\pi$
$938$	0	0
$939$	−6.04712	−0.197340
$940$	0	0
$941$	48.9170	1.59465	0.797324	−	0.603552i	$-0.206248\pi$
$941$	48.9170	1.59465	0.797324	+	0.603552i	$0.206248\pi$
$942$	0	0
$943$	−36.0544	−1.17409
$944$	0	0
$945$	0	0
$946$	0	0
$947$	7.11340	0.231155	0.115577	−	0.993298i	$-0.463128\pi$
$947$	7.11340	0.231155	0.115577	+	0.993298i	$0.463128\pi$
$948$	0	0
$949$	−7.50878	−0.243745
$950$	0	0
$951$	−14.3864	−0.466512
$952$	0	0
$953$	22.2372	0.720334	0.360167	−	0.932888i	$-0.382720\pi$
$953$	22.2372	0.720334	0.360167	+	0.932888i	$0.382720\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	26.4560	0.855202
$958$	0	0
$959$	0.757842	0.0244720
$960$	0	0
$961$	70.8485	2.28543
$962$	0	0
$963$	4.04303	0.130285
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−13.9152	−0.447482	−0.223741	−	0.974649i	$-0.571827\pi$
$967$	−13.9152	−0.447482	−0.223741	+	0.974649i	$0.571827\pi$
$968$	0	0
$969$	−30.3522	−0.975053
$970$	0	0
$971$	24.2947	0.779655	0.389827	−	0.920888i	$-0.372535\pi$
$971$	24.2947	0.779655	0.389827	+	0.920888i	$0.372535\pi$
$972$	0	0
$973$	−33.9707	−1.08905
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−46.3314	−1.48227	−0.741136	−	0.671355i	$-0.765713\pi$
$977$	−46.3314	−1.48227	−0.741136	+	0.671355i	$0.765713\pi$
$978$	0	0
$979$	12.7596	0.407800
$980$	0	0
$981$	−0.773815	−0.0247060
$982$	0	0
$983$	−41.0582	−1.30955	−0.654777	−	0.755822i	$-0.727237\pi$
$983$	−41.0582	−1.30955	−0.654777	+	0.755822i	$0.727237\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−39.9785	−1.27253
$988$	0	0
$989$	26.2210	0.833780
$990$	0	0
$991$	20.2765	0.644105	0.322053	−	0.946722i	$-0.395627\pi$
$991$	20.2765	0.644105	0.322053	+	0.946722i	$0.395627\pi$
$992$	0	0
$993$	1.06688	0.0338565
$994$	0	0
$995$	0	0
$996$	0	0
$997$	24.3634	0.771596	0.385798	−	0.922583i	$-0.373926\pi$
$997$	24.3634	0.771596	0.385798	+	0.922583i	$0.373926\pi$
$998$	0	0
$999$	5.78488	0.183026

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7500.2.a.n.1.1		12
5.2	odd	4		7500.2.d.g.1249.1		24
5.3	odd	4		7500.2.d.g.1249.24		24
5.4	even	2		7500.2.a.m.1.12		12
25.2	odd	20		1500.2.o.c.649.2		24
25.9	even	10		1500.2.m.d.901.6		24
25.11	even	5		1500.2.m.c.601.1		24
25.12	odd	20		300.2.o.a.169.5	✓	24
25.13	odd	20		1500.2.o.c.349.2		24
25.14	even	10		1500.2.m.d.601.6		24
25.16	even	5		1500.2.m.c.901.1		24
25.23	odd	20		300.2.o.a.229.5	yes	24
75.23	even	20		900.2.w.c.829.3		24
75.62	even	20		900.2.w.c.469.3		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.2.o.a.169.5	✓	24	25.12	odd	20
300.2.o.a.229.5	yes	24	25.23	odd	20
900.2.w.c.469.3		24	75.62	even	20
900.2.w.c.829.3		24	75.23	even	20
1500.2.m.c.601.1		24	25.11	even	5
1500.2.m.c.901.1		24	25.16	even	5
1500.2.m.d.601.6		24	25.14	even	10
1500.2.m.d.901.6		24	25.9	even	10
1500.2.o.c.349.2		24	25.13	odd	20
1500.2.o.c.649.2		24	25.2	odd	20
7500.2.a.m.1.12		12	5.4	even	2
7500.2.a.n.1.1		12	1.1	even	1	trivial
7500.2.d.g.1249.1		24	5.2	odd	4
7500.2.d.g.1249.24		24	5.3	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -4.13266 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -4.13266 q^{7} +1.00000 q^{9} -3.76632 q^{11} -0.698665 q^{13} -5.29251 q^{17} +5.73494 q^{19} -4.13266 q^{21} +5.46649 q^{23} +1.00000 q^{27} -7.02438 q^{29} -10.0920 q^{31} -3.76632 q^{33} +5.78488 q^{37} -0.698665 q^{39} -6.59552 q^{41} +4.79668 q^{43} +9.67379 q^{47} +10.0789 q^{49} -5.29251 q^{51} -3.39638 q^{53} +5.73494 q^{57} +0.745297 q^{59} +11.8079 q^{61} -4.13266 q^{63} -4.79759 q^{67} +5.46649 q^{69} -3.14256 q^{71} +10.7473 q^{73} +15.5649 q^{77} -9.12492 q^{79} +1.00000 q^{81} +8.44098 q^{83} -7.02438 q^{87} -3.38783 q^{89} +2.88735 q^{91} -10.0920 q^{93} +10.4351 q^{97} -3.76632 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{3} + 8 q^{7} + 12 q^{9}+O(q^{10}) \) 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

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