LMFDB - Embedded newform 304.7.e.d.113.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [304,7,Mod(113,304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(304, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("304.113");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 304 = 2^{4} \cdot 19 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	304.e (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:	$69.9364414204$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 483x^{6} + 75582x^{4} + 4242376x^{2} + 71047680 $ x^8 + 483x^6 + 75582x^4 + 4242376*x^2 + 71047680
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{11}\cdot 29 $
Twist minimal:	no (minimal twist has level 19)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			113.1
Root			$5.43437i$ of defining polynomial
Character	$\chi$	$=$	304.113
Dual form			304.7.e.d.113.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-33.7437i q^{3} -51.7597 q^{5} +313.488 q^{7} -409.636 q^{9} +O(q^{10})$	$q-33.7437i q^{3} -51.7597 q^{5} +313.488 q^{7} -409.636 q^{9} -460.660 q^{11} +177.130i q^{13} +1746.56i q^{15} +6127.64 q^{17} +(-5407.00 - 4220.22i) q^{19} -10578.2i q^{21} -10910.4 q^{23} -12945.9 q^{25} -10776.5i q^{27} +11145.5i q^{29} +15953.5i q^{31} +15544.4i q^{33} -16226.1 q^{35} -76514.9i q^{37} +5977.04 q^{39} -125282. i q^{41} +20603.1 q^{43} +21202.7 q^{45} -150655. q^{47} -19374.2 q^{49} -206769. i q^{51} +113881. i q^{53} +23843.6 q^{55} +(-142406. + 182452. i) q^{57} +231084. i q^{59} -314764. q^{61} -128416. q^{63} -9168.22i q^{65} -379771. i q^{67} +368156. i q^{69} +218789. i q^{71} +379291. q^{73} +436843. i q^{75} -144411. q^{77} +331511. i q^{79} -662264. q^{81} -191832. q^{83} -317165. q^{85} +376090. q^{87} -268881. i q^{89} +55528.3i q^{91} +538330. q^{93} +(279865. + 218437. i) q^{95} +1.70707e6i q^{97} +188703. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 108 q^{5} + 140 q^{7} - 1052 q^{9}+O(q^{10}) $ 8 * q + 108 * q^5 + 140 * q^7 - 1052 * q^9	$ 8 q + 108 q^{5} + 140 q^{7} - 1052 q^{9} + 2024 q^{11} + 6008 q^{17} - 20552 q^{19} + 50252 q^{23} + 78492 q^{25} - 210800 q^{35} - 43724 q^{39} - 260800 q^{43} - 191012 q^{45} + 100248 q^{47} - 301872 q^{49} + 52480 q^{55} - 186860 q^{57} - 54548 q^{61} + 137408 q^{63} + 479968 q^{73} - 1755300 q^{77} - 4279648 q^{81} - 483040 q^{83} + 2111780 q^{85} - 2802652 q^{87} + 1507528 q^{93} + 2383888 q^{95} - 528224 q^{99}+O(q^{100}) $ 8 * q + 108 * q^5 + 140 * q^7 - 1052 * q^9 + 2024 * q^11 + 6008 * q^17 - 20552 * q^19 + 50252 * q^23 + 78492 * q^25 - 210800 * q^35 - 43724 * q^39 - 260800 * q^43 - 191012 * q^45 + 100248 * q^47 - 301872 * q^49 + 52480 * q^55 - 186860 * q^57 - 54548 * q^61 + 137408 * q^63 + 479968 * q^73 - 1755300 * q^77 - 4279648 * q^81 - 483040 * q^83 + 2111780 * q^85 - 2802652 * q^87 + 1507528 * q^93 + 2383888 * q^95 - 528224 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/304\mathbb{Z}\right)^\times$.

$n$	$97$	$191$	$229$
$\chi(n)$	$-1$	$1$	$1$

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
$3$		−	33.7437i		−	1.24977i	−0.780718	−	0.624883i	$-0.785147\pi$
$3$		−	33.7437i		−	1.24977i	0.780718	−	0.624883i	$-0.214853\pi$
$4$		0			0
$5$	−51.7597			−0.414078			−0.207039	−	0.978333i	$-0.566383\pi$
$5$	−51.7597			−0.414078			−0.207039	+	0.978333i	$0.566383\pi$
$6$		0			0
$7$	313.488			0.913960			0.456980	−	0.889477i	$-0.348931\pi$
$7$	313.488			0.913960			0.456980	+	0.889477i	$0.348931\pi$
$8$		0			0
$9$	−409.636			−0.561915
$10$		0			0
$11$	−460.660			−0.346101			−0.173050	−	0.984913i	$-0.555362\pi$
$11$	−460.660			−0.346101			−0.173050	+	0.984913i	$0.555362\pi$
$12$		0			0
$13$			177.130i			0.0806238i	0.999187	+	0.0403119i	$0.0128352\pi$
$13$			177.130i			0.0806238i	−0.999187	+	0.0403119i	$0.987165\pi$
$14$		0			0
$15$			1746.56i			0.517500i
$16$		0			0
$17$	6127.64			1.24723			0.623615	−	0.781732i	$-0.285663\pi$
$17$	6127.64			1.24723			0.623615	+	0.781732i	$0.285663\pi$
$18$		0			0
$19$	−5407.00	−	4220.22i	−0.788308	−	0.615281i
$19$	−5407.00	−	4220.22i	−0.788308	−	0.615281i
$20$		0			0
$21$		−	10578.2i		−	1.14224i
$22$		0			0
$23$	−10910.4			−0.896717			−0.448358	−	0.893854i	$-0.647991\pi$
$23$	−10910.4			−0.896717			−0.448358	+	0.893854i	$0.647991\pi$
$24$		0			0
$25$	−12945.9			−0.828540
$26$		0			0
$27$		−	10776.5i		−	0.547503i
$28$		0			0
$29$			11145.5i			0.456988i	0.973545	+	0.228494i	$0.0733802\pi$
$29$			11145.5i			0.456988i	−0.973545	+	0.228494i	$0.926620\pi$
$30$		0			0
$31$			15953.5i			0.535514i	0.963486	+	0.267757i	$0.0862824\pi$
$31$			15953.5i			0.535514i	−0.963486	+	0.267757i	$0.913718\pi$
$32$		0			0
$33$			15544.4i			0.432545i
$34$		0			0
$35$	−16226.1			−0.378450
$36$		0			0
$37$		−	76514.9i		−	1.51057i	−0.655397	−	0.755285i	$-0.727498\pi$
$37$		−	76514.9i		−	1.51057i	0.655397	−	0.755285i	$-0.272502\pi$
$38$		0			0
$39$	5977.04			0.100761
$40$		0			0
$41$		−	125282.i		−	1.81776i	−0.417058	−	0.908880i	$-0.636939\pi$
$41$		−	125282.i		−	1.81776i	0.417058	−	0.908880i	$-0.363061\pi$
$42$		0			0
$43$	20603.1			0.259136			0.129568	−	0.991571i	$-0.458641\pi$
$43$	20603.1			0.259136			0.129568	+	0.991571i	$0.458641\pi$
$44$		0			0
$45$	21202.7			0.232677
$46$		0			0
$47$	−150655.			−1.45107			−0.725537	−	0.688183i	$-0.758409\pi$
$47$	−150655.			−1.45107			−0.725537	+	0.688183i	$0.758409\pi$
$48$		0			0
$49$	−19374.2			−0.164678
$50$		0			0
$51$		−	206769.i		−	1.55874i
$52$		0			0
$53$			113881.i			0.764935i	0.923969	+	0.382467i	$0.124926\pi$
$53$			113881.i			0.764935i	−0.923969	+	0.382467i	$0.875074\pi$
$54$		0			0
$55$	23843.6			0.143313
$56$		0			0
$57$	−142406.	+	182452.i	−0.768958	+	0.985200i
$58$		0			0
$59$			231084.i			1.12516i	0.826743	+	0.562580i	$0.190191\pi$
$59$			231084.i			1.12516i	−0.826743	+	0.562580i	$0.809809\pi$
$60$		0			0
$61$	−314764.			−1.38674			−0.693370	−	0.720582i	$-0.743875\pi$
$61$	−314764.			−1.38674			−0.693370	+	0.720582i	$0.743875\pi$
$62$		0			0
$63$	−128416.			−0.513568
$64$		0			0
$65$		−	9168.22i		−	0.0333845i
$66$		0			0
$67$		−	379771.i		−	1.26269i	−0.775501	−	0.631346i	$-0.782503\pi$
$67$		−	379771.i		−	1.26269i	0.775501	−	0.631346i	$-0.217497\pi$
$68$		0			0
$69$			368156.i			1.12069i
$70$		0			0
$71$			218789.i			0.611295i	0.952145	+	0.305648i	$0.0988729\pi$
$71$			218789.i			0.611295i	−0.952145	+	0.305648i	$0.901127\pi$
$72$		0			0
$73$	379291.			0.974999			0.487499	−	0.873123i	$-0.337909\pi$
$73$	379291.			0.974999			0.487499	+	0.873123i	$0.337909\pi$
$74$		0			0
$75$			436843.i			1.03548i
$76$		0			0
$77$	−144411.			−0.316322
$78$		0			0
$79$			331511.i			0.672383i	0.941794	+	0.336192i	$0.109139\pi$
$79$			331511.i			0.672383i	−0.941794	+	0.336192i	$0.890861\pi$
$80$		0			0
$81$	−662264.			−1.24617
$82$		0			0
$83$	−191832.			−0.335496			−0.167748	−	0.985830i	$-0.553649\pi$
$83$	−191832.			−0.335496			−0.167748	+	0.985830i	$0.553649\pi$
$84$		0			0
$85$	−317165.			−0.516450
$86$		0			0
$87$	376090.			0.571129
$88$		0			0
$89$		−	268881.i		−	0.381408i	−0.981648	−	0.190704i	$-0.938923\pi$
$89$		−	268881.i		−	0.381408i	0.981648	−	0.190704i	$-0.0610771\pi$
$90$		0			0
$91$			55528.3i			0.0736869i
$92$		0			0
$93$	538330.			0.669267
$94$		0			0
$95$	279865.	+	218437.i	0.326421	+	0.254774i
$96$		0			0
$97$			1.70707e6i			1.87041i	0.354109	+	0.935204i	$0.384784\pi$
$97$			1.70707e6i			1.87041i	−0.354109	+	0.935204i	$0.615216\pi$
$98$		0			0
$99$	188703.			0.194479
$100$		0			0
$101$	712272.			0.691324			0.345662	−	0.938359i	$-0.387654\pi$
$101$	712272.			0.691324			0.345662	+	0.938359i	$0.387654\pi$
$102$		0			0
$103$			1.46263e6i			1.33852i	0.743030	+	0.669259i	$0.233388\pi$
$103$			1.46263e6i			1.33852i	−0.743030	+	0.669259i	$0.766612\pi$
$104$		0			0
$105$			547527.i			0.472974i
$106$		0			0
$107$		−	681335.i		−	0.556173i	−0.960556	−	0.278086i	$-0.910300\pi$
$107$		−	681335.i		−	0.556173i	0.960556	−	0.278086i	$-0.0897001\pi$
$108$		0			0
$109$		−	750959.i		−	0.579878i	−0.957045	−	0.289939i	$-0.906365\pi$
$109$		−	750959.i		−	0.579878i	0.957045	−	0.289939i	$-0.0936350\pi$
$110$		0			0
$111$	−2.58189e6			−1.88786
$112$		0			0
$113$			828261.i			0.574027i	0.957927	+	0.287013i	$0.0926624\pi$
$113$			828261.i			0.574027i	−0.957927	+	0.287013i	$0.907338\pi$
$114$		0			0
$115$	564717.			0.371310
$116$		0			0
$117$		−	72559.1i		−	0.0453038i
$118$		0			0
$119$	1.92094e6			1.13992
$120$		0			0
$121$	−1.55935e6			−0.880214
$122$		0			0
$123$	−4.22747e6			−2.27177
$124$		0			0
$125$	1.47882e6			0.757158
$126$		0			0
$127$			3.56446e6i			1.74013i	0.492935	+	0.870066i	$0.335924\pi$
$127$			3.56446e6i			1.74013i	−0.492935	+	0.870066i	$0.664076\pi$
$128$		0			0
$129$		−	695225.i		−	0.323859i
$130$		0			0
$131$	−1.28645e6			−0.572242			−0.286121	−	0.958194i	$-0.592366\pi$
$131$	−1.28645e6			−0.572242			−0.286121	+	0.958194i	$0.592366\pi$
$132$		0			0
$133$	−1.69503e6	−	1.32299e6i	−0.720481	−	0.562342i
$134$		0			0
$135$			557789.i			0.226709i
$136$		0			0
$137$	−3.79392e6			−1.47546			−0.737728	−	0.675098i	$-0.764101\pi$
$137$	−3.79392e6			−1.47546			−0.737728	+	0.675098i	$0.764101\pi$
$138$		0			0
$139$	−3.53764e6			−1.31725			−0.658626	−	0.752470i	$-0.728862\pi$
$139$	−3.53764e6			−1.31725			−0.658626	+	0.752470i	$0.728862\pi$
$140$		0			0
$141$			5.08365e6i			1.81350i
$142$		0			0
$143$		−	81596.9i		−	0.0279040i
$144$		0			0
$145$		−	576887.i		−	0.189229i
$146$		0			0
$147$			653755.i			0.205808i
$148$		0			0
$149$	644521.			0.194840			0.0974200	−	0.995243i	$-0.468941\pi$
$149$	644521.			0.194840			0.0974200	+	0.995243i	$0.468941\pi$
$150$		0			0
$151$		−	1.00793e6i		−	0.292752i	−0.989229	−	0.146376i	$-0.953239\pi$
$151$		−	1.00793e6i		−	0.292752i	0.989229	−	0.146376i	$-0.0467609\pi$
$152$		0			0
$153$	−2.51010e6			−0.700837
$154$		0			0
$155$		−	825749.i		−	0.221744i
$156$		0			0
$157$	−6.99464e6			−1.80745			−0.903725	−	0.428113i	$-0.859178\pi$
$157$	−6.99464e6			−1.80745			−0.903725	+	0.428113i	$0.859178\pi$
$158$		0			0
$159$	3.84277e6			0.955990
$160$		0			0
$161$	−3.42027e6			−0.819563
$162$		0			0
$163$	−3.86384e6			−0.892187			−0.446094	−	0.894986i	$-0.647185\pi$
$163$	−3.86384e6			−0.892187			−0.446094	+	0.894986i	$0.647185\pi$
$164$		0			0
$165$		−	804572.i		−	0.179107i
$166$		0			0
$167$		−	5.91450e6i		−	1.26990i	−0.772554	−	0.634949i	$-0.781021\pi$
$167$		−	5.91450e6i		−	1.26990i	0.772554	−	0.634949i	$-0.218979\pi$
$168$		0			0
$169$	4.79543e6			0.993500
$170$		0			0
$171$	2.21490e6	+	1.72875e6i	0.442962	+	0.345736i
$172$		0			0
$173$		−	3.74467e6i		−	0.723228i	−0.932328	−	0.361614i	$-0.882226\pi$
$173$		−	3.74467e6i		−	0.723228i	0.932328	−	0.361614i	$-0.117774\pi$
$174$		0			0
$175$	−4.05840e6			−0.757252
$176$		0			0
$177$	7.79763e6			1.40619
$178$		0			0
$179$		−	1.09184e7i		−	1.90371i	−0.306552	−	0.951854i	$-0.599175\pi$
$179$		−	1.09184e7i		−	1.90371i	0.306552	−	0.951854i	$-0.400825\pi$
$180$		0			0
$181$			44238.1i			0.00746038i	0.999993	+	0.00373019i	$0.00118736\pi$
$181$			44238.1i			0.00746038i	−0.999993	+	0.00373019i	$0.998813\pi$
$182$		0			0
$183$			1.06213e7i			1.73310i
$184$		0			0
$185$			3.96039e6i			0.625493i
$186$		0			0
$187$	−2.82276e6			−0.431667
$188$		0			0
$189$		−	3.37831e6i		−	0.500396i
$190$		0			0
$191$	1.09262e7			1.56808			0.784042	−	0.620708i	$-0.213155\pi$
$191$	1.09262e7			1.56808			0.784042	+	0.620708i	$0.213155\pi$
$192$		0			0
$193$		−	1.58192e6i		−	0.220046i	−0.993929	−	0.110023i	$-0.964908\pi$
$193$		−	1.58192e6i		−	0.220046i	0.993929	−	0.110023i	$-0.0350924\pi$
$194$		0			0
$195$	−309370.			−0.0417228
$196$		0			0
$197$	−2.68965e6			−0.351801			−0.175901	−	0.984408i	$-0.556284\pi$
$197$	−2.68965e6			−0.351801			−0.175901	+	0.984408i	$0.556284\pi$
$198$		0			0
$199$	−967242.			−0.122737			−0.0613686	−	0.998115i	$-0.519547\pi$
$199$	−967242.			−0.122737			−0.0613686	+	0.998115i	$0.519547\pi$
$200$		0			0
$201$	−1.28149e7			−1.57807
$202$		0			0
$203$			3.49398e6i			0.417669i
$204$		0			0
$205$			6.48455e6i			0.752694i
$206$		0			0
$207$	4.46928e6			0.503879
$208$		0			0
$209$	2.49079e6	+	1.94408e6i	0.272834	+	0.212949i
$210$		0			0
$211$		−	3.68691e6i		−	0.392478i	−0.980556	−	0.196239i	$-0.937127\pi$
$211$		−	3.68691e6i		−	0.392478i	0.980556	−	0.196239i	$-0.0628729\pi$
$212$		0			0
$213$	7.38275e6			0.763976
$214$		0			0
$215$	−1.06641e6			−0.107302
$216$		0			0
$217$			5.00123e6i			0.489438i
$218$		0			0
$219$		−	1.27987e7i		−	1.21852i
$220$		0			0
$221$			1.08539e6i			0.100556i
$222$		0			0
$223$			1.00457e7i			0.905872i	0.891543	+	0.452936i	$0.149623\pi$
$223$			1.00457e7i			0.905872i	−0.891543	+	0.452936i	$0.850377\pi$
$224$		0			0
$225$	5.30312e6			0.465569
$226$		0			0
$227$			9.74935e6i			0.833486i	0.909024	+	0.416743i	$0.136828\pi$
$227$			9.74935e6i			0.833486i	−0.909024	+	0.416743i	$0.863172\pi$
$228$		0			0
$229$	1.25492e7			1.04499			0.522493	−	0.852644i	$-0.325002\pi$
$229$	1.25492e7			1.04499			0.522493	+	0.852644i	$0.325002\pi$
$230$		0			0
$231$			4.87298e6i			0.395329i
$232$		0			0
$233$	5.50788e6			0.435428			0.217714	−	0.976013i	$-0.430140\pi$
$233$	5.50788e6			0.435428			0.217714	+	0.976013i	$0.430140\pi$
$234$		0			0
$235$	7.79785e6			0.600857
$236$		0			0
$237$	1.11864e7			0.840322
$238$		0			0
$239$	−1.03934e7			−0.761314			−0.380657	−	0.924716i	$-0.624302\pi$
$239$	−1.03934e7			−0.761314			−0.380657	+	0.924716i	$0.624302\pi$
$240$		0			0
$241$		−	2.34911e7i		−	1.67823i	−0.543952	−	0.839116i	$-0.683073\pi$
$241$		−	2.34911e7i		−	1.67823i	0.543952	−	0.839116i	$-0.316927\pi$
$242$		0			0
$243$			1.44912e7i			1.00991i
$244$		0			0
$245$	1.00280e6			0.0681893
$246$		0			0
$247$	747529.	−	957745.i	0.0496063	−	0.0635563i
$248$		0			0
$249$			6.47312e6i			0.419291i
$250$		0			0
$251$	−200237.			−0.0126626			−0.00633132	−	0.999980i	$-0.502015\pi$
$251$	−200237.			−0.0126626			−0.00633132	+	0.999980i	$0.502015\pi$
$252$		0			0
$253$	5.02596e6			0.310354
$254$		0			0
$255$			1.07023e7i			0.645441i
$256$		0			0
$257$			1.33812e7i			0.788309i	0.919044	+	0.394154i	$0.128962\pi$
$257$			1.33812e7i			0.788309i	−0.919044	+	0.394154i	$0.871038\pi$
$258$		0			0
$259$		−	2.39865e7i		−	1.38060i
$260$		0			0
$261$		−	4.56560e6i		−	0.256789i
$262$		0			0
$263$	2.16755e7			1.19152			0.595760	−	0.803162i	$-0.296851\pi$
$263$	2.16755e7			1.19152			0.595760	+	0.803162i	$0.296851\pi$
$264$		0			0
$265$		−	5.89446e6i		−	0.316743i
$266$		0			0
$267$	−9.07304e6			−0.476671
$268$		0			0
$269$		−	9.47444e6i		−	0.486740i	−0.969934	−	0.243370i	$-0.921747\pi$
$269$		−	9.47444e6i		−	0.486740i	0.969934	−	0.243370i	$-0.0782529\pi$
$270$		0			0
$271$	9.62807e6			0.483762			0.241881	−	0.970306i	$-0.422236\pi$
$271$	9.62807e6			0.483762			0.241881	+	0.970306i	$0.422236\pi$
$272$		0			0
$273$	1.87373e6			0.0920914
$274$		0			0
$275$	5.96367e6			0.286758
$276$		0			0
$277$	−1.47609e7			−0.694501			−0.347251	−	0.937772i	$-0.612885\pi$
$277$	−1.47609e7			−0.694501			−0.347251	+	0.937772i	$0.612885\pi$
$278$		0			0
$279$		−	6.53513e6i		−	0.300914i
$280$		0			0
$281$			2.57329e6i			0.115976i	0.998317	+	0.0579882i	$0.0184686\pi$
$281$			2.57329e6i			0.115976i	−0.998317	+	0.0579882i	$0.981531\pi$
$282$		0			0
$283$	−1.68438e6			−0.0743156			−0.0371578	−	0.999309i	$-0.511830\pi$
$283$	−1.68438e6			−0.0743156			−0.0371578	+	0.999309i	$0.511830\pi$
$284$		0			0
$285$	7.37087e6	−	9.44367e6i	0.318408	−	0.407949i
$286$		0			0
$287$		−	3.92744e7i		−	1.66136i
$288$		0			0
$289$	1.34104e7			0.555581
$290$		0			0
$291$	5.76029e7			2.33757
$292$		0			0
$293$		−	3.83169e7i		−	1.52331i	−0.647985	−	0.761653i	$-0.724388\pi$
$293$		−	3.83169e7i		−	1.52331i	0.647985	−	0.761653i	$-0.275612\pi$
$294$		0			0
$295$		−	1.19609e7i		−	0.465904i
$296$		0			0
$297$			4.96431e6i			0.189491i
$298$		0			0
$299$		−	1.93256e6i		−	0.0722967i
$300$		0			0
$301$	6.45883e6			0.236840
$302$		0			0
$303$		−	2.40347e7i		−	0.863993i
$304$		0			0
$305$	1.62921e7			0.574218
$306$		0			0
$307$			1.27696e7i			0.441328i	0.975350	+	0.220664i	$0.0708225\pi$
$307$			1.27696e7i			0.441328i	−0.975350	+	0.220664i	$0.929178\pi$
$308$		0			0
$309$	4.93547e7			1.67283
$310$		0			0
$311$	−2.29592e7			−0.763264			−0.381632	−	0.924314i	$-0.624638\pi$
$311$	−2.29592e7			−0.763264			−0.381632	+	0.924314i	$0.624638\pi$
$312$		0			0
$313$	−2.79625e7			−0.911892			−0.455946	−	0.890008i	$-0.650699\pi$
$313$	−2.79625e7			−0.911892			−0.455946	+	0.890008i	$0.650699\pi$
$314$		0			0
$315$	6.64678e6			0.212657
$316$		0			0
$317$		−	1.15462e7i		−	0.362462i	−0.983441	−	0.181231i	$-0.941992\pi$
$317$		−	1.15462e7i		−	0.362462i	0.983441	−	0.181231i	$-0.0580081\pi$
$318$		0			0
$319$		−	5.13428e6i		−	0.158164i
$320$		0			0
$321$	−2.29908e7			−0.695086
$322$		0			0
$323$	−3.31321e7	−	2.58599e7i	−0.983200	−	0.767397i
$324$		0			0
$325$		−	2.29312e6i		−	0.0668000i
$326$		0			0
$327$	−2.53401e7			−0.724712
$328$		0			0
$329$	−4.72285e7			−1.32622
$330$		0			0
$331$		−	3.61243e7i		−	0.996128i	−0.867140	−	0.498064i	$-0.834044\pi$
$331$		−	3.61243e7i		−	0.996128i	0.867140	−	0.498064i	$-0.165956\pi$
$332$		0			0
$333$			3.13433e7i			0.848813i
$334$		0			0
$335$			1.96568e7i			0.522853i
$336$		0			0
$337$			3.86365e6i			0.100950i	0.998725	+	0.0504752i	$0.0160736\pi$
$337$			3.86365e6i			0.100950i	−0.998725	+	0.0504752i	$0.983926\pi$
$338$		0			0
$339$	2.79486e7			0.717399
$340$		0			0
$341$		−	7.34914e6i		−	0.185342i
$342$		0			0
$343$	−4.29551e7			−1.06447
$344$		0			0
$345$		−	1.90556e7i		−	0.464051i
$346$		0			0
$347$	3.99827e7			0.956939			0.478469	−	0.878104i	$-0.341192\pi$
$347$	3.99827e7			0.956939			0.478469	+	0.878104i	$0.341192\pi$
$348$		0			0
$349$	1.63712e7			0.385127			0.192563	−	0.981285i	$-0.438320\pi$
$349$	1.63712e7			0.385127			0.192563	+	0.981285i	$0.438320\pi$
$350$		0			0
$351$	1.90885e6			0.0441418
$352$		0			0
$353$	−6.15479e7			−1.39923			−0.699615	−	0.714520i	$-0.746645\pi$
$353$	−6.15479e7			−1.39923			−0.699615	+	0.714520i	$0.746645\pi$
$354$		0			0
$355$		−	1.13245e7i		−	0.253124i
$356$		0			0
$357$		−	6.48197e7i		−	1.42463i
$358$		0			0
$359$	7.15774e7			1.54701			0.773504	−	0.633792i	$-0.218502\pi$
$359$	7.15774e7			1.54701			0.773504	+	0.633792i	$0.218502\pi$
$360$		0			0
$361$	1.14254e7	+	4.56374e7i	0.242858	+	0.970062i
$362$		0			0
$363$			5.26183e7i			1.10006i
$364$		0			0
$365$	−1.96320e7			−0.403725
$366$		0			0
$367$	−5.78244e6			−0.116980			−0.0584902	−	0.998288i	$-0.518629\pi$
$367$	−5.78244e6			−0.116980			−0.0584902	+	0.998288i	$0.518629\pi$
$368$		0			0
$369$			5.13200e7i			1.02143i
$370$		0			0
$371$			3.57004e7i			0.699120i
$372$		0			0
$373$		−	7.08862e7i		−	1.36595i	−0.730441	−	0.682976i	$-0.760686\pi$
$373$		−	7.08862e7i		−	1.36595i	0.730441	−	0.682976i	$-0.239314\pi$
$374$		0			0
$375$		−	4.99009e7i		−	0.946270i
$376$		0			0
$377$	−1.97421e6			−0.0368441
$378$		0			0
$379$			9.66939e7i			1.77616i	0.459692	+	0.888079i	$0.347960\pi$
$379$			9.66939e7i			1.77616i	−0.459692	+	0.888079i	$0.652040\pi$
$380$		0			0
$381$	1.20278e8			2.17476
$382$		0			0
$383$		−	8.74413e7i		−	1.55640i	−0.628019	−	0.778198i	$-0.716134\pi$
$383$		−	8.74413e7i		−	1.55640i	0.628019	−	0.778198i	$-0.283866\pi$
$384$		0			0
$385$	7.47470e6			0.130982
$386$		0			0
$387$	−8.43978e6			−0.145612
$388$		0			0
$389$	1.24502e7			0.211509			0.105754	−	0.994392i	$-0.466274\pi$
$389$	1.24502e7			0.211509			0.105754	+	0.994392i	$0.466274\pi$
$390$		0			0
$391$	−6.68547e7			−1.11841
$392$		0			0
$393$			4.34096e7i			0.715168i
$394$		0			0
$395$		−	1.71589e7i		−	0.278419i
$396$		0			0
$397$	−9.66655e6			−0.154490			−0.0772449	−	0.997012i	$-0.524612\pi$
$397$	−9.66655e6			−0.154490			−0.0772449	+	0.997012i	$0.524612\pi$
$398$		0			0
$399$	−4.46425e7	+	5.71966e7i	−0.702797	+	0.900433i
$400$		0			0
$401$		−	1.73149e7i		−	0.268526i	−0.990946	−	0.134263i	$-0.957133\pi$
$401$		−	1.73149e7i		−	0.268526i	0.990946	−	0.134263i	$-0.0428667\pi$
$402$		0			0
$403$	−2.82585e6			−0.0431752
$404$		0			0
$405$	3.42786e7			0.516010
$406$		0			0
$407$			3.52474e7i			0.522809i
$408$		0			0
$409$		−	9.98449e7i		−	1.45934i	−0.683801	−	0.729669i	$-0.739674\pi$
$409$		−	9.98449e7i		−	1.45934i	0.683801	−	0.729669i	$-0.260326\pi$
$410$		0			0
$411$			1.28021e8i			1.84398i
$412$		0			0
$413$			7.24422e7i			1.02835i
$414$		0			0
$415$	9.92918e6			0.138921
$416$		0			0
$417$			1.19373e8i			1.64626i
$418$		0			0
$419$	−3.42446e7			−0.465533			−0.232766	−	0.972533i	$-0.574778\pi$
$419$	−3.42446e7			−0.465533			−0.232766	+	0.972533i	$0.574778\pi$
$420$		0			0
$421$			9.97444e6i			0.133673i	0.997764	+	0.0668363i	$0.0212905\pi$
$421$			9.97444e6i			0.133673i	−0.997764	+	0.0668363i	$0.978709\pi$
$422$		0			0
$423$	6.17137e7			0.815381
$424$		0			0
$425$	−7.93280e7			−1.03338
$426$		0			0
$427$	−9.86746e7			−1.26742
$428$		0			0
$429$	−2.75338e6			−0.0348734
$430$		0			0
$431$		−	1.17535e7i		−	0.146803i	−0.997302	−	0.0734014i	$-0.976615\pi$
$431$		−	1.17535e7i		−	0.146803i	0.997302	−	0.0734014i	$-0.0233854\pi$
$432$		0			0
$433$			8.93309e7i			1.10037i	0.835043	+	0.550184i	$0.185442\pi$
$433$			8.93309e7i			1.10037i	−0.835043	+	0.550184i	$0.814558\pi$
$434$		0			0
$435$	−1.94663e7			−0.236492
$436$		0			0
$437$	5.89923e7	+	4.60440e7i	0.706889	+	0.551733i
$438$		0			0
$439$		−	6.34878e7i		−	0.750406i	−0.926943	−	0.375203i	$-0.877573\pi$
$439$		−	6.34878e7i		−	0.750406i	0.926943	−	0.375203i	$-0.122427\pi$
$440$		0			0
$441$	7.93636e6			0.0925349
$442$		0			0
$443$	3.65478e7			0.420388			0.210194	−	0.977660i	$-0.432590\pi$
$443$	3.65478e7			0.420388			0.210194	+	0.977660i	$0.432590\pi$
$444$		0			0
$445$			1.39172e7i			0.157933i
$446$		0			0
$447$		−	2.17485e7i		−	0.243505i
$448$		0			0
$449$			5.68613e7i			0.628170i	0.949395	+	0.314085i	$0.101698\pi$
$449$			5.68613e7i			0.628170i	−0.949395	+	0.314085i	$0.898302\pi$
$450$		0			0
$451$			5.77123e7i			0.629128i
$452$		0			0
$453$	−3.40113e7			−0.365872
$454$		0			0
$455$		−	2.87413e6i		−	0.0305121i
$456$		0			0
$457$	−1.53349e8			−1.60669			−0.803347	−	0.595512i	$-0.796949\pi$
$457$	−1.53349e8			−1.60669			−0.803347	+	0.595512i	$0.796949\pi$
$458$		0			0
$459$		−	6.60345e7i		−	0.682862i
$460$		0			0
$461$	−2.18233e7			−0.222750			−0.111375	−	0.993778i	$-0.535525\pi$
$461$	−2.18233e7			−0.222750			−0.111375	+	0.993778i	$0.535525\pi$
$462$		0			0
$463$	4.91907e7			0.495610			0.247805	−	0.968810i	$-0.420291\pi$
$463$	4.91907e7			0.495610			0.247805	+	0.968810i	$0.420291\pi$
$464$		0			0
$465$	−2.78638e7			−0.277129
$466$		0			0
$467$	8.46888e7			0.831525			0.415763	−	0.909473i	$-0.363515\pi$
$467$	8.46888e7			0.831525			0.415763	+	0.909473i	$0.363515\pi$
$468$		0			0
$469$		−	1.19054e8i		−	1.15405i
$470$		0			0
$471$			2.36025e8i			2.25889i
$472$		0			0
$473$	−9.49102e6			−0.0896870
$474$		0			0
$475$	6.99987e7	+	5.46346e7i	0.653144	+	0.509785i
$476$		0			0
$477$		−	4.66499e7i		−	0.429829i
$478$		0			0
$479$	−1.59447e8			−1.45080			−0.725402	−	0.688325i	$-0.758346\pi$
$479$	−1.59447e8			−1.45080			−0.725402	+	0.688325i	$0.758346\pi$
$480$		0			0
$481$	1.35531e7			0.121788
$482$		0			0
$483$			1.15412e8i			1.02426i
$484$		0			0
$485$		−	8.83575e7i		−	0.774495i
$486$		0			0
$487$		−	1.05220e8i		−	0.910983i	−0.890240	−	0.455492i	$-0.849464\pi$
$487$		−	1.05220e8i		−	0.910983i	0.890240	−	0.455492i	$-0.150536\pi$
$488$		0			0
$489$			1.30380e8i			1.11503i
$490$		0			0
$491$	−1.76955e8			−1.49492			−0.747461	−	0.664306i	$-0.768727\pi$
$491$	−1.76955e8			−1.49492			−0.747461	+	0.664306i	$0.768727\pi$
$492$		0			0
$493$			6.82955e7i			0.569969i
$494$		0			0
$495$	−9.76722e6			−0.0805296
$496$		0			0
$497$			6.85878e7i			0.558699i
$498$		0			0
$499$	3.81855e7			0.307324			0.153662	−	0.988123i	$-0.450893\pi$
$499$	3.81855e7			0.307324			0.153662	+	0.988123i	$0.450893\pi$
$500$		0			0
$501$	−1.99577e8			−1.58708
$502$		0			0
$503$	−1.08285e8			−0.850873			−0.425436	−	0.904988i	$-0.639879\pi$
$503$	−1.08285e8			−0.850873			−0.425436	+	0.904988i	$0.639879\pi$
$504$		0			0
$505$	−3.68670e7			−0.286262
$506$		0			0
$507$		−	1.61816e8i		−	1.24164i
$508$		0			0
$509$		−	6.17811e7i		−	0.468492i	−0.972177	−	0.234246i	$-0.924738\pi$
$509$		−	6.17811e7i		−	0.468492i	0.972177	−	0.234246i	$-0.0752621\pi$
$510$		0			0
$511$	1.18903e8			0.891110
$512$		0			0
$513$	−4.54792e7	+	5.82686e7i	−0.336869	+	0.431601i
$514$		0			0
$515$		−	7.57055e7i		−	0.554250i
$516$		0			0
$517$	6.94007e7			0.502218
$518$		0			0
$519$	−1.26359e8			−0.903866
$520$		0			0
$521$		−	2.19732e8i		−	1.55375i	−0.629657	−	0.776873i	$-0.716805\pi$
$521$		−	2.19732e8i		−	1.55375i	0.629657	−	0.776873i	$-0.283195\pi$
$522$		0			0
$523$		−	1.28062e8i		−	0.895186i	−0.894237	−	0.447593i	$-0.852281\pi$
$523$		−	1.28062e8i		−	0.895186i	0.894237	−	0.447593i	$-0.147719\pi$
$524$		0			0
$525$			1.36945e8i			0.946388i
$526$		0			0
$527$			9.77573e7i			0.667909i
$528$		0			0
$529$	−2.90001e7			−0.195899
$530$		0			0
$531$		−	9.46605e7i		−	0.632245i
$532$		0			0
$533$	2.21912e7			0.146555
$534$		0			0
$535$			3.52657e7i			0.230299i
$536$		0			0
$537$	−3.68427e8			−2.37919
$538$		0			0
$539$	8.92490e6			0.0569950
$540$		0			0
$541$	7.84005e7			0.495139			0.247569	−	0.968870i	$-0.420368\pi$
$541$	7.84005e7			0.495139			0.247569	+	0.968870i	$0.420368\pi$
$542$		0			0
$543$	1.49276e6			0.00932373
$544$		0			0
$545$			3.88694e7i			0.240115i
$546$		0			0
$547$		−	1.44687e8i		−	0.884030i	−0.897008	−	0.442015i	$-0.854264\pi$
$547$		−	1.44687e8i		−	0.884030i	0.897008	−	0.442015i	$-0.145736\pi$
$548$		0			0
$549$	1.28939e8			0.779230
$550$		0			0
$551$	4.70364e7	−	6.02637e7i	0.281176	−	0.360247i
$552$		0			0
$553$			1.03925e8i			0.614531i
$554$		0			0
$555$	1.33638e8			0.781720
$556$		0			0
$557$	3.19606e8			1.84948			0.924739	−	0.380602i	$-0.124283\pi$
$557$	3.19606e8			1.84948			0.924739	+	0.380602i	$0.124283\pi$
$558$		0			0
$559$			3.64944e6i			0.0208925i
$560$		0			0
$561$			9.52502e7i			0.539483i
$562$		0			0
$563$			1.52462e8i			0.854351i	0.904169	+	0.427175i	$0.140491\pi$
$563$			1.52462e8i			0.854351i	−0.904169	+	0.427175i	$0.859509\pi$
$564$		0			0
$565$		−	4.28706e7i		−	0.237692i
$566$		0			0
$567$	−2.07612e8			−1.13895
$568$		0			0
$569$		−	1.12723e8i		−	0.611892i	−0.952049	−	0.305946i	$-0.901027\pi$
$569$		−	1.12723e8i		−	0.611892i	0.952049	−	0.305946i	$-0.0989728\pi$
$570$		0			0
$571$	−2.03701e8			−1.09417			−0.547085	−	0.837077i	$-0.684263\pi$
$571$	−2.03701e8			−1.09417			−0.547085	+	0.837077i	$0.684263\pi$
$572$		0			0
$573$		−	3.68690e8i		−	1.95974i
$574$		0			0
$575$	1.41245e8			0.742965
$576$		0			0
$577$	−3.35128e8			−1.74455			−0.872275	−	0.489016i	$-0.837356\pi$
$577$	−3.35128e8			−1.74455			−0.872275	+	0.489016i	$0.837356\pi$
$578$		0			0
$579$	−5.33799e7			−0.275006
$580$		0			0
$581$	−6.01371e7			−0.306630
$582$		0			0
$583$		−	5.24605e7i		−	0.264745i
$584$		0			0
$585$			3.75564e6i			0.0187593i
$586$		0			0
$587$	−2.06651e8			−1.02170			−0.510850	−	0.859670i	$-0.670669\pi$
$587$	−2.06651e8			−1.02170			−0.510850	+	0.859670i	$0.670669\pi$
$588$		0			0
$589$	6.73272e7	−	8.62606e7i	0.329492	−	0.422150i
$590$		0			0
$591$			9.07587e7i			0.439669i
$592$		0			0
$593$	2.33592e8			1.12019			0.560097	−	0.828427i	$-0.310764\pi$
$593$	2.33592e8			1.12019			0.560097	+	0.828427i	$0.310764\pi$
$594$		0			0
$595$	−9.94274e7			−0.472014
$596$		0			0
$597$			3.26383e7i			0.153393i
$598$		0			0
$599$			1.58646e8i			0.738156i	0.929398	+	0.369078i	$0.120326\pi$
$599$			1.58646e8i			0.738156i	−0.929398	+	0.369078i	$0.879674\pi$
$600$		0			0
$601$			2.01145e8i			0.926587i	0.886205	+	0.463294i	$0.153333\pi$
$601$			2.01145e8i			0.926587i	−0.886205	+	0.463294i	$0.846667\pi$
$602$		0			0
$603$			1.55568e8i			0.709526i
$604$		0			0
$605$	8.07117e7			0.364477
$606$		0			0
$607$			3.28217e8i			1.46756i	0.679388	+	0.733779i	$0.262245\pi$
$607$			3.28217e8i			1.46756i	−0.679388	+	0.733779i	$0.737755\pi$
$608$		0			0
$609$	1.17900e8			0.521989
$610$		0			0
$611$		−	2.66856e7i		−	0.116991i
$612$		0			0
$613$	−1.04114e8			−0.451988			−0.225994	−	0.974129i	$-0.572563\pi$
$613$	−1.04114e8			−0.451988			−0.225994	+	0.974129i	$0.572563\pi$
$614$		0			0
$615$	2.18813e8			0.940691
$616$		0			0
$617$	1.55494e8			0.662002			0.331001	−	0.943630i	$-0.392614\pi$
$617$	1.55494e8			0.662002			0.331001	+	0.943630i	$0.392614\pi$
$618$		0			0
$619$	2.47327e8			1.04280			0.521398	−	0.853314i	$-0.325411\pi$
$619$	2.47327e8			1.04280			0.521398	+	0.853314i	$0.325411\pi$
$620$		0			0
$621$			1.17575e8i			0.490955i
$622$		0			0
$623$		−	8.42910e7i		−	0.348592i
$624$		0			0
$625$	1.25737e8			0.515018
$626$		0			0
$627$	6.56006e7	−	8.40484e7i	0.266137	−	0.340978i
$628$		0			0
$629$		−	4.68856e8i		−	1.88403i
$630$		0			0
$631$	1.10573e7			0.0440111			0.0220055	−	0.999758i	$-0.492995\pi$
$631$	1.10573e7			0.0440111			0.0220055	+	0.999758i	$0.492995\pi$
$632$		0			0
$633$	−1.24410e8			−0.490506
$634$		0			0
$635$		−	1.84495e8i		−	0.720550i
$636$		0			0
$637$		−	3.43175e6i		−	0.0132769i
$638$		0			0
$639$		−	8.96240e7i		−	0.343496i
$640$		0			0
$641$			1.52484e8i			0.578961i	0.957184	+	0.289481i	$0.0934826\pi$
$641$			1.52484e8i			0.578961i	−0.957184	+	0.289481i	$0.906517\pi$
$642$		0			0
$643$	−1.39087e8			−0.523184			−0.261592	−	0.965179i	$-0.584247\pi$
$643$	−1.39087e8			−0.523184			−0.261592	+	0.965179i	$0.584247\pi$
$644$		0			0
$645$			3.59846e7i			0.134103i
$646$		0			0
$647$	4.55618e8			1.68224			0.841121	−	0.540847i	$-0.181896\pi$
$647$	4.55618e8			1.68224			0.841121	+	0.540847i	$0.181896\pi$
$648$		0			0
$649$		−	1.06451e8i		−	0.389419i
$650$		0			0
$651$	1.68760e8			0.611683
$652$		0			0
$653$	2.89552e8			1.03989			0.519945	−	0.854200i	$-0.325953\pi$
$653$	2.89552e8			1.03989			0.519945	+	0.854200i	$0.325953\pi$
$654$		0			0
$655$	6.65864e7			0.236953
$656$		0			0
$657$	−1.55371e8			−0.547867
$658$		0			0
$659$		−	4.78112e8i		−	1.67060i	−0.549791	−	0.835302i	$-0.685293\pi$
$659$		−	4.78112e8i		−	1.67060i	0.549791	−	0.835302i	$-0.314707\pi$
$660$		0			0
$661$			2.72696e8i			0.944224i	0.881539	+	0.472112i	$0.156508\pi$
$661$			2.72696e8i			0.944224i	−0.881539	+	0.472112i	$0.843492\pi$
$662$		0			0
$663$	3.66251e7			0.125672
$664$		0			0
$665$	8.77343e7	+	6.84775e7i	0.298335	+	0.232853i
$666$		0			0
$667$		−	1.21601e8i		−	0.409789i
$668$		0			0
$669$	3.38980e8			1.13213
$670$		0			0
$671$	1.44999e8			0.479951
$672$		0			0
$673$			3.70225e8i			1.21456i	0.794487	+	0.607282i	$0.207740\pi$
$673$			3.70225e8i			1.21456i	−0.794487	+	0.607282i	$0.792260\pi$
$674$		0			0
$675$			1.39512e8i			0.453628i
$676$		0			0
$677$		−	2.52922e6i		−	0.00815119i	−0.999992	−	0.00407559i	$-0.998703\pi$
$677$		−	2.52922e6i		−	0.00815119i	0.999992	−	0.00407559i	$-0.00129731\pi$
$678$		0			0
$679$			5.35147e8i			1.70948i
$680$		0			0
$681$	3.28979e8			1.04166
$682$		0			0
$683$			9.64579e7i			0.302744i	0.988477	+	0.151372i	$0.0483692\pi$
$683$			9.64579e7i			0.302744i	−0.988477	+	0.151372i	$0.951631\pi$
$684$		0			0
$685$	1.96372e8			0.610954
$686$		0			0
$687$		−	4.23457e8i		−	1.30599i
$688$		0			0
$689$	−2.01718e7			−0.0616720
$690$		0			0
$691$	1.13291e8			0.343369			0.171685	−	0.985152i	$-0.445079\pi$
$691$	1.13291e8			0.343369			0.171685	+	0.985152i	$0.445079\pi$
$692$		0			0
$693$	5.91562e7			0.177746
$694$		0			0
$695$	1.83107e8			0.545445
$696$		0			0
$697$		−	7.67682e8i		−	2.26716i
$698$		0			0
$699$		−	1.85856e8i		−	0.544183i
$700$		0			0
$701$	−1.17038e7			−0.0339761			−0.0169881	−	0.999856i	$-0.505408\pi$
$701$	−1.17038e7			−0.0339761			−0.0169881	+	0.999856i	$0.505408\pi$
$702$		0			0
$703$	−3.22909e8	+	4.13716e8i	−0.929426	+	1.19079i
$704$		0			0
$705$		−	2.63128e8i		−	0.750931i
$706$		0			0
$707$	2.23289e8			0.631842
$708$		0			0
$709$	−6.14873e8			−1.72523			−0.862615	−	0.505861i	$-0.831175\pi$
$709$	−6.14873e8			−1.72523			−0.862615	+	0.505861i	$0.831175\pi$
$710$		0			0
$711$		−	1.35799e8i		−	0.377823i
$712$		0			0
$713$		−	1.74058e8i		−	0.480204i
$714$		0			0
$715$			4.22343e6i			0.0115544i
$716$		0			0
$717$			3.50712e8i			0.951465i
$718$		0			0
$719$	6.26651e8			1.68593			0.842964	−	0.537969i	$-0.180808\pi$
$719$	6.26651e8			1.68593			0.842964	+	0.537969i	$0.180808\pi$
$720$		0			0
$721$			4.58518e8i			1.22335i
$722$		0			0
$723$	−7.92676e8			−2.09740
$724$		0			0
$725$		−	1.44289e8i		−	0.378633i
$726$		0			0
$727$	3.38552e8			0.881094			0.440547	−	0.897730i	$-0.354785\pi$
$727$	3.38552e8			0.881094			0.440547	+	0.897730i	$0.354785\pi$
$728$		0			0
$729$	6.19454e6			0.0159892
$730$		0			0
$731$	1.26248e8			0.323202
$732$		0			0
$733$	6.16356e8			1.56502			0.782510	−	0.622638i	$-0.213939\pi$
$733$	6.16356e8			1.56502			0.782510	+	0.622638i	$0.213939\pi$
$734$		0			0
$735$		−	3.38382e7i		−	0.0852207i
$736$		0			0
$737$			1.74945e8i			0.437019i
$738$		0			0
$739$	3.03227e8			0.751335			0.375668	−	0.926754i	$-0.377413\pi$
$739$	3.03227e8			0.751335			0.375668	+	0.926754i	$0.377413\pi$
$740$		0			0
$741$	−3.23178e7	−	2.52244e7i	−0.0794306	−	0.0619963i
$742$		0			0
$743$		−	5.99361e8i		−	1.46124i	−0.682784	−	0.730621i	$-0.739231\pi$
$743$		−	5.99361e8i		−	1.46124i	0.682784	−	0.730621i	$-0.260769\pi$
$744$		0			0
$745$	−3.33602e7			−0.0806789
$746$		0			0
$747$	7.85814e7			0.188520
$748$		0			0
$749$		−	2.13591e8i		−	0.508319i
$750$		0			0
$751$		−	6.18350e8i		−	1.45987i	−0.683516	−	0.729936i	$-0.739550\pi$
$751$		−	6.18350e8i		−	1.45987i	0.683516	−	0.729936i	$-0.260450\pi$
$752$		0			0
$753$			6.75675e6i			0.0158253i
$754$		0			0
$755$			5.21702e7i			0.121222i
$756$		0			0
$757$	−2.74435e8			−0.632632			−0.316316	−	0.948654i	$-0.602446\pi$
$757$	−2.74435e8			−0.632632			−0.316316	+	0.948654i	$0.602446\pi$
$758$		0			0
$759$		−	1.69595e8i		−	0.387870i
$760$		0			0
$761$	−1.68374e8			−0.382051			−0.191026	−	0.981585i	$-0.561181\pi$
$761$	−1.68374e8			−0.382051			−0.191026	+	0.981585i	$0.561181\pi$
$762$		0			0
$763$		−	2.35417e8i		−	0.529985i
$764$		0			0
$765$	1.29922e8			0.290201
$766$		0			0
$767$	−4.09321e7			−0.0907147
$768$		0			0
$769$	9.93895e7			0.218555			0.109278	−	0.994011i	$-0.465146\pi$
$769$	9.93895e7			0.218555			0.109278	+	0.994011i	$0.465146\pi$
$770$		0			0
$771$	4.51532e8			0.985202
$772$		0			0
$773$		−	7.96234e8i		−	1.72386i	−0.507027	−	0.861930i	$-0.669256\pi$
$773$		−	7.96234e8i		−	1.72386i	0.507027	−	0.861930i	$-0.330744\pi$
$774$		0			0
$775$		−	2.06533e8i		−	0.443695i
$776$		0			0
$777$	−8.09394e8			−1.72543
$778$		0			0
$779$	−5.28716e8	+	6.77399e8i	−1.11843	+	1.43295i
$780$		0			0
$781$		−	1.00787e8i		−	0.211570i
$782$		0			0
$783$	1.20109e8			0.250203
$784$		0			0
$785$	3.62040e8			0.748425
$786$		0			0
$787$		−	3.17143e8i		−	0.650625i	−0.945607	−	0.325313i	$-0.894530\pi$
$787$		−	3.17143e8i		−	0.650625i	0.945607	−	0.325313i	$-0.105470\pi$
$788$		0			0
$789$		−	7.31410e8i		−	1.48912i
$790$		0			0
$791$			2.59650e8i			0.524637i
$792$		0			0
$793$		−	5.57542e7i		−	0.111804i
$794$		0			0
$795$	−1.98901e8			−0.395854
$796$		0			0
$797$			7.89712e8i			1.55989i	0.625848	+	0.779945i	$0.284753\pi$
$797$			7.89712e8i			1.55989i	−0.625848	+	0.779945i	$0.715247\pi$
$798$		0			0
$799$	−9.23158e8			−1.80982
$800$		0			0
$801$			1.10143e8i			0.214319i
$802$		0			0
$803$	−1.74724e8			−0.337448
$804$		0			0
$805$	1.77032e8			0.339363
$806$		0			0
$807$	−3.19702e8			−0.608311
$808$		0			0
$809$	−3.58300e8			−0.676708			−0.338354	−	0.941019i	$-0.609870\pi$
$809$	−3.58300e8			−0.676708			−0.338354	+	0.941019i	$0.609870\pi$
$810$		0			0
$811$			2.10199e8i			0.394064i	0.980397	+	0.197032i	$0.0631303\pi$
$811$			2.10199e8i			0.394064i	−0.980397	+	0.197032i	$0.936870\pi$
$812$		0			0
$813$		−	3.24887e8i		−	0.604589i
$814$		0			0
$815$	1.99991e8			0.369435
$816$		0			0
$817$	−1.11401e8	−	8.69495e7i	−0.204279	−	0.159441i
$818$		0			0
$819$		−	2.27464e7i		−	0.0414058i
$820$		0			0
$821$	8.92910e8			1.61353			0.806767	−	0.590869i	$-0.201215\pi$
$821$	8.92910e8			1.61353			0.806767	+	0.590869i	$0.201215\pi$
$822$		0			0
$823$	−3.24504e7			−0.0582131			−0.0291066	−	0.999576i	$-0.509266\pi$
$823$	−3.24504e7			−0.0582131			−0.0291066	+	0.999576i	$0.509266\pi$
$824$		0			0
$825$		−	2.01236e8i		−	0.358381i
$826$		0			0
$827$		−	1.76093e8i		−	0.311334i	−0.987810	−	0.155667i	$-0.950247\pi$
$827$		−	1.76093e8i		−	0.311334i	0.987810	−	0.155667i	$-0.0497527\pi$
$828$		0			0
$829$			6.29583e8i			1.10507i	0.833490	+	0.552534i	$0.186339\pi$
$829$			6.29583e8i			1.10507i	−0.833490	+	0.552534i	$0.813661\pi$
$830$		0			0
$831$			4.98087e8i			0.867964i
$832$		0			0
$833$	−1.18718e8			−0.205391
$834$		0			0
$835$			3.06133e8i			0.525836i
$836$		0			0
$837$	1.71923e8			0.293196
$838$		0			0
$839$		−	6.78394e7i		−	0.114867i	−0.998349	−	0.0574336i	$-0.981708\pi$
$839$		−	6.78394e7i		−	0.114867i	0.998349	−	0.0574336i	$-0.0182917\pi$
$840$		0			0
$841$	4.70601e8			0.791162
$842$		0			0
$843$	8.68323e7			0.144943
$844$		0			0
$845$	−2.48210e8			−0.411386
$846$		0			0
$847$	−4.88839e8			−0.804480
$848$		0			0
$849$			5.68371e7i			0.0928771i
$850$		0			0
$851$			8.34805e8i			1.35455i
$852$		0			0
$853$	−6.22120e8			−1.00237			−0.501184	−	0.865341i	$-0.667102\pi$
$853$	−6.22120e8			−1.00237			−0.501184	+	0.865341i	$0.667102\pi$
$854$		0			0
$855$	−1.14643e8	−	8.94798e7i	−0.183421	−	0.143162i
$856$		0			0
$857$		−	2.10361e8i		−	0.334213i	−0.985939	−	0.167107i	$-0.946558\pi$
$857$		−	2.10361e8i		−	0.334213i	0.985939	−	0.167107i	$-0.0534424\pi$
$858$		0			0
$859$	−3.19350e8			−0.503833			−0.251917	−	0.967749i	$-0.581061\pi$
$859$	−3.19350e8			−0.503833			−0.251917	+	0.967749i	$0.581061\pi$
$860$		0			0
$861$	−1.32526e9			−2.07631
$862$		0			0
$863$		−	1.58867e7i		−	0.0247173i	−0.999924	−	0.0123586i	$-0.996066\pi$
$863$		−	1.58867e7i		−	0.0247173i	0.999924	−	0.0123586i	$-0.00393398\pi$
$864$		0			0
$865$			1.93823e8i			0.299473i
$866$		0			0
$867$		−	4.52515e8i		−	0.694346i
$868$		0			0
$869$		−	1.52714e8i		−	0.232712i
$870$		0			0
$871$	6.72691e7			0.101803
$872$		0			0
$873$		−	6.99279e8i		−	1.05101i
$874$		0			0
$875$	4.63594e8			0.692012
$876$		0			0
$877$		−	9.80406e8i		−	1.45347i	−0.686916	−	0.726737i	$-0.741036\pi$
$877$		−	9.80406e8i		−	1.45347i	0.686916	−	0.726737i	$-0.258964\pi$
$878$		0			0
$879$	−1.29295e9			−1.90378
$880$		0			0
$881$	5.62161e8			0.822115			0.411058	−	0.911609i	$-0.365159\pi$
$881$	5.62161e8			0.822115			0.411058	+	0.911609i	$0.365159\pi$
$882$		0			0
$883$	9.19482e8			1.33555			0.667776	−	0.744362i	$-0.267246\pi$
$883$	9.19482e8			1.33555			0.667776	+	0.744362i	$0.267246\pi$
$884$		0			0
$885$	−4.03603e8			−0.582271
$886$		0			0
$887$		−	8.30961e8i		−	1.19072i	−0.803459	−	0.595360i	$-0.797009\pi$
$887$		−	8.30961e8i		−	1.19072i	0.803459	−	0.595360i	$-0.202991\pi$
$888$		0			0
$889$			1.11742e9i			1.59041i
$890$		0			0
$891$	3.05079e8			0.431299
$892$		0			0
$893$	8.14591e8	+	6.35796e8i	1.14389	+	0.892819i
$894$		0			0
$895$			5.65134e8i			0.788283i
$896$		0			0
$897$	−6.52116e7			−0.0903540
$898$		0			0
$899$	−1.77810e8			−0.244724
$900$		0			0
$901$			6.97823e8i			0.954049i
$902$		0			0
$903$		−	2.17945e8i		−	0.295994i
$904$		0			0
$905$		−	2.28975e6i		−	0.00308918i
$906$		0			0
$907$			4.99166e7i			0.0668995i	0.999440	+	0.0334498i	$0.0106494\pi$
$907$			4.99166e7i			0.0668995i	−0.999440	+	0.0334498i	$0.989351\pi$
$908$		0			0
$909$	−2.91772e8			−0.388466
$910$		0			0
$911$			7.57244e8i			1.00157i	0.865572	+	0.500784i	$0.166955\pi$
$911$			7.57244e8i			1.00157i	−0.865572	+	0.500784i	$0.833045\pi$
$912$		0			0
$913$	8.83694e7			0.116115
$914$		0			0
$915$		−	5.49754e8i		−	0.717638i
$916$		0			0
$917$	−4.03287e8			−0.523006
$918$		0			0
$919$	6.93924e8			0.894058			0.447029	−	0.894520i	$-0.352482\pi$
$919$	6.93924e8			0.894058			0.447029	+	0.894520i	$0.352482\pi$
$920$		0			0
$921$	4.30893e8			0.551557
$922$		0			0
$923$	−3.87542e7			−0.0492849
$924$		0			0
$925$			9.90557e8i			1.25157i
$926$		0			0
$927$		−	5.99148e8i		−	0.752133i
$928$		0			0
$929$	7.33299e8			0.914606			0.457303	−	0.889311i	$-0.348816\pi$
$929$	7.33299e8			0.914606			0.457303	+	0.889311i	$0.348816\pi$
$930$		0			0
$931$	1.04756e8	+	8.17631e7i	0.129817	+	0.101323i
$932$		0			0
$933$			7.74727e8i			0.953902i
$934$		0			0
$935$	1.46105e8			0.178744
$936$		0			0
$937$	−1.23695e7			−0.0150360			−0.00751800	−	0.999972i	$-0.502393\pi$
$937$	−1.23695e7			−0.0150360			−0.00751800	+	0.999972i	$0.502393\pi$
$938$		0			0
$939$			9.43558e8i			1.13965i
$940$		0			0
$941$			6.92901e8i			0.831577i	0.909461	+	0.415788i	$0.136494\pi$
$941$			6.92901e8i			0.831577i	−0.909461	+	0.415788i	$0.863506\pi$
$942$		0			0
$943$			1.36687e9i			1.63002i
$944$		0			0
$945$			1.74860e8i			0.207203i
$946$		0			0
$947$	1.33520e9			1.57216			0.786079	−	0.618126i	$-0.212108\pi$
$947$	1.33520e9			1.57216			0.786079	+	0.618126i	$0.212108\pi$
$948$		0			0
$949$			6.71840e7i			0.0786081i
$950$		0			0
$951$	−3.89612e8			−0.452992
$952$		0			0
$953$		−	1.20779e9i		−	1.39545i	−0.716368	−	0.697723i	$-0.754197\pi$
$953$		−	1.20779e9i		−	1.39545i	0.716368	−	0.697723i	$-0.245803\pi$
$954$		0			0
$955$	−5.65537e8			−0.649308
$956$		0			0
$957$	−1.73250e8			−0.197668
$958$		0			0
$959$	−1.18935e9			−1.34851
$960$		0			0
$961$	6.32990e8			0.713225
$962$		0			0
$963$			2.79100e8i			0.312522i
$964$		0			0
$965$			8.18798e7i			0.0911161i
$966$		0			0
$967$	−6.74016e7			−0.0745402			−0.0372701	−	0.999305i	$-0.511866\pi$
$967$	−6.74016e7			−0.0745402			−0.0372701	+	0.999305i	$0.511866\pi$
$968$		0			0
$969$	−8.72610e8	+	1.11800e9i	−0.959067	+	1.22877i
$970$		0			0
$971$		−	4.59641e8i		−	0.502067i	−0.967978	−	0.251033i	$-0.919230\pi$
$971$		−	4.59641e8i		−	0.502067i	0.967978	−	0.251033i	$-0.0807704\pi$
$972$		0			0
$973$	−1.10901e9			−1.20392
$974$		0			0
$975$	−7.73783e7			−0.0834844
$976$		0			0
$977$		−	3.29203e8i		−	0.353004i	−0.984300	−	0.176502i	$-0.943522\pi$
$977$		−	3.29203e8i		−	0.353004i	0.984300	−	0.176502i	$-0.0564783\pi$
$978$		0			0
$979$			1.23863e8i			0.132006i
$980$		0			0
$981$			3.07620e8i			0.325842i
$982$		0			0
$983$		−	8.54966e8i		−	0.900095i	−0.893005	−	0.450047i	$-0.851407\pi$
$983$		−	8.54966e8i		−	0.900095i	0.893005	−	0.450047i	$-0.148593\pi$
$984$		0			0
$985$	1.39216e8			0.145673
$986$		0			0
$987$			1.59366e9i			1.65747i
$988$		0			0
$989$	−2.24787e8			−0.232371
$990$		0			0
$991$			6.98814e8i			0.718027i	0.933332	+	0.359013i	$0.116887\pi$
$991$			6.98814e8i			0.718027i	−0.933332	+	0.359013i	$0.883113\pi$
$992$		0			0
$993$	−1.21897e9			−1.24493
$994$		0			0
$995$	5.00642e7			0.0508227
$996$		0			0
$997$	−1.30205e9			−1.31384			−0.656920	−	0.753960i	$-0.728141\pi$
$997$	−1.30205e9			−1.31384			−0.656920	+	0.753960i	$0.728141\pi$
$998$		0			0
$999$	−8.24563e8			−0.827042

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.7.e.d.113.1		8
4.3	odd	2		19.7.b.b.18.5	yes	8
12.11	even	2		171.7.c.d.37.4		8
19.18	odd	2	inner	304.7.e.d.113.8		8
76.75	even	2		19.7.b.b.18.4	✓	8
228.227	odd	2		171.7.c.d.37.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.7.b.b.18.4	✓	8	76.75	even	2
19.7.b.b.18.5	yes	8	4.3	odd	2
171.7.c.d.37.4		8	12.11	even	2
171.7.c.d.37.5		8	228.227	odd	2
304.7.e.d.113.1		8	1.1	even	1	trivial
304.7.e.d.113.8		8	19.18	odd	2	inner

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:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	304.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-33.7437i q^{3} -51.7597 q^{5} +313.488 q^{7} -409.636 q^{9} +O(q^{10})\)	\(q-33.7437i q^{3} -51.7597 q^{5} +313.488 q^{7} -409.636 q^{9} -460.660 q^{11} +177.130i q^{13} +1746.56i q^{15} +6127.64 q^{17} +(-5407.00 - 4220.22i) q^{19} -10578.2i q^{21} -10910.4 q^{23} -12945.9 q^{25} -10776.5i q^{27} +11145.5i q^{29} +15953.5i q^{31} +15544.4i q^{33} -16226.1 q^{35} -76514.9i q^{37} +5977.04 q^{39} -125282. i q^{41} +20603.1 q^{43} +21202.7 q^{45} -150655. q^{47} -19374.2 q^{49} -206769. i q^{51} +113881. i q^{53} +23843.6 q^{55} +(-142406. + 182452. i) q^{57} +231084. i q^{59} -314764. q^{61} -128416. q^{63} -9168.22i q^{65} -379771. i q^{67} +368156. i q^{69} +218789. i q^{71} +379291. q^{73} +436843. i q^{75} -144411. q^{77} +331511. i q^{79} -662264. q^{81} -191832. q^{83} -317165. q^{85} +376090. q^{87} -268881. i q^{89} +55528.3i q^{91} +538330. q^{93} +(279865. + 218437. i) q^{95} +1.70707e6i q^{97} +188703. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 108 q^{5} + 140 q^{7} - 1052 q^{9}+O(q^{10}) \) 8 * q + 108 * q^5 + 140 * q^7 - 1052 * q^9	\( 8 q + 108 q^{5} + 140 q^{7} - 1052 q^{9} + 2024 q^{11} + 6008 q^{17} - 20552 q^{19} + 50252 q^{23} + 78492 q^{25} - 210800 q^{35} - 43724 q^{39} - 260800 q^{43} - 191012 q^{45} + 100248 q^{47} - 301872 q^{49} + 52480 q^{55} - 186860 q^{57} - 54548 q^{61} + 137408 q^{63} + 479968 q^{73} - 1755300 q^{77} - 4279648 q^{81} - 483040 q^{83} + 2111780 q^{85} - 2802652 q^{87} + 1507528 q^{93} + 2383888 q^{95} - 528224 q^{99}+O(q^{100}) \) 8 * q + 108 * q^5 + 140 * q^7 - 1052 * q^9 + 2024 * q^11 + 6008 * q^17 - 20552 * q^19 + 50252 * q^23 + 78492 * q^25 - 210800 * q^35 - 43724 * q^39 - 260800 * q^43 - 191012 * q^45 + 100248 * q^47 - 301872 * q^49 + 52480 * q^55 - 186860 * q^57 - 54548 * q^61 + 137408 * q^63 + 479968 * q^73 - 1755300 * q^77 - 4279648 * q^81 - 483040 * q^83 + 2111780 * q^85 - 2802652 * q^87 + 1507528 * q^93 + 2383888 * q^95 - 528224 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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