LMFDB - Embedded newform 7500.2.d.d.1249.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7500,2,Mod(1249,7500)]

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, 1]))

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

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

chi := DirichletCharacter("7500.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7500 = 2^{2} \cdot 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7500.d (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$59.8878015160$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.324000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 9x^{6} + 26x^{4} + 24x^{2} + 1 $ x^8 + 9x^6 + 26x^4 + 24*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 300)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.7
Root			$1.95630i$ of defining polynomial
Character	$\chi$	$=$	7500.1249
Dual form			7500.2.d.d.1249.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +0.511170i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +0.511170i q^{7} -1.00000 q^{9} -1.82709 q^{11} +6.12165i q^{13} -2.58347i q^{17} +4.86324 q^{19} -0.511170 q^{21} -6.63282i q^{23} -1.00000i q^{27} +7.99711 q^{29} -4.88558 q^{31} -1.82709i q^{33} +7.43757i q^{37} -6.12165 q^{39} +5.73054 q^{41} -2.05126i q^{43} -7.95439i q^{47} +6.73870 q^{49} +2.58347 q^{51} +1.32912i q^{53} +4.86324i q^{57} +10.0927 q^{59} +4.77116 q^{61} -0.511170i q^{63} +14.3335i q^{67} +6.63282 q^{69} -5.36974 q^{71} -3.48067i q^{73} -0.933955i q^{77} +1.31999 q^{79} +1.00000 q^{81} +0.912590i q^{83} +7.99711i q^{87} -11.7758 q^{89} -3.12920 q^{91} -4.88558i q^{93} +7.39984i q^{97} +1.82709 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^9	$ 8 q - 8 q^{9} - 2 q^{11} + 10 q^{19} - 8 q^{21} + 8 q^{29} - 18 q^{31} - 10 q^{39} + 8 q^{49} - 8 q^{51} - 2 q^{59} - 4 q^{61} + 18 q^{69} - 40 q^{71} + 6 q^{79} + 8 q^{81} - 30 q^{89} - 20 q^{91} + 2 q^{99}+O(q^{100}) $ 8 * q - 8 * q^9 - 2 * q^11 + 10 * q^19 - 8 * q^21 + 8 * q^29 - 18 * q^31 - 10 * q^39 + 8 * q^49 - 8 * q^51 - 2 * q^59 - 4 * q^61 + 18 * q^69 - 40 * q^71 + 6 * q^79 + 8 * q^81 - 30 * q^89 - 20 * q^91 + 2 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/7500\mathbb{Z}\right)^\times$.

$n$	$2501$	$3751$	$6877$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.511170i	0.193204i	0.995323	+	0.0966021i	$0.0307974\pi$
$7$	0.511170i	0.193204i	−0.995323	+	0.0966021i	$0.969203\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−1.82709	−0.550889	−0.275444	−	0.961317i	$-0.588825\pi$
$11$	−1.82709	−0.550889	−0.275444	+	0.961317i	$0.588825\pi$
$12$	0	0
$13$	6.12165i	1.69784i	0.528522	+	0.848920i	$0.322746\pi$
$13$	6.12165i	1.69784i	−0.528522	+	0.848920i	$0.677254\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 2.58347i	− 0.626583i	−0.949657	−	0.313291i	$-0.898568\pi$
$17$	− 2.58347i	− 0.626583i	0.949657	−	0.313291i	$-0.101432\pi$
$18$	0	0
$19$	4.86324	1.11570	0.557852	−	0.829941i	$-0.311626\pi$
$19$	4.86324	1.11570	0.557852	+	0.829941i	$0.311626\pi$
$20$	0	0
$21$	−0.511170	−0.111547
$22$	0	0
$23$	− 6.63282i	− 1.38304i	−0.722358	−	0.691519i	$-0.756942\pi$
$23$	− 6.63282i	− 1.38304i	0.722358	−	0.691519i	$-0.243058\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	7.99711	1.48503	0.742513	−	0.669831i	$-0.233634\pi$
$29$	7.99711	1.48503	0.742513	+	0.669831i	$0.233634\pi$
$30$	0	0
$31$	−4.88558	−0.877476	−0.438738	−	0.898615i	$-0.644574\pi$
$31$	−4.88558	−0.877476	−0.438738	+	0.898615i	$0.644574\pi$
$32$	0	0
$33$	− 1.82709i	− 0.318056i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.43757i	1.22273i	0.791349	+	0.611364i	$0.209379\pi$
$37$	7.43757i	1.22273i	−0.791349	+	0.611364i	$0.790621\pi$
$38$	0	0
$39$	−6.12165	−0.980248
$40$	0	0
$41$	5.73054	0.894961	0.447480	−	0.894294i	$-0.352321\pi$
$41$	5.73054	0.894961	0.447480	+	0.894294i	$0.352321\pi$
$42$	0	0
$43$	− 2.05126i	− 0.312814i	−0.987693	−	0.156407i	$-0.950009\pi$
$43$	− 2.05126i	− 0.312814i	0.987693	−	0.156407i	$-0.0499912\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 7.95439i	− 1.16027i	−0.814522	−	0.580133i	$-0.803001\pi$
$47$	− 7.95439i	− 1.16027i	0.814522	−	0.580133i	$-0.196999\pi$
$48$	0	0
$49$	6.73870	0.962672
$50$	0	0
$51$	2.58347	0.361758
$52$	0	0
$53$	1.32912i	0.182569i	0.995825	+	0.0912846i	$0.0290973\pi$
$53$	1.32912i	0.182569i	−0.995825	+	0.0912846i	$0.970903\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	4.86324i	0.644152i
$58$	0	0
$59$	10.0927	1.31396	0.656981	−	0.753908i	$-0.271833\pi$
$59$	10.0927	1.31396	0.656981	+	0.753908i	$0.271833\pi$
$60$	0	0
$61$	4.77116	0.610884	0.305442	−	0.952211i	$-0.401196\pi$
$61$	4.77116	0.610884	0.305442	+	0.952211i	$0.401196\pi$
$62$	0	0
$63$	− 0.511170i	− 0.0644014i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	14.3335i	1.75111i	0.483117	+	0.875556i	$0.339505\pi$
$67$	14.3335i	1.75111i	−0.483117	+	0.875556i	$0.660495\pi$
$68$	0	0
$69$	6.63282	0.798497
$70$	0	0
$71$	−5.36974	−0.637271	−0.318635	−	0.947877i	$-0.603225\pi$
$71$	−5.36974	−0.637271	−0.318635	+	0.947877i	$0.603225\pi$
$72$	0	0
$73$	− 3.48067i	− 0.407382i	−0.979035	−	0.203691i	$-0.934706\pi$
$73$	− 3.48067i	− 0.407382i	0.979035	−	0.203691i	$-0.0652937\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 0.933955i	− 0.106434i
$78$	0	0
$79$	1.31999	0.148510	0.0742551	−	0.997239i	$-0.476342\pi$
$79$	1.31999	0.148510	0.0742551	+	0.997239i	$0.476342\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0.912590i	0.100170i	0.998745	+	0.0500849i	$0.0159492\pi$
$83$	0.912590i	0.100170i	−0.998745	+	0.0500849i	$0.984051\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	7.99711i	0.857381i
$88$	0	0
$89$	−11.7758	−1.24824	−0.624118	−	0.781330i	$-0.714541\pi$
$89$	−11.7758	−1.24824	−0.624118	+	0.781330i	$0.714541\pi$
$90$	0	0
$91$	−3.12920	−0.328030
$92$	0	0
$93$	− 4.88558i	− 0.506611i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	7.39984i	0.751340i	0.926754	+	0.375670i	$0.122587\pi$
$97$	7.39984i	0.751340i	−0.926754	+	0.375670i	$0.877413\pi$
$98$	0	0
$99$	1.82709	0.183630
$100$	0	0
$101$	13.0962	1.30312	0.651561	−	0.758596i	$-0.274114\pi$
$101$	13.0962	1.30312	0.651561	+	0.758596i	$0.274114\pi$
$102$	0	0
$103$	− 11.0302i	− 1.08684i	−0.839463	−	0.543418i	$-0.817130\pi$
$103$	− 11.0302i	− 1.08684i	0.839463	−	0.543418i	$-0.182870\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.08083i	0.781203i	0.920560	+	0.390602i	$0.127733\pi$
$107$	8.08083i	0.781203i	−0.920560	+	0.390602i	$0.872267\pi$
$108$	0	0
$109$	−11.3231	−1.08456	−0.542280	−	0.840198i	$-0.682439\pi$
$109$	−11.3231	−1.08456	−0.542280	+	0.840198i	$0.682439\pi$
$110$	0	0
$111$	−7.43757	−0.705943
$112$	0	0
$113$	3.76860i	0.354520i	0.984164	+	0.177260i	$0.0567234\pi$
$113$	3.76860i	0.354520i	−0.984164	+	0.177260i	$0.943277\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 6.12165i	− 0.565946i
$118$	0	0
$119$	1.32059	0.121058
$120$	0	0
$121$	−7.66174	−0.696522
$122$	0	0
$123$	5.73054i	0.516706i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	4.17193i	0.370199i	0.982720	+	0.185100i	$0.0592608\pi$
$127$	4.17193i	0.370199i	−0.982720	+	0.185100i	$0.940739\pi$
$128$	0	0
$129$	2.05126	0.180604
$130$	0	0
$131$	14.7908	1.29228	0.646140	−	0.763219i	$-0.276382\pi$
$131$	14.7908	1.29228	0.646140	+	0.763219i	$0.276382\pi$
$132$	0	0
$133$	2.48594i	0.215559i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1.22693i	0.104824i	0.998626	+	0.0524119i	$0.0166909\pi$
$137$	1.22693i	0.104824i	−0.998626	+	0.0524119i	$0.983309\pi$
$138$	0	0
$139$	−9.08570	−0.770639	−0.385320	−	0.922783i	$-0.625909\pi$
$139$	−9.08570	−0.770639	−0.385320	+	0.922783i	$0.625909\pi$
$140$	0	0
$141$	7.95439	0.669880
$142$	0	0
$143$	− 11.1848i	− 0.935320i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	6.73870i	0.555799i
$148$	0	0
$149$	−1.90097	−0.155733	−0.0778666	−	0.996964i	$-0.524811\pi$
$149$	−1.90097	−0.155733	−0.0778666	+	0.996964i	$0.524811\pi$
$150$	0	0
$151$	−4.60292	−0.374580	−0.187290	−	0.982305i	$-0.559970\pi$
$151$	−4.60292	−0.374580	−0.187290	+	0.982305i	$0.559970\pi$
$152$	0	0
$153$	2.58347i	0.208861i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 3.96076i	− 0.316103i	−0.987431	−	0.158052i	$-0.949479\pi$
$157$	− 3.96076i	− 0.316103i	0.987431	−	0.158052i	$-0.0505212\pi$
$158$	0	0
$159$	−1.32912	−0.105406
$160$	0	0
$161$	3.39050	0.267209
$162$	0	0
$163$	20.9795i	1.64324i	0.570035	+	0.821620i	$0.306929\pi$
$163$	20.9795i	1.64324i	−0.570035	+	0.821620i	$0.693071\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	18.0373i	1.39577i	0.716210	+	0.697885i	$0.245875\pi$
$167$	18.0373i	1.39577i	−0.716210	+	0.697885i	$0.754125\pi$
$168$	0	0
$169$	−24.4746	−1.88266
$170$	0	0
$171$	−4.86324	−0.371901
$172$	0	0
$173$	25.5975i	1.94614i	0.230503	+	0.973072i	$0.425963\pi$
$173$	25.5975i	1.94614i	−0.230503	+	0.973072i	$0.574037\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	10.0927i	0.758616i
$178$	0	0
$179$	8.41832	0.629215	0.314607	−	0.949222i	$-0.398127\pi$
$179$	8.41832	0.629215	0.314607	+	0.949222i	$0.398127\pi$
$180$	0	0
$181$	−20.0676	−1.49161	−0.745807	−	0.666162i	$-0.767936\pi$
$181$	−20.0676	−1.49161	−0.745807	+	0.666162i	$0.767936\pi$
$182$	0	0
$183$	4.77116i	0.352694i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	4.72023i	0.345177i
$188$	0	0
$189$	0.511170	0.0371822
$190$	0	0
$191$	3.86324	0.279534	0.139767	−	0.990184i	$-0.455365\pi$
$191$	3.86324	0.279534	0.139767	+	0.990184i	$0.455365\pi$
$192$	0	0
$193$	10.6925i	0.769662i	0.922987	+	0.384831i	$0.125740\pi$
$193$	10.6925i	0.769662i	−0.922987	+	0.384831i	$0.874260\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4.16442i	0.296703i	0.988935	+	0.148351i	$0.0473966\pi$
$197$	4.16442i	0.296703i	−0.988935	+	0.148351i	$0.952603\pi$
$198$	0	0
$199$	27.5965	1.95627	0.978134	−	0.207978i	$-0.0666882\pi$
$199$	27.5965	1.95627	0.978134	+	0.207978i	$0.0666882\pi$
$200$	0	0
$201$	−14.3335	−1.01100
$202$	0	0
$203$	4.08789i	0.286913i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	6.63282i	0.461013i
$208$	0	0
$209$	−8.88558	−0.614628
$210$	0	0
$211$	1.93489	0.133203	0.0666016	−	0.997780i	$-0.478784\pi$
$211$	1.93489	0.133203	0.0666016	+	0.997780i	$0.478784\pi$
$212$	0	0
$213$	− 5.36974i	− 0.367928i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 2.49736i	− 0.169532i
$218$	0	0
$219$	3.48067	0.235202
$220$	0	0
$221$	15.8151	1.06384
$222$	0	0
$223$	− 18.4077i	− 1.23267i	−0.787485	−	0.616334i	$-0.788617\pi$
$223$	− 18.4077i	− 1.23267i	0.787485	−	0.616334i	$-0.211383\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	19.2596i	1.27830i	0.769081	+	0.639152i	$0.220714\pi$
$227$	19.2596i	1.27830i	−0.769081	+	0.639152i	$0.779286\pi$
$228$	0	0
$229$	26.4417	1.74732	0.873658	−	0.486540i	$-0.161741\pi$
$229$	26.4417	1.74732	0.873658	+	0.486540i	$0.161741\pi$
$230$	0	0
$231$	0.933955	0.0614497
$232$	0	0
$233$	− 15.7608i	− 1.03253i	−0.856430	−	0.516264i	$-0.827323\pi$
$233$	− 15.7608i	− 1.03253i	0.856430	−	0.516264i	$-0.172677\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.31999i	0.0857424i
$238$	0	0
$239$	−18.3653	−1.18795	−0.593975	−	0.804483i	$-0.702442\pi$
$239$	−18.3653	−1.18795	−0.593975	+	0.804483i	$0.702442\pi$
$240$	0	0
$241$	−25.2904	−1.62910	−0.814549	−	0.580095i	$-0.803016\pi$
$241$	−25.2904	−1.62910	−0.814549	+	0.580095i	$0.803016\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	29.7710i	1.89429i
$248$	0	0
$249$	−0.912590	−0.0578331
$250$	0	0
$251$	−17.4297	−1.10015	−0.550076	−	0.835115i	$-0.685401\pi$
$251$	−17.4297	−1.10015	−0.550076	+	0.835115i	$0.685401\pi$
$252$	0	0
$253$	12.1188i	0.761900i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	29.6169i	1.84745i	0.383052	+	0.923727i	$0.374873\pi$
$257$	29.6169i	1.84745i	−0.383052	+	0.923727i	$0.625127\pi$
$258$	0	0
$259$	−3.80186	−0.236236
$260$	0	0
$261$	−7.99711	−0.495009
$262$	0	0
$263$	− 16.7350i	− 1.03192i	−0.856611	−	0.515962i	$-0.827434\pi$
$263$	− 16.7350i	− 1.03192i	0.856611	−	0.515962i	$-0.172566\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 11.7758i	− 0.720669i
$268$	0	0
$269$	12.7897	0.779800	0.389900	−	0.920857i	$-0.372509\pi$
$269$	12.7897	0.779800	0.389900	+	0.920857i	$0.372509\pi$
$270$	0	0
$271$	28.6724	1.74172	0.870861	−	0.491529i	$-0.163562\pi$
$271$	28.6724	1.74172	0.870861	+	0.491529i	$0.163562\pi$
$272$	0	0
$273$	− 3.12920i	− 0.189388i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 5.33733i	− 0.320689i	−0.987061	−	0.160344i	$-0.948739\pi$
$277$	− 5.33733i	− 0.320689i	0.987061	−	0.160344i	$-0.0512605\pi$
$278$	0	0
$279$	4.88558	0.292492
$280$	0	0
$281$	−6.42725	−0.383418	−0.191709	−	0.981452i	$-0.561403\pi$
$281$	−6.42725	−0.383418	−0.191709	+	0.981452i	$0.561403\pi$
$282$	0	0
$283$	− 10.4636i	− 0.621999i	−0.950410	−	0.311000i	$-0.899336\pi$
$283$	− 10.4636i	− 0.621999i	0.950410	−	0.311000i	$-0.100664\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.92928i	0.172910i
$288$	0	0
$289$	10.3257	0.607394
$290$	0	0
$291$	−7.39984	−0.433786
$292$	0	0
$293$	4.63761i	0.270932i	0.990782	+	0.135466i	$0.0432532\pi$
$293$	4.63761i	0.270932i	−0.990782	+	0.135466i	$0.956747\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.82709i	0.106019i
$298$	0	0
$299$	40.6038	2.34818
$300$	0	0
$301$	1.04854	0.0604371
$302$	0	0
$303$	13.0962i	0.752358i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 17.5664i	− 1.00257i	−0.865282	−	0.501285i	$-0.832861\pi$
$307$	− 17.5664i	− 1.00257i	0.865282	−	0.501285i	$-0.167139\pi$
$308$	0	0
$309$	11.0302	0.627484
$310$	0	0
$311$	18.3649	1.04138	0.520689	−	0.853746i	$-0.325675\pi$
$311$	18.3649	1.04138	0.520689	+	0.853746i	$0.325675\pi$
$312$	0	0
$313$	− 25.3537i	− 1.43308i	−0.697548	−	0.716538i	$-0.745726\pi$
$313$	− 25.3537i	− 1.43308i	0.697548	−	0.716538i	$-0.254274\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	23.6513i	1.32839i	0.747560	+	0.664195i	$0.231225\pi$
$317$	23.6513i	1.32839i	−0.747560	+	0.664195i	$0.768775\pi$
$318$	0	0
$319$	−14.6115	−0.818084
$320$	0	0
$321$	−8.08083	−0.451028
$322$	0	0
$323$	− 12.5640i	− 0.699080i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 11.3231i	− 0.626171i
$328$	0	0
$329$	4.06605	0.224168
$330$	0	0
$331$	32.9027	1.80849	0.904247	−	0.427010i	$-0.140433\pi$
$331$	32.9027	1.80849	0.904247	+	0.427010i	$0.140433\pi$
$332$	0	0
$333$	− 7.43757i	− 0.407576i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 5.20406i	− 0.283483i	−0.989904	−	0.141742i	$-0.954730\pi$
$337$	− 5.20406i	− 0.283483i	0.989904	−	0.141742i	$-0.0452702\pi$
$338$	0	0
$339$	−3.76860	−0.204682
$340$	0	0
$341$	8.92640	0.483392
$342$	0	0
$343$	7.02282i	0.379197i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 3.34293i	− 0.179458i	−0.995966	−	0.0897290i	$-0.971400\pi$
$347$	− 3.34293i	− 0.179458i	0.995966	−	0.0897290i	$-0.0286001\pi$
$348$	0	0
$349$	28.9138	1.54772	0.773859	−	0.633358i	$-0.218324\pi$
$349$	28.9138	1.54772	0.773859	+	0.633358i	$0.218324\pi$
$350$	0	0
$351$	6.12165	0.326749
$352$	0	0
$353$	20.3359i	1.08237i	0.840903	+	0.541185i	$0.182024\pi$
$353$	20.3359i	1.08237i	−0.840903	+	0.541185i	$0.817976\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	1.32059i	0.0698931i
$358$	0	0
$359$	−37.2014	−1.96342	−0.981708	−	0.190394i	$-0.939024\pi$
$359$	−37.2014	−1.96342	−0.981708	+	0.190394i	$0.939024\pi$
$360$	0	0
$361$	4.65109	0.244794
$362$	0	0
$363$	− 7.66174i	− 0.402137i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 3.68079i	− 0.192136i	−0.995375	−	0.0960679i	$-0.969373\pi$
$367$	− 3.68079i	− 0.192136i	0.995375	−	0.0960679i	$-0.0306266\pi$
$368$	0	0
$369$	−5.73054	−0.298320
$370$	0	0
$371$	−0.679409	−0.0352732
$372$	0	0
$373$	20.3589i	1.05414i	0.849821	+	0.527072i	$0.176710\pi$
$373$	20.3589i	1.05414i	−0.849821	+	0.527072i	$0.823290\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	48.9555i	2.52134i
$378$	0	0
$379$	6.39635	0.328558	0.164279	−	0.986414i	$-0.447470\pi$
$379$	6.39635	0.328558	0.164279	+	0.986414i	$0.447470\pi$
$380$	0	0
$381$	−4.17193	−0.213735
$382$	0	0
$383$	− 2.46938i	− 0.126179i	−0.998008	−	0.0630896i	$-0.979905\pi$
$383$	− 2.46938i	− 0.126179i	0.998008	−	0.0630896i	$-0.0200954\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2.05126i	0.104271i
$388$	0	0
$389$	−14.5718	−0.738818	−0.369409	−	0.929267i	$-0.620440\pi$
$389$	−14.5718	−0.738818	−0.369409	+	0.929267i	$0.620440\pi$
$390$	0	0
$391$	−17.1357	−0.866588
$392$	0	0
$393$	14.7908i	0.746098i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	24.7242i	1.24087i	0.784258	+	0.620435i	$0.213044\pi$
$397$	24.7242i	1.24087i	−0.784258	+	0.620435i	$0.786956\pi$
$398$	0	0
$399$	−2.48594	−0.124453
$400$	0	0
$401$	6.50743	0.324966	0.162483	−	0.986711i	$-0.448050\pi$
$401$	6.50743	0.324966	0.162483	+	0.986711i	$0.448050\pi$
$402$	0	0
$403$	− 29.9078i	− 1.48981i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 13.5891i	− 0.673587i
$408$	0	0
$409$	−11.0490	−0.546339	−0.273169	−	0.961966i	$-0.588072\pi$
$409$	−11.0490	−0.546339	−0.273169	+	0.961966i	$0.588072\pi$
$410$	0	0
$411$	−1.22693	−0.0605201
$412$	0	0
$413$	5.15910i	0.253863i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 9.08570i	− 0.444929i
$418$	0	0
$419$	−18.6090	−0.909107	−0.454554	−	0.890719i	$-0.650201\pi$
$419$	−18.6090	−0.909107	−0.454554	+	0.890719i	$0.650201\pi$
$420$	0	0
$421$	−13.6494	−0.665230	−0.332615	−	0.943063i	$-0.607931\pi$
$421$	−13.6494	−0.665230	−0.332615	+	0.943063i	$0.607931\pi$
$422$	0	0
$423$	7.95439i	0.386755i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	2.43887i	0.118025i
$428$	0	0
$429$	11.1848	0.540008
$430$	0	0
$431$	34.2271	1.64866	0.824330	−	0.566109i	$-0.191552\pi$
$431$	34.2271	1.64866	0.824330	+	0.566109i	$0.191552\pi$
$432$	0	0
$433$	9.02161i	0.433551i	0.976221	+	0.216775i	$0.0695540\pi$
$433$	9.02161i	0.433551i	−0.976221	+	0.216775i	$0.930446\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 32.2570i	− 1.54306i
$438$	0	0
$439$	−19.0068	−0.907146	−0.453573	−	0.891219i	$-0.649851\pi$
$439$	−19.0068	−0.907146	−0.453573	+	0.891219i	$0.649851\pi$
$440$	0	0
$441$	−6.73870	−0.320891
$442$	0	0
$443$	− 1.68124i	− 0.0798779i	−0.999202	−	0.0399390i	$-0.987284\pi$
$443$	− 1.68124i	− 0.0798779i	0.999202	−	0.0399390i	$-0.0127163\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 1.90097i	− 0.0899126i
$448$	0	0
$449$	14.1334	0.666997	0.333499	−	0.942751i	$-0.391771\pi$
$449$	14.1334	0.666997	0.333499	+	0.942751i	$0.391771\pi$
$450$	0	0
$451$	−10.4702	−0.493024
$452$	0	0
$453$	− 4.60292i	− 0.216264i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	24.8188i	1.16097i	0.814269	+	0.580487i	$0.197138\pi$
$457$	24.8188i	1.16097i	−0.814269	+	0.580487i	$0.802862\pi$
$458$	0	0
$459$	−2.58347	−0.120586
$460$	0	0
$461$	−33.8628	−1.57715	−0.788574	−	0.614941i	$-0.789180\pi$
$461$	−33.8628	−1.57715	−0.788574	+	0.614941i	$0.789180\pi$
$462$	0	0
$463$	2.73777i	0.127235i	0.997974	+	0.0636175i	$0.0202638\pi$
$463$	2.73777i	0.127235i	−0.997974	+	0.0636175i	$0.979736\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	34.6764i	1.60463i	0.596899	+	0.802316i	$0.296399\pi$
$467$	34.6764i	1.60463i	−0.596899	+	0.802316i	$0.703601\pi$
$468$	0	0
$469$	−7.32684	−0.338322
$470$	0	0
$471$	3.96076	0.182502
$472$	0	0
$473$	3.74784i	0.172326i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 1.32912i	− 0.0608564i
$478$	0	0
$479$	−5.46003	−0.249475	−0.124738	−	0.992190i	$-0.539809\pi$
$479$	−5.46003	−0.249475	−0.124738	+	0.992190i	$0.539809\pi$
$480$	0	0
$481$	−45.5302	−2.07600
$482$	0	0
$483$	3.39050i	0.154273i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	23.6023i	1.06952i	0.845003	+	0.534762i	$0.179599\pi$
$487$	23.6023i	1.06952i	−0.845003	+	0.534762i	$0.820401\pi$
$488$	0	0
$489$	−20.9795	−0.948725
$490$	0	0
$491$	13.2009	0.595748	0.297874	−	0.954605i	$-0.403722\pi$
$491$	13.2009	0.595748	0.297874	+	0.954605i	$0.403722\pi$
$492$	0	0
$493$	− 20.6603i	− 0.930492i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 2.74485i	− 0.123123i
$498$	0	0
$499$	−25.5183	−1.14235	−0.571177	−	0.820827i	$-0.693513\pi$
$499$	−25.5183	−1.14235	−0.571177	+	0.820827i	$0.693513\pi$
$500$	0	0
$501$	−18.0373	−0.805848
$502$	0	0
$503$	− 29.1880i	− 1.30143i	−0.759323	−	0.650714i	$-0.774470\pi$
$503$	− 29.1880i	− 1.30143i	0.759323	−	0.650714i	$-0.225530\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 24.4746i	− 1.08695i
$508$	0	0
$509$	−28.7214	−1.27305	−0.636527	−	0.771254i	$-0.719630\pi$
$509$	−28.7214	−1.27305	−0.636527	+	0.771254i	$0.719630\pi$
$510$	0	0
$511$	1.77921	0.0787078
$512$	0	0
$513$	− 4.86324i	− 0.214717i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	14.5334i	0.639178i
$518$	0	0
$519$	−25.5975	−1.12361
$520$	0	0
$521$	34.1775	1.49734	0.748672	−	0.662941i	$-0.230692\pi$
$521$	34.1775	1.49734	0.748672	+	0.662941i	$0.230692\pi$
$522$	0	0
$523$	15.0569i	0.658390i	0.944262	+	0.329195i	$0.106777\pi$
$523$	15.0569i	0.658390i	−0.944262	+	0.329195i	$0.893223\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	12.6217i	0.549811i
$528$	0	0
$529$	−20.9943	−0.912794
$530$	0	0
$531$	−10.0927	−0.437987
$532$	0	0
$533$	35.0804i	1.51950i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	8.41832i	0.363277i
$538$	0	0
$539$	−12.3122	−0.530325
$540$	0	0
$541$	43.8477	1.88516	0.942579	−	0.333983i	$-0.108393\pi$
$541$	43.8477	1.88516	0.942579	+	0.333983i	$0.108393\pi$
$542$	0	0
$543$	− 20.0676i	− 0.861184i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	10.3709i	0.443428i	0.975112	+	0.221714i	$0.0711652\pi$
$547$	10.3709i	0.443428i	−0.975112	+	0.221714i	$0.928835\pi$
$548$	0	0
$549$	−4.77116	−0.203628
$550$	0	0
$551$	38.8919	1.65685
$552$	0	0
$553$	0.674738i	0.0286928i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	24.3856i	1.03325i	0.856212	+	0.516625i	$0.172812\pi$
$557$	24.3856i	1.03325i	−0.856212	+	0.516625i	$0.827188\pi$
$558$	0	0
$559$	12.5571	0.531109
$560$	0	0
$561$	−4.72023	−0.199288
$562$	0	0
$563$	− 28.7426i	− 1.21136i	−0.795709	−	0.605679i	$-0.792902\pi$
$563$	− 28.7426i	− 1.21136i	0.795709	−	0.605679i	$-0.207098\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0.511170i	0.0214671i
$568$	0	0
$569$	−17.4992	−0.733604	−0.366802	−	0.930299i	$-0.619547\pi$
$569$	−17.4992	−0.733604	−0.366802	+	0.930299i	$0.619547\pi$
$570$	0	0
$571$	−19.1736	−0.802388	−0.401194	−	0.915993i	$-0.631405\pi$
$571$	−19.1736	−0.802388	−0.401194	+	0.915993i	$0.631405\pi$
$572$	0	0
$573$	3.86324i	0.161389i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 9.28517i	− 0.386547i	−0.981145	−	0.193273i	$-0.938090\pi$
$577$	− 9.28517i	− 0.386547i	0.981145	−	0.193273i	$-0.0619104\pi$
$578$	0	0
$579$	−10.6925	−0.444365
$580$	0	0
$581$	−0.466489	−0.0193532
$582$	0	0
$583$	− 2.42843i	− 0.100575i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	17.7035i	0.730701i	0.930870	+	0.365350i	$0.119051\pi$
$587$	17.7035i	0.730701i	−0.930870	+	0.365350i	$0.880949\pi$
$588$	0	0
$589$	−23.7597	−0.979003
$590$	0	0
$591$	−4.16442	−0.171301
$592$	0	0
$593$	25.8975i	1.06348i	0.846907	+	0.531742i	$0.178462\pi$
$593$	25.8975i	1.06348i	−0.846907	+	0.531742i	$0.821538\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	27.5965i	1.12945i
$598$	0	0
$599$	−10.0259	−0.409646	−0.204823	−	0.978799i	$-0.565662\pi$
$599$	−10.0259	−0.409646	−0.204823	+	0.978799i	$0.565662\pi$
$600$	0	0
$601$	−1.69846	−0.0692818	−0.0346409	−	0.999400i	$-0.511029\pi$
$601$	−1.69846	−0.0692818	−0.0346409	+	0.999400i	$0.511029\pi$
$602$	0	0
$603$	− 14.3335i	− 0.583704i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	13.3043i	0.540005i	0.962860	+	0.270002i	$0.0870245\pi$
$607$	13.3043i	0.540005i	−0.962860	+	0.270002i	$0.912976\pi$
$608$	0	0
$609$	−4.08789	−0.165650
$610$	0	0
$611$	48.6939	1.96995
$612$	0	0
$613$	− 7.02832i	− 0.283871i	−0.989876	−	0.141936i	$-0.954667\pi$
$613$	− 7.02832i	− 0.283871i	0.989876	−	0.141936i	$-0.0453326\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	9.94014i	0.400175i	0.979778	+	0.200087i	$0.0641226\pi$
$617$	9.94014i	0.400175i	−0.979778	+	0.200087i	$0.935877\pi$
$618$	0	0
$619$	25.0631	1.00737	0.503686	−	0.863887i	$-0.331977\pi$
$619$	25.0631	1.00737	0.503686	+	0.863887i	$0.331977\pi$
$620$	0	0
$621$	−6.63282	−0.266166
$622$	0	0
$623$	− 6.01945i	− 0.241164i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 8.88558i	− 0.354856i
$628$	0	0
$629$	19.2147	0.766140
$630$	0	0
$631$	10.1433	0.403800	0.201900	−	0.979406i	$-0.435288\pi$
$631$	10.1433	0.403800	0.201900	+	0.979406i	$0.435288\pi$
$632$	0	0
$633$	1.93489i	0.0769049i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	41.2520i	1.63446i
$638$	0	0
$639$	5.36974	0.212424
$640$	0	0
$641$	14.6798	0.579818	0.289909	−	0.957054i	$-0.406375\pi$
$641$	14.6798	0.579818	0.289909	+	0.957054i	$0.406375\pi$
$642$	0	0
$643$	8.52311i	0.336119i	0.985777	+	0.168059i	$0.0537500\pi$
$643$	8.52311i	0.336119i	−0.985777	+	0.168059i	$0.946250\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 38.2446i	− 1.50355i	−0.659421	−	0.751774i	$-0.729198\pi$
$647$	− 38.2446i	− 1.50355i	0.659421	−	0.751774i	$-0.270802\pi$
$648$	0	0
$649$	−18.4403	−0.723846
$650$	0	0
$651$	2.49736	0.0978794
$652$	0	0
$653$	− 8.39715i	− 0.328606i	−0.986410	−	0.164303i	$-0.947463\pi$
$653$	− 8.39715i	− 0.328606i	0.986410	−	0.164303i	$-0.0525375\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	3.48067i	0.135794i
$658$	0	0
$659$	47.2314	1.83987	0.919937	−	0.392066i	$-0.128240\pi$
$659$	47.2314	1.83987	0.919937	+	0.392066i	$0.128240\pi$
$660$	0	0
$661$	−25.3165	−0.984698	−0.492349	−	0.870398i	$-0.663862\pi$
$661$	−25.3165	−0.984698	−0.492349	+	0.870398i	$0.663862\pi$
$662$	0	0
$663$	15.8151i	0.614206i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 53.0434i	− 2.05385i
$668$	0	0
$669$	18.4077	0.711682
$670$	0	0
$671$	−8.71734	−0.336529
$672$	0	0
$673$	− 15.8549i	− 0.611160i	−0.952166	−	0.305580i	$-0.901150\pi$
$673$	− 15.8549i	− 0.611160i	0.952166	−	0.305580i	$-0.0988504\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 34.5194i	− 1.32669i	−0.748314	−	0.663345i	$-0.769136\pi$
$677$	− 34.5194i	− 1.32669i	0.748314	−	0.663345i	$-0.230864\pi$
$678$	0	0
$679$	−3.78258	−0.145162
$680$	0	0
$681$	−19.2596	−0.738029
$682$	0	0
$683$	25.0403i	0.958140i	0.877777	+	0.479070i	$0.159026\pi$
$683$	25.0403i	0.958140i	−0.877777	+	0.479070i	$0.840974\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	26.4417i	1.00881i
$688$	0	0
$689$	−8.13643	−0.309973
$690$	0	0
$691$	−27.5020	−1.04623	−0.523113	−	0.852263i	$-0.675230\pi$
$691$	−27.5020	−1.04623	−0.523113	+	0.852263i	$0.675230\pi$
$692$	0	0
$693$	0.933955i	0.0354780i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 14.8047i	− 0.560767i
$698$	0	0
$699$	15.7608	0.596130
$700$	0	0
$701$	−5.38643	−0.203443	−0.101721	−	0.994813i	$-0.532435\pi$
$701$	−5.38643	−0.203443	−0.101721	+	0.994813i	$0.532435\pi$
$702$	0	0
$703$	36.1707i	1.36420i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6.69440i	0.251769i
$708$	0	0
$709$	−20.4847	−0.769320	−0.384660	−	0.923058i	$-0.625681\pi$
$709$	−20.4847	−0.769320	−0.384660	+	0.923058i	$0.625681\pi$
$710$	0	0
$711$	−1.31999	−0.0495034
$712$	0	0
$713$	32.4052i	1.21358i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 18.3653i	− 0.685864i
$718$	0	0
$719$	34.1442	1.27336	0.636681	−	0.771127i	$-0.280307\pi$
$719$	34.1442	1.27336	0.636681	+	0.771127i	$0.280307\pi$
$720$	0	0
$721$	5.63830	0.209981
$722$	0	0
$723$	− 25.2904i	− 0.940560i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	30.6343i	1.13616i	0.822972	+	0.568082i	$0.192314\pi$
$727$	30.6343i	1.13616i	−0.822972	+	0.568082i	$0.807686\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−5.29936	−0.196004
$732$	0	0
$733$	36.9642i	1.36530i	0.730744	+	0.682651i	$0.239173\pi$
$733$	36.9642i	1.36530i	−0.730744	+	0.682651i	$0.760827\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 26.1885i	− 0.964667i
$738$	0	0
$739$	−33.1345	−1.21887	−0.609435	−	0.792836i	$-0.708604\pi$
$739$	−33.1345	−1.21887	−0.609435	+	0.792836i	$0.708604\pi$
$740$	0	0
$741$	−29.7710	−1.09367
$742$	0	0
$743$	18.9855i	0.696512i	0.937400	+	0.348256i	$0.113226\pi$
$743$	18.9855i	0.696512i	−0.937400	+	0.348256i	$0.886774\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 0.912590i	− 0.0333899i
$748$	0	0
$749$	−4.13068	−0.150932
$750$	0	0
$751$	−7.10651	−0.259320	−0.129660	−	0.991559i	$-0.541389\pi$
$751$	−7.10651	−0.259320	−0.129660	+	0.991559i	$0.541389\pi$
$752$	0	0
$753$	− 17.4297i	− 0.635173i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	8.77180i	0.318817i	0.987213	+	0.159408i	$0.0509586\pi$
$757$	8.77180i	0.318817i	−0.987213	+	0.159408i	$0.949041\pi$
$758$	0	0
$759$	−12.1188	−0.439883
$760$	0	0
$761$	20.7407	0.751848	0.375924	−	0.926650i	$-0.377325\pi$
$761$	20.7407	0.751848	0.375924	+	0.926650i	$0.377325\pi$
$762$	0	0
$763$	− 5.78806i	− 0.209542i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	61.7841i	2.23089i
$768$	0	0
$769$	13.7109	0.494427	0.247214	−	0.968961i	$-0.420485\pi$
$769$	13.7109	0.494427	0.247214	+	0.968961i	$0.420485\pi$
$770$	0	0
$771$	−29.6169	−1.06663
$772$	0	0
$773$	− 26.8751i	− 0.966628i	−0.875447	−	0.483314i	$-0.839433\pi$
$773$	− 26.8751i	− 0.966628i	0.875447	−	0.483314i	$-0.160567\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 3.80186i	− 0.136391i
$778$	0	0
$779$	27.8690	0.998511
$780$	0	0
$781$	9.81100	0.351065
$782$	0	0
$783$	− 7.99711i	− 0.285794i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1.45322i	0.0518016i	0.999665	+	0.0259008i	$0.00824540\pi$
$787$	1.45322i	0.0518016i	−0.999665	+	0.0259008i	$0.991755\pi$
$788$	0	0
$789$	16.7350	0.595782
$790$	0	0
$791$	−1.92640	−0.0684948
$792$	0	0
$793$	29.2074i	1.03718i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 49.4322i	− 1.75098i	−0.483237	−	0.875490i	$-0.660539\pi$
$797$	− 49.4322i	− 1.75098i	0.483237	−	0.875490i	$-0.339461\pi$
$798$	0	0
$799$	−20.5499	−0.727003
$800$	0	0
$801$	11.7758	0.416078
$802$	0	0
$803$	6.35950i	0.224422i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	12.7897i	0.450218i
$808$	0	0
$809$	42.7171	1.50185	0.750927	−	0.660385i	$-0.229607\pi$
$809$	42.7171	1.50185	0.750927	+	0.660385i	$0.229607\pi$
$810$	0	0
$811$	−34.9012	−1.22555	−0.612774	−	0.790258i	$-0.709946\pi$
$811$	−34.9012	−1.22555	−0.612774	+	0.790258i	$0.709946\pi$
$812$	0	0
$813$	28.6724i	1.00558i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 9.97578i	− 0.349008i
$818$	0	0
$819$	3.12920	0.109343
$820$	0	0
$821$	−0.260235	−0.00908227	−0.00454114	−	0.999990i	$-0.501445\pi$
$821$	−0.260235	−0.00908227	−0.00454114	+	0.999990i	$0.501445\pi$
$822$	0	0
$823$	− 23.1099i	− 0.805560i	−0.915297	−	0.402780i	$-0.868044\pi$
$823$	− 23.1099i	− 0.805560i	0.915297	−	0.402780i	$-0.131956\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 6.37150i	− 0.221559i	−0.993845	−	0.110779i	$-0.964665\pi$
$827$	− 6.37150i	− 0.221559i	0.993845	−	0.110779i	$-0.0353347\pi$
$828$	0	0
$829$	25.1774	0.874446	0.437223	−	0.899353i	$-0.355962\pi$
$829$	25.1774	0.874446	0.437223	+	0.899353i	$0.355962\pi$
$830$	0	0
$831$	5.33733	0.185150
$832$	0	0
$833$	− 17.4092i	− 0.603194i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	4.88558i	0.168870i
$838$	0	0
$839$	40.6296	1.40269	0.701346	−	0.712821i	$-0.252583\pi$
$839$	40.6296	1.40269	0.701346	+	0.712821i	$0.252583\pi$
$840$	0	0
$841$	34.9538	1.20530
$842$	0	0
$843$	− 6.42725i	− 0.221366i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 3.91645i	− 0.134571i
$848$	0	0
$849$	10.4636	0.359111
$850$	0	0
$851$	49.3320	1.69108
$852$	0	0
$853$	26.1351i	0.894848i	0.894322	+	0.447424i	$0.147659\pi$
$853$	26.1351i	0.894848i	−0.894322	+	0.447424i	$0.852341\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 30.0649i	− 1.02700i	−0.858091	−	0.513498i	$-0.828349\pi$
$857$	− 30.0649i	− 1.02700i	0.858091	−	0.513498i	$-0.171651\pi$
$858$	0	0
$859$	−10.0107	−0.341562	−0.170781	−	0.985309i	$-0.554629\pi$
$859$	−10.0107	−0.341562	−0.170781	+	0.985309i	$0.554629\pi$
$860$	0	0
$861$	−2.92928	−0.0998297
$862$	0	0
$863$	12.4877i	0.425087i	0.977152	+	0.212543i	$0.0681747\pi$
$863$	12.4877i	0.425087i	−0.977152	+	0.212543i	$0.931825\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	10.3257i	0.350679i
$868$	0	0
$869$	−2.41174	−0.0818126
$870$	0	0
$871$	−87.7444	−2.97311
$872$	0	0
$873$	− 7.39984i	− 0.250447i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 35.1337i	− 1.18638i	−0.805062	−	0.593191i	$-0.797868\pi$
$877$	− 35.1337i	− 1.18638i	0.805062	−	0.593191i	$-0.202132\pi$
$878$	0	0
$879$	−4.63761	−0.156423
$880$	0	0
$881$	−6.88011	−0.231797	−0.115898	−	0.993261i	$-0.536975\pi$
$881$	−6.88011	−0.231797	−0.115898	+	0.993261i	$0.536975\pi$
$882$	0	0
$883$	− 18.7383i	− 0.630594i	−0.948993	−	0.315297i	$-0.897896\pi$
$883$	− 18.7383i	− 0.630594i	0.948993	−	0.315297i	$-0.102104\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 29.7928i	− 1.00034i	−0.865926	−	0.500172i	$-0.833270\pi$
$887$	− 29.7928i	− 1.00034i	0.865926	−	0.500172i	$-0.166730\pi$
$888$	0	0
$889$	−2.13257	−0.0715240
$890$	0	0
$891$	−1.82709	−0.0612098
$892$	0	0
$893$	− 38.6841i	− 1.29451i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	40.6038i	1.35572i
$898$	0	0
$899$	−39.0705	−1.30308
$900$	0	0
$901$	3.43375	0.114395
$902$	0	0
$903$	1.04854i	0.0348934i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	14.5112i	0.481838i	0.970545	+	0.240919i	$0.0774488\pi$
$907$	14.5112i	0.481838i	−0.970545	+	0.240919i	$0.922551\pi$
$908$	0	0
$909$	−13.0962	−0.434374
$910$	0	0
$911$	33.8659	1.12203	0.561014	−	0.827806i	$-0.310411\pi$
$911$	33.8659	1.12203	0.561014	+	0.827806i	$0.310411\pi$
$912$	0	0
$913$	− 1.66739i	− 0.0551824i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	7.56063i	0.249674i
$918$	0	0
$919$	31.6348	1.04354	0.521769	−	0.853087i	$-0.325272\pi$
$919$	31.6348	1.04354	0.521769	+	0.853087i	$0.325272\pi$
$920$	0	0
$921$	17.5664	0.578834
$922$	0	0
$923$	− 32.8716i	− 1.08198i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	11.0302i	0.362278i
$928$	0	0
$929$	−44.3560	−1.45527	−0.727637	−	0.685962i	$-0.759382\pi$
$929$	−44.3560	−1.45527	−0.727637	+	0.685962i	$0.759382\pi$
$930$	0	0
$931$	32.7719	1.07406
$932$	0	0
$933$	18.3649i	0.601240i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	26.2950i	0.859021i	0.903062	+	0.429511i	$0.141314\pi$
$937$	26.2950i	0.859021i	−0.903062	+	0.429511i	$0.858686\pi$
$938$	0	0
$939$	25.3537	0.827386
$940$	0	0
$941$	14.9037	0.485846	0.242923	−	0.970046i	$-0.421894\pi$
$941$	14.9037	0.485846	0.242923	+	0.970046i	$0.421894\pi$
$942$	0	0
$943$	− 38.0097i	− 1.23776i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 10.5532i	− 0.342932i	−0.985190	−	0.171466i	$-0.945150\pi$
$947$	− 10.5532i	− 0.342932i	0.985190	−	0.171466i	$-0.0548504\pi$
$948$	0	0
$949$	21.3074	0.691668
$950$	0	0
$951$	−23.6513	−0.766946
$952$	0	0
$953$	− 26.8445i	− 0.869579i	−0.900532	−	0.434789i	$-0.856823\pi$
$953$	− 26.8445i	− 0.869579i	0.900532	−	0.434789i	$-0.143177\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 14.6115i	− 0.472321i
$958$	0	0
$959$	−0.627171	−0.0202524
$960$	0	0
$961$	−7.13111	−0.230036
$962$	0	0
$963$	− 8.08083i	− 0.260401i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 50.5675i	− 1.62614i	−0.582165	−	0.813071i	$-0.697794\pi$
$967$	− 50.5675i	− 1.62614i	0.582165	−	0.813071i	$-0.302206\pi$
$968$	0	0
$969$	12.5640	0.403614
$970$	0	0
$971$	26.3331	0.845070	0.422535	−	0.906346i	$-0.361140\pi$
$971$	26.3331	0.845070	0.422535	+	0.906346i	$0.361140\pi$
$972$	0	0
$973$	− 4.64434i	− 0.148891i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	6.36917i	0.203768i	0.994796	+	0.101884i	$0.0324870\pi$
$977$	6.36917i	0.203768i	−0.994796	+	0.101884i	$0.967513\pi$
$978$	0	0
$979$	21.5155	0.687639
$980$	0	0
$981$	11.3231	0.361520
$982$	0	0
$983$	− 56.8285i	− 1.81255i	−0.422690	−	0.906274i	$-0.638914\pi$
$983$	− 56.8285i	− 1.81255i	0.422690	−	0.906274i	$-0.361086\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	4.06605i	0.129424i
$988$	0	0
$989$	−13.6056	−0.432634
$990$	0	0
$991$	13.3244	0.423263	0.211631	−	0.977350i	$-0.432122\pi$
$991$	13.3244	0.423263	0.211631	+	0.977350i	$0.432122\pi$
$992$	0	0
$993$	32.9027i	1.04413i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 25.5938i	− 0.810565i	−0.914192	−	0.405282i	$-0.867173\pi$
$997$	− 25.5938i	− 0.810565i	0.914192	−	0.405282i	$-0.132827\pi$
$998$	0	0
$999$	7.43757	0.235314

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7500.2.d.d.1249.7		8
5.2	odd	4		7500.2.a.g.1.2		4
5.3	odd	4		7500.2.a.d.1.3		4
5.4	even	2	inner	7500.2.d.d.1249.2		8
25.2	odd	20		300.2.m.a.121.1	✓	8
25.9	even	10		1500.2.o.a.349.3		16
25.11	even	5		1500.2.o.a.649.4		16
25.12	odd	20		300.2.m.a.181.1	yes	8
25.13	odd	20		1500.2.m.b.901.2		8
25.14	even	10		1500.2.o.a.649.1		16
25.16	even	5		1500.2.o.a.349.2		16
25.23	odd	20		1500.2.m.b.601.2		8
75.2	even	20		900.2.n.a.721.2		8
75.62	even	20		900.2.n.a.181.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.2.m.a.121.1	✓	8	25.2	odd	20
300.2.m.a.181.1	yes	8	25.12	odd	20
900.2.n.a.181.2		8	75.62	even	20
900.2.n.a.721.2		8	75.2	even	20
1500.2.m.b.601.2		8	25.23	odd	20
1500.2.m.b.901.2		8	25.13	odd	20
1500.2.o.a.349.2		16	25.16	even	5
1500.2.o.a.349.3		16	25.9	even	10
1500.2.o.a.649.1		16	25.14	even	10
1500.2.o.a.649.4		16	25.11	even	5
7500.2.a.d.1.3		4	5.3	odd	4
7500.2.a.g.1.2		4	5.2	odd	4
7500.2.d.d.1249.2		8	5.4	even	2	inner
7500.2.d.d.1249.7		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7500 = 2^{2} \cdot 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7500.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +0.511170i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +0.511170i q^{7} -1.00000 q^{9} -1.82709 q^{11} +6.12165i q^{13} -2.58347i q^{17} +4.86324 q^{19} -0.511170 q^{21} -6.63282i q^{23} -1.00000i q^{27} +7.99711 q^{29} -4.88558 q^{31} -1.82709i q^{33} +7.43757i q^{37} -6.12165 q^{39} +5.73054 q^{41} -2.05126i q^{43} -7.95439i q^{47} +6.73870 q^{49} +2.58347 q^{51} +1.32912i q^{53} +4.86324i q^{57} +10.0927 q^{59} +4.77116 q^{61} -0.511170i q^{63} +14.3335i q^{67} +6.63282 q^{69} -5.36974 q^{71} -3.48067i q^{73} -0.933955i q^{77} +1.31999 q^{79} +1.00000 q^{81} +0.912590i q^{83} +7.99711i q^{87} -11.7758 q^{89} -3.12920 q^{91} -4.88558i q^{93} +7.39984i q^{97} +1.82709 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^9	\( 8 q - 8 q^{9} - 2 q^{11} + 10 q^{19} - 8 q^{21} + 8 q^{29} - 18 q^{31} - 10 q^{39} + 8 q^{49} - 8 q^{51} - 2 q^{59} - 4 q^{61} + 18 q^{69} - 40 q^{71} + 6 q^{79} + 8 q^{81} - 30 q^{89} - 20 q^{91} + 2 q^{99}+O(q^{100}) \) 8 * q - 8 * q^9 - 2 * q^11 + 10 * q^19 - 8 * q^21 + 8 * q^29 - 18 * q^31 - 10 * q^39 + 8 * q^49 - 8 * q^51 - 2 * q^59 - 4 * q^61 + 18 * q^69 - 40 * q^71 + 6 * q^79 + 8 * q^81 - 30 * q^89 - 20 * q^91 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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